# Yarkovsky and YORP effects

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Post-publication activity

Curator: David Vokrouhlicky

Prof. David Vokrouhlicky accepted the invitation on 22 February 2010 (self-imposed deadline: 22 August 2010).

The Yarkovsky effect describes a small but significant force that affects the orbital motion of meteoroids and asteroids smaller than 30-40 kilometers in diameter. It is caused by sunlight; when these bodies heat up in the Sun, they eventually re-radiate the energy away as heat, which in turn creates a tiny thrust. This recoil acceleration is much weaker than solar and planetary gravitational forces, but it can produce substantial orbital changes over timescales ranging from millions to billions of years. The same physical phenomenon also creates a thermal torque that, complemented by a torque produced by scattered sunlight, can modify the rotation rates and obliquities of small bodies as well. This rotational variant has been coined the Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect. During the past decade or so, the Yarkovsky and YORP effects have been used to explore and potentially resolve a number of unsolved mysteries in planetary science dealing with small bodies.

# Historical notes

Interesting problems in science usually have a long history. It is rare, though, that they have a prehistory or perhaps even a mythology of sorts. Yet this is the case for the Yarkovsky effect. I.O. Yarkovsky, a Russian civil engineer born in a family of Polish descent, noted in a privately-published pamphlet (Yarkovsky 1901 and Figure 1) that heating a prograde-rotating planet should produce a transverse acceleration in its motion. While the context of Yarkovsky’s work was mistaken and he was only roughly able to estimate the magnitude of the effect, he succeeded in planting the germ of an idea that later blossomed into a full-fledged theory of how the orbits of small objects revolving about the Sun are modified by the absorption and reemission of sunlight.

Figure 1: Title page of I.O. Yarkovsky's pamphlet, privately published in Bryansk in 1901. Here, for the first time, the concept of his effect appeared, though in a context that is now obsolete (i.e., he assumed that ether existed between solar system bodies).

The legend of Yarkovsky would likely have been lost, along with interest in his pamphlet, if it had not been for the pioneering Estonian astronomer E.J. Öpik. Öpik had read the original document, and he re-introduced its ideas into the modern literature decades after Yarkovsky’s death (Öpik 1951). More recently, interest in Yarkovsky’s lost pamphlet prompted Dutch amateur astronomer G. Beekman to search for it in Russian archives, where he found it only a few years ago (Beekman 2006 and Figure 1). Thanks to his efforts, it is now available for everyone to read for the first time in over 100 years. With this said, though, some memory of Yarkovsky’s idea had been independently kept in the Soviet Union as shown by work of V.V. Radzievskii and others (e.g., Radzievskii 1952).

Meteorite transport was the primary application that led Öpik to re-introduce the Yarkovsky effect, with the idea being that meter-sized and smaller stones could be delivered to Earth from the main asteroid belt located between the orbits of Mars and Jupiter. Astronomers in the 1970s and 1980s also considered these effects, but it is fair to say that the topic hibernated at the outskirts of planetary science during this period. This is because other important components, such as how the dynamics of small bodies are affected by mean-motion and secular resonances, were not known in detail until the 1980s and were not easy to treat numerically until the 1990s. Even worse, modeling the transport of small objects from the asteroid belt to the Earth by thermal forces alone depended on parameters and/or rotation states for small bodies that were almost completely unknown. In the minds of many dynamicists of the day, the Yarkovsky effect was a small and somewhat unwanted complication on a difficult problem, such that it was easier to ignore it for the time being.

Still, some quantitative progress was made during this time. In the late 1970s and 1980s, new insights on the role of radiative (thermal) forces and torques came from the somewhat unexpected field of space geodesy. First, an analysis of tiny orbital residuals in the geodynamical satellite LAGEOS led D.P. Rubincam to develop applications of thermal forces that could characterize the motion of this spacecraft (e.g., Rubincam 1987). Second, data describing the rotation of balloon-type spacecraft since the late 1960s revealed that radiation torques provided a significant brake on their rotation rates. Lessons from both led D.P. Rubincam to revive applications of radiation forces and torques in planetary science (e.g., Rubincam 1995, 2000). In fact, this revival can be appreciated even more by noting that the applications from space geodesy allowed Rubincam to recognize two different variants of the Yarkovsky effect: diurnal and seasonal, of which the latter had not been previously known in planetary science.

The modern boom of thermal force and torque applications in planetary sciences started in the late 1990s largely through the theoretical and numerical modeling work of D.P. Rubincam and P. Farinella. Their work inspired other researchers to investigate additional applications, many of which reached far beyond the original field of meteorite delivery. The issues now affected by thermal forces run the gamut from the fine details of the impact hazard onto the Earth, through formation and dispersal of binary asteroids, up to understanding the orbital structure of asteroid families.

The Yarkovsky and YORP effects are sometimes found to work in a complicated synergy to provide unexpected results. Some examples include the production of asteroid spin vectors with similar magnitudes and orientations (e.g., certain observed members of the Koronis asteroid family; Slivan 2002). The Yarkovsky effect is also becoming a routine part of orbit determination of well-tracked, small near-Earth asteroids. The precise measurement of orbital displacement produced by the Yarkovsky effect over time can even allow ground-based observers to deduce asteroid physical properties like bulk density (Chesley et al. 2003). In-depth review papers on the Yarkovsky and YORP effects can be found in Bottke et al. (2002, 2006).

# Basic concepts: Radiation force and torque

In basic terms, the Yarkovsky effect is defined as a thermal radiation force that causes objects to undergo semimajor axis drift as a function of their size, orbit, and material properties. The YORP effect is a thermal torque that can increase or decrease a body’s spin rate and can modify its spin axis orientation. The name YORP effect comes from an abbreviation of the names Yarkovsky-O’Keefe-Radzievskii-Paddack, a listing of those scientists who worked on or inspired the idea that radiation-driven accelerations could modify the rotation rates of small planetary bodies (see Rubincam 2000).

The theory is as follows. Sunlight impinging on the surface of a body in space is removed from the incident beam, with the energy reprocessed in several ways: (i) light is directly scattered in the optical band, (ii) light is absorbed and conducted as heat to the deeper layers of the body, and (iii) heat is re-emitted as infrared radiation in the thermal band. This is because the effective temperatures in the relevant range of heliocentric distances, a fraction of AU to several AU, are equal to few hundreds of degrees K. Thus, the Yarkovsky and YORP effects can be considered mechanical forces and torques produced by these radiative processes.

The Yarkovsky and YORP effects are formulated by calculating the net radiation energy and momentum exchange produced by all parts of a small body. In practice, this means computing these components on a small (infinitesimal) surface element on an arbitrary-shaped body. The total effect is then the surface integral over all elements. As far as the physical assumptions are concerned, the spectral dependence is simplified into two representative bands, "optical" and "thermal", assuming these components characterize the solar spectrum emissivity and the body’s emissivity.

Note that the relative motion of the body with respect to the Sun is neglected for all but the smallest bodies. The relevant "aberration-type" correction (the Poynting-Robertson effect) is mainly important for dynamics of sub-mm particles in interplanetary (or circumplanetary) space.

## Description of surface processes

Energy budget. -- Energy conservation, needed for surface temperature $$T$$ solution, occurs as a surface boundary condition in the mathematical description of the Yarkovsky and YORP effects. At the most general level it can be given as

$\tag{1} \epsilon(\mu_0) \, \sigma \, T^4 + K \, \mathbf n_{\perp} \cdot \nabla T = (1-A_h(\mu_0)) \, \mu_0 \, F\; ,$

where $$\mathbf n_{\perp}$$ is the outward-oriented normal vector to the chosen surface element $$d{\mathbf S} = {\mathbf n}_{\perp}\, dS\ ,$$ $$\mu_0$$ is the cosine of local zenith distance of the Sun ($$\mu_0 = \mathbf n_{\perp} \cdot \mathbf n_0 \ ,$$ if $$\mathbf n_0$$ is the unit vector pointing toward the Sun), $$\nabla T$$ is the gradient of the surface temperature, $$K$$ is the surface thermal conductivity, $$\sigma$$ is the Stefan-Boltzmann constant and $$F$$ is the sunlight flux. The two functions $$\epsilon (\mu_0)$$ (hemispherical emissivity) and $$A_h(\mu_0)$$ (hemispherical albedo) can be written as:

$\tag{2} A_h(\mu_0)= \frac{1}{\mu_0} \int_{\Omega_+} d \Omega \, \mu \; r_{\mathrm{sca}}(\mu, \mu_0, \phi)\; ,$

and

$\tag{3} \epsilon(\mu_0) = \frac{1}{\pi} \int_{\Omega_+} d \Omega \, \mu \; r_{\mathrm{th}}(\mu, \mu_0, \phi)\; ,$

with $$r_{\mathrm{sca}}(\mu, \mu_0, \phi)$$ and $$r_{\mathrm{th}}(\mu, \mu_0, \phi)$$ being the bi-directional scattering function in optical and bi-directional emissivity function in thermal wavelengths. The underlying assumption, as in the related formalism used for asteroid photometry and radiometry observations, is that the scattered and thermally emitted radiation fields are composed of contributions from many microscopic scatterers and emitters (grain size and smaller). In that case the bi-directional functions depend on spherical angles of the hemisphere $$\Omega_+$$ exterior to the surface facet, namely cosine of colatitude $$\mu$$ and longitude $$\phi$$ measured from $$\mathbf m =( \mathbf n_0 - \mu_0\, \mathbf n_{\perp} ) / \sqrt{1 - \mu^2_0}$$ in the horizon plane of $$d {\mathbf S}\ .$$

Note that $$r_{\mathrm{sca}}$$ and $$r_{\mathrm{th}}$$ do not include effects of the macroscopic surface irregularities, craters, boulders and larger-scale features, that need to be separately included in the computations. In most of the literature on the Yarkovsky and YORP effects $$r_{\mathrm{sca}}$$ and $$r_{\mathrm{th}}$$ have been approximated by Lambert’s law, namely $$r_{\mathrm{sca}}=\mu_0 A/\pi$$ with a characteristic albedo value (constant) $$A_h(\mu_0) = A$$ and $$r_{\mathrm{th}} = \epsilon$$ with a characteristic emissivity value (constant) $$\epsilon (\mu_0) = \epsilon\ .$$ While the Yarkovsky effect description requires finite (non-zero) thermal conductivity $$K\ ,$$ the most simple approximation of the YORP effect (often referred to as the Rubincam approximation) assumes $$K = 0\ .$$ Then the energy budget equation (1) simplifies to $$\epsilon\,\sigma T^4 = (1-A) \mu_0 F$$ and can be used to express the temperature $$T$$ without the need to solve the heat diffusion problem.

Linear momentum budget. -- Solar photons hitting the body produce scattered and thermally emitted radiation that carries away linear momentum. This causes the objects to move according to Newton’s third law of action-reaction. The dynamical consequences (both on translation and rotation) of the incident photons are small, but those related to the scattered and thermally emitted components can be large. The infinitesimal force $$d {\mathbf f}_{\mathrm{sca}}$$ due to the scattered part is given by

$\tag{4} d \mathbf f_{\mathrm{sca}} = -\frac{F}{c}\, (K^{\mathrm{sca}}_1\, \mathbf n_{\perp} + K^{\mathrm{sca}}_2\, \mathbf m ) \, dS\; ,$

where $$c$$ is the light velocity and

$\tag{5} K^{\mathrm{sca}}_1(\mu_0) = \int_{\Omega_+} d\Omega \, \mu^2 \; r_{\mathrm{sca}}(\mu,\mu_0,\phi)\; ,$

$\tag{6} K^{\mathrm{sca}}_2(\mu_0) = \int_{\Omega_+} d\Omega \, \mu \sqrt{1-\mu^2} \, \cos \phi \; r_{\mathrm{sca}}(\mu,\mu_0,\phi) \; .$

$\tag{7} d \mathbf f_{\mathrm{th}} = - \frac{\sigma T^4}{c}\, ( K^{\mathrm{th}}_1\, \mathbf n_{\perp} + K^{\mathrm{th}}_2\, \mathbf m ) \, dS \; ,$

with

$\tag{8} K^{\mathrm{th}}_1(\mu_0) = \frac{1}{\pi} \int_{\Omega_+ } d \Omega \, \mu^2 \; r_{\mathrm{th}}(\mu,\mu_0,\phi)\; ,$

$\tag{9} K^{\mathrm{th}}_2(\mu_0) = \frac{1}{\pi} \int_{\Omega_+ } d \Omega \, \mu \sqrt{ 1 - \mu^2 } \, \cos \phi \; r_{\mathrm{th}}(\mu, \mu_0, \phi)\; .$

In the simplest situation, using Lambert’s law to model scattering and thermal emission, one has $$K^{\mathrm{sca}}_1(\mu_0)= \frac{2}{3}\, \mu_0 A$$ and $$K^{\mathrm{sca}}_2(\mu_0)= 0\ ,$$ and similarly $$K^{\mathrm{th}}_1(\mu_0)= \frac{2}{3}\, \epsilon$$ and $$K^{\mathrm{th}}_2(\mu_0)= 0\ .$$ The thermal force, for instance, is then given by

$\tag{10} d \mathbf f_{\mathrm{th}} = - \frac{2}{3} \frac{\epsilon \sigma T^4}{c}\, \mathbf n_{\perp} \, dS \; .$

The YORP effect computation may also avoid the need to solve the surface temperature $$T$$ if one uses Rubincam’s approximation.

Total force and torque. -- While this theory has been used to describe the radiative effects on an infinitesimal surface element $$d {\mathbf S}\ ,$$ additional mathematical labor is needed to estimate the effect for the entire body. In principle this is simply a surface integral

$\tag{11} \mathbf f_{\mathrm{sca}} = \int_{ S } d \mathbf f_{\mathrm{sca}} , \qquad \mathbf f_{\mathrm{th}} = \int_{ S' } d \mathbf f_{\mathrm{th}}$

for the radiation force due to the scattered and thermally emitted radiation fields, and

$\tag{12} \mathbf T_{\mathrm{sca}} = \int_{ S } \mathbf r \times d \mathbf f_{\mathrm{sca}} , \qquad \mathbf T_{\mathrm{th}} = \int_{ S' } \mathbf r \times d \mathbf f_{\mathrm{th}}$

for the corresponding radiation torques. Here $${\mathbf r}$$ is the position vector of $$d {\mathbf S}$$ with respect to an appropriately chosen reference center in the body, while the integration domains correspond to the entire surface for the thermal effects ($$S'$$) and the surface part illuminated by the Sun for the effects due to the scattered radiation ($$S$$).

A detailed description of $$S$$ can be difficult to compute with accuracy, partly because it depends on the mutual geometric configuration of the Sun as well as the asteroid’s spin axis and prime-meridian directions, but also because it needs to account for the possibility that irregular-shaped objects spinning in the Sun may have regions capable of shadowing adjacent regions. The latter phenomena, referred to here as mutual occlusions, is important for the prediction of the YORP effect, and it creates additional complications that need to be attacked using detailed and computationally-expensive numerical approaches.

The thermal force $${\mathbf f}_{\mathrm{th}}$$ is the basis of the Yarkovsky effect, while the combined effect of the torques $${\mathbf T}_{\mathrm{sca}}$$ and $${\mathbf T}_{\mathrm{th}}$$ is the basis of the YORP torque. The integrals in (11) and (12) could be evaluated using a wide variety of analytical and numerical methods as one proceeds from simpler to more general shape models. The analytical methods have certain advantages, namely one gets a complete understanding of how the results are dependant on various parameters. On the other hand, they require the use of simple shape models (e.g., a sphere or a spheroid), which limits their effectiveness when dealing with real asteroid shapes. Numerical methods are often better suited to compute the Yarkovsky and YORP effects for a body of an arbitrary shape. To date, these kinds of results have been most effective in helping us understand how surface irregularities affect a body’s evolution. The most common numerical method used to compute how the Yarkovsky and YORP affect irregularly-shaped bodies is to represent their surface components with a large number of small triangular facets. In some extreme cases, computations have been carried out for asteroid models made up of millions of facets.

# Basic concepts: Temperature solution

Determination of the Yarkovsky force and the YORP torque on a rotating body requires computation of its surface temperature $$T\ .$$ This can be calculated in the frame co-moving with the body from heat conduction theory, whose basic relation, often called the Fourier heat equation, is written as:

$\tag{13} \rho \, C_p \frac{ \partial T }{ \partial t } = \nabla \cdot (K \, \nabla T)\; .$

Here $$\rho$$ is the bulk density and $$C_p$$ is the specific heat capacity at constant pressure. This is basically an energy conservation relation, and it states that the energy change in a volume element, given by the time derivative of the temperature, reflects the heat flux $$-K \nabla T$$ through its boundary. Motion of the continuum as well as other energy sources and sinks (e.g., sublimation or phase transitions) are neglected in this approach. In general, the physical constants $$\rho\ ,$$ $$C_p$$ and $$K$$ depend both on (i) the temperature $$T\ ,$$ and (ii) the position of the volume element in the body. When temperature changes are small enough, such that $$\rho\ ,$$ $$C_p$$ and $$K$$ can be reliably approximated by the values at the body's mean temperature, and they do not change in a given region, Eq. (13) simplifies to

$\tag{14} \frac{ \partial T }{ \partial t } - \kappa \nabla^2 T = 0\; ,$

where

$\tag{15} \kappa = \frac{ K }{ \rho \, C_p }\; .$

In many applications it is useful to introduce scaling of the space and time variables such that the physical constants drop out of the heat conduction equation (14). For instance, if the temperature is periodic with a fundamental frequency $$\nu\ ,$$ such as the rotation frequency expressing diurnal cycles of $$T\ ,$$ $$t$$ may be suitably replaced with a non-dimensional quantity $$\zeta = \exp(\iota \nu t )\ .$$ This rescaling naturally involves introduction of the scale length $$l_s = \sqrt{ \kappa/\nu }$$ such that (14) takes the form

$\tag{16} \iota \, \zeta \frac{ \partial T }{ \partial \zeta } - \nabla'^2 T = 0\; ,$

where the scaled nabla operator reads$\nabla' = l_s \nabla$. As an aside, it turns out that $$l_s\!$$ is the penetration depth of thermal wave with frequency $$\nu\!$$ under the surface of the body. Scaling the temperature is not a fundamental property of the heat conduction equation but is motivated by the linearization of the boundary condition (1).

The heat diffusion equation has to be complemented with the appropriate number of boundary constraints in the space and time domains. Solutions of the Yarkovsky force and YORP torque have so far always employed the assumption of periodicity, either diurnal (where the fundamental frequency is due to rotation of the body) or seasonal (where the fundamental frequency is the mean motion about the Sun). This assumption removes the time boundary. The space boundary is specified by the surface of the body, with the energy condition (1) providing the fundamental constraint for $$T\ .$$

Additional constraints come from requirement of regularity of $$T$$ over the entire volume of the body. Note that either form of the heat conduction equation given above, Eqs. (14) and (16), is linear and thus the general solution can be given as a superposition of fundamental modes, of which some typically diverge in the volume and must be excluded.

When the body is not homogeneous,but consists of, say, discrete layers of different thermal properties, additional constraints have to be imposed on their boundaries. While the general solution of the heat diffusion problem can formally be given as a superposition of their fundamental modes, there are two obstacles that prevent one from trying to calculate an explicit solution for $$T$$ on the body’s surface.

First, the fundamental modes need to be algebraically manageable for sufficiently simple geometries such as semi-space (plane-parallel), spherical or cylindrical cases. No formulation is available for a body with a highly irregular shape, which is unfortunate given that these shapes are typical for small objects in the Solar system. Second, even in the case of simple geometry, the quartic emission term in the boundary condition (1) violates the linearity of the problem.

The first issue is essentially insurmountable and, if the irregular shape is really needed (as in the case of the YORP effect), it needs to be attacked by numerical methods. The second issue is usually faced by the linearization method, namely a split of $$T$$ into a mean value $$T_0$$ and an increment $$\Delta T$$ $$(T = T_0 + \Delta T)$$ such that $$|\Delta T| \ll T_0$$ is assumed. Then $$T^4\simeq T_0^4 + 4\,T_0^3 \Delta T$$ in (1).

Analytical solutions for the surface temperature have been obtained using the linearized approximation for bodies having simple shapes. These are very useful for basic characterization of the Yarkovsky and/or YORP effects and have been implemented in a number of numerical integration packages (such as OrbFit or SWIFT). The last few years have also seen a number of semi-numerical or numerical methods developed with applications to particular situations.

# Basic concepts: Orbital and rotational effects

## Orbital consequences of the Yarkovsky effect

The dynamical effects of the radiation forces described above in the optical wavelengths are typically small for meter-sized and larger bodies. They even vanish for circular orbits. We therefore restrict our discussion to the orbital effects of the thermal forces (i.e., $${\mathbf f}_{\mathrm{th}}$$ in (11)). These thermal forces are much smaller than attraction of the center (Sun or planet), allowing one to use the framework of perturbation theory to describe their orbital consequences. Focusing on the most important of them, namely how they change the semimajor axis $$a$$, one can write: $\tag{17} \frac{da}{dt} = \frac{2}{mn^2 a} \,(\mathbf{f} \cdot \mathbf{v}) = \frac{2}{mn} (\mathbf{f} \cdot \mathbf{e}_\tau + \mathcal{O}(e) )\; ,$

where $$M$$ and $$m$$ are masses of the body and the center, $$n$$ is the mean motion $$\left(a^3 n^2 = G\left(M+m\right)\right)$$ and $${\mathbf e}_\tau$$ is a unit vector in the osculating orbital plane and transverse to direction to the center. For small perturbations, the first-order effects are obtained by inserting the unperturbed (Keplerian) motions into the right hand side of (17). The long-term (accumulated) orbital effects are further characterized by averaging over the mean longitude in orbit. This removes the short-period effects from the analysis, which are unimportant for most situations, even for the detection issue discussed below.

A distinct feature of the Yarkovsky effect is its ability to modify the semimajor axis of the body. An analytical estimate of $$da/dt$$ for a spherical body not only provides one with reasonable accuracy but also and yields insights into how numerous parameters such as size, thermal inertia, heliocentric distance and/or obliquity affect the Yarkovsky drift rate. Assuming rotation about the principal axis of the inertia tensor, it is natural to split the effects of the thermal force $${\mathbf f}_{\mathrm{th}}$$ into (i) a component aligned with the spin axis direction (independent on the rotation cycle), and (ii) two components in the equatorial plane of the body (dependent on the rotation cycle). When these values are input into perturbation theory equations, the spin-aligned component produces a secular effect (i.e., one that is not averaged out over time):

Figure 2: Schematic representation of the two variants of the Yarkovsky effect: (i) the diurnal component (left), and (ii) the seasonal component (right). A circular orbit and optimum values of the obliquity are assumed for simplicity, $$\gamma = 0^\circ$$ on the left figure, and $$\gamma = 90^\circ$$ on the right figure. Sunlight always heats the body on the nearside (noon), but due to finite thermal inertia the maximum surface temperature, and thus maximum of the recoil effect of the thermal radiation, is displaced from the solar direction. In the diurnal variant (left), the body's rotation forces the maximum emissivity to be be skewed toward the afternoon side on the body and thus the recoil force is always directed along the gold arrows. A net positive along-track force makes the body systematically accelerated and thus spiraling outward away from the Sun. The effect would have an opposite sign/sense if the body had a retrograde rotation (i.e., with obliquity $$\gamma = 180^\circ$$). In the seasonal variant (right), thermal relaxation occurs on the timescale comparable to the revolution period of the body about the center. The seasonal force -- again shown with the gold arrows -- is directed along the spin axis and is due to north/south temperature difference on the body. One way to think about this is that the hottest part of summer in the Northern Hemisphere is more likely to be in July-August rather than in June. The net, orbit-averaged, along-track force is always negative and the seasonal variant of the Yarkovsky effect makes the orbit migrate consistently towards the center. For extreme values of the obliquity, $$\gamma = 0^\circ$$ or $$\gamma = 180^\circ\ ,$$ the seasonal component is zero because of symmetry between the north and south hemispheres

$\tag{18} \left(\frac{da}{dt}\right)_{\mathrm{seasonal}}= \frac{4}{9}\,(1-A)\,\Phi\, F_n(R', \Theta )\, \sin^2 \gamma\; ,$

while the equatorial components provide a secular effect

$\tag{19} \left(\frac{da}{dt}\right)_{\mathrm{diurnal}} = - \frac{8}{9}\,(1-A)\,\Phi\, F_\omega(R', \Theta )\, \cos \gamma\; .$

In the linear approximation the total effect $$(da/dt)_{\mathrm{tot}}$$ is a superposition of (18) and (19); $$A$$ is an effective albedo value close to the Bond albedo, $$\Phi = \pi R^2 F/(mc)$$ is the characteristic radiation pressure factor, $$R$$ is the radius of the body, and $$\gamma$$ is obliquity of the spin axis (i.e., angle between the spin axis direction and normal to the osculating orbital plane about the center). The $$F_\nu$$-functions are described below. The dependence on different fundamental frequencies $$\nu$$ in the problem, the mean motion frequency $$n$$ in (18), and the rotational frequency $$\omega$$ in (19) gives rise to the terms seasonal and diurnal Yarkovsky effects, respectively.

Dependence on the obliquity. -- Examining the dependence on $$\gamma$$ in (18) and (19), one has: (i) $$(da/dt)_{\mathrm{diurnal}}\varpropto \cos\gamma$$ in the diurnal case, and (ii) $$(da/dt)_{\mathrm{seasonal}}\propto \sin^2 \gamma$$ in the seasonal case. As a consequence, the diurnal component of the Yarkovsky effect can produce both outward secular migration of the orbit (i.e., $$da/dt$$ positive because the $$F_\omega$$ function is negative) for prograde-rotating bodies with $$\gamma < 90^\circ$$ and inward secular migration of the orbit for retrograde-rotating bodies with $$\gamma > 90^\circ\ .$$ The maximum drift occurs when the spin axis is perpendicular to the orbital plane, $$\gamma = 0^\circ$$ or $$\gamma = 180^\circ\ ,$$ and vanishes when the spin axis is in the orbital plane, $$\gamma = 90^\circ\ .$$ In addition, because of the dependence on $$\cos\gamma\ ,$$ a population evolving in semimajor axis with isotropically distributed spin axes will have an average semimajor axis change of zero. This implies the net, long-term effect is the population diffusing into smaller and larger semimajor axis values while keeping the same mean value. On the other hand, the seasonal variant of the Yarkovsky effect always causes orbital decay towards the center (i.e., $$da/dt$$ negative). It is maximum when the spin axis is in the orbital plane and null when the spin axis is perpendicular to it. Figure 2 provides a clear intuitive basis for these conclusions.

Figure 3: Top panel: mean drift $$\Delta a$$ due to the Yarkovsky forces in a constant interval of time 1 My for a sample of asteroids at 2-3 AU heliocentric distance. The bodies have been given random orientation of spin axes in space and different values of the surface thermal conductivity K = 0.001 W/m/K, 0.01 W/m/K, 0.1 W/m/K and 1 W/m/K (see the labels and colors of different curves). The mean change of the semimajor axis, as determined by the initial and final values, has been evaluated for bodies of different size $$D\ ,$$ from meters to ten kilometers (abscissa). $$\Delta a$$ generally decreases toward larger objects, roughly following the $$\varpropto 1/D$$ rule. Bottom panel: mean drift $$\Delta a$$ due to the Yarkovsky forces over a variable time interval equal to the estimated collisional lifetime of a main-belt asteroid with size $$d\ .$$ The longer lifetime for larger bodies mostly compensates for the intrinsic decrease of $$\Delta a$$ from the top panel.

Dependence on the size. -- The magnitude of the semimajor axis secular change is given by the F-functions which read (the frequency ν stands either for ω or n)

$\tag{20} F_\nu(R',\Theta) = - \frac{\kappa_1(R')\,\Theta_\nu}{1 + \kappa_2(R')\,\Theta_\nu + \kappa_3(R')\, \Theta^2_\nu}\; .$

The frequency dependence appears explicitly in the appropriate thermal parameter $$\Theta_\nu = \Gamma \sqrt{\nu}/(\epsilon\sigma T^3_{\star} )$$ and implicitly in the scaling of the size of the body $$R' = R/l_s\ .$$ Recall the penetration depth $$l_s = \sqrt{ K /(\rho C_p \nu) } = K /(\Gamma \sqrt{\nu} )$$ of the thermal wave depends on the frequency and differs thus for the diurnal and seasonal waves. Because the rotation frequency $$\omega$$ is typically much larger than the mean motion $$n\ ,$$ the penetration depth of the seasonal wave is larger than that of the diurnal wave.

To determine the limiting cases for small/large bodies, one needs to scale our equations using the appropriate penetration depth $$l_s$$ or use the non-dimensional parameter $$R'\ .$$ In the limit of large bodies for which $$R'\gg 1$$ (bodies larger than few meters across for relevant thermal parameters and heliocentric distances), the $$\kappa$$-functions in (20) become constant $$\kappa_1\rightarrow$$ ½, $$\kappa_2\rightarrow 1\ ,$$ and $$\kappa_3\rightarrow$$ ½ (for $$R'\rightarrow \infty$$). So the size dependence drops from the $$F$$-factor in (20) and it is only included in the radiation pressure factor $$\Phi = \pi R^2 F/(mc)\ .$$ For a spherical body, $$m \varpropto R^3$$ and thus finally $$da/dt\varpropto 1/R.$$ As a rule of thumb the Yarkovsky effect is optimum for bodies that have $$R' \sim 1\ ,$$ so at sizes larger than a few kilometers the effect is inversely proportional to size.

The longest characteristic timescale available for accumulation of orbital effects is of the order $$\sim 1$$ Gy. This implies the Yarkovsky forces are essentially negligible for asteroids larger than $$\sim 30$$ kilometers. In the opposite limit of very small bodies, where $$R'\ll 1\ ,$$ one obtains $$\kappa_1\varpropto R'\ ,$$ $$\kappa_2\varpropto 1/R'$$ and $$\kappa_3\varpropto 1/R'^2$$ when $$R'\rightarrow 0$$ in (20). As a result, for finite $$\Theta_\nu$$ we have $$F_\nu\varpropto R^3/\Theta$$ in this limit. Combined with the $$R$$-dependence in $$\Phi\ ,$$ we finally obtain $$da/dt\varpropto R^2$$ in the limit of very small objects. This formal result supports our intuition that the Yarkovsky effect should vanish for very small particles. This is because efficient thermal conduction throughout the volume of the body makes the temperature very close to the equilibrium value everywhere. This also damps the amplitude of the recoil effect due to thermally-emitted photons. In practice, the role of the Yarkovsky forces below millimeter size becomes negligible. For smaller grains the Poynting-Robertson effect takes over, forcing particles to migrate inward toward the Sun.

Figure 3 shows the characteristic drift rates in semimajor axes for bodies of different sizes and surface thermal conductivities. The bottom part of the figure, where the drift rate is multiplied by a characteristic collisional lifetime of a body in the main asteroid belt, indicates that meteorite precursors can move by a characteristic value of $$\sim 0.1$$ AU. This implies that meteoroids can move from buffer zones 0.1 AU wide or so to dynamical resonances in the main asteroid belt, where their eccentricities can be pumped up to planet-crossing orbits.

In the case of multi-kilometer size asteroids, the spread of $$\Delta a$$ is smaller, with a characteristic value of $$\sim 0.04$$ AU. This is a reasonable representation of how far asteroid families extend in semimajor axis away from their central or mean values. As a result, over a billion years or so, small asteroids in many families have migrated far enough to significantly change their initial spread of fragments in $$a\ .$$

Dependence on the heliocentric distance. -- In the case of the diurnal effect, typically characterized by large-$$\Theta_\omega$$ and large-$$R'$$ regime, one has $$da/dt\varpropto \Phi/(n\Theta_\nu)\ .$$ Because the subsolar temperature $$T_\star$$ decreases with the heliocentric distance $$d$$ as $$T_\star\varpropto 1/d^{\,1/2}\ ,$$ we have $$\Theta_\omega \varpropto d^{\,3/2}$$ and thus $$n\Theta_\omega$$ is roughly independent of the heliocentric distance. Then the dependence of da/dt on d simply derives from the flux decrease $$\Phi\varpropto 1/d^{\,2}$$ and thus also $$da/dt\varpropto 1/d^{\,2}\ .$$ A slightly more complicated situation may occur for the seasonal variant of the Yarkovsky effect. This is because both $$l_n$$ and $$\Theta_n$$ increase proportionally to $$d^{\,3/4}$$ for near circular orbits. This dependence in the $$F_n$$-function combines with the $$\varpropto 1/d^{\,2}$$ decrease of the solar flux in $$\Phi\ ,$$ producing an overall shallower decrease of the averaged $$da/dt$$ due to the seasonal effect with $$d.$$

## Rotational consequences of the YORP effect

With rare exceptions, the problem of combining rotational dynamics with radiation torques (12) on a small body in space has been analyzed using simplifying assumptions, namely the assumption of rotation about the principal axis of the inertia tensor. Denoting $$C$$ its largest eigenvalue and $${\mathbf s}$$ the corresponding eigenvector, the spin axis and the rotational angular momentum vector can be written as simply $${\mathbf L} = C \omega\, {\mathbf s}\ .$$ Its time derivative is equal to the acting torques $${\mathbf T}\ .$$ The Euler equation can then be split into a pair of equations

$\tag{21} \frac{d\omega}{dt} = \frac{{\mathbf T}\cdot {\mathbf s}}{C}$

and

$\tag{22} \frac{d{\mathbf s}}{dt} = \frac{{\mathbf T}-{\mathbf s}\left({\mathbf s}\cdot {\mathbf T}\right)}{C\omega}\; ,$

to individually track evolution of the rotation frequency $$\omega$$ and the spin vector $${\mathbf s}\ .$$ Conventionally $${\mathbf s}$$ is parametrized using the obliquity value $$\gamma\ ,$$ given by $$\cos\gamma={\mathbf s}\cdot{\mathbf N}\ ,$$ where $${\mathbf N}$$ is a unit vector along the orbital angular momentum, and the precession angle $$\psi\ .$$ The YORP effects on $$\psi$$ are less important, mainly because the dominant secular effects from gravitational solar torques have always an uncertainty larger than the YORP contribution. This allows us to focus on $$\gamma$$, where

$\tag{23} \frac{d(\cos\gamma)}{dt} = \frac{{\mathbf T}\cdot {\mathbf e}}{C \omega}$

holds with an auxiliary vector $${\mathbf e}={\mathbf N}-{\mathbf s}\left({\mathbf s}\cdot{\mathbf N}\right)\ .$$

To obtain long-term (secular) effects, one uses a first order perturbation theory with the unperturbed uniform rotation solution inserted into the right-hand sides of (21) and (23) and then averages over both the rotation and revolution cycles. Analytical results for the YORP effect are more difficult to achieve than for the Yarkovsky effect, so numerical models allow to glean insights into what happens to a body. They are both needed, however, to understand basic properties of the YORP effect and so we can adequately compare model results with observations. Similar to the Yarkovsky effect case, a number of different approaches have been developed along these lines over the last decade.

Secular rates in $$\omega$$ and $$\gamma$$ are given as (zero surface conductivity and circular orbit assumed)

$\tag{24} \frac{d\omega}{dt} = \frac{\Lambda}{C}\,\sum_{p\geq 1}\,{\mathcal A}_p\, P_{2p}\left(\cos\gamma\right)\; ,$

and

$\tag{25} \frac{d\gamma}{dt} = \frac{\Lambda}{C\omega}\,\sum_{p\geq 1}\,{\mathcal B}_p\, P^1_{2p}\left(\cos\gamma\right)\; ,$

where $$\Lambda = 2(1-A)F R^3/(3c)$$ (with the same notation as above for the Yarkovsky effect), $$({\mathcal A}_p,{\mathcal B}_p)$$ are complicated functions that depend on the surface irregularities found on the body, $$P_{2p}\left(\cos\gamma\right)$$ are Legendre polynomials and $$P^1_{2p}\left(\cos\gamma\right)$$ Legendre associated functions of the first order. The series in (24) and (25) may converge very slowly, indicating they are dependant on very high-order terms ($$p$$ large). In practice, this means a YORP solution may depend on increasingly fine structures found on the body's surface (e.g., craters, boulders, etc.). This trend has been confirmed by numerical tests and represents one of the most fundamental obstacles for accurate YORP prediction. Moreover, the analytical estimates above do not take into account the effects of mutual occlusions of the surface facets, which can be very important in numerical runs.

Figure 4: Estimated mean YORP effect on the $$\sim 30$$ m diameter near-Earth asteroid 1998 KY26: (i) secular change of the rotation rate $$d\omega/dt$$ (left panel), and (ii) the obliquity $$d\gamma/dt$$ (right panel). The results have been computed numerically using the radar-derived shape and for different values of the obliquity at the abscissa. The YORP effect's ability to change rotation rate (left) does not depend on the surface thermal conductivity $$K$$ in the plane-parallel numerical model used, while the effect in obliquity (right) is $$K$$-dependent (results for several values of $$K$$ are shown by different curves; numerical values in W/m/K). Both numerically derived curves, $$d\omega/dt$$ and $$d\gamma/dt\ ,$$ show qualitative features explained by the analytic model, Eqs. (24) and (25).

Dependence on the obliquity. -- Secular changes $$d\omega/dt$$ and $$d\gamma/dt$$ in (24) and (25) show a different parity when $$\gamma$$ is transformed to $$180^\circ-\gamma\ :$$ (i) $$d\omega/dt$$ is even, while (ii) $$d\gamma/dt$$ is an odd function of $$\gamma\ .$$ When the quadrupole term ($$p=1$$) dominates, $$d\gamma/dt$$ vanishes for $$\gamma=0^\circ\ ,$$ $$90^\circ$$ and $$180^\circ$$ and its value is either positive or negative for the entire range of prograde-rotating objects (and takes the opposite value for the retrograde-rotating objects). If YORP was only affecting the obliquity, it would thus secularly evolve towards asymptotic values with the spin vector $${\mathbf s}$$ either perpendicular to the orbital plane or in the orbital plane. In the same way, $$d\omega/dt$$ vanishes in the nodes of the second-degree Legendre polynomial, i.e., $$\gamma\sim 55^\circ$$ and $$\gamma\sim 125^\circ\ .$$ Near the asymptotic states of the obliquity, $$d\omega/dt$$ takes maximum value which can either be positive (long-term acceleration of the rotation rate) or negative (long-term deceleration of the rotation rate). An example of the obliquity dependence of $$d\omega/dt$$ and $$d\gamma/dt$$ due to YORP for a small near-Earth asteroid 1998 KY26 is shown in Figure 4.

The gross picture from the quadrupole contribution in (24) and (25) can be modified if high-order terms are important. Additionally, when the effects of finite surface thermal conductivity are included, the results for (i) the rotation rate effect $$d\omega/dt$$ are not changed, but (ii) those for the obliquity effect $$d\gamma/dt$$ are modified.

Dependence on the size. -- Noting the proper dependence of $$\Lambda$$ and $$C$$ on the characteristic radius $$R$$ (i.e., $$\Lambda\propto R^3$$ and $$C\propto R^5$$), Eqs. (24) and (25) indicate that both $$d\omega/dt$$ and $$d\gamma/dt$$ are $$\propto 1/R^2\ .$$ Thus, the YORP effect is strongly dependant on the size of the body. Tests indicate that $$\sim (10-20)$$ km size asteroids at a few AU heliocentric distance need Gy timescale to modify their rotation state by YORP, as derived from scaling $$\omega/(d\omega/dt).$$ YORP is essentially an unimportant effect for larger objects.

A less clear and more interesting situation is how the YORP effect modifies the spin vector of small bodies. For example, if a small object is spinning up, it could presumably continue to spin up until it disrupts. The problem, however, is that the cosmic ray exposure lifetimes of stony meteoroids in space are typically many tens of millions of years in length, many orders of magnitude longer than the rotational bursting timescale obtained from the YORP equations. This problem has not been studied in detail, but observations imply that YORP must become less effective and shut down at some level for small bodies. An improved analytical model, with non-radial thermal flux components included, still predicts a divergence in $$d\omega/dt$$ and $$d\gamma/dt$$ as $$\propto 1/R\ ,$$ but some physical assumptions are violated as we pass certain thresholds. This intriguing problem has not yet been studied in detail, and the solution to how and why YORP shuts down is currently unknown.

# Direct detection of Yarkovsky and YORP effects

Accurate ground based observations have now allowed astronomers to directly detect both the Yarkovsky and YORP effects on asteroids. This was an important validation both of the theory presented above and its underlying concepts. It is also useful because most of the applications of the Yarkovsky and YORP effects involve statistical studies of small bodies populations in the Solar system rather than detailed descriptions of the dynamics of individual objects. Moreover, further observations during the next decade or so will allow many more direct detections of both the Yarkovsky and YORP effects. The Yarkovsky effect will also likely become an inevitable part of the orbit determination for small near-Earth asteroids and an important element in any analysis of their potential impact hazard to Earth.

## Detection of Yarkovsky effect

Figure 5: Orbital solution of near-Earth asteroid (6489) Golevka from astrometric data before May 2003 projected into the plane of radar observables: (i) range ("roughly distance to the radar antenna") at the abscissa, and (ii) range-rate ("roughly relative radial velocity of the asteroid with respect to the radar antenna") on the ordinate. The origin referred to here is the center of the nominal solution that only includes gravitational perturbations. The blue ellipse represents a 90 % confidence level in the orbit due to uncertainties in astrometric observations as well as small body and planetary masses. The center of the red ellipse is the predicted solution with the Yarkovsky forces included; note the range offset of $$\sim 17$$ km and the range rate offset of $$\sim 5$$ mm/s. The actual Arecibo observations from May 24, 26 and 27 of the year 2003 are shown by the diamond (the measurement uncertainty in range is smaller than the symbol). The observations fall within the uncertainty region of the orbital solution containing the Yarkovsky forces (red ellipse; 90 % confidence level).

Detecting the Yarkovsky effect among real asteroids is not straightforward, partly because it is not the only force capable of modifying the motion of small bodies in the Solar System, but also because its strength is weak and observations have inherently finite accuracy. In practice, one needs (i) very precise astrometric observations, with a superior measurements provided by planetary radar (if available), (ii) observations spanning at least a decade or so, and (iii) the target asteroid must be small enough that the effects are measureable over the time interval (i.e., which means the object is smaller than few kilometers in size). This means plausible Yarkovsky candidate targets must be winnowed from a list of criteria: favorable orbital geometry, close approaches to the Earth where radar data can be acquired, and the availability of other observations, such as photometry providing information about the rotation period or spin axis orientation of the near-Earth asteroid. The first asteroid to meet all these criteria was (6489) Golevka (Figure 5 and Chesley et al. 2003).

The possibility of decorrelating orbital perturbations produced by the Yarkovsky effect from other effects derives from the Yarkovsky effect's ability to secularly change the semimajor axis $$a$$ of an asteroid's heliocentric orbit. Kepler's third law directly translates a nonzero average $$da/dt$$ value onto a nonzero secular change $$dn/dt$$ of the orbital mean motion $$n\ ,$$ producing a quadratic advance $$\Delta \lambda$$ in the longitude in orbit $$\lambda\ .$$ An order of magnitude estimate, neglecting eccentricity corrections, provides $$\Delta \lambda \sim \int \delta n\,dt \sim \int (dn/dt)\,t\,dt \sim \frac{3n}{4a}\,(da/dt)\,(\Delta T)^2$$ in time $$\Delta T\ .$$ The same effect can also be expressed as a transverse displacement along the orbit of the order $$\sim a\,\Delta \lambda$$ which again propagates quadratically in time. It is this $$\propto (\Delta T)^2$$ progression in either of the two quantities that makes the Yarkovsky effect distinct from other perturbations.

At present, ground-based astrometric observations of the sky-plane position in optical wavelengths are only accurate to $$0.1-0.5$$ arcseconds at best, with a factor few worse in regular astrometric survey observations. This is not much better than the expected Yarkovsky displacement over a decade or so, unless the body approaches the Earth at a very close distance. The accuracy of radar astrometry is much better; the best observations have uncertainty of only few tens of meters. This can be several orders of magnitude better than the Yarkovsky displacement and thus radar astrometry is potentially much better at detecting weak orbital perturbations such as the Yarkovsky forces than optical observations. Its only drawback is that, at present, the number of near-Earth asteroids that have come close enough to Earth to be observed by radar is limited; about 450 or so compared to more than 8500 near-Earth asteroids observed in optical wavebands.

Figure 6: Asteroids for which Yarkovsky forces have been estimated using orbit determination methods. The second column gives the $$(da/dt)$$ value with formal uncertainty, the third column gives absolute magnitude as determined from astrometric observations (typically uncertain to $$\sim (0.3-0.5)$$ magnitude), the fourth column gives obliquity $$\gamma$$ if determined from photometric observations and/or radar observations, and the fifth column gives mean heliocentric distance $$\bar{r}\ .$$

Figure 6 provides a list of objects where the Yarkovsky effect has been detected using orbit determination methods (S.R. Chesley and D. Cotto-Figueroa, personal communication 2009). The uncertainty in $$(da/dt)$$ for these bodies is fractionally less than one half of the derived value.

The role of Yarkovsky forces in impact hazard computations. -- For near-Earth asteroids that potentially could hit the Earth in the near future, the Yarkovsky forces must be part of any orbit determination procedure. Their importance looms even larger if one needs a high-accuracy ephemeris on a longer-timescale, say tens to hundreds of years. A special class of problems, in this respect, relates to how asteroid impact hazards for Earth are calculated; the orbital uncertainties produced by the Yarkovsky effect mean that we must observe potentially hazardous asteroids for long time periods before we can really rule an impact completely out. So far, the role of Yarkovsky forces in evaluating the impact probability has been studied for three asteroids, but more cases should be expected soon: (i) (29075) 1950 DA, (ii) (99942) Apophis, and (iii) (101955) 1999 RQ36. The Yarkovsky effect has been recognized as the most important element in their orbital uncertainty and thus their impact probability with Earth.

Future outlook. -- Future radar and ground-based astrometric observations will continue to provide possibilities for further detections of the Yarkovsky effect using orbital determination of the near-Earth asteroids. This is because, by definition, their addition to the available database will meet both of the necessary requirements: (i) they extend the timebase $$\Delta T\ ,$$ and (ii) very likely they will be even more accurate than the available astrometric observations to-date. The outlook is especially optimistic if the powerful next-generation sky surveys, such as PanSTARRS or LSST, begin to detect large numbers of near-Earth asteroids. With their wide field cameras and observing schedule covering the high-latitude regions, they will not only regularly observe the currently known population of objects, but will also discover many more small asteroids. Moreover, their large CCD cameras will allow astrometry with roughly an order of magnitude better accuracy than is currently available.

Additionally, the space-based astrometric project Gaia promises to further advance our abilities to detect the Yarkovsky effect. With an internal accuracy of $$\sim 1$$ milliarcseconds for moving objects (even better for targets brighter than $$\sim 13$$ magnitude in V band) and 5 years of operation, it will provide a superb astrometric database for a myriad of near-Earth asteroids; its location near the L2 Lagrange point of the Sun-Earth system will allow much more complete observations of the asteroid population than it is possible from the Earth's surface.

## Detection of YORP effect

Figure 7: Additional sidereal phase $$\Delta \phi$$ required to link photometric observations of a small near-Earth asteroid (54509) YORP during its yearly apparitions from July 2001 till August 2005. The nominal model assumes an asteroid shape determined by the reconstruction of radar echoes, pole position at ($$180^\circ\ ,$$ $$-85^\circ$$) ecliptic longitude and latitude and a constant rotation rate $$42582.41\pm 0.02$$ deg/day. This nominal model, however, is unable to match the rotation phase observed at subsequent apparitions (symbols). The difference is well fit by a quadratic function $$\Delta\phi$$ with $$(d\omega/dt)=(350\pm 35)\times 10^{-8}$$ rad/day$$^2\ .$$ This value fits the theoretically predicted YORP value for this object reasonably well.

Using sufficient astronomical observations, one can determine both the sidereal rotation period and orientation of rotation poles for small bodies in the Solar System. In principle, the solution to both components may reveal secular effects due to the YORP torques. However, YORP torques are extremely weak. Fortunately, it is relatively straightforward to get an accurate determination of the rotation period of an asteroid, thereby allowing direct determination of the YORP effect over sufficiently long time intervals. Still, the YORP effect in $$d\omega/dt$$ is a strong function of size, even more so than the Yarkovsky forces, such that the possibility to detect YORP is limited to small near-Earth asteroids.

The YORP effect may be detected using two techniques (or their combination): (i) optical photometry, and (ii) radar. Both provide instantaneous information about the orientation of the body with respect to the detector (and the Sun in the first case). A time series of this relative configuration might be, after appropriate geometric transformations are taken into account, transformed into a time dependence of the rotation phase $$\phi$$ of the body in the inertial space ("sidereal rotation"). If the body rotated about the principal axis of the inertia tensor with frequency $$\omega\ ,$$ $$\phi$$ would be linear function of time, i.e. $$\phi=\omega\,t + C\ .$$ The YORP effect contributes in $$d\omega/dt$$ by a non-zero average (secular term) and periodic terms. The latter are, however, typically too small to be detected, so the basic perturbation produced by the YORP effect is a quadratic advance in the sidereal phase of rotation $$\Delta \phi= \frac{1}{2}\,(d\omega/dt)\,(\Delta T)^2$$ in time $$\Delta T$$ (similar to the Yarkovsky effect in the longitude in orbit discussed above).

Figure 8: Asteroids for which YORP torques have been detected using observations of the sidereal rotation phase. The second column gives $$(d\omega/dt)$$ value with formal uncertainty, the third column gives absolute magnitude as determined from astrometric observations (typically uncertain to $$\sim (0.3-0.5)$$ magnitude), the fourth column gives rotation period in hours, the fifth column gives obliquity $$\gamma\ ,$$ and the sixth column gives mean heliocentric distance $$\bar{r}\ .$$ For the last two targets, observational limits constrain the available theory which in turn predicts the larger value.

The first target for which YORP torques have been directly detected is a small coorbital asteroid (54509) YORP, formerly 2000 PH5, that also has the largest detected value of $$(d\omega/dt)$$ yet known (Figure 8). In four years (2001-2005) of accurate radar and lightcurve observations, the additional phase advance was $$\Delta \phi\sim 225^\circ$$ (Figure 7; see also Lowry et al. 2007 and Taylor et al. 2007). Larger asteroids have $$(d\omega/dt)$$ significantly smaller, so longer intervals of time $$(\Delta T)$$ between the first and the last observations are required. For instance, in the case of (1620) Geographos, with $$(d\omega/dt)\simeq 1.2\times 10^{-8}$$ rad/day$$^2\ ,$$ the time interval of 39 years between the first observations in 1969 and the last observations in 2008 provided $$\Delta \phi\sim 70^\circ\ .$$ To appreciate the accuracy of these observations, one may also express them in terms of the detected annual change of the rotation period that amounts to only $$1.25$$ ms/y for (54509) YORP and $$2.7$$ ms/y for (1620) Geographos. Figure 8 summarizes some useful information about those asteroids where the YORP effect has been determined.

# Planetary Applications: Yarkovsky effect

Meteorite and NEA transport. -- Since the time of E. Chladni, scientists have recognized that meteorites originate in outer space. What asteroids, or groups of asteroids, were their sources and how they were transported from their source regions to Earth was a longstanding problem in solar system dynamics. Research into the Yarkovsky effect has helped us to understand the transport problem, with important implications for solving where precisely they come from in the main asteroid belt.

The Yarkovsky effect, with its ability to secularly change the semimajor axes of meteoroids (the precursors of meteorites, which are believed to be fragments of larger asteroids located in the main belt between the orbits of Mars and Jupiter), was originally proposed to be the main element driving meteorites to the Earth (see Öpik 1951). However, direct transport from the main belt, say as a small body slowly spiraling inward toward the Sun by the Yarkovsky effect, required very long timescales and unrealistic values of the thermal parameters and/or rotation rates for meter size bodies. Moreover, AM/PM fall statistics and measured pre-atmospheric trajectories in rare cases (like the Pribram meteorite), indicated many meteorites had orbits with semimajor axis still close to the main belt values.

Advances in our understanding of asteroid dynamics, in particular our knowledge of secular and mean-motion resonances, allowed us to recognize in the late 1970s and early 1980s that resonances are transport routes that move main belt objects onto planet-crossing orbits. The Yarkovsky effect would presumably not be necessary for meteorite transport if collisional breakups of parent asteroids directly injected their precursors into nearby resonances. The problem with this scenario, however, is that the transport timescales of meteorite precursors via the resonances were too short to match cosmic-ray exposure ages of the most common ordinary chondrite classes, many which were dominated by ages of ten to nearly hundred My.

When neither of the two hypotheses worked alone, astronomers concluded that a constructive synergy of both might be the correct answer to the meteorite transfer problem. In this model, meteoroids or their immediate precursor objects are collisionally born in the inner and/or central parts of the main belt, where they are then transported to resonances by the Yarkovsky effect. En route, some of their precursors may undergo fragmentation, which can produce new swarms of daughter meteoroids which eventually reach planet-crossing orbits. With this extended model, one can explain the distribution of the cosmic-ray exposure ages of stony meteorites as a combination of several timescales: the time it takes a meteoroid to travel to a resonances and the time it takes for that resonance to deliver the meteoroid to a Earth-crossing orbit, and the time it takes the meteoroid on a planet-crossing orbit to hit Earth (e.g., Vokrouhlický and Farinella 2000).

Farinella and Vokrouhlický (1999) noted that larger, kilometer sized asteroids in the near-Earth population can also be resupplied via the Yarkovsky effect, with thermal drift slowly feeding the resonances. Because Yarkovsky drift in semimajor axis is smaller than for meteoroids (Figure 3), and their collisional lifetimes are longer, the delivery mechanism can involve both strong resonances and weaker but more numerous high-order resonances, which criss-cross the asteroid belt. Many of these tiny resonances are found in the the inner main belt. Numerical models show they allow multi-kilometer objects to escape over a timescale of hundreds of My to Gys. After evolving out from the main belt region, these orbits first become Mars crossers before being further transported by planetary encounters to the near-Earth population.

Spreading of asteroid families. -- Asteroid families are clusters of fragments produced when two asteroids slam into one another at hypervelocities. They are traditionally found in the space of proper orbital elements, with only the youngest ones recognizable in the space of osculating orbital elements. It has long been recognized, however, that the orbital structure of major families do not match the spread predicted by the expected ejection velocity fields. Advances in our understanding of asteroid dynamics in 1990's provided two possibilities to solve this problem: (i) weak mean-motion and secular resonances modify the values of proper eccentricity $$e$$ and inclination $$i$$ among asteroids, and (ii) the Yarkovsky effect modifies the value of proper semimajor axis $$a\ .$$ Both turn out to be important in explaining observations.

Interestingly, because the dominating variant of the Yarkovsky effect is the diurnal component, families undergo both positive and negative changes in $$a$$ (depending on the obliquity value of the individual asteroids). During this evolution, the orbits may become trapped in some of the weaker resonances, thereby inducing evolution of the proper eccentricity or inclination. If they evolve far enough to reach one of the stronger mean motion resonances, they can be pushed out of the main belt and eliminated from the family. The latter case produces many of the observed sharp truncations seem among certain families.

By analyzing the peculiar structure of the Koronis family, Bottke et al. (2001) provided the first clear example of how family structures could be affected by resonant and Yarkovsky dynamical effects. A number of additional examples have been published since that time. Moreover, by using these dynamical effects like a clock, it is possible to use the Yarkovsky effect to estimate the ages of the asteroid families. This information has been used in studies of space weathering as well as the characterization of epochs of elevated dust or asteroid accretion on the Earth.

Age constraints for older families involves matching their structure proper orbital element space. More detailed results can be obtained for young families with ages less than $$\sim 10$$ My. In this case, direct backward integration of the orbits of all family members can be used to monitor the best possible convergence of secular angles, namely the longitude of node and pericenter. This analysis also requires thermal forces to be included in the dynamical model.

# Planetary applications: YORP effect

Asteroids in spin-orbit resonances. -- In an attempt to generalize Cassini's second and third laws, G. Colombo developed a mathematical model in the 1960's that describes the evolution of a body's spin axis rotating about a principal axis of its inertia tensor. Colombo included two fundamental elements in his approach: (i) gravitational torques due to a massive center (e.g., Sun), and (ii) regular precession of the orbital plane of the body by exterior perturbers (e.g., planets). Because (i) produces a regular precession of the spin axis, a secular spin-orbit resonance (with a stable fixed point called Cassini state 2) may occur between its frequency and the frequency by which the orbital plane rotates in the inertial space. Such a resonance may occur only for a certain range of obliquity and rotation period values, and thus there is only a small probability that the spin state of any given asteroid is located in the Cassini state 2 associated with one of the frequencies by which its orbital plane precesses in space.

With this as background, the discovery of five prograde-rotating Koronis member asteroids with similar spin vectors (i.e., spin axes nearly parallel in inertial space and similar rotation periods) was an enormous surprise to asteroid experts. Additionally, the sample of retrograde-rotating asteroids in the same observation campaign showed these objects had anomalously large obliquities ($$\geq 154^\circ$$) and either very short or very long rotation periods (Slivan 2002). This puzzling situation, however, was solved using a model where gravitational spin dynamics was augmented by introducing the long-term effects of YORP torques (Vokrouhlický et al. 2003). The YORP effect was shown to bring, on a $$\sim (2-3)$$ Gy timescale, prograde states close to Cassini state 2 associated with the prominent $$s_6$$ frequency in the orbital precession, providing thus a natural explanation for their alignment in inertial space. No such trapping zone exists for retrograde-rotating bodies, which evolve toward extreme values in both their obliquities and rotation periods.

The possibility exists for asteroid spin states to be trapped in similar spin-orbit resonant states for bodies residing on low-inclination orbits, especially in the central and outer parts of the main asteroid belt.

Distribution of spin rate and pole orientation of small asteroids. -- The distribution of rotation frequencies of large asteroids in the main belt matches a Maxwellian function quite well with a mean rotation period of $$\sim (8-12)$$ hr, depending on the size of the bin used. However, data for asteroids smaller than $$\sim 20$$ km show significant deviations from this law, with many asteroids either having very slow or very fast rotation rates. Recently, a significant quantity of rotation-frequency data has become available from a sample of small main belt asteroids (note that data are also available for small asteroids on planet-crossing orbits, but the interpretation may be complicated by the effects of planetary close approaches). After eliminating possible binary systems, solitary kilometer-size asteroids in the main asteroid belt were shown to have a roughly uniform distribution of rotation frequencies. The only statistically significant deviation was an excess of slow rotators (periods less than a day or so).

These results are well explained with a simple model of a relaxed YORP evolution. In this view asteroid spin rates were driven by the YORP effect toward extreme (large or small) values on a characteristic (YORP-) timescale dependent on the size. Asteroids evolving toward a state of rapid rotation shed mass and rotational angular momentum, yielding slower spinning objects. Those spinning down from YORP lose so much rotational angular momentum that they can enter into a tumbling phase. These objects may later emerge from this state naturally, with a new spin vector, or may gain rotation angular momentum by subcatastrophic impacts. After a few cycles the spin rates settle to an approximately uniform distribution and the memory of its initial value is erased.

In a similar fashion, the spin poles of large and small asteroids also differ from one another. The distribution of pole orientation of large asteroids in the main belt is roughly isotropic, with only a moderate excess of prograde rotating bodies. On the other hand, rotation poles of small asteroids (sizes $$\leq 30$$ km) are strongly concentrated toward ecliptic north and south poles. This result can again be explained with the above model of YORP evolution, with YORP torques driving obliquities toward extreme values.

Structure of asteroid families. -- While the structure of asteroid families in proper element space can be modified by long-term diffusion in semimajor axis due to the Yarkovsky effect, the process may be affected by the YORP effect in an intriguing way. This is because the strength of the Yarkovsky effect depends on the obliquity value, which itself is modified by the YORP effect. Numerical simulations have determined that in majority of cases YORP effect should asymptotically tilt the spin axis toward extreme obliquity values, $$0^\circ$$ or $$180^\circ\ ,$$ for which the diurnal variant of the Yarkovsky effect is maximized (Eq. (19). In this way, radial migration due to the thermal forces is accelerated for small members (few kilometer in size) in families. As a result these small asteroids may be preferentially brought to extreme values in semimajor axis. This pattern is observed in several families and can be used to constrain their age.

Origin of binary asteroids and asteroidal pairs. -- Binary and multiple systems of asteroids are of great value to planetary science because studies of their mutual motion can provide much more complete information about their components than observations of single objects. For that reason considerable attention has been devoted to these systems in recent times.

The origin of binary and multiple systems may, in many cases, be also related to YORP. Restricting our attention to small binary systems in the main belt and among near-Earth objects, which may make up 15 to 20 % of their respective populations, it is clear a robust formation mechanism is needed to explain observations. While several processes can lead to the formation of binary and multiple systems, such as asteroid collisions or tidal disruption events during planetary encounters, important clues are provided by the available observations: (i) the primary (larger) component of the binary nearly always rotates very fast (rotation periods between 2 to 4 hr), and (ii) the estimated total angular momentum of the system is very close to rotational angular momentum of a critically rotating parent body. The conditions (i) and (ii) above hold for the majority of small systems both among near-Earth and main belt asteroids. The most reasonable scenario for their formation is therefore based on idea of rotational fission and the YORP effect. YORP secularly accelerate an asteroid's rotation rate, which in turn drives it toward the fission limit. The relevant characteristic timescale to bring a "typically" rotating kilometer-size asteroid in the inner main belt to the fission limit is $$\sim (20-50)$$ My, shorter that its collisional lifetime. While YORP effect is likely the driving mechanism, details of the fission mechanics are not known and they are subject to current research.

Binary systems are closely related to asteroidal pairs, namely couples of asteroids on very similar orbits. Significant correlation between the rotation rate of the primary (larger) component in the pair and the mass ratio of the two components strongly suggests asteroid pairs were also formed by rotational fission of the parent object (Pravec et al. 2010). Details of fission process, such as elongation of the parent asteroid and the relative mass of the material shed from the primary, and the inherent stability of the two objects in orbit, help explain the probable reason why a binary system is formed in one case and an unbound asteroid pair in the other.

# Further topics

While covering many of the important applications of the Yarkovsky and YORP effect, this page neglects discussion of several topics of potential importance for the future. They include:

• Binary YORP (BYORP) -- This idea was introduced by Ćuk and Burns (2005) and since then gained considerable attention. This model assumes that smaller (kilometer-size) components in binary asteroids may evolve fast due to the YORP effect, once being tidally locked in the spin orbit 1/1 mean motion resonance. This configuration may efficiently transfer rotational angular momentum into orbital angular momentum, provided tidal effects are able to maintain the synchronous state with low amplitude of librations, and thus drive the orbital evolution of the binary system into new states.
• Thermal forces for dynamics of ring particles -- Similar to artificial spacecraft, meter-size boulders in circumplanetary rings can undergo secular orbital decay or expansion due to thermal forces by planetary or solar heating. In this case, however, the complication is that ring particles are not isolated but reside in compact systems where collisions are frequent. This makes it difficult to analytically estimate the mutual effects of bodies in such systems.

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