Lunar Eclipse: Umbra and penumbra radii

[Calico snail]

Homework Statement

Find the radii of the umbra and penumbra circles drawn drawn perpendicular to the EMS axis, formed at a distance equal to that of the moon from the Earth.

Homework Equations

(From: http://www.opticiansfriend.com/articles/equations.html#Shadows)

D2 / L1 = (P + U) / (L1 + L2)
D1 / L1 = P / L2
Total Shadow = 2P + U

D1 = Diameter of light source
D2 = Diameter of object between light source and Shadow
L1 = Length from light source to L1
L2 = Length from L1 to Umbra
U = Umbra
P = Penumbra

The Attempt at a Solution

I substituted 12,576 km for D2 as the diameter of Earth and 149.6 million km for the distance between the Earth and sun. My calculations revealed that P+U= 1305010.784. Am I misguided in the belief that these equations give my the area of the circles from which I can extrapolate the radii?

 

[SteamKing]

The equations don't suggest that either P or U are areas.

If you show your calculations, it might give a better clue what your value for P + U means.

 

[Calico snail]

I found a completely different equation:
penumbral radius: Rp = 1.02 * (0.998340 * Pm + Ss + Ps)
umbral radius: Ru = 1.02 * (0.998340 * Pm - Ss + Ps)

where: Pm = Equatorial horizontal parallax of the Moon,
Ss = Geocentric semi-diameter of the Sun, and
Ps = Equatorial horizontal parallax of the Sun.

Should this equation work? My instructor gave me a question eons beyond my current level of understanding.

 

[dauto]

D₂ should be the diameter of the object casting the shadow - namely the Moon.
Draw a picture and try to understand the equation instead of blindly plugging in data on a poorly understood equation. Chances of getting it right will improve significantly.

 

[Calico snail]

D₂ should be the diameter of the object casting the shadow - namely the Moon.
Draw a picture and try to understand the equation instead of blindly plugging in data on a poorly understood equation. Chances of getting it right will improve significantly.

Are you saying that the moon casts a shadow in a lunar eclipse? My problem is that my teacher has not even begun to mention anything about this in class, and has not provided any resources for me. I'm just trying to learn how to calculate the radius of the umbral and penumbral circle in a lunar eclipse.

 

[haruspex]

Are you saying that the moon casts a shadow in a lunar eclipse? My problem is that my teacher has not even begun to mention anything about this in class, and has not provided any resources for me. I'm just trying to learn how to calculate the radius of the umbral and penumbral circle in a lunar eclipse.

You didn't indicate that this was for a lunar eclipse, except where you said you used the diameter of the Earth as D2.
Clearly P and U must be distances, not areas. The LHS of each equation is a distance divided by a distance, so is dimensionless . The RHS must therefore be dimensionless. So P and U must have the same dimension as L1 and L2.
However, the way the author has defined U and P is a bit surprising (to me). The clue is in the line Total = 2P+U, not P+U, nor 2P+2U.
It isn't that hard to figure out these equations. Draw a decent diagram and look for similar triangles.

I'm not sure what the second set of equations you posted is saying. There's no explanation given for those magic numbers. As I understand it, lunar parallax is an angle, so it makes no sense to add it to a distance. http://en.wikipedia.org/wiki/Parallax#Lunar_parallax