What is the relationship between orbital energy and gravitational waves?

In summary: For non-circular orbits:The change in gravitational binding energy is equal to the change in gravitational potential energy.
  • #1
kmarinas86
979
1
The emission of gravational waves changes orbital energy right?

If orbital energy [itex]O[/itex] is a function of radius, what would [itex]O_{final}-O_{initial}[/itex] be, using variables such as [itex]G[/itex], [itex]M[/itex], and [itex]r[/itex]?
 
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  • #2
Can you clarify the question?

Hi, kmarinas86 (presumably also the Wikipedia user with a similar name?),

Can you be more specific about the scenario you have in mind? Are you talking about two isolated massive objects in quasi-Keplerian motion, treated according to the quadrupole approximation in weak-field gtr?

Chris Hillman
 
  • #3
Assume a circular orbit for simplicity, then work out the kinetic and potential energy of an object of mass m orbiting a distance r from a static object of mass M (M>>m for simplicity). Then consider it in a circular orbit at a distance R with R<r, and then work out the difference. In the absense of any other effects, the loss of energy will be due to gravitational radiation, to a decent approximation I'd imagine.
 
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  • #4
AlphaNumeric said:
Assume a circular orbit for simplicity, then work out the kinetic and potential energy of an object of mass m orbiting a distance r from a static object of mass M (M>>m for simplicity). Then consider it in a circular orbit at a distance R with R<r, and then work out the difference. In the absense of any other effects, the loss of energy will be due to gravitational radiation, to a decent approximation I'd imagine.

[itex]v_{circular,r}=\sqrt{GM/r}[/itex]

[itex]v_{circular,R}=\sqrt{GM/R}[/itex]

[itex]KE_r=.5mv_{circular,r}^2=.5mGM/r[/itex]

[itex]KE_R=.5mv_{circular,R}^2=.5mGM/R[/itex]

[itex]KE_R-KE_r=.5mGM\left(1/R-1/r)[/itex]

That was simple enough.

According to what you say, wouldn't gravitational radiation also be emitted equal to half the change in gravitational potential such that the there is:

[itex]\Delta KE=.5mGM\left(1/R-1/r)[/itex]

[itex]\Delta gravitational\ radiation=.5mGM\left(1/R-1/r)[/itex]

But what about heat?

For non-circular orbits:

I know (from other sources) that for a collapsing gravitating object, half of the change in the gravitational binding energy will lead to kinetic energy inside the object and the other half will leave as electromagnetic radiation (per the virial theorem). How do I distinguish and understand the value of binding energy in comparison to the value of gravitational potential energy? Is there a clear path from [itex]GM^2/r[/itex] to [itex]GMm/r[/itex] ? Now this a bit tricky (I think). For pairs of objects originally at infinite distance from each other, the change in kinetic energy is [itex].5GMm/r_{in\ between}[/itex], but for forming stars, the gravitational contribution to kinetic energy (I guess) is [itex].5GM^2/r_{star}[/itex]... I know it's a lot of questions. But I hope you can answer some of them at least.
 
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Related to What is the relationship between orbital energy and gravitational waves?

What are gravitational waves?

Gravitational waves are ripples in the curvature of spacetime, caused by the acceleration of massive objects. They were first predicted by Albert Einstein in his theory of general relativity.

How are gravitational waves created?

Gravitational waves are created when massive objects, such as black holes or neutron stars, accelerate and produce changes in the curvature of spacetime. These changes propagate outward as waves at the speed of light.

How do we detect gravitational waves?

Gravitational waves are detected using highly sensitive instruments called interferometers. These devices use lasers to measure tiny changes in the distance between two points, caused by the passing of a gravitational wave.

What is the energy of gravitational waves?

The energy of gravitational waves is directly proportional to the mass and acceleration of the objects that created them. Gravitational waves carry a very small amount of energy, but over large distances and time scales, they can have a significant impact on the universe.

What is the importance of studying gravitational waves?

Studying gravitational waves allows us to observe and understand the universe in a completely new way. They provide information about the nature of space and time, as well as the behavior of massive objects. Gravitational waves can also help us to detect and study objects that are invisible to traditional telescopes, such as black holes.

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