YORP-Induced Long-Term Evolution of the Spin State of Small Asteroids and Meteoroids: Rubincam's Approximation

The rotation states of small asteroids and meteoroids are determined primarily by their collisions, gravitational torques due to the Sun and planets (in the case of close encounters), and internal dissipative effects (that relax the free-precession energy toward the fundamental state of principal-axis rotation). Rubincam has recently pointed out that thermal reemission on irregular-shaped bodies also results in a torque that may secularly change both the rotation rate and the orientation of the spin axis (the so-called YORP effect). Here we pursue investigation of this effect. Keeping the zero thermal-relaxation approximation of Rubincam and the assumption of the principal-axis rotation, we study the YORP effect both for precisely determined shapes of near-Earth asteroids and also for a large statistical sample of automatically generated shapes by the Gaussian-sphere technique of Muinonen. We find that the asymptotic state of the YORP evolution is characterized by an arbitrary value of the obliquity, with higher but nearly equal likelihood of 0°/180° and 90° states. At the adopted approximation, the most typical feature of this end state of the YORP evolution is secular deceleration of the rotation rate, which means that at some instant collisions will randomize the rotation state. In a minority of cases, the final state of the obliquity evolution leads to a permanent acceleration of the body's rotation, eventually resulting in rotational fission. The YORP-induced slow evolution may also play an important role in driving the rotation state of small asteroids toward the resonances between the forced precession due to the solar torque and perturbations of the orbital node and inclination. We find that for small Themis asteroids these resonances are isolated in the relevant range of frequencies, and the YORP evolving rotation may be either temporarily captured or rapidly jump across these resonances. In contrast, the possible values of the forced precession for small Flora asteroids may be resonant with clustered, nonisolated lines of the orbital perturbation. The individual rotation histories of small Flora asteroids may be thus very complicated and basically unpredictable. We comment on possible astronomical consequences of these results.     

Extreme sensitivity of the YORP effect to small-scale topography

Radiation recoil (YORP) torques are shown to be extremely sensitive to small-scale surface topography, using numerical simulations. Starting from a set of “base objects” representative of the near-Earth object population, random realizations of three types of small-scale topography are added: Gaussian surface fluctuations, craters, and boulders. For each, the expected relative errors in the spin and obliquity components of the YORP torque caused by the observationally unresolved small-scale topography are computed. Gaussian power, at angular scales below an observational limit, produces expected errors of order 100% if observations constrain the surface to a spherical harmonic order l?10. For errors under 10%, the surface must be constrained to at least l=20. A single crater with diameter roughly half the object's mean radius, placed at random locations, results in expected errors of several tens of percent. The errors scale with crater diameter D as D² for D>0.3 and as D³ for D<0.3 mean radii. Objects that are identical except for the location of a single large crater can differ by factors of several in YORP torque, while being photometrically indistinguishable at the level of hundredths of a magnitude. Boulders placed randomly on identical base objects create torque errors roughly 3 times larger than do craters of the same diameter, with errors scaling as the square of the boulder diameter. A single boulder comparable to Yoshinodai on 25143 Itokawa, moved by as little as twice its own diameter, can alter the magnitude of the torque by factors of several, and change the sign of its spin component at all obliquities. Most of the total torque error produced by multiple unresolved craters is contributed by the handful of largest craters; but both large and small boulders contribute comparably to the total boulder-induced error. A YORP torque prediction derived from groundbased data can be expected to be in error by of order 100% due to unresolved topography. Small surface changes caused by slow spin-up or spin-down may have significant stochastic effects on the spin evolution of small bodies. For rotation periods between roughly 2 and 10 h, these unpredictable changes may reverse the sign of the YORP torque. Objects in this spin regime may random-walk up and down in spin rate before the rubble-pile limit is exceeded and fissioning or loss of surface objects occurs. Similar behavior may be expected at rotation rates approaching the limiting values for tensile-strength dominated objects.     

Averaged rotational dynamics of an asteroid in tumbling rotation under the YORP torque

The secular effect of YORP torque on the rotational dynamics of an asteroid in non-principal axis rotation is studied. The general rotational equations of motion are derived and approximated with an illumination function expanded up to second order. The resulting equations of motion can be averaged over the fast rotation angles to yield secular equations for the angular momentum, dynamic inertia and obliquity. We study the properties of these secular equations and compare results to previous research. Finally, an application to several real asteroid shapes is made, in particular we study the predicted rotational dynamics of the asteroid Toutatis, which is known to be in a non-principal axis state.     

Effect of density inhomogeneity on YORP: The case of Itokawa

The effect of density inhomogeneity on the YORP effect for a given shape model is investigated. A density inhomogeneity will cause an offset between the center of figure and the center of mass and a re-orientation of the principal axes away from those associated with the shape alone. Both of these effects can alter the predicted YORP rate of change in angular velocity and obliquity. We apply these corrections to the Itokawa shape model and find that its YORP angular velocity rate is sensitive to offsets between its center of mass and center of figure, with a shift on the order of 15 m being able to change the sign of the YORP effect for that asteroid. Given the non-detection of YORP for Itokawa as of 2008, this can shed light on the density distribution within that body. The theory supports a shift of the asteroid center of mass towards Itokawa's neck region, where there is an accumulation of finer gravels, or towards the asteroid's “Head” region. Detection of the YORP effect for Itokawa should provide some strong constraints on its density distribution. This theory could also be applied to asteroids visited by future spacecraft to constrain density inhomogeneities.     

The dynamical evolution of uniformly rotating asteroids subject to YORP

A detailed derivation is given of the effect of solar radiation on the rotational dynamics of asteroids, commonly called the YORP effect. The current derivation goes beyond previous discussions published in the literature and provides a comprehensive secular dynamical analysis of the effect of solar radiation torques acting on a uniformly rotating body, and the evolution of its rotation state over time. Our predicted model has the global radiation properties of the asteroid as explicit parameters, and hence can be specified independent of these parameters. The resulting secular equations for the rotation rate and rotation pole are characterized by three parameters of the body's shape and explicitly includes the effect of thermal inertia on the evolution of these rotation state parameters. With this detailed model, in conjunction with estimated asteroid shapes and poles, we compute the expected YORP torques and dynamic response of several asteroids and the change in rotation rate for specific shapes as a function of obliquity. Finally, we define a convenient dimensionless parameter that is only a function of the body geometry and that can be used to characterize the effects of YORP.     

The YORP effect with finite thermal conductivity

The Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect has been recently suggested to significantly change, on a long-term, rotation state of small asteroids and meteoroids. Though YORP is closely related to the Yarkovsky (orbital) effect, it differs from the latter in two aspects: (i) YORP needs bodies of irregular shape to be effective, and (ii) YORP acts on bodies of zero surface thermal conductivity. To simplify computations, YORP has been so far investigated in the zero surface thermal conductivity limit only. Here we analyze the role of the surface conductivity and we find it substantially changes previous conclusions. Most importantly, unlike in the zero-conductivity limit, (i) YORP preferentially tilts obliquity toward two asymptotic states perpendicular to the orbital plane, and (ii) YORP asymptotically decelerates and accelerates rotation rate in about equal number of cases. Our work also indicates that direct detection of the YORP effect for a small asteroid may significantly constrain its mass.     

小行星YORP效应的可探测候选体

YORP (Yarkovsky-O''Keefe-Radzievskii-Paddack)效应是小行星长期动力学演化的机制之一. 与碰撞、引力摄动等因素相比, YORP效应作用量级小, 短时标观测效应不明显, 这给直接测量YORP效应带来了很大的困难. 利用小行星光变数据库中已知的小行星数据, 统计了小行星的自转速率分布, 使用核密度估计以及Kolmogorov-Smirnov检验分别分析了近地小行星和主带小行星自转速率的分布特性, 分别给出了在近地小行星和主带小行星中寻找受YORP效应影响减速自转的最佳样本群; 基于7颗已被探测到YORP旋转加速度的近地小行星, 利用YORP强度估计方法和光变探测条件建立了筛选模型, 给出了未来可直接通过光变数据探测\lk YORP效应的10颗近地小行星.     

Axial rotation of near-earth asteroids: The influence of the YORP effect

The distribution of axial rotation velocities of near-Earth asteroids (NEAs) substantially differs from that of the Main-Belt asteroids by an excess of both quickly and slowly rotating objects. Among the possible causes of this difference is the influence of the solar radiation—the so-called YORP effect—that arises from the absorption of solar energy and its reemission in the thermal range by a rotating body of irregular shape. It is known that the magnitude of this effect depends on the asteroid size and the quantity of received solar energy (the insolation). Analysis of the observational data showed that the mean diameter of NEAs decreases from the middle of the distribution to the edges, i.e., the excess of both slowly (ω ≤ 2 rev/day) and quickly (ω = 8–11 rev/day) rotating objects is formed due to the asteroids with sizes smaller than those in the middle of the distribution, which agrees well with the influence of the YORP effect. Moreover, the dependence of the axial rotation velocity of NEAs on the relative insolation shows that, for the NEAs referred to, both excesses are found in orbits where, on average, they receive 8–10% more solar energy than the NEAs in the middle of the distribution. This result also agrees with the character of the influence of the YORP effect and can be considered as an additional argument in its support. Thus, the study showed that one can infer that the currently available observational data suggest the possible influence of the YORP effect on the axial rotation of the near-Earth asteroids having sizes of D ～ 2 km and less. This is the first attempt to find the influence of the YORP effect on the axial rotation of the NEA family as a whole.     

Influence of the YORP effect on rotation rates of near‐Earth asteroids

The distribution of near‐Earth asteroid (NEA) rotation rates differs considerably from the similar distribution of Main Belt asteroids (MBAs) by the presence of excesses of fast and slow rotators, which are not observed or not so prominent in the distribution for MBAs. Among possible reasons for the difference, there can be influence of solar radiation on spin rate of small NEAs, the so‐called “YORP effect,” which appears due to reflection, absorption, and IR re‐emission of the sunlight by an irregularly shaped rotating asteroid. It is known that the YORP‐effect action strongly depends on the amount of solar energy obtained by the body (insolation), its size, and albedo. The analysis of observation data has shown that: (1) the mean diameter of NEAs decreases from the middle of the distribution to its ends, that is, the excesses of slow rotators (ω ≤ 2 rev day^?1) and fast rotators (ω ≥ 8 rev day^?1) are composed of smaller NEAs than in the middle of the distribution; (2) NEAs of both excesses are in the orbits where their insolation is about 8–10% larger than that of NEAs in the middle of the distribution; and (3) the objects in both excesses have a little lower albedo on average than that of objects in the middle of the distribution. All these results qualitatively agree well with the YORP‐effect action and may be considered as independent arguments in favor of it.     

The response of the African summer monsoon to remote and local forcing due to precession and obliquity

In this paper, we examine the orbital signal in Earth's climate with a coupled model of intermediate complexity (ECBilt). The orbital influence on climate is studied by isolating the obliquity and precession signal in several time-slice experiments. Focus is on monsoonal systems with emphasis on the African summer monsoon. The model shows that both the precession and the obliquity signal in the African summer monsoon consists of an intensified precipitation maximum and further northward extension during minimum precession and maximum obliquity than during maximum precession and minimum obliquity. In contrast to obliquity, precession also influences the seasonal timing of the occurrence of the maximum precipitation. The response of the African monsoon to orbital-induced insolation forcing can be divided into a response to insolation forcing at high northern latitudes and a response to insolation forcing at low latitudes, whereby the former dominates. The results also indicate that the amplitude of the precipitation response to obliquity depends on precession, while the precipitation response to precession is independent of obliquity. Our model experiments provide an explanation for the precession and obliquity signals in sedimentary records of the Mediterranean (e.g., Lourens et al. [Paleoceanography 11 (1996) 391, Nature 409 (2001) 1029]), through monsoon-induced variations in Nile river outflow and northern Africa aridity.     

The oblateness effect on the solar radiation incident at the top of the atmospheres of the outer planets

Calculations of the daily solar radiation incident at the top of the atmospheres of Jupiter, Saturn, Uranus, and Neptune, with and without the effect of the oblateness, are presented in a series of figures illustrating the seasonal and latitudinal variation of the ratio of both insolations. It is shown that for parts of the summer, the daily insolation of an oblate planet is increased, the zone of enhanced solar radiation being strongly dependent upon the obliquity, whereas the rate of increase is fixed by both the flattening and the obliquity. In winter, the oblateness effect results in a more extensive polar region, the daily solar radiation of an oblate planet always being reduced when compared to a spherical planet. In addition, we also numerically studied the mean daily solar radiation. As previously stated by A.W. Brinkman and J. McGregor (1979, Icarus, 38, 479–482), it is found that in summer the horizon plane is tilted toward the Sun for latitudes less than the subsolar point, but is titled away from the Sun beyond this latitude. It follows that the mean summer daily insolation is increased between the equator and the subsolar point, but decreased poleward of the above-mentioned limit. In winter, however, the horizon plane is always tilted away from the Sun, causing the mean winter daily insolation to be reduced. The partial gain of the mean summertime insolation being much smaller than the loss during winter season evidently yields a mean annual daily insolation which is decreased at all latitudes.     

The insolation at Pluto

Calculations of the daily solar radiation incident at the top of Pluto's atmosphere and its variability with latitude and season and of the latitudinal variation of the mean annual daily insolation are presented. The large eccentricity of Pluto produces significant north-south seasonal asymmetries in the daily insolation. As for Uranus, having a similarly large obliquity, the equator receives less annual average energy than the poles.     

Zero secular torque on asteroids from impinging solar photons in the YORP effect: A simple proof

YORP torques, where “YORP” stands for “Yarokovsky-O’Keefe-Radzievskii-Paddack,” arise mainly from sunlight reflected off a Solar System object and the infrared radiation emitted by it. We show here, through the most elementary demonstration that we can devise, that secular torques from impinging solar photons are generally negligible and thus cause little secular evolution of an asteroid’s obliquity or spin rate.     

Some aspects of the solar radiation incident at the top of the atmospheres of Mercury and Venus

A formalism has been developed for the calculation of the insolation on the planets Mercury and Venus neglecting any atmospheric absorption. For Mercury, the instantaneous insolation curves are repeated in a 2-tropical year cycle, the distribution of the solar radiation being perfectly symmetric between both hemispheres. In addition to latitudinal variations, one observes a longitudinal effect expressed by different instantaneous insolation distributions during the course of the time; on the equator, the relative diurnal insolation variability may attain a factor of 3. The small obliquity of Venus results in a nearly symmetric solar radiation distributions with respect to the equator except at the poles, where an important seasonal effect has been found. It has to be noted that no longitudinal dependence exists. Finally, the insolation curves are repeated in a nearly half-year cycle.     

Effects of thermal radiation on the dynamics of binary NEAs

The Yarkovsky force, produced when thermal radiation is re-emitted asymmetrically, causes significant orbital evolution of asteroids in the 10 m-10 km size range. When acting on a non-spherical body, the momentum carried by this radiation generally produces a torque, called the YORP effect, which may be important in re-orienting asteroidal spins. Here we explore a related effect, the “binary YORP” (BYORP), that can modify the orbit of a synchronously rotating secondary in a binary system. It arises because a locked secondary is effectively an asymmetric appendage of the primary. It should be particularly important for Near-Earth Asteroids (NEAs) owing to their small sizes, proximity to the Sun, and benign collisional environment. To estimate BYORP's strength, we subjected 100 random Gaussian spheroids to the thermal radiation model of Rubincam (2000, Radiative spin-up and spin-down of small asteroids, Icarus, 148, 2-11). For most shapes, a significant torque arose on the secondary's orbit, typically modifying the orbit's size, eccentricity and inclination in less than 10⁵ years, for components of 1 and 0.3 km radii, separated by 2 km, at 1 AU, each of density 1750 kg m⁻³. Together YORP and BYORP are capable of synchronizing secondaries and circularizing orbits, making tidal dissipation unnecessary to explain the evolved state of observed NEA pairs. However, BYORP's rapid timescale poses a problem for the abundance of observed NEA binaries, since their formation rate is thought to be much slower. We consider and reject the following resolutions of this quandary: (i) the approximation using Gaussian spheroids inadequately models YORP; (ii) most secondaries are not synchronous, but inhabit other spin-orbit resonances (very unlikely); (iii) tidal dissipation is much more efficient than previously estimated, and thus capable of stabilizing observed systems; and (iv) moderately close encounters with planets can re-orient secondaries, turning BYORP into a slower, random-walk process. Finally, we speculate that most observed binary NEAs are in a stable state in which the obliquity-changing torques of YORP (acting on the primary) and BYORP cancel out, and that those systems must be close to 55° inclination, where the momentum-changing torques of both YORP and BYORP tend to be very small. Some retrograde systems might develop such that the nodes precess at a Sun-synchronous rate, while some prograde ones might move into the “evection” resonance. All three of these hypotheses can be tested directly by comparison with the i, Ω and ? observed for NEA binaries.     

Insolation patterns on synchronous exoplanets with obliquity

A previous paper [Dobrovolskis, A.R., 2007. Icarus 192, 1-23] showed that eccentricity can have profound effects on the climate, habitability, and detectability of extrasolar planets. This complementary study shows that obliquity can have comparable effects.The known exoplanets exhibit a wide range of orbital eccentricities, but those within several million kilometers of their suns are generally in near-circular orbits. This fact is widely attributed to the dissipation of tides in the planets. Tides in a planet affect its spin even more than its orbit, and such tidally evolved planets often are assumed to be in synchronous rotation, so that their rotation periods are identical to their orbital periods. The canonical example of synchronous spin is the way that our Moon always keeps nearly the same hemisphere facing the Earth.Tides also tend to reduce the planet’s obliquity (the angle between its spin and orbital angular velocities). However, orbit precession can cause the rotation to become locked in a “Cassini state”, where it retains a nearly constant non-zero obliquity. For example, our Moon maintains an obliquity of about 6.7° with respect to its orbit about the Earth. In comparison, stable Cassini states can exist for practically any obliquity up to ∼90° or more for planets of binary stars, or in multi-planet systems with high mutual inclinations, such as are produced by scattering or by the Kozai mechanism.This work considers planets in synchronous rotation with circular orbits, but arbitrary obliquity β; this affects the distribution of insolation over the planet’s surface, particularly near its poles. For β=0, one hemisphere bakes in perpetual sunshine, while the opposite hemisphere experiences eternal darkness. As β increases, the region of permanent daylight and the antipodal realm of endless night both shrink, while a more temperate area of alternating day and night spreads in longitude, and especially in latitude. The regions of permanent day or night disappear at β=90°. The insolation regime passes through several more transitions as β continues to increase toward 180°, but the surface distribution of insolation remains non-uniform in both latitude and longitude.Thus obliquity, like eccentricity, can protect certain areas of the planet from the worst extremes of temperature and solar radiation, and can improve the planet’s habitability. These results also have implications for the direct detectability of extrasolar planets, and for the interpretation of their thermal emissions.     

Long term evolution and chaotic diffusion of the insolation quantities of Mars

As the obliquity of Mars is strongly chaotic, it is not possible to give a solution for its evolution over more than a few million years. Using the most recent data for the rotational state of Mars, and a new numerical integration of the Solar System, we provide here a precise solution for the evolution of Mars' spin over 10 to 20 Myr. Over 250 Myr, we present a statistical study of its possible evolution, when considering the uncertainties in the present rotational state. Over much longer time span, reaching 5 Gyr, chaotic diffusion prevails, and we have performed an extensive statistical analysis of the orbital and rotational evolution of Mars, relying on Laskar's secular solution of the Solar System, based on more than 600 orbital and 200,000 obliquity solutions over 5 Gyr. The density functions of the eccentricity and obliquity are specified with simple analytical formulas. We found an averaged eccentricity of Mars over 5 Gyr of 0.0690 with standard deviation 0.0299, while the averaged value of the obliquity is 37.62° with a standard deviation of 13.82°, and a maximal value of 82.035°. We find that the probability for Mars' obliquity to have reached more than 60° in the past 1 Gyr is 63.0%, and 89.3% in 3 Gyr. Over 4 Gyr, the position of Mars' axis is given by a uniform distribution on a spherical cap limited by the obliquity 58.62°, with the addition of a random noise allowing a slow diffusion beyond this limit. We can also define a standard model of Mars' insolation parameters over 4 Gyr with the most probable values 0.068 for the eccentricity and 41.80° for the obliquity.     

A Detectable Candidate for the YORP Effect of Asteroids

The YORP (Yarkovsky-O’Keefe-Radzievskii-Paddack) effect is one of the mechanisms of the long-term dynamical evolution of asteroids. Compared with factors such as collision and gravitational perturbation, the YORP is of small magnitude, and the short-time scale observation effect is inconspicuous, which brings great difficulties to the direct measurement of the YORP. From the Asteroid Lightcurve Database, asteroids having a high confidence rotation period were selected for this study. Two subsample groups for identifying potential asteroids slowed by the YORP effect are provided by using the kernel density estimation method and the Kolmogorov-Smirnov test to analyze the rotation rate distribution characteristics of near-Earth asteroids and main belt asteroids; a screening model is proposed based on the light-curve data of seven YORP asteroids with YORP rotation acceleration, combined with the YORP intensity estimation method and the detection conditions of the YORP effect. Finally, ten candidates that can directly detect the YORP effect through light-curve data in the future are listed based on the screening model.     

Insolation changes on pluto caused by orbital element variations

In this paper, we compare changes in the insolation at Pluto, corresponding to three epochs during the dynamical history of the planet: t = – 1, 0 and 0.5, where t is the time in millions of years A.D. The two extreme values of t coincide respectively with a maximum (126 ) and a minimum (102 ) value of the obliquity (). The other orbital elements i.e. the eccentricity (e) and the longitude of the perihelion ( _p ) which affect solar radiation and which are apt to significant periodic changes are also calculated for the times under consideration. In a series of figures, the combined influence of the evolving dynamic parameters on the daily insolation and on the mean (summer, winter, annual) daily insolation is illustrated.