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SCHUBART (in reply to a question by Roosen): My estimate of the total mass of the asteroids refers to the observable objects. The mass contribution of the unobservable small asteroids with a diameter of less than 1 km is unknown.

KIANG: I may point out, that many decades ago attempts were made by Harzer to determine the total mass of the asteroid ring from gravitational effects using the perturbations on the orbit of Mars. A rather large, but extremely uncertain value of about one-tenth the mass of Earth was obtained.

SCHUBART: Harzer made his determination before the effects of relativity became known. It is, therefore, not based on real effects due to the asteroids; compare also von Brunn's (1910) work. We should not use gravitational determinations of the total mass, unless they are confirmed with modern computing techniques.

DISCUSSION REFERENCES

Brunn, A. von. 1910, Über die Masse des Planetoidenringes. Schr. Naturforschenden Gesellschaft Danzig, Neue Folge 12(4), 101-148.

Gehrels, T., Roemer, E., Taylor, R. C., and Zellner, B. H. 1970, Minor Planets and Related Objects. IV. Asteroid (1566) Icarus. Astron. J. 75, 186-195.

Schubart, J. 1970. The Planetary Masses and the Orbits of the First Four Minor Planets. Paper presented at IAU Colloquium no. 9. (1971, Celest. Mech., to be published.)

THE METHOD OF DETERMINING INFRARED DIAMETERS

DA V/D A. ALLEW
Hale Observatories

Over the past decade, largely because of the pioneer work of F. J. Low in Tucson, it has become possible to make accurate and reliable astronomical measurements at infrared wavelengths as long as 20 pm. At such wavelengths we see solar system bodies not by reflected sunlight but by their own thermal emission. The larger asteroids, though subtending small angles, give signals at 10 pum comparable in intensity with those from the brightest stars. It is now possible to determine the absolute flux from such asteroids to an accuracy of about 10 percent. Because an asteroid might reasonably be expected to have no atmosphere and no internal source of heat, the thermal radiation it emits must just balance the solar radiation it absorbs, and the measured flux will depend only upon its size. Infrared measurements therefore provide an opportunity to determine the diameters of the brighter asteroids.

ASSUMPTIONS

Certain assumptions must be made before the infrared flux can be converted into a diameter. The ideal asteroid must not rotate (when viewed from the Sun) and must be a smooth spherical blackbody at the observing wavelength (~ 10 pm). In addition, the proportion of solar energy scattered by each element of surface must be represented by a single albedo, A (the Bond albedo). Then a measurement of the optical flux from the asteroid gives us the product d’A = p, where d is the diameter. The absorbed solar energy is proportional to 1 - A = 1 - psd”, and the infrared flux is a complex function of d” that has previously been derived (Allen, 1970). Each infrared measurement can be converted directly to a diameter. In view of the assumptions made in defining this dimension, it will be called the infrared diameter. The relationship between infrared and true diameter is discussed below.

RESULTS

The infrared facility at the University of Minnesota (Ney and Stein, 1968) has been used to determine the infrared diameters of Ceres and Juno (Murdock, unpublished) and Vesta (Allen, 1970). The results are given in table I. It will be seen that in each case the infrared diameter exceeds Barnard's TABLE I.—Infrared Diameters of Three Asteroids

Asteroid Number of Infrared Barnard's Density, Reference
determinations diameter, kn diameter, km g-cm−3 to mass
Ceres 6 1160 + 80 770 | 1.6 + 0.7 |Schubart (1970)
Juno 4 290 + 20 220 - -
Vesta 11 570 + 10 380 || 2.5 + 0.7 | Hertz (1968)

micrometer measures by about 50 percent, but that the corresponding densities seem more reasonable than those obtained by using Barnard's values (5.5 for Ceres and 8.5 for Vesta).

DISCUSSION OF RESULTS

An asteroid will not, in general, be ideal, and the infrared diameter may bear little relationship to the linear dimension. For the larger bodies, at least, the figures will be similar. Table II shows the effect on the infrared diameter caused by a breakdown of the various assumptions.

TABLE II.-Breakdown of Assumptions

Assumption Effect on diameter, Sense”
percent

Blackbody < 5 +
Shape, roughness 15 to 35 -
Rotation:

Dust 15 +

Rock 50 +
Albedo 8 +
Absolute calibration 5 +

*+: linear diameter is greater than infrared diameter; -: linear diameter is less than infrared diameter.

Blackness

Observations at two or three wavelengths in the 8 to 14 pm atmospheric window and around 20 pum have shown that Ceres, Juno, and Vesta are graybodies to within experimental uncertainty. Observations of the Moon and Mercury show these bodies to be close to black. It seems unlikely that asteroids have emissivities below 0.95.

Roughness

The Moon and Mercury at full phase emit 30 percent more flux than would a smooth sphere because mountain slopes predominantly seen near the limb are

warmer than level terrain. In the extreme an asteroid might emit as a flat disk; the infrared diameter would then be 35 percent too large.

Rotation

If an asteroid rotates, it emits some of its thermal radiation on the night side and the signal received at Earth is reduced. The exact reduction depends on the period and on the nature of the surface. The table shows the magnitude of the effect for a rotation period equal to that of Vesta and for two types of surface—solid rock and porous dust. We expect the largest asteroids to retain a cover of dust, as does the Moon, but smaller bodies with weaker surface gravities will probably behave as solid rock, and the infrared diameter will be much too low.

Albedo

Even if the reflected sunlight is not well represented by the Bond albedo, the effect on an asteroid's diameter is slight. The figure in the table refers to a factor 2 error in albedo.

Variability

There is no evidence for variation of the infrared flux from the three asteroids discussed above. Matson' has, however, found some to vary considerably. In such cases, simultaneous optical and infrared measurements are needed to determine whether the variations are caused by changing albedo or shape or both.

CONCLUSIONS

With current detectors it is possible to measure reliable infrared diameters for several dozen asteroids. Notwithstanding the errors and uncertainties, these may be the most reliable dimensions currently available. When more accurate diameters are measured (Dollfus?), the infrared data will give us information on the roughness and thermal properties of the asteroids.

REFERENCES

Allen, D. A. 1970, Infrared Diameter of Vesta. Nature 227, 158.

Hertz, H. G. 1968, Mass of Vesta. Science 160,299.

Ney, E. P., and Stein, W. A. 1968, Observations of the Crab Nebula. Astrophys. J. 152, L21.

Schubart, J. 1970, The Mass of Ceres, IAU Circ. 2268.

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