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THE BINARY MODEL
The lightcurves of Gehrels (Dunlap and Gehrels, 1969) are a heterogeneous lot obtained with different telescopes, photometers, and skies. The best quality in observations occurs at the largest and smallest amplitudes. The largest amplitude was observed on April 29 and May 1, 1968, with the 152 cm reflector at Cerro Tololo. The zero point of magnitude was obtained only on the second night. The smallest amplitude was observed on February 4, 1965, with the 213 cm reflector at Kitt Peak.
The author has carried out an analysis that can be called only a reconnaissance. An unusual amount of work has been required compared to the usual solution for an eclipsing binary. Interim light elements were derived to plot the lightcurves of 1965 and 1968 against phase. There is no evidence for any differences between successive half-periods, so each night's observations were plotted on a single half-period. A notable feature of the 1968 observations is an asymmetry such that the maxima occur 0.012 period early. The descent into the minimum is slower than the rise from it. In 1957 this asymmetry appears to be at the limit of detection but in the opposite direction. The 1957 observations are thin and were made at the Radcliffe Observatory, Pretoria, at the 188 cm reflector.
The most obvious explanation for the asymmetry is libration of the components about the radius vector joining them. This hypothesis can be tested by extensive observation at the next opposition in which Earth is near the plane of Hektor's revolution (or rotation).
The libration was taken into account in the rectification. The formula used for rectification of the intensity was
IR = —— (15) [1 - 2 cos” (0-60)%
where I is the observed intensity, Po the rectified intensity, z the photometric ellipticity, 6 the phase angle, and 60 the phase angle at which we look along the major axes of the components. We use here the standard preliminary model of two similar ellipsoids similarly situated. Rectification for phase was effected by the formula
26 sin? 0 (16)
sin - - 1 1 - z cos” (6 - 90)
where 6 is the rectified phase angle. Solution for z in the standard graphical plot of 12 versus cos” (0-90) employed
Execution of the usual procedures using the tables of X functions of Merrill (1950) produced the following solution:
where lo is the brightness at minimum in units of that outside eclipse, k is the ratio of radii of the components, a, is the largest semiaxis of the larger component in units of the distance between the asteroids, and i, is the inclination of the relative orbit in the rectified model in which the components are spheres. The most extreme solution pointing to the highest density of the components is that for equal bodies, k = 1.00. In this case a, = 0.38 and i, = 70°1. The lightcurve for February 4, 1965, shows an apparent elliptic variability only with z = 0.154 + 0.002. The unit of intensity corresponds to an absolute magnitude of 7.63 compared with 7.70 for May 1, 1968. Combination of the results from 1968 and 1965 yields for the inclinations of the orbit and the ratios of principal axes the following results:
Here k = 1.00 has been chosen to present the most extreme case. The mean density of the components comes to 9.6 g cm-3. Correction for finite sizes of the components in ellipsoidal shape yields a drop of a few tenths, and adoption of k = 0.60 would push the density down to that of iron, about 8 g cm-3. Before one blithely proposes that Hektor is a binary composed of two solid iron ellipsoids, it seems only prudent to question the rectification, which is large. The extreme form of doing this is to consider a contact binary as a
In the case of a contact binary, the rectification cannot be considered separately. In this reconnaissance, a start was made by assuming (1) that sin? 6, = 1 where 6, denotes the rectified phase angle at external contact; (2) that the ratio of radii k = 1; and (3) that the fractional depth of greatest eclipse oo = 1; i.e., that central eclipse occurred. This representation failed, so that two progressions away from this model were next considered. One such sequence of models had k = 1 but oo decreased successively. No models fitting the observations could be found. Next, a sequence of grazing annular and total eclipses was tried (ao = 1), so that k was varied. This sequence yielded a satisfactory representation of the observations at k = 0.80:
Correction for finite sizes of components reduced p to 2.0 g-cm-3.
The two directions of Hektor in 1965 and 1968 lie some 110°4 apart, whence it appears that the pole of revolution of the binary must lie near the plane of Hektor's heliocentric orbit. The librations will cause the epochs of the shallower minima to be not very reliable in derivations of this pole.
Better light elements have been derived by bootstrapping across successively longer intervals on the assumption that the pole lies in the plane of the orbit. The corresponding sidereal period is
P = 042884.483 + 0.0000002
The ellipsoids can be replaced individually by two point masses of spheres to yield the same moment of inertia about the smallest semiaxis. Then the period of libration can be calculated in the field of the other represented as a point mass. The results indicate a period of libration of two to four periods of orbital revolution. The near equality of the heights of maxima in 1968 during two long nights at a 2 day interval strongly suggests periods of about 1 day for the librations.
Observations can only be planned efficiently if the above analysis is completed by using many values of the photometric ellipticity z for 1968. It is to be expected that there will be two groups of solutions—those at high densities described above and belonging to a narrow range of z near 0.745 and those at low densities and belonging to a wider range of z near 0.60. It is to be hoped that this latter group will reach up to or approach more conventional densities like that of meteoritic stone.
Observations will be needed in 1972, 1973, and 1974 to settle the choice between the cigar-shaped and binary models by seeking the periodicities in the librations and to improve the accuracy of the photometric solutions either by observation of annular and total eclipses or by observation of very deep partial eclipses on either side of the orbital plane (or equatorial plane). This implies an extensive campaign in 1973 at one observatory coupled with an international campaign during the dark of the Moon closest to opposition. The best available range of photometric solutions will be required for intelligent planning of the extensive campaign at one observatory. The international campaign would be aimed at covering the suspected 24 hr periods of the librations. Good lightcurves at single epochs would be desirable in 1972 and 1974.
It is a pleasure to acknowledge useful and extensive discussions with B. G. Marsden and F. A. Franklin.
Chandrasekhar, S. 1965, On the Equilibrium and Stability of the Riemann Ellipsoids. Astrophys.J. 142, 890-921.
Cook, A. F., and Franklin, F. 1970, An Explanation of the Light Curve of Iapetus. Icarus 13, 282-291.
Dunlap, J. L., and Gehrels, T. 1969, Minor Planets III. Lightcurves of a Trojan Asteroid. Astron. J. 74, 796-803.
Jardetzky, S. 1958, Theories of Figures of Celestial Bodies. Interscience Pub., Inc. New York.
Merrill, J. E. 1950, Tables for Solution of Light Curves of Eclipsing Binaries. Contrib. no. 23, Princeton Univ. Observ.
HARTMANN: I wish to make a comment on irregular shapes of asteroids. Cook's evidence that Hektor would not retain an irregular shape rests on the crushing strength he assigns to the material. It appears that Cook's value of 1 MN-m” (10 bars), based on the Lost City chondrite, is unusually low. Wood (1963) lists compressive strengths of eight chondrites; they range from about 6 to 370 MN-m-2 (60 to 3700 bars) although Wood notes that some more crumbly chondrites are known. The one iron listed has a strength of about 370 MN-m-”. Thus, according to the 0.7 MN-m-2 (7 bar) stress found by Cook for a Jacobi ellipsoid of Hektor's dimensions and chondritic density, the asteroid could be quite irregular.
How large an asteroid can be irregular? A simple estimate comes from the size of a nonrotating spherical asteroid whose central pressure P is just equal to the crushing strength. Under this condition the central core begins to be crushed and hence lacks rigidity. Larger asteroids would have a nonrigid interior and could thus deform to an equilibrium shape. For typical chondritic strengths we have
Thus, the diameter = 46 to 880 km (if p = 3.7).
It is concluded that asteroids substantially larger than Hektor (42 by 112 km) can be irregular in shape. Such irregularity is indeed evidenced by lightcurves and is theoretically expected because many if not most asteroids are probably fragmentary pieces that have resulted from collisions.