Bill J Gray wrote:
>
> Hi Paul,
>
> "...when at inferior conjunction, Venus too is approximately as
> bright as magnitude -3."
>
> I commented on Venus and Mercury vanishing without actually
> having checked the matter (I made a rash assumption that they would
> get vanishingly faint, too). For these objects, the magnitude is
> based on empirical formulae from the _Explanatory Supplement to the
> Astronomical Almanac_, which give a passably decent magnitude
> even when the objects are near the sun.
It might be interesting to compare various such formulae (i = phase
angle,
e = Earth's Saturnicentric latitude)):
Explanatory Supplement to the Astronomical Almanac ("New Expl. Suppl.")
V(1,0) delta-m(i)
Mercury -0.36 3.80*(i/100) - 2.73*(i/100)^2 + 2.00*(i/100)^3
Venus -4.29 0.09*(i/100) + 2.39*(i/100)^2 - 0.65*(i/100)^3
Mars -1.52 0.016*i
Jupiter -9.25 0.005*i
Saturn -7.19 0.044*i - 2.60*sin(e) + 1.25*sin^2(e)
Uranus -7.19 0.0028*i
Neptune -6.87
Pluto -1.01 0.041*i
Explanatory Supplement to the Astronomical Ephemeris ("Old Expl.
Suppl.")
V(1,0) delta-m(i)
Mercury -0.003 1.815*(i/100) + 1.023*(i/100)^2
Venus -4.00 1.322*(i/100) + 0.4247*(i/100)^3
Mars -1.30 0.01486*i
Jupiter -8.93 0.0
Saturn -8.68 0.044*i - 2.60*sin(e) + 1.25*sin^2(e)
Uranus -6.85 0.0
Neptune -7.05 0.0
C.W. Allen "Astrophysical Quantities", 3rd ed., 1973
V(1,0) delta-m(i)
Mercury -0.36 0.027*i + 2.2E-13*i^6
Venus -4.34 0.013*i + 4.2E-7*i^3
Earth -3.9
Mars -1.51 0.016*i
Jupiter -9.25 0.014*i
Saturn -9.0 0.044*i - 2.6*sin(e) + 1.2*sin^2(e)
Uranus -7.15 0.001*i
Neptune -6.90 0.001*i
Pluto -1.0 0.0
Ceres +3.40 0.05*i
Pallas +4.53 0.04*i
Juno +5.62 0.03*i
Vesta +3.54 0.03*i
Eros +11.44 0.02*i
Moon +0.23 0.026*i + 4.0E-9*i^4
Io -1.9 0.04*i
Europa -1.5 0.03*i
Ganymede -2.2 0.03*i
Callisto -1.2 0.07*i
Titan -1.1 0.009*i
Allen also gives this table for the Moon's phase function:
i mi-m0 Brightness
0 0.00 1.000
5 0.08 0.929
10 0.23 0.809
20 0.51 0.625
30 0.79 0.483
40 1.06 0.377
50 1.35 0.288
60 1.62 0.225
70 1.91 0.172
80 2.24 0.127
90 2.63 0.089
100 3.04 0.061
110 3.48 0.041
120 3.93 0.027
130 4.44 0.017
140 5.07 0.009
150 5.9 0.004
160 7.5 0.001
There are two different definitions for "albedo":
p = geometric albedo: the albedo of a perfectly diffusing disk with the
same diameter and the same distance as the planet and having the
same apparent brightness as the planet (which is assumed to be in
"full" phase)
A = Bond albedo: ratio of total light reflected from spherical planet to
total light incident on it
q = A/p = 2 * Integrai_from_0_to_pi_of( phi(i) )
where phi(i) = phase function
We have the following simple cases:
Perfectly diffusing disk: q = 1.00
Perfectly diffusing sphere (Lambert's Law): q = 1.50
Lommer-Seelinger law sphere: q = 1.64
phi(i) = 0.5*(1+cos(i)) ==> prop. to ill. area: q = 2.00
Metallic reflection sphere: q = 4.00
Note that the case "q = 2.00", i.e. phase function proportional to
illuminated area, is the phase function assumed when the Astronomical
Almanac gives the times for Venus' "greatest brilliancy".
q below 1.0 indicates a "rough" surface without an atmosphere.
There's an approximate relation between q and the first term in the
phase function:
q 2.0 1.5 1.0 0.5 0.2
delta-m/i 0.006 0.010 0.018 0.034 0.057
Finally, for the real planets, we have (from Allen):
Geom. Ratio Bond
albedo albedo
p q A
Mercury 0.096 0.58 0.056
Venus 0.6 1.2 0.72
Earth 0.37 1.05 0.39
Mars 0.154 1.02 0.16
Jupiter 0.44 1.6 0.70
Saturn 0.47 1.6 0.75
Uranus 0.57 1.6 0.90
Neptune 0.51 1.6 0.82
Pluto 0.12 1.2 0.145
Ceres 0.12 0.3 0.035
Pallas 0.12 0.4 0.05
Juno 0.28 0.5 0.14
Vesta 0.44 0.6 0.27
Eros 0.30 0.8 0.23
Moon 0.112 0.60 0.067
Io 0.9 0.6 0.55
Europa 0.8 0.6 0.5
Ganymede 0.5 0.6 0.3
Callisto 0.26 0.6 0.15
Titan 0.21 (not given)
> "...The lunar magnitude ought not to be a problem here: when the
> Moon is really new... its magnitude is still as bright as magnitude
> -3, due to Earthshine."
>
> Ah, a good point. My current formula for lunar magnitude doesn't
> account for this.
It's simple to account for: convert the magnitude to a linear
brightness scale, add the intensity for Earthshine, and convert
back to magnitude. If one should do it in a really strict way,
one shouldn't assume Earthshine is constant but rather that it
varies with the _Earth's_ phase function! But doing that is
overkill I think: even when the Moon is a very thin crescent,
the sunlit parts of the lunar disk outshines the Earthshine.
> The _Explanatory Supplement_ gives magnitude formulae for all
> planets and some planetary satellites, covering the range of phase
> angles seen from Earth. It doesn't give such formulae for the moon,
> less "significant" satellites, outer planets as seen at small
> phase angles (for example, Mars as seen from Jupiter), or Earth
> as seen from any place off Earth. The _Explanatory Supplement_
> didn't have to worry about such oddities.
Allen's "Astrophysical Quantities" gives some such quantities;
see above.
> For small, rocky satellites, I used the same formulae as for
> asteroids. For the Moon, Earth, and other objects, I derived
> a formula based on a diffusely-reflecting sphere.
I derived such a formula too, many years ago. One interesting result
I found was that, if viewed from infinite distance and if illuminated
by a point light source at infinite distance, the brightness at
"half" phase was exactly pi times fainter than the brightness
at "full" phase, assuming a sphere following Lambert's law, i.e.
a perfectly duffusing sphere.
The real moon is some 11 times fainter when half than when full.
> This is a lousy
> assumption for the Moon, but was the best I could do at the time.
> Through the wonders of on-line resources, I may now be able to do
> an on-line literature search
Or you can read this..... :-)
> (something I couldn't do back when I
> derived the diffuse-sphere formula) and find something better.
Before the Internet was easily accessible, you had to visit libraries
or buy books. E.g. Allen's "Astrophysical Quantities".
> Even if I can't, adding in brightness due to earthshine shouldn't
> be too difficult.
>
> -- Bill
>
>
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