ISSN 0038-0946, Solar System Research, 2009, Vol. 43, No. 1, pp. 79–81. © Pleiades Publishing, Inc., 2009. Published in Russian in Astronomicheskii Vestnik, 2009, Vol. 43, No. 1, pp. 83–86.

Influence of the Largest Asteroids on the Orbital Motions of Terrestrial Planets: Application to the Earth and Mars1 J. Souchaya, D. Gaucheza, and A. Nedelcub a

Observatoire de Paris/DANOF, URA 1125 du CNRS, 61 Avenue de l’Observatoire, F-75014, Paris, France b Institul Astronomic Academiei Romane, Cutitul de Argint, Bucuresti, Romania e-mail: [email protected]; [email protected] Received April 30, 2008

Abstract—The level of precision of modern numerical ephemeris of the Solar System necessitates taking into account the gravitational influence of the largest asteroids on the terrestrial planets. This can be done in a straightforward manner when assuming that the mass of the asteroid is well known. Nevertheless, this is rarely the case, even for the largest asteroids. In this paper, we use recent determinations of the masses of Ceres, Pallas, and Vesta to both qualitatively and quantitatively determine the action of these asteroids on the orbital parameters of the Earth and Mars. This is done by the numerical integration by comparing the orbital motions of the perturbed planet when adding or not the perturbing asteroid to the classical 9 bodies problem (the Sun + the eight planets). Some preliminary results are discussed. PACS numbers: 96.30.Ys, 96.15.De, 96.30. Gc DOI: 10.1134/S0038094609010080 1

1. INTRODUCTION Very up-to-date ephemerides of the planets are taking into account the gravitational effects of the asteroids on these planets. This is, for instance, the case of DE414 (Konopliv et al., 2006), EMLP2006 (Pitjeva, 2006), and INPOP06 (Fienga et al., 2007), which take into consideration the presence of roughly 300 asteroids in the theoretical framework of the parametrized post-Newtonian metric for general relativity including the solar oblateness and the perturbation of a massive ring of small asteroids. As a complementary study, it looks interesting to individually evaluate the effects of the largest asteroids such as Ceres, Pallas, and Vesta on the nearest planets, i.e., the Earth and Mars.

2

1 The

(4)

cos i 1 – e ∂R ∂R dω ------ – ------------------------------------ ------, ------- = ----------------2 2 2 ∂e ∂i dt na e na 1 – e sin i

(5)

dM 2 ∂R 1 – e ∂R - ------. -------- = n + – ------ ------ – -----------dt na ∂a na 2 e ∂e

(6)

2

2

The perturbing function itself is expressed as a combination of the orbital elements of the two bodies considered and involves, in particular, combinations of their mean longitudes. The corresponding analytical developments are cumbersome and complex. On the contrary, it is possible to evaluate in a straightforward manner the perturbations from the numerical integration. The procedure consists in computing the 9-bodies problem (the Sun and the eight planets), then adding the perturbing asteroid to compute a 10-bodies problem. The subtraction of the two signals concerning one parameter of the orbital motion of the planet studied is a clear and direct measurement of the influence of the asteroid on the selected planet and selected parameter. We adopt this procedure in the following, where we show some partial results. Our numerical integrator is a Runge–Kutta of the 12th order. Notice that we consider the Earth and the Moon as a unique body located at their barycenter. We thus neglect the second-order effect due to the differential gravitational action of the asteroid on the two bodies.

2. THEORETICAL BASIS The individual effect of a given asteroid on a planet, such as the Earth or Mars, can be measured as a perturbation of its osculating orbital elements (a, e, i, Ω, ω, and M) as a function of the perturbing function R, according to Lagrange’s formula, in the following way: 2 ∂R da ------ = ------ -------- , (1) na ∂M dt – 1 – e ∂R 1 – e ∂R de - ------- + ------------ -------- , ------ = -------------------2 2 dt na e ∂ω na e ∂M –1 ∂R cos i ∂R di ----- = -------------------------- ------- + ------------------------------------ ------- , 2 2 ∂Ω 2 2 ∂ω dt na 1 – e na 1 – e sin i

1 ∂R dΩ ------- = ------------------------------------ ------, 2 2 ∂i dt na 1 – e sin i

2

(2) (3)

text was submitted by the autors in English.

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SOUCHAY et al. 0.030 0.025

0.3 0.2 ∆a, km

∆a, km

0.020

Ceres

0.015 0.010

0.1

0.005

0

0

–0.1

–0.005 2000 2010 2020 2030 2040 2050 Julian years

Vesta

Pallas

–0.2 2000 2010 2020 2030 2040 2050 Julian years

Fig. 1. Variations in the semimajor axis of the Earth–Moon barycenter due to the gravitational effect of Ceres.

Fig. 2. Variations in the semimajor axis of Mars due to the individual gravitational effect of Ceres, Pallas, and Vesta.

3. EFFECTS OF CERES, PALLAS, AND VESTA ON THE EARTH–MOON BARYCENTER AND MARS ORBITAL MOTION Note that a lot of recent estimations of the masses have been carried out, reducing the error bars and converging to more accurate nominal values (see Pitjeva, 2007). Finally, for our computations done in the following, we have chosen the values of the asteroid masses as given in the DE405 ephemerides, that is to say, 4.7 × 10–10 MSun, 1.0 × 10–10 MSun, and 1.3 × 10–10 MSun for Ceres, Pallas, and Vesta, respectively. As a representative example, we show in Fig. 1 the variations of the Earth semimajor axis due to Ceres. The variations are very regular with a peak-to-peak amplitude of 20 m. Since the mass of Mars is considerably smaller than the mass of the Earth (the ratio is 0.105) and closer to the perturbing asteroids, we can expect that the variations of the orbital elements due to these latter ones are considerably larger. This is, for instance, the case in Fig. 2, where the effects of the three asteroids Ceres, Pallas,

and Vesta on Mars’ semimajor axis are shown separately. We can observe the peak-to-peak amplitudes of roughly 400 m for Ceres and Vesta, and 200 m for Pallas in a 50-year time span without any significant linear trend. On the contrary, Mars other orbital elements e, i and the longitude of perihelion ω are dominated by linear trends whose values are given in the table. The Mars inclination is affected at the level of 0.01′′/cy, its eccentricity at the level of 10–8, and its longitude of the perihelion at the level of 0.1′′/cy. Note that Ceres has a leading effect on i and ω, while Vesta is leading on e.

Linear variations of the Mars inclination, eccentricity, and longitude of the node due to Ceres, Pallas, and Vesta. The overall slopes are calculated two ways: the first one (A) by only adding the individual effects; the second one (B) by including the three asteroids in the integration (12 bodies problem) iMars, rate

eMars, rate

ωMars, rate

Ceres

–0.008084

0.543 × 10–8

–1.1964

Pallas

–0.004244

–1.086 × 10–8

1.0323

Vesta

–0.005620

–1.413 × 10–8

–0.7769

All (A)

–0.017948

–1.957 × 10–8

–0.9410

All (B)

–0.017948

–1.957 × 10–8

–0.9408

Asteroid

4. EFFECTS OF CERES, PALLAS, AND VESTA ON THE EARTH–MARS DISTANCE AND ON THE EARTH–MARS ORIENTATION Two important fundamental measurements, both in an astrometric and a space navigation point of view, are directly dependent on the variations in the orbital parameters of the Earth and Mars due to the asteroids as described in the last section: these are the Earth–Mars distance and the direction of the Earth–Mars vector. In Fig. 3, we show the variations of the Earth–Mars distance due to the combined effects of Ceres, Pallas, and Vesta for a 50-year time span. We can observe a gradually increasing amplitude of the signal which can be suitably fitted by a set of Fourier and Poisson components with well-defined frequencies. A suitable FFT analysis indicates that the leading component has a 777.8-d period with an amplitude of 0.78 km for the Fourier component and a 17.75 km/cy for the Poisson component. Moreover, the residuals after fitting several leading components, corresponding to the flat curve in Fig. 3, are significantly smaller (by a factor 10) than the original signal. This suggests that a deepened analysis should lead to a very good characterization of the signature of the effects of the asteroids. In Fig. 4, we represent the evolution of the Earth– Mars variations of the orientation due to the three leading asteroids, expressed in arcseconds, for a 1000-y SOLAR SYSTEM RESEARCH

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5 4 3 2 1 0 –1 –2 –3 –4 –5

mas

Amplitude, km

INFLUENCE OF THE LARGEST ASTEROIDS ON THE ORBITAL MOTIONS

0

2000

4000

6000 8000 Julian days

Fig. 3. Influence of the combined effects of the largest asteroids Ceres, Pallas, and Vesta on the distance from the EMB to Mars, for 22.5 years from J2000.0. The curve in bold with a large amplitude is the original one. The thin curve represents the fit. The curve in bold shows the residuals after the subtraction of the leading sinusoidal and Poisson terms.

time interval. We can observe a very regular linear trend of the amplitude of the variations, which reaches 1" peak-to-peak in the given interval. These increasing variations both for the distance and the orientation are undoubtedly due to the linear trends observed for the orbital parameters of the Earth and Mars (see table), which although appearing rather small lead to substantial variations after several decades, with respect to the performance of modern space navigation and astrometric measurements. 4. CONCLUSIONS In this short study, we have shown that it is possible to easily characterize the individual effects of any given asteroid on the orbital motions of the Earth and Mars, and also, in parallel, to evaluate their influence on the Earth–Mars distance and the Earth-to-Mars orientation vector, which are two important parameters in terms of

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1000 900 800 700 700 500 400 300 200 100 0 2000 2200 2400 2600 2800 3000 Julian years

Fig. 4. Influence of the combined effects of the largest asteroids Ceres, Pallas, and Vesta on the direction of Mars as seen from the EMB.

space navigation and astrometry. This study is a starting point for a deeper, more complete study consisting of fitting all of the signals with suitable linear, sinusoidal, and Poisson components and analyzing the analytical signature of these components as a combination of the orbital parameters (as the mean longitudes) of the perturbing body (large asteroid) and perturbed body (the Earth or Mars). REFERENCES Fienga, A., INPOP06, 2007 (in prep.). Konopliv, A.S., Yoder, C.F, Standish, E.M., Yuan, D.-N., and Sjorgen, W.L., Icarus, 2006, vol. 182, p. 23K Pitjeva, E.V., IAUJD, 2006, vol. 16E, p. 14P. Pitjeva, E.V., Journées Systemes de Reference, Meudon, 2007. Standish, E.M. and Fienga, A., Astronomy and Astrophysics, 2002, vol. 384, p. 322.