The giant planets in the Solar System are rotating rapidly, which results in flattening at the poles. This characteristic can provide constraints on their internal structure. Here we follow the discussion by Guillot (2006), based on a work initiated by Lagrange, Clairaut, Darwin and Poincare, and detailed by Zharkov and Trubitsyn (1978).
We have seen that the gravitational potential V (r, q) may be expressed at a function of the gravitational moments and Legendre polynomials (see 7.1.1). Similarly, the centrifugal potential W(r, q) may be expressed as:
where P2 is the second-degree Legendre polynomial. The equipotentials for a planet are defined as the surfaces at which the total potential U = V + W is constant. These are also pressure surfaces where pressure, density, and temperature are constant.
We introduce the factor q, defined as the ratio between the centrifugal acceleration at the equator and the gravitational acceleration's principal term:
Re is the equatorial radius of the object. From the parameter q, which has been measured for the four giant planets, it is possible to calculate the axial moment of symmetry from the following equation:
MRl M 5
We find a value for C/[MR^] that lies between 0.22 and 0.26, significantly less than the corresponding value for a sphere of homogeneous density (0.40), which indicates that the giant planets are differentiated, with a central region that is more dense.
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