A Three-dimensional Numerical Solution for the Shape of a Rotationally Distorted Polytrope of Index Unity
Abstract
We present a new three-dimensional numerical method for calculating the non-spherical shape and internal structure of a model of a rapidly rotating gaseous body with a polytropic index of unity. The calculation is based on a finite-element method and accounts for the full effects of rotation. After validating the numerical approach against the asymptotic solution of Chandrasekhar that is valid only for a slowly rotating gaseous body, we apply it to models of Jupiter and a rapidly rotating, highly flattened star (α Eridani). In the case of Jupiter, the two-dimensional distributions of density and pressure are determined via a hybrid inverse approach by adjusting an a priori unknown coefficient in the equation of state until the model shape matches the observed shape of Jupiter. After obtaining the two-dimensional distribution of density, we then compute the zonal gravity coefficients and the total mass from the non-spherical model that takes full account of rotation-induced shape change. Our non-spherical model with a polytropic index of unity is able to produce the known mass of Jupiter with about 4% accuracy and the zonal gravitational coefficient J 2 of Jupiter with better than 2% accuracy, a reasonable result considering that there is only one parameter in the model. For α Eridani, we calculate its rotationally distorted shape and internal structure based on the observationally deduced rotation rate and size of the star by using a similar hybrid inverse approach. Our model of the star closely approximates the observed flattening.
- Publication:
-
The Astrophysical Journal
- Pub Date:
- February 2013
- DOI:
- 10.1088/0004-637X/763/2/116
- Bibcode:
- 2013ApJ...763..116K
- Keywords:
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- gravitation;
- planets and satellites: individual: Jupiter;
- planets and satellites: interiors;
- stars: individual: α Eridani