# American Institute of Mathematical Sciences

July  2022, 21(7): 2327-2336. doi: 10.3934/cpaa.2022035

## On the spherical geopotential approximation for Saturn

 Department of Mathematics, Brown University, Box 1917, Providence, RI 02912, USA

Received  October 2021 Revised  December 2021 Published  July 2022 Early access  February 2022

Fund Project: 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.

Citation: Susanna V. Haziot. On the spherical geopotential approximation for Saturn. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2327-2336. doi: 10.3934/cpaa.2022035
##### References:
 [1] A. Affholder, F. Guyot, B. Sauterey, R. Ferrière and S. Mazevet, Bayesian analysis of Enceladus's plume data to assess methanogenesis, Nat. Astron., 8 (2021), 805-814. [2] P. Benard, An oblate-spheroid geopotential approximation for global meteorology, Quart. J. Roy. Meterol. Soc., 140 (2014), 170-184. [3] H. Busemann and W. Feller, Krümmungseigenschaften konvexer Flächen, Acta Math., 66 (1936), 1-47.  doi: 10.1007/BF02546515. [4] A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flow, J. Differ. Equ., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019. [5] A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. A, 477 (2021), 25 pp. [6] A. Constantin, D. G. Crowdy, V. S. Krishnamurthy and M. H. Wheeler, Stuart-type polar vortices on a rotating sphere, Discrete Cont. Dyn. Syst., 41 (2021), 201-215.  doi: 10.3934/dcds.2020263. [7] G. Faure and T. M. Mensing, Introduction to Planetary Science: The Geological Perspective, Springer, Dordrecht, The Netherlands, 2007. [8] R. A. Freedman and W. J. Kaufmann III, Universe: The Solar System, Freeman, New York, NY, 2002. [9] B. Galperin and P.L. Read, Zonal Jets: Phenomenology, Genesis and Physics, Cambridge University Press, Cambridge, 2019. [10] E. Gregersen, The Outer Solar System, Britannica Educational Publishing, New York, NY, 2009. [11] P. Gruber, Convex and Discrete Geometry, Springer-Verlag, Berlin-Heidelberg, 2007. [12] W. K. Hartmann, Moons and Planets, Brooks/Cole, Belmont, CA, 2005. [13] A. P. Ingersoll, et al., Atmospheres of the giant planets in The New Solar System, Sky Publishing, Cambridge, MA, 1999. [14] J. E. Klepeis, Hydrogen-helium mixtures at megabars pressure: Implications for Jupiter and Saturn, Science, 254 (1991), 986-989. [15] J. Lunine, Saturn's Titan: A Strict Test for Life's Cosmic Ubiquity, Proceed. Amer. Phil. Soc., 153 (2009), 403-418. [16] NASA/Jet Propulsion Laboratory, Life on Titan? New clues to what's consuming hydrogen, acetylene on Saturn's moon, Sci. Daily, 2010. [17] A. Staniforth, Spherical and spheroidal geopotential approximations, Quart. J. Roy. Meterol. Soc., 140 (2014), 2685-2692.

show all references

##### References:
 [1] A. Affholder, F. Guyot, B. Sauterey, R. Ferrière and S. Mazevet, Bayesian analysis of Enceladus's plume data to assess methanogenesis, Nat. Astron., 8 (2021), 805-814. [2] P. Benard, An oblate-spheroid geopotential approximation for global meteorology, Quart. J. Roy. Meterol. Soc., 140 (2014), 170-184. [3] H. Busemann and W. Feller, Krümmungseigenschaften konvexer Flächen, Acta Math., 66 (1936), 1-47.  doi: 10.1007/BF02546515. [4] A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flow, J. Differ. Equ., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019. [5] A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. A, 477 (2021), 25 pp. [6] A. Constantin, D. G. Crowdy, V. S. Krishnamurthy and M. H. Wheeler, Stuart-type polar vortices on a rotating sphere, Discrete Cont. Dyn. Syst., 41 (2021), 201-215.  doi: 10.3934/dcds.2020263. [7] G. Faure and T. M. Mensing, Introduction to Planetary Science: The Geological Perspective, Springer, Dordrecht, The Netherlands, 2007. [8] R. A. Freedman and W. J. Kaufmann III, Universe: The Solar System, Freeman, New York, NY, 2002. [9] B. Galperin and P.L. Read, Zonal Jets: Phenomenology, Genesis and Physics, Cambridge University Press, Cambridge, 2019. [10] E. Gregersen, The Outer Solar System, Britannica Educational Publishing, New York, NY, 2009. [11] P. Gruber, Convex and Discrete Geometry, Springer-Verlag, Berlin-Heidelberg, 2007. [12] W. K. Hartmann, Moons and Planets, Brooks/Cole, Belmont, CA, 2005. [13] A. P. Ingersoll, et al., Atmospheres of the giant planets in The New Solar System, Sky Publishing, Cambridge, MA, 1999. [14] J. E. Klepeis, Hydrogen-helium mixtures at megabars pressure: Implications for Jupiter and Saturn, Science, 254 (1991), 986-989. [15] J. Lunine, Saturn's Titan: A Strict Test for Life's Cosmic Ubiquity, Proceed. Amer. Phil. Soc., 153 (2009), 403-418. [16] NASA/Jet Propulsion Laboratory, Life on Titan? New clues to what's consuming hydrogen, acetylene on Saturn's moon, Sci. Daily, 2010. [17] A. Staniforth, Spherical and spheroidal geopotential approximations, Quart. J. Roy. Meterol. Soc., 140 (2014), 2685-2692.
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 $\mathcal{E}$, 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
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 $P$
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)
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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