Determination of Masses of Mercury and Venus from Observations of Five Minor Planets
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ACTA ASTRONOMICA Vol. 45 (1995) pp. 665±672 Determination of Masses of Mercury and Venus from Observations of Five Minor Planets by G. Sitarski Space Research Center, Polish Academy of Sciences, ul. Bartycka 18a, 00-716 Warsaw, Poland e-mail: [email protected] Received June 8, 1995 ABSTRACT We collected 1985 astrometric observations of the ®ve minor planets: (1566) Icarus, (1620) Geographos, (1862) Apollo, (2101) Adonis and (2212) Hephaistos. The asteroids can closely approach Mercury and Venus. The observations cover the time interval from 1930 for Apollo up to 1995 for Adonis. Equations of motion of the asteroids have been integrated by the recurrent power series including all the planetary perturbations, perturbations caused by the four biggest minor planets and relativistic effects. Joining all the observational equations for the ®ve asteroids we created a set of thirty two normal equations to determine by the least squares method thirty corrections to orbital elements of the ®ve asteroids along with two corrections to the planetary masses. Thus we found the following values of the reciprocal masses for Mercury and Venus in the solar unit mass: 1 1 m m = = 6019522 9964 408522 85 8 80 M V Key words: Planets and satelites: individual: Mercury, Venus ± Minor planets, asteroids 1. Introduction Determination of planetary masses is a dif®cult problem whentheplanet has no satellite. It is the known caseof Pluto that its mass was overestimated until the Plu- ton's moon, Charon, was discovered in 1978. Mercury and Venus have no moons, therefore, masses of these planets can be determined from their disturbing effects in the motion of asteroids closely approaching the planets and observed during long time intervals. Lieske and Null (1969) determined the mass of Mercury from perturbations on (1566) Icarus using the astrometric observations of the asteroid made in 1949±1969; the reciprocal mass of Mercury was then found by the authors as 5934000 65000 in the solarunit mass. Valuesof the planetarymassesas given in different sources (e.g.,JPL DE200, or Astrophysical Data 1992, or IAU/WGAS 666 A. A. Circ. 74) slightly differ from each other. Table 1 contains the planetary masses taken from IAU/WGAS Circ. 74 (Fukushima 1994). Table1 Values of reciprocal planetary masses in the solar unit mass. Mercury 6023600 . Jupiter 1047.3486 Venus 408523 .71 Saturn 3497.898 Earth 332946 .05 Uranus 22902.98 Earth+Moon 328900 .56 Neptune 19412.24 Mars 3098708 . Pluto 135000000 . There is a number of minor planets which can closely approach Mercury and Venus. We chose ®ve such asteroids and used their astrometric observations to correct the masses of Mercury and Venus. Orbital elements of the ®ve minor planets are given in Table 2, and their heliocentric orbits are presented graphically in Fig. 1. Fig. 1. Heliocentric orbits of the ®ve minor planets projected on the ecliptic plane. The dashed curves denote parts of the orbits placed below the ecliptic plane. Vol. 45 667 Table2 Orbital elements of the ®ve minor planets for 1995 Feb. 12.0 ET. Mean anomaly M and angular i a q elements (referred to J2000.0) are in degrees, semimajor axis and perihelion distance are in a.u., period P is in years. a q e i P Asteroid M (1566) 334.84354 1.0780417 0.1867942 0.8267282 31.2207 88.1547 22.8804 1.12 (1620) 163.19005 1.2455288 0.8276824 0.3354771 276.7354 337.3667 13.3403 1.39 (1862) 22.25153 1.4711238 0.6475240 0.5598439 285.6275 35.9347 6.3561 1.78 (2101) 27.33005 1.8739368 0.4406684 0.7648435 42.2841 350.7457 1.3523 2.57 (2212) 69.83467 2.1680971 0.3609173 0.8335327 208.3823 28.4395 11.7818 3.19 We have integrated equations of motion of the minor planets including per- turbations caused by the nine planets and also some subtle disturbing effects: perturbations caused by the four biggest asteroids (Ceres, Pallas, Vesta, and Hy- giea) and relativistic effects. Corrections to planetary masses have been computed together with corrections of orbital elements of the disturbed asteroids. The method is presented in the next Section. 2. Method of Computations Equations of motion of an asteroid have the following vectorial form: R 2 3 2 r _ _ ( r + k + r_ = ) r 1 2 r 2 3 1 c r r where r is the radius vector of the asteroid, k is Gaussian gravitation constant, 2 4 _ c = r_ = r r r R c is the speed of light in a.u./day, 3 1 00069809 10 , , is planetary disturbing function. The solar term in Eq. 1 is modi®ed by including the relativistic effects (Sitarski 1983, 1992). r Let us assume that a small correction is caused by an inaccuracy of initial m orbital elements of the asteroid and of the masses of Mercury M and of Venus m r V as well. Then should satisfy the following differential equation: r r 2 3 2 2 _ _ r + k + r_ k r r = r 1 2 r 2 3 3 5 c r r r r r r R r r M V V 2 M 2 ) r k = + m k + m ( V 3 3 M 3 3 2 r r r r M M V V r r M where M is the radius vector of Mercury, is the distance of Mercury from the Sun, M is the distance of the disturbed asteroid from Mercury: terms in Eq. 2 with subscripts V correspond to Venus. 668 A. A. Substituting to Eq. (2) 6 X r = G E + G m + G m M V k k 7 8 = k 1 E E k where k are corrections to the six orbital elements , we obtain a set of G i = differential equations for the functions i for 1 8 which are to be integrated G together with the equations of motion. Numerical values of i allow us to compute values of coef®cients in the observational equations (Sitarski 1971, 1979b). E To determine six corrections of orbital elements k and two corrections of m m V planetary masses M , by the least squares method, we can solve the set of eight normal equations for each asteroid separately. However, we can join observationalequations for the ®ve asteroids creating a new set of thirty two normal equations with thirty two unknowns. Solving the new set of normal equations we correct the masses of Mercury and Venus along with the orbits of the asteroids. Thus we can improve masses of the planets using all the observations of the ®ve disturbed minor planets together. The above method has been successfully applied when correcting the mass of Ceres from observations of two minor planets (Sitarski and Todorovic-Juchniewicz 1992). 3. Observational Material We chose ®ve minor planets having a possibility to approach closely Mercury and Venus: (1566) Icarus, (1620) Geographos, (1862) Apollo, (2101) Adonis, and (2212) Hephaistos. Numbers of astrometric observations of the asteroids in differ- ent observational seasons are given in Table 3. In Fig. 2 we present graphically Table3 Distribution of astrometric observations of the ®ve minor planets. Asteroid Interval Number of obs. (1566) Icarus 1949 June 27 ± 1994 June 17 734 (1620) Geographos 1951 Aug. 31 ± 1994 Nov. 30 949 (1862) Apollo 1930 Dec. 13 ± 1989 Dec. 29 144 (2101) Adonis 1936 Feb. 12 ± 1995 Mar. 6 68 (2212) Hephaistos 1978 Sep. 27 ± 1994 Nov. 8 90 T o t a l 1930 Dec. 13 ± 1995 Mar. 6 1985 Vol. 45 669 Fig. 2. Plots of distances of the ®ve minor planets from Mercury and Venus during the observation periods of the asteroids. In the time-axes we marked distributions of the minor planets' observations. 670 A. A. minimum distances of the asteroids from Mercury and Venus, marking also the distribution of observations on the time-axes of the plots. We can see that Apollo and Adonis have the longest observation intervals but the number and distribution of observations is unfavorable to the planetary mass determination. Apollo was discovered in 1932 and rediscovered again in 1973, but three positions of that as- teroid made in December1930 and one made in March 1957 were measured on old plates taken at Lowell Observatory. Adonis discovered in 1936 was rediscovered in 1977, and no additional observation was made during the forty-year observation break or before 1936. Icarus and Geographos have a great number of observations covering well the observation intervals. Hephaistos has relatively short observation interval, however, in April 1991 it approached Mercury to within 0.049 a.u., and it was the shortest distance between Mercury and the ®ve considered asteroids. 4. Numerical Results For orbital elements of the ®ve minor planets we chose the common osculating epoch1995 Feb.12.0ET. We haveintegrated backwardsthe equations of motion (1) as well as the variation Eqs. (2) by the recurrentpowerseries (Sitarski 1979a). Thus we obtained observationalequations (for each minor planet separately) necessary to compute corrections of orbital elements of the asteroids and corrections of masses of Mercury and Venus by the least squares method. Table 4 shows the minimum distances less than 0.1 a.u. reached between four asteroids and two planets during the observation intervals. We can see that ± from the point of view of close approaches to the planets ± e.g., Apollo is very good for determination of the mass of Venus whereas it is justi®ed to use Hephaistos for correction of the mass of Mercury. In the case of Geographos, the minor planet approachedVenus several times, and although the minimum distances to that planet were about 0.2 a.u.