Schwarzschild Geodesics - Exact Solution Using Elliptic Functions

Exact Solution Using Elliptic Functions

The fundamental equation of the orbit is easier to solve if it is expressed in terms of the inverse radius u = 1/r

$\left( \frac{du}{d\varphi} \right)^{2} = \frac{1}{b^{2}} - \left( 1 - u r_{s} \right) \left( \frac{1}{a^{2}} + u^{2} \right)$

The right-hand side of this equation is a cubic polynomial, which has three roots, denoted here as u₁, u₂, and u₃

$\left( \frac{du}{d\varphi} \right)^{2} = r_{s} \left( u - u_{1} \right) \left( u - u_{2} \right) \left( u - u_{3} \right)$

The sum of the three roots equals the coefficient of the u2 term

$u_{1} + u_{2} + u_{3} = \frac{1}{r_{s}}$

A cubic polynomial with real coefficients can either have three real roots, or one real root and two complex conjugate roots. If all three roots are real numbers, the roots are labeled so that u₁ < u₂ < u₃. If instead there is only one real root, then that is denoted as u₃; the complex conjugate roots are labeled u₁ and u₂. Using Descartes' rule of signs, there can be at most one negative root; u₁ is negative if and only if b < a. As discussed below, the roots are useful in determining the types of possible orbits.

Given this labeling of the roots, the solution of the fundamental orbital equation is

$u = u_{1} + \left( u_{2} - u_{1} \right) \, \mathrm{sn}^{2}\left( \frac{1}{2} \varphi \sqrt{r_{s} \left( u_{3} - u_{1} \right)} + \delta \right)$

where sn represents the sinus amplitudinus function (one of the Jacobi elliptic functions) and δ is a constant of integration reflecting the initial position. The elliptic modulus k of this elliptic function is given by the formula

$k = \sqrt{\frac{u_{2} - u_{1}}{u_{3} - u_{1}}}$

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