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Re: [Phys-L] field of an ellipsoidal distribution



There are two or three ways of looking at the Somigliana equation.

1) For a fluid planet in equilibrium, the equation is
firmly based on the physics, and you can calculate
some of the parameters from first principles. For
uniform density, you can calculate all of the
parameters.

2a) As applied to the earth, you could consider it to
be invalid. It's "equation hunting" ... grabbing
some random equation and applying it outside its
domain of validity.

The earth is clearly not a fluid, as you can see in
the dry land and (especially) the mountain ranges.
It's not in equilibrium, as you can see in plate
tectonics et cetera.

2b) On the other hand, the equation is a good fit to
the data! You can treat it as a purely empirical
fitting function, with three adjustable parameters.
As such, it is a better fit than any of the other
three-parameter model functions I know of.

Let's be clear: Ideally the eccentricity that plugs
into the Somigliana equation "should" be independently
determined by measuring the polar radius and the
equatorial radius ... but it's not. Obviously it
can't be, because there are high mountains at the
south pole but not at the north. So instead, we
treat the e-value as just another adjustable parameter.
We tweak all the parameters to get the best fit to
the data.

============

Note that all of the fitted functions we are talking
about give you the /sea level/ gravity, as a function
of latitude alone. For best results, you need to add
another term, proportional to elevation.

============================

The big lesson here is that if you are trying to cook
up a fitted function, you will get better results if
you use something with more-or-less the right functional
form.

For example, as the saying goes:
No polynomial ever had an asymptote, horizontal or vertical.
Therefore sometimes you are better off fitting to a
rational function rather than a polynomial.

Obviously if the physics is periodic, you are better
off fitting a Fourier series rather than a plain old
polynomial.

If the physics tells you it's an odd function, you are
better off keeping only the odd-degree terms in the
polynomial, or only the "sine" terms in the Fourier
series.

And so on.

So, even if the Somigliana equation doesn't capture
all of the physics, it captures some ... and that
works to your advantage.