In this paper, we show by means of a diffeomorphism that when approximating the planet Saturn by a sphere, the errors associated with the spherical geopotential approximation are so significant that this approach is rendered unsuitable for any rigorous mathematical analysis.
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Figure 2. The approximate dimensions, in kilometers, of Saturn's layers, measured along the equator; see [14]
Figure 4. The spherical and geopotential coordinate systems for fixed longitude $ \varphi $. Here $ \theta $ denotes the geocentric latitude of the point $ Q $. The normal vector of the ellipsoid $ \mathcal{E} $ at $ Q $ intersects the equatorial plane at the focus point $ A $, at an angle $ \beta $, called the geodetic latitude angle of $ P $
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A cross section of Saturn depicting the composition of its interior
The approximate dimensions, in kilometers, of Saturn's layers, measured along the equator; see [14]
Saturn is approximated by an ellispoid
The spherical and geopotential coordinate systems for fixed longitude
Saturn viewed as an ellipsoid with a point
The spherical geopotential approximation for fixed longitude