A satellite describes an elliptic orbit about a planet of mass M. The minimum and maximum values of the distance r from the satellite to the center of the planet are, respectively, r0 and r1. Use the principles of conservation of energy and conservation of angular momentum to derive the relation

Question

A satellite describes an elliptic orbit about a planet of mass M. The minimum and maximum values of the distance r from the satellite to the center of the planet are, respectively, ${ r }_{ 0 }$ and ${ r }_{ 1 }$ . Use the principles of conservation of energy and conservation of angular momentum to derive the relation

$\frac { 1 }{ { r }_{ 0 } } +\frac { 1 }{ { r }_{ 1 } } =\frac { 2GM }{ { h }^{ 2 } }$

where h is the angular momentum per unit mass of the satellite and G is the constant of gravitation.

[latex]\frac { 1 }{ { r }_{ 0 } } +\frac { 1 }{ { r }_{ 1 } } =\frac { 2GM }{ { h }^{ 2 } }[/latex]

where h is the angular momentum per unit mass of the satellite and G is the constant of gravitation.

Accepted Answer

Angular momentum:

[latex]\begin{aligned}h & =r_{0}, \quad u_{0}=r_{1} u_{1} \b & =r_{0} u_{0}=r_{1} u_{1} \ u_{0} & =\frac{h}{r_{0}} \quad u_{1}=\frac{h}{r_{1}}\quad\quad ext{(1)}\end{aligned}[/latex]

Conservation of energy:

[latex]\begin{aligned}T_{A} & =\frac{1}{2} m u_{0}^{2} \V_{A} & =-\frac{G M m}{r_{0}} \T_{B} & =\frac{1}{2} m u_{1}^{2} \V_{B} & =-\frac{G M m}{r_{1}} \T_{A}+V_{A} & =T_{B}+V_{B} \\frac{1}{2} m u_{0}^{2}-\frac{G M m}{r_{0}} & =\frac{1}{2} m u_{1}^{2}-\frac{G M m}{r_{1}} \ u_{0}^{2}- u_{1}^{2} & =2 G M\left[\frac{1}{r_{0}}-\frac{1}{r_{1}} ight]=2 G M\left[\frac{r_{1}-r_{0}}{r_{1} r_{0}} ight]\end{aligned}[/latex]

Substituting for [latex] u_{0}[/latex] and [latex] u_{1}[/latex] from Eq. (1)

[latex]\begin{aligned}h^{2}\left[\frac{1}{r_{0}^{2}}-\frac{1}{r_{1}^{2}} ight] & =2 G M\left[\frac{r_{1}-r_{0}}{r_{1} r_{0}} ight] \h^{2}\left[\frac{r_{1}^{2}-r_{0}^{2}}{r_{1}^{2} r_{0}^{2}} ight] & =\frac{h^{2}}{r_{1}^{2} r_{0}^{2}}\left(r_{1}-r_{0} ight)\left(r_{1}+r_{0} ight)=2 G M\left[\frac{r_{1}-r_{0}}{r_{1} r_{0}} ight] \h^{2}\left\lgroup\frac{1}{r_{0}}+\frac{1}{r_{1}} ight group & =2 G M \quad\quad\quad\quad\left\lgroup\frac{1}{r_{0}}+\frac{1}{r_{1}} ight group=\frac{2 G M}{h^{2}} \quad ext { Q.E.D. }\end{aligned}[/latex]

Question 13.192: A satellite describes an elliptic orbit about a planet of ma...

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