The Paczynski Wiita potential is an approximate correction to the classical point mass gravitational potential that reproduces several effects of general relativity. The form of the potential is

$ \phi \left( r \right ) = \frac{G M}{r - r_c} $

where $ G $ is the universal constant of gravitation, $ M $ is the is the mass, $ r $ is the distance from said mass and $ r_c = \frac{2 G M}{c^2} $ is the Schwarzschild radius. In the next section we derive the effects of general relativity reproduced by this potential. The rationale behind this potential can be easily understood. It diverges at Schwartzschild's radius, and tends to the non relativistic results at large radii.

Innermost Stable Circular Orbit Edit

Let us consider a circular orbit around a point mass with such potential. The orbital frequency is given by

$ \omega^2 r = \frac{G M}{\left( r - r_c\right )^2} \Rightarrow \omega = \sqrt{\frac{G M}{\left(r -r_c \right )^2 r }} $

The energy of such orbit is given by

$ \frac{U}{m} = -\frac{G M}{r-r_c} + \frac{1}{2} r^2 \omega^2 = - \frac{G M}{2} \frac{r-2 r_c}{r-c_c} $ A minimum of the energy is obtained at

$ r = 3 r_c $ . At smaller radii the energy is higher. Hence, this is the radius of the innermost stable orbit.

Precession Edit

Now we discuss orbits which are not necessarily circular. Let us write the equations of motion for an arbitrary orbit. Since this is a central potential, angular momentum is conserved.

$ L = r^2 \dot{\theta} $

The equation of motion in the radial direction is

$ \ddot{r} - r \dot{\theta}^2 = - \frac{G M}{\left(r - r_c \right )^2} \approx - \frac{G M}{r^2} \left( 1 + 2 \frac{r_c}{r}\right) $

Hence we proceed in the usual manner. We substitute $ r = \frac{1}{s} $ and use the chain rule to replace time derivatives with derivatives with respect to the angle $ \frac{d}{dt} = \dot{\theta} \frac{d}{d \theta} = L s^2 \frac{d}{d \theta} $ to obtain

$ - L^2 \frac{d^2 s}{d \theta^2} -L^2 s = - G M - 2 G M r_s s $

Solving the equation yields

$ s \left( \theta \right ) = \frac{1}{r \left(\theta \right )}= \frac{G M}{L^2 - 2 G M r_c}\left[1 + \epsilon \cos \left( \sqrt{1-\frac{G M r_c}{L^2}} \left(\theta - \theta_0 \right )\right ) \right] $

We can see that even we then angle goes from $ \theta = 0 $ to $ 2 \pi $ the phase inside the cosine does not go full circle. The difference between them is precession.

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