# info on earthshine wanted

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### Lionel Armstrong

Jan 4, 1995, 4:37:47 PM1/4/95
to
I am looking for data regarding the amount of light reflected by the
earth as seen from the lunar surface at a location on lunar "nightside".
I will need a raange of the intensity of light reflected by the earth in
various phases from a thin crescent to a full earth. The amount should be
quite high due to the fact that it will be reflected back to the earth
and still be detectable by the naked eye. My guess is that it would be
slightly higher than the amount of sunlight reflected by the moon as seen
from earth.

Any information that anyone can provide will be appreciated very much.

THANX Lionel Armstrong
no...@freenet.vancouver.bc.ca

### Paul Schlyter

Jan 5, 1995, 3:57:00 AM1/5/95
to
In article <3ef4fb\$7...@freenet.vancouver.bc.ca>,
Lionel Armstrong <no...@freenet.vancouver.bc.ca> wrote:

According to good ol' Allen's "Astrophysical Quantities" (except the
phase function which is taked from "Planets and Satellites" by Kuiper and
Middlehurst):

Radius Albedo V(1,0) Phase function, mag's
Geometric Bond (a = phase angle/100 deg)

Earth 1.00 0.37 0.39 -3.84 1.30*a + 0.19*a^2 + 0.48*a^3
Moon 0.27 0.112 0.067 +0.23 3.05*a - 1.02*a^2 + 1.05*a^3

Mean magnitude of full moon, as seen from Earth: -12.73
Mean magntiude of Sun, as seen from Earth: -26.74

V(1,0) is the magnitude of the planet, when situated 1 a.u. from the Sun
and 1 a.u. from the observer, and seen from a phase angle of 0 degrees
(i.e. at "full" phase).

The "Albedo" column may need some explanation: The BOND albedo is the
ratio of total relfected to total incident light - it's this albedo
that's of interest when computing the radiation budget of the planet.

The GEOMETRIC albedo on the other hand is the ratio of the brightness
of the planet, as seen from zero phase angle (i.e. when "full"), to
the brightness of an equally large perfectly diffusing sphere (i.e. a
sphere having a Bond albedo of 100% and diffusing the light according
to Lamberts law, i.e. as proportional to the cosine of the angle to
the normal, at each area element).

Note the pronounced "opposition effect" of the Moon -- its geometric
albedo is nearly twice as large as its Bond albedo. Therefore the
full moon will be considerably brighter than the Moon a few days
before or after full, and as much as 10 times brighter than the Moon
in the first or last quarters.

The Phase function tells how the brightness varies over the phase
cycle, and gives the magnitude difference of the brightess at any
phase to the brightness at "full" phase. At "full" phase the
difference is of course zero magnitudes. a is the phase angle and is
found by dividing the phase angle in degrees by 100 degrees. At
"full" phase, a is zero. At "half" phase, the phase angle is 90
degrees and a becomes 90/100 = 0.9, and at "new" phase a becomes
180/100 = 1.8 -- now these formulae shouldn't be trusted much beyond
120-130 degrees.

Now, a simple computation: how bright does the moon's Earthshine
shine, at precisely "new" phase?

First:

1. How bright does the full Earth shine, as seen from the Moon? V(1,0)
for the Earth is 4.07 magnitudes brighter than V(1,0) for the Moon.
The Moon shines at mag -12.73 when full - thus the Earth shines at
-12.73 - 4.07 = -16.80 when full, as seen from the Moon (i.e. when
the Moon is "new").

2. How bright does Earthshine shine? When illuminated by the Sun
(mag -26.74), the full Moon shines at -12.73. Earthshine is really
a "full moon illuminated by the Earth". Since the full Earth, as
seen from the Moon, shines -16.80 - -26.74 = 9.94 magnitudes fainter
than the Sun, likewise the Earthshine ought to shine 9.94 magnitudes
fainter than the full Moon. Thus Earthshine shines at -12.73 + 9,94
= -2.79

as bright as totally eclipsed moon during a pretty bright lunar eclipse.

At other lunar phases the brightness of the Earthshine should follow the
Earth's phase function:

Moon's elong. Moon's Magnitude of Earthshine
frun Sun phase All moon Unillum. part only

0 0.00 -2.79 -2.79
10 0.01 -2.66 -2.65
20 0.03 -2.52 -2.49
30 0.07 -2.37 -2.29
40 0.12 -2.21 -2.07
50 0.18 -2.03 -1.82
60 0.25 -1.84 -1.53
70 0.33 -1.62 -1.19
80 0.41 -1.38 -0.80
90 0.50 -1.12 -0.36
100 0.59 -0.82 +0.14
110 0.67 -0.49 +0.72
120 0.75 -0.13 +1.38
130 0.82 +0.28 +2.15

Of course the brightness of the Earthshine also varies with the amount of
cloud cover on Earth, in a way that's not predictable.

--
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Paul Schlyter, Swedish Amateur Astronomer's Society (SAAF)
Nybrogatan 75 A, S-114 40 Stockholm, SWEDEN
e-mail: pau...@saaf.se p...@ausys.se