GR problem - satelite in orbit (Schwarzschild geometry)

[quasar987]

Homework Statement

An observer in a rocket is in a circular equatorial orbit arounda planet and the period of the orbit is the same as the period of revolution of the planet. The planet has mass M = 1033kg and radius R = 1000km. The observer sends a signal every 20 seconds according to its clock towards an observer on the equator. According to the observer on the planet, each day lasts 10 hours. The Schwarzschild metric is appropriate to describe the geometry outside the planet.

Calculate the radius of the rocket's orbit.

Homework Equations

Schwarzschild's metric and the geodesic equations.

The Attempt at a Solution

I tried crying.

I seriously haven't been able to write anything on this. Normally, in a classical Newtonian problem, I would get an expression of the speed "of" circular orbits as a function of the orbit radius knowing the time of 1 revolution is 10 hours and solve for r.

But here there is no gravitational force. I just know that the orbit is a geodesic.

 

[George Jones]

Are you familiar with the constants of motion for Schwarzschild orbits?

With the effective potential for Schwarzschild orbits?

 

[quasar987]

Kinda. E and J are constants... like in the KEpler problem. Tell me more. :)

 

[George Jones]

Kinda. E and J are constants... like in the KEpler problem. Tell me more. :)

I meant familiar with the specific expressions for E, J, and the effective potential.

Baby's bath time.

Even though I'm not the the governor of California, I'll be back!

 

[quasar987]

I seem to have like 5 different expression for each J and E in my notes. Picking 2 that seem linked:

$$J=\frac{d\phi}{dp}r^2$$

$$E=1-\left(\frac{dr}{dt}\right)^2$$

 

[George Jones]

I seem to have like 5 different expression for each J and E in my notes. Picking 2 that seem linked:

$$J=\frac{d\phi}{dp}r^2$$

$$E=1-\left(\frac{dr}{dt}\right)^2$$

I was looking for

$$E = \left( 1 - \frac{2M}{r} \right) \frac{dt}{d\tau}$$

$$J = r^2 sin^2 \theta \frac{d\phi}{d\tau}.$$

That these quantities are conserved follows from Lagrrange' equations. Note that the metric is independent of $t$ and $\phi.$

Assume that the orbit is in the plane $\theta = \pi/2,$ and use the conserved quantities to eliminate $dt/d\tau$ and $d\phi/d\tau$ in the metric.

 

[quasar987]

Assume that the orbit is in the plane $\theta = \pi/2,$ and use the conserved quantities to eliminate $dt/d\tau$ and $d\phi/d\tau$ in the metric.

I take it you meant "in the geodesic equations".

 

[George Jones]

I take it you meant "in the geodesic equations".

I really did mean in the metric. Speaking more loosely, use the conserved quantities to eliminate $dt$ and $d\phi$ in the Schwarzschild metric. This will lead to the useful concept of effective potential.

 

[George Jones]

Once $dt$ and $d\phi$ have been eliminated, solve for

$$\left( \frac{dr}{d\tau} \right)^2$$

as a function of $r$, and the constants $E$ and $L$. Clearly, setting this to zero is necessary for circular orbits, but it is not sufficient. Think Newtonian orbits.

This is the standard path to orbital motion about Schwarzschild.

 

[quasar987]

Ah, I see!

Too bad the exam was yesterday and I missed the orbit question. :(

But now I know, thx.