Estimating the ratio of flux of sunlight to moonlight (at full moon)

[cragwolf]

I'm trying to estimate the ratio of the flux of sunlight to moonlight (at full moon). What I'm interested in is if I'm doing the physics right. I know what the solar flux is at the surface of the Earth (actually, just outside the atmosphere): roughly 1400 Watts/m^2. I can actually calculate that by knowing the energy output of the Sun, its radius, and its distance to the Earth.

Now, the Moon should get roughly the same energy flux (OK, a little bit less since it's slightly further away at full moon, but it does not make much difference). The Moon is initially treated like a disk that intercepts part of the expanding sphere of sunlight. This disk has area pi * R^2, where R is the radius of the Moon.

The Moon then reflects this sunlight, but now I treat it like a sphere, and only half the sphere (the half we on Earth can see) is involved in reflecting the sunlight. So we have this expanding hemisphere of moonlight that now the Earth intercepts.

So the Earth is thus capturing a flux of:

1400 * (pi * R^2) / (2 * pi * d^2)

where d is the distance of the Moon from the Earth.

Oh, but I forgot the albedo of the Moon, which is 0.07, so the flux is actually:

1400 * 0.07 * (pi * R^2) / (2 * pi * d^2)

Or roughly:

1400 * (1 / 1400000)

In other words, moonlight is 1400000 times fainter than sunlight. But when I look up the magnitudes:

Sun = -26.7
Moon = -12.6

which is a factor of 14.1 magnitudes of difference, or roughly a ratio of 436000 to 1. So I'm off by a factor of about 3. This difference has me worried, and I'm wondering if I did the physics wrong.

 

[marcus]

In other words, moonlight is 1400000 times fainter than sunlight. But when I look up the magnitudes:

Sun = -26.7
Moon = -12.6

which is a factor of 14.1 magnitudes of difference, or roughly a ratio of 436000 to 1. So I'm off by a factor of about 3. This difference has me worried, and I'm wondering if I did the physics wrong.

not all the sunlight goes out in that expanding hemisphere
BTW I remember you from an especially great time at PF----2003-2004. We had Astronomy Game then, IIRC.
Maybe I am mistaking you for someone whose name sounds like cragwolf, but if not then you probably remember back that far.

anyway if all the light went out in an expanding hemisphere, then nobody would ever see the crescent moon. We would see the HALF moon (the moon at quarter phase). But you are denying light to anybody who makes an obtuse angle with the sun (from moon's standpoint)

so the crescent (before halfmoon) and the waning would be invisible.

think of a glancing scattering----it shows that the expanding hemisphere is not where all the light goes

 

[cragwolf]

Indeed you're right, about me and the Moon. Thanks for the hint, I'm going to look at it again when I get home from work.