Hi Truman,

The algorithm used is a "standard" one. You can find it in Meeus'
_Astronomical Algorithms_ or in the _Explanatory Supplement to the
Astronomical Almanac_. It looks like this:

mag = H + 5 log( r * delta) - 2.5 log( (1-G)phi1 + G phi2)

where

phi1 = exp( -3.33 tan( beta/2) ^ .63)
phi2 = exp( -1.87 tan( beta/2) ^ 1.22)
beta = phase angle
r = distance from sun to asteroid, in AU
delta = distance from earth to asteroid, in AU
H = magnitude of the asteroid as seen from the Sun if the asteroid
were one AU from the sun
G = "slope parameter" for that asteroid, usually about .15

and 'exp' and 'log' are both base-e operations.

I don't know of the reasoning behind the above formula;
it was probably derived empirically. It's considered valid
for phase angles from zero to 120 degrees (i.e., for objects
that get between the earth and sun, you can get some not
particularly valid results.)

As for how the phase angle is computed: if you've got r,
delta, and the distance from the earth to the sun, you have
three sides of a "normal" plane triangle. With that, you
can compute any of the three angles of that triangle. One of
them (the asteroid-earth-sun angle) is the elongation. Another,
the earth-asteroid-sun angle, is the phase angle, beta. I
don't think the third has any "common" name.

-- Bill