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differential equations - Orbit followed by a particle around Schwarzschild Black Hole



The following is a equation which describes various possible orbits of a particle around the Schwarzschild black hole spacetime in general relativity. I want to solve it from -Pi to Pi but fails, any suggestions?


d = 6 m; m = 2;

s = 
  NDSolve[
    {u'[Ï•] - Sqrt[2 m u[Ï•]^3 - u[Ï•]^2 + 1/d^2] == 0, u[0] == 0}, 
    u, {Ï•, -0.4, 2.5}]


P1 = PolarPlot[Evaluate[{1/(u[Ï•])} /. s], {Ï•, -0.4, 2.5},AspectRatio->1/GoldenRatio]


enter image description here



Answer




The equation can be solved with Method -> "StiffnessSwitching", together with a higher WorkingPrecision:


d = 6 m; m = 2;
s = NDSolve[{u'[Ï•] - Sqrt[2 m u[Ï•]^3 - u[Ï•]^2 + 1/d^2] == 0, u[0] == 0}, 
  u, {Ï•, -Pi, Pi}, Method -> "StiffnessSwitching", WorkingPrecision -> 64]

Plot[Evaluate[u[Ï•] /. s // Re], {Ï•, -Pi, Pi}]


Mathematica graphics


Plot[Evaluate[1/u[Ï•] /. s // Re], {Ï•, -Pi, Pi}]

Mathematica graphics


PolarPlot[Evaluate[1/u[Ï•] /. s // Re], {Ï•, -Pi, Pi}, Exclusions -> Ï• == 0,
  PlotRange -> {{-30, 30}, Automatic}]

Mathematica graphics


I added Re because even with WorkingPrecision -> 64 the solution still involves very small imaginary numeric error e.g.



s[[1, 1, -1]][-Pi]
(* -0.0736592722609752787242287643805033631850830491206634648394182567 - 
 4.668866930386631320069097140978040*10^-31 I *)


Inspired by Solution 1, I found the problem can be resolved by adding a Re in the equation:


d = 6 m; m = 2;
s = NDSolve[{u'[Ï•] - Re@Sqrt[2 m u[Ï•]^3 - u[Ï•]^2 + 1/d^2] == 0, 
   u[0] == 0}, u, {Ï•, -Pi, Pi}]


The resulting graph is the same as above so I'd like to omit it here.



I still have a feeling that the equation is wrong, at least incomplete, for example, don't you think the following


d = 6 m; m = 2;
s = NDSolve[{u'[Ï•]^2 == (Re@Sqrt[2 m u[Ï•]^3 - u[Ï•]^2 + 1/d^2])^2, 
   u[0] == 0}, u, {Ï•, -Pi, Pi}]

PolarPlot[Evaluate[1/u[Ï•] /. s], {Ï•, -Pi, Pi}, Exclusions -> Ï• == 0, 
 PlotRange -> {{-30, 30}, Automatic}]


Mathematica graphics


seems to be more natural?


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