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# Orbits in Kerr metric

Posted Dec11-10 at 08:35 AM by stevebd1
Updated Dec19-10 at 04:30 AM by stevebd1

Note: The following only applies at the equatorial plane.

local gravitational acceleration for an object in orbit at the equatorial plane-

\begin{flalign} &a(U)=-\gamma^2[g-2v\theta_{\hat{\phi}}-v^2k_{(\text{lie})}]\\[6mm] &\vec{g}=-\vec{a}(n)\\[6mm] &a(n)_{\hat{r}}=\frac{M\left[(r^2+a^2)^2-4a^2Mr\right]}{\sqrt{\Delta}\,r^2(r^3+a^2r+2Ma^2)}\\[6mm] &\theta_{\hat{\phi}\,\hat{r}}=\frac{Ma(3r^2+a^2)}{r ^2(r^3+a^2r+2Ma^2)}\\[6mm] &k_{(\text{lie})\,\hat{r}}=\frac{\sqrt{\Delta}(r^3-Ma^2)}{r^2(r^3+a^2r+2Ma^2)} \end{flalign}

where v is tangential velocity, $\theta_{\hat{\phi}}$ is described as the shear vector and k(lie) is the ZAMO Lie relative curvature vector.

(There appears to be a slight conflict of signs as to obtain a(U)=0 for a stable orbit, g should simply equal a(n) or v should be negative, using the long hand versions of the above equations on page 10 of the paper linked below, $\theta_{\hat{\phi}}$ and k(lie) do indeed come out negative which makes a lot more sense.)

For a stable orbit (i.e. a(U)=0)-

\begin{flalign} &v_\pm=\frac{r^2+a^2\mp 2a\sqrt{Mr}}{\sqrt{\Delta} \left[a\pm r\sqrt{r/M}\right]}\ \equiv (\Omega_{s\pm}-\omega)\frac{R}{\alpha}\\[6mm] &\text{where}\\[4mm] &\Omega_{s\pm}=\frac{\pm\sqrt{M}}{r^{3/2}\pm a\sqrt{M}}\\[6mm] &\omega=2Mra/\Sigma^2\\[6mm] &R=\Sigma/\rho \sin \theta\\[6mm] &\alpha=\sqrt{\Delta}\ \rho/\Sigma \end{flalign}

$\text{and}\ \Sigma^2=(r^2+a^2)^2-a^2\Delta \sin^2\theta,\ \Delta= r^{2}+a^{2}-2Mr$ $\text {and}\ \rho^2=r^2+a^2 \cos^2\theta$

where Ωs is the angular velocity for a stable orbit (p/m denotes prograde/retrograde orbits), ω is the frame dragging rate, R is the reduced circumference and α is the reduction factor.

As observed from infinity-

$$a(U)_{\infty}=\frac{M(r^2+a^2)^2}{\Sigma_{\phi}^3}-\omega_{\phi}^2\frac{\Sigma_{\phi}}{r}-2v\frac{Ma(3r^2+a^2)}{\Sigma_{\phi}^2r}-v^2\frac{(r^3-Ma^2)}{\Sigma_{\phi}r^2}\equiv a(U)\cdot\alpha$$

$\text{where}\ \Sigma_{\phi}^2=(r^3+a^2r+2Ma^2)r\ \text{and}\ \omega_{\phi}=2Mra/\Sigma_{\phi}^2$

(the subtext φ implies applicable in the azimuth plane only)

and for a stable orbit-

\begin{flalign} &v_{\pm,\infty}=(\Omega_{s\pm}-\omega)\frac{\Sigma_{\phi}}{r}\\[6mm] &\text{where the period of circular orbit as observed from infinity is-}\\[6mm] &T=\frac{2 \pi}{\Omega_s_\pm}=\frac{2\pi(r^{3/2}\pm a\sqrt{M})}{\sqrt{M}} \end{flalign}

multiply by α to obtain local orbit period.

Reduction factor (redshift) of the object in orbit-

\begin{flalign} &A=\sqrt{g_{tt} + 2\Omega g_{\phi t}+\Omega^2 g_{\phi \phi}}\\[6mm] &\text{where}\\[4mm] &g_{tt}=1+2Mr/\rho^2\\[6mm] &g_{t\phi}=2Mra\sin^2\theta/\rho^2\\[6mm] &g_{\phi\phi}=-(r^2+a^2+[2Mra^2\sin^2\theta]/\rho^2)\sin^2\theta\\[6mm] &\text{which is equivalent to-}\\[4mm] &A=\alpha\cdot\sqrt{1-v_t^2} \end{flalign}

where vt is the tangential velocity of the orbiting object. Note that A=α when Ω=ω.

source- http://arxiv.org/abs/gr-qc/0407004
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