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On the spherical geopotential approximation for Saturn

The author is supported by NSF grant DMS-2102961

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  • 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.

    Mathematics Subject Classification: Primary: 86-10.

    Citation:

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  • Figure 1.  A cross section of Saturn depicting the composition of its interior

    Figure 2.  The approximate dimensions, in kilometers, of Saturn's layers, measured along the equator; see [14]

    Figure 3.  Saturn is approximated by an ellispoid $ \mathcal{E} $, with polar radius $ d_P' $ and equatorial radius $ d_E' $. The polar radius is equal to only about $ 90.2\% $ of the equatorial one. In comparison, for Earth the polar radius is $ 99.67\% $ of the equatorial one

    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 $

    Figure 5.  Saturn viewed as an ellipsoid with a point $ P $ on its surface expressed in terms of the geocentric latitude $ \alpha $. The angle $ \alpha $ is related to the geodetic latitude $ \beta $ by (2.1)

    Figure 6.  The spherical geopotential approximation for fixed longitude $ \varphi $. The point $ P $ on the ellipsoid is mapped into $ \hat{P} $ on the sphere of radius $ R' $. This is obtained by setting the geocentric latitude angle of $ \hat{P} $ to be equal to the geodetic latitude of $ P $

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