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Normal gravity of a sphere and the small flattening limit of an ellipsoid #194
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And here are some images showing how the small angle-limit formulas compare with the standard equations for the parameters
For the case of Earth, you can see the approximate formulas are good for flattenings less than about Inspecting how the standard equations degrade in precision with decreasing flattening (for bodies the size of Earth and Ceres), the small-angle equations should be used in the following situations:
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PR #197 adds the small flattening equations to the I propose that we create a new method, such as
that will return an ellipsoid class instance with the same parameters of the Sphere class, but with flattening set to 0. We could also entertain making
but at this moment, I see no reason to do so, given the limited functionality of the TriaxialEllipsoid class. |
We previously had a discussion about how to define normal gravity on a sphere. At the time, I think that I was probably a little confused, but now that I see how Boule works and defines things, and after have looked through Chapter 2 of Physical Geodesy again, I think that the way we treat the Sphere class is not entirely consistent with the Ellipsoid class.
Start with the properties of a reference ellipsoid, as found in the boule documentation:
However, for a sphere, the assumptions are different:
Because of this, if you take an Ellipsoid, and gradually decrease the flattening to zero, you will not asymptotically approach the results of a sphere. This is because the gravity potential is not constant on the surface of the sphere, but it is on an ellipsoid.
Now, I agree that the assumptions for the Sphere class are correct if we assume that the body is a fluid in hydrostatic equilibrium. However, we can add arbitrary density anomalies in the mantle in order to generate a gravity potential (gravitation+centrifugal) at the surface that is constant, just like in the case for Ellipsoid. The mathematical problem is that if you put
flattening=0
in the Ellipsoid equations, you will get divide by zero errors, but as I will show below, these can be avoided by taking the limit where the semiminor axis approaches the semimajor axis.What I propose is the following: use the same assumptions for Sphere as for Ellipsoid, and then with the equations in Ellipsoid, find the asymptotic limits as the flattening goes to zero. I note that the following equations could probably have been more easily derived starting with spherical harmonics instead of ellipsoidal harmonics, but I will follow the geodesy tradition of doing things the hard way!
reference normal gravity potential
Start with eq. 2-123,
and use the small-angle approximation (only the first two terms are necessary here)
When$b$ approaches $a$ and $E$ goes towards zero, we have
For the case of a sphere with$b=a$ , the reference potential is
normal gravitation potential
Eq. 2-124 for the normal gravitation potential (where$\beta$ is the reduced latitude)
has the offending term
Using the higher-order small angle approximation of the arctan function above (all three terms are necessary), we find that
and
As$b$ approaches $a$ the normal gravity potential approaches
For the case of a sphere, with$a=b$ , $u=r$ and $\beta =\phi$ we have
the normal gravity potential 2-124 is
where$\phi$ is the spherical geocentric latitude and $r$ is geocentric spherical radius.
normal gravity potential
The normal gravity potential
also has the offending$q/q_0$ term. Substituting the above approximation into 2-126, as $b$ approaches $a$ , the normal gravity potential goes to
For the case of a sphere with$a=b$ , $u=r$ , and $\beta=\phi$ , the normal gravity potential is
normal gravity
The normal gravity on the ellipsoid is given by 2-146, which is Somilgiana's formula. For this we need the normal gravity at the pole and equator. Each of these terms has the offending ratio
where
Using the small-angle approximation (all three terms are necessary), we find that
and that
The gravity at the equator, eq 2-141, is
and the gravity at the pole, eq 2-142, is
For the case of a sphere with$a=b$ , the above two equations reduce to
and
which yields for the normal gravity
When computing the normal gravity above the ellipsoid, we use the equations in the appendix of Li and Götze (2001). These equations have two offending terms when the flattening goes to zero. One is nearly the same as computed above:
The second involves the term concerning$\cos \beta'$ , which can be rewritten as
where$R=r^{\prime \prime 2} / E^2$ and $D=d^{\prime \prime 2} / E^2$ . Using the approximation
for the inner square root in the above expression, in the limit where$E$ approaches zero, we find
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