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level. For some bright stars, small apparent diameters of the order of 0.005 are large enough to smear out the first minimum of light. Larger apparent diameters will change the slope of the lightcurve. Figure 1 shows one of our curves obtained at Meudon Observatory on the star e-Capricorni, 4.7 mag, with a telescope of 60 cm diameter. A telescope of 200 cm produces the same noise level with a star of 7.3 mag. The brightest asteroids are expected to have apparent diameters larger than 0.05, thus permitting a relaxation of the time resolution on the order of 30 times; the same noise level is obtained with a gain of brightness of 3.8 mag, and the technique reaches a magnitude of about 11, provided by at least 15 asteroids. Further decrease of time resolution could be tried and larger telescopes used. This interesting technique has not yet been used for asteroids.
The asteroidal diameter determinations currently available are summarized in table II.
The double-image micrometer, the interferometer, the diskmeter, and the lunar occultation photometry are suitable techniques available for refinement of these determinations and for their extension to a larger number of objects.
Barnard, E. 1902, On the Dimensions of the Planets and Satellites. Astron. Nachr. 157,
VEVERKA: You quoted the diameter measurements of the first four asteroids made by Barnard with a filar micrometer. The apparent diameter of Juno was given as 0.3. How meaningful, in your opinion, are such measurements when even at best “seeing” causes a smearing of 0.2 to 0'3? I realize that Barnard's measurements are internally quite consistent, but I would like to have an estimate of the absolute uncertainty involved, including systematic errors.
DOLLFUS: One of the purposes of my presentation was precisely to warn about the difficulty of the method. Barnard was a very experienced observer for filar micrometry, and was well aware of all the difficulties of the problem. However, we should not overestimate the accuracy of his measurements, and systematic error of 0.10 could be considered as highly probable. This is why the Barnard results, despite the high training and reputation of the author, should be checked when possible by other techniques. GEHRELS: This does raise a problem. People like Barnard who did this type of work must have had a feeling of their precision and if I take it strictly from what you are saying then it would also concern Barnard's measurements of Ceres and Vesta. Should we not be allowed some confidence in these measures? DOLLFUS: He did observations during several long periods of time but the gist lies in the systematic errors and against them accumulation of data does not really improve the result. The point is that the measures are distorted; one can in some respects simulate the conditions at the laboratory and study the systematic errors. GEHRELS: In these measurements can you make a comment on which way the error might go? Should we expect measurements that are too large or too small? DOLLFUS: Uncorrected filar measurements are too large; at the limit, a point source gives the apparent diameter of the diffraction pattern blurred by atmospheric turbulence. KUIPER: Certainly Barnard should be considered a very keen observer. He was aware of what the atmosphere did to spoil the image and he would not use a night if it was not good. I have confidence in his diameters; the absolute errors would be somewhat similar to the case of close binaries, and guessed to be 10 percent or perhaps from 20 to 25 percent, but they are real measures. DOLLFUS: I was not clear in my statement. For disks with diameters of the order of the size of the diffraction pattern, a correction is to be applied. This is precisely the case of the asteroids; but it is difficult to judge what is the contribution of the diffraction if one includes the blurring due to atmospheric turbulence. KUIPER: The diffraction was 0.12 and the overall blurring not more than 0.2. What counts is the square root of the sum of the squares, and therefore the diffraction is the least important in the final value. Certainly it has all the uncertainties that we are all aware of and Barnard was aware of. Values of the order of 0.5 would be essentially correct. Values smaller than 0"3 I would consider dubious. CHAPMAN: There has been one recent test of the methods that have been used and described by Dollfus and Kuiper for determining asteroid diameters. Dollfus has summarized the modern attempts that have been made to determine the diameter of Neptune using these same methods. (See Dollfus, 1970.) All attempts to measure the diameter during the last half century prior to the recent stellar occultation by Neptune agreed with each other but differed from the correct value by more than 10 percent. The 2.5 diameter of Neptune is an order of magnitude larger than the largest asteroid diameters. Considering, in addition, that smearing by diffraction and seeing is of the same order as asteroid diameters, the failure of these methods to reach higher accuracy than 10 percent on Neptune (even taking account of the problems of limb darkening) casts doubt on any direct diameter measurements of asteroids. DOLLFUS: The problem of the diameter determinations of Neptune is of a different nature: The limitations in Neptune measurements are altogether the lack of brightness and the limb darkening. Most of the earlier diameter measurements of Neptune concluded too small diameters because of the limb-darkening effect on a disk of very low brightness. However, our last telescopic measurements were obtained with the new Pic-du-Midi telescope, which gives three times more brightness than the previous one; a value of 2.23 was obtained, almost 10 percent larger than the previous measurements. This last value was considered by the observers as far better than the older. The value was later compared with the star occultation result; the agreement is within 3 percent and unexpectedly accurate in view of the difficulty of measuring so dark an object. The case of minor planets is not the same. Observations are no longer limited by the lack of brightness; the average surface brightness is at least 30 times larger, with no limb darkening; the measurements, in this respect, are far easier. However, the diameters being smaller, careful laboratory analysis of the systematic errors introduced by the spreading of the images should be carried out for final evaluation.
Dollfus, A. 1970, Diamétres des Planètes et Satellites. Surfaces and Interiors of Planets and Satellites, ch. 2, pp. 45-139. Academic Press, Inc. London and New York.
ASTEROID MASSES AND DENSITIES
Before 1966, when Hertz (1966) published his first direct determination of the mass of Vesta, all our knowledge on asteroid masses was based on estimates. The masses of the first four minor planets resulted from the measured diameters by Barnard (1900) (see the paper by Dollfus in this volume") and from estimated mean densities. The diameters of the smaller objects were derived from their brightness and an estimate of their reflectivity (usually the reflectivity of the Moon was adopted). In 1901, Bauschinger and Neugebauer (1901) derived a value for the total mass of the first 458 asteroids. All the diameters were computed from the brightness with an assumed value for the reflectivity. The diameter of Ceres found in this way is very close to Barnard's (1900) value. The mean density of the 458 asteroids was put equal to that of Earth, and their total mass resulted as 3 X 10-” solar mass. Stracke (1942) used the same method with an increased material, but the addition of more than 1000 faint asteroids did not bring a significant change in the estimate of the total mass. The report on the McDonald asteroid survey (Kuiper et al., 1958) does not contain another estimate of the total mass of the asteroid ring, but it points to the possibility of a very rapid increase in the number of asteroids with decreasing absolute brightness. If this increase is strong enough, each interval of 1 mag in absolute magnitude can contribute the same amount to the total mass. In the range of magnitudes covered by the Palomar-Leiden survey (PLS) (van Houten et al., 1970), there are no indications for such a strong increase.
The attempts to find gravitational evidence on asteroid masses started with the total mass, but von Brunn (1910) demonstrated that at his time it was not possible to detect gravitational effects caused by the total mass of the asteroids. In the paper mentioned above, Stracke (1942) expressed the hope that accurate orbital theories of the first four minor planets and of Eros can answer the question of the gravitational effects of the total mass, if these theories are compared with the observations of a sufficiently long interval of time.