- #1
epovo
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- TL;DR Summary
- I followed Schutz derivation and I don't get his result
Schutz finds that the orbital period for a circular orbit in Schwarzschild is
$$ P = 2 \pi \sqrt {\frac { r^3} {M} }$$
He gets this from
$$ \frac {dt} {d\phi} = \frac {dt / d\tau} {d\phi/d\tau} $$
Where previously he had ## \frac {d\phi}{d\tau} = \tilde L / r^2## and ## \frac {dt}{d\tau} = \frac {\tilde E} { 1 - 2M/r}## and where
## \tilde L^2= \frac {Mr } { 1-3M/r}## and ##\tilde E = \frac {(1- 2M/r)^2} {1-3M/r} ##
After doing the algebra I don't get that expression for the period (I get a much more complicated expression).
I punched in some numbers for M and r in a spreadsheet and the period given by the expression above does not match the calculations I have done. It does not even seem to be a very good approximation. Help, please!
$$ P = 2 \pi \sqrt {\frac { r^3} {M} }$$
He gets this from
$$ \frac {dt} {d\phi} = \frac {dt / d\tau} {d\phi/d\tau} $$
Where previously he had ## \frac {d\phi}{d\tau} = \tilde L / r^2## and ## \frac {dt}{d\tau} = \frac {\tilde E} { 1 - 2M/r}## and where
## \tilde L^2= \frac {Mr } { 1-3M/r}## and ##\tilde E = \frac {(1- 2M/r)^2} {1-3M/r} ##
After doing the algebra I don't get that expression for the period (I get a much more complicated expression).
I punched in some numbers for M and r in a spreadsheet and the period given by the expression above does not match the calculations I have done. It does not even seem to be a very good approximation. Help, please!