GR problem - satelite in orbit (Schwarzschild geometry)

In summary, an observer in a rocket is in a circular equatorial orbit around a planet. The period of the orbit is the same as the period of revolution of the planet. Every day lasts 10 hours according to the observer on the planet. The Schwarzschild metric is appropriate to describe the geometry outside the planet. The radius of the rocket's orbit is calculated.
  • #1

quasar987

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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.
 
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  • #2
Are you familiar with the constants of motion for Schwarzschild orbits?

With the effective potential for Schwarzschild orbits?
 
  • #3
Kinda. E and J are constants... like in the KEpler problem. Tell me more. :)
 
  • #4
quasar987 said:
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!
 
  • #5
I seem to have like 5 different expression for each J and E in my notes. Picking 2 that seem linked:

[tex]J=\frac{d\phi}{dp}r^2[/tex]

[tex]E=1-\left(\frac{dr}{dt}\right)^2[/tex]
 
  • #6
quasar987 said:
I seem to have like 5 different expression for each J and E in my notes. Picking 2 that seem linked:

[tex]J=\frac{d\phi}{dp}r^2[/tex]

[tex]E=1-\left(\frac{dr}{dt}\right)^2[/tex]

I was looking for

[tex]E = \left( 1 - \frac{2M}{r} \right) \frac{dt}{d\tau}[/tex]

[tex]J = r^2 sin^2 \theta \frac{d\phi}{d\tau}.[/tex]

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

Assume that the orbit is in the plane [itex]\theta = \pi/2,[/itex] and use the conserved quantities to eliminate [itex]dt/d\tau[/itex] and [itex]d\phi/d\tau[/itex] in the metric.
 
  • #7
George Jones said:
Assume that the orbit is in the plane [itex]\theta = \pi/2,[/itex] and use the conserved quantities to eliminate [itex]dt/d\tau[/itex] and [itex]d\phi/d\tau[/itex] in the metric.

I take it you meant "in the geodesic equations".
 
  • #8
quasar987 said:
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 [itex]dt[/itex] and [itex]d\phi[/itex] in the Schwarzschild metric. This will lead to the useful concept of effective potential.
 
  • #9
Once [itex]dt[/itex] and [itex]d\phi[/itex] have been eliminated, solve for

[tex]\left( \frac{dr}{d\tau} \right)^2[/tex]

as a function of [itex]r[/itex], and the constants [itex]E[/itex] and [itex]L[/itex]. 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.
 
  • #10
Ah, I see!

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

But now I know, thx.
 

1. What is the GR problem?

The GR problem, also known as the Einstein field equations, is a set of equations in general relativity that describe how matter and energy affect the curvature of spacetime. It is used to calculate the motion of objects in a gravitational field, such as a satellite in orbit.

2. What is a satellite in orbit?

A satellite in orbit is an object that is in constant motion around a larger body, such as a planet or star. It follows a curved path due to the gravitational pull of the larger body.

3. How does the Schwarzschild geometry affect the motion of a satellite in orbit?

The Schwarzschild geometry, also known as the Schwarzschild solution, is a mathematical solution to the GR problem that describes the gravitational field outside a spherical, non-rotating mass. In the case of a satellite in orbit, the Schwarzschild geometry determines the shape of the orbit and the rate at which the satellite moves around the larger body.

4. What factors influence the orbit of a satellite in Schwarzschild geometry?

The orbit of a satellite in Schwarzschild geometry is influenced by several factors, including the mass and size of the larger body, the mass and velocity of the satellite, and the distance between the two bodies. These factors affect the strength of the gravitational force and the curvature of spacetime, which in turn determine the shape and stability of the orbit.

5. Can the GR problem accurately predict the motion of a satellite in orbit?

Yes, the GR problem has been extensively tested and has been shown to accurately predict the motion of satellites in orbit. It has been used in numerous space missions, such as the orbit of the Moon around Earth and the orbit of the planets around the Sun, with high levels of precision and accuracy.

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