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# Appendix A: Hartle's formalism

We have solved the equations derived by Hartle and Thorne which describe the structure of slowly rotating relativistic stars (Hartle 1967; Hartle & Thorne 1968). When the static equilibrium configuration is perturbed by rotation, the geometry of space-time is changed. For an appropriate choice of coordinates, the perturbed geometry is described by (c=G=1) (A1)
where P2 is the Legendre polynomial of order 2; , which is called the angular velocity of the local inertial frame, is a function of r and is proportional to the star's angular velocity . In the above equations , , , and are all functions of r proportional to . The quantity is the angular velocity of the fluid relative to the local inertial frame.

The surface of constant density in the rotating configuration is given by the radius (A2)
where (A3)
(in fact, this equation defines ) (A4) (A5)
In order to have an invariant parametrization of a surface of constant density, it can be embedded in a three-dimensional flat space, with coordinates , and . The surface in the flat space is, to order , the spheroid (A6)
The mean radius of this spheroid is (A7)
After some algebraic manipulations, Einstein equations for a perfect fluid can be written as: (A8) (A9) (A10) (A11) (A12) (A13) (A14) (A15) (A16) (A17)
We have introduced the intermediate variables u and in order to split the equation of second order for into a system of two equations of first order: (A18)
where (A19) (A20)
We must also integrate the homogeneous equations for and in order to obtain the general solution (A21) (A22)
The expression for can be deduced from the relativistic thermal equilibrium condition and the equation of hydrostatic equilibrium (A23)
The boundary conditions at the center are the following: (A24) (A25) (A26) (A27) (A28) (A29) (A30) (A31) (A32) (A33)
The homogeneous equations for h2 and v2 must verify the following conditions at the center (A34) (A35)
with A and B non null constants related by (A36)
The general solution can be written (A37) (A38)
Outside the star and have the analytic form: (A39) (A40)
where K is a constant, is the associated Legendre polynomial of the second kind and J is given by: (A41)
K and A' are obtained by matching and to the external solution. But before doing this we must scale all the variables by the appropriate factor so that the potential at the surface and reach the correct values given by the expressions: (A42) (A43)
We have integrated the above system of equations with the corresponding boundary conditions from the center to the surface by using standard ODE solvers (fourth order Runge-Kutta). The star's surface is fixed by the condition g/cm3.

The correction to the gravitational mass induced by rotation is then (A44)
We have calculated the "amu" mass ( ) and the "proper" mass : (A45) (A46)
with (A47)
The mean adiabatic index is given by (A48)
Other quantities of interest are: the moment of inertia (A49)
the binding energy (A50)
the change in the binding energy due to the rotation is (A51)
The total number of baryons, A, in the nonrotating star is related to by (A52)
where is the rest mass per baryon.

The quadrupolar momentum is defined as (A53)
Finally, the eccentricity at the surface, e, of the spheroid is defined by (A54)
Let us notice the fact that quantities such as , ( ), and Q scale as and e does as .  Up: Thermal history and structure

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