- #1

RJLiberator

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## Homework Statement

In analogy to the Bohr theory of the hydrogen atom, develop a quantum theory of Earth satellites, obtaining expressions for the orbit radius, r_n, and the energy, E_n in terms of the quantum number n and other relevant parameters. Use the "Old" Quantum Theory. A satellite of mass 1000kg is in a circular orbit of radius 7000 km. To what value of n does this correspond? What is the satellite energy? Determine the differences in the radius, Δr_n, and in the energy ΔE_n, for successive orbits at this radius. (Hint: Differentiate with respect to n and set Δn = 1. Why is this legitimate since n is not a continuous variable?)

## Homework Equations

[tex] V_{Sat} = \sqrt{\frac{GM_e}{r}}[/tex]

L=angular momentum of satelite:

[tex]L_{sat} = rm_{sat}v_{sat}[/tex]

T = period

[tex]T = 2*\pi\sqrt{\frac{r^3}{GM_e}}[/tex]

**Bohr–Sommerfeld model**,

[tex]\int_0^T p_rdq_r\ =nh[/tex]

h = Planck's constant = 6.626*10^(-34) J*s

## The Attempt at a Solution

So, based on the above equations and m = 1000kg, r = 7000km, I can find V_s = 7548.6 m/s and the period, T = 5826.6 s.If I have angular momentum correctly, then that is equal to 5.284*10^(13) kg*m^2/s .

I can easily set up the bounds of integration from 0 to the period, T.

But I've never used the Bohr-Sommerfeld model. I don't know what is meant by the wiki's description of "where

*p_r*is the radial momentum canonically conjugate to the coordinate

*q*which is the radial position and

*T*is one full orbital period."

Once I am able to calculate the integral, it will be easy to solve for n, and thus should be easy to find the Energy.

I believe that we are allowed to treat n

*like*a continuous variable due to how minute it will be compared to the other numbers. n will likely be so small, that it will act like a continuous variable, thus we can differentiate it.

Any help on how to get started with the Bohr-Sommerfeld model in relation to this problem? What is "dq" ?