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

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

Read more about this topic: Schwarzschild Geodesics

Famous quotes containing the words exact, solution and/or functions:

“If we define a sign as an exact reference, it must include symbol because a symbol is an exact reference too. The difference seems to be that a sign is an exact reference to something definite and a symbol an exact reference to something indefinite.”
—William York Tindall (1903–1981)

“To the questions of the officiously meddling police Falter replied absently and tersely; but, when he finally grew tired of this pestering, he pointed out that, having accidentally solved “the riddle of the universe,” he had yielded to artful exhortation and shared that solution with his inquisitive interlocutor, whereupon the latter had died of astonishment.”
—Vladimir Nabokov (1899–1977)

“Adolescents, for all their self-involvement, are emerging from the self-centeredness of childhood. Their perception of other people has more depth. They are better equipped at appreciating others’ reasons for action, or the basis of others’ emotions. But this maturity functions in a piecemeal fashion. They show more understanding of their friends, but not of their teachers.”
—Terri Apter (20th century)