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> From: Paul Schlyter <pausch_at_saaf.se>
>> This "opposition effect" is also responsible for the full moon being
>> about ten times brighter than the half moon, and not merely twice as
>> bright as could be naively expected, or PI times as bright as if it
>> would be if the lunar surface was a perfect diffusor (a "Lambertian"
>> reflector).
>
> Do you have any references for this -- or better yet some data.
What I do have are phase functions for the Moon from e.g. Allen's
"Astrophysical Quantities" (1974) and G. Kuiper's "The Planets"
(somewhere around 1960.
> I have constructed a lunar exoatmospheric irradiance model, which is
> combined with a solar and lunar ephemeris model. I compensate for:
>
> 1. Amount of lunar surface illuminated by sun
Of course ... a thin crescent must be much fainter than a full moon.
> 2. Variation in earth-moon and moon-sun distance during lunar
> cycle.
> There is a total variation in the irradiance of about
> 26 percent from this.
That's standard. The magnitude is usually expressed like this:
m = V(1,0) + 5 * log(r*R) + phi(phang)
where V(1,0) is the "absolute magnitude" of the celestial body as seen
from a phase angle of zero (i.e. "full") and 1 a.u. from both the Sun and
the observer, R = heliocentric distance in a.u., r = distance from
observer
in a.u., and phi(phang) is the "phase function" where phang = the phase
angle (0 = "full", 90_deg = "half", 180_deg = "new"). Allen gives these
photometric quantities for various celestial bodies as:
V(1,0) phi(L), L in degrees
Mercury -0.36 0.027*L + 2.2E-13*L^6
Venus -4.34 0.013*L + 4.2E-7*L^3
Earth -3.9 ?
Mars -1.51 0.016*L
Moon +0.23 0.026*L + 4.0E-9*L^4
Thus phi(90_deg) will be 2.5 for Mercury, 1.5 for Venus and 2.6 for the
Moon - this figure tells how many mangitudes fainter the body will be
when "half" compared to when "full", assuming other things like
distances being equal.
> 3. Variation in the albedo of different portions of the moon
> surface. The irradiance is about 20 percent brighter
> at first quarter (waxing) than at third quarter
> (waning) due to differences in the lunar surface.
> 4. Spectral variation in the lunar albedo (averaged over the
> whole lunar surface).
These are minor corrections. Ignoring 3. will yield an error of max 0.2
magnitudes, and ignoring 4. will yield an even smaller error.
> The correction factor for the third factor as a function of lunar
> elongation is:
>
> DATA ELONG / 0., 10., 20., 30., 40., 50., 60., 70.,
> + 80., 90.,100.,110.,120.,130.,140.,150.,160.,170.,180.,190.,
> + 200.,210.,220.,230.,240.,250.,260.,270.,280.,290.,300.,310.,
> + 320.,330.,340.,350.,360./
> DATA CORREC /0.255,0.259,0.263,0.267,0.271,0.275,
> + 0.284,0.307,0.326,0.361,0.393,0.429,0.465,0.499,0.545,
> + 0.613,0.706,0.829,1.000,0.829,0.662,0.574,0.484,0.422,
> + 0.384,0.354,0.332,0.314,0.305,0.295,0.294,0.292,0.286,
> + 0.267,0.248,0.229,0.210/
What units are you using here? I would expect the new Moon to be much
fainter than about 1/4 - 1/5 the brightness of the full moon...
> If you have some quantitative values for this brightening due to the
> opposition effect, which I assume is a sharply peaked value, falling off
> sharply if the lunar elongation is more than a couple of degrees off
from
> full (180 deg), and any reference, I would appreciate it.
Measuring the lunar brightness at a phase angle of exactly zero (i.e. when
"full") presents one major problem: the Moon will then be eclipsed....
Paul Schlyter, Swedish Amateur Astronomer's Society (SAAF)
Grev Turegatan 40, S-114 38 Stockholm, SWEDEN
e-mail: pausch_at_saaf.se psr_at_home.ausys.se paul_at_inorbit.com
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