Planetary visual magnitudes and phase integrals

[Malenfant]

I've just completed a script for Celestia that calculates the apparent magnitudes of planets.

Problem is, according to this:
http://en.wikipedia.org/wiki/Absolute_magnitude

one factor in the calculation is the phase integral of the planet. They give an equation for it there (assuming an ideal, diffuse reflecting sphere), but elsewhere I've seen phase integrals equal to constants: 1.5 for a Lambert Sphere, 4/3 for a Raleigh atmosphere, and 1.63 for a Lommel-Seeliger sphere.

Oddly enough when I run the script for Mars (comparing the apparent magnitude I get with the values in Astronomy magazines), I find that a phase integral of 1.0 seems to work pretty well (a result of around -2.3 today). The diffuse sphere equation gives an app mag that is too low (about -1.84) and the Lambert, Raleigh, and Lommel-Seeliger ones are too bright ((about -2.8 ).

But when I compare to other planets I find that 1.0 doesn't work for Venus or Mercury, but does for Jupiter (at least for the times I tried).


What I'm wondering is how apparent magnitudes for the planets are actually calculated for these magazines and astronomy almanacs etc - I'm sure they must be calculated rather than derived from observations so that they can get a prediction of what the magnitude will be at a given date. I'm guessing that they must use different Phase Integrals for each planet, but is there anywhere that these are listed? And would they really be constants, or would they be equations that depend on the phase angle (as the wikipedia article implies)?

[al'be:do]

They definitely depend on the Phase at a given time.
That's why it's called phase integral.

Here's the definition of the phase angle:
http://en.wikipedia.org/wiki/Phase_angle_(astronomy)

One thing you told cannot be. The phase integral MUST be between 0 and 1 and never can be greater or smaller than that interval.

So the one value that fit well in your computations was probably a value for a fullly lit "disc".
A very important value is the geometric albedo of the planet (Moon:0.12, Earth around 0.25 or so). This is a constant value that also influences the result.

Try to calculate the planetary phase at the given time and reintegrate the whole stuff. I hope you'll get a better value then.

I can take a look into my render paper collection if there are other common reflection models besides the ones you listed and if they are helpful for your computations.

al'be:do

[Malenfant]

I solved this problem a while back

See here: http://www.celestiaproject.net/forum/viewtopic.php?t=8160

The final script can be downloaded at:
http://www.evildrganymede.net/art/celestia/index.htm