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Nongravitational Forces in Planetary Systems

The document discusses the role of nongravitational forces in shaping the dynamics and properties of small planetary bodies in the solar system, including comets, asteroids, and exoplanetary systems. It highlights how these forces, such as sublimation recoil and radiation pressure, are crucial for understanding phenomena that gravity alone cannot explain. The paper aims to simplify the understanding of these complex forces for non-specialists and provides order-of-magnitude descriptions and examples of their applications.

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Nongravitational Forces in Planetary Systems David Jewitt Department of Earth, Planetary and Space Sciences, University of California at Los Angeles, Los Angeles, CA 90095-1567, USA; djewitt@gmail.com Received 2024 September 12; revised 2024 November 24; accepted 2024 November 25; published 2025 January 16 Abstract Nongravitational forces play surprising and, sometimes, centrally important roles in shaping the motions and properties of small planetary bodies. In the solar system, the morphologies of comets, the delivery of meteorites, and the shapes and dynamics of asteroids and binaries are all affected by nongravitational forces. In exoplanetary systems and debris disks, nongravitational forces affect the lifetimes of circumstellar particles and feed refractory debris to the photospheres of the central stars. Unlike the gravitational force, which is a simple function of the well- known separations and masses of bodies, the nongravitational forces are frequently functions of poorly known or even unmeasurable physical properties. Here, we present order-of-magnitude descriptions of nongravitational forces, with examples of their application. Unified Astronomy Thesaurus concepts: Small Solar System bodies (1469); Asteroid satellites (2207); Long period comets (933); Main-belt comets (2131); Short period comets (1452); Debris disks (363); Comets (280); Asteroids (72) 1. Introduction Gravity alone provides an ample description of the dynamics and of many physical properties of planetary-mass bodies. However, scientific interest is increasingly focused on smaller bodies in orbit around the Sun and other stars, and small bodies are additionally susceptible to a host of other forces. These so- called nongravitational forces include recoil and torque from anisotropic mass loss, radiation pressure, Poynting–Robertson drag, Yarkovsky force, YORP torque, and forces from magnetic interactions with the solar wind. Important examples of phenomena that cannot be understood using gravity alone are numerous, ranging from the motion of dust particles in comets, the Zodiacal cloud and debris disks, to the orbital drift of asteroids and their delivery into planet-crossing orbits, to the centripetal shaping and disintegration of comets and asteroids, to the formation of asteroid binaries and pairs. Unfortunately, the research literature tends to present nongravitational forces either in excruciating and largely unhelpful detail or, more usually, without meaningful discus- sion of any sort. Indeed, nongravitational forces are often hidden as lines of code in elaborate numerical models, where their practical role is to help to improve the fit to data by adding extra degrees of freedom. Sadly, they sometimes do so without giving a parallel improvement in our understanding of the relevant physics. The objective of the current paper is to provide a highly simplified but nevertheless informative account of the different nongravitational forces that are important in the solar system. We also add some pointers to sample applications from the literature, in a style suitable for the nonspecialist. With this in mind, in the following, we either neglect geometrical factors representing body shape, or approximate the relevant bodies as spheres, with bulk density ρ [kg m−3 ] and radius a [m]. The orbits of all bodies are assumed to be circular, so that the heliocentric distance is also the semimajor axis. Of course, real bodies are not spherical, and real orbits are not circles. These simplifying approximations remain useful, however, by giving a guide to the order of magnitude of the forces and timescales involved. When the heliocentric distance, rH [m], is expressed in astronomical unit, we give it the symbol rau. For example, the flux of sunlight, Fe [W m−2 ], which plays an obvious role in several of the nongravitational forces, is given by ( )     p = = F L r F S r 4 or, equivalently, , 1 H 2 au 2 where Le = 4 × 1026 W is the luminosity of the Sun, and Se = 1360 W m−2 is the solar constant. 2. Sublimation Recoil By far the largest nongravitational force considered here is that due to the sublimation of ice from small bodies heated by the Sun. Sublimated volatiles freely expand into the near vacuum of interplanetary space, carrying with them momentum and exerting a recoil force on the ice-containing parent body. Since sublimation is exponentially dependent on temperature, most sublimation occurs on the hot dayside of the nucleus, and the resulting recoil force is primarily antisolar. This sublimation recoil force is  =  k MV R th, where  M [kg s−1 ] is the sublimation rate, Vth [m s−1 ] is the bulk speed of the outflowing gas, and kR is a dimensionless constant that describes the angular dependence of the flow. Purely collimated flow would have kR = 1, while isotropic outgassing would have kR = 0, meaning no net force on the nucleus. The best estimate of kR is based on measurements from the unusually well-studied comet 67P/Churyumov–Gerasimenko, for which kR ∼ 1/2 (D. Jewitt et al. 2020). This is also the value expected from uniform sublimation across the sunward facing hemisphere of a spherical nucleus. We can also write  m = M m Qg H , where μ is the molecular weight of the sublimated ice, mH = 1.67 × 10−27 kg is the mass of the hydrogen atom, and Qg is the gas production rate in molecules per second. Setting a =  M S, where M = 4πρa3 /3 The Planetary Science Journal, 6:12 (26pp), 2025 January https://doi.org/10.3847/PSJ/ad9824 © 2025. The Author(s). Published by the American Astronomical Society. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1
is the body mass, gives a nongravitational acceleration (NGA) ( ) a m pr = k m a Q V 3 4 2 S R g H 3 th for a spherical body of radius, a, and density ρ. The outflow speed of the gas, Vth, is roughly equal to the thermal speed of the constituent molecules ( ) / pm = V k T m 8 . 3 th B H 1 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ Here, kB = 1.38 × 10−23 J K−1 is Boltzmann's constant, and temperature T refers to the surface temperature of the sublimating surface. This is typically depressed below the local radiative equilibrium temperature because a fraction of the absorbed solar energy that would otherwise drive radiative cooling is instead used to break bonds between molecules in the process of sublimation. While / µ - T rH 1 2 in radiative equilibrium, the temperature of sublimating ice is an even weaker function of heliocentric distance because of this depression. We take Vth = 500 m s−1 at 1 au and assume for simplicity that this speed applies across the terrestrial planet region. The temperature and sublimation rate per unit area, fs(T) [kg m−2 s−1 ], are connected by the energy balance equation for a surface element of area oriented with its normal offset from the direction of illumination by angle θ; ( ) ( ) ( ) ( )  q es - = + S r A T H T f T 1 cos . 4 s au 2 4 Here, A and ε are the Bond albedo and emissivity of the surface, σ = 5.67 × 10−8 W m−2 K−4 is the Stefan–Boltzmann constant, and H(T) [J kg−1 ] is the latent heat of sublimation for the ice in question. An additional, generally small term for heat conducted beneath the surface has been neglected from Equation (4). Additional information is needed to solve Equation (4). The temperature dependence of fs can be obtained from the Clausius–Clapeyron equation for the slope of the solid/vapor phase boundary (the relation dP/dT = PH/(NAkT2 ), where NA is Avagadro's number, and k is the Boltzmann constant, which is only applicable when H is independent of T) or, better, from laboratory measurements of the sublimation vapor pressure over ice as a function of temperature. The optical properties A and ε have a minor effect on the solution provided A = 1 and ε ? 0. In practice, A = 0 and ε = 0.9 are widely assumed. In Figure 1 we show solutions to Equation (4) for the three most abundant cometary ices of water, carbon dioxide, and carbon monoxide. These three, having approximate latent heats H = 2.8 × 106 J kg−1 , 0.57 × 106 J kg−1 , and 0.29 × 106 J kg−1 , respectively, are representative of low, medium, and high volatility solids. All three plotted curves trend asymptotically toward µ - f r s au 2 as rau → 0. This is because the exponential temperature dependence of fs is stronger than T4 , so that the second term on the right of Equation (4) dominates the first at the high temperatures found at small rau. Setting the radiative term in Equation (4) equal to zero gives ( ) ( )  ~ f S H T r 5 s au 2 where we have set A = 0 and θ = 0 for simplicity. Equation (5) gives a useful approximation even at rau = 1 au, where fs = 6 × 10−4 , 2.4 × 10−3 , and 4.7 × 10−3 kg m−2 s−1 , Figure 1. Equilibrium sublimation mass fluxes as a function of heliocentric distance for H2O (solid black lines), CO2 (long-dashed red lines), and CO (short-dashed blue lines) ices, computed from Equation (4). Two models are shown for each ice. The upper model for each ice shows sublimation at the subsolar temperature, taken as the highest temperature on a spherical body, while the lower model shows sublimation at the local isothermal blackbody temperature, which is the lowest possible temperature. The location of the asteroid belt is marked as a shaded blue rectangle, with inner and outer edges at 2.1 and 3.2 au. 2 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
respectively, for water, carbon dioxide, and carbon monoxide ices on a flat plate oriented normal to the Sun. Ices like carbon monoxide are so volatile that the µ - f r s au 2 regime extends over the entire planetary region while, for the intermediate volatile carbon dioxide ice, the sublimation rate inflects closer to the Sun, but still beyond the orbit of Jupiter. Across the ∼1 au width of the asteroid belt, shown in Figure 1 as a blue shaded region, the sublimation flux of water varies appreciably, by a factor of ∼4 in the high temperature limit and by a factor ∼200 in the low temperature limit. This strong variation explains the sudden rise of the outgassing activity observed in comets as they cross asteroid belt distances (e.g., N. Biver et al. 2002) and the strong concentration of outgassing from main-belt comets near perihelion (H. H. Hsieh et al. 2015). The total mass-loss rate from a sublimating body is given by the integral  ò = M f dS s over the surface, S, taking account of the object shape because it controls the local solar incidence angle θ. In practice, evaluation of this integral is impossible because the shapes of most bodies are not well determined, and the distribution of surface ice is patchy. Moreover, ice can sublimate from beneath the surface, resulting in a complicated and generally underconstrained thermophysical problem for which simple solutions do not exist. Therefore, while it is useful to consider the broad trends shown in Figure 1, it is not in general possible to calculate a global value of fs or  M from first principles. 2.1. Examples 1. Comets. Nongravitational motions are especially obvious in the orbits of short-period comets (SPCs), where the cumulative effects can be measured over many orbits. The NGA is conventionally written (B. G. Marsden et al. 1973) ( )( ) ( ) / a = + + g r A A A 6 S H 1 2 2 2 3 2 1 2 in which A1, A2, and A3 are the components of the NGA in the directions radial to the Sun, perpendicular to A1 but in the orbital plane, and perpendicular to the orbital plane, respectively. The acceleration components are conven- tionally expressed in au day−2 , where 1 au day−2 = 20.1 m s−2 . For comets, g(rH) conventionally takes a contrived analytic form based on solutions to Equation (4) for water ice (B. G. Marsden et al. 1973). For asteroids, ( ) / = g r r 1 H au 2 , where rau is the heliocentric distance in astronomical unit. The cumulative distribu- tions of A1 for Jupiter-family comets and long-period comets (LPCs) are shown separately in Figure 2 using NGA parameters from JPL Horizons.1 The astrometric measurements from which NGA is determined are fraught with systematic errors (for example, in a comet with a coma, the center of light and the nucleus may not coincide) and some measure of judgment is required in the rejection of unreliable data in determining the likely best fit. To try to reduce this problem, we have plotted in Figure 2 only measurements having signal-to-noise ratio (SNR) > 10. Figure 2 shows that the αs distributions are significantly different for LPCs and SPCs, with the LPCs showing a larger acceleration (rms value 16 × 10−7 au day−2 , corresponding to NGA α = 32 × 10−6 m s−2 ) than the SPCs (rms 4 × 10−7 au day−2 or NGA α = 8 × 10−6 m s−2 ). This difference could indicate a systematic difference in the sizes or densities Figure 2. Cumulative distribution of the A1 (radial) component of the nongravitational acceleration (NGA) for 101 short-period comets (SPCs; blue line, 2 „ TJ „ 3) and 33 long-period comets (LPCs; red line, TJ < 2). Only comets with A1 measured to >10σ significance are plotted. The LPCs show systematically larger A1, consistent with having smaller and/or less dense nuclei, and with having larger outgassing rates per unit area. 1 https://ssd.jpl.nasa.gov/ 3 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
of SPC and LPC nuclei, or a difference in the outgassing rates per unit area of surface, or some combination of these effects. A more prosaic explanation is a measure- ment bias; small NGAs cannot generally be accurately measured in LPCs because they are observed at only one apparition, whereas SPCs can be measured over multi- ple orbits and longer periods of time. Some 16/101 SPCs (but 0/33 LPCs) exhibit negative A1 parameters, which indicate unexpected sunward acceleration. These determinations, which might indicate that the astro- metric errors are underestimated, cannot be explained by recoil from dayside sublimation. 2. Dark Comets. Some small, apparently inert bodies exhibit substantial NGA, yet present no evidence for outgassing. Notable examples include the interstellar interloper 1I/ ’Oumuamua (M. Micheli et al. 2018) and the ∼150 m radius object 523599 (2003 RM; D. Farnocchia et al. 2023), as well as a number of tiny asteroids (essentially, boulders) in the 3–15 m radius range (D. Z. Seligman et al. 2023). The magnitude of the NGA in these evocatively named “Dark Comets” is too large to reflect the action of radiation pressure or Yarkovsky force but consistent with small sublimation rates. For example, the measured acceleration of 2003 RM is α = 2 × 10−12 au day−2 (α = 4 × 10−11 m s−2 ). From Equation (2) with kR = 0.5, μ = 18 (for H2O), ρ = 103 kg m−3 , and Vth = 500 m s−1 , we find Qg = 4 × 1025 s−1 (equivalent to 1 kg s−1 ), which might be small enough to have escaped detection. The Tisserand parameter with respect to Jupiter of 2003 RM is formally that of a comet (TJ = 2.96), in which case the presence of ice and outgassing should not be surprising. However, D. Farnocchia et al. (2023) preferred an origin in the outer asteroid belt and derived a much smaller Qg ∼ 1023 s−1 . The boulders described by D. Z. Seligman et al. (2023) can be accelerated by sublimation at even smaller rates. For example, the strongest (8σ) detection of NGA is in the 4 m radius boulder 2010 RF12, with A3 = (−0.17 ± 0.02) × 10−10 au day−2 (αS = 3.3 × 10−10 m s−2 ). Substitution into Equation (2) gives Qg ∼ 1019 s−1 (roughly 3 × 10−7 kg s−1 ), which is small enough to have escaped detection by any existing direct technique. Radiation pressure acceleration should be of the same order as αs but cannot account for A3, which acts perpendicularly to the projected radial line. The main puzzle presented by the accelerated boulders is their implied short mass-loss lifetimes. 2020 RF12, with mass ∼ 3 × 105 kg, can sustain mass loss at 3 × 10−7 kg s−1 for only 1012 s (3 × 104 yr). The conduction timescale is even shorter (∼6 months for a compact rock with diffusivity κ = 10−6 m2 s−1 and ∼500 yr if it is a porous dust ball with the very low diffusivity κ = 10−9 m2 s−1 ; see Appendix). As a result, the internal temperatures of this and other boulders near 1 au would quickly equilibrate to orbit-averaged values that are too high (∼300 K) for water ice to survive. Even if the mass loss is intermittent, it is hard to see how such tiny boulders could retain ice on dynamically relevant (Myr) timescales. 3. Radiation Pressure A radiant energy flux density Fν [J m−2 s−1 Hz−1 ] corresponds to a photon flux Fν/(hν) [photon m−2 s−1 Hz−1 ], where h = 6.63 × 10−34 J s is Planck's constant, and hν is the energy of a photon having frequency ν. The momentum of a single photon is h/λ = hν/c, where λ is the wavelength, and c = 3 × 108 m s−1 is the speed of light. Then, considering the Sun as the source of photons, the flux of momentum in photons of frequency ν → ν + dν is dPr,ν = Fν/(hν) × (hν/c)dν. When integrated over all frequencies, this gives a pressure [ ] ( )  = - P F c N m 7 r 2 in which  ò n = n ¥ F F d 0 . Equation (7) is the radiation pressure. For example, at rH = 1 au, where the flux of sunlight is given by the solar constant, Se = 1360 W m−2 , Equation (7) gives Pr = 4.5 × 10−6 N m−2 (about 4 million times less than the pressure exerted by the weight of a sheet of paper). This tiny pressure is about 105 times smaller than the pressure due to water ice sublimation at 1 au, and is insignificant on macroscopic bodies, but can dominate the motion of small particles. The force exerted by radiation impinging on a spherical grain of radius a, is /  p =  Q a F c pr 2 , where Qpr is a size and wavelength-dependent dimensionless multiplier,2 and Fe [W m−2 ] is the local flux of sunlight. The radiation pressure acceleration is proportional to the cross section per unit mass of the accelerated particle, and so is inversely related to the particle size. For a spherical particle of density ρ, this results in an acceleration ( )  a r = Q F ca 3 4 . 8 rad pr The inverse dependence shows that small, low-density particles can be more strongly accelerated by radiation than large, high-density particles. In the case of the radiation field around the Sun, we note that the flux is given by ( ) /   p = F L r 4 H 2 , where Le = 4 × 1026 W is the solar luminosity and rH is the heliocentric distance in meters. Substituting gives ( )  a r p = Q ca L r 3 4 4 . 9 H rad pr 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ The numerical multiplier in Equations (8) and (9) is specific to the assumed spherical particle shape. Nevertheless, the equations give a useful approximation to the acceleration induced by radiation pressure as a function of particle size, and 2 Qpr is the ratio of the effective cross section for radiation pressure to the geometric cross section of the particle. It is a function of the composition, shape, structure, and size of a particle relative to λ, the wavelength of radiation with which it interacts. The limits are Qpr = 1 as a/λ → 0 while Qpr → constant as a/λ → ∞. In between, Qpr varies with a/λ in a complicated way, reflecting interactions between electromagnetic waves as they pass, and pass through, the particle (J. A. Burns et al. 1979; C. F. Bohren & D. R. Huff- man 1983). In planetary science and astronomy, the zeroth order approximation is to set Qpr = 0 for a < λ and Qpr = 1 for a λ. For many natural particle size distributions in which the smallest particles are the most abundant, this approximation gives rise to the rule-of-thumb that observations primarily sample particles with a ∼ λ. Calculation of Qpr for homogeneous spheres uses the Mie Theory. For other shapes and for porous and fractal particles of relevance to natural solar system particles, there is no analytic theory, and Qpr must be calculated numerically (e.g., K. Silsbee & B. T. Draine 2016). 4 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
density. It is useful to normalize αrad by the local solar gravitational acceleration ( )   = g GM r . 10 H 2 The ratio βrad = αrad/ge is given by ( )   b pr = Q ca L GM 3 16 , 11 rad pr ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ which is a function of both particle (ρ, a, Qpr) and stellar (Le, Me) properties. For example, consider spherical dust grains with ρ = 1000 kg m−3 and Qpr = 1. Substitution into Equation (11) gives ( ) b = m a 0.6 , 12 rad where aμ is the radius expressed in microns. βrad is independent of heliocentric distance because gravity and radiation pressure both vary inversely with the square of rH. Equation (12) is a useful guide for homogeneous spheres, but it is important to note that the β parameter for other particle shapes cannot be so simply estimated because Qpr can assume very different values. Likewise, βrad ? 1 is never encountered, regardless of how small the particles are (K. Silsbee & B. T. Draine 2016). 3.1. Examples 1. Comet Tails. Radiation pressure acceleration has been detected in bodies as large as the ∼4–10 m diameter asteroid 2011 MD, for which βrad ∼ 10−7 (M. Micheli et al. 2014; M. Mommert et al. 2014), but the classic application is to the motion of small particles in comets. The dust particle trajectory after release from a comet nucleus is determined by the ejection velocity and βrad. In the theory of comet tails by M. J. Finson & R. F. Probstein (1968), dust particles are assumed to be released from the nucleus with zero relative speed and then to be accelerated by solar gravity and radiation pressure. Two limiting cases are often considered: synchrones show the locus of positions of dust particles having a range of sizes but released from the nucleus at the same time, while syndynes show the locus of positions of particles of one size (i.e., one βrad) but released over a range of times. Synchrones project on the sky as straight lines, with a position angle that depends on the time of ejection. Syndynes are curved by the orbital motions of the particles. See Figure 3 for examples. Syndyne/synchrone analysis enables a simple assessment of cometary dust properties. For example, depending on the observing geometry, a linear tail morphology may suggest a synchrone-like structure indicative of impulsive ejection. Linear tails are observed in outbursting comets and in dust released by energetic impacts between asteroids (D. Jewitt 2012), where the position angle of the tail gives a useful estimate of the ejection date. On the other hand, broad, fan-like tail structures are indicative of protracted emission for which syndyne models give constraints on the particle sizes. When observed from a position close to the comet orbital plane, however, syndynes and synchrones overlap so strongly that their diagnostic power is largely lost. With the aid of fast computers, more sophisticated synthetic models of comets can be used to explore a wide range of dust and ejection conditions. Such Monte Carlo models involve many parameters (including the form of the size distribution (usually assumed to be a power law), the largest and smallest particle radii, their ejection velocities as a function of particle size, the angular distribution of the ejection from the nucleus and its time dependence) and so are nonunique. Nevertheless, Monte Carlo coma models are useful in narrowing the range of Figure 3. Disintegrated LPC C/2021 A1 (Leonard) on UT 2022 March 31 when at rH = 1.756 au outbound. The left panel shows synchrones for particles ejected from 80–160 days before the date of observation. The right panel shows syndynes for particles with β = 0.0003, 0.001, 0.003, 0.01, and 0.03, as marked. The linear morphology of the tail is better matched by the straight synchrone models than by the curved syndynes. Ejection occurred 110 ± 10 days before the image was taken, i.e., on UT 2021 December 11 ± 10. Calculation by Yoonyoung Kim in D. Jewitt et al. (2023). 5 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
allowed parameters needed in order to match a given observation. Much of what we know about the solid particles ejected from comets has been determined this way. In the example of Figure 3, inferences from the syndynes and synchrones were supported by a full Monte Carlo simulation. A 0.6 km radius nucleus disintegrated suddenly into a broad distribution of particle sizes resembling a –3.5 index power law, with no fragments larger than 60 m. 2. β Meteoroids. The radiation pressure force opposes gravity such that, at heliocentric distance rH, the net sunward attraction is ( ) /  b - GM r 1 H rad 2 and the effec- tive escape velocity is ( ( ) ) / /  b = - V GM r 2 1 e H rad 1 2. Comparing this to the local Keplerian speed, ( ) / /  = V GM rH K 1 2 shows that particles in circular orbits with βrad > 1/2 (a  1 μm by Equation (12)) should be gravitationally unbound and promptly leave the solar system along nearly radial trajectories.3 Such particles, called β-meteoroids (H. A. Zook & O. E. Berg 1975), have been detected. They are submicron-sized dust particles that stream away from the Sun faster than the gravitational escape speed under the action of radiation pressure. The β particles are produced when larger dust particles sublimate and shrink upon approach to the Sun, as well as by collisional shattering of larger grains from the Zodiacal cloud that are spiraling in under the action of Poynting–Robertson drag (I. Mann & A. Czechow- ski 2021; see Section 4). Their source region is concentrated at 10–20 Re (0.05–0.1 au), where black- body temperatures lie in the range 880–1240 K and both collision and sublimation rates are high (I. Mann et al. 1994, J. R. Szalay et al. 2021). The effective mass-loss rate in β meteoroids is estimated between 102 kg s−1 (J. R. Szalay et al. 2021) and 103 kg s−1 (E. Grun et al. 1985). If sustained over the age of the solar system, a dust mass ∼ 1019 –1020 kg must have been expelled by radiation pressure as β meteoroids, corresponding to a few percent of the current mass in the asteroid belt. 3. Kuiper Belt Dust. While β meteoroids are best known from spacecraft measurements obtained in the inner and middle solar system, more distant counterparts may originate as well in dust sources in the outer solar system. In this regard, recent impact counter measurements from the New Horizons spacecraft have been used to suggest (albeit at only 2σ significance) an excess concentration of dust grains (radius ∼ 0.6 μm, β ∼ 1) beyond the 47 au edge of the classical Kuiper Belt (A. Doner et al. 2024). These dust grains might be produced by collisional shattering in the main Kuiper Belt followed by radiation pressure acceleration outward as Kuiper Belt β meteor- oids. Meanwhile, β meteoroids ejected from other stars presumably contribute to the flux of interstellar dust particles entering the solar system (M. Landgraf 2000). 4. Extended Debris Disk Dust. Many stars are encircled by disks in which the dust lifetime is shorter than the main- sequence age of the star. In these so-called debris disks, the dust must be replenished, presumably by collisional destruction of unseen parent bodies in what are, effectively, extrasolar Kuiper belts (A. M. Hughes et al. 2018). Since the luminosity of a main-sequence star scales in proportion to a high power of its mass (e.g., µ   L Mx with x = 3.5–4), the importance of radiation pressure should grow with stellar mass. A classic example is provided by the disk of α Lyrae (Vega), a ∼700 Myr old A0V star, 2.2 Me in mass with luminosity 47 Le. All else being equal, βrad is ∼20 times larger at α Lyrae than around the Sun, leading to the immediate expulsion of particles 20 times larger (i.e., tens of microns instead of ∼1 μm). The large extent of the α Lyrae dust (which reaches ∼1000 au from the star) is likely due to the action of radiation pressure (K. Y. L. Su et al. 2005). The implied dust mass flux (1012 kg s−1 ) is too large to be sustained over the life of Vega. K. Y. L. Su et al. (2005) suggested that the dust could be the recent product of a massive collision in the disk. 5. Lifting of Dust. Dust can be lifted from the surface of a small body of mass Ma and radius r when the solar radiation pressure acceleration, /  b GM rH rad 2 , exceeds the local gravitational attraction to the body, GMa/r2 . The critical particle size for ejection obtained by setting these accelerations equal is ( ) ~ a r r 10 1 km 1 13 c au 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ where ac is in microns, rau is in astronomical unit, and r is in kilometers (D. Jewitt 2012). For example, particles on a 1 km radius body located at 1 au can be swept away provided ac ∼ 10 μm, or smaller. Close to the Sun (e.g., at 0.14 au, the perihelion distance of (3200) Phaethon), much larger particles (ac  1 mm) can be expelled. Y. P. Bach & M. Ishiguro (2021) noted that thermal radiation pressure from a sufficiently hot surface might also expel particles; although, no established examples of this process currently exist. Both loss processes are highly directional (e.g., solar radiation pressure will force particles back into the surface except near the terminator) and, whether by solar or thermal radiation pressure, particle ejection must overcome the cohesive forces that bind small particles to the surface. 4. Poynting–Robertson Drag Formally a relativistic effect with a rather involved derivation (H. P. Robertson 1937), the Poynting–Robertson drag is more simply described as a consequence of aberration. As seen from a body moving in a circular orbit, the direction from which sunlight travels is aberrated relative to the radial direction from the Sun by an angle ( ) / q = - V c tan 1 K , where ( ) / /  = V GM rH K 1 2 is the orbital speed, and c is the speed of light. For solar system objects of interest, VK = c, and we may approximate θ ∼ VK/c. For example, the Earth (rH = 1 au) has orbital speed VK = 30 km s−1 giving VK/c ∼ 10−4 (about 20″). This tiny angle gives a nonradial component of the radiation pressure force, which acts steadily against the direction of motion, and so can do work against the orbit. The result is a drag force (the Poynting–Robertson drag force) that results in inexorable orbital decay. To estimate the timescale for this orbital decay, τPR, we assume a spherical body of radius a moving in a circular orbit and use Equation (9) to write the Poynting–Robertson 3 For a noncircular orbit, larger particles with βrad > (1 − e)/2 can be ejected by radiation pressure. 6 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
acceleration tangential to the orbit as ( )  a r p = Q ca L r V c 3 4 4 . 14 H PR pr 2 K ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ Then, the timescale for this acceleration to collapse the orbit is τPR ∼ VK/αPR or ( )  t r p ~ ac Q L 4 3 4 r . 15 H PR 2 pr 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ For example, the timescale for a 1 mm radius particle of density ρ = 1000 kg m−3 to spiral into the Sun from Earth orbit (rH = 1 au) under the action of the Poynting–Robertson drag is τPR ∼ 8 × 1013 s, or about 3 Myr (assuming Qpr = 1). Setting τPR = 4.5 Gyr in Equation (15) gives a ∼ 1.5 m: all primordial material smaller than a few meters in size is removed by Poynting–Robertson drag if it survives that long against other destructive processes (e.g., impact destruction, gravitational scattering). 4.1. Examples 1. Zodiacal Dust. Interplanetary dust is released by sublimating comets and, to a lesser extent, by collisions between asteroids, forming a diffuse inner solar system structure known as the Zodiacal cloud. Dust particles in the Zodiacal cloud have a range of sizes with a mass- weighted mean near s̄ = 100–200 μm, for which the Poynting–Robertson time is τpr ∼ 0.5 Myr at 1 au. Smaller particles are quickly depleted by Poynting– Robertson drag while the survival of larger particles is limited more by collisional shattering. The total rate of production required to maintain the Zodiacal cloud in steady state is ∼103 –104 kg s−1 , albeit with order-of- magnitude uncertainty (D. Nesvorný et al. 2011; J. K. Rigley & M. C. Wyatt 2022, P. Pokorný et al. 2024). The steady Poynting–Robertson rain of Zodiacal dust toward the Sun is reversed when a fraction of those particles shrink small enough (through sublimation, and/ or through collisional shattering) to be expelled by radiation pressure as β meteoroids. 2. Interplanetary Dust Particles. Zodiacal dust particles enter the Earth's atmosphere at an average rate ∼ 1 kg s−1 (S. G. Love & D. E. Brownlee 1993). Entering particles smaller than a few tens of microns are decelerated by friction so high in the atmosphere that they do not melt and, when collected from the lower stratosphere, are known as interplanetary dust particles (IDPs). The rate of IDP delivery presumably varies on long timescales as asteroids collide and produce variable amounts of dust, which is then cleared in part by Poynting–Robertson drag. However, stratigraphic mea- surements of 3 He, which is delivered to Earth by IDPs, show only factor-of-2 variations in the rate in the last 100 Myr (K. A. Farley et al. 2021). On the way from their source to the Earth, the IDPs are impacted by energetic solar and cosmic-ray nuclei, causing damage tracks and the transmutation of elements. L. P. Keller & G. J. Flynn (2022) used particle track densities to calculate the effective space exposure ages of 10 μm sized IDPs. They found that about 25% of IDPs show high cosmic-ray track densities that are indicative of exposure ages > 106 yr. By contrast, the Poynting– Robertson lifetime of 10 μm particles at 1 au is only about 15 kyr (from Equation (15)). The 2 orders-of- magnitude age discrepancy could indicate that these high track density IDPs originate at much larger distances, because t µ rH pr 2 (Equation (15)). For example, 10 μm particles released at rH  10 au would have Poynting– Robertson timescales τpr > 106 yr, consistent with those measured. The only known substantial dust source beyond 10 au is the Kuiper Belt, which L. P. Keller & G. J. Flynn (2022) proposed is the source of the heavily cosmic-ray-damaged dust. Independent IDP exposure age estimates based on the production of unstable Al26 and Be10 nuclei lead to qualitatively consistent conclusions: IDPs originate over a vast range of heliocentric distances, with some from the Kuiper Belt (J. Feige et al. 2024). The reported collisional production rate of dust in the Kuiper Belt (∼104 kg s−1 ) exceeds the ∼103 kg s−1 released from the SPCs and 50 kg s−1 from LPCs (A. R. Poppe 2016). For this reason, it should not be surprising to find an IDP contribution from the Kuiper Belt. Indeed, early simulations of dust transport to the inner solar system suggested a maximum 25% contrib- ution (A. Moro-Martìn & R. Malhotra 2003), consistent with the new estimate based on track densities in the IDPs. 3. White Dwarf Contamination. White dwarfs are degen- erate post-main-sequence, roughly solar-mass stars col- lapsed to Earth-like dimensions. Some white dwarfs exhibit excess infrared emission from circumstellar disks having characteristic radii  1 Re. Refractory material falling from these disks may enter the Roche sphere of the star, become tidally shredded, and then sublimate to form a gaseous metal disk that contaminates the photo- sphere with rock-forming elements. Poynting–Robertson drag moves material inward to the sublimation radius to feed the metal gas disk. A straightforward application of Equation (15) gives timescales too long, and peak accretion rates (∼10 kg s−1 ) orders of magnitude too small to account for the observed degree of white dwarf pollution (J. Farihi et al. 2010). However, R. R. Rafikov (2011) showed that the effective τPR is much shorter, and the mass delivery rate much greater (∼105 kg s−1 ), because Poynting–Robertson drag need only move particles across a thin transition zone at the inner edge of the refractory disk into the sublimation radius. 5. Dissipative Forces 5.1. Tidal Dissipation The shapes of bodies in mutual orbit are cyclically deformed by varying gravitational forces, resulting in the dissipation of rotational and orbital energy as heat. Over time, this dissipation can have important dynamical consequences. Although gravity is the driver, the fundamental origin of energy loss is nongravitational, being rooted in the physics of inelastic materials. The character of tidal dissipation is best seen by neglecting numerical multipliers and geometrical terms associated with the shapes of the orbiting bodies. The shapes of most solar system bodies are unknown, so this neglect is at least partly reasonable. We consider, for simplicity, a binary object consisting of a primary of mass mp and a secondary of mass 7 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
ms, separated by a distance r, with mp ? ms. The radii of the primary and secondary objects are ap and as, respectively. We also assume that the densities, ρ, of the primary and secondary are the same so that, neglecting shape specific factors, r ~ m a p p 3 and r ~ m a s s 3 . The gravitational pull of each body elongates the other along an axis connecting their centers, with the secondary being most deformed because of the greater mass and gravity of the primary. If the deformation were instantaneous, this tidal bulge would stay perfectly aligned along the line of centers. However, the primary, in general, rotates with an angular frequency, ωp, that is different from the orbital frequency of the satellite, ωs, and the tidal response is not instantaneous. Empirically, most satellites orbit beyond the corotation radius, at which the orbital period is equal to the rotation period of the primary body. With ωp > ωs, the tidal bulge of the primary is carried ahead of the line of centers, and the resulting torque on the primary acts to slow its rotation while expanding the orbit of the secondary. In the opposite case (ωp < ωs), the tidal bulge lags behind, and the tidal torque increases ωp while contracting the orbit of the secondary. The latter is the case for Mars’ satellite Phobos, which orbits at 2.76 RM, far inside the corotation radius at 6.03 RM (1 RM = 3.4 × 106 m). The orbit of Phobos will collapse into the planet because of tidal dissipation on a (surprisingly short) timescale of a few tens of Myr (B. A. Black & T. Mittal 2015). Mars’ other known satellite, Deimos, orbits beyond the corotation radius at 6.92 RM and is being slowly pushed away from the planet by tides. The finite response time and inelasticity of the material are central to the mechanism of tidal evolution because the misalignment of the tidal bulge creates an asymmetry upon which gravity can exert a torque. The gravitational force experienced from distance r is F = Gmpms/r2 or, equivalently, / r ~ F G a a r p s 2 3 3 2. The gravity of the satellite is slightly different on the near and far sides of the primary, by an amount ( ) d r ~ ~ F dF dr a G a a r . 16 p p s 2 4 3 3 This small differential force periodically stretches and relaxes the bodies as they rotate and orbit, doing work in the process. The average stress due to gravitational deformation is / d = S F ap 2 [N m−2 ]. The relation between the stress and the strain (strain is the fractional change in the length scale, s = δap/ap) is called the bulk modulus, defined as μ = S/s. Substituting, the deformation is ( ) d d m r m ~ = a F a G a a r . 17 p p p s 2 3 3 3 The work done by a force δF applied over a distance δap is δW ∼ δFδap. Substituting from Equations (16) and (17) gives ( ) d r m = W G a a r a . 18 p s p 2 3 3 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ If the material were perfectly elastic, energy added to the body by stretching would be returned upon relaxation back to the original shape. But in real materials, owing to internal friction, a fraction of the energy is dissipated as heat. The fraction lost per stretching cycle is conventionally defined as Q−1 = δE/E (i.e., Q is the inverse of what one would expect) and referred to as the tidal “quality factor.” High Q corresponds to low dissipation per cycle and vice versa. Given this, and recognizing that δW in Equation (18) is the work done per cycle (not per second), we write the tidal power as ( ) d d r w m ~ W t G a a r Q a . 19 p s p p 2 3 3 3 2 ⎜ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎞ ⎠ The key point in all of this is that the tidal power (Equation (19)) scales with r−6 because the differential tidal force, δF, and the deformation it causes, δap, are both proportional to r−3 . Finally, the timescale for dissipation to substantially change the rotational energy is τt ∼ W/(δW/δt), where W is the rotational energy. We take the rotational energy of the primary to be w ~ W Ip p 2 where r ~ ~ I m a a p p p p 2 5 is the moment of inertia. Then, ( ) t m w r ~ Q G a m m r a 20 t p p p s p 2 3 2 2 6 ⎜ ⎟ ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ is the order-of-magnitude time needed for torques from the satellite to substantially change the rotation of the primary. Tides raised on the satellite by the gravity of the primary are larger, and act to change the rotation on an even shorter timescale, as can be seen by swapping as and ap in Equation (20). Obviously, this derivation is highly simplified, but it serves to show the functional dependence of the tidal evolution timescale on material (ρ, μ, Q) and geometrical (ap, as, r) properties. 5.2. Internal Dissipation Even without an external force from a binary companion, internal energy dissipation in an inelastic material can modify the spins of individual asteroids. The loss of rotational energy occurs when a body is rotating in a nonprincipal axis state, such that its rotational energy is not a minimum for its shape. Then, any element of the body is subject to a cyclically varying stress as it rotates, which induces a variable strain, allowing internal friction to act. The visual model is a deflected gyroscope, which both rotates around its axis and precesses and nutates at the same time. We can obtain a useful expression for the functional dependence of the damping timescale by the same dimensional method as used for the tidal effects in Section 5.1. Precession and nutation of the rotation axis induce time-variable stresses that do work by cyclically compressing and relaxing the material. The internal stress, proportional to the energy density in the body, is S ∼ ρa2 ω2 [N m−2 ], where a is the nominal size of the body, and ω [s−1 ] the relevant angular frequency. Stress and strain (δa/a) are related through the modulus μ = S/(δa/ a), giving δa ∼ Sa/μ. Then, by analogy with Section 5.1, the power dissipated as heat by the stress is δW/δt ∼ Sa2 δa(ω/Q), which becomes ( ) d d r w m = W t a Q . 21 2 7 5 This compares with the instantaneous rotational energy W ∼ ρa5 ω2 , and the ratio of W to δW/δt gives the damping 8 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
timescale ( ) t m r w ~ A Q a 22 damp 2 3 where A is a numerical multiplier introduced to account for aspects of material physics and asteroid shape that are not treated here. J. A. Burns & V. S. Safronov (1973) derived A ∼ 100, but values spanning a range of nearly 3 orders of magnitude have been reported in the literature, from A = 1 to 4 (M. Efroimsky & A. Lazarian 2000), to A = 200 to 800 (I. Sharma et al. 2005), depending on details of the assumed visco-elastic properties and asteroid body shape. Given the huge uncertainties in A, as well as those in μ and Q, the timescale offered by Equation (22) should be understood as no more than a qualitative guide, but the equation at least serves to show that damping by internal friction should be strongest in large bodies rotating rapidly. This is consistent with the observation that a majority of the known nonprincipal axis (i.e., undamped) rotators are kilometer-sized or smaller asteroids (P. Pravec et al. 2005). Assuming μ = 1011 N m−2 , Q = 100, ρ = 1000 kg m−3 , and a geometric middle value A = 30, we rewrite Equation (22) for the damping time in Myr as ( ) t ~ P a 10 23 r damp 3 2 with a in kilometers and rotation period Pr in hours. A 1 km body with an excited rotation at Pr = 3 hr would damp on a timescale τdamp ∼ 300 Myr, by this relation. The 0.1 km scale interstellar object 1I/’Oumuamua, with period ∼ 8 hr, has τdamp longer than the age of the Universe, consistent with photometric evidence that it might be in an excited rotational state (M. Drahus et al. 2018). Larger asteroids whose rotation should be damped according to Equation (23) can nevertheless be excited by impact, radiation torques, or gravitational torques from close approaches to planets. The nuclei of some comets also exhibit excited rotation, notably the 15 km long nucleus of 1P/Halley (N. H. Samarasinha & M. F. A’Hearn 1991). In the comets, rotational excitation is a natural product of strong sublimation torques. Again, we emphasize that the material properties μ, Q, and A (Equation (22)) are extremely poorly known (see Section 5.3, Figure 4), and damping times very different from those given by Equation (23) are possible. 5.3. Examples 1. Material Properties. Neither Q nor μ can be calculated from first principles for planetary bodies of interest. Empirically, the quality factor varies widely (e.g., Q ∼ 10 for the Earth, Q ∼ 100 for the Moon; P. Goldreich & S. Soter 1966). Values for gas giant planets are uncertain, but probably 1 or 2 orders of magnitude lower than the Q ∼ 106 initially estimated by these authors (see J. Fuller et al. 2024). A value Q ∼ 100 is often assumed (albeit with little firm evidence) to apply to small solar system bodies. Separately, the Young's moduli of terrestrial rocks span the range from μ ∼ 109 N m−2 for sandstone and ∼1011 N m−2 for basalt to 4 × 1011 N m−2 for diamond. Many small bodies have an aggregate, “rubble pile” structure (K. J. Walsh 2018) that should give them mechanical properties quite different from solid rocks. In particular, the porous and fragmented nature of asteroid rubble piles would seem to point by analogy toward μ values at the lower end of this terrestrial rock range, in Figure 4. μQ as a function of radius estimated from binary asteroids. The subkilometer objects (orange triangles) are near-Earth asteroid binaries, for which a median age 107 yr is assumed. The ∼3 km asteroids in the main belt (yellow circles) are assumed to have median collisional age 109 yr while the 100 km objects (green diamonds) are assumed to be 4.5 Gyr, as old as the solar system. Note the enormous scatter in μQ. Data are from P. A. Taylor & J.-L. Margot (2011). 9 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
which case μQ ∼ 1011 N m−2 might be expected. This value remains little more than a guess, and evidence from asteroid binaries suggests that even smaller μQ values may prevail, as we next describe. Useful empirical constraints on the product μQ can be obtained from the application of Equation (20) to measurements of binary asteroids. For example, if the age of a binary asteroid is known and the other physical properties (separation, period, component sizes, and density) in the equation are measured or can be estimated, then substitution gives μQ. Figure 4 shows μQ obtained in this way as a function of primary asteroid radius for binaries in three size groups (P. A. Taylor & J.-L. Margot 2011). The median value for subkilometer near-Earth binaries (assuming τt = 107 yr, the median dynamical lifetime of this population) is μQ = 5 × 108 N m−2 . This is far smaller even than μQ ∼ 1011 N m−2 estimated above. The median value for 3 km scale main-belt binaries (assuming τt = 109 yr, the approximate collisional lifetime of objects this size) is μQ = 7 × 1011 N m−2 . Finally, μQ = 1.4 × 1011 N m−2 for 100 km scale asteroids, computed assuming that they have survived for the 4.5 Gyr age of the solar system. The considerable differences between these estimates may, in part, reflect a real size dependence of μQ (see F. Nimmo & I. Matsuyama 2019), but they also testify to an element of guesswork in assigning values to some of the parameters in Equation (20). For example, it is difficult to assign more than a statistical age (e.g., based on the assumed collisional or dynamical lifetime) to any given binary object. 2. Rotational Dissipation. By Equation (20), the tidal evolution timescale grows particularly strongly (as r6 ) with the separation. In a given system, doubling the separation increases the tidal timescale by a factor 26 = 64, almost 2 orders of magnitude. This strong distance dependence is evident in real solar system objects. For example, a majority of satellites close to their parent planets rotate synchronously, while the more distant satellites do not. Figure 5 plots the available measure- ments of the satellites of Saturn, showing the trend for close-in satellites to be rotationally synchronized while the rotations of distant satellites are unrelated to their orbital periods. The 360 km diameter satellite Hyperion stands out. Although Hyperion should be synchronized according to Equation (20), its rotation is instead chaotic (i.e., the period changes irregularly) owing to impulsive torques exerted on its irregular body shape over the course of its eccentric (e = 0.12) orbit, as predicted by J. Wisdom et al. (1984) and established by J. J. Klavetter (1989a) and J. J. Klavetter (1989b). The rotation of the Earth's Moon is obviously synchronized with its orbital motion, and measurements show that the Earth's rotation is currently slowing (at ∼10−5 s yr−1 , or ∼3 hr per billion years; R. Mitchell & U. Kirscher 2018) as a result of tidal friction. Inserting r/ap = 60, mp/ms = 81, μQ = 1011 N m−2 , ρ = 5 × 103 kg m−3 , and ap = 6400 km into Equation (20) gives τt ∼ 0.5 Gyr. Although this is only accurate to an order of magnitude, it is clear that tidal interaction with the Moon is responsible for slowing the spin of the Earth. Likewise, terrestrial tides on the Moon slowed its rotation on a much shorter (∼10 Myr) timescale, explaining its current synchronous rotation state. 6. Yarkovsky Force An isothermal sphere at a finite temperature would radiate photons isotropically, experiencing no net recoil force from the Figure 5. Ratio of the rotation to orbit periods as a function of semimajor axis (in units of planet radius) for the measured satellites of Saturn. The inner (regular) satellites rotate synchronously, while the outer (irregular) satellites do not. Hyperion is shown with an error bar to symbolically mark the fact that its rotation is chaotic. 10 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
loss of photon momentum. However, real solar system bodies are neither spherical nor isothermal, being hotter on the Sun- facing (day) side than on the opposite (night) side. In addition, the distribution of slopes, defined by both large- and small- scale features (e.g., boulders) on the surface, imbues the asteroid with a chirality, so that reflected and radiated photons exert a shape and surface texture dependent torque. Moreover, body rotation, combined with the finite thermal response time of the surface material, carries peak heat from local noon into the afternoon, giving a “thermal lag” (see the Appendix). As a result of this thermal lag, the afternoon temperatures will always be higher than the corresponding morning temperatures on a rotating body. The chirality and the azimuthal thermal lag asymmetry in the temperature lead to a net force, the so-called diurnal Yarkovsky force. The component of the force parallel to the direction of orbital motion can do work on the orbit, causing it to shrink or expand depending on whether the sense of the rotation is retrograde (i.e., opposite to the sense of orbital motion) or prograde (in the same sense as the orbital motion); see Figure 6. The above describes the diurnal Yarkovsky effect, which results from axial rotation of the asteroid and is maximized at 0° and 180° obliquity (a geometry assumed, for simplicity, in this discussion and in Figure 6). For example, given a porous regolith with diffusivity κ = 10−9 to 10−8 m2 s−1 , the thermal skin depth (Equation (A2)) on an asteroid rotating with period P = 5 hr is ℓ ∼ (Pκ)1/2 ∼ 4 to 13 mm. At nonzero obliquities, the Yarkovsky force is reduced by a factor equal to the cosine of the obliquity. The annual or seasonal Yarkovsky effect has a similar physical origin in thermal lag, but results from motion around the orbit and is maximized instead at 90° obliquity (Figure 7). One important, systematic difference between the seasonal and diurnal effects is that the former always opposes the orbital motion, causing a shrinkage of the orbit, whereas diurnal Yarkovsky can expand or contract it depending on the rotation direction. A second difference is that, whereas the diurnal Yarkovsky effect results from thermal lag on asteroid rotation periods (typically hours), the seasonal effect results from thermal lag on the orbital timescale (typically years). The ratio of seasonal to diurnal timescales is years/hours ∼104 and, since the thermal skin depth scales with the square root of the time (Equation (A2)), the seasonal Yarkovsky effect depends on the thermophysical properties of a surface skin ∼102 times deeper than the diurnal Yarkovsky. Instead of the diurnal Yarkovsky effect being driven by temperature variations in the top ∼4–13 mm of regolith, the seasonal effect is driven by an upper layer ∼0.4–1.3 m thick, all else being equal. The relative magnitudes of the diurnal and seasonal Yarkovsky effects depend on many quantities, including the object size, spin vector, and the depth dependence of the thermophysical properties of the surface layers (see the Appendix). The diurnal effect typically dominates for bodies with regoliths (i.e., low diffusivity surfaces); we ignore the seasonal effect from further discussion here for simplicity. We also ignore the reflected component of the torque, because the albedos of asteroids are low (C-type albedos are a few percent, while even S-type albedos are only ∼20%) and reflected torques are secondary. Even so, the application to most real asteroids is problematic, because the force depends on many unknown or poorly constrained quantities. These include the magnitude and direction of the spin vector, the body shape, the surface roughness, and the thermophysical parameters responsible for the thermal lag (see Equation (A1)). For all of these reasons, the Yarkovsky force cannot in general be calculated for a given body. Instead, it can be inferred from careful measurements of the action of the force. The Yarkovsky force has been reviewed by W. F. Bottke et al. (2006) and W. D. Vokrouhlický et al. (2015). Here we offer an order-of-magnitude derivation that captures the essence of the process, and we follow with some examples of its application. The temperature of the Sun is ∼6000 K, and its blackbody spectrum is peaked in the optical near 0.5 μm, whereas the isothermal blackbody temperature at 1 au is roughly 300 K, and the spectral peak lies in the infrared near 10 μm. The infrared photons carry 20 times less energy and 20 times less momentum than the absorbed solar photons, but they are 20 times more numerous so as to maintain energy balance on the body. Therefore, we can use the radiation pressure exerted by optical photons ( ( /  p L r c 4 H 2 )) to estimate the Yarkovsky force due to thermal emission. Specifically, we write the magnitude of the diurnal Yarkovsky force as ( )  p p =  k a c L r 4 24 Y Y H 2 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ where 0 „ kY „ 1 is a dimensionless coefficient (to be determined) representing the fraction of the radiation pressure force acting parallel to the orbital motion. (From the discussion in the preceding paragraphs, it should be evident that kY is a function of the spin vector and thermophysical parameters, and we are deliberately separating these from consideration here to simplify the presentation. Thermal considerations are briefly described in the Appendix.) Dividing by the mass of the body, assumed spherical, the resulting acceleration is ( )  a pr = k L car 3 16 . 25 Y Y H 2 We obtain an estimate for the Yarkovsky timescale from τY ∼ VK/αY, VK being the Kepler speed. Again assuming a circular orbit, we find ( ) ( ) / /   t pr ~ ac k L GM r 16 3 . 26 Y Y H 1 2 3 2 The order-of-magnitude radial drift can be estimated from d(rH)/dt ∼ rH/τY, or ( ) ( ) /   pr ~ - dr dt k L ac GM r 3 16 , 27 H Y H 1 2 measured in m s−1 , and drH/dt can be positive or negative, depending on the sense of asteroid rotation (see Figure 6). The maximum possible drift rate at 1 au is given by setting kY = 1, rH = 1 au, with the other parameters as above. For a = 1 km, this gives d(rH)/dt ∼ 0.5 km yr−1 or 4 × 10−3 au Myr−1 . (We will show below that a more typical value is kY ∼ 0.05, so that d(rH)/dt ∼ 2 × 10−4 au Myr−1 is a better estimate for a 1 km body at 1 au.) The asteroid belt is dynamically structured (e.g., at resonance locations) on scales <0.05 au, which can be crossed by kilometer-sized asteroids on timescales <250 Myr. We conclude that, depending on the size of the asteroid and the actual value of kY, the Yarkovsky force is capable of modifying the orbital properties on timescales that are very short 11 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
Figure 6. Schematic plan view of a body orbiting the Sun, to illustrate the diurnal Yarkovsky force. The dotted line shows the orbit of the body having angular rate, ωK, while the body itself rotates about an axis (black dot) fixed perpendicular to the plane, with angular rate, ω. Rotation carries midday heat from the subsolar point into the afternoon, where its loss by radiation is marked IR. In the prograde case (left panel), ωK and ω are parallel, and the recoil force, F, acts with the orbital motion, causing the orbit to expand. In the retrograde case (right panel), ωK and ω are antiparallel. The recoil force opposes the motion and causes the orbit to shrink. Figure 7. Plan view to illustrate the seasonal Yarkovsky effect. The rotation vector (small black arrow, marked ω), lies in the orbit plane and remains fixed in inertial space. Peak insolation is reached when the spin vector points directly at the Sun, as at position A and shown by the strong curly red arrows. Heat is conducted into the interior and slowly leaks out as the asteroid moves around the orbit toward position B. The recoil force from the leakage of this residual heat (shown with faint curly red arrows), F, acts opposite to the motion. Half an orbit later, the cycle repeats with the opposite hemisphere at position C, from which heat is retained to position D giving a recoil still opposing the orbital motion. The seasonal Yarkovsky force always opposes the orbital motion regardless of the sense of body rotation, resulting in orbital shrinkage. 12 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
compared to the age of the solar system. A practical limit to the influence of Yarkovsky drag is set by collisions with other bodies, which can reset the spin and so change the magnitude and even direction of the radiative torque (P. Farinella et al. 1998; P. A. Wiegert 2015). 6.1. Examples 1. Orbital Drift. To obtain an estimate of the Yarkovsky constant, kY, we must use empirical data connecting measurements of the drift rate with the asteroid size, orbit, and density. A. H. Greenberg et al. (2020) and M. Fenucci et al. (2024) reported drift rate detections for 247 and 348 asteroids, respectively. Figure 8 shows measurements of the magnitude of the radial drift rate as a function of radius for 58 near-Earth asteroids from M. Fenucci et al. (2024), having semimajor axes near 1 au and formal SNR > 10. Lines in Figure 8 show Equation (27) with assumed values kY = 0.01, 0.1 and 1. The data are reasonably well matched by the expected 1/a size dependence of the drift rate (Equation (27)) and by kY ∼ 0.02 to 0.13. We take kY = 0.05 as a representative value. kY ∼ 0.05 implies a lag angle ( ) q = ~ - sin 0.05 1 0.05 radians, or about 3°. It should be remembered that the data are biased toward asteroids with the largest kY, because small values of kY give small and difficult-to-measure (low SNR) drift rates. Given a random distribution of asteroid spin vectors, the above model predicts that the ratio of positive (corresponding to prograde rotators) to negative (retrograde rotator) values of drH/dt should be close to unity. Instead, in both the studies by A. H. Greenberg et al. (2020) and by M. Fen- ucci et al. (2024), fully 70% of the high-quality determinations have drH/dt < 0, giving a ratio negative drift/positive drift = 2.3, and indicating an excess of retrograde rotators. (Figure 9). This inference matches measurements of the spins of near-Earth asteroids, which show a preponderance of retrograde rotators (retrograde/ prograde ratio ∼ 2; A. La Spina et al. 2004). The bias in favor of retrograde rotation in the near-Earth population is thought to occur because inward-drifting main-belt asteroids are more easily able to reach the ν6 secular resonance responsible for their deflection into near-Earth space. 2. Origin of Meteorites. Ultimately, Yarkovsky drift helps to supply the meteorites. The asteroid belt is crossed by numerous mean-motion and secular resonances, near which the orbits of bodies are unstable. Orbital eccentricities of resonant asteroids are excited until they become planet-crossing and short-lived. The mean- motion resonances would be nearly empty if it were not for Yarkovsky drift, which is responsible for feeding resonance regions with nearby asteroids through semi- major axis drift (P. Farinella et al. 1998). About 80% of the near-Earth objects are supplied from the ν6 secular resonance and, spectrally, two-thirds of these are S-type asteroids (J. A. Sanchez et al. 2024). The S-types are related to the thermally metamorphosed LL chondrite meteorites, themselves fragments of bodies from the inner belt. 3. Dispersal of Asteroid Families. Shattering collisions between asteroids produce families of objects having initially similar orbital elements. However, family asteroids experience a size-dependent drift from their initial orbital semimajor axes under the action of the Yarkovsky force. By Equation (27), the semimajor axis drift in time Δt is ΔrH ∝ Δt/a, and ΔrH can be positive or negative depending on the sense of rotation of the asteroid. Small asteroids drift the farthest, all else being equal, giving rise to a V-shaped distribution in a plot of Figure 8. Absolute value of the radial drift rate plotted as a function of asteroid radius in meters. Lines show the model drift rate calculated from Equation (27) with ρ = 103 kg m−3 , rH = 1 au, and kY = 0.01 (red dashed line), 0.1 (solid black line), and 1 (dashed blue line). Evidently, the data are broadly consistent with kY ∼ 0.05. Data are from the compilation by M. Fenucci et al. (2024). 13 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
semimajor axis versus 1/a. This effect is observed in the main asteroid belt, where it has been used to identify families and estimate their ages. An example is shown in Figure 10 which compares data for the Erigone family with Equation (27) (we assumed kY = 0.05, ρ = 2000 kg m−3 and initial semimajor axis rH = 2.37 au). There is some ambiguity in defining the edges of the distribution, but spreading ages near 70–100 Myr are Figure 9. Histogram of near-Earth asteroid radial drift rates showing an excess with negative values. The ratio of (inward-drifting) retrograde rotators to (outward drifting) prograde rotators is ∼2.3:1. Data refer to subkilometer asteroids from M. Fenucci et al. (2024). Four objects with drH/dt outside the plotted range have been excluded for clarity of the plot. Asteroids with formal SNR < 10 have not been considered. Figure 10. Yarkovsky spreading diagram for the Erigone main-belt asteroid family. The vertical dashed black line shows rH = 2.37 au, the semimajor axis of the presumed family parent body. Solid blue and red dashed lines show trajectories for family ages 70 and 100 Myr. Data are from B. T. Bolin et al. (2018), and models are from Equation (27). 14 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
plausible, while much larger and smaller ages are not. B. T. Bolin et al. (2018) estimated an age of 90 Myr. The method is not perfect. The initial orbits may not have exactly the same semimajor axis, the effect of asteroid obliquity is not included (Equation (27) gives the maximum value, for obliquity = 0°) and the individual obliquities are in any case not known, the sizes and densities of asteroids are approximate, and external perturbations (e.g., from planets) can act to spread the orbits (especially for the older clusters). Nevertheless, Yarkovsky spreading gives an impressive explanation for the V-shaped diagram and a useful measure of the family age. 4. Post-main-sequence Evolution of the Sun. In 5 Gyr, the Sun will exhaust its core hydrogen and bloat into a red giant, with a luminosity ∼103.5 times its present value (E. Vassiliadis & P. R. Wood 1993). When this occurs, the Yarkovsky force will likewise jump by a factor ∼103.5 causing accelerated orbital drift to compete with many other processes to destabilize small orbiting bodies of the solar system. Destabilized planetary systems are likely implicated in feeding debris to the metal-polluted central white dwarf remaining after the red giant phase (D. Veras et al. 2019). 5. Binary Yarkovsky. The Yarkovsky force can affect a binary in two ways. First, in a binary with small separation, thermal radiation from each component can heat the other, resulting in a kind of radiation drag distinct from the Yarkovsky drag caused by direct sunlight. Second, if the orbital inclination is also small, the two components will pass through each other's shadow in each orbit. Since the thermal response of the surface material is lagged (see the Appendix), the entry into and exit from the shadow introduces an additional asymmetry into the radiative force that results in yet another torque. These subtle effects, the former somewhat unfortunately named “planetary Yarkovsky” and the latter “Yarkovsky- Schach” drag, may rival tidal dissipation (Section 5.1) and BYORP (Section 8.2) in the orbital evolution of some close binary asteroids (W.-H. Zhou et al. 2024). 7. Lorentz Force A charged particle interacts with a moving magnetic field through the Lorentz force, L. For simplicity, we ignore the fact that L is a vector acting perpendicular to both the velocity of the particle and the direction of the field, and we consider only the magnitude of the force, =  BqV L . Here, B is the magnetic flux density, q is the charge on the particle, and V is the velocity of the particle with respect to the field. To evaluate the Lorenz force, we first need to know the charge on a particle. Coulomb's Law gives the force between two charges, q, separated by distance, r, as ( ) / pe =  q r 4 2 0 2 , where ò0 = 8.854 × 10−12 F m−1 is the permittivity of free space. The work done in bringing a charge q from r = ∞ to the surface of a particle of radius r = a is just ( ) / ò pe = = ¥  E dr q a 4 a 2 0 . The Volt is a measure of the work done, E, in moving a charge, q, through a potential difference, U. Then, U = E/q gives the relation ( ) p =  q Ua 4 28 0 for the charge on a particle of radius a when its potential, U, is known. Numerous effects contribute to the charging of dust particles in the solar system, but the dominant effect is photoionization from solar ultraviolet, which leads to U ∼ 5–10 Volts (S. P. Wyatt 1969; corresponding to photon wavelengths ∼1000–2000 Å). This potential can vary depending on fluctuations in the solar ultraviolet flux, itself a function of the ∼22 yr solar magnetic cycle. The potential, U, depends on the photon energy and the ionization threshold of the material, and is therefore approximately independent of distance from the Sun. However, the photon flux varies as - rH 2 and so the charging time increases as rH 2 ; it is ∼103 times longer in the Kuiper Belt at 30 au than near the Earth at 1 au, elevating the relative importance of leakage currents that ultimately can limit the accumulated charge. In the following, we take U = 10 Volts independent of distance. At distances rau ? 1, the Sun's Parker spiral (e.g., M. J. Owens & R. J. Forsyth 2013) is wound so tightly as to be nearly azimuthal. In the planetary region, the azimuthal magnetic flux density approximately follows ( ) = - B r B r H H 1 1 , with B1 ∼ 600 T m and rH expressed in meters (as determined from Figure 11 of A. Balogh & G. Erdõs 2013), while the field is swept with the solar wind at speed V ∼ 500 km s−1 , approximately independent of heliocentric distance. The field does exhibit substantial fluctuations through the solar cycle and also varies with heliographic latitude; we ignore these effects here for simplicity. Substituting for q from Equation (28) and for B and dividing by the particle mass gives the particle acceleration due to the Lorentz force as ( ) a e r = B UV a r 3 . 29 L H 1 0 2 Further dividing by the local gravitational acceleration (Equation (10)) defines the magnetic β parameter βL = αL/ge ( )  b e r = B UV GM r a 3 . 30 L H 1 0 2 ⎛ ⎝ ⎞ ⎠ Substitution gives ( ) b ~ m r a 0.1 31 L au 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ with rau being the heliocentric distance expressed in astronom- ical unit, and aμ is the particle radius expressed in microns. Figure 11 shows β and βL as functions of heliocentric distance for particles with density ρ = 103 kg m−3 and 1, 10, 102 , and 103 μm in radius. Equations (30) and (31) show that the dynamical importance of the Lorentz force is maximized for the smallest particles and the largest heliocentric distances. 7.1. Examples 1. Interstellar Dust. Spacecraft with velocity-measuring impact detectors record dust particles traveling faster than the local gravitational escape speed from the Sun (E. Grun et al. 1985). These are of interstellar origin but are, on average, substantially larger than the interstellar dust known from astronomical measurements of extinc- tion and polarization. The impact energies correspond to particles several tenths of a micron (up to ∼0.4 μm) in 15 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
size, whereas the extinction and polarization signatures of the interstellar medium are due to dust with sizes down to a ∼ 10 nm. Some small particles are deflected by radiation pressure but, as a → 0, Qpr = 1, leaving the Sun's magnetic field to deflect the smallest and most abundant (I. Mann & A. Czechowski 2021). For example, consider dust approaching the heliopause at 25 km s−1 , where B ∼ 0.5 nT (L. F. Burlaga et al. 2022). These parameters give βL > 1 for a < 0.1 μm, so smaller particles are deflected from the planetary region by the heliospheric magnetic field, consistent with dust counter measurements (V. J. Sterken et al. 2013). The penetration distance of interstellar dust is a function not only of particle size but also of time, because of the Sun's 22 yr magnetic cycle (M. Landgraf 2000). 2. Comet Dust. The Lorentz force acts roughly perpendicu- larly to the Sun–comet line, allowing for the possibility that the dust morphology of a comet might be altered by the Lorentz force relative to that expected on the basis of radiation pressure and solar gravity alone. The ratio of the Lorentz acceleration (Equation (31)) to the radiation pressure acceleration (Equation (12)) in the geometric optics limit is ( ) b b ~ m r a 0.2 . 32 L au ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ Near 1 au, βL/β = 1 for the micron-sized and larger particles that dominate the scattering, providing justifica- tion for the neglect of Lorentz force in models of comets in the terrestrial planet region. Equation (32) suggests that the effects of Lorentz force should be more significant at larger heliocentric distances where, unfortunately, we possess few relevant observations. Three exceptions are the LPCs C/2014 B1 (Schwartz), C/2010 U3 (Boattini), and C/1995 O1 (Hale–Bopp). C/2014 B1 (Schwartz) was dominated by quite large (aμ > 100) particles at rau ∼ 10 (D. Jewitt et al. 2019). It has βL/β  0.02 by Equation (32), and showed no effect from Lorentz force. Observations of C/ 2010 U3 (Boattini) at larger distance (rau ∼ 20 au) showed smaller particles (aμ = 10; M.-T. Hui et al. 2019). Equation (32) gives βL/β ∼ 0.4, consistent with a larger but still not dominant influence of the Lorentz force. C/1995 O1 (Hale–Bopp) was observed at rau ∼ 20 and reported to show a coma of small particles with aμ ∼ 1 (E. A. Kramer et al. 2014). The coma could not be well matched by models using only radiation pressure. With the above values, Equation (32) gives βL/β ∼ 4, consistent with a Lorentz force dominated morphology. In the Kuiper Belt, at rH = 40 au, the motion of all particles smaller than about 8 μm, including the micron- sized particles detected by the New Horizons particle counter (A. Doner et al. 2024), should be affected by the Lorentz force. At very small particle sizes, a = λ, the geometric optics limit implicit in Equation (32) breaks down, and the radiation force is reduced by a factor Qpr < 1, increasing βL/β. The motion of particles a  100 nm is strongly affected by Lorentz forces (I. Mann & A. Czechowski 2021). 3. Dust in Planetary Magnetospheres. The giant planets sustain strong dynamo-generated magnetic fields and provide numerous examples where magnetic forces are important. For example, the equatorial surface field of Jupiter is BJ ∼ 400 μT, about 105 times stronger than the Figure 11. βrad (solid red lines) and βL (dashed blue lines) for particles with radii 1, 10, 102 , and 103 μm and density ρ = 103 kg m−3 , as functions of heliocentric distance. βL > βrad only for the smallest particles at the largest heliocentric distances. 16 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
solar wind field near Earth, and magnetic forces are of correspondingly greater significance. Strong magneto- spheric fields are able to retain plasma that would otherwise be quickly stripped away by the solar wind, creating a local gas environment that is distinct from the open interplanetary medium. The magnetospheres of Jupiter and Saturn are filled with dense plasma (∼108 –109 m−3 , or 102 –103 times the solar wind density) from the outgassing satellites Io and Enceladus, respec- tively. Dust near these satellites is charged negatively by contact with the plasma (because the electrons travel faster in thermal equilibrium and so deliver a larger charging current than the heavier, slower ions). The charge reverses to positive values at larger distances where the effects of plasma charging are overcome by those of UV photoelectron loss (F. Spahn et al. 2019). Charged dust particles are accelerated by the planetary field, which sweeps past at the corotation speed. One consequence of magnetospheric dust interactions is that Jupiter (E. Grun et al. 1993) and Saturn (S. Kempf et al. 2005) eject streams of nanometer-sized particles (mass ∼ 10−24 kg, magnetic βL ? 1) at speeds above the gravitational escape speed from either planet (H.-W. Hsu et al. 2012). Another is that the B-ring of Saturn displays transient, 104 km scale, quasiradial dust structures known as spokes, whose abundance is seasonally modulated by the ring illumination but also by Saturn's rotating magnetic field (B. A. Smith et al. 1981; C. A. Porco & G. E. Danielson 1982). The origin of spokes is still not fully understood, but it is clear that they consist of micron-sized dust particles briefly elevated above the Saturn ring plane and with motions that reflect the combined influence of local electrostatic and magnetic forces, as well as planetary gravity (M. Horányi et al. 2004). 7.2. Summary The relative magnitudes of the accelerations are compared in Figure 12. For this purpose, we evaluate each acceleration at rH = 1 au and for a body radius a = 1 km, and we divide by the local solar gravitational acceleration (ge = 0.006 m s−2 at rH = 1 au). 8. Torques Most asteroids and cometary nuclei are irregular in shape. On such objects, the nongravitational forces described above generally do not pass through the center of mass, resulting in a torque. Left unchecked, the torque will drive inexorably toward rotational instability, where the centripetal forces exceed the gravitational and cohesive forces holding the body together. A strengthless oblate ellipsoid of density ρ and with equatorial axes a = b and polar axis c „ a has critical period ( ) / / t p r = G a c 3 . 33 crit 1 2 1 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ For a sphere (a = c) of density ρ = 103 kg m−3 , Equation (33) gives τcrit = 3.3 hr. A highly oblate body of the same density but with a / c = 2 would have a critical period 21/2 times longer, or 4.7 hr. A majority of the measured rotation periods of solar system bodies are within a factor of a few of τcrit (B. D. Warner et al. 2009), suggesting the importance of centripetal forces. However, it must be admitted that observa- tional biases against the measurement of periods either much shorter or much longer than a few hours are strong in most existing data sets. In general, applied torques change both the direction and the magnitude of the spin vector, inducing excited or nonprincipal axis rotation. Observationally, precessional effects tend to be subtle, requiring longer sequences of observation than are commonly available. Therefore, in the following discussion, we concentrate on the more easily measured changes to the magnitude of the spin (i.e., to the rotational period) for which we already possess abundant evidence. Torques due to sublimation and to radiation are of particular significance for the destruction of comets and asteroids, respectively. We consider them separately next. In both cases, a simple dimensional treatment is informative. We note that the timescale for an applied torque, T, to change the spin is just τs ∼ L/T, where L is the angular momentum of the body. For a homogeneous sphere of radius a and density ρ, the angular momentum is L = 16π2 ρa5 /(15P), where P is the instantaneous rotation period. 8.1. Sublimation Torque The magnitude of the sublimation torque on a comet is equal to the momentum lost per second in sublimated material Figure 12. Relative magnitudes of the NGAs discussed in the text, relative to solar gravity, for rH = 1 au and object radius a = 1 km. 17 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
multiplied by the moment arm, which we write as kTa, where a is the nucleus radius and 0 „ kT „ 1 is a dimensionless multiplier. kT = 0 corresponds to perfectly isotropic sublima- tion with no net torque. kT = 1 corresponds to perfectly collimated ejection tangential to the surface of the body. Then, with a mass-loss rate μmHQg [kg s−1 ] and an outflow speed Vth, the magnitude of the torque is T = μmHQgkTVtha. The timescale for the torque to change the spin is then (D. Jewitt 2021) ( ) t p r m ~ a m Q k V P 16 15 1 . 34 s g T 2 4 H th Note that, for a fixed Qg, Equation (34) has a very strong (a4 ) nucleus radius dependence. However, it is natural to expect that the production rate, Qg, should scale with the nucleus area, Qg ∝ a2 and, if it does, τs ∝ a2 is anticipated for the nuclei of comets, all else being equal. The rotational lightcurves of some comets have enabled the measurement of spin changes (e.g., R. Kokotanekova et al. 2018), allowing τs to be directly estimated. With additional observational constraints on ρ, a, Qg, and P, and with the use of Equation (34), the dimensionless moment arm, kT, can be estimated (D. Jewitt 2021). The median value is kT = 0.007, meaning that only 0.7% of the outflow momentum is used to torque the nucleus. Even this tiny fraction is sufficient to quickly modify the spins of small comets. Measured values of τs as a function of a are plotted in Figure 13. The relation ( ) t ~ a 100 35 s 2 with a in kilometers and τs in years, matches the data well (D. Jewitt 2021). (This relation strictly applies to JFCs with perihelia in the 1–2 au range.) Spin-up by outgassing torques ends with rotational instability and breakup. The fragmented nucleus of 332P/Ikeya–Murakami at rH = 1.6 au gives a good example (Figure 14). A cloud of fragments expands from main nucleus C, whose rotation in about 2 hr suggests rotational instability (D. Jewitt et al. 2016). A necessary condition for Equation (34) to remain valid is the persistence of ice at or near the physical surface of the nucleus for timescales τs. Near-surface ice persists because the speed with which the ice sublimation surface erodes into the nucleus exceeds the speed with which heat conducts into the interior, causing fresh ice to be continually excavated. The particular problem is that sublimation exposes refractory particles too large to be ejected by gas drag. These should eventually clog the surface, producing a “rubble mantle” and inhibiting or shutting down further sublimation. How this works in detail is not known, even after several years of in situ investigation of the nucleus of 67P/Churyumov–Gerasimenko by the Rosetta spacecraft (N. Attree et al. 2023). Orbital evolution may play a role, particularly when the perihelion distance migrates to smaller values, leading to higher temperatures and sublimation fluxes. In the case of the active asteroids, ice is exposed only intermittently, perhaps in response to occasional impacts that clear a surface refractory mantle. The duty cycle (ratio of the time spent in sublimation to the total elapsed time) is less than 10−4 or even 10−5 , allowing ice to persist for long times but rendering Equation (35) inapplicable to these objects. The significance of the short timescales indicated by Equation (35) is that small nuclei cannot survive long once they reach the vicinity of the Sun. This may explain the paucity of subkilometer comet nuclei relative to power-law extrapola- tions from larger sizes. It also complicates any attempt to relate the populations and properties of comets near the Sun to their Figure 13. Measured timescale for changing the rotation period, τs, as a function of the nucleus radius for SPCs with perihelia in the 1–2 au range. Filled red circles show spin-change detections, while yellow diamonds show lower limits to the allowed spin-up timescales. Comets are identified by their numerical labels. The dashed line shows τs = 100a2 . Data are from D. Jewitt (2021). 18 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
similarly sized counterparts in the Kuiper Belt and Oort cloud source regions. 8.2. YORP Torque Nonspherical bodies warmed by the Sun radiate infrared photons, which carry away momentum and can exert a torque (e.g., D. P. Rubincam 2000). Here we offer a simplified treatment that captures the physical essence of the torque, followed by some examples of its application to real bodies. The torque is proportional to the surface area, a2 , multiplied by the moment arm, which we write as ¢ k a T , where   ¢ k 0 1 T is another dimensionless constant that must be empirically determined. Again setting timescale τYORP ∼ L/T, we obtain, ( )  t pr = ¢ a c k P r S 16 15 . 36 T YORP 2 au 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ The dimensionless moment arm, ¢ kT , is such a sensitive function of the body shape, surface roughness, and thermo- physical parameters, all of which are unknown for most asteroids, that it cannot in general be calculated. Instead, we rely on measurements of the few asteroids where changes in the rotation periods can be measured and the other parameters in Equation (36) are constrained. These asteroids are listed in Table 1, where measurements of a, P, and dP/dt are from the Figure 14. Labeled fragments released from component C of comet 332P/Ikeya–Murakami in 2015 and separating from it at speeds 0.06–4 m s−1 . C itself rotates with a probable period near 2 hr, suggesting rotational instability as the cause of the release of fragments. Data are from D. Jewitt et al. (2016). Table 1 YORP Timescalesa Object D b P c dω/dtd τe (1620) Geographos 2.56 5.22 1.14 ± 0.03 6.9 ± 0.2 (1685) Toro 3.5 10.20 0.33 ± 0.03 12.3 ± 1.0 (1862) Apollo 1.55 3.07 4.94 ± 0.09 2.7 ± 0.05 (2100) Ra-Shalom 2.30 19.82 <0.6 >3.5 (3103) Eger 1.78 5.71 <1.5 >4.8 (10115) 1992 SK 1.0 7.32 8.3 ± 0.6 0.68 ± 0.05 (25143) Itokawa 0.32 12.13 3.54 ± 0.38 1.0 ± 0.1 (54509) YORP 0.11 0.20 350 ± 35 0.59 ± 0.05 (85989) 1999 JD6 1.53 7.66 <1.2 >4.5 (85990) 1999 JV6 0.44 6.54 <7.2 >0.9 (68346) 2001 KZ66 0.80 4.99 8.43 ± 0.69 1.0 ± 0.1 (101955) Bennu 0.49 4.30 6.34 ± 0.91 1.5 ± 0.2 (138852) 2000 WN10 0.3 4.46 5.5 ± 0.7 1.7 ± 0.2 (161989) Cacus 1.0 3.76 1.86 ± 0.09 5.9 ± 0.3 Notes. a Asteroid data from J. Ďurech et al. (2024). b Diameter in kilometers. c Rotation period in hours. d Rate of change of the angular frequency ×10−8 [radian day−1 ]. Values from J. Ďurech et al. (2024) that are statistically insignificant are listed as 3σ upper limits. e YORP time, Myr. 19 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
compilation by J. Ďurech et al. (2024; see also B. Rozitis & S. F. Green 2013 and references therein). Figure 15 shows the YORP timescale computed from τYORP = P/(dP/dt), with dP/dt = − (P2 /2π)dω/dt and dω/dt from the Table. τYORP is shown as a function of the asteroid radius with each object labeled by its identifying number. The asteroids used in Figure 15 all have semimajor axis ∼1 au. The Figure clearly shows that τYORP and a are correlated, with larger asteroids showing longer YORP timescales. Adopting the form of Equation (36) and the data from Figure 15, we estimate [ ] ( ) t ~ a r 4.5 Myr 37 YORP 2 au 2 where a is the radius in kilometers, and rau is the semimajor axis in astronomical unit. By this equation, across the asteroid main belt from 2.1–3.3 au, a 1 km radius asteroid would have τYORP ∼ 20–50 Myr. Setting τYORP = 4500 Myr in Equation (37) and solving for a, we find that YORP can influence the spins of main- belt asteroids up to a ∼ 10–14 km. This is in broad agreement with the asteroid rotational barrier (Figure 16), which is prominent for asteroids up to about 10 km in diameter (implying spin-up and breakup for smaller bodies) but less evident beyond about 30 km, where primordial spin is likely preserved (e.g., P. Pravec et al. 2002). It should be noted that the dashed blue line in Figure 15 shows a least-squares fit to the data, weighted by the estimated uncertainties on τYORP (but not on a). It gives τYORP ∝ a1.87±0.04 . Although the value of the index is formally (3σ) smaller than the expected value, τYORP ∝ a2 from Equation (36) (shown in the Figure as a solid red line), the difference is likely not important given the existence of systematic errors in the sample. A major limitation to the order-of-magnitude estimates of the torque timescales, both for sublimation (Equation (35)) and for YORP (Equation (37)), concerns the stability of the torque. In comets, the magnitude and direction of the outgassing torque depend on the surface distribution and angular dependence of the outgassing. As the surface erodes, we expect the spatial and angular distribution of outgassing sources to change, altering the torque. For most comets, we possess few or no observational constraints on the areal distribution of sources or their evolution. Indeed, in situ measurements from 67P/Churyumov–Gerasi- menko show the difficulty in modeling this process even given the most detailed data (N. Attree et al. 2023). Likewise, the YORP torque is highly sensitive to the surface shape and texture and can change in magnitude and even direction in response to minor surface changes (T. S. Statler 2009). Empirical but indirect evidence for this comes from asteroid (162173) Ryugu (a ∼ 0.5 km), which has the characteristic diamond shape indicative of rotational instability but a current rotation period near 7.6 hr. The rotation of Ryugu may have slowed in response to a changing YORP torque, leaving its equatorial ridge as evidence of its previously rapid spin. Spin evolution in the presence of a changing torque may be more akin to a random walk process than to the steady change implicit in Equations (35) and (37) (W. F. Bottke et al. 2015; T. S. Statler 2015). The relevant spin-change timescales would then be much longer than estimated here. Unfortunately, we possess too little information to address this problem with any confidence. 8.3. Examples 1. Asteroid Spin Barrier. The distribution of asteroid spin frequencies as a function of absolute magnitude, H, is shown in Figure 16. H is related to the asteroid diameter Figure 15. Empirical timescale for YORP spin-change for asteroids near 1 au as a function of the asteroid radius [km]. Data with upward pointing arrows represent 3σ lower limits to τYORP based on nondetections of rotational acceleration. They are not included in the fit. The dashed blue line shows a weighted least-squares fit to the detections, τYORP ∝ a1.87±0.04 . The solid red line shows Equation (37) for rau = 1. Data from J. Ďurech et al. (2024). 20 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
by D = 1329 × 10−0.2H /p1/2 , where p is the geometric albedo. (We assumed p = 0.1 to calculate the diameters shown along the top axis of the Figure; in fact, the albedos of a majority of the plotted asteroids have not been measured.) Few asteroids larger than D ∼ 0.2 km have periods shorter than ∼2.4 hr (marked in the Figure by a horizontal, dashed red line) whereas, among smaller asteroids, more rapid rotation is common (P. Pravec et al. 2002; J. Beniyama et al. 2022). This is (admittedly indirect) evidence for spin-up by YORP of a “bunch of grapes” internal structure in which centripetal accelera- tions in asteroids with periods  2.4 hr result in mass shedding or breakup into smaller, more cohesive component units. Measured periods of some tiny asteroids are as short as a few seconds (J. Beniyama et al. 2022). 2. Asteroid Reshaping and Pair Formation. Separate evidence for rotational reshaping of asteroids is provided by the morphologies of some spacecraft-visited asteroids, which show approximate rotational symmetry but with an equatorial skirt consisting of material that has evidently migrated from higher latitudes (D. J. Scheeres 2015). New observations of asteroids have also revealed cases of episodic (311P/Panstarrs (2013 P5); D. Jewitt et al. 2013) and catastrophic (P/2013 R3, Catalina-Pan- STARRS; D. Jewitt et al. 2017) mass loss that indicate mass shedding and rotational breakup in real-time. Like 311P, the 2 km radius asteroid (6478) Gault displayed episodic mass loss consistent with mass shedding instability and also has a rotation period (2.55 hr) suggestively close to the rotational barrier (J. X. Luu et al. 2021). See Figures 17 and 18. Extreme end-cases of continued spin-up under the influence of YORP torque include rubble-pile disaggregation (D. J. Scheeres 2018), which might have been observed in P/2013 R3 (Figure 18), and the formation of asteroid binaries and pairs. These are independent asteroids with orbital element similarities that are statistically improbable by chance alone (S. A. Jacobson & D. J. Scheeres 2011; K. J. Walsh 2018). Asteroid pairs show a systematic relation between the angular frequency of the primary and the secondary/primary mass ratio, such that high mass ratio pairs have distinctly long-period primaries (P. Pravec et al. 2019). This is a result of the combined action of primary spin-up by YORP torques and tidal transfer of rotational energy from the primary, needed to expand the orbit of the secondary. Sudden mass transfer events on asteroids should lead to excited rotation; relevant observations are difficult and presently lacking. 3. Spin Alignment. It might be expected that collisionally produced asteroid families should have a very broad or even random distribution of spin vectors. S. M. Slivan (2002) discovered that the spin vectors of the Koronis family asteroids are instead clustered, with obliquities preferentially near 45° and 170°. This clustering was subsequently modeled as a consequence of interaction between gravitational and YORP torques (D. Vokrouhli- cký et al. 2003). Subsequent work showed that the general asteroid obliquity distribution is size dependent (J. Hanuš et al. 2011). Objects of diameter  30 km are more likely to have YORP-aligned obliquities near 0° and 180° (see Figure 7(a) of J. Hanuš et al. 2011). The efficacy of alignment by radiation torques is dependent not just on size but also on many poorly constrained physical and thermophysical parameters (O. Golubov Figure 16. Distribution of asteroid rotational frequencies [rotations day−1 ] as a function of absolute magnitude, H. Approximate asteroid diameters are marked at the top of the Figure, while their rotation periods are indicated on the right-hand axis. The dashed red line marking the “spin barrier” at P−1 ∼ 10 day−1 shows that few asteroids with diameter D  0.2 km have periods <2.4 hr. Data are from B. D. Warner et al. (2009; updated 2023 October 1 from the F-D Basic file at http://www. MinorPlanet.info/php/lcdb.php). 21 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
et al. 2021). The timescale for changing the obliquity can be shorter than the timescale for changing the spin (T. S. Statler 2015). 4. Post-main-sequence Evolution. As with the Yarkovsky force, the YORP effect will also be magnified as the Sun depletes its core hydrogen and its luminosity surges in the giant phase, by a factor up to ∼103.5 (E. Vassiliadis & P. R. Wood 1993). Rotational disruption of extrasolar asteroids by YORP spin-up may be an important contributor to debris that contaminates the photospheres of some white dwarf stars (D. Veras et al. 2014), where median mass accretion rates are in the range 105 –106 kg s−1 (J. Williams et al. 2024). 5. Tangential YORP (TYORP). The YORP torque described above is a result of gross deviations from rotational symmetry in the body shapes of asteroids. This is sometimes referred to as normal YORP or NYORP. Another radiation-driven torque, known as tangential YORP (TYORP), can be supplied by rocks and other small-scale surface structures (O. Golubov & Y. N. Krugly 2012). TYORP is different from NYORP in that it can exist even on a spherical asteroid, provided its surface is littered with rocks of the appropriate scale. Consider a rock of size, a, on the surface of an asteroid rotating with period, P. The conduction cooling time of the rock is τc ∼ a2 /κ (Equation (A2)), where κ is the Figure 17. Two active asteroids showing episodic ejections likely due to YORP-driven rotational instability. Each tail corresponds to a particular synchrone; the tail position angle is a measure of the date of ejection. 311P/Panstarrs (top) exhibited nine ejections over about 9 months (six visible on 2013 September 10 plus three later; D. Jewitt et al. 2013) while (6478) Gault displayed three tails in 2019 (J. X. Luu et al. 2021). 22 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
thermal diffusivity. If τc = P, the rock temperature will quickly equilibrate to the changing insolation through the day. Its temperatures at sunrise and sunset will be equal. If τc ? P, the rock temperature will be determined by the average of the day/night insolation, with little temporal variation. Only when τc ∼ P will the rock temperature exhibit substantial diurnal variation, being hotter near local sunset than near local sunrise because of cooling in the night. This temperature asymmetry can produce a torque with a large moment arm, because radiation and recoil from the warm Sun-facing side of a rock at sunset is not balanced by radiation and recoil from an equally warm Sun-facing side at sunrise. The critical rock size is aTY ∼ (Pκ)1/2 . For example, with P = 5 hr, κ = 10−6 m2 s−1 (as appropriate for a nonporous, consolidated material), aTY ∼ 0.15 m. Decimeter-sized rocks exert particularly strong tangential YORP torques. The magnitude of TYORP again depends on many unknown properties of the asteroid surface, notably the size and spatial distributions of surface rocks, and the degree to which each rock is separated from its neighbors enough to be thermally independent. But under some circumstances, the magnitude of TYORP may rival that of NYORP (O. Golubov & Y. N. Krugly 2012). More- over, the TYORP and NYORP torque vectors need not act in the same direction, raising the possibility that one might cancel the other, leading to no net torque on a body that otherwise might be expected to exhibit rotational acceleration. 6. Binary YORP (BYORP). Angular momentum added by photons can have particularly dramatic dynamical effects on some asteroid binaries (M. Ćuk & J. A. Burns 2005). By itself, tidal dissipation in close binaries commonly leads to synchronous rotation of the secondary (in which the orbit period and rotation period of the secondary are the same). In the synchronous state, the secondary can act effectively as an asymmetric extension of the primary, providing a large lever arm for the action of the radiation torque (M. Ćuk & J. A. Burns 2005). This torque is known as BYORP (B for binary). In nonsynchronous binaries, the effect of BYORP is averaged to zero. The evolution of binary asteroids under the combined action of tidal dissipation and BYORP radiative torque can lead to complicated orbital and spin evolution (S. A. Jacobson & D. J. Scheeres 2011), with evolutionary timescales as short as ∼104 yr (E. Steinberg & R. Sari 2011). If the tidal torque on the secondary is small compared to the radiative torque, the satellite can escape synchronous rotation and follow its own spin evolution, potentially leading to rotational breakup in orbit. The fragments from such a breakup would have short mutual collision times and quickly reaccumulate into a new rubble-pile satellite that repeatedly transforms into a ring before collapsing back into a single body. 8.4. Summary The timescales for the action of water ice sublimation torque (τS, Equation (34)) and YORP torque (τYORP, Equation (36)) are compared as functions of heliocentric distance in Figure 19. In the Figure, solid and dashed lines denote assumed object Figure 18. Rotationally disrupted asteroids 331P/Gibbs (top, from D. Jewitt et al. 2021) and P/2013 R3 (bottom, from D. Jewitt et al. 2017). Major fragments in each object are labeled. The largest of these in 331P is the 100 m scale fragment A, which is also a contact binary. Sizes of the fragments in P/2013 R3 are less certain because of heavy dust contamination, but estimated to be 100 m. Both objects are active asteroids, possessing cometary designations but having asteroid orbits (D. Jewitt 2012). 23 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
radii of 1 km and 0.1 km, respectively. The sublimation model was computed assuming ( ) q cos = 1/2 in Equation (4), which lies in between the high- and low-temperature limit models in Figure 1, and is scaled to timescale τs = 102 yr on a 1 km nucleus at 1 au, in agreement with the data (Figure 13). Figure 19 shows that, for a given object size and distances  5 or 6 au, the timescales for spin-change are ∼105 times shorter for sublimation torques than for YORP torques. Beyond Saturn, the diminishing water ice sublimation rate pushes the sublimation timescale to exceed that from YORP, and, in practice, both processes become so slow in the middle and outer solar system as to become largely irrelevant. Acknowledgments I thank Marco Fenucci for providing a digital table of his Yarkovsky drift measurements, Pedro Lacerda, Jane Luu, Joe Masiero, Darryl Seligman, David Vokrouhlicky, Emerson Whittaker, and an anonymous referee for comments. Appendix Thermophysics To gain some physical insight into how the thermophysical parameters of an asteroid might affect Yarkovsky and YORP, we consider the one-dimensional heat conduction equation ( ) r ¶ ¶ = ¶ ¶ c T t k T z , A1 p 2 2 where T is the temperature, k is the thermal conductivity, ρ is the bulk density, and cp is the specific heat capacity of the asteroid surface materials. Dimensionalizing this equation gives the timescale for heat to conduct over a distance ℓ as ( ) t k k r ~ = ℓ k c where A2 C p 2 and κ [m2 s−1 ] is the thermal diffusivity.4 For a body with rotation period P, we set τC = P and use Equation (A2) to find the distance over which heat can conduct in one rotation period as ℓ = (κP)1/2 , which defines the diurnal thermal skin depth on the asteroid. An exact solution of Equation (A1) would show that T varies with depth as a damped sinusoid, with (κP)1/2 being the e-folding length scale of the damping, but our order- of-magnitude approximation is sufficient here. On a spherical body with radius a, the heat contained within the skin depth, ℓ, is ( ) ( ) / p r k = H a P c T 4 A3 p 2 1 2 where cp [J K−1 kg−1 ] is the specific heat capacity of the shell. Loss of heat by radiation into space occurs at the rate ( ) p s = dH dt a T 4 A4 2 4 where σ = 5.67 × 10−8 W m−2 K−4 is the Stefan–Boltzmann radiation constant, and we assume emissivity ε = 1. The order- Figure 19. The YORP (blue) and water ice sublimation (red) timescales plotted as a function of rH. Timescales for objects 1 km radius (solid lines) and 0.1 km in radius (dashed lines) are shown. The shaded band marks the approximate location of the main asteroid belt, across which sublimation supplies a ∼105 times stronger torque if near-surface ice is present. 4 Diffusivity appears directly in Equation (A1) and is the natural measure of thermal response of a material through Equation (A2). In planetary science, the use of thermal inertia is instead widely preferred. Thermal inertia is defined by ( ) / r = I k cp 1 2, which has the somewhat uncomfortable units [J m−2 s−1/2 K−1 ]. Solid dielectrics (e.g., nonporous rocks) have I ∼ 103 J m−2 s−1/2 K−1 , while the finest dust, as found in the regoliths of small outer solar system bodies, has I ∼ 1 to 10 J m−2 s−1/2 K−1 (C. Ferrari 2018). Asteroid inertias in the range 10 „ I „ 100 J m−2 s−1/2 K−1 are common (E. M. MacLennan & J. P. Emery 2021). 24 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
of-magnitude cooling timescale is τc ∼ H/(dH/dt), so that we can use Equations (A3) and (A4) to write the thermal parameter, Θ = τc/P (see J. R. Spencer et al. 1989) ( ) / Q r s k = c T P . A5 p 3 1 2 ⎛ ⎝ ⎞ ⎠ Asteroids with very rapid rotation (Θ ? 1) retain their heat through the night and should remain nearly longitudinally isothermal, producing little net diurnal Yarkovsky force. Similarly, asteroids with Θ = 1 will experience little diurnal Yarkovsky drift because rotation is too slow to carry the diurnal temperature maximum away from the midday meridian. Intuitively, on the other hand, asteroids with Θ  1 should retain heat into the afternoon but lose it by the morning and so will experience a net force from asymmetric radiation.5 As a rough example, consider an asteroid with a very porous surface regolith having κ = 10−9 m2 s−1 and with nominal ρ = 103 kg m−3 , cp = 103 J K−1 kg−1 , P = 2 × 104 s (i.e., about 6 hr), and orbiting near 1 au, where T ∼ 300 K. Substitution gives skin depth ℓ ∼ 5 mm, Θ ∼ 0.15, and the peak heat of noon will be retained for about one-sixth of a rotation, corresponding to a lag angle θ ∼ 60°. The Yarkovsky force on a small asteroid with these parameters is likely to be significant. By comparison, an asteroid having the same properties but consisting of solid rock (for which κ = 10−6 m2 s−1 would have ℓ ∼ 15 cm, Θ ∼ 4.5 and by virtue of taking several rotations in order to cool) would experience minimal diurnal temperature variation and reduced Yarkovsky acceleration. These simple considerations are only illustrative. Real asteroids are complicated and usually poorly characterized, with irregular shapes and boulder-strewn, variegated surfaces. On such bodies, the distribution of surface temperature and the magnitudes of the Yarkovsky and YORP effects cannot be accurately calculated, only worked out after the fact by measuring radial drift and changes in the rotation period. 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F., et al. 1981, Sci, 212, 163 Spahn, F., Sachse, M., Seiß, M., et al. 2019, SSRv, 215, 11 5 With more effort than is warranted here, it can be shown that the effect is proportional to Θ/(1 + 2Θ + 2Θ2 ), a function that is maximized for Θ ∼ 0.7 (D. P. Rubincam 1995). 25 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
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Nongravitational Forces in Planetary Systems

  • 1.
    Nongravitational Forces inPlanetary Systems David Jewitt Department of Earth, Planetary and Space Sciences, University of California at Los Angeles, Los Angeles, CA 90095-1567, USA; djewitt@gmail.com Received 2024 September 12; revised 2024 November 24; accepted 2024 November 25; published 2025 January 16 Abstract Nongravitational forces play surprising and, sometimes, centrally important roles in shaping the motions and properties of small planetary bodies. In the solar system, the morphologies of comets, the delivery of meteorites, and the shapes and dynamics of asteroids and binaries are all affected by nongravitational forces. In exoplanetary systems and debris disks, nongravitational forces affect the lifetimes of circumstellar particles and feed refractory debris to the photospheres of the central stars. Unlike the gravitational force, which is a simple function of the well- known separations and masses of bodies, the nongravitational forces are frequently functions of poorly known or even unmeasurable physical properties. Here, we present order-of-magnitude descriptions of nongravitational forces, with examples of their application. Unified Astronomy Thesaurus concepts: Small Solar System bodies (1469); Asteroid satellites (2207); Long period comets (933); Main-belt comets (2131); Short period comets (1452); Debris disks (363); Comets (280); Asteroids (72) 1. Introduction Gravity alone provides an ample description of the dynamics and of many physical properties of planetary-mass bodies. However, scientific interest is increasingly focused on smaller bodies in orbit around the Sun and other stars, and small bodies are additionally susceptible to a host of other forces. These so- called nongravitational forces include recoil and torque from anisotropic mass loss, radiation pressure, Poynting–Robertson drag, Yarkovsky force, YORP torque, and forces from magnetic interactions with the solar wind. Important examples of phenomena that cannot be understood using gravity alone are numerous, ranging from the motion of dust particles in comets, the Zodiacal cloud and debris disks, to the orbital drift of asteroids and their delivery into planet-crossing orbits, to the centripetal shaping and disintegration of comets and asteroids, to the formation of asteroid binaries and pairs. Unfortunately, the research literature tends to present nongravitational forces either in excruciating and largely unhelpful detail or, more usually, without meaningful discus- sion of any sort. Indeed, nongravitational forces are often hidden as lines of code in elaborate numerical models, where their practical role is to help to improve the fit to data by adding extra degrees of freedom. Sadly, they sometimes do so without giving a parallel improvement in our understanding of the relevant physics. The objective of the current paper is to provide a highly simplified but nevertheless informative account of the different nongravitational forces that are important in the solar system. We also add some pointers to sample applications from the literature, in a style suitable for the nonspecialist. With this in mind, in the following, we either neglect geometrical factors representing body shape, or approximate the relevant bodies as spheres, with bulk density ρ [kg m−3 ] and radius a [m]. The orbits of all bodies are assumed to be circular, so that the heliocentric distance is also the semimajor axis. Of course, real bodies are not spherical, and real orbits are not circles. These simplifying approximations remain useful, however, by giving a guide to the order of magnitude of the forces and timescales involved. When the heliocentric distance, rH [m], is expressed in astronomical unit, we give it the symbol rau. For example, the flux of sunlight, Fe [W m−2 ], which plays an obvious role in several of the nongravitational forces, is given by ( )     p = = F L r F S r 4 or, equivalently, , 1 H 2 au 2 where Le = 4 × 1026 W is the luminosity of the Sun, and Se = 1360 W m−2 is the solar constant. 2. Sublimation Recoil By far the largest nongravitational force considered here is that due to the sublimation of ice from small bodies heated by the Sun. Sublimated volatiles freely expand into the near vacuum of interplanetary space, carrying with them momentum and exerting a recoil force on the ice-containing parent body. Since sublimation is exponentially dependent on temperature, most sublimation occurs on the hot dayside of the nucleus, and the resulting recoil force is primarily antisolar. This sublimation recoil force is  =  k MV R th, where  M [kg s−1 ] is the sublimation rate, Vth [m s−1 ] is the bulk speed of the outflowing gas, and kR is a dimensionless constant that describes the angular dependence of the flow. Purely collimated flow would have kR = 1, while isotropic outgassing would have kR = 0, meaning no net force on the nucleus. The best estimate of kR is based on measurements from the unusually well-studied comet 67P/Churyumov–Gerasimenko, for which kR ∼ 1/2 (D. Jewitt et al. 2020). This is also the value expected from uniform sublimation across the sunward facing hemisphere of a spherical nucleus. We can also write  m = M m Qg H , where μ is the molecular weight of the sublimated ice, mH = 1.67 × 10−27 kg is the mass of the hydrogen atom, and Qg is the gas production rate in molecules per second. Setting a =  M S, where M = 4πρa3 /3 The Planetary Science Journal, 6:12 (26pp), 2025 January https://doi.org/10.3847/PSJ/ad9824 © 2025. The Author(s). Published by the American Astronomical Society. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1
  • 2.
    is the bodymass, gives a nongravitational acceleration (NGA) ( ) a m pr = k m a Q V 3 4 2 S R g H 3 th for a spherical body of radius, a, and density ρ. The outflow speed of the gas, Vth, is roughly equal to the thermal speed of the constituent molecules ( ) / pm = V k T m 8 . 3 th B H 1 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ Here, kB = 1.38 × 10−23 J K−1 is Boltzmann's constant, and temperature T refers to the surface temperature of the sublimating surface. This is typically depressed below the local radiative equilibrium temperature because a fraction of the absorbed solar energy that would otherwise drive radiative cooling is instead used to break bonds between molecules in the process of sublimation. While / µ - T rH 1 2 in radiative equilibrium, the temperature of sublimating ice is an even weaker function of heliocentric distance because of this depression. We take Vth = 500 m s−1 at 1 au and assume for simplicity that this speed applies across the terrestrial planet region. The temperature and sublimation rate per unit area, fs(T) [kg m−2 s−1 ], are connected by the energy balance equation for a surface element of area oriented with its normal offset from the direction of illumination by angle θ; ( ) ( ) ( ) ( )  q es - = + S r A T H T f T 1 cos . 4 s au 2 4 Here, A and ε are the Bond albedo and emissivity of the surface, σ = 5.67 × 10−8 W m−2 K−4 is the Stefan–Boltzmann constant, and H(T) [J kg−1 ] is the latent heat of sublimation for the ice in question. An additional, generally small term for heat conducted beneath the surface has been neglected from Equation (4). Additional information is needed to solve Equation (4). The temperature dependence of fs can be obtained from the Clausius–Clapeyron equation for the slope of the solid/vapor phase boundary (the relation dP/dT = PH/(NAkT2 ), where NA is Avagadro's number, and k is the Boltzmann constant, which is only applicable when H is independent of T) or, better, from laboratory measurements of the sublimation vapor pressure over ice as a function of temperature. The optical properties A and ε have a minor effect on the solution provided A = 1 and ε ? 0. In practice, A = 0 and ε = 0.9 are widely assumed. In Figure 1 we show solutions to Equation (4) for the three most abundant cometary ices of water, carbon dioxide, and carbon monoxide. These three, having approximate latent heats H = 2.8 × 106 J kg−1 , 0.57 × 106 J kg−1 , and 0.29 × 106 J kg−1 , respectively, are representative of low, medium, and high volatility solids. All three plotted curves trend asymptotically toward µ - f r s au 2 as rau → 0. This is because the exponential temperature dependence of fs is stronger than T4 , so that the second term on the right of Equation (4) dominates the first at the high temperatures found at small rau. Setting the radiative term in Equation (4) equal to zero gives ( ) ( )  ~ f S H T r 5 s au 2 where we have set A = 0 and θ = 0 for simplicity. Equation (5) gives a useful approximation even at rau = 1 au, where fs = 6 × 10−4 , 2.4 × 10−3 , and 4.7 × 10−3 kg m−2 s−1 , Figure 1. Equilibrium sublimation mass fluxes as a function of heliocentric distance for H2O (solid black lines), CO2 (long-dashed red lines), and CO (short-dashed blue lines) ices, computed from Equation (4). Two models are shown for each ice. The upper model for each ice shows sublimation at the subsolar temperature, taken as the highest temperature on a spherical body, while the lower model shows sublimation at the local isothermal blackbody temperature, which is the lowest possible temperature. The location of the asteroid belt is marked as a shaded blue rectangle, with inner and outer edges at 2.1 and 3.2 au. 2 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 3.
    respectively, for water,carbon dioxide, and carbon monoxide ices on a flat plate oriented normal to the Sun. Ices like carbon monoxide are so volatile that the µ - f r s au 2 regime extends over the entire planetary region while, for the intermediate volatile carbon dioxide ice, the sublimation rate inflects closer to the Sun, but still beyond the orbit of Jupiter. Across the ∼1 au width of the asteroid belt, shown in Figure 1 as a blue shaded region, the sublimation flux of water varies appreciably, by a factor of ∼4 in the high temperature limit and by a factor ∼200 in the low temperature limit. This strong variation explains the sudden rise of the outgassing activity observed in comets as they cross asteroid belt distances (e.g., N. Biver et al. 2002) and the strong concentration of outgassing from main-belt comets near perihelion (H. H. Hsieh et al. 2015). The total mass-loss rate from a sublimating body is given by the integral  ò = M f dS s over the surface, S, taking account of the object shape because it controls the local solar incidence angle θ. In practice, evaluation of this integral is impossible because the shapes of most bodies are not well determined, and the distribution of surface ice is patchy. Moreover, ice can sublimate from beneath the surface, resulting in a complicated and generally underconstrained thermophysical problem for which simple solutions do not exist. Therefore, while it is useful to consider the broad trends shown in Figure 1, it is not in general possible to calculate a global value of fs or  M from first principles. 2.1. Examples 1. Comets. Nongravitational motions are especially obvious in the orbits of short-period comets (SPCs), where the cumulative effects can be measured over many orbits. The NGA is conventionally written (B. G. Marsden et al. 1973) ( )( ) ( ) / a = + + g r A A A 6 S H 1 2 2 2 3 2 1 2 in which A1, A2, and A3 are the components of the NGA in the directions radial to the Sun, perpendicular to A1 but in the orbital plane, and perpendicular to the orbital plane, respectively. The acceleration components are conven- tionally expressed in au day−2 , where 1 au day−2 = 20.1 m s−2 . For comets, g(rH) conventionally takes a contrived analytic form based on solutions to Equation (4) for water ice (B. G. Marsden et al. 1973). For asteroids, ( ) / = g r r 1 H au 2 , where rau is the heliocentric distance in astronomical unit. The cumulative distribu- tions of A1 for Jupiter-family comets and long-period comets (LPCs) are shown separately in Figure 2 using NGA parameters from JPL Horizons.1 The astrometric measurements from which NGA is determined are fraught with systematic errors (for example, in a comet with a coma, the center of light and the nucleus may not coincide) and some measure of judgment is required in the rejection of unreliable data in determining the likely best fit. To try to reduce this problem, we have plotted in Figure 2 only measurements having signal-to-noise ratio (SNR) > 10. Figure 2 shows that the αs distributions are significantly different for LPCs and SPCs, with the LPCs showing a larger acceleration (rms value 16 × 10−7 au day−2 , corresponding to NGA α = 32 × 10−6 m s−2 ) than the SPCs (rms 4 × 10−7 au day−2 or NGA α = 8 × 10−6 m s−2 ). This difference could indicate a systematic difference in the sizes or densities Figure 2. Cumulative distribution of the A1 (radial) component of the nongravitational acceleration (NGA) for 101 short-period comets (SPCs; blue line, 2 „ TJ „ 3) and 33 long-period comets (LPCs; red line, TJ < 2). Only comets with A1 measured to >10σ significance are plotted. The LPCs show systematically larger A1, consistent with having smaller and/or less dense nuclei, and with having larger outgassing rates per unit area. 1 https://ssd.jpl.nasa.gov/ 3 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 4.
    of SPC andLPC nuclei, or a difference in the outgassing rates per unit area of surface, or some combination of these effects. A more prosaic explanation is a measure- ment bias; small NGAs cannot generally be accurately measured in LPCs because they are observed at only one apparition, whereas SPCs can be measured over multi- ple orbits and longer periods of time. Some 16/101 SPCs (but 0/33 LPCs) exhibit negative A1 parameters, which indicate unexpected sunward acceleration. These determinations, which might indicate that the astro- metric errors are underestimated, cannot be explained by recoil from dayside sublimation. 2. Dark Comets. Some small, apparently inert bodies exhibit substantial NGA, yet present no evidence for outgassing. Notable examples include the interstellar interloper 1I/ ’Oumuamua (M. Micheli et al. 2018) and the ∼150 m radius object 523599 (2003 RM; D. Farnocchia et al. 2023), as well as a number of tiny asteroids (essentially, boulders) in the 3–15 m radius range (D. Z. Seligman et al. 2023). The magnitude of the NGA in these evocatively named “Dark Comets” is too large to reflect the action of radiation pressure or Yarkovsky force but consistent with small sublimation rates. For example, the measured acceleration of 2003 RM is α = 2 × 10−12 au day−2 (α = 4 × 10−11 m s−2 ). From Equation (2) with kR = 0.5, μ = 18 (for H2O), ρ = 103 kg m−3 , and Vth = 500 m s−1 , we find Qg = 4 × 1025 s−1 (equivalent to 1 kg s−1 ), which might be small enough to have escaped detection. The Tisserand parameter with respect to Jupiter of 2003 RM is formally that of a comet (TJ = 2.96), in which case the presence of ice and outgassing should not be surprising. However, D. Farnocchia et al. (2023) preferred an origin in the outer asteroid belt and derived a much smaller Qg ∼ 1023 s−1 . The boulders described by D. Z. Seligman et al. (2023) can be accelerated by sublimation at even smaller rates. For example, the strongest (8σ) detection of NGA is in the 4 m radius boulder 2010 RF12, with A3 = (−0.17 ± 0.02) × 10−10 au day−2 (αS = 3.3 × 10−10 m s−2 ). Substitution into Equation (2) gives Qg ∼ 1019 s−1 (roughly 3 × 10−7 kg s−1 ), which is small enough to have escaped detection by any existing direct technique. Radiation pressure acceleration should be of the same order as αs but cannot account for A3, which acts perpendicularly to the projected radial line. The main puzzle presented by the accelerated boulders is their implied short mass-loss lifetimes. 2020 RF12, with mass ∼ 3 × 105 kg, can sustain mass loss at 3 × 10−7 kg s−1 for only 1012 s (3 × 104 yr). The conduction timescale is even shorter (∼6 months for a compact rock with diffusivity κ = 10−6 m2 s−1 and ∼500 yr if it is a porous dust ball with the very low diffusivity κ = 10−9 m2 s−1 ; see Appendix). As a result, the internal temperatures of this and other boulders near 1 au would quickly equilibrate to orbit-averaged values that are too high (∼300 K) for water ice to survive. Even if the mass loss is intermittent, it is hard to see how such tiny boulders could retain ice on dynamically relevant (Myr) timescales. 3. Radiation Pressure A radiant energy flux density Fν [J m−2 s−1 Hz−1 ] corresponds to a photon flux Fν/(hν) [photon m−2 s−1 Hz−1 ], where h = 6.63 × 10−34 J s is Planck's constant, and hν is the energy of a photon having frequency ν. The momentum of a single photon is h/λ = hν/c, where λ is the wavelength, and c = 3 × 108 m s−1 is the speed of light. Then, considering the Sun as the source of photons, the flux of momentum in photons of frequency ν → ν + dν is dPr,ν = Fν/(hν) × (hν/c)dν. When integrated over all frequencies, this gives a pressure [ ] ( )  = - P F c N m 7 r 2 in which  ò n = n ¥ F F d 0 . Equation (7) is the radiation pressure. For example, at rH = 1 au, where the flux of sunlight is given by the solar constant, Se = 1360 W m−2 , Equation (7) gives Pr = 4.5 × 10−6 N m−2 (about 4 million times less than the pressure exerted by the weight of a sheet of paper). This tiny pressure is about 105 times smaller than the pressure due to water ice sublimation at 1 au, and is insignificant on macroscopic bodies, but can dominate the motion of small particles. The force exerted by radiation impinging on a spherical grain of radius a, is /  p =  Q a F c pr 2 , where Qpr is a size and wavelength-dependent dimensionless multiplier,2 and Fe [W m−2 ] is the local flux of sunlight. The radiation pressure acceleration is proportional to the cross section per unit mass of the accelerated particle, and so is inversely related to the particle size. For a spherical particle of density ρ, this results in an acceleration ( )  a r = Q F ca 3 4 . 8 rad pr The inverse dependence shows that small, low-density particles can be more strongly accelerated by radiation than large, high-density particles. In the case of the radiation field around the Sun, we note that the flux is given by ( ) /   p = F L r 4 H 2 , where Le = 4 × 1026 W is the solar luminosity and rH is the heliocentric distance in meters. Substituting gives ( )  a r p = Q ca L r 3 4 4 . 9 H rad pr 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ The numerical multiplier in Equations (8) and (9) is specific to the assumed spherical particle shape. Nevertheless, the equations give a useful approximation to the acceleration induced by radiation pressure as a function of particle size, and 2 Qpr is the ratio of the effective cross section for radiation pressure to the geometric cross section of the particle. It is a function of the composition, shape, structure, and size of a particle relative to λ, the wavelength of radiation with which it interacts. The limits are Qpr = 1 as a/λ → 0 while Qpr → constant as a/λ → ∞. In between, Qpr varies with a/λ in a complicated way, reflecting interactions between electromagnetic waves as they pass, and pass through, the particle (J. A. Burns et al. 1979; C. F. Bohren & D. R. Huff- man 1983). In planetary science and astronomy, the zeroth order approximation is to set Qpr = 0 for a < λ and Qpr = 1 for a λ. For many natural particle size distributions in which the smallest particles are the most abundant, this approximation gives rise to the rule-of-thumb that observations primarily sample particles with a ∼ λ. Calculation of Qpr for homogeneous spheres uses the Mie Theory. For other shapes and for porous and fractal particles of relevance to natural solar system particles, there is no analytic theory, and Qpr must be calculated numerically (e.g., K. Silsbee & B. T. Draine 2016). 4 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 5.
    density. It isuseful to normalize αrad by the local solar gravitational acceleration ( )   = g GM r . 10 H 2 The ratio βrad = αrad/ge is given by ( )   b pr = Q ca L GM 3 16 , 11 rad pr ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ which is a function of both particle (ρ, a, Qpr) and stellar (Le, Me) properties. For example, consider spherical dust grains with ρ = 1000 kg m−3 and Qpr = 1. Substitution into Equation (11) gives ( ) b = m a 0.6 , 12 rad where aμ is the radius expressed in microns. βrad is independent of heliocentric distance because gravity and radiation pressure both vary inversely with the square of rH. Equation (12) is a useful guide for homogeneous spheres, but it is important to note that the β parameter for other particle shapes cannot be so simply estimated because Qpr can assume very different values. Likewise, βrad ? 1 is never encountered, regardless of how small the particles are (K. Silsbee & B. T. Draine 2016). 3.1. Examples 1. Comet Tails. Radiation pressure acceleration has been detected in bodies as large as the ∼4–10 m diameter asteroid 2011 MD, for which βrad ∼ 10−7 (M. Micheli et al. 2014; M. Mommert et al. 2014), but the classic application is to the motion of small particles in comets. The dust particle trajectory after release from a comet nucleus is determined by the ejection velocity and βrad. In the theory of comet tails by M. J. Finson & R. F. Probstein (1968), dust particles are assumed to be released from the nucleus with zero relative speed and then to be accelerated by solar gravity and radiation pressure. Two limiting cases are often considered: synchrones show the locus of positions of dust particles having a range of sizes but released from the nucleus at the same time, while syndynes show the locus of positions of particles of one size (i.e., one βrad) but released over a range of times. Synchrones project on the sky as straight lines, with a position angle that depends on the time of ejection. Syndynes are curved by the orbital motions of the particles. See Figure 3 for examples. Syndyne/synchrone analysis enables a simple assessment of cometary dust properties. For example, depending on the observing geometry, a linear tail morphology may suggest a synchrone-like structure indicative of impulsive ejection. Linear tails are observed in outbursting comets and in dust released by energetic impacts between asteroids (D. Jewitt 2012), where the position angle of the tail gives a useful estimate of the ejection date. On the other hand, broad, fan-like tail structures are indicative of protracted emission for which syndyne models give constraints on the particle sizes. When observed from a position close to the comet orbital plane, however, syndynes and synchrones overlap so strongly that their diagnostic power is largely lost. With the aid of fast computers, more sophisticated synthetic models of comets can be used to explore a wide range of dust and ejection conditions. Such Monte Carlo models involve many parameters (including the form of the size distribution (usually assumed to be a power law), the largest and smallest particle radii, their ejection velocities as a function of particle size, the angular distribution of the ejection from the nucleus and its time dependence) and so are nonunique. Nevertheless, Monte Carlo coma models are useful in narrowing the range of Figure 3. Disintegrated LPC C/2021 A1 (Leonard) on UT 2022 March 31 when at rH = 1.756 au outbound. The left panel shows synchrones for particles ejected from 80–160 days before the date of observation. The right panel shows syndynes for particles with β = 0.0003, 0.001, 0.003, 0.01, and 0.03, as marked. The linear morphology of the tail is better matched by the straight synchrone models than by the curved syndynes. Ejection occurred 110 ± 10 days before the image was taken, i.e., on UT 2021 December 11 ± 10. Calculation by Yoonyoung Kim in D. Jewitt et al. (2023). 5 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 6.
    allowed parameters neededin order to match a given observation. Much of what we know about the solid particles ejected from comets has been determined this way. In the example of Figure 3, inferences from the syndynes and synchrones were supported by a full Monte Carlo simulation. A 0.6 km radius nucleus disintegrated suddenly into a broad distribution of particle sizes resembling a –3.5 index power law, with no fragments larger than 60 m. 2. β Meteoroids. The radiation pressure force opposes gravity such that, at heliocentric distance rH, the net sunward attraction is ( ) /  b - GM r 1 H rad 2 and the effec- tive escape velocity is ( ( ) ) / /  b = - V GM r 2 1 e H rad 1 2. Comparing this to the local Keplerian speed, ( ) / /  = V GM rH K 1 2 shows that particles in circular orbits with βrad > 1/2 (a  1 μm by Equation (12)) should be gravitationally unbound and promptly leave the solar system along nearly radial trajectories.3 Such particles, called β-meteoroids (H. A. Zook & O. E. Berg 1975), have been detected. They are submicron-sized dust particles that stream away from the Sun faster than the gravitational escape speed under the action of radiation pressure. The β particles are produced when larger dust particles sublimate and shrink upon approach to the Sun, as well as by collisional shattering of larger grains from the Zodiacal cloud that are spiraling in under the action of Poynting–Robertson drag (I. Mann & A. Czechow- ski 2021; see Section 4). Their source region is concentrated at 10–20 Re (0.05–0.1 au), where black- body temperatures lie in the range 880–1240 K and both collision and sublimation rates are high (I. Mann et al. 1994, J. R. Szalay et al. 2021). The effective mass-loss rate in β meteoroids is estimated between 102 kg s−1 (J. R. Szalay et al. 2021) and 103 kg s−1 (E. Grun et al. 1985). If sustained over the age of the solar system, a dust mass ∼ 1019 –1020 kg must have been expelled by radiation pressure as β meteoroids, corresponding to a few percent of the current mass in the asteroid belt. 3. Kuiper Belt Dust. While β meteoroids are best known from spacecraft measurements obtained in the inner and middle solar system, more distant counterparts may originate as well in dust sources in the outer solar system. In this regard, recent impact counter measurements from the New Horizons spacecraft have been used to suggest (albeit at only 2σ significance) an excess concentration of dust grains (radius ∼ 0.6 μm, β ∼ 1) beyond the 47 au edge of the classical Kuiper Belt (A. Doner et al. 2024). These dust grains might be produced by collisional shattering in the main Kuiper Belt followed by radiation pressure acceleration outward as Kuiper Belt β meteor- oids. Meanwhile, β meteoroids ejected from other stars presumably contribute to the flux of interstellar dust particles entering the solar system (M. Landgraf 2000). 4. Extended Debris Disk Dust. Many stars are encircled by disks in which the dust lifetime is shorter than the main- sequence age of the star. In these so-called debris disks, the dust must be replenished, presumably by collisional destruction of unseen parent bodies in what are, effectively, extrasolar Kuiper belts (A. M. Hughes et al. 2018). Since the luminosity of a main-sequence star scales in proportion to a high power of its mass (e.g., µ   L Mx with x = 3.5–4), the importance of radiation pressure should grow with stellar mass. A classic example is provided by the disk of α Lyrae (Vega), a ∼700 Myr old A0V star, 2.2 Me in mass with luminosity 47 Le. All else being equal, βrad is ∼20 times larger at α Lyrae than around the Sun, leading to the immediate expulsion of particles 20 times larger (i.e., tens of microns instead of ∼1 μm). The large extent of the α Lyrae dust (which reaches ∼1000 au from the star) is likely due to the action of radiation pressure (K. Y. L. Su et al. 2005). The implied dust mass flux (1012 kg s−1 ) is too large to be sustained over the life of Vega. K. Y. L. Su et al. (2005) suggested that the dust could be the recent product of a massive collision in the disk. 5. Lifting of Dust. Dust can be lifted from the surface of a small body of mass Ma and radius r when the solar radiation pressure acceleration, /  b GM rH rad 2 , exceeds the local gravitational attraction to the body, GMa/r2 . The critical particle size for ejection obtained by setting these accelerations equal is ( ) ~ a r r 10 1 km 1 13 c au 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ where ac is in microns, rau is in astronomical unit, and r is in kilometers (D. Jewitt 2012). For example, particles on a 1 km radius body located at 1 au can be swept away provided ac ∼ 10 μm, or smaller. Close to the Sun (e.g., at 0.14 au, the perihelion distance of (3200) Phaethon), much larger particles (ac  1 mm) can be expelled. Y. P. Bach & M. Ishiguro (2021) noted that thermal radiation pressure from a sufficiently hot surface might also expel particles; although, no established examples of this process currently exist. Both loss processes are highly directional (e.g., solar radiation pressure will force particles back into the surface except near the terminator) and, whether by solar or thermal radiation pressure, particle ejection must overcome the cohesive forces that bind small particles to the surface. 4. Poynting–Robertson Drag Formally a relativistic effect with a rather involved derivation (H. P. Robertson 1937), the Poynting–Robertson drag is more simply described as a consequence of aberration. As seen from a body moving in a circular orbit, the direction from which sunlight travels is aberrated relative to the radial direction from the Sun by an angle ( ) / q = - V c tan 1 K , where ( ) / /  = V GM rH K 1 2 is the orbital speed, and c is the speed of light. For solar system objects of interest, VK = c, and we may approximate θ ∼ VK/c. For example, the Earth (rH = 1 au) has orbital speed VK = 30 km s−1 giving VK/c ∼ 10−4 (about 20″). This tiny angle gives a nonradial component of the radiation pressure force, which acts steadily against the direction of motion, and so can do work against the orbit. The result is a drag force (the Poynting–Robertson drag force) that results in inexorable orbital decay. To estimate the timescale for this orbital decay, τPR, we assume a spherical body of radius a moving in a circular orbit and use Equation (9) to write the Poynting–Robertson 3 For a noncircular orbit, larger particles with βrad > (1 − e)/2 can be ejected by radiation pressure. 6 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 7.
    acceleration tangential tothe orbit as ( )  a r p = Q ca L r V c 3 4 4 . 14 H PR pr 2 K ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ Then, the timescale for this acceleration to collapse the orbit is τPR ∼ VK/αPR or ( )  t r p ~ ac Q L 4 3 4 r . 15 H PR 2 pr 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ For example, the timescale for a 1 mm radius particle of density ρ = 1000 kg m−3 to spiral into the Sun from Earth orbit (rH = 1 au) under the action of the Poynting–Robertson drag is τPR ∼ 8 × 1013 s, or about 3 Myr (assuming Qpr = 1). Setting τPR = 4.5 Gyr in Equation (15) gives a ∼ 1.5 m: all primordial material smaller than a few meters in size is removed by Poynting–Robertson drag if it survives that long against other destructive processes (e.g., impact destruction, gravitational scattering). 4.1. Examples 1. Zodiacal Dust. Interplanetary dust is released by sublimating comets and, to a lesser extent, by collisions between asteroids, forming a diffuse inner solar system structure known as the Zodiacal cloud. Dust particles in the Zodiacal cloud have a range of sizes with a mass- weighted mean near s̄ = 100–200 μm, for which the Poynting–Robertson time is τpr ∼ 0.5 Myr at 1 au. Smaller particles are quickly depleted by Poynting– Robertson drag while the survival of larger particles is limited more by collisional shattering. The total rate of production required to maintain the Zodiacal cloud in steady state is ∼103 –104 kg s−1 , albeit with order-of- magnitude uncertainty (D. Nesvorný et al. 2011; J. K. Rigley & M. C. Wyatt 2022, P. Pokorný et al. 2024). The steady Poynting–Robertson rain of Zodiacal dust toward the Sun is reversed when a fraction of those particles shrink small enough (through sublimation, and/ or through collisional shattering) to be expelled by radiation pressure as β meteoroids. 2. Interplanetary Dust Particles. Zodiacal dust particles enter the Earth's atmosphere at an average rate ∼ 1 kg s−1 (S. G. Love & D. E. Brownlee 1993). Entering particles smaller than a few tens of microns are decelerated by friction so high in the atmosphere that they do not melt and, when collected from the lower stratosphere, are known as interplanetary dust particles (IDPs). The rate of IDP delivery presumably varies on long timescales as asteroids collide and produce variable amounts of dust, which is then cleared in part by Poynting–Robertson drag. However, stratigraphic mea- surements of 3 He, which is delivered to Earth by IDPs, show only factor-of-2 variations in the rate in the last 100 Myr (K. A. Farley et al. 2021). On the way from their source to the Earth, the IDPs are impacted by energetic solar and cosmic-ray nuclei, causing damage tracks and the transmutation of elements. L. P. Keller & G. J. Flynn (2022) used particle track densities to calculate the effective space exposure ages of 10 μm sized IDPs. They found that about 25% of IDPs show high cosmic-ray track densities that are indicative of exposure ages > 106 yr. By contrast, the Poynting– Robertson lifetime of 10 μm particles at 1 au is only about 15 kyr (from Equation (15)). The 2 orders-of- magnitude age discrepancy could indicate that these high track density IDPs originate at much larger distances, because t µ rH pr 2 (Equation (15)). For example, 10 μm particles released at rH  10 au would have Poynting– Robertson timescales τpr > 106 yr, consistent with those measured. The only known substantial dust source beyond 10 au is the Kuiper Belt, which L. P. Keller & G. J. Flynn (2022) proposed is the source of the heavily cosmic-ray-damaged dust. Independent IDP exposure age estimates based on the production of unstable Al26 and Be10 nuclei lead to qualitatively consistent conclusions: IDPs originate over a vast range of heliocentric distances, with some from the Kuiper Belt (J. Feige et al. 2024). The reported collisional production rate of dust in the Kuiper Belt (∼104 kg s−1 ) exceeds the ∼103 kg s−1 released from the SPCs and 50 kg s−1 from LPCs (A. R. Poppe 2016). For this reason, it should not be surprising to find an IDP contribution from the Kuiper Belt. Indeed, early simulations of dust transport to the inner solar system suggested a maximum 25% contrib- ution (A. Moro-Martìn & R. Malhotra 2003), consistent with the new estimate based on track densities in the IDPs. 3. White Dwarf Contamination. White dwarfs are degen- erate post-main-sequence, roughly solar-mass stars col- lapsed to Earth-like dimensions. Some white dwarfs exhibit excess infrared emission from circumstellar disks having characteristic radii  1 Re. Refractory material falling from these disks may enter the Roche sphere of the star, become tidally shredded, and then sublimate to form a gaseous metal disk that contaminates the photo- sphere with rock-forming elements. Poynting–Robertson drag moves material inward to the sublimation radius to feed the metal gas disk. A straightforward application of Equation (15) gives timescales too long, and peak accretion rates (∼10 kg s−1 ) orders of magnitude too small to account for the observed degree of white dwarf pollution (J. Farihi et al. 2010). However, R. R. Rafikov (2011) showed that the effective τPR is much shorter, and the mass delivery rate much greater (∼105 kg s−1 ), because Poynting–Robertson drag need only move particles across a thin transition zone at the inner edge of the refractory disk into the sublimation radius. 5. Dissipative Forces 5.1. Tidal Dissipation The shapes of bodies in mutual orbit are cyclically deformed by varying gravitational forces, resulting in the dissipation of rotational and orbital energy as heat. Over time, this dissipation can have important dynamical consequences. Although gravity is the driver, the fundamental origin of energy loss is nongravitational, being rooted in the physics of inelastic materials. The character of tidal dissipation is best seen by neglecting numerical multipliers and geometrical terms associated with the shapes of the orbiting bodies. The shapes of most solar system bodies are unknown, so this neglect is at least partly reasonable. We consider, for simplicity, a binary object consisting of a primary of mass mp and a secondary of mass 7 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 8.
    ms, separated bya distance r, with mp ? ms. The radii of the primary and secondary objects are ap and as, respectively. We also assume that the densities, ρ, of the primary and secondary are the same so that, neglecting shape specific factors, r ~ m a p p 3 and r ~ m a s s 3 . The gravitational pull of each body elongates the other along an axis connecting their centers, with the secondary being most deformed because of the greater mass and gravity of the primary. If the deformation were instantaneous, this tidal bulge would stay perfectly aligned along the line of centers. However, the primary, in general, rotates with an angular frequency, ωp, that is different from the orbital frequency of the satellite, ωs, and the tidal response is not instantaneous. Empirically, most satellites orbit beyond the corotation radius, at which the orbital period is equal to the rotation period of the primary body. With ωp > ωs, the tidal bulge of the primary is carried ahead of the line of centers, and the resulting torque on the primary acts to slow its rotation while expanding the orbit of the secondary. In the opposite case (ωp < ωs), the tidal bulge lags behind, and the tidal torque increases ωp while contracting the orbit of the secondary. The latter is the case for Mars’ satellite Phobos, which orbits at 2.76 RM, far inside the corotation radius at 6.03 RM (1 RM = 3.4 × 106 m). The orbit of Phobos will collapse into the planet because of tidal dissipation on a (surprisingly short) timescale of a few tens of Myr (B. A. Black & T. Mittal 2015). Mars’ other known satellite, Deimos, orbits beyond the corotation radius at 6.92 RM and is being slowly pushed away from the planet by tides. The finite response time and inelasticity of the material are central to the mechanism of tidal evolution because the misalignment of the tidal bulge creates an asymmetry upon which gravity can exert a torque. The gravitational force experienced from distance r is F = Gmpms/r2 or, equivalently, / r ~ F G a a r p s 2 3 3 2. The gravity of the satellite is slightly different on the near and far sides of the primary, by an amount ( ) d r ~ ~ F dF dr a G a a r . 16 p p s 2 4 3 3 This small differential force periodically stretches and relaxes the bodies as they rotate and orbit, doing work in the process. The average stress due to gravitational deformation is / d = S F ap 2 [N m−2 ]. The relation between the stress and the strain (strain is the fractional change in the length scale, s = δap/ap) is called the bulk modulus, defined as μ = S/s. Substituting, the deformation is ( ) d d m r m ~ = a F a G a a r . 17 p p p s 2 3 3 3 The work done by a force δF applied over a distance δap is δW ∼ δFδap. Substituting from Equations (16) and (17) gives ( ) d r m = W G a a r a . 18 p s p 2 3 3 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ If the material were perfectly elastic, energy added to the body by stretching would be returned upon relaxation back to the original shape. But in real materials, owing to internal friction, a fraction of the energy is dissipated as heat. The fraction lost per stretching cycle is conventionally defined as Q−1 = δE/E (i.e., Q is the inverse of what one would expect) and referred to as the tidal “quality factor.” High Q corresponds to low dissipation per cycle and vice versa. Given this, and recognizing that δW in Equation (18) is the work done per cycle (not per second), we write the tidal power as ( ) d d r w m ~ W t G a a r Q a . 19 p s p p 2 3 3 3 2 ⎜ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎞ ⎠ The key point in all of this is that the tidal power (Equation (19)) scales with r−6 because the differential tidal force, δF, and the deformation it causes, δap, are both proportional to r−3 . Finally, the timescale for dissipation to substantially change the rotational energy is τt ∼ W/(δW/δt), where W is the rotational energy. We take the rotational energy of the primary to be w ~ W Ip p 2 where r ~ ~ I m a a p p p p 2 5 is the moment of inertia. Then, ( ) t m w r ~ Q G a m m r a 20 t p p p s p 2 3 2 2 6 ⎜ ⎟ ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ is the order-of-magnitude time needed for torques from the satellite to substantially change the rotation of the primary. Tides raised on the satellite by the gravity of the primary are larger, and act to change the rotation on an even shorter timescale, as can be seen by swapping as and ap in Equation (20). Obviously, this derivation is highly simplified, but it serves to show the functional dependence of the tidal evolution timescale on material (ρ, μ, Q) and geometrical (ap, as, r) properties. 5.2. Internal Dissipation Even without an external force from a binary companion, internal energy dissipation in an inelastic material can modify the spins of individual asteroids. The loss of rotational energy occurs when a body is rotating in a nonprincipal axis state, such that its rotational energy is not a minimum for its shape. Then, any element of the body is subject to a cyclically varying stress as it rotates, which induces a variable strain, allowing internal friction to act. The visual model is a deflected gyroscope, which both rotates around its axis and precesses and nutates at the same time. We can obtain a useful expression for the functional dependence of the damping timescale by the same dimensional method as used for the tidal effects in Section 5.1. Precession and nutation of the rotation axis induce time-variable stresses that do work by cyclically compressing and relaxing the material. The internal stress, proportional to the energy density in the body, is S ∼ ρa2 ω2 [N m−2 ], where a is the nominal size of the body, and ω [s−1 ] the relevant angular frequency. Stress and strain (δa/a) are related through the modulus μ = S/(δa/ a), giving δa ∼ Sa/μ. Then, by analogy with Section 5.1, the power dissipated as heat by the stress is δW/δt ∼ Sa2 δa(ω/Q), which becomes ( ) d d r w m = W t a Q . 21 2 7 5 This compares with the instantaneous rotational energy W ∼ ρa5 ω2 , and the ratio of W to δW/δt gives the damping 8 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 9.
    timescale ( ) t m r w ~A Q a 22 damp 2 3 where A is a numerical multiplier introduced to account for aspects of material physics and asteroid shape that are not treated here. J. A. Burns & V. S. Safronov (1973) derived A ∼ 100, but values spanning a range of nearly 3 orders of magnitude have been reported in the literature, from A = 1 to 4 (M. Efroimsky & A. Lazarian 2000), to A = 200 to 800 (I. Sharma et al. 2005), depending on details of the assumed visco-elastic properties and asteroid body shape. Given the huge uncertainties in A, as well as those in μ and Q, the timescale offered by Equation (22) should be understood as no more than a qualitative guide, but the equation at least serves to show that damping by internal friction should be strongest in large bodies rotating rapidly. This is consistent with the observation that a majority of the known nonprincipal axis (i.e., undamped) rotators are kilometer-sized or smaller asteroids (P. Pravec et al. 2005). Assuming μ = 1011 N m−2 , Q = 100, ρ = 1000 kg m−3 , and a geometric middle value A = 30, we rewrite Equation (22) for the damping time in Myr as ( ) t ~ P a 10 23 r damp 3 2 with a in kilometers and rotation period Pr in hours. A 1 km body with an excited rotation at Pr = 3 hr would damp on a timescale τdamp ∼ 300 Myr, by this relation. The 0.1 km scale interstellar object 1I/’Oumuamua, with period ∼ 8 hr, has τdamp longer than the age of the Universe, consistent with photometric evidence that it might be in an excited rotational state (M. Drahus et al. 2018). Larger asteroids whose rotation should be damped according to Equation (23) can nevertheless be excited by impact, radiation torques, or gravitational torques from close approaches to planets. The nuclei of some comets also exhibit excited rotation, notably the 15 km long nucleus of 1P/Halley (N. H. Samarasinha & M. F. A’Hearn 1991). In the comets, rotational excitation is a natural product of strong sublimation torques. Again, we emphasize that the material properties μ, Q, and A (Equation (22)) are extremely poorly known (see Section 5.3, Figure 4), and damping times very different from those given by Equation (23) are possible. 5.3. Examples 1. Material Properties. Neither Q nor μ can be calculated from first principles for planetary bodies of interest. Empirically, the quality factor varies widely (e.g., Q ∼ 10 for the Earth, Q ∼ 100 for the Moon; P. Goldreich & S. Soter 1966). Values for gas giant planets are uncertain, but probably 1 or 2 orders of magnitude lower than the Q ∼ 106 initially estimated by these authors (see J. Fuller et al. 2024). A value Q ∼ 100 is often assumed (albeit with little firm evidence) to apply to small solar system bodies. Separately, the Young's moduli of terrestrial rocks span the range from μ ∼ 109 N m−2 for sandstone and ∼1011 N m−2 for basalt to 4 × 1011 N m−2 for diamond. Many small bodies have an aggregate, “rubble pile” structure (K. J. Walsh 2018) that should give them mechanical properties quite different from solid rocks. In particular, the porous and fragmented nature of asteroid rubble piles would seem to point by analogy toward μ values at the lower end of this terrestrial rock range, in Figure 4. μQ as a function of radius estimated from binary asteroids. The subkilometer objects (orange triangles) are near-Earth asteroid binaries, for which a median age 107 yr is assumed. The ∼3 km asteroids in the main belt (yellow circles) are assumed to have median collisional age 109 yr while the 100 km objects (green diamonds) are assumed to be 4.5 Gyr, as old as the solar system. Note the enormous scatter in μQ. Data are from P. A. Taylor & J.-L. Margot (2011). 9 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 10.
    which case μQ∼ 1011 N m−2 might be expected. This value remains little more than a guess, and evidence from asteroid binaries suggests that even smaller μQ values may prevail, as we next describe. Useful empirical constraints on the product μQ can be obtained from the application of Equation (20) to measurements of binary asteroids. For example, if the age of a binary asteroid is known and the other physical properties (separation, period, component sizes, and density) in the equation are measured or can be estimated, then substitution gives μQ. Figure 4 shows μQ obtained in this way as a function of primary asteroid radius for binaries in three size groups (P. A. Taylor & J.-L. Margot 2011). The median value for subkilometer near-Earth binaries (assuming τt = 107 yr, the median dynamical lifetime of this population) is μQ = 5 × 108 N m−2 . This is far smaller even than μQ ∼ 1011 N m−2 estimated above. The median value for 3 km scale main-belt binaries (assuming τt = 109 yr, the approximate collisional lifetime of objects this size) is μQ = 7 × 1011 N m−2 . Finally, μQ = 1.4 × 1011 N m−2 for 100 km scale asteroids, computed assuming that they have survived for the 4.5 Gyr age of the solar system. The considerable differences between these estimates may, in part, reflect a real size dependence of μQ (see F. Nimmo & I. Matsuyama 2019), but they also testify to an element of guesswork in assigning values to some of the parameters in Equation (20). For example, it is difficult to assign more than a statistical age (e.g., based on the assumed collisional or dynamical lifetime) to any given binary object. 2. Rotational Dissipation. By Equation (20), the tidal evolution timescale grows particularly strongly (as r6 ) with the separation. In a given system, doubling the separation increases the tidal timescale by a factor 26 = 64, almost 2 orders of magnitude. This strong distance dependence is evident in real solar system objects. For example, a majority of satellites close to their parent planets rotate synchronously, while the more distant satellites do not. Figure 5 plots the available measure- ments of the satellites of Saturn, showing the trend for close-in satellites to be rotationally synchronized while the rotations of distant satellites are unrelated to their orbital periods. The 360 km diameter satellite Hyperion stands out. Although Hyperion should be synchronized according to Equation (20), its rotation is instead chaotic (i.e., the period changes irregularly) owing to impulsive torques exerted on its irregular body shape over the course of its eccentric (e = 0.12) orbit, as predicted by J. Wisdom et al. (1984) and established by J. J. Klavetter (1989a) and J. J. Klavetter (1989b). The rotation of the Earth's Moon is obviously synchronized with its orbital motion, and measurements show that the Earth's rotation is currently slowing (at ∼10−5 s yr−1 , or ∼3 hr per billion years; R. Mitchell & U. Kirscher 2018) as a result of tidal friction. Inserting r/ap = 60, mp/ms = 81, μQ = 1011 N m−2 , ρ = 5 × 103 kg m−3 , and ap = 6400 km into Equation (20) gives τt ∼ 0.5 Gyr. Although this is only accurate to an order of magnitude, it is clear that tidal interaction with the Moon is responsible for slowing the spin of the Earth. Likewise, terrestrial tides on the Moon slowed its rotation on a much shorter (∼10 Myr) timescale, explaining its current synchronous rotation state. 6. Yarkovsky Force An isothermal sphere at a finite temperature would radiate photons isotropically, experiencing no net recoil force from the Figure 5. Ratio of the rotation to orbit periods as a function of semimajor axis (in units of planet radius) for the measured satellites of Saturn. The inner (regular) satellites rotate synchronously, while the outer (irregular) satellites do not. Hyperion is shown with an error bar to symbolically mark the fact that its rotation is chaotic. 10 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 11.
    loss of photonmomentum. However, real solar system bodies are neither spherical nor isothermal, being hotter on the Sun- facing (day) side than on the opposite (night) side. In addition, the distribution of slopes, defined by both large- and small- scale features (e.g., boulders) on the surface, imbues the asteroid with a chirality, so that reflected and radiated photons exert a shape and surface texture dependent torque. Moreover, body rotation, combined with the finite thermal response time of the surface material, carries peak heat from local noon into the afternoon, giving a “thermal lag” (see the Appendix). As a result of this thermal lag, the afternoon temperatures will always be higher than the corresponding morning temperatures on a rotating body. The chirality and the azimuthal thermal lag asymmetry in the temperature lead to a net force, the so-called diurnal Yarkovsky force. The component of the force parallel to the direction of orbital motion can do work on the orbit, causing it to shrink or expand depending on whether the sense of the rotation is retrograde (i.e., opposite to the sense of orbital motion) or prograde (in the same sense as the orbital motion); see Figure 6. The above describes the diurnal Yarkovsky effect, which results from axial rotation of the asteroid and is maximized at 0° and 180° obliquity (a geometry assumed, for simplicity, in this discussion and in Figure 6). For example, given a porous regolith with diffusivity κ = 10−9 to 10−8 m2 s−1 , the thermal skin depth (Equation (A2)) on an asteroid rotating with period P = 5 hr is ℓ ∼ (Pκ)1/2 ∼ 4 to 13 mm. At nonzero obliquities, the Yarkovsky force is reduced by a factor equal to the cosine of the obliquity. The annual or seasonal Yarkovsky effect has a similar physical origin in thermal lag, but results from motion around the orbit and is maximized instead at 90° obliquity (Figure 7). One important, systematic difference between the seasonal and diurnal effects is that the former always opposes the orbital motion, causing a shrinkage of the orbit, whereas diurnal Yarkovsky can expand or contract it depending on the rotation direction. A second difference is that, whereas the diurnal Yarkovsky effect results from thermal lag on asteroid rotation periods (typically hours), the seasonal effect results from thermal lag on the orbital timescale (typically years). The ratio of seasonal to diurnal timescales is years/hours ∼104 and, since the thermal skin depth scales with the square root of the time (Equation (A2)), the seasonal Yarkovsky effect depends on the thermophysical properties of a surface skin ∼102 times deeper than the diurnal Yarkovsky. Instead of the diurnal Yarkovsky effect being driven by temperature variations in the top ∼4–13 mm of regolith, the seasonal effect is driven by an upper layer ∼0.4–1.3 m thick, all else being equal. The relative magnitudes of the diurnal and seasonal Yarkovsky effects depend on many quantities, including the object size, spin vector, and the depth dependence of the thermophysical properties of the surface layers (see the Appendix). The diurnal effect typically dominates for bodies with regoliths (i.e., low diffusivity surfaces); we ignore the seasonal effect from further discussion here for simplicity. We also ignore the reflected component of the torque, because the albedos of asteroids are low (C-type albedos are a few percent, while even S-type albedos are only ∼20%) and reflected torques are secondary. Even so, the application to most real asteroids is problematic, because the force depends on many unknown or poorly constrained quantities. These include the magnitude and direction of the spin vector, the body shape, the surface roughness, and the thermophysical parameters responsible for the thermal lag (see Equation (A1)). For all of these reasons, the Yarkovsky force cannot in general be calculated for a given body. Instead, it can be inferred from careful measurements of the action of the force. The Yarkovsky force has been reviewed by W. F. Bottke et al. (2006) and W. D. Vokrouhlický et al. (2015). Here we offer an order-of-magnitude derivation that captures the essence of the process, and we follow with some examples of its application. The temperature of the Sun is ∼6000 K, and its blackbody spectrum is peaked in the optical near 0.5 μm, whereas the isothermal blackbody temperature at 1 au is roughly 300 K, and the spectral peak lies in the infrared near 10 μm. The infrared photons carry 20 times less energy and 20 times less momentum than the absorbed solar photons, but they are 20 times more numerous so as to maintain energy balance on the body. Therefore, we can use the radiation pressure exerted by optical photons ( ( /  p L r c 4 H 2 )) to estimate the Yarkovsky force due to thermal emission. Specifically, we write the magnitude of the diurnal Yarkovsky force as ( )  p p =  k a c L r 4 24 Y Y H 2 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ where 0 „ kY „ 1 is a dimensionless coefficient (to be determined) representing the fraction of the radiation pressure force acting parallel to the orbital motion. (From the discussion in the preceding paragraphs, it should be evident that kY is a function of the spin vector and thermophysical parameters, and we are deliberately separating these from consideration here to simplify the presentation. Thermal considerations are briefly described in the Appendix.) Dividing by the mass of the body, assumed spherical, the resulting acceleration is ( )  a pr = k L car 3 16 . 25 Y Y H 2 We obtain an estimate for the Yarkovsky timescale from τY ∼ VK/αY, VK being the Kepler speed. Again assuming a circular orbit, we find ( ) ( ) / /   t pr ~ ac k L GM r 16 3 . 26 Y Y H 1 2 3 2 The order-of-magnitude radial drift can be estimated from d(rH)/dt ∼ rH/τY, or ( ) ( ) /   pr ~ - dr dt k L ac GM r 3 16 , 27 H Y H 1 2 measured in m s−1 , and drH/dt can be positive or negative, depending on the sense of asteroid rotation (see Figure 6). The maximum possible drift rate at 1 au is given by setting kY = 1, rH = 1 au, with the other parameters as above. For a = 1 km, this gives d(rH)/dt ∼ 0.5 km yr−1 or 4 × 10−3 au Myr−1 . (We will show below that a more typical value is kY ∼ 0.05, so that d(rH)/dt ∼ 2 × 10−4 au Myr−1 is a better estimate for a 1 km body at 1 au.) The asteroid belt is dynamically structured (e.g., at resonance locations) on scales <0.05 au, which can be crossed by kilometer-sized asteroids on timescales <250 Myr. We conclude that, depending on the size of the asteroid and the actual value of kY, the Yarkovsky force is capable of modifying the orbital properties on timescales that are very short 11 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 12.
    Figure 6. Schematicplan view of a body orbiting the Sun, to illustrate the diurnal Yarkovsky force. The dotted line shows the orbit of the body having angular rate, ωK, while the body itself rotates about an axis (black dot) fixed perpendicular to the plane, with angular rate, ω. Rotation carries midday heat from the subsolar point into the afternoon, where its loss by radiation is marked IR. In the prograde case (left panel), ωK and ω are parallel, and the recoil force, F, acts with the orbital motion, causing the orbit to expand. In the retrograde case (right panel), ωK and ω are antiparallel. The recoil force opposes the motion and causes the orbit to shrink. Figure 7. Plan view to illustrate the seasonal Yarkovsky effect. The rotation vector (small black arrow, marked ω), lies in the orbit plane and remains fixed in inertial space. Peak insolation is reached when the spin vector points directly at the Sun, as at position A and shown by the strong curly red arrows. Heat is conducted into the interior and slowly leaks out as the asteroid moves around the orbit toward position B. The recoil force from the leakage of this residual heat (shown with faint curly red arrows), F, acts opposite to the motion. Half an orbit later, the cycle repeats with the opposite hemisphere at position C, from which heat is retained to position D giving a recoil still opposing the orbital motion. The seasonal Yarkovsky force always opposes the orbital motion regardless of the sense of body rotation, resulting in orbital shrinkage. 12 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 13.
    compared to theage of the solar system. A practical limit to the influence of Yarkovsky drag is set by collisions with other bodies, which can reset the spin and so change the magnitude and even direction of the radiative torque (P. Farinella et al. 1998; P. A. Wiegert 2015). 6.1. Examples 1. Orbital Drift. To obtain an estimate of the Yarkovsky constant, kY, we must use empirical data connecting measurements of the drift rate with the asteroid size, orbit, and density. A. H. Greenberg et al. (2020) and M. Fenucci et al. (2024) reported drift rate detections for 247 and 348 asteroids, respectively. Figure 8 shows measurements of the magnitude of the radial drift rate as a function of radius for 58 near-Earth asteroids from M. Fenucci et al. (2024), having semimajor axes near 1 au and formal SNR > 10. Lines in Figure 8 show Equation (27) with assumed values kY = 0.01, 0.1 and 1. The data are reasonably well matched by the expected 1/a size dependence of the drift rate (Equation (27)) and by kY ∼ 0.02 to 0.13. We take kY = 0.05 as a representative value. kY ∼ 0.05 implies a lag angle ( ) q = ~ - sin 0.05 1 0.05 radians, or about 3°. It should be remembered that the data are biased toward asteroids with the largest kY, because small values of kY give small and difficult-to-measure (low SNR) drift rates. Given a random distribution of asteroid spin vectors, the above model predicts that the ratio of positive (corresponding to prograde rotators) to negative (retrograde rotator) values of drH/dt should be close to unity. Instead, in both the studies by A. H. Greenberg et al. (2020) and by M. Fen- ucci et al. (2024), fully 70% of the high-quality determinations have drH/dt < 0, giving a ratio negative drift/positive drift = 2.3, and indicating an excess of retrograde rotators. (Figure 9). This inference matches measurements of the spins of near-Earth asteroids, which show a preponderance of retrograde rotators (retrograde/ prograde ratio ∼ 2; A. La Spina et al. 2004). The bias in favor of retrograde rotation in the near-Earth population is thought to occur because inward-drifting main-belt asteroids are more easily able to reach the ν6 secular resonance responsible for their deflection into near-Earth space. 2. Origin of Meteorites. Ultimately, Yarkovsky drift helps to supply the meteorites. The asteroid belt is crossed by numerous mean-motion and secular resonances, near which the orbits of bodies are unstable. Orbital eccentricities of resonant asteroids are excited until they become planet-crossing and short-lived. The mean- motion resonances would be nearly empty if it were not for Yarkovsky drift, which is responsible for feeding resonance regions with nearby asteroids through semi- major axis drift (P. Farinella et al. 1998). About 80% of the near-Earth objects are supplied from the ν6 secular resonance and, spectrally, two-thirds of these are S-type asteroids (J. A. Sanchez et al. 2024). The S-types are related to the thermally metamorphosed LL chondrite meteorites, themselves fragments of bodies from the inner belt. 3. Dispersal of Asteroid Families. Shattering collisions between asteroids produce families of objects having initially similar orbital elements. However, family asteroids experience a size-dependent drift from their initial orbital semimajor axes under the action of the Yarkovsky force. By Equation (27), the semimajor axis drift in time Δt is ΔrH ∝ Δt/a, and ΔrH can be positive or negative depending on the sense of rotation of the asteroid. Small asteroids drift the farthest, all else being equal, giving rise to a V-shaped distribution in a plot of Figure 8. Absolute value of the radial drift rate plotted as a function of asteroid radius in meters. Lines show the model drift rate calculated from Equation (27) with ρ = 103 kg m−3 , rH = 1 au, and kY = 0.01 (red dashed line), 0.1 (solid black line), and 1 (dashed blue line). Evidently, the data are broadly consistent with kY ∼ 0.05. Data are from the compilation by M. Fenucci et al. (2024). 13 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 14.
    semimajor axis versus1/a. This effect is observed in the main asteroid belt, where it has been used to identify families and estimate their ages. An example is shown in Figure 10 which compares data for the Erigone family with Equation (27) (we assumed kY = 0.05, ρ = 2000 kg m−3 and initial semimajor axis rH = 2.37 au). There is some ambiguity in defining the edges of the distribution, but spreading ages near 70–100 Myr are Figure 9. Histogram of near-Earth asteroid radial drift rates showing an excess with negative values. The ratio of (inward-drifting) retrograde rotators to (outward drifting) prograde rotators is ∼2.3:1. Data refer to subkilometer asteroids from M. Fenucci et al. (2024). Four objects with drH/dt outside the plotted range have been excluded for clarity of the plot. Asteroids with formal SNR < 10 have not been considered. Figure 10. Yarkovsky spreading diagram for the Erigone main-belt asteroid family. The vertical dashed black line shows rH = 2.37 au, the semimajor axis of the presumed family parent body. Solid blue and red dashed lines show trajectories for family ages 70 and 100 Myr. Data are from B. T. Bolin et al. (2018), and models are from Equation (27). 14 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 15.
    plausible, while muchlarger and smaller ages are not. B. T. Bolin et al. (2018) estimated an age of 90 Myr. The method is not perfect. The initial orbits may not have exactly the same semimajor axis, the effect of asteroid obliquity is not included (Equation (27) gives the maximum value, for obliquity = 0°) and the individual obliquities are in any case not known, the sizes and densities of asteroids are approximate, and external perturbations (e.g., from planets) can act to spread the orbits (especially for the older clusters). Nevertheless, Yarkovsky spreading gives an impressive explanation for the V-shaped diagram and a useful measure of the family age. 4. Post-main-sequence Evolution of the Sun. In 5 Gyr, the Sun will exhaust its core hydrogen and bloat into a red giant, with a luminosity ∼103.5 times its present value (E. Vassiliadis & P. R. Wood 1993). When this occurs, the Yarkovsky force will likewise jump by a factor ∼103.5 causing accelerated orbital drift to compete with many other processes to destabilize small orbiting bodies of the solar system. Destabilized planetary systems are likely implicated in feeding debris to the metal-polluted central white dwarf remaining after the red giant phase (D. Veras et al. 2019). 5. Binary Yarkovsky. The Yarkovsky force can affect a binary in two ways. First, in a binary with small separation, thermal radiation from each component can heat the other, resulting in a kind of radiation drag distinct from the Yarkovsky drag caused by direct sunlight. Second, if the orbital inclination is also small, the two components will pass through each other's shadow in each orbit. Since the thermal response of the surface material is lagged (see the Appendix), the entry into and exit from the shadow introduces an additional asymmetry into the radiative force that results in yet another torque. These subtle effects, the former somewhat unfortunately named “planetary Yarkovsky” and the latter “Yarkovsky- Schach” drag, may rival tidal dissipation (Section 5.1) and BYORP (Section 8.2) in the orbital evolution of some close binary asteroids (W.-H. Zhou et al. 2024). 7. Lorentz Force A charged particle interacts with a moving magnetic field through the Lorentz force, L. For simplicity, we ignore the fact that L is a vector acting perpendicular to both the velocity of the particle and the direction of the field, and we consider only the magnitude of the force, =  BqV L . Here, B is the magnetic flux density, q is the charge on the particle, and V is the velocity of the particle with respect to the field. To evaluate the Lorenz force, we first need to know the charge on a particle. Coulomb's Law gives the force between two charges, q, separated by distance, r, as ( ) / pe =  q r 4 2 0 2 , where ò0 = 8.854 × 10−12 F m−1 is the permittivity of free space. The work done in bringing a charge q from r = ∞ to the surface of a particle of radius r = a is just ( ) / ò pe = = ¥  E dr q a 4 a 2 0 . The Volt is a measure of the work done, E, in moving a charge, q, through a potential difference, U. Then, U = E/q gives the relation ( ) p =  q Ua 4 28 0 for the charge on a particle of radius a when its potential, U, is known. Numerous effects contribute to the charging of dust particles in the solar system, but the dominant effect is photoionization from solar ultraviolet, which leads to U ∼ 5–10 Volts (S. P. Wyatt 1969; corresponding to photon wavelengths ∼1000–2000 Å). This potential can vary depending on fluctuations in the solar ultraviolet flux, itself a function of the ∼22 yr solar magnetic cycle. The potential, U, depends on the photon energy and the ionization threshold of the material, and is therefore approximately independent of distance from the Sun. However, the photon flux varies as - rH 2 and so the charging time increases as rH 2 ; it is ∼103 times longer in the Kuiper Belt at 30 au than near the Earth at 1 au, elevating the relative importance of leakage currents that ultimately can limit the accumulated charge. In the following, we take U = 10 Volts independent of distance. At distances rau ? 1, the Sun's Parker spiral (e.g., M. J. Owens & R. J. Forsyth 2013) is wound so tightly as to be nearly azimuthal. In the planetary region, the azimuthal magnetic flux density approximately follows ( ) = - B r B r H H 1 1 , with B1 ∼ 600 T m and rH expressed in meters (as determined from Figure 11 of A. Balogh & G. Erdõs 2013), while the field is swept with the solar wind at speed V ∼ 500 km s−1 , approximately independent of heliocentric distance. The field does exhibit substantial fluctuations through the solar cycle and also varies with heliographic latitude; we ignore these effects here for simplicity. Substituting for q from Equation (28) and for B and dividing by the particle mass gives the particle acceleration due to the Lorentz force as ( ) a e r = B UV a r 3 . 29 L H 1 0 2 Further dividing by the local gravitational acceleration (Equation (10)) defines the magnetic β parameter βL = αL/ge ( )  b e r = B UV GM r a 3 . 30 L H 1 0 2 ⎛ ⎝ ⎞ ⎠ Substitution gives ( ) b ~ m r a 0.1 31 L au 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ with rau being the heliocentric distance expressed in astronom- ical unit, and aμ is the particle radius expressed in microns. Figure 11 shows β and βL as functions of heliocentric distance for particles with density ρ = 103 kg m−3 and 1, 10, 102 , and 103 μm in radius. Equations (30) and (31) show that the dynamical importance of the Lorentz force is maximized for the smallest particles and the largest heliocentric distances. 7.1. Examples 1. Interstellar Dust. Spacecraft with velocity-measuring impact detectors record dust particles traveling faster than the local gravitational escape speed from the Sun (E. Grun et al. 1985). These are of interstellar origin but are, on average, substantially larger than the interstellar dust known from astronomical measurements of extinc- tion and polarization. The impact energies correspond to particles several tenths of a micron (up to ∼0.4 μm) in 15 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 16.
    size, whereas theextinction and polarization signatures of the interstellar medium are due to dust with sizes down to a ∼ 10 nm. Some small particles are deflected by radiation pressure but, as a → 0, Qpr = 1, leaving the Sun's magnetic field to deflect the smallest and most abundant (I. Mann & A. Czechowski 2021). For example, consider dust approaching the heliopause at 25 km s−1 , where B ∼ 0.5 nT (L. F. Burlaga et al. 2022). These parameters give βL > 1 for a < 0.1 μm, so smaller particles are deflected from the planetary region by the heliospheric magnetic field, consistent with dust counter measurements (V. J. Sterken et al. 2013). The penetration distance of interstellar dust is a function not only of particle size but also of time, because of the Sun's 22 yr magnetic cycle (M. Landgraf 2000). 2. Comet Dust. The Lorentz force acts roughly perpendicu- larly to the Sun–comet line, allowing for the possibility that the dust morphology of a comet might be altered by the Lorentz force relative to that expected on the basis of radiation pressure and solar gravity alone. The ratio of the Lorentz acceleration (Equation (31)) to the radiation pressure acceleration (Equation (12)) in the geometric optics limit is ( ) b b ~ m r a 0.2 . 32 L au ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ Near 1 au, βL/β = 1 for the micron-sized and larger particles that dominate the scattering, providing justifica- tion for the neglect of Lorentz force in models of comets in the terrestrial planet region. Equation (32) suggests that the effects of Lorentz force should be more significant at larger heliocentric distances where, unfortunately, we possess few relevant observations. Three exceptions are the LPCs C/2014 B1 (Schwartz), C/2010 U3 (Boattini), and C/1995 O1 (Hale–Bopp). C/2014 B1 (Schwartz) was dominated by quite large (aμ > 100) particles at rau ∼ 10 (D. Jewitt et al. 2019). It has βL/β  0.02 by Equation (32), and showed no effect from Lorentz force. Observations of C/ 2010 U3 (Boattini) at larger distance (rau ∼ 20 au) showed smaller particles (aμ = 10; M.-T. Hui et al. 2019). Equation (32) gives βL/β ∼ 0.4, consistent with a larger but still not dominant influence of the Lorentz force. C/1995 O1 (Hale–Bopp) was observed at rau ∼ 20 and reported to show a coma of small particles with aμ ∼ 1 (E. A. Kramer et al. 2014). The coma could not be well matched by models using only radiation pressure. With the above values, Equation (32) gives βL/β ∼ 4, consistent with a Lorentz force dominated morphology. In the Kuiper Belt, at rH = 40 au, the motion of all particles smaller than about 8 μm, including the micron- sized particles detected by the New Horizons particle counter (A. Doner et al. 2024), should be affected by the Lorentz force. At very small particle sizes, a = λ, the geometric optics limit implicit in Equation (32) breaks down, and the radiation force is reduced by a factor Qpr < 1, increasing βL/β. The motion of particles a  100 nm is strongly affected by Lorentz forces (I. Mann & A. Czechowski 2021). 3. Dust in Planetary Magnetospheres. The giant planets sustain strong dynamo-generated magnetic fields and provide numerous examples where magnetic forces are important. For example, the equatorial surface field of Jupiter is BJ ∼ 400 μT, about 105 times stronger than the Figure 11. βrad (solid red lines) and βL (dashed blue lines) for particles with radii 1, 10, 102 , and 103 μm and density ρ = 103 kg m−3 , as functions of heliocentric distance. βL > βrad only for the smallest particles at the largest heliocentric distances. 16 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 17.
    solar wind fieldnear Earth, and magnetic forces are of correspondingly greater significance. Strong magneto- spheric fields are able to retain plasma that would otherwise be quickly stripped away by the solar wind, creating a local gas environment that is distinct from the open interplanetary medium. The magnetospheres of Jupiter and Saturn are filled with dense plasma (∼108 –109 m−3 , or 102 –103 times the solar wind density) from the outgassing satellites Io and Enceladus, respec- tively. Dust near these satellites is charged negatively by contact with the plasma (because the electrons travel faster in thermal equilibrium and so deliver a larger charging current than the heavier, slower ions). The charge reverses to positive values at larger distances where the effects of plasma charging are overcome by those of UV photoelectron loss (F. Spahn et al. 2019). Charged dust particles are accelerated by the planetary field, which sweeps past at the corotation speed. One consequence of magnetospheric dust interactions is that Jupiter (E. Grun et al. 1993) and Saturn (S. Kempf et al. 2005) eject streams of nanometer-sized particles (mass ∼ 10−24 kg, magnetic βL ? 1) at speeds above the gravitational escape speed from either planet (H.-W. Hsu et al. 2012). Another is that the B-ring of Saturn displays transient, 104 km scale, quasiradial dust structures known as spokes, whose abundance is seasonally modulated by the ring illumination but also by Saturn's rotating magnetic field (B. A. Smith et al. 1981; C. A. Porco & G. E. Danielson 1982). The origin of spokes is still not fully understood, but it is clear that they consist of micron-sized dust particles briefly elevated above the Saturn ring plane and with motions that reflect the combined influence of local electrostatic and magnetic forces, as well as planetary gravity (M. Horányi et al. 2004). 7.2. Summary The relative magnitudes of the accelerations are compared in Figure 12. For this purpose, we evaluate each acceleration at rH = 1 au and for a body radius a = 1 km, and we divide by the local solar gravitational acceleration (ge = 0.006 m s−2 at rH = 1 au). 8. Torques Most asteroids and cometary nuclei are irregular in shape. On such objects, the nongravitational forces described above generally do not pass through the center of mass, resulting in a torque. Left unchecked, the torque will drive inexorably toward rotational instability, where the centripetal forces exceed the gravitational and cohesive forces holding the body together. A strengthless oblate ellipsoid of density ρ and with equatorial axes a = b and polar axis c „ a has critical period ( ) / / t p r = G a c 3 . 33 crit 1 2 1 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ For a sphere (a = c) of density ρ = 103 kg m−3 , Equation (33) gives τcrit = 3.3 hr. A highly oblate body of the same density but with a / c = 2 would have a critical period 21/2 times longer, or 4.7 hr. A majority of the measured rotation periods of solar system bodies are within a factor of a few of τcrit (B. D. Warner et al. 2009), suggesting the importance of centripetal forces. However, it must be admitted that observa- tional biases against the measurement of periods either much shorter or much longer than a few hours are strong in most existing data sets. In general, applied torques change both the direction and the magnitude of the spin vector, inducing excited or nonprincipal axis rotation. Observationally, precessional effects tend to be subtle, requiring longer sequences of observation than are commonly available. Therefore, in the following discussion, we concentrate on the more easily measured changes to the magnitude of the spin (i.e., to the rotational period) for which we already possess abundant evidence. Torques due to sublimation and to radiation are of particular significance for the destruction of comets and asteroids, respectively. We consider them separately next. In both cases, a simple dimensional treatment is informative. We note that the timescale for an applied torque, T, to change the spin is just τs ∼ L/T, where L is the angular momentum of the body. For a homogeneous sphere of radius a and density ρ, the angular momentum is L = 16π2 ρa5 /(15P), where P is the instantaneous rotation period. 8.1. Sublimation Torque The magnitude of the sublimation torque on a comet is equal to the momentum lost per second in sublimated material Figure 12. Relative magnitudes of the NGAs discussed in the text, relative to solar gravity, for rH = 1 au and object radius a = 1 km. 17 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 18.
    multiplied by themoment arm, which we write as kTa, where a is the nucleus radius and 0 „ kT „ 1 is a dimensionless multiplier. kT = 0 corresponds to perfectly isotropic sublima- tion with no net torque. kT = 1 corresponds to perfectly collimated ejection tangential to the surface of the body. Then, with a mass-loss rate μmHQg [kg s−1 ] and an outflow speed Vth, the magnitude of the torque is T = μmHQgkTVtha. The timescale for the torque to change the spin is then (D. Jewitt 2021) ( ) t p r m ~ a m Q k V P 16 15 1 . 34 s g T 2 4 H th Note that, for a fixed Qg, Equation (34) has a very strong (a4 ) nucleus radius dependence. However, it is natural to expect that the production rate, Qg, should scale with the nucleus area, Qg ∝ a2 and, if it does, τs ∝ a2 is anticipated for the nuclei of comets, all else being equal. The rotational lightcurves of some comets have enabled the measurement of spin changes (e.g., R. Kokotanekova et al. 2018), allowing τs to be directly estimated. With additional observational constraints on ρ, a, Qg, and P, and with the use of Equation (34), the dimensionless moment arm, kT, can be estimated (D. Jewitt 2021). The median value is kT = 0.007, meaning that only 0.7% of the outflow momentum is used to torque the nucleus. Even this tiny fraction is sufficient to quickly modify the spins of small comets. Measured values of τs as a function of a are plotted in Figure 13. The relation ( ) t ~ a 100 35 s 2 with a in kilometers and τs in years, matches the data well (D. Jewitt 2021). (This relation strictly applies to JFCs with perihelia in the 1–2 au range.) Spin-up by outgassing torques ends with rotational instability and breakup. The fragmented nucleus of 332P/Ikeya–Murakami at rH = 1.6 au gives a good example (Figure 14). A cloud of fragments expands from main nucleus C, whose rotation in about 2 hr suggests rotational instability (D. Jewitt et al. 2016). A necessary condition for Equation (34) to remain valid is the persistence of ice at or near the physical surface of the nucleus for timescales τs. Near-surface ice persists because the speed with which the ice sublimation surface erodes into the nucleus exceeds the speed with which heat conducts into the interior, causing fresh ice to be continually excavated. The particular problem is that sublimation exposes refractory particles too large to be ejected by gas drag. These should eventually clog the surface, producing a “rubble mantle” and inhibiting or shutting down further sublimation. How this works in detail is not known, even after several years of in situ investigation of the nucleus of 67P/Churyumov–Gerasimenko by the Rosetta spacecraft (N. Attree et al. 2023). Orbital evolution may play a role, particularly when the perihelion distance migrates to smaller values, leading to higher temperatures and sublimation fluxes. In the case of the active asteroids, ice is exposed only intermittently, perhaps in response to occasional impacts that clear a surface refractory mantle. The duty cycle (ratio of the time spent in sublimation to the total elapsed time) is less than 10−4 or even 10−5 , allowing ice to persist for long times but rendering Equation (35) inapplicable to these objects. The significance of the short timescales indicated by Equation (35) is that small nuclei cannot survive long once they reach the vicinity of the Sun. This may explain the paucity of subkilometer comet nuclei relative to power-law extrapola- tions from larger sizes. It also complicates any attempt to relate the populations and properties of comets near the Sun to their Figure 13. Measured timescale for changing the rotation period, τs, as a function of the nucleus radius for SPCs with perihelia in the 1–2 au range. Filled red circles show spin-change detections, while yellow diamonds show lower limits to the allowed spin-up timescales. Comets are identified by their numerical labels. The dashed line shows τs = 100a2 . Data are from D. Jewitt (2021). 18 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 19.
    similarly sized counterpartsin the Kuiper Belt and Oort cloud source regions. 8.2. YORP Torque Nonspherical bodies warmed by the Sun radiate infrared photons, which carry away momentum and can exert a torque (e.g., D. P. Rubincam 2000). Here we offer a simplified treatment that captures the physical essence of the torque, followed by some examples of its application to real bodies. The torque is proportional to the surface area, a2 , multiplied by the moment arm, which we write as ¢ k a T , where   ¢ k 0 1 T is another dimensionless constant that must be empirically determined. Again setting timescale τYORP ∼ L/T, we obtain, ( )  t pr = ¢ a c k P r S 16 15 . 36 T YORP 2 au 2 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ The dimensionless moment arm, ¢ kT , is such a sensitive function of the body shape, surface roughness, and thermo- physical parameters, all of which are unknown for most asteroids, that it cannot in general be calculated. Instead, we rely on measurements of the few asteroids where changes in the rotation periods can be measured and the other parameters in Equation (36) are constrained. These asteroids are listed in Table 1, where measurements of a, P, and dP/dt are from the Figure 14. Labeled fragments released from component C of comet 332P/Ikeya–Murakami in 2015 and separating from it at speeds 0.06–4 m s−1 . C itself rotates with a probable period near 2 hr, suggesting rotational instability as the cause of the release of fragments. Data are from D. Jewitt et al. (2016). Table 1 YORP Timescalesa Object D b P c dω/dtd τe (1620) Geographos 2.56 5.22 1.14 ± 0.03 6.9 ± 0.2 (1685) Toro 3.5 10.20 0.33 ± 0.03 12.3 ± 1.0 (1862) Apollo 1.55 3.07 4.94 ± 0.09 2.7 ± 0.05 (2100) Ra-Shalom 2.30 19.82 <0.6 >3.5 (3103) Eger 1.78 5.71 <1.5 >4.8 (10115) 1992 SK 1.0 7.32 8.3 ± 0.6 0.68 ± 0.05 (25143) Itokawa 0.32 12.13 3.54 ± 0.38 1.0 ± 0.1 (54509) YORP 0.11 0.20 350 ± 35 0.59 ± 0.05 (85989) 1999 JD6 1.53 7.66 <1.2 >4.5 (85990) 1999 JV6 0.44 6.54 <7.2 >0.9 (68346) 2001 KZ66 0.80 4.99 8.43 ± 0.69 1.0 ± 0.1 (101955) Bennu 0.49 4.30 6.34 ± 0.91 1.5 ± 0.2 (138852) 2000 WN10 0.3 4.46 5.5 ± 0.7 1.7 ± 0.2 (161989) Cacus 1.0 3.76 1.86 ± 0.09 5.9 ± 0.3 Notes. a Asteroid data from J. Ďurech et al. (2024). b Diameter in kilometers. c Rotation period in hours. d Rate of change of the angular frequency ×10−8 [radian day−1 ]. Values from J. Ďurech et al. (2024) that are statistically insignificant are listed as 3σ upper limits. e YORP time, Myr. 19 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 20.
    compilation by J.Ďurech et al. (2024; see also B. Rozitis & S. F. Green 2013 and references therein). Figure 15 shows the YORP timescale computed from τYORP = P/(dP/dt), with dP/dt = − (P2 /2π)dω/dt and dω/dt from the Table. τYORP is shown as a function of the asteroid radius with each object labeled by its identifying number. The asteroids used in Figure 15 all have semimajor axis ∼1 au. The Figure clearly shows that τYORP and a are correlated, with larger asteroids showing longer YORP timescales. Adopting the form of Equation (36) and the data from Figure 15, we estimate [ ] ( ) t ~ a r 4.5 Myr 37 YORP 2 au 2 where a is the radius in kilometers, and rau is the semimajor axis in astronomical unit. By this equation, across the asteroid main belt from 2.1–3.3 au, a 1 km radius asteroid would have τYORP ∼ 20–50 Myr. Setting τYORP = 4500 Myr in Equation (37) and solving for a, we find that YORP can influence the spins of main- belt asteroids up to a ∼ 10–14 km. This is in broad agreement with the asteroid rotational barrier (Figure 16), which is prominent for asteroids up to about 10 km in diameter (implying spin-up and breakup for smaller bodies) but less evident beyond about 30 km, where primordial spin is likely preserved (e.g., P. Pravec et al. 2002). It should be noted that the dashed blue line in Figure 15 shows a least-squares fit to the data, weighted by the estimated uncertainties on τYORP (but not on a). It gives τYORP ∝ a1.87±0.04 . Although the value of the index is formally (3σ) smaller than the expected value, τYORP ∝ a2 from Equation (36) (shown in the Figure as a solid red line), the difference is likely not important given the existence of systematic errors in the sample. A major limitation to the order-of-magnitude estimates of the torque timescales, both for sublimation (Equation (35)) and for YORP (Equation (37)), concerns the stability of the torque. In comets, the magnitude and direction of the outgassing torque depend on the surface distribution and angular dependence of the outgassing. As the surface erodes, we expect the spatial and angular distribution of outgassing sources to change, altering the torque. For most comets, we possess few or no observational constraints on the areal distribution of sources or their evolution. Indeed, in situ measurements from 67P/Churyumov–Gerasi- menko show the difficulty in modeling this process even given the most detailed data (N. Attree et al. 2023). Likewise, the YORP torque is highly sensitive to the surface shape and texture and can change in magnitude and even direction in response to minor surface changes (T. S. Statler 2009). Empirical but indirect evidence for this comes from asteroid (162173) Ryugu (a ∼ 0.5 km), which has the characteristic diamond shape indicative of rotational instability but a current rotation period near 7.6 hr. The rotation of Ryugu may have slowed in response to a changing YORP torque, leaving its equatorial ridge as evidence of its previously rapid spin. Spin evolution in the presence of a changing torque may be more akin to a random walk process than to the steady change implicit in Equations (35) and (37) (W. F. Bottke et al. 2015; T. S. Statler 2015). The relevant spin-change timescales would then be much longer than estimated here. Unfortunately, we possess too little information to address this problem with any confidence. 8.3. Examples 1. Asteroid Spin Barrier. The distribution of asteroid spin frequencies as a function of absolute magnitude, H, is shown in Figure 16. H is related to the asteroid diameter Figure 15. Empirical timescale for YORP spin-change for asteroids near 1 au as a function of the asteroid radius [km]. Data with upward pointing arrows represent 3σ lower limits to τYORP based on nondetections of rotational acceleration. They are not included in the fit. The dashed blue line shows a weighted least-squares fit to the detections, τYORP ∝ a1.87±0.04 . The solid red line shows Equation (37) for rau = 1. Data from J. Ďurech et al. (2024). 20 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 21.
    by D =1329 × 10−0.2H /p1/2 , where p is the geometric albedo. (We assumed p = 0.1 to calculate the diameters shown along the top axis of the Figure; in fact, the albedos of a majority of the plotted asteroids have not been measured.) Few asteroids larger than D ∼ 0.2 km have periods shorter than ∼2.4 hr (marked in the Figure by a horizontal, dashed red line) whereas, among smaller asteroids, more rapid rotation is common (P. Pravec et al. 2002; J. Beniyama et al. 2022). This is (admittedly indirect) evidence for spin-up by YORP of a “bunch of grapes” internal structure in which centripetal accelera- tions in asteroids with periods  2.4 hr result in mass shedding or breakup into smaller, more cohesive component units. Measured periods of some tiny asteroids are as short as a few seconds (J. Beniyama et al. 2022). 2. Asteroid Reshaping and Pair Formation. Separate evidence for rotational reshaping of asteroids is provided by the morphologies of some spacecraft-visited asteroids, which show approximate rotational symmetry but with an equatorial skirt consisting of material that has evidently migrated from higher latitudes (D. J. Scheeres 2015). New observations of asteroids have also revealed cases of episodic (311P/Panstarrs (2013 P5); D. Jewitt et al. 2013) and catastrophic (P/2013 R3, Catalina-Pan- STARRS; D. Jewitt et al. 2017) mass loss that indicate mass shedding and rotational breakup in real-time. Like 311P, the 2 km radius asteroid (6478) Gault displayed episodic mass loss consistent with mass shedding instability and also has a rotation period (2.55 hr) suggestively close to the rotational barrier (J. X. Luu et al. 2021). See Figures 17 and 18. Extreme end-cases of continued spin-up under the influence of YORP torque include rubble-pile disaggregation (D. J. Scheeres 2018), which might have been observed in P/2013 R3 (Figure 18), and the formation of asteroid binaries and pairs. These are independent asteroids with orbital element similarities that are statistically improbable by chance alone (S. A. Jacobson & D. J. Scheeres 2011; K. J. Walsh 2018). Asteroid pairs show a systematic relation between the angular frequency of the primary and the secondary/primary mass ratio, such that high mass ratio pairs have distinctly long-period primaries (P. Pravec et al. 2019). This is a result of the combined action of primary spin-up by YORP torques and tidal transfer of rotational energy from the primary, needed to expand the orbit of the secondary. Sudden mass transfer events on asteroids should lead to excited rotation; relevant observations are difficult and presently lacking. 3. Spin Alignment. It might be expected that collisionally produced asteroid families should have a very broad or even random distribution of spin vectors. S. M. Slivan (2002) discovered that the spin vectors of the Koronis family asteroids are instead clustered, with obliquities preferentially near 45° and 170°. This clustering was subsequently modeled as a consequence of interaction between gravitational and YORP torques (D. Vokrouhli- cký et al. 2003). Subsequent work showed that the general asteroid obliquity distribution is size dependent (J. Hanuš et al. 2011). Objects of diameter  30 km are more likely to have YORP-aligned obliquities near 0° and 180° (see Figure 7(a) of J. Hanuš et al. 2011). The efficacy of alignment by radiation torques is dependent not just on size but also on many poorly constrained physical and thermophysical parameters (O. Golubov Figure 16. Distribution of asteroid rotational frequencies [rotations day−1 ] as a function of absolute magnitude, H. Approximate asteroid diameters are marked at the top of the Figure, while their rotation periods are indicated on the right-hand axis. The dashed red line marking the “spin barrier” at P−1 ∼ 10 day−1 shows that few asteroids with diameter D  0.2 km have periods <2.4 hr. Data are from B. D. Warner et al. (2009; updated 2023 October 1 from the F-D Basic file at http://www. MinorPlanet.info/php/lcdb.php). 21 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 22.
    et al. 2021).The timescale for changing the obliquity can be shorter than the timescale for changing the spin (T. S. Statler 2015). 4. Post-main-sequence Evolution. As with the Yarkovsky force, the YORP effect will also be magnified as the Sun depletes its core hydrogen and its luminosity surges in the giant phase, by a factor up to ∼103.5 (E. Vassiliadis & P. R. Wood 1993). Rotational disruption of extrasolar asteroids by YORP spin-up may be an important contributor to debris that contaminates the photospheres of some white dwarf stars (D. Veras et al. 2014), where median mass accretion rates are in the range 105 –106 kg s−1 (J. Williams et al. 2024). 5. Tangential YORP (TYORP). The YORP torque described above is a result of gross deviations from rotational symmetry in the body shapes of asteroids. This is sometimes referred to as normal YORP or NYORP. Another radiation-driven torque, known as tangential YORP (TYORP), can be supplied by rocks and other small-scale surface structures (O. Golubov & Y. N. Krugly 2012). TYORP is different from NYORP in that it can exist even on a spherical asteroid, provided its surface is littered with rocks of the appropriate scale. Consider a rock of size, a, on the surface of an asteroid rotating with period, P. The conduction cooling time of the rock is τc ∼ a2 /κ (Equation (A2)), where κ is the Figure 17. Two active asteroids showing episodic ejections likely due to YORP-driven rotational instability. Each tail corresponds to a particular synchrone; the tail position angle is a measure of the date of ejection. 311P/Panstarrs (top) exhibited nine ejections over about 9 months (six visible on 2013 September 10 plus three later; D. Jewitt et al. 2013) while (6478) Gault displayed three tails in 2019 (J. X. Luu et al. 2021). 22 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 23.
    thermal diffusivity. Ifτc = P, the rock temperature will quickly equilibrate to the changing insolation through the day. Its temperatures at sunrise and sunset will be equal. If τc ? P, the rock temperature will be determined by the average of the day/night insolation, with little temporal variation. Only when τc ∼ P will the rock temperature exhibit substantial diurnal variation, being hotter near local sunset than near local sunrise because of cooling in the night. This temperature asymmetry can produce a torque with a large moment arm, because radiation and recoil from the warm Sun-facing side of a rock at sunset is not balanced by radiation and recoil from an equally warm Sun-facing side at sunrise. The critical rock size is aTY ∼ (Pκ)1/2 . For example, with P = 5 hr, κ = 10−6 m2 s−1 (as appropriate for a nonporous, consolidated material), aTY ∼ 0.15 m. Decimeter-sized rocks exert particularly strong tangential YORP torques. The magnitude of TYORP again depends on many unknown properties of the asteroid surface, notably the size and spatial distributions of surface rocks, and the degree to which each rock is separated from its neighbors enough to be thermally independent. But under some circumstances, the magnitude of TYORP may rival that of NYORP (O. Golubov & Y. N. Krugly 2012). More- over, the TYORP and NYORP torque vectors need not act in the same direction, raising the possibility that one might cancel the other, leading to no net torque on a body that otherwise might be expected to exhibit rotational acceleration. 6. Binary YORP (BYORP). Angular momentum added by photons can have particularly dramatic dynamical effects on some asteroid binaries (M. Ćuk & J. A. Burns 2005). By itself, tidal dissipation in close binaries commonly leads to synchronous rotation of the secondary (in which the orbit period and rotation period of the secondary are the same). In the synchronous state, the secondary can act effectively as an asymmetric extension of the primary, providing a large lever arm for the action of the radiation torque (M. Ćuk & J. A. Burns 2005). This torque is known as BYORP (B for binary). In nonsynchronous binaries, the effect of BYORP is averaged to zero. The evolution of binary asteroids under the combined action of tidal dissipation and BYORP radiative torque can lead to complicated orbital and spin evolution (S. A. Jacobson & D. J. Scheeres 2011), with evolutionary timescales as short as ∼104 yr (E. Steinberg & R. Sari 2011). If the tidal torque on the secondary is small compared to the radiative torque, the satellite can escape synchronous rotation and follow its own spin evolution, potentially leading to rotational breakup in orbit. The fragments from such a breakup would have short mutual collision times and quickly reaccumulate into a new rubble-pile satellite that repeatedly transforms into a ring before collapsing back into a single body. 8.4. Summary The timescales for the action of water ice sublimation torque (τS, Equation (34)) and YORP torque (τYORP, Equation (36)) are compared as functions of heliocentric distance in Figure 19. In the Figure, solid and dashed lines denote assumed object Figure 18. Rotationally disrupted asteroids 331P/Gibbs (top, from D. Jewitt et al. 2021) and P/2013 R3 (bottom, from D. Jewitt et al. 2017). Major fragments in each object are labeled. The largest of these in 331P is the 100 m scale fragment A, which is also a contact binary. Sizes of the fragments in P/2013 R3 are less certain because of heavy dust contamination, but estimated to be 100 m. Both objects are active asteroids, possessing cometary designations but having asteroid orbits (D. Jewitt 2012). 23 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
  • 24.
    radii of 1km and 0.1 km, respectively. The sublimation model was computed assuming ( ) q cos = 1/2 in Equation (4), which lies in between the high- and low-temperature limit models in Figure 1, and is scaled to timescale τs = 102 yr on a 1 km nucleus at 1 au, in agreement with the data (Figure 13). Figure 19 shows that, for a given object size and distances  5 or 6 au, the timescales for spin-change are ∼105 times shorter for sublimation torques than for YORP torques. Beyond Saturn, the diminishing water ice sublimation rate pushes the sublimation timescale to exceed that from YORP, and, in practice, both processes become so slow in the middle and outer solar system as to become largely irrelevant. Acknowledgments I thank Marco Fenucci for providing a digital table of his Yarkovsky drift measurements, Pedro Lacerda, Jane Luu, Joe Masiero, Darryl Seligman, David Vokrouhlicky, Emerson Whittaker, and an anonymous referee for comments. Appendix Thermophysics To gain some physical insight into how the thermophysical parameters of an asteroid might affect Yarkovsky and YORP, we consider the one-dimensional heat conduction equation ( ) r ¶ ¶ = ¶ ¶ c T t k T z , A1 p 2 2 where T is the temperature, k is the thermal conductivity, ρ is the bulk density, and cp is the specific heat capacity of the asteroid surface materials. Dimensionalizing this equation gives the timescale for heat to conduct over a distance ℓ as ( ) t k k r ~ = ℓ k c where A2 C p 2 and κ [m2 s−1 ] is the thermal diffusivity.4 For a body with rotation period P, we set τC = P and use Equation (A2) to find the distance over which heat can conduct in one rotation period as ℓ = (κP)1/2 , which defines the diurnal thermal skin depth on the asteroid. An exact solution of Equation (A1) would show that T varies with depth as a damped sinusoid, with (κP)1/2 being the e-folding length scale of the damping, but our order- of-magnitude approximation is sufficient here. On a spherical body with radius a, the heat contained within the skin depth, ℓ, is ( ) ( ) / p r k = H a P c T 4 A3 p 2 1 2 where cp [J K−1 kg−1 ] is the specific heat capacity of the shell. Loss of heat by radiation into space occurs at the rate ( ) p s = dH dt a T 4 A4 2 4 where σ = 5.67 × 10−8 W m−2 K−4 is the Stefan–Boltzmann radiation constant, and we assume emissivity ε = 1. The order- Figure 19. The YORP (blue) and water ice sublimation (red) timescales plotted as a function of rH. Timescales for objects 1 km radius (solid lines) and 0.1 km in radius (dashed lines) are shown. The shaded band marks the approximate location of the main asteroid belt, across which sublimation supplies a ∼105 times stronger torque if near-surface ice is present. 4 Diffusivity appears directly in Equation (A1) and is the natural measure of thermal response of a material through Equation (A2). In planetary science, the use of thermal inertia is instead widely preferred. Thermal inertia is defined by ( ) / r = I k cp 1 2, which has the somewhat uncomfortable units [J m−2 s−1/2 K−1 ]. Solid dielectrics (e.g., nonporous rocks) have I ∼ 103 J m−2 s−1/2 K−1 , while the finest dust, as found in the regoliths of small outer solar system bodies, has I ∼ 1 to 10 J m−2 s−1/2 K−1 (C. Ferrari 2018). Asteroid inertias in the range 10 „ I „ 100 J m−2 s−1/2 K−1 are common (E. M. MacLennan & J. P. Emery 2021). 24 The Planetary Science Journal, 6:12 (26pp), 2025 January Jewitt
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    of-magnitude cooling timescaleis τc ∼ H/(dH/dt), so that we can use Equations (A3) and (A4) to write the thermal parameter, Θ = τc/P (see J. R. Spencer et al. 1989) ( ) / Q r s k = c T P . A5 p 3 1 2 ⎛ ⎝ ⎞ ⎠ Asteroids with very rapid rotation (Θ ? 1) retain their heat through the night and should remain nearly longitudinally isothermal, producing little net diurnal Yarkovsky force. Similarly, asteroids with Θ = 1 will experience little diurnal Yarkovsky drift because rotation is too slow to carry the diurnal temperature maximum away from the midday meridian. Intuitively, on the other hand, asteroids with Θ  1 should retain heat into the afternoon but lose it by the morning and so will experience a net force from asymmetric radiation.5 As a rough example, consider an asteroid with a very porous surface regolith having κ = 10−9 m2 s−1 and with nominal ρ = 103 kg m−3 , cp = 103 J K−1 kg−1 , P = 2 × 104 s (i.e., about 6 hr), and orbiting near 1 au, where T ∼ 300 K. Substitution gives skin depth ℓ ∼ 5 mm, Θ ∼ 0.15, and the peak heat of noon will be retained for about one-sixth of a rotation, corresponding to a lag angle θ ∼ 60°. The Yarkovsky force on a small asteroid with these parameters is likely to be significant. By comparison, an asteroid having the same properties but consisting of solid rock (for which κ = 10−6 m2 s−1 would have ℓ ∼ 15 cm, Θ ∼ 4.5 and by virtue of taking several rotations in order to cool) would experience minimal diurnal temperature variation and reduced Yarkovsky acceleration. These simple considerations are only illustrative. Real asteroids are complicated and usually poorly characterized, with irregular shapes and boulder-strewn, variegated surfaces. On such bodies, the distribution of surface temperature and the magnitudes of the Yarkovsky and YORP effects cannot be accurately calculated, only worked out after the fact by measuring radial drift and changes in the rotation period. 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