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The poles of asteroids 39 (Sather and Taylor, 1972), 624 (Dunlap and Gehrels, 1969), 1566 (Gehrels et al., 1970), and 1620 (Dunlap and Gehrels, 1971) are determined from photometric astrometry. My present photometric astrometry determination of the pole of Eros was done utilizing the observational data of Beyer (1953) for the opposition of 1951-52. This is a preliminary value from the one opposition only.

CRITICAL SUMMARY AND CONCLUSIONS

The precision of Zessewitsche's pole is dependent on the accuracy of the micrometer measurements of the position angle and the value for the inclination of the equator. Zessewitsche recognized the lack of precision in the latter. The poles of Watson, Rosenhagen, and Krug depend on the precision of values determined for the absolute brightness and amplitude obtained in many oppositions. This precision is poor. Stobbe and Beyer used more precise data within oppositions, but had to choose a “best” pole from among several possible poles.

With increased observational precision in recent years, we might expect more reliable pole determinations. We see from table I, however, that little agreement exists. Discounting the apparently incorrect poles of Chang and Tempesti, we still see great disagreement between the poles of Cailliatte and Gehrels.

It may be concluded, then, that we must challenge the fundamental validity of the amplitude-aspect relationship upon which great doubt has already been cast by J. L. Dunlap” and seek a more reliable way to determine a precise pole position. Photometric astrometry shows great promise, and we are currently engaged in a program using and further improving that method.

ACKNOWLEDGMENTS

This program is supported by the National Aeronautics and Space Administration.

REFERENCES

Beyer, M. 1953, Der Lichtwechsel und die Lage der Rotationsachse des Planeten 433 Eros wahrend der Opposition 1951-52. Astron. Nachr. 281, 121-130.

Bos, W. H. van den, and Finsen, W. S. 1931, Physical Observations of Eros. Astron. Nachr. 241, 329-334. r

Cailliatte, C. 1956, Contribution à l'Etude des Astéroïdes Variables. Bull. Astron. Paris 20, 283-341.

Cailliatte, C. 1960, Contribution a I'Étude des Astéroïdes Variables (Suite). Publ. Observ. Lyon 6(1), pp. 259-272.

Chang, Y. C., and Chang, C. S. 1962, Photometric Investigations of Seven Variable Asteroids. Acta Astron. Sinica 10, 101-111.

2See p. 151.

Chang, Y. C., and Chang, C. S. 1963, Photometric Observations of Variable Asteroids, II.
Acta Astron. Sinica 11, 139-149.
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. 76,
in press.
Gehrels, T. 1967, The Rotation of Vesta. Astron. J. 72, 929-938.
Gehrels, T., and Owings, D. 1962, Photometric Studies of Asteroids. IX. Additional
Light-Curves. Astrophys.J. 135,906-924.
Gehrels, T., Roemer, E., Taylor, R. C., and Zellner, B. H. 1970, Asteroid (1566) Icarus.
Astron. J. 75, 186-195.
Huruhata, M. 1940, The Rotation Period of Eros. Harvard Col. Observ. Circ. 442, pp. 1-6.
Krug, W., and Schrutka-Rechtenstamm, G. 1936, Untersuchungen über Gestalt und Grösse
des Planetoiden Eros. Z. Astrophys. 13, 1-12.
Roach, F. E., and Stoddard, L. G. 1938, A Photoelectric Light-Curve of Eros. Astrophys.
J. 88,305-312.
Rosenhagen, J. 1932, Einige Bemerkungen zur Helligkeit und zum Lichtwechsel des
Planeten Eros. Mitt. Wien. Sternw. 1(2), 45-52.
Sather, R. E., and Taylor, R. C. 1972, Minor Planets and Related Objects. IX. Astron. J.,
in preparation.
Stobbe, J. 1940, Der Lichtwechsel des Eros. Astron. Nachr. 270, 1-24.
Tempesti, P., and Burchi, R. 1969, A Photometric Research on the Minor Planet 12
Victoria. Mem. Soc. Astron. Ital. 40, 415-432.
Watson, F. 1937, The Physical Nature of Eros. Harvard Col. Observ. Circ.419, pp. 1-14.
Zessewitsche, W. 1932, Die Bestimmung der Winkelelemente der inneren Bewegung von
Eros. Astron. Nachr. 246,441-450.
Zessewitsche, W. 1937, On the Rotation of Eros. Observatory 60,289-292.

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A simplified lightcurve inversion method is applied for the special case where observations are taken in the equatorial plane of the asteroid. The solution is obtained in terms of a spotted two-surface model using Lambert's law and geometrical reflectivities.

The general problem of interpreting the lightcurve of a rotating body in terms of its shape and surface spottiness has been discussed in detail by Russell (1906). However, in the special case where the rotational axis is perpendicular to the line of sight, the analysis may be greatly simplified. Although the ambiguity between the shape and spot contributions to the light variation remains unresolved, it is possible to examine the type of surface reflectivity law and to set some limits on the range of albedo variation that will be consistent with the observed lightcurve.

Without loss of generality (insofar as being able to reproduce the observed lightcurve is concerned) we can assume the asteroid to be spherical in shape. The surface is taken to consist of bright and dark areas that reflect either geometrically (a cos ?) or diffusely according to Lambert's law (c. cos” 7) where y, defined by

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is the angle between the outward normal of a surface element and the line of sight in the polar coordinates centered on the asteroid. The polar angle and longitude of the sub-Earth point are designated by 60 and 40, respectively. In the special case when the observer is in the equatorial plane of the asteroid, the integration over the visible hemisphere is greatly simplified, and we can write the brightness variation of the asteroid as

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where g(@o) is the ratio of reflected to incident light; B is the normal albedo of the geometrically reflecting surface area; A is the normal albedo of the diffusely reflecting area; and h(p) is the spot distribution function that gives, as a function of longitude, the fractional area that reflects diffusely according to the Lambert law. Because no information regarding the latitude distribution of bright and dark areas appears in the lightcurve when the asteroid is viewed from within its equatorial plane, h(o) is taken to be constant with latitude. Assuming that h(p) can be expressed in the form

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equation (1) can be integrated to obtain a Fourier series in bo. By comparing the resulting Fourier coefficients with the corresponding terms obtained from a Fourier analysis of the observed lightcurve, we find that the coefficients for the cos nto terms are related by

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where the Cn are the Fourier coefficients obtained from the observed lightcurve and the an are the coefficients defined in equation (2). The same relationships apply for the sin nôo terms. The above set of relations contains the available information regarding the relative proportion and longitude distribution of geometrically and diffusely reflecting surface areas and the range of albedo combinations that are compatible with the observed lightcurve. The limits for the allowed albedo range are imposed by the physical requirement that the spot distribution function h(p) must not become negative or exceed unity.

Because of the infinity of possible solutions, it appears best to consider families of solutions for constant A/B ratios. By specifying a ratio for A/B and by setting ao = 0.5, we define a model for which the surface is evenly divided between geometrically and diffusely reflecting areas. (See fig. 1.) This allows for the greatest amplitude fluctuation for h(?) and marks the approximate center of the allowed albedo range for the specified A/B ratio. The locus of these points falls along the broken line shown in figure 2. Then, keeping the A/B ratio fixed, increasing (or decreasing) A and B simultaneously until h(?) becomes negative (or greater than unity), establishes the range of albedo combinations that are compatible with the physical restriction imposed on h(p).

This procedure defines two separate albedo regions—one corresponding to bright spots, the other to dark spots. Because all albedo combinations in a given enclosure are equally capable of reproducing the observed lightcurve to the same degree of accuracy, it is clearly impossible to differentiate between bright spot and dark spot models on the basis of the observed lightcurve alone.

The size and location of the limiting enclosures depends both on the size of the lightcurve coefficients and on the proportionality factors appearing in relations (3) through (7). Generally speaking, each additional Fourier term that is included to approximate the observed lightcurve tends to diminish the size of the allowed albedo region. However, because of observational scatter, the higher order Fourier terms become increasingly unreliable. This is an important factor because the n = 1 and n = 2 terms contain contributions from both geometrically and diffusely reflecting areas. It is only on the basis of terms

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O” 60° 120° 180° 240° 3OO” 360°
LONGITUDE

Figure 1.-The fractional area that reflects diffusely according to Lambert's law. For 4 Vesta, h(p) includes Fourier terms up to n = 4. For 39 Laetitia, terms up to n = 2 are included. In both cases ao = 0.5.

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