Calculating asteroid diameters.
In fact, Earth-based telescopes never show the disks of asteroids clearly. Most of what we know about their sizes has come within just the last two decades, through such innovative techniques as radar, stellar-occultation timings, speckle interferometry, and a rare spacecraft flyby.
Even without hard data, however, a good estimate can often be made. Because we know how bright the Sun is, it is a simple matter to calculate how bright an object should appear a certain distance away, provided we know its surface area and the fraction of sunlight it reflects, called the albedo. Turning the problem around, we can work out the surface area, hence the diameter, from the observed brightness and a reasonable estimate of the albedo. That's exactly what my computer program sets out to do.
The key relationship is contained in the following formula from page 14 of Planetary Satellites, Joseph A. Burns, editor (University of Arizona Press, 1977):
V = |V.sub.s~ - 2.5 logp - 5 logs + 5 logr|delta~,
where V is the apparent visual magnitude of the asteroid and |V.sub.s~ is that of the Sun. Also, p is the asteroid's albedo, s is its radius, and r and |delta~ are its distances from the Sun and Earth, respectively, with s, r, and |delta~ expressed in astronomical units (1 a.u. equals 149,597,870 kilometers).
While it might be tempting to rearrange this formula and solve for s directly, to do so would underestimate the size in most cases because it ignores the asteroid's phase. Just as the Moon appears much dimmer at first quarter than at full phase, we seldom see an asteroid illuminated face on by the Sun and thus at peak brightness.
Instead, we can use the formula to relate the asteroid's size to its so-called absolute magnitude, H, defined as the object's brightness if it were seen 1 a.u. away from an observer located on the Sun. No phase is involved, and the calculation is simplified because both r and |delta~ are 1. Setting H = V and |V.sub.s~ = -26.78 for the Sun's visual magnitude at 1 a.u., we find:
H = -26.78 - 2.5logp - 5logs.
Now, to obtain the asteroid's diameter, d, we need to multiply the radius, s, by 2 and also by 149,597,870. Rearranging the formula gives:
logd = 3.12 - 0.2H - 0.5logp,
where d is the asteroid's diameter in kilometers -- just what we want to find.
As a check on the formula's validity, I found an excellent reference source for diameters, absolute magnitudes, and albedos. It is a table in the book Asteroids II, edited by Richard P. Binzel, Tom Gehrels, and Mildred Shapley Matthews (University of Arizona Press, 1989). A computer version of the same data is on the National Space Science Data Center's CD-ROM.
For 1,790 of the known asteroids, in addition to H this book lists a quantity called G, the "slope parameter" that tells how the asteroid's brightness changes with its phase. In each case astronomers have chosen H and G to work together to best represent the observational data. More up-to-date and refined values are contained in the annual Ephemerides of Minor Planets (EMP) published by the Institute of Theoretical Astronomy, St. Petersburg, Russia. Many asteroids also show variations in brightness due to rotation, so the listed H is an average value. When G is not known, the value 0.15 gives a reasonable approximation.
My computer program, written in Basic, uses the information commonly given on the International Astronomical Union Circulars when an asteroid is discovered: its visual magnitude and its distance from both the Earth and Sun. From this information and a value of G that you input, it calculates the phase angle and then H, using a formula given in the EMP. Finally, with the help of an albedo estimate that you also enter, it derives the object's diameter in kilometers. Keep in mind that many small asteroids are highly irregular or elongated in shape. For this reason, and because of uncertainties in the albedo, the calculated diameter may be in error by as much as 50 percent.
To test the program on your own computer, consider the Earth-approaching asteroid 1993 B|X.sub.3~ recently discovered by Robert H. McNaught on a plate taken with the United Kingdom Schmidt Telescope in Australia. According to IAU Circular 5706, on February 2nd McNaught's object shone at magnitude 17.1 in Hydra, 0.071 a.u. from Earth and 1.034 a.u. from the Sun. On that date the Earth-Sun distance was 0.986 a.u. The program computes a phase angle of 46|degrees~ and an absolute magnitude of 21.0. Assuming the albedo is 0.04, as is typical for such an object, the diameter comes out as 0.4 km (four football fields).
The program will also work for a planetary moon or a comet nucleus -- in fact, any solar-system body that does not have an atmosphere. All you need are its apparent visual magnitude, the Sun-Earth-object distances, and a reasonable estimate of the albedo.
What Is the Albedo?
MOST MINOR-PLANET albedos were pure guesses until the launch of the Infrared Astronomical Satellite (IRAS) in 1983. The IRAS mission dramatically increased the number of asteroids observed in the infrared, and its all-sky survey is regarded as the most standardized set of such observations.
A large, dark asteroid and a small, highly reflective one might have identical apparent magnitudes, but their thermal properties will be quite different. Through the technique of thermal modeling, astronomers have been able to deduce an object's albedo -- a strong clue to its surface composition. (Albedos are defined in different ways depending on the context; this article uses what is called the "visual geometric albedo" throughout.)
An albedo of 0.04 (4 percent) is about average for more than 3/4 of the known asteroids. These are of the C type, believed to be related to the carbonaceous chondrite meteorites. An albedo in the range of 0.02 to 0.06 is also typical of the extinct comet nuclei that are undoubtedly responsible for a large percentage of the near-Earth asteroids. Most of the remaining main-belt asteroids are believed related to the stony and metal-rich meteorites, and are thus designated as types S and M, respectively. Their albedos range from 0.10 to 0.22 (type S) and 0.10 to 0.18 (type M).
Two other major types, P and D, circle at least 5 a.u. from the Sun and are noted for their reddish color. They include many of the Trojan asteroids -- those that trail or lead Jupiter in its orbit and are gravitationally controlled by that planet. The recently discovered "asteroid" 5145 Pholus (1992 AD) is probably an extreme example of this group. Orbiting between Saturn and Neptune, this enigmatic object is redder than any other known asteroid. P and D types have albedos from 0.02 to 0.07.
Values greater than 0.30 are very rare. The highest albedo in the Asteroids II data is 0.56, for 437 Rhodia. Seen up close, such an object would look as if it were made of chalk.
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|Title Annotation:||Astronomical Computing; includes related article; astronomical techniques|
|Author:||Rowe, Basil H.|
|Publication:||Sky & Telescope|
|Date:||Jun 1, 1993|
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