calc_f_kerr¶

luminet.black_hole_math.calc_f_kerr(bh_mass, a, r)¶

Calculate the \(f\)-function from Page and Thorne [2] (Equation 12)

The \(f\)-function is used when calculating the relationship between intrinsic flux and radius for an accretion disk:

\[F_s(r) = \frac{\dot{M}_0}{4\pi}e^{-(\nu + \psi + \mu)}f\]

Here, \(\nu\), \(\psi\) and \(\mu\) are metric coefficients (functions of \(r\)) of the Kerr metric. \(\dot{M}_0\) is the radius-independent, time-averaged rate at which mass flows inward. Defining the innermost stable orbit as \(r_{ms}\), \(x=\sqrt{r/M}=\sqrt{r^*}\), \(x_0=\sqrt{r_{ms}/M}\) and \(a^*=a/M\), the \(f\)-function is defined as:

\[\begin{split}\begin{align*} f = &\frac{3}{2M}\frac{1}{x^2(x^3 - 3x + 2a^*)}\Bigg[ x - x_0 - \frac{3}{2}a^*\ln\left(\frac{x}{x_0}\right) \\ &- \frac{3(x_1 - a^*)^2}{x_1(x_1-x_2)(x_1-x_3)}\ln\left(\frac{x-x_1}{x_0-x_1}\right) \\ &- \frac{3(x_2 - a^*)^2}{x_2(x_2-x_1)(x_2-x_3)}\ln\left(\frac{x-x_2}{x_0-x_2}\right) \\ &- \frac{3(x_3 - a^*)^2}{x_3(x_3-x_1)(x_3-x_2)}\ln\left(\frac{x-x_3}{x_0-x_3}\right) \Bigg] \end{align*}\end{split}\]

, where

\[\begin{split}\begin{align*} x_1 &= 2\cos(\frac{1}{3}\cos^{-1}(a_*) - \frac{\pi}{3}) \\ x_2 &= 2\cos(\frac{1}{3}\cos^{-1}(a_*) + \frac{\pi}{3}) \\ x_3 &= -2\cos(\frac{1}{3}\cos^{-1}(a_*)) \\ \end{align*}\end{split}\]

For a Swarzschild black hole, \(a=0\) and these simplify to:

\[\begin{split}\begin{align*} x_1 &= \sqrt{3} \\ x_2 &= 0 \\ x_3 &= - \sqrt{3} \\ f &= \frac{3}{2M}\frac{1}{{r^{*}}^{1.5}(r^*-3)}\left[x - x_0 + \frac{\sqrt{3}}{2}\ln\left( \frac{(\sqrt{6} - \sqrt{3})(\sqrt{r^*}+\sqrt{3})}{(\sqrt{6} + \sqrt{3})(\sqrt{r^*} - \sqrt{3})} \right) \right] \end{align*}\end{split}\]

Parameters:

bh_mass (float) – Mass of the black hole.
a (float) – Specific angular momentum of the black hole. Should always be between \(-1\) and \(1\). \(a > 0\) if the accretion disk orbits in the same direction as the hole rotates; \(a < 0\) if it orbits in the opposite direction.
r (float) – Radius of the orbit

Attention

Luminet [1] has a mistake in Equation 15. The factor in fromt of the \(log\) should be \(\sqrt{3}/2\) instead of \(\sqrt{3}/3\). This can be verified by solving Page and Thorne [2] Equation 15n. The resulting images of the paper are correct though.