Why does geopotential not take into account the rotation of the Earth?

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In summary, the rotation of the Earth has no direct effect on the potential energy of a satellite orbiting the Earth.
  • #1
baxter
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Hi

Geopotential U is the gravitational potential of the Earth without taking into account the rotation of the Earth whereas gravity's potential ##U_T## considers the rotation of the Earth such as :

##U_T = U + \frac{1}{2}\omega ^2r^2cos(\psi)^2## (with common spherical coordinates)

However, for a satellite, the potential of the Earth undergone by the satellite is ##U## and not ##U_T##.

My question : why?

In fact, I agree the satellite does not belong the the surface of the Earth but it seems that the geopotential ##U## does not consider the rotation of the Earth ? It would say that the rotation of the Earth does not affect the gravitational force of the Earth on the satellite ?
 
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  • #2
The Newtonian answer is simple: The satellite isn't sitting on the surface of the Earth. The Earth's rotation rate has zero direct effect on the satellite.

The rotation of the Earth does have an indirect effect on the satellite because the Earth is not spherically symmetric. The Earth instead has a somewhat lumpy density distribution. The equatorial bulge is by far the biggest non-spherical effect of the Earth's rotation on satellites. This bulge is a direct consequence of the Earth's rotation. It is what makes Sun synchronous orbits possible. A prolate ellipsoid model yields a good but not perfect model of the Earth's gravity field. There are other deviations from a spherically distributed density. For example, the northern hemisphere has considerably more land mass than does the southern hemisphere, and the Pacific is much bigger than is the Atlantic.

In a Newtonian universe, there would be nothing left after removing all of these effects that result from the Earth's orientation. The Earth's angular velocity itself has no effect on the satellite's motion.

That's not quite true in general relativity. There is a frame dragging effect in general relativity that is tiny (very, very tiny!) in the case of the Earth. This frame dragging effect is much more sizable for an object closely orbiting a rapidly rotating neutron star.
 
  • #3
baxter said:
Hi

Geopotential U is the gravitational potential of the Earth without taking into account the rotation of the Earth whereas gravity's potential ##U_T## considers the rotation of the Earth such as :

##U_T = U + \frac{1}{2}\omega ^2r^2cos(\psi)^2## (with common spherical coordinates)

However, for a satellite, the potential of the Earth undergone by the satellite is ##U## and not ##U_T##.

My question : why?

In fact, I agree the satellite does not belong the the surface of the Earth but it seems that the geopotential ##U## does not consider the rotation of the Earth ? It would say that the rotation of the Earth does not affect the gravitational force of the Earth on the satellite ?

It depends on what you want to know.

U relates to gravitational force solely due to the Earth (or the potential energy since the mass of the second object isn't actually known yet).

U_t relates to the net gravitational force felt by a person on Earth. The rotation of the Earth is relevant only because that rotation is imparting motion to the person and the motion of the person (or object) creates a centrifugal force that opposes gravity.

Both would be useful bits of information on Earth.

U_t would depend on the satellite's motion; not the rotation of the Earth. U_t wouldn't yield much useful information for a satellite in a circular orbit, as the net "weight" felt by the satellite would have to be zero in order for the satellite to be in a circular orbit.

"Weightlessness" doesn't mean the lack of a gravitational force. It means the centrifugal force created by the person's (or satellite's) velocity cancels out the gravitational force.
 
  • #4
Thanks for your explanations, they are very useful ;)
 
  • #5


Thank you for your question. The geopotential, also known as the Earth's gravitational potential, is a measure of the gravitational potential energy of a unit mass at a particular point on the Earth's surface. It is used to describe the variation in the Earth's gravity field due to the distribution of mass within the Earth. This potential energy is directly related to the acceleration due to gravity at that point.

The reason why geopotential does not take into account the rotation of the Earth is because it is only concerned with the gravitational force between the Earth and the satellite, and not the centrifugal force caused by the Earth's rotation. The centrifugal force is a fictitious force that arises due to the rotation of a non-inertial reference frame, such as the surface of the Earth. It does not affect the gravitational force between two objects.

In other words, the geopotential assumes a non-rotating reference frame, where the Earth is a perfect sphere with a uniform distribution of mass. This simplification allows for easier calculations and modeling of the Earth's gravity field. However, for more precise calculations, the Earth's rotation and its effects on the gravity field must be taken into account, which is where the gravity potential ##U_T## comes into play.

I hope this clarifies why geopotential does not take into account the rotation of the Earth. As scientists, we continuously strive to improve our understanding and models of the Earth's complex systems, and the inclusion of rotation in geopotential calculations is just one example of this ongoing effort.
 

1. What is geopotential?

Geopotential is a measure of the potential energy per unit mass of an object due to its position relative to the Earth's gravity field. It is often used in geodesy and geophysics to describe the variation in the strength of the Earth's gravity field.

2. How is geopotential different from gravitational potential?

Geopotential takes into account the Earth's rotation, while gravitational potential does not. This means that geopotential is a more accurate measure of the strength of the Earth's gravity field.

3. How do scientists measure geopotential?

Scientists use a variety of techniques to measure geopotential, including satellite-based measurements, ground-based measurements using gravimeters, and mathematical models that take into account the Earth's shape and rotation.

4. What is the relationship between geopotential and gravity?

Geopotential and gravity are closely related, as geopotential is a measure of the potential energy per unit mass of an object due to its position in the Earth's gravity field. As the strength of the Earth's gravity varies, so does the geopotential.

5. How does geopotential affect the Earth's topography?

Geopotential has a significant impact on the Earth's topography, as it affects the shape of the Earth's surface and the distribution of land and water. The variation in geopotential also contributes to the formation of features such as mountains and valleys.

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