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# On the spherical geopotential approximation for Saturn

The author is supported by NSF grant DMS-2102961

• 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: • • 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 

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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