Calculating gravity in rotating frame(ECR)

In summary, the paper discusses the use of a spherical harmonic expansion of gravitational force and the addition of a phi potential to account for the Earth's rotation. The phi potential includes contributions from both the centrifugal force and Coriolis acceleration, but the Coriolis component is constant while the centrifugal component depends on position. When rotating from ECR to ECI, a Coriolis term must be added as it is not included in the phi potential.
  • #1
Banjo
2
0
In reading through URL=ftp://164.214.2.65/pub/gig/tr8350.2/wgs84fin.pdf]this[/URL] pdf I came across the potential described on pg 51. I have no problem with the spherical harmonic expansion of the gravitational force, however I'm a bit confused about the phi potential added to account for the Earth's rotation. I think this only gives the the "centrifugal" force. Assuming the gravity vector that results is in ECR (Earth Centered Rotating, if I was going to rotate the acceleration given by this formula into ECI (Earth Centered Inertial)(from ECR), wouldn't I also need to add a coriolis term? or at least not add the coriolis term when doing the rotation? I have been told otherwise which is why I ask. BTW the position must be in ECR as the formula for V, the gravitational potential, is in terms of geodetic lattitude and longitude.
 
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  • #2
The phi potential as defined in the paper you linked to is related to the Coriolis force, as it contains contributions from the centrifugal force and Coriolis acceleration. The difference between the two is that the Coriolis component of the phi potential is constant (i.e. it does not depend on the position of the body), while the centrifugal component depends on the position. Therefore, when performing a rotation from ECR to ECI, you would need to add a Coriolis term, as this term is not included in the phi potential.
 
  • #3


The calculation of gravity in a rotating frame is a complex topic and can be confusing, so it's understandable that you have some questions about it. Let's break down the different components and clarify their roles.

First, we have the spherical harmonic expansion of the gravitational force. This is a mathematical representation of the Earth's gravitational field, which takes into account the varying density of the Earth's mass distribution. This is necessary because the Earth is not a perfect sphere and the gravitational force varies slightly depending on your location on its surface.

Next, we have the phi potential, which is added to account for the Earth's rotation. This potential accounts for the centrifugal force, which is the perceived outward force due to the Earth's rotation. This is necessary because in a rotating frame, objects appear to be pushed outward, and this needs to be accounted for in the calculation of gravity.

Now, you are correct in saying that if you were to rotate the acceleration given by this formula into the Earth Centered Inertial (ECI) frame, you would also need to add a Coriolis term. The Coriolis force is a result of the Earth's rotation and is responsible for the apparent deflection of objects moving in a rotating frame. So, if you were to rotate the acceleration into the ECI frame, you would need to add the Coriolis term to account for this deflection.

However, if you are already working in the Earth Centered Rotating (ECR) frame, the Coriolis term is not necessary. The formula for V, the gravitational potential, is in terms of geodetic latitude and longitude, which are coordinates in the ECR frame. So, if you are using these coordinates, you do not need to add the Coriolis term.

In summary, the phi potential accounts for the centrifugal force in the ECR frame, and if you were to rotate the acceleration into the ECI frame, you would also need to add the Coriolis term. But if you are already working in the ECR frame, the Coriolis term is not necessary. I hope this helps clarify things for you.
 

1. How is gravity calculated in a rotating frame?

In a rotating frame, the apparent gravity experienced by an object is a combination of the gravitational force and the centrifugal force. The formula for calculating gravity in a rotating frame, also known as the ECR (Earth-Centered Rotating) frame, is given by:
G = GM/r^2 + ω^2r,
where G is the gravitational constant, M is the mass of the central body, r is the distance from the center of the central body, and ω is the angular velocity of the rotating frame.

2. How does the rotation of the Earth affect the calculation of gravity?

The rotation of the Earth affects the calculation of gravity in two ways. First, it introduces the centrifugal force, which is a fictitious force that appears to act on objects due to their rotation. Second, it causes a variation in the value of g, the acceleration due to gravity, at different points on the Earth's surface. This is due to the variation in the distance from the center of the Earth to the surface at different latitudes.

3. Can the ECR frame be used for any rotating body?

Yes, the ECR frame can be used for any rotating body, not just the Earth. The formula for calculating gravity in a rotating frame is applicable to any rotating frame, as long as the central body's mass and the angular velocity of the frame are known.

4. How does the ECR frame differ from the inertial frame?

The ECR frame is a non-inertial frame, meaning it is accelerating due to the rotation of the Earth. In contrast, the inertial frame is a reference frame that is not accelerating. In the ECR frame, the laws of motion and gravity are different from those in the inertial frame, as they need to account for the centrifugal force. However, as the Earth's rotation is relatively slow, the ECR frame can be approximated as an inertial frame for most practical purposes.

5. Is the ECR frame necessary for calculating gravity on Earth?

No, the ECR frame is not necessary for calculating gravity on Earth. The simpler Newtonian formula, F = GMm/r^2, is sufficient for most practical purposes. The ECR frame is necessary when dealing with more complex scenarios, such as calculating the motion of satellites or objects near the Earth's poles, where the effect of the Earth's rotation on gravity cannot be neglected.

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