<figure><img src="image_1.png" /></figure>

HAL
open science

# Detecting the Yarkovsky effect with the Gaia mission: list of the most promising candidates

S. Mouret, F. Mignard

► To cite this version:

S. Mouret, F. Mignard. Detecting the Yarkovsky effect with the Gaia mission: list of the most promising candidates. Monthly Notices of the Royal Astronomical Society, 2011, 413, pp.741-748. 10.1111/j.1365-2966.2010.18168.x. http://arxiv.org/abs/1007.22185

HAL Id: hal-00722185

https://hal.science/hal-00722185

Submitted on 17 Sep 2021

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Detecting the Yarkovsky effect with the Gaia mission: list of the most promising candidates

S. Mouret<fnref n="1" />,<fnref n="2" />* and F. Mignard<fnref n="3" />

Accepted 2010 December 6. Received 2010 November 26; in original form 2010 June 16

## ABSTRACT

The European Space Agency astrometric Gaia mission, due for a launch in late 2012, will observe a large number of asteroids (>250 000 to $V = 20$ mag) over five years with an unprecedented positional accuracy (at the submilliarcsecond level). Among them, there will be a subset of near-Earth asteroids (NEAs), all sensitive to the tiny non-gravitational force due to the Yarkovsky effect, hardly detectable with ground based astrometry. Here we investigate the potential of Gaia to detect the Yarkovsky effect. From realistic simulated data on the currently known NEAs observable by Gaia, we performed a variance analysis from the observation residuals on a data model linearized with respect to the initial position and velocity of the asteroid and its semimajor axis drift rate ($da/dt$) – the main secular effect due to the Yarkovsky effect. The partial derivatives necessary to evaluate the expected accuracy with Gaia observations of ($da/dt$) are obtained by a numerical integration of the variational equations. We thus give the list of the most promising 64 NEAs for the detection of the Yarkovsky effect by Gaia, with an expected precision on ($da/dt$) better than $5 \times 10^{-4}$ au Myr$^{-1}$ (from underestimated astrometric precision). We also add for each asteroid, the physical parameters to be precisely estimated from complementary ground-based observations (photometric, radar) to accurately model the Yarkovsky effect.

**Key words:** astrometry – minor planets, asteroids: general.

## 1 INTRODUCTION

Orbiting the Earth–Sun Lagrangian point $L_2$, the European Space Agency astrometric satellite Gaia will observe over five years a huge number of minor planets (>250 000) with an unprecedented precision – at the submilliarcsecond. New perspectives in asteroid science have been set forth (Mignard et al. 2007). This includes in particular the weak dynamical effects impacting the motion of asteroids and which, until now, have been neglected because of insufficient observational accuracy. This might not be the case with Gaia thanks to its exquisite astrometric capability. Therefore, it is quite relevant to investigate the potential of Gaia in detecting non-gravitational forces such as the Yarkovsky effect (Delbò, Tanga & Mignard 2008), and to select the best potential candidates in order to improve our knowledge of their physical parameters entering the dynamical model.

Discovered by the Polish civil engineer Ivan O. Yarkovsky (1844–1902) (Yarkovsky 1901), but brought out of oblivion by Öpik (1951), the eponymous effect may be defined as a radiative re-

coil of anisotropic re-emission of heat (thermal infrared photons) received from the Sun in the visible. It mainly produces a secular variation in the orbital semimajor axis of km-sized asteroids, and to a lesser degree in the eccentricity. Therefore, this effect cannot be ignored in explaining some aspects of the dynamics of minor planets.

The study of its impact in the motion of asteroids has truly begun with Rubincam’s paper (1995): having shown that the observed decay of the Earth satellite LAGEOS was due to the Yarkovsky thermal drag (Rubincam 1988) – a seasonal variant that he introduced – the emphasis of his investigation was logically focused on meteorite delivery. Nevertheless, the first to have pointed out the importance of the Yarkovsky effect in the dynamics of such asteroids is P. Farinella (Farinella, Vokrouhlický & Hartmann 1998; Vokrouhlický & Farinella 1999). Indeed, this effect is thought to play a role in shaping some characteristics of the asteroid dynamics (Bottke et al. 2002).

Considered in the long term, it could explain the origin of NEOs, in a scenario previously hypothesized by Peterson (1976) and Afonso, Gomes & Florczak (1995), and showed in Morbidelli & Vokrouhlický (2003): the Yarkovsky effect is capable of delivering asteroids with diameters smaller than 20 km from their original

<footnote n="1">¹Lohrmann Observatory, Dresden Technical University, Institute for Planetary Geodesy, 01062 Dresden, Germany</footnote>

<footnote n="2">²IMCCE, UMR CNRS 8028, Paris observatory, 77 av. Denfert-Rochereau, 75014 Paris, France</footnote>

<footnote n="3">³UNS, OCA/Cassiniée, UMR CNRS 6062, Observatory of the Côte d'Azur, Le Mont Gros, BP 4229, 06304 Nice Cedex 4, France</footnote>

<footnote>*E-mail: serge.mouret@tu-dresden.de</footnote>
orbits into powerful resonances, which eventually turn their orbits into Earth-crossing ones. The Yarkovsky effect is also claimed to be an important cause of the dispersion of asteroid dynamical families that collisions cannot explain (Bottke et al. 2001, 2002). It, thus, becomes a decisive factor in estimating the age of asteroid families by means of backward numerical integrations (Nesvorný & Bottke 2004). Furthermore, the Yarkovsky effect has been shown to be significantly involved in the meteorite delivery process to Earth accounting for meteorite cosmic ray exposure ages (Vokrouhlický & Farinella 2000). In addition, a complementary effect, the so-called YORP (a second-order variation on the Yarkovsky effect), is also capable of changing the spin rates and axes of irregular asteroids with diameters not larger than a few km (Rubincam 2000). It could explain the large number of asteroids with a very fast and very slow rotational speed (Kaasalainen et al. 2007; Lowry et al. 2007).

At the short time-scale (~tens of years), direct detections of the Yarkovsky effect are not easy, and were achieved only for a few near-Earth asteroids (NEAs) of which the best known are (6489) Golevka from radar measurements (Chesley et al. 2003) and (152563) 1992 BF from astrometric observations over a longer period of time (~50 yr) (Chesley, Vokrouhlický & Matson 2006). However, this tiny non-gravitational acceleration turns out to be a real problem in deriving precise NEA orbits. In certain cases, a difficulty arises to identify NEAs (Vokrouhlický, Chesley & Matson 2008), predict close Earth approaches and, more important, to estimate direct impact probabilities with the Earth (Giorgini et al. 2002, Milani et al. 2009) because the orbit is too poorly known. Moreover, NEAs, on account of small semimajor axis and high eccentricities, constitute a very valuable set of targets to estimate certain global parameters such as the solar quadrupole $J_2$, or the gravitational parametrized post-Newtonian (PPN) parameters $\beta$ and $\gamma$. The improvement of astrometric accuracy, notably with Gaia, offers us the opportunity to attempt these estimations from asteroid observations. To achieve this goal, one needs an accurate dynamical modelling of the NEAs, including the small non-gravitational effects, like Yarkovsky's.

This non-gravitational perturbation depends on physical characteristics of the asteroid (diameter, mass, surface density, thermal inertia, spin, shape, etc.), some of which could be inferred from accurate observations as demonstrated by Chesley et al. (2003) who constrained the bulk density of (6489) Golevka to $2.7_{-0.6}^{+0.4} \text{ g cm}^{-3}$ by analysing the non-gravitational effects on its orbit from radar ranging. Mass is probably the most wanted parameter and an important constraint on the internal structure and composition of an asteroid. However, as NEAs are generally small and consequently light, mass derivation by analysing their gravitational perturbations on other Solar system bodies is impossible, even with Gaia astrometric accuracy. Most of them are not binaries or with detectable moons, thus excluding another means of estimating mass. Today, the Yarkovsky effect seems to be the only possibility to assess the mass (Chesley et al. 2003), if spacecraft flybys are not considered.

In Section 2, we describe the simulations used to estimate the potential of Gaia in detecting the Yarkovsky effect for known NEAs expected to be observed by the space probe, the dynamical modelling as well as the NEAs tested. In Section 3, we list the best Gaia expected precisions on the semimajor axis drift rate – main signature of this effect – and the corresponding NEAs supplemented by many information in view of the Gaia observation characteristics, our knowledge about the physical features of each asteroid and numerous issues itemized in Section 1. Finally, prospects are discussed in the framework of the Gaia mission.

<table><caption>**Table 1.** Distribution of 1686 NEAs with respect to the number of predicted observations by Gaia. In parentheses, the number of asteroids with a diameter <1 km is given.</caption><thead><tr><th colspan="2">Number of<br>observations <i>n</i></th><th rowspan="2">asteroids</th><th colspan="2">Number of<br>observations <i>n</i></th></tr><tr><th><i>n</i> &lt; 10</th><th>asteroids</th><th><i>n</i> &lt; 60</th><th>asteroids</th></tr></thead><tbody><tr><td>10 ≤ <i>n</i> &lt; 20</td><td>321</td><td>(129)</td><td>60 ≤ <i>n</i> &lt; 70</td><td>16 (1)</td></tr><tr><td>20 ≤ <i>n</i> &lt; 30</td><td>121</td><td>(43)</td><td>70 ≤ <i>n</i> &lt; 80</td><td>16 (1)</td></tr><tr><td>30 ≤ <i>n</i> &lt; 40</td><td>65</td><td>(10)</td><td>80 ≤ <i>n</i> &lt; 100</td><td>11 (1)</td></tr><tr><td>40 ≤ <i>n</i> &lt; 50</td><td>47</td><td>(10)</td><td><i>n</i> &gt; 100</td><td>7 (–)</td></tr></tbody></table>

# 2 DYNAMICAL MODELLING AND SIMULATIONS

## 2.1 The observation of known NEAs by Gaia

Fitted with its two telescopes, Gaia will carry out a 5-yr continuous survey of the sky, visiting and revisiting celestial areas according to a well-defined pattern determined by its complex scanning law. It will orbit the Earth–Sun Lagrangian point $L_2$ on a Lissajous orbit, and all the sufficiently point-like sources brighter than $V = 20$ mag will be repeatedly observed. An overview of the mission, the observing principle and its implications for Solar system objects are given in Mignard et al. (2007). From a software developed for the mission preparation and caring for the updated observational characteristics of the satellite, we performed a systematic exploration of the Gaia transit times from 2012 January 1, over a period of five years for 6930 known NEAs. Their orbital elements and absolute magnitude – necessary to filter out the non-detectable crossings of faint sources – were taken from the ASTORB catalogue of Bowell.<fnref n="1" />

1686 NEAs – about one-quarter of the initial sample – are expected to meet the Gaia detection limit to be observed. They are objects hard to observe because they are simultaneously fast moving and faint. Looking at the distribution of the number of transits in Table 1, one sees that about 38 per cent will be observed at least 10 times, and about 12 per cent with a subkm diameter, based on an estimate from the MPC (light-curve parameters) or concluded from the diameter as a function of the absolute magnitude $H$ and the geometric visible albedo $p_v$ as (Bowell et al. 1989)

$$d = \frac{1329}{\sqrt{p_v}} \times 10^{-0.2H} \quad (\text{km}). \quad (1)$$

The albedo being poorly known in the NEA population, the unbiased average value ($p_v = 0.14$) was used (Stuart & Binzel 2004).

The position of the objects observed by Gaia will be expressed in longitude $\lambda$ and latitude $\beta$ on a reference great circle computed from the mean positions of instantaneous scanning circles over one period. However, the along-scan (AL) precision $\sigma_{\lambda}$ will be much better than in the across-scan direction; therefore, only the Gaia longitude will be considered in our simulations. More details can be read in Mignard et al. (2007).

The focal plane of the Gaia telescopes will be filled with a mosaic of CCDs. An asteroid entering in the field of view will first pass across one of the two sky mappers, and then the astrometric field (AF) composed of nine columns of CCDs, providing as many independent and nearly simultaneous 1D positions. The expected astrometric accuracy for a stellar source for one full transit (nine individual CCD measurements) is shown in Fig. 1, using the current

<footnote n="1">¹ ftp://ftp.lowell.edu/pub/elgb/astorb.html</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** The expected astrometric precision of individual observations $\sigma_{\lambda}$ for one transit (nine CCD measurements) in the case of non-moving objects. The sawtooth feature at the bright end results from the on-board handling of the CCD saturation. The trade-off between the number of teeth and their amplitude is not yet final.</figcaption></figure>

instrument parameters. The AF precision of a single observation for one CCD in the case of non-moving point-like sources is currently estimated to 0.1 mas for $V \le 13$ mag and reaches 3 mas at $V = 20$ mag (De Bruijne 2009). Regarding moving objects, the astrometric precision is not yet precisely known. We tried to define a realistic preliminary AL precision as a function of the apparent magnitude $V$ and also by considering the AL apparent velocity of the object based on current simulations in estimating the centroiding error produced by the apparent velocity over one CCD (Dell'Oro, private communication). The maximum apparent velocity of a main-belt asteroid (MBA), of the order of 30 mas s$^{-1}$, should not produce an error greater than 15–20 per cent of the nominal error, while for fast NEAs, velocities larger than 100 mas s$^{-1}$, this can be of the order of the nominal error or little more larger. Therefore, we have defined two plots in Fig. 2 representing the AL astrometric precision with respect to the apparent magnitude $V$. The first plot (continuous line) is the case where the AL apparent velocity is less than 33.3 mas s$^{-1}$ (the fastest MBA apparent velocity found from a very huge sample), we added an additional error of 20 per cent at the nominal precision – that for non-moving object – and 130 per cent (arbitrary value) in the second plot (dashed line) for the fastest NEAs. However, the

<figure><img src="image_4.png" /><figcaption>**Figure 2.** The expected AL accuracy of individual observations $\sigma_{\lambda}$ of a moving object as a function of the apparent magnitude, for slow moving main-belt objects (solid curve) and fast-moving NEAs (dashed curve). The astrometric accuracy is for the measurement over one CCD, equivalent to a frame of 4.5 s.</figcaption></figure>

AL accuracy should be better than those we consider because the asteroid position will be derived from signals collected by several CCDs (10 for the optimal cases) and not only one as here. Nevertheless, fast apparent moving objects such as certain NEAs will not cross all the Gaia focal plane.

## 2.2 Modelling the Yarkovsky force

Regarding the Yarkovsky effect, a wide range of modelling is possible: from a simple estimation of the secular drift rate in the semimajor axis ($da/dt$) by a transverse force, in order to detect the effect, to sophisticated non-linear models incorporating the irregular shape and spin state of the asteroid (Chesley et al. 2003) in order to evaluate asteroid physical parameters. An intermediate level of refinement can be found in Vokrouhlický (1998, 1999) assuming a spherical body, and a force as a function of many parameters: diameter, mass, spin state, surface thermal conductivity, surface porosity, specific heat capacity and spin obliquity vector with respect to the Sun.

This paper aims primarily to list the most valuable asteroids to detect the Yarkovsky effect and not to prepare the data analysis, so a simplified dynamical model is acceptable. We then chose to just model the transverse force $F_t$ which produces a secular drift ($da/dt$) of the semimajor axis – constant parameter to be fitted – and depends on the inverse of the square heliocentric distance $r$ (Chesley & Vokrouhlický 2008), thus imitating the solar flux responsible for the Yarkovsky effect:

$$F_t = \frac{n a^2 (1 - e^2)}{2 r^2} \left( \frac{da}{dt} \right) t, \quad (2)$$

where $n$ is the mean motion of the asteroid, $e$ the eccentricity, $v$ the true anomaly and $t$ is the unit vector in the transverse direction defined by

$$t = \frac{(r \times v) \times r}{\|(r \times v) \times r\|} \quad (3)$$

with $r$ and $v$, the respective heliocentric position and velocity vectors of the asteroid.

Despite its simplicity, this representation gives a good approximation of the effect and requires only the knowledge of the asteroid orbital elements. This allows us to overcome the poor knowledge of the NEA physical parameters on which the Yarkovsky effect depends. In addition, estimating the Gaia expected precision on ($da/dt$) enables us to select the best candidates to detect this effect, and to investigate the possibility to constrain further physical parameters by analysing the perturbations in the asteroid motion.

## 2.3 Variance analysis

From simulated data, we performed a variance analysis for the position and velocity vectors $u_0 = (r_0, \dot{r}_0)$ of each asteroid at the initial epoch $T_0$ and the semimajor axis drift rate $\dot{a}$ due to a transverse force (see equation 2). The reference time $T_0$ is taken at the centre of the Gaia observational time-span (2012–2017), that is $T_0 = \text{JD} 245 6841.125$ (2014 July 3).

We have considered a linear least-square problem to fit the unknown parameters on the observation residuals:

$$ \begin{align} \mathbf{W}^{1/2}(\mathbf{O} - \mathbf{C}) &= \mathbf{W}^{1/2} \mathbf{A} \begin{pmatrix} \delta \mathbf{u}_0 \\ \delta \dot{a} \end{pmatrix} \nonumber \\ \Rightarrow \quad \begin{pmatrix} \delta \mathbf{u}_0 \\ \delta \dot{a} \end{pmatrix} &= (\mathbf{A}' \mathbf{W} \mathbf{A})^{-1} \mathbf{A}' \mathbf{W} (\mathbf{O} - \mathbf{C}), \tag{4} \end{align} $$
where (O – C) are the observed minus computed positions in Gaia longitude λ. The corrections to the initial state vector (u₀, ˙a) are (δu₀, δ ˙a).

The weighting matrix W is defined by

$$
\mathbf{W} = \begin{pmatrix} \cdots & 0 \\ \sigma_i^{-2} & \cdots \\ 0 & \cdots \end{pmatrix},
$$

where σᵢ is the error on the asteroid position at the ith simulated
observation date from the apparent magnitude and velocity of the
asteroid converted to Gaia astrometric precision (see Fig. 2).

The matrix **A** contains the partial derivatives of the longitudes λ with respect to the state vector (u₀, ˙a),

$$
\mathbf{A} = \begin{pmatrix} \vdots \\ \frac{\partial \lambda_i}{\partial u_0} & \frac{\partial \lambda_i}{\partial \dot{a}} \\ \vdots \\ \vdots \end{pmatrix}.
$$

The elements of the matrix $(\partial\lambda_i/\partial u_0)$, $(\partial\lambda_i/\partial\dot{a})$ are decomposed
with respect to the rectangular coordinates:

$$
\frac{\partial \lambda_i}{\partial \mathbf{u}_0} = \sum_{q=1}^{3} \frac{\partial \lambda_i}{\partial r_q} \frac{\partial r_q}{\partial \mathbf{u}_0}, \quad \frac{\partial \lambda_i}{\partial \dot{a}} = \sum_{q=1}^{3} \frac{\partial \lambda_i}{\partial r_q} \frac{\partial r_q}{\partial \dot{a}}.
$$

They are then evaluated by analytically computing the quantities $(\partial\lambda_i/\partial r_q)$, while the equations of motion taking into account the planetary perturbations (in the force function $\mathbf{F}$) are numerically integrated,

$$
\ddot{r}(t) = \mathbf{F}(t) \tag{5}
$$

together with the variational equations,

$$
\begin{align*}
\frac{d^2}{dt^2} \left( \frac{\partial r_q}{\partial \mathbf{r}_0} \right) &= \sum_{n=1}^{3} \left( \frac{\partial F_q}{\partial r_n} \frac{\partial r_n}{\partial \mathbf{r}_0} + \frac{\partial F_q}{\partial \dot{r}_n} \frac{\partial \dot{r}_n}{\partial \mathbf{r}_0} \right), \\
\frac{d^2}{dt^2} \left( \frac{\partial r_q}{\partial \dot{a}} \right) &= \frac{\partial F_q}{\partial \dot{a}} + \sum_{n=1}^{3} \left( \frac{\partial F_q}{\partial r_n} \frac{\partial r_n}{\partial \dot{a}} + \frac{\partial F_q}{\partial \dot{r}_n} \frac{\partial \dot{r}_n}{\partial \dot{a}} \right).
\end{align*}
$$

The formal precisions of the fitted parameters are then given by
the diagonal elements of the inverse normal matrix (A'WA)^{-1}. The
matrix inversion is obtained by the singular value decomposition,
which allows us to evaluate the conditioning of the normal matrix,
and so the stability of the solution.

3 RESULTS

The most promising candidates for a Yarkovsky effect detection by Gaia are listed in Tables 2 and 3; Table 4 refers to non-NEAs. The Yarkovsky effect has not been seen yet for the NEAs of Table 2 unlike those in Table 3 (Chesley & Vokrouhlický 2008). The asteroids with diameters greater than 20 km were deleted from the list. The first two columns give the international astronomical union (IAU) number, when available, and the name of the asteroid. If the latter is a binary system, the letter B is written. We list the semimajor axis *a* and the eccentricity *e* of the NEA, a diameter estimate – the symbol (*) depicts an estimation from the absolute magnitude and albedo as to equation (1) – which is an important clue about the sensitivity to the Yarkovsky effect. The expected precision on the semimajor axis drift rate σ (da/dt) from simulated Gaia data is

given in au a million year. In order to better appreciate the signature
of the Yarkovsky force in the Gaia data, we estimated the detection
level as the ratio between (da/dt) and the Gaia formal precision
for this parameter. In Table 2, (da/dt) was numerically estimated
from formulas used in the modelling of the Yarkovsky perturbations
by Vokrouhlický (1998, 1999), while in Table 3, (da/dt) was taken
from Chesley & Vokrouhlický (2008). Realistic values were used
for the asteroid physical properties required by the model: the diam-
eter, either taken from a direct measurement or estimated from the
absolute magnitude and albedo (see Section 2.1), the thermal con-
ductivity, assumed to 0.01 W m⁻¹ K⁻¹, the albedo to 0.14 (Stuart &
Binzel 2004), the surface and bulk densities, respectively, to 1.7 and
2.5 g cm⁻<fnref n="3" />, and the specific heat capacity to 600 J kg⁻¹ K⁻¹. For the
sake of simplicity, we considered a null obliquity which maximizes
the Yarkovsky effect. Regarding the rotation period, when it is not
known, the set of known ones is used to compute a mean of the
drift rate (da/dt). Given the short lifetime of Gaia operations and
the fact that the Yarkovsky forces produce a perturbation in the or-
bital longitude that propagates as ∝ t<fnref n="2" /> and that the effects from the
change of the initial conditions u₀ is as ∝ t, we have added a supple-
mentary column to indicate the minimum and maximum absolute
values found for the correlations between (da/dt) and the initial
vector u₀. Additional information about the Gaia observations of
each asteroid can be read in the next columns with the total number

The observational arc in years
and the mean apparent magnitude mean V of the object. We then
mention the possibility to have future radar measurements of each
NEA from Vokrouhlický et al. (2005) for a potential combination
with the Gaia astrometric measurements (see Section 4), as well
as the current knowledge of the spin state and shape, respectively,
from the Collaborative Asteroid Lightcurve Link (CALL)<fnref n="2" /> and the
data base of asteroid models from Inversion Techniques (DAMIT).<fnref n="3" />
This final quantity is required to accurately model the Yarkovsky
effect, and to investigate the possibility to constrain some of the
parameters on which it depends, by analysing its non-gravitational
signature on the dynamics of each asteroid. However, we need to
have robust solutions for these physical parameters, and even if
Gaia will perform such a derivation for several thousands of as-
teroids, the geometry of NEA observations could prevent us from
deriving precise estimates (Delbò et al. 2008), hence, the necessity
to complete the Gaia observations by ground-based observations
(photometric, radar) (Thuillot, Tanga & Hestroffer 2009). Indeed,
we noticed a problem with the conditioning of the normal matrix
for all the objects selected.

The last three columns of Tables 2 and 3 pertain to the modelling
of the dynamics of each NEA in relation with close Earth encoun-
ters. The first one refers to the number of times the asteroid will pass
within 0.2 au from the Earth between 2020 and 2040 – the detailed
close approaches are given in Table 5 – while the other two indicate
if the NEA is useful to derive the solar quadrupole J₂ and the PPN
parameters β from the Gaia data (Mignard et al. 2007).

The dynamical model used will be essential to accurately measure
the Yarkovsky effect from the asteroid dynamics. At present, the
uncertainty in the gravitational perturbations by massive asteroids
can be an important source of error because of the large uncertainty
of their masses. So, we searched whether close approaches between
massive asteroids and NEAs listed in Tables 2 and 3 could take

<footnote n="2">² http://www.minorplanetobserver.com/astlc/LightcurveParameters.htm</footnote>

<footnote n="3">³ http://astro.troja.mff.cuni.cz/projects/asteroids3D/web.php</footnote>
<table><caption>**Table 2 – continued**</caption><thead><tr><th>Asteroid<br>n° IAU</th><th>name</th><th>Orbital<br>elem.<br>a e<br>(au)</th><th>Diam.<br>d<br>(km)</th><th>σ(da/dt)<br>(au Myr<sup>-1</sup>)</th><th>ā/σ(ā)</th><th>Correlations<br>with u<sub>0</sub><br>(min-max)</th><th>Gaia obs.<br>characteristics<br>nb. obs.<br>arc<br>(yr)</th><th>Mean V</th><th>Radar<br>candidate</th><th>Available<br>data<br>Spin<br>Shape</th><th>Earth<br>enc.</th><th>Gaia<br>fitting<br>β J<sub>2</sub></th></tr></thead><tbody><tr><td>53110</td><td>1999 AR7</td><td>1.64 0.21</td><td>1.6* 4.57 × 10<sup>-4</sup></td><td>0.3</td><td>0.08–0.67</td><td>34-5 4.5</td><td>19.2</td><td></td><td></td><td></td><td>-</td><td></td></tr><tr><td>85818</td><td>1998 XM4</td><td>1.66 0.42</td><td>2.8* 4.68 × 10<sup>-4</sup></td><td>0.2</td><td>0.41–0.53</td><td>48-4 4.1</td><td>18.7</td><td></td><td></td><td></td><td>1</td><td></td></tr><tr><td>3200</td><td>Phaethon</td><td>1.27 0.89</td><td>5.1 4.82 × 10<sup>-4</sup></td><td>0.2</td><td>0.18–0.52</td><td>78-14 4.9</td><td>18.2</td><td></td><td></td><td>X</td><td>-</td><td>X X</td></tr><tr><td>6569</td><td>Ondaatje</td><td>1.63 0.22</td><td>1.5 4.82 × 10<sup>-4</sup></td><td>0.4</td><td>0.19–0.82</td><td>57-0 4.5</td><td>19.4</td><td></td><td></td><td></td><td>-</td><td></td></tr><tr><td>144332</td><td>2004 DV24</td><td>1.42 0.29</td><td>1.7* 4.93 × 10<sup>-4</sup></td><td>0.2</td><td>0.03–0.60</td><td>44-11 4.7</td><td>19.3</td><td></td><td></td><td></td><td>1</td><td></td></tr><tr><td>2008</td><td>EA32</td><td>0.62 0.30</td><td>1.8* 4.93 × 10<sup>-4</sup></td><td>0.3</td><td>0.16–0.84</td><td>43-21 2.9</td><td>18.1</td><td></td><td></td><td></td><td>1</td><td></td></tr></tbody></table>

<table><caption>**Table 3.** List of the most promising sub-20 km NEAs listed in Chesley & Vokrouhlický (2008) to detect anew the Yarkovsky effect.</caption><thead><tr><th>Asteroid<br>n° IAU</th><th>name</th><th>Orbital<br>elem.<br>a e<br>(au)</th><th>Diam.<br>d<br>(km)</th><th>σ da/dt<br>(au Myr<sup>-1</sup>)</th><th>ā/σ(ā)</th><th>Correlations<br>with u<sub>0</sub><br>(min-max)</th><th>Gaia obs.<br>characteristics<br>nb. obs.<br>arc<br>(yr)</th><th>Mean V</th><th>Radar<br>candidate</th><th>Available<br>data<br>Spin<br>Shape</th><th>Earth<br>enc.</th><th>Gaia<br>fitting<br>β J<sub>2</sub></th></tr></thead><tbody><tr><td>1685</td><td>Toro</td><td>1.37 0.44</td><td>4.1 9.28 × 10<sup>-5</sup></td><td>0.6</td><td>0.32–0.96</td><td>93-4 4.6</td><td>17.2</td><td></td><td></td><td>X</td><td>3</td><td></td></tr><tr><td><b>2100</b></td><td><b>Ra-Shalom</b></td><td>0.83 0.44</td><td>2.8 1.33 × 10<sup>-4</sup></td><td><b>5.3</b></td><td>0.17–0.87</td><td>52-13 4.2</td><td>17.7</td><td></td><td>X</td><td>X X</td><td>4</td><td>X X</td></tr><tr><td>1620</td><td>Geographos</td><td>1.25 0.34</td><td>2.5 1.84 × 10<sup>-4</sup></td><td>0.6</td><td>0.16–0.79</td><td>72-4 4.9</td><td>18.2</td><td></td><td>X</td><td>X X</td><td>2</td><td></td></tr><tr><td><b>2063</b></td><td><b>Bacchus</b></td><td>1.08 0.35</td><td>1.1 2.48 × 10<sup>-4</sup></td><td><b>4.3</b></td><td>0.20–0.56</td><td>109-9 4.1</td><td>18.8</td><td></td><td></td><td>X</td><td>2</td><td></td></tr></tbody></table>

<table><caption>**Table 4.** List of the best 10 Gaia expected precisions on da/dt for non-NEAs.</caption><thead><tr><th rowspan="2">n° IAU</th><th rowspan="2">Asteroid name</th><th colspan="2">Orbital elem.</th><th colspan="2">Diam.</th><th rowspan="2">σ da/dt<br>(au Myr<sup>-1</sup>)</th><th colspan="3">Gaia obs. characteristics</th></tr><tr><th>a<br>(au)</th><th>e</th><th>d<br>(km)</th><th>σ<br>(au)</th><th>nb. obs.</th><th>Arc<br>(yr)</th><th>Mean V</th></tr></thead><tbody><tr><td>1139</td><td>Atami</td><td>B</td><td>1.95</td><td>0.26</td><td>9.3</td><td>1.13 × 10<sup>-4</sup></td><td>74</td><td>4.8</td><td>16.4</td></tr><tr><td>1747</td><td>Wright</td><td></td><td>1.71</td><td>0.11</td><td>6.3</td><td>1.14 × 10<sup>-4</sup></td><td>102</td><td>4.6</td><td>16.8</td></tr><tr><td>3800</td><td>Karayusuf</td><td></td><td>1.58</td><td>0.08</td><td>3.0</td><td>1.41 × 10<sup>-4</sup></td><td>83</td><td>5.0</td><td>17.9</td></tr><tr><td>1103</td><td>Sequoia</td><td></td><td>1.93</td><td>0.09</td><td>6.5</td><td>1.54 × 10<sup>-4</sup></td><td>93</td><td>4.8</td><td>16.5</td></tr><tr><td></td><td>2010 CP8</td><td></td><td>1.72</td><td>0.21</td><td>5.6*</td><td>1.59 × 10<sup>-4</sup></td><td>90</td><td>4.5</td><td>17.5</td></tr><tr><td>6618</td><td>1936 SO</td><td></td><td>1.88</td><td>0.04</td><td>5.3</td><td>1.71 × 10<sup>-4</sup></td><td>89</td><td>4.7</td><td>17.0</td></tr><tr><td>4764</td><td>Joneberhart</td><td></td><td>1.93</td><td>0.05</td><td>4.6</td><td>1.79 × 10<sup>-4</sup></td><td>98</td><td>5.0</td><td>17.6</td></tr><tr><td>2629</td><td>Rudra</td><td></td><td>1.74</td><td>0.23</td><td>4.5*</td><td>1.87 × 10<sup>-4</sup></td><td>73</td><td>4.3</td><td>17.7</td></tr><tr><td>5427</td><td>Jensmartin</td><td></td><td>1.93</td><td>0.07</td><td>5.3</td><td>2.02 × 10<sup>-4</sup></td><td>68</td><td>4.9</td><td>17.2</td></tr><tr><td>244</td><td>Sita</td><td></td><td>2.17</td><td>0.14</td><td>10.9</td><td>2.11 × 10<sup>-4</sup></td><td>62</td><td>4.8</td><td>16.4</td></tr></tbody></table>

place during the Gaia mission. We found that a few perturbers – mainly (1) Ceres – are involved in a significant encounter, meaning with a deflection angle of the perturbed NEA > 1 mas, during this period. Besides, all these perturbers will have a mass well derived by Gaia and, very accurately for both of them, (1) Ceres and (4) Vesta, by Dawn (Russell et al. 2007). Thus, the perturbing asteroids should not be a problem in the modelling of the Yarkovsky effect.

Despite our using an underestimated astrometric performance in the variance analysis – (see Section 2.1) – we found 64 promising NEAs to detect the Yarkovsky effect with a precision on (da/dt) smaller than $5 \times 10^{-4}$ au Myr$^{-1}$ – this precision corresponds to the current levels obtained from radar measurements over much larger period of time (Chesley & Vokrouhlický 2008) – and 10 showing a ratio $ā/σ(ā) \ge 2$. The last mentioned can be considered as the best candidates for a clean detection. A realistic extrapolation of the results given in Tables 2 and 3 in the case of an optimal precision (nominal error from Fig. 1) would consist in dividing the formal precisions on (da/dt) by ~3 given that the mean apparent magnitude of each selected NEA is greater than 16 mag, and that the comparison of the two expected Gaia astrometric precisions for non-moving objects, one used as nominal error in Fig. 2 for moder-

ate apparent velocities and the other one directly plotted in Fig. 1, shows a ratio of ~3 (in fact consistent with nine CCD crossings) from the apparent magnitude $V = 14$ mag. More than the precision, the number of candidates for a Yarkovsky detection is really interesting because a larger choice of asteroids to be studied is given. In addition, Gaia will discover new NEAs, some of them will probably contribute to expand the list of the best candidates. Furthermore, the impact of systematic errors in the measurement of the Yarkovsky effects will be limited because of the short observational time-span of the mission and the accurate dynamical model that will be used for fitting – many masses of perturbing asteroids estimated etc.

The NEAs are well known to be the most sensitive object to the Yarkovsky effect, because they are small objects (low mass) with a small perihelion – strong solar radiation. However, other asteroids can have a motion affected by this effect, albeit smaller. Therefore, the variance analysis on the (da/dt) was extended to the other asteroids that Gaia will observe (>230 000). Of course, the asteroids not having too small a number of observations for the least-square treatment were automatically rejected. The best 10 precisions on (da/dt) are listed in Table 4, which are better than $2.11 \times 10^{-4}$ au Myr$^{-1}$, but do not allow us to foresee a significant detection:
<table><caption>**Table 5.** List of the close approaches (minimum distance ≤0.2 au) from 2020 to 2040 between the Earth and the NEAs listed in Tables 2 and 3. The asteroid designation is supplemented by the expected precision on (da/dt), the encounter date and the impact parameter min. dist.</caption><thead><tr><th rowspan="2">n° IAU</th><th rowspan="2">Asteroid name</th><th rowspan="2">σ(da/dt)<br>(au Myr<sup>-1</sup>)</th><th colspan="2">Earth encounters</th></tr><tr><th>Date<br>(d/m/yr)</th><th>Min. dist.<br>(au)</th></tr></thead><tbody><tr><td rowspan="2">1620</td><td rowspan="2">Geographos</td><td rowspan="2">1.84 × 10<sup>-4</sup></td><td>12/08/2026</td><td>0.170</td></tr><tr><td>12/03/2040</td><td>0.114</td></tr><tr><td rowspan="3">1685</td><td rowspan="3">Toro</td><td rowspan="3">9.28 × 10<sup>-5</sup></td><td>20/01/2024</td><td>0.133</td></tr><tr><td>20/01/2032</td><td>0.129</td></tr><tr><td>21/01/2040</td><td>0.139</td></tr><tr><td rowspan="2">1943</td><td rowspan="2">Anteros</td><td rowspan="2">1.61 × 10<sup>-4</sup></td><td>30/05/2026</td><td>0.132</td></tr><tr><td>23/05/2038</td><td>0.066</td></tr><tr><td rowspan="2">2062</td><td rowspan="2">Aten</td><td rowspan="2">8.66 × 10<sup>-5</sup></td><td>31/12/2032</td><td>0.191</td></tr><tr><td>20/01/2034</td><td>0.133</td></tr><tr><td rowspan="2">2063</td><td rowspan="2">Bacchus</td><td rowspan="2">2.48 × 10<sup>-4</sup></td><td>30/03/2024</td><td>0.120</td></tr><tr><td>14/09/2031</td><td>0.126</td></tr><tr><td rowspan="4">2100</td><td rowspan="4">Ra-Shalom</td><td rowspan="4">1.33 × 10<sup>-4</sup></td><td>31/08/2022</td><td>0.187</td></tr><tr><td>12/08/2025</td><td>0.173</td></tr><tr><td>15/10/2035</td><td>0.191</td></tr><tr><td>05/10/2038</td><td>0.156</td></tr><tr><td rowspan="2">4769</td><td rowspan="2">Castalia</td><td rowspan="2">2.59 × 10<sup>-4</sup></td><td>22/08/2023</td><td>0.110</td></tr><tr><td>04/04/2027</td><td>0.129</td></tr><tr><td>4953</td><td>1990 MU</td><td>1.88 × 10<sup>-4</sup></td><td>06/06/2027</td><td>0.031</td></tr><tr><td>5381</td><td>Sekhmet</td><td>1.09 × 10<sup>-4</sup></td><td>20/05/2038</td><td>0.199</td></tr><tr><td>12711</td><td>Tukmit</td><td>3.76 × 10<sup>-4</sup></td><td>15/07/2022</td><td>0.129</td></tr><tr><td rowspan="3">33342</td><td rowspan="3">1998 WT24</td><td rowspan="3">2.78 × 10<sup>-4</sup></td><td>24/12/2026</td><td>0.101</td></tr><tr><td>26/11/2029</td><td>0.050</td></tr><tr><td>20/12/2040</td><td>0.027</td></tr><tr><td rowspan="7">66146</td><td rowspan="7">1998 TU3</td><td rowspan="7">1.14 × 10<sup>-4</sup></td><td>05/11/2024</td><td>0.089</td></tr><tr><td>26/08/2026</td><td>0.090</td></tr><tr><td>07/11/2031</td><td>0.083</td></tr><tr><td>01/09/2033</td><td>0.132</td></tr><tr><td>06/11/2038</td><td>0.118</td></tr><tr><td>07/09/2040</td><td>0.161</td></tr><tr><td>26/05/2020</td><td>0.160</td></tr><tr><td rowspan="5">66391</td><td rowspan="5">1999 KW4</td><td rowspan="5">1.24 × 10<sup>-4</sup></td><td>02/06/2034</td><td>0.197</td></tr><tr><td>29/05/2035</td><td>0.112</td></tr><tr><td>25/05/2036</td><td>0.015</td></tr><tr><td>26/05/2037</td><td>0.112</td></tr><tr><td>02/10/2027</td><td>0.063</td></tr><tr><td rowspan="2">68267</td><td rowspan="2">2001 EA16</td><td rowspan="2">3.98 × 10<sup>-4</sup></td><td>25/09/2040</td><td>0.132</td></tr><tr><td>18/05/2027</td><td>0.126</td></tr><tr><td rowspan="3">85818</td><td rowspan="3">1998 XM4</td><td rowspan="3">4.68 × 10<sup>-4</sup></td><td>02/03/2023</td><td>0.190</td></tr><tr><td>15/05/2024</td><td>0.197</td></tr><tr><td>18/05/2028</td><td>0.149</td></tr><tr><td rowspan="3">86667</td><td rowspan="3">2000 FO10</td><td rowspan="3">3.26 × 10<sup>-4</sup></td><td>19/05/2032</td><td>0.135</td></tr><tr><td>17/05/2036</td><td>0.172</td></tr><tr><td>17/09/2027</td><td>0.145</td></tr><tr><td rowspan="3">87684</td><td rowspan="3">2000 SY2</td><td rowspan="3">1.92 × 10<sup>-4</sup></td><td>13/09/2031</td><td>0.051</td></tr><tr><td>10/09/2035</td><td>0.105</td></tr></tbody></table>

the ratio between numerical estimates (da/dt) as in Table 2 and the Gaia expected precisions σ (da/dt) are always smaller than one – 0.6 is the maximum value found.

## 4 PERSPECTIVES

Constraining physical parameters by analysing the Yarkovsky effect on the dynamics of minor planets is a promising, and so far unsub-

<table><caption>**Table 5 – continued**</caption><thead><tr><th rowspan="2">n° IAU</th><th rowspan="2">Asteroid name</th><th rowspan="2">σ(da/dt)<br>(au Myr<sup>-1</sup>)</th><th colspan="2">Earth encounters</th></tr><tr><th>Date<br>(d/m/yr)</th><th>Min. dist.<br>(au)</th></tr></thead><tbody><tr><td>88710</td><td>2001 SL9</td><td>2.78 × 10<sup>-4</sup></td><td>27/04/2026</td><td>0.199</td></tr><tr><td rowspan="2">96590</td><td rowspan="2">1998 XB</td><td rowspan="2">9.50 × 10<sup>-5</sup></td><td>28/11/2029</td><td>0.182</td></tr><tr><td>14/11/2035</td><td>0.172</td></tr><tr><td>137170</td><td>1999 HF1</td><td>1.07 × 10<sup>-4</sup></td><td>16/03/2022</td><td>0.181</td></tr><tr><td rowspan="2">137924</td><td rowspan="2">2000 BD19</td><td rowspan="2">1.38 × 10<sup>-4</sup></td><td>10/02/2020</td><td>0.112</td></tr><tr><td>13/02/2029</td><td>0.109</td></tr><tr><td rowspan="5">138127</td><td rowspan="5">2000 EE14</td><td rowspan="5">2.81 × 10<sup>-4</sup></td><td>06/03/2021</td><td>0.168</td></tr><tr><td>06/03/2028</td><td>0.171</td></tr><tr><td>26/02/2029</td><td>0.179</td></tr><tr><td>07/03/2035</td><td>0.174</td></tr><tr><td>27/02/2036</td><td>0.174</td></tr><tr><td rowspan="5">138852</td><td rowspan="5">2000 WN10</td><td rowspan="5">1.46 × 10<sup>-4</sup></td><td>10/11/2020</td><td>0.137</td></tr><tr><td>10/11/2021</td><td>0.142</td></tr><tr><td>10/11/2022</td><td>0.148</td></tr><tr><td>10/11/2023</td><td>0.155</td></tr><tr><td>10/11/2024</td><td>0.163</td></tr><tr><td rowspan="4">144332</td><td rowspan="4">2004 DV24</td><td rowspan="4">4.93 × 10<sup>-4</sup></td><td>10/09/2035</td><td>0.094</td></tr><tr><td>07/08/2033</td><td>0.128</td></tr><tr><td>18/08/2038</td><td>0.186</td></tr><tr><td>13/01/2023</td><td>0.085</td></tr><tr><td rowspan="2">153415</td><td rowspan="2">2001 QP153</td><td rowspan="2">3.14 × 10<sup>-4</sup></td><td>09/01/2030</td><td>0.097</td></tr><tr><td>01/03/2039</td><td>0.123</td></tr><tr><td rowspan="4">162385</td><td rowspan="4">2000 BM19</td><td rowspan="4">3.23 × 10<sup>-4</sup></td><td>17/12/2021</td><td>0.036</td></tr><tr><td>02/12/2024</td><td>0.088</td></tr><tr><td>12/11/2027</td><td>0.138</td></tr><tr><td>03/11/2023</td><td>0.059</td></tr><tr><td rowspan="2">164121</td><td rowspan="2">2003 YT1</td><td rowspan="2">2.35 × 10<sup>-4</sup></td><td>05/11/2030</td><td>0.148</td></tr><tr><td>09/09/2031</td><td>0.120</td></tr><tr><td rowspan="2">163899</td><td rowspan="2">2003 SD220</td><td rowspan="2">1.61 × 10<sup>-4</sup></td><td>08/07/2020</td><td>0.163</td></tr><tr><td>15/07/2029</td><td>0.048</td></tr><tr><td rowspan="2">164121</td><td rowspan="2">2003 YT1</td><td rowspan="2">2.35 × 10<sup>-4</sup></td><td>22/07/2038</td><td>0.172</td></tr><tr><td>20/06/2027</td><td>0.196</td></tr><tr><td rowspan="2">163899</td><td rowspan="2">2003 SD220</td><td rowspan="2">1.61 × 10<sup>-4</sup></td><td>22/06/2029</td><td>0.046</td></tr><tr><td>29/11/2034</td><td>0.177</td></tr><tr><td rowspan="2">164121</td><td rowspan="2">2003 YT1</td><td rowspan="2">2.35 × 10<sup>-4</sup></td><td>07/12/2037</td><td>0.120</td></tr><tr><td>09/02/2020</td><td>0.153</td></tr><tr><td rowspan="2">164121</td><td rowspan="2">2003 YT1</td><td rowspan="2">2.35 × 10<sup>-4</sup></td><td>01/01/2026</td><td>0.180</td></tr><tr><td>21/08/2034</td><td>0.178</td></tr><tr><td rowspan="2">164121</td><td rowspan="2">2003 YT1</td><td rowspan="2">2.35 × 10<sup>-4</sup></td><td>13/02/2035</td><td>0.177</td></tr><tr><td>26/08/2035</td><td>0.114</td></tr><tr><td rowspan="2">164121</td><td rowspan="2">2003 YT1</td><td rowspan="2">2.35 × 10<sup>-4</sup></td><td>23/02/2036</td><td>0.092</td></tr><tr><td>29/11/2034</td><td>0.177</td></tr></tbody></table>

stantiated, goal with Gaia. Preliminary variance analysis using Gaia simulated data gives the best precision for the fit of the diameter and density, the mass having been introduced in terms of volume and density (Mouret 2007). The force was modelled with a linearized version of the Yarkovsky effect described in detail in Vokrouhlický (1998, 1999). This modelling is inversely proportional to the mass of the asteroid, which can then be assessed from the data. This is an important parameter leading to the density and the associated constraints on internal structure and composition, when combined with the volume. Besides, spectroscopic measurements would enable us to investigate the possibility to link the taxonomic class
for small objects like NEAs to their density. Using surface force
opens a new way to determine masses, complementary to the clas-
sical perturbation technique: the efficiency of the former increases
with smaller bodies precisely when the standard method becomes
inefficient. The use of the Yarkovsky effect will extend the power
of Gaia in deriving asteroid masses, covering the lower range not
overlapping with more than 150 masses that should be derived from
gravitational perturbations during close encounters between aster-
oids (Mouret, Hestroffer & Mignard 2007). Realistic distribution for
the other physical parameters (diameter, thermal inertia, spin state)
impacting the magnitude and the direction of the Yarkovsky force
will be used in the dynamical modelling to process the observations.

Detections of the Yarkovsky effect have already been achieved
from ground-based observations, mainly radar (Chesley et al. 2003;
Vokrouhlický et al. 2008). We plan to investigate the contribution
of the observations from the ground (radar, astrometric) to the Gaia
data in improving the modelling of the Yarkovsky effect and the fit
precision of physical parameters. Besides, the interest for ground-
based observations is strengthened by the fact that the star catalogue
from Gaia will allow us to improve their reduction.

During the two years before the launch of Gaia, new NEAs
will be discovered increasing the number of potential targets for a
Yarkovsky effect detection. Therefore, the list of the most promising
candidates will be regularly updated and available in electronic
form.

5 CONCLUSIONS

Using realistic simulated data for Gaia observations, we estimated
the potential of the mission to detect the Yarkovsky effect from cur-
rently known asteroids. We thus computed the expected precisions
on the semimajor axis drift rate (da/dt) and listed the most valuable
NEAs. A precision on (da/dt) better than 5 × 10⁻⁴ au Myr⁻¹ is ex-
pected for 64 NEAs (six for a precision smaller than 10⁻⁴ au Myr⁻¹),
and 10 show an expected signal-to-noise ratio greater than 2. How-
ever, the result should be better because the astrometric precision
used in the simulations was underestimated. For each asteroid, our
knowledge of major physical parameters necessary to accurately
model this effect is indicated so that observation campaigns are
prepared to complete the Gaia data. The interest for each object
in modelling the non-gravitational forces is given in terms of close
Earth-encounter predictions and fits of the solar quadrupole J₂ and
the PPN parameter β from the Gaia observations. Furthermore,
Gaia will discover new objects, and some will probably supple-
ment the number of targets for detecting the Yarkovsky effect.
Our promising results are incentive to continue the investigation
by studying the contribution of radar measurements to the Gaia
data and the possibility to estimate NEA masses by analysing the
Yarkovsky effect.

ACKNOWLEDGMENTS

We thank A. Dell'Oro for precious information about the Gaia astro-
metric precision for moving objects, as well as D. Vokrouhlický for

his valuable and constructive comments. S. Mouret acknowledges
the support of the Deutsche Forschungsgemeinschaft (German Re-
search Foundation).

REFERENCES

Afonso G. B., Gomes R. S., Florczak M. A., 1995, Planet. Space Sci., 43, 787

Bottke W. F., Vokrouhlický D., Brož M., Nesvorný D., Morbidelli A., 2001, *Sci*, 294, 1693

Bottke W. F., Jr, Vokrouhlický D., Rubincam D. P., Brož M., 2002, in Binzel
R. P., Bottke W. F., Cellino A., Paolicchi P., eds, Asteroids III. Univ.
Arizona Press, Tucson, p. 395

Bowell E., Hapke B., Domingue D., Lumme K., Peltoniemi J., Harris
A. W., 1989, in Binzel R. P., Gehrels T., Matthews M. S., eds, Asteroids
II. Univ. Arizona Press, Tucson, p. 524

Chesley S. R., Vokrouhlický D., 2008, AAS/Division of Dynamical Astron-
omy Meeting, Vol. 39, Searching for Yarkovsky Among the NEAs. Am.
Astron. Soc., Washington, p. 02.04

Chesley S. R. et al., 2003, *Sci*, 302, 1739

Chesley S. R., Vokrouhlický D., Matson R. D., 2006, BAAS, 38, 591

De Bruijne J., 2009, Technical note, Gaia-CA-TN-ESA-JDB-053-01,
Along- and Across-scan Location-estimation Performance

Delbò M., Tanga P., Mignard F., 2008, Planet. Space Sci., 56, 1823

Farinella P., Vokrouhlický D., Hartmann W. K., 1998, Icarus, 132, 378

Giorgini J. D. et al., 2002, *Sci*, 296, 132

Kaasalainen M., Óurech J., Warner B. D., Krugly Y. N., Gaftonyuk N. M., 2007, Nat, 446, 420

Lowry S. C. et al., 2007, *Sci*, 316, 272

Mignard F. et al., 2007, Earth Moon and Planets, 101, 97

Milani A., Chesley S. R., Sansaturio M. E., Bernardi F., Valsecchi G. B.,
Arratia O., 2009, Icarus, 203, 460

Morbidelli A., Vokrouhlický D., 2003, Icarus, 163, 120

Mouret S., 2007, PhD thesis, IMCCE, Paris observatory

Mouret S., Hestroffer D., Mignard F., 2007, A&A, 472, 1017

Nesvorný D., Bottke W. F., 2004, Icarus, 170, 324

Öpik E. J., 1951, Proc. R. Irish Acad. 165, 54A

Peterson C., 1976, Icarus, 29, 91

Rubincam D. P., 1988, J. Geophys. Res., 93, 13805

Rubincam D. P., 1995, J. Geophys. Res., 100, 1585

Rubincam D. P., 2000, Icarus, 148, 2

Russell C. T. et al., 2007, Earth Moon and Planets, 101, 65

Stuart J. S., Binzel R. P., 2004, Icarus, 170, 295

Thuillot W., Tanga P., Hestroffer D., 2009, in Heydari-Malayeri M., Reylé
C., Samadi R., eds, SF2A-2009: Proc. Ann. Meeting French Soc. Astron.
Astrophys. EDP Sci. Publ., Les Ulis, p. 83

Vokrouhlický D., 1998, A&A, 335, 1093

Vokrouhlický D., 1999, A&A, 344, 362

Vokrouhlický D., Farinella P., 1999, in Henrard J., Ferraz-Mello S., eds, The Impact of Modern Dynamics in Astronomy. Kluwer, Dordrecht, p. 365

Vokrouhlický D., Farinella P., 2000, Nat, 407, 606

Vokrouhlický D., Čapek D., Chesley S. R., Ostro S. J., 2005, Icarus, 173, 166

Vokrouhlický D., Chesley S. R., Matson R. D., 2008, AJ, 135, 2336

Yarkovsky I. O., 1901, The density of luminiferous ether and the resistance
it offers to motion. Bryansk

This paper has been typeset from a TeX/LATeX file prepared by the author.