An exact solution for arbitrarily rotating gaseous polytropes with index unity

Abstract

Many gaseous planets and stars are rapidly rotating and can be approximately described by a polytropic equation of state with index unity. We present the first exact analytic solution, under the assumption of the oblate spheroidal shape, for an arbitrarily rotating gaseous polytrope with index unity in hydrostatic equilibrium, giving rise to its internal structure and gravitational field. The new exact solution is derived by constructing the non-spherical Green's function in terms of the oblate spheroidal wavefunction. We then apply the exact solution to a generic object whose parameter values are guided by the observations of the rapidly rotating star α Eridani with its eccentricity E α = 0.7454, the most oblate star known. The internal structure and gravitational field of the object are computed from its assumed rotation rate and size. We also compare the exact solution to the three-dimensional numerical solution based on a finite-element method taking full account of rotation-induced shape change and find excellent agreement between the exact solution and the finite-element solution with about 0.001 per cent discrepancy.Key words: planets and satellites: gaseous planets -planets and satellites: interiors -stars: interiors. I N T RO D U C T I O NA fully compressible polytropic gas with index unity obeying the polytropic equation of state (EOS)where p * is the pressure, K is a constant and ρ * is the density, has been widely employed to study the physical properties of gaseous planets, exoplanets and stars (see for example, Chandrasekhar 1933; Roberts 1962;Hubbard 1973;Stevenson 1982;Dintrans & Ouyed 2001;Horedt 2004;Kong et al. 2014). In this paper, the superscript * is adopted to represent a dimensional variable and its corresponding dimensionless variable is denoted without the superscript. Many astrophysical gaseous bodies are rapidly rotating, causing significant departure from sphericity: the eccentricity at the one-bar surface is E S = 0.4316 for Saturn (Seidelmann et al. 2007) while the star α Eridani is marked by a much larger departure from sphericity with the approximate eccentricity E α = 0.7454 (Carciofi et al. 2008). The rotational effect on the shape and physical structure of a slowly rotating polytrope was first studied by Chandrasekhar (1933) using a perturbation analysis. For an isolated, non-rotating and selfgravitating body, the density distribution ρ * within the interior of E-mail: kzhang@ex.ac.uk the polytrope is spherically symmetric and described by the LaneEmden equation, a second-order ordinary differential equation that can be readily solved to determine the one-dimensional density distribution. For a polytropic body that is slowly rotating with small angular velocity such that its departure from sphericity is slight, Chandrasekhar (1933) introduced a small parameter ∼ 2 and was able to solve for the density distribution ρ * of the slowly rotating polytrope via a perturbation method in terms of the small expansion parameter . Without developing a small parameter expansion, R...

The shape of a rapidly rotating polytrope with index unity

“…1-3 we plot the shapes of three of the obtained configurations. Because Kong et al (2015) discuss a model of α Eri with an ellipsoidal surface, we believe that a comparison of the numerical solution and their analytic approximation for the observed parameters of α Eri would be in order. Such a comparison is shown in the first of the presented figures.…”

“…1 was computed with the observed mass M = 4.9M , the equatorial radius Re = 12.0R , and the angular velocity Ω = 2.9725 × 10 −5 s −1 , as quoted by Kong et al (2015, Table 1). The values of solar parameters used in the code are: M = 1.98855 × 10 33 g, R = 6.95700 × 10 10 cm.…”

“…1. For comparison, we also plot an ellipse with the semi-axes that were obtained by Kong et al (2015) and an ellipse fitted to the obtained shape, i.e., an ellipse with semi-axes equal to the values of Rp and Re of our numerical model. There is a clear discrepancy between all these shapes.…”

“…A modern account of these works can be found in Horedt (2004). Kong et al (2015) assume that the shape of the boundary surface of such polytropes is an ellipsoid of revolution and construct analytic solutions of the corresponding equation for the distribution of the density. In this note we show that they satisfy only approximately the coupled system of Euler and Poisson equations describing equilibrium configurations of rigidly rotating polytropes with n = 1.…”

“…As expected, the deviation from the ellipsoidal shape is significant for rapidly rotating configurations. We also pay special attention to the model that was computed in Kong et al (2015) assuming observational parameters of α Eri.…”