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I am trying to solve the Schwarzschild geodesic equations and trying to plot them. I am new to the subject, so I am struggling with the initial conditions that I need to feed my computer.

For reference I have these system of differential equations whose solution I want to plot:

$$\dot{\phi} = \frac{l}{r^2}$$ $$\dot{t} = \frac{e}{1-\frac{2GM}{rc^2}}$$ $$\dot{r} = e^2- \left( 1+\frac{l^2}{r^2} \right) \left(1-\frac{2GM}{rc^2} \right)$$

Since I am considering the equatorial plane ($\theta = \frac{\pi}{2}$), what initial values of angular momentum and Energy (or range) should I choose to get valid orbits of particles around the spacetime. Initially, I want to feed valid Energy and Angular momentum values, which should give some consistent solutions. Once I am confident with my model, I can feed arbitrary values as well.

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  • $\begingroup$ This might be helpful: Conserved Energy and Angular Momentum in the Schwarzschild Metric. Also, Divergent reflections around the photon sphere of a black hole $\endgroup$
    – PM 2Ring
    Commented Sep 21, 2021 at 7:36
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    $\begingroup$ I suggest setting your specific energy & angular momentum to values appropriate for a circular (or near-circular) Newtonian orbit, and see how GR affects it. At large radius, the GR orbit should be very similar to the Newtonian. $\endgroup$
    – PM 2Ring
    Commented Sep 21, 2021 at 7:38
  • $\begingroup$ Having done this several times in several languages, using several techniques, I can say in good faith that it is harder to find good initial conditions than to write the solver itself! As mentioned in the other comments, a good understanding of conservation is needed. $\endgroup$
    – m4r35n357
    Commented Sep 21, 2021 at 16:07
  • $\begingroup$ yes, that's what is troubling me, I was thinking of trying out some safe values so that I don't encounter Infinities or absurd values in my solution in between. I was actually hoping to try out some real data as @PM2Ring mentioned but in that case I will encounter numbers of astronomical scale and i'll have to scale accordingly, the problem is I couldn't find any paper where they discussed how they put the data to scale and how they preprocessed the data $\endgroup$
    – unstableEquilibrium
    Commented Sep 22, 2021 at 8:38
  • $\begingroup$ One option to help keep the numbers small and to simplify the equations is to use $c=1$, and measure distances as multiples of the Schwarzschild radius, $r_s=\frac{2GM}{c^2}$. Also consider the technique mentioned in that Nature article: use the reciprocal distance $u=\frac{r_s}{r}$ instead of $r$. $\endgroup$
    – PM 2Ring
    Commented Sep 22, 2021 at 12:38

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The specific energy $\mathcal{E}$ and angular momentum $\mathcal{L}$ for bound geodesics in Schwarzschild are given by

$$\mathcal{E}= \frac{\sqrt{(p-2)^2-4e^2}}{\sqrt{p(p-3-e^2)}}, $$

and

$$ \mathcal{L}= \frac{p}{\sqrt{p-3-e^2}}, $$

where $e$ is the eccentricity and $p$ is the semi-latusrectum. There are stable orbits only for $p > 6+2e$.

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  • $\begingroup$ Are these all written taking c = 1 ? And is p scaled down by the factor of c here? $\endgroup$
    – unstableEquilibrium
    Commented Sep 21, 2021 at 8:50
  • $\begingroup$ These are written with $G=M=c=1$. $p$ has units of length, so should be scaled by a factor $GM/c^2$ $\endgroup$
    – TimRias
    Commented Sep 21, 2021 at 9:14

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