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)

$$\begin{eqnarray} {\rm d}s^{2} & = & -{\rm e}^{\nu} \left[ 1\, +\, 2(h_{\mbox{\ti... ...ega {\rm d}t)^{\mbox{\tiny 2}}]\, +\, O(\Omega^{\mbox{\tiny 3}}),\end{eqnarray}$$ (A1)

where P₂ is the Legendre polynomial of order 2; $\omega$, 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 $\Omega$. In the above equations $h_{\mbox{\tiny 0}}$, $h_{\mbox{\tiny 2}}$,$m_{\mbox{\tiny 0}}$, $m_{\mbox{\tiny 2}}$ and $v_{\mbox{\tiny 2}}$are all functions of r proportional to $\Omega^{\mbox{\tiny 2}}$. The quantity $\overline{\omega}\, \equiv \, \Omega\, -\, \omega$ 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

$$r\, +\, \xi_{\mbox{\tiny 0}}(r)\, +\, \xi_{\mbox{\tiny 2}} (r) P_{2}(\cos\, \theta),$$ (A2)

where

$$\xi_{\mbox{\tiny 0}}\ =\ -p_{\mbox{\tiny 0}}^{\ast} ( E\, +\, p)/({\rm d}p/{\rm d}r),$$ (A3)

(in fact, this equation defines $p_{\mbox{\tiny 0}}^{\ast}$)

$$\xi_{\mbox{\tiny 2}}\ =\ -p_{\mbox{\tiny 2}}^{\ast} (E\, +\, p)/({\rm d}p/{\rm d}r),$$ (A4)

$$p_{\mbox{\tiny 2}}^{\ast}\ =\ -h_{\mbox{\tiny 2}}\, -\, \frac{r^{2} \overline{\omega}^{2}} {3 {\rm e}^{\nu}}\cdot$$ (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 $r^{\ast}$, $\theta^{\ast}$ and $\phi^{\ast}$. The surface in the flat space is, to order $\Omega^{2}$, the spheroid

$$\begin{eqnarray} r^{\ast}(\theta^{\ast}) & = & r\, +\, \xi_{\mbox{\tiny 0}}(r) \... ...}}(r)\, -\, h_{\mbox{\tiny 2}}(r)]\} P_{2}(\cos\, \theta^{\ast}).\end{eqnarray}$$ (A6)

The mean radius of this spheroid is

$$\overline{r}^{*} = r + \xi_{\mbox{\tiny 0}} (r).$$ (A7)

After some algebraic manipulations, Einstein equations for a perfect fluid can be written as:

$$\frac{{\rm d}m}{{\rm d}r}\ =\ 4 \pi r^{2} \rho,$$ (A8)

$$\frac{{\rm d}p}{{\rm d}r}\ =\ -\, \frac{( \rho\, +\, p)( m\, +\, 4 \pi r^3 p)}{r ( r\, -\, 2m )},$$ (A9)

$$\frac{{\rm d}\nu}{{\rm d}r}\ =\ 2\frac{m\, +\, 4 \pi r^{3} p}{r( r\, -\, 2 m )},$$ (A10)

$$\frac{{\rm d}\xi}{{\rm d}r}\ =\ \frac{u}{r^{4}}\, -\, \frac{4 \pi r^{4} ( \rho\, +\, p )} {r\, -\, 2m},$$ (A11)

$$\frac{{\rm d}u}{{\rm d}r}\ =\ \frac{16 \pi r^{5} \xi (\rho\, +\, p)}{r\, -\, 2m},$$ (A12)

$$\frac{{\rm d}m_{\mbox{\tiny 0}}}{{\rm d}r}\ =\ 4 \pi r^{2} (... ...\, \frac{8 \pi r^{5} (\rho\, +\, p ) \xi^{2}}{3 ( r\, -\, 2m)},$$ (A13)

$$\begin{eqnarray} \frac{{\rm d}p_{\mbox{\tiny 0}}^{\ast}}{{\rm d}r} & = & \frac{u... ... +\, \frac{(r\, -\, 3m-\, 4 \pi r^{3} p) \xi}{r\, -\, 2m} \right],\end{eqnarray}$$ (A14)

$$\begin{eqnarray} \frac{{\rm d}( \delta E_{\rm B} )}{4 \pi r^{2} {\rm d}r} & = & ... ...ac{8 \pi r^{5} (\rho\, +\, p) \xi^{2}}{3 r ( r\, -\, 2m)} \right],\end{eqnarray}$$ (A15)

$$\begin{eqnarray} \frac{{\rm d}v_{\mbox{\tiny 2}}}{{\rm d}r} & = & \frac{1}{r(r\,... ...i^{2}}{3(r\, -\, 2m)} +\, \frac{u^{2}}{6r^{4}} \right] \right\},\end{eqnarray}$$ (A16)

$$\begin{eqnarray} \frac{{\rm d}h_{\mbox{\tiny 2}}}{{\rm d}r} & = & h_{\mbox{\tiny... ...ht. \\ & & \left. +\, \frac{1}{2(m\, +\, 4 \pi r^{3} p)} \right].\end{eqnarray}$$ (A17)

We have introduced the intermediate variables u and $\xi$ in order to split the equation of second order for $\overline{\omega}$ into a system of two equations of first order:

$$u\ =\ r^{4}j \frac{{\rm d} \overline{\omega}}{{\rm d}r},$$ (A18)

where

$$j\ =\ {\rm e}^{- \nu/2} \sqrt{1\, -\, \frac{2m}{r}},$$ (A19)

$$\xi\ =\ j \overline{\omega}.$$ (A20)

We must also integrate the homogeneous equations for $h_{\mbox{\tiny 2}}$ and $v_{\mbox{\tiny 2}}$in order to obtain the general solution

$$\frac{{\rm d} v_{\mbox{\tiny 2}}^{\rm h}}{{\rm d}r}\ =\ -\, ... ...iny 2}}^{\rm h} \frac{m\, +\, 4 \pi r^{3} p}{r ( r\, -\, 2m)},$$ (A21)

$$\begin{eqnarray} \frac{{\rm d}h_{\mbox{\tiny 2}}^{\rm h}}{{\rm d}r} & = & h_{\mb... ... -\, \frac{2 v_{\mbox{\tiny 2}}^{\rm h}}{m\, +\, 4 \pi r^{3} p}.\end{eqnarray}$$ (A22)

The expression for $\frac{{\rm d} \rho}{{\rm d} P}$ can be deduced from the relativistic thermal equilibrium condition and the equation of hydrostatic equilibrium

$$\left( \frac{{\rm d} \rho}{{\rm d} p} \right)_{\rm H.E.}\! =... ...! \left( \frac{\partial p}{\partial T} \right)_{\rho} \right].$$ (A23)

The boundary conditions at the center are the following:

$$\nu_{\rm c}\ =\ 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \... ...\ \ \left( \frac{{\rm d} \nu} {{\rm d}r} \right)_{\rm c}\ =\ 0,$$ (A24)

$$\xi_{\rm c}\ =\ j_{\rm c} \overline{\omega}_{\rm c}\ =\ \ove... ...\ \ \left( \frac{{\rm d} \xi}{{\rm d}r} \right)_{\rm c}\ =\ 0,$$ (A25)

$$u_{\rm c}\ =\ 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \... ...\ \ \ \ \left( \frac{{\rm d}u}{{\rm d}r} \right)_{\rm c}\ =\ 0,$$ (A26)

$$m_{\mbox{\tiny 0}\rm c}\ =\ 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \... ...ac{{\rm d} m_{\mbox{\tiny 0}}}{{\rm d}r} \right)_{\rm c}\ =\ 0,$$ (A27)

$$p_{\mbox{\tiny 0}\rm c}^{\ast}\ =\ 0,\ \ \ \ \ \ \ \ \ \ \ \... ...{\mbox{\tiny 0} \rm c}^{\ast}}{{\rm d}r} \right)_{\rm c}\ =\ 0,$$ (A28)

$$\delta E_{\rm B_{c}}\ =\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \... ...frac{{\rm d} \delta E_{\rm B}}{{\rm d}r} \right)_{\rm c}\ =\ 0,$$ (A29)

$$v_{\mbox{\tiny 2}\rm c}\ =\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ... ...ac{{\rm d}v_{\mbox{\tiny 2}}} {{\rm d}r} \right)_{\rm c}\ =\ 0,$$ (A30)

$$h_{\mbox{\tiny 2}\rm c}\ =\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ... ...ac{{\rm d}h_{\mbox{\tiny 2}}} {{\rm d}r} \right)_{\rm c}\ =\ 0,$$ (A31)

$$v_{\mbox{\tiny 2}\rm c}^{\rm h}\ =\ 0,\ \ \ \ \ \ \ \ \ \ \ ... ...d}v_{\mbox{\tiny 2}}^{\rm h}} {{\rm d}r} \right)_{\rm c}\ =\ 0,$$ (A32)

$$h_{\mbox{\tiny 2}\rm c}^{\rm h}\ =\ 0,\ \ \ \ \ \ \ \ \ \ \ ... ...d}h_{\mbox{\tiny 2}}^{\rm h}} {{\rm d}r} \right)_{\rm c}\ =\ 0.$$ (A33)

The homogeneous equations for h₂ and v₂ must verify the following conditions at the center

$$\lim_{r \rightarrow 0} h_{\mbox{\tiny 2}}^{\rm h}\ =\ A r^{2},$$ (A34)

$$\lim_{r \rightarrow 0} v_{\mbox{ 2}}^{\rm h}\ =\ B r^{4},$$ (A35)

with A and B non null constants related by

$$B\, +\, 2 \pi \left( p_{c}\, +\, \frac{p}{3} \right) A\ =\ 0.$$ (A36)

The general solution can be written

$$v_{\mbox{\tiny 2}}\ =\ A' v_{\mbox{\tiny 2}}^{\rm h}\, +\, v_{\mbox{\tiny 2}}^{\rm p},$$ (A37)

$$h_{\mbox{\tiny 2}}\ =\ A' h_{\mbox{\tiny 2}}^{\rm h}\, +\, h_{\mbox{\tiny 2}}^{\rm p}.$$ (A38)

Outside the star $h_{\mbox{\tiny 2}}$ and $v_{\mbox{\tiny 2}}$ have the analytic form:

$$h_{\mbox{\tiny 2}}\ =\ J^{2} \left( \frac{1}{M r^{3}}\, +\, ... ...right)\, +\, K Q_{2}^{ \ 2} \left( \frac{r}{M}\, -\, 1 \right),$$ (A39)

$$v_{\mbox{\tiny 2}}\ =\ - \frac{J^{2}}{r^{4}}\, +\, \frac{2 ... ...\, 2M)]^{1/2}} Q_{2}^{ \ 1} \left( \frac{r}{M}\, -\, 1 \right),$$ (A40)

where K is a constant, $Q_{n}^{ \ m}$ is the associated Legendre polynomial of the second kind and J is given by:

$$J\ =\ \frac{ue^{- \nu/2}}{6 \sqrt{1\, -\, 2 \frac{M}{R}}}.$$ (A41)

K and A' are obtained by matching $h_{\mbox{\tiny 2}}$and $v_{\mbox{\tiny 2}}$ 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 $\overline{\omega}$reach the correct values given by the expressions:

$$\nu_{\mbox{\tiny S}}\ = 2\frac{\log \left (\sqrt{1\, -\, \frac{2M}{R}} \right)}{2},$$ (A42)

$$\Omega_{\rm k}\ =\ \overline{\omega}\, +\, \frac{2J}{R^{3}}\ =\ \sqrt{\frac{M}{R^{3}}}.$$ (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 $P(r=R) = P(\rho_{\rm surf}), \rho_{\rm surf} \approx 5 \ 10^{9}$ g/cm³.

The correction to the gravitational mass induced by rotation is then

$$\delta M = m_{0} (R) + J^{2}/R^{3}.$$ (A44)

We have calculated the "amu" mass ($M_{\rm A}$) and the "proper" mass $(M_{\rm P})$:

$$M_{\rm A}=\ 4 \pi \int_{0}^{R} r^{2} {\rm e}^{\Lambda} \rho_{\mbox{\tiny 0}} {\rm d} r,$$ (A45)

$$M_{\rm P}=\ 4 \pi \int_{0}^{R} r^{2} {\rm e}^{\Lambda} \rho {\rm d} r,$$ (A46)

with

$${\rm e}^{\Lambda}\ =\ \left( 1\, -\, \frac{2m}{r} \right)^{-1/2}.$$ (A47)

The mean adiabatic index is given by

$$\left< \Gamma_{1} \right\gt\ =\ \frac{\int_{0}^{R} r^{2} p (... ...m d}r} {\int_{0}^{R} r^{2} p ( 1\, -\, 2 m/r)^{-1/2} {\rm d}r}.$$ (A48)

Other quantities of interest are: the moment of inertia

$$I\ =\frac{J}{\Omega},$$ (A49)

the binding energy

$$E_{\rm B}\ =\ M_{\rm A}\, -\, M,$$ (A50)

the change in the binding energy due to the rotation is

$$\delta E_{\rm B}\ =\ \delta M_{\rm A}\, -\, \delta M.$$ (A51)

The total number of baryons, A, in the nonrotating star is related to $M_{\rm A}$ by

$$M_{\rm A}\ =\ \mu A,$$ (A52)

where $\mu$ is the rest mass per baryon.

The quadrupolar momentum is defined as

$$Q\ =\ \frac{8 K M^{3}}{5}\, +\, \frac{J^{2}}{M}.$$ (A53)

Finally, the eccentricity at the surface, e, of the spheroid is defined by

$$\begin{eqnarray} e & = & [({\rm radius \,\, at \,\, equator})^{2}/({\rm radius \... ...}}\, -\, h_{\mbox{\tiny 2}}\, +\, \xi_{\mbox{\tiny 0}}/r)]^{1/2}.\end{eqnarray}$$ (A54)

Let us notice the fact that quantities such as $\delta M$, $\delta R$($\equiv \xi_{\mbox{\tiny 0}} (R)$), $\delta E_{\rm B}$ and Q scale as $\Omega^{2}$ and e does as $\Omega$.