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Figure D-1.-Diameter distribution of craters of Mars compared with diameter distribution predicted for craters formed from asteroids in the PLS (upper two solid curves) and present-day Mars-crossing asteroids (lowest curve). Upper solid and dashed curves show how saturation effects would truncate the Mars crater distribution. Nonetheless, the Mars crater counts from 50 to 200 km show a marginal trace of the “flat spot” in the asteroid size distribution, suggesting Mars craters are caused by impacts of Marscrossing asteroids over the last 4 aeons.

by Hills," had already been disturbed, which is reasonable because the original mass distribution probably would have been changed by fragmentation by the time Mars had accreted. It does leave one interesting point: We apparently do not see all the way back to the beginning of Mars’ history (i.e., to its accretionary phase), as seems to be the case with the Moon.

LINDBLAD: Alfvén (Alfvén and Arrhenius, 1970) has suggested that an asteroidal belt may exist in the Jovian satellite system. By analogy with the formation of our planetary system it appears likely that an asteroid or “minor-moon” belt exists between the orbits of the Jovian satellites Amalthea (no. 5) and Io (no. 1). It is suggested that a special effort be made during a Grand Tour Jupiter flyby mission to detect such a belt, and measure parameters of interest; i.e., particle spatial density, size, and composition.

"See p. 225.

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A Jupiter-Saturn-Pluto mission with a Jupiter flyby trajectory passing about 100 000 km inside the orbit of the satellite Io would be desirable for the Jovian “asteroid belt” experiment.

According to a recent JPL study, it is feasible to intercept the Jovian satellites Io, Ganymede, and Callisto in one single 1977 flyby mission. However, it is known that these satellites are rather similar in their physical characteristics. Alfvén (1971, private communication) has pointed out the desirability of including in the flight plan (if feasible) also a near encounter with one of the outer Jovian satellites (nos. 6, 7, and 10). It is suggested that feasibility studies for a Jupiter flyby mission including one of the outer Jovian satellites as well be considered.

A similar situation may exist in the Saturnian satellite system where a concentration of small particles may exist between satellites 5 and 6.

DISCUSSION REFERENCES

Alfvén, H., and Arrhenius, G. 1970, Structure and Evolution of the Solar System. Astrophys. Space Sci. 8, 338.

Kaula, W. M. 1966, Theory of Satellite Geodesy. Blaisdell Pub. Co. Waltham, Mass.

Kuiper, G. P. 1956, On the Origin of the Satellites and the Trojans. Vistas in Astronomy (ed., Arthur Beer), vol. 2, pp. 1631-1666. Pergamon Press. New York.

TROJANS AND COMETS OF THE JUPITER GROUP

EUGENE RABE
Cincinnati Observatory

In a recent paper (Rabe, 1970), I suggested that at least some comets of the Jupiter group may have originated from the relatively dense Trojan clouds that, according to a recent survey by the van Houtens and Gehrels (1970), seem to be associated with the equilateral points L4 and L5 of the Jupiter orbit. This suggestion was inspired by the finding that the periodic comet SlaughterBurnham has been captured (or perhaps recaptured) into unstable or temporary “Trojan” librations lasting approximately 2500 yr and by the circumstance that the Jacobi “constants” of most Jupiter group comets have values between 3.0 and 2.5, or just in that range which would also be occupied by all known and unknown Trojans associated with Jupiter in stable or unstable librations involving heliocentric eccentricities up to about 0.5 and inclinations as large as 30°. Moreover, it is well known that nearly all comets of the Jupiter group are able to approach Jupiter rather closely, thus providing the possibility for such drastic orbital changes as temporary capture into, or escape from, librational motion of the Trojan type. All, presumably stable, orbits of actual Trojan planets presently known have eccentricities e not exceeding 0.15, but we also know that stable short-period librations with much larger e values do exist in the restricted Sun-Jupiter problem, so that even in the real, nonrestricted situation there should be a possibility for corresponding librations with these more substantial short-period components. These librations may be unstable but may have long lifetimes nevertheless. The case of P/Slaughter-Burnham has proved that such motions are indeed possible, even with an e as large as 0.52.

The Jacobi integral is valid only in the restricted three-body problem, but the experience from many numerical integrations indicates that in the real Sun-Jupiter elliptic problem, with e' -- 0.05, the Jacobi “constant” C from the approximating Tisserand criterion,

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tends to vary only within relatively narrow limits, as long as the osculating elements used in evaluating equation (1) do not belong to some moment at which the small body in question is inside of Jupiter's gravitational “sphere of action.” In equations (1) and (2), a is the semimajor axis of the Trojan's or comet's heliocentric orbit expressed in units of the mean Sun-Jupiter distance, © is related to e through e = sin (), and I denotes the orbital inclination relative to the Jupiter orbit as computed from the respective ecliptical inclinations i and i' and nodes Q and Q' according to equation (2). The prime symbol denotes the elements of Jupiter.

For the overwhelming majority of all minor planets, CP 3 and a < 1. To the exceptions with C33 belong all the known Trojans (because of their small deviations from ao = a' = 1 in combination with nonvanishing I and e), several members of the Hilda family with their also relatively large a values of the order of 0.8, and some asteroids with smaller values of a but with such exceptionally large values of e or/and I, as to enable the second term of the right-hand side of equation (1) to become sufficiently small. If the auxiliary angle Y is introduced through

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equation (1) can be solved to express cos y as a function of the nearly constant C and of the variable a. Assuming that at some time the value a0 = 1 can be attained by a, in consequence of Jupiter's perturbing action, the resulting function cos y reduces to

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This equation contains the well-known statement that the attainment of a0 = 1, and thus a “crossover” from a < 1 to a > 1 or vice versa, is impossible if C>3. Clearly, CP 3 leads to cos Yo P 1 and thus to either cos @ P 1 or cos I X 1, if not to both inequalities together. Actually, because terms of the order of Jupiter's mass, p * 0.001, have been neglected in equation (1), even if applied to the restricted problem with e' = 0, the subsequent equation (4) proves the impossibility of crossovers only for C values that exceed 3.000 by amounts larger than some quantity of order pl. Conservatively, one may require CP 3.01. Considering also Jupiter's orbital eccentricity e' - 0.05, however, in conjunction with the resulting slight variability of C (as evidenced in many relevant numerical integrations), the critical C limit for the possible occurrence of ao = 1 has to be increased even more. Indications are that the effective limit (in the absence of significant perturbations from other major planets) lies near C = 3.03. It also appears that for any asteroids or comets within 3.00 < C < 3.03, any crossovers would tend to happen through temporary capture into satellite rather than Trojan status, as evidenced by the C values that one finds for the many sets of elements a, e, and I obtained by Hunter (1967) in his work on satellite/asteroid transfers. For any comets with

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