• Support PF! Buy your school textbooks, materials and every day products Here!

GR problem - satelite in orbit (Schwarzschild geometry)

  • Thread starter quasar987
  • Start date
  • #1
quasar987
Science Advisor
Homework Helper
Gold Member
4,773
8

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.
 

Answers and Replies

  • #2
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,270
808
Are you familiar with the constants of motion for Schwarzschild orbits?

With the effective potential for Schwarzschild orbits?
 
  • #3
quasar987
Science Advisor
Homework Helper
Gold Member
4,773
8
Kinda. E and J are constants... like in the KEpler problem. Tell me more. :)
 
  • #4
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,270
808
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
quasar987
Science Advisor
Homework Helper
Gold Member
4,773
8
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
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,270
808
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
quasar987
Science Advisor
Homework Helper
Gold Member
4,773
8
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
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,270
808
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
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,270
808
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
quasar987
Science Advisor
Homework Helper
Gold Member
4,773
8
Ah, I see!

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

But now I know, thx.
 

Related Threads on GR problem - satelite in orbit (Schwarzschild geometry)

Replies
6
Views
4K
  • Last Post
Replies
9
Views
3K
Replies
2
Views
5K
Replies
16
Views
1K
Replies
5
Views
1K
Replies
1
Views
915
Replies
6
Views
544
Replies
1
Views
689
Top