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Coyne, G. V., and Gehrels, T. 1967, Interstellar Polarization. Astron. J. 72,888.
Dunlap, J. L., and Gehrels, T. 1969, Lightcurves of a Trojan Asteroid. Astron. J. 74, 796-803.
Dunlap, J. L., and Gehrels, T. 1971, Minor Planets and Related Objects VIII. Astron. J., to be published. -
Houten, C. J. van. 1963, Uber den Rotationslichtwechsel der kleinen Planeten. Sterne Weltraum 2,228-230.
Roach, F. E., and Stoddard, L. G. 1938, A Photoelectric Lightcurve of Eros. Astrophys.J. 88.305-312.
BANDERMANN: Is there any obvious reason why the Aml of successive minima in the lightcurves is usually larger than the Aml of successive maxima”
DUNLAP: Perhaps small differences in surface reflectivity are relatively more important at low than at high levels of brightness.
AL N: I think laboratory work of this kind is very important. It is so healthy to see in the laboratory what is correct in theory and how many different solutions we can have. The theoretical models that are used always imply a number of assumptions that may not be applicable in nature.
624 HEKTOR: A BINARY ASTEROID?
A. F. COOK
Dunlap and Gehrels (1969) have published lightcurves of the Trojan asteroid 624 Hektor. They proposed a conventional explanation in which Hektor is regarded as having the shape of a cigar. Two circumstances suggest, but do not prove, that Hektor is a binary asteroid. (1) The cigar shape at the conventional density of stony meteorites (3.7 g cm-3) appears to produce stresses that may well exceed the crushing strength of meteoritic stone. (2) The lightcurves exhibit an asymmetry changing with time that suggests librations of two ellipsoidal components. Observations are clearly required to look for these periodicities when we shall again be nearly in the plane of Hektor's revolution (or rotation) in 1973. An additional supporting lightcurve is desirable in 1972 and also in 1974. The periods of libration are probably nearly 1 day, if they exist, so that observations should be made from more than one geographic longitude in 1973. The present paper is an exposition on these considerations.
THE CIGAR-SHAPED MODEL
Dunlap and Gehrels (1969) employed a geometric albedo of 0.28 + 0.14 and a cigar shape consisting of a right circular cylinder capped by two hemispheres at the ends. The radius of the cylinder and of the hemispheres is 21 km, and the height of the cylinder is 70 km. Mathematical convenience is served by replacement of this model by an ellipsoid of Jacobi with the same ratio of end-on to side-view cross sections. The ratio of the intermediate semiaxis to the largest semiaxis is as follows:
A convenient graph for finding the ratio of the smallest semiaxis c to the largest has been published by Chandrasekhar (1965). His figure 2 (p. 902) yields
The density of this ellipsoid at which equilibrium occurs so that no stresses are applied, i.e., so that the pressure is everywhere isotropic, can also be found
from another graph by Chandrasekhar (1965, fig. 3, p. 903). The abscissa in this case is Arccos (c/a) = 77°, whence the ordinate is Q2 TGpe
P is the period of rotation (2.492 X 10° s according to Dunlap and Gehrels, 1969), and G is the universal constant of gravitation. Solution for the density pe of the asteroid in equilibrium yields 1.7 g-cm-3. It follows that if Hektor is a single body, either it is of lower density than a carbonaceous chondrite of type I or it is not in equilibrium.
STRESS IN THE CIGAR-SHAPED MODEL
Computation of a representative stress at the density of meteoritic stone is required to assess the viability of the Jacobi ellipsoid as a large meteoritic stone. Jardetzky (1958) provides the appropriate mathematical discussion. His equation (2.2.13) on page 31 can be transformed to read
- = L., - - L. = L - – - L. = (1)
G 2 b2 " " a nGo where p without subscript refers to the actual density, and the potential takes the form
where x is taken along the largest semiaxis, z along the shortest, and y along the intermediate one; the origin lies at the center of the ellipsoid; and C’ is an arbitrary constant. Poisson's equation takes the form
Solution of equations (1), (2), and (3) for L2, Ly, and Ly yields
where Ci/G is the pseudopotential. The pressure p, at the surface is given by
where x, y, z, refer to a point on the surface. At the equilibrium value of the density pe. p. vanishes all over the surface.
We compute the difference due to a different value of p and consider only the
differences in pressures along the principal axes, whence
In terms of a, b, c, and Q2/1Gp, these expressions become
Next we subtract the hydrostatic part or mean to find
1 Ap, - - 92(p- p.) (1 + —H |a *e 3 (p − pe 1 + a2/b2+ a2/c2 These are the hydrostatic pressures that would be required on the surface to keep the internal pressures isotropic. In their absence, an anisotropic pressure will appear at the center with signs opposite to those in equations (12). At p = 3.7 g cm-3 as for meteoritic stone, we have
(or 7.6 × 10−8 a2, -3.3 x 10-8 a”, and -4.6 x 10−8 a” dyne-cm-2, respectively). This loading resembles that in a conventional unidirectional compression test of
P - 12 nN-m-2 (14)
(or 1.2 x 10−7 a2 dyne-cm-2). The cross sections in side view and end-on of Dunlap and Gehrels' (1969) model impose a = 77 km whence P = 0.7 MN-m” (7 bars), compared with a crushing strength of about 1 MN-m-2 (10 bars) for the Lost City meteorite (R. E. McCrosky, private communication, 1971). A geometric albedo of 0.14 (as for the brightest parts of the Moon) makes Hektor larger in dimension by a factor of V2, whence P = 1.4 MN-m-2 (14 bars). Finally, a geometric albedo of 0.07 (upper limit for the dark part of Iapetus according to Cook and Franklin, 1970) introduces a further factor of V2 in dimension and raises P to about 3 MN-m-” (30 bars). A large body like Hektor will have weak inclusions and thus have a lower bulk strength than a small body like the Lost City meteorite. Moreover, meteoritic bombardment will tend to induce failures as well. All this casts doubt on the model of Dunlap and Gehrels (1969) and suggests that a binary model may be more satisfactory.