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 wave function. We then apply the exact solution to a young extrasolar gas giant planet, β Pictoris b. Based on the newly measured parameters – the rotation period of the planet, its mass and radius, we are able to compute its non-spherical shape, internal density/pressure distribution and gravitational zonal coefficients.