Quantization of Earth's angular momentum

[Kavorka]

Homework Statement

If the angular momentum of Earth in its motion around the Sun were quantized like a hydrogen electron, what would Earth's quantum number be? How much energy would be released in a transition to the next lowest level? Would that energy release be detectable? What would be the radius of that orbit?

Homework Equations

L = mvr = nħ
F = GMm/r² = mv²/r
E_n = -E₀/n²

The Attempt at a Solution

I found Earth's quantum number by solving mvr = nħ for n, with m = 5.972 x 10²⁴ kg, r = 149.7 x 10⁹ m and v = 29.78 x 10³ m/s.

n = 2.523 x 10⁷⁴

The second part is what gets me.
In order to express the energy of a energy level in terms of a planet, I used
F = GMm/r² = mv²/r
and E = KE + PE_g = mv²/2 - GMm/r
to get:

E = -GMm/2r

I then plugged in v = (GM/r)^1/2 to r = nħ/mv getting r = n²ħ²/m²GM

And then plugged this into E getting:

E = -G²M²m³/2n²ħ²

The energy of a transition would equal: E_n+1 - E_n = -E₀/(n+1)² + E₀/n²

Which I could solve with a super huge/super small online calculator. I already assume that the energy will be super small and the orbit won't noticeably change. My issue is...what do I plug in for m? Or should I do it a completely different way? Please help!

 

[Kavorka]

Oh duh, I went through all that work and didn't realize I need the mass of the sun as well as the Earth.