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In James Hartle'sGravity, pp. 318-319, Example 15.1, he considers the case of a particle that starts falling from infinity into a Kerr black hole, initially with no kinetic energy (e = 1) and initially moving radially (ℓ = 0). The particle's motion is constrained along the equatorial plane. He says we can integrate:

[tex]\frac{d\phi}{dr} = \frac{d\phi/d\tau}{dr/d\tau} = -\frac{2Ma}{r\Delta} \Bigg[ \frac{2M}{r} \bigg( 1 - \frac{a^2}{r^2}\bigg)\Bigg] ^{-1/2} \tag{C-1}[/tex]...to get the orbit's shape ∅(r).

(M is the mass of the Kerr Black Hole, a is its Angular Momentum per Mass M. e is the particle's energy-per-unit-rest-mass-at-infinity and ℓ is the particle's angular-momentum-per-unit-rest-mass-at-infinity. r is the coordinate radius of the Kerr metric and Δ = r^2 - 2Mr + a^2)

1. The problem statement, all variables and given/known data

I want to generalize this problem for orbits of a particle with e and ℓ values that I can freely set. Although I'm generalizing the problem for different values of e and ℓ, I still only want to work with particles constrained in the Kerr equatorial plane. I've worked out the equations for this generalization, and I've placed them under theRelevant equationsof this template.

The main thing I actually want to accomplish by generalizing this problem is to find out how I can obtain a particle with a spiral-like orbit by altering any of the Kerr Black hole parameters (M and a) or any of the particle parameters (e and ℓ).

2. Relevant equations

The following equations are derived from the geodesic equation for the Kerr metric, taken from Hartle'sGravity, page 318.

[tex]

\frac{d\phi}{d\tau} = \frac{1}{r^2 - 2Mr + a^2} \cdot \Bigg[ \bigg(1 - \frac{2M}{r} \bigg) ℓ + \bigg( \frac{2Ma}{r} \bigg) e \Bigg] \tag{1}

[/tex] [tex]

\frac{e^2 - 1}{2} = \frac{1}{2} \bigg( \frac{dr}{d\tau} \bigg)^2 - \frac{M}{r} + \frac{ℓ^2 - a^2 (e^2 - 1)}{2r^2} - \frac{M(ℓ-ae)^2}{r^3} \tag{2}

[/tex]From these equations, I derived:

[tex]

\frac{dr}{d\tau} = -\sqrt{e^2 - 1 + \frac{2M}{r} - \frac{ℓ^2 - a^2 (e^2 - 1)}{r^2} + \frac{2M(ℓ-ae)^2}{r^3}} \tag{3}

[/tex]Transforming the derivatives into finite differences, we get:

[tex]

\phi_{i+1} - \phi_i = \Delta \phi = \Delta \tau \cdot \Bigg[ \frac{d\phi}{d\tau} \Bigg] \tag{4}

[/tex][tex]

r_{i+1} - r_{i} = \Delta r = \Delta \tau \cdot \Bigg[ \frac{dr}{d\tau}\Bigg] \tag{5}[/tex]

3. The attempt at a solution

I gave several variables initial values:

I substituted Equations 1 and 3 into Equations 4 and 5 (respectively) and handled the integrations numerically as shown in the following code:Code (Text):tau(1,1) = 0;

rOFtau(1,1) = 30*M;

phi(1,1) = 0;

Dr = 1;

I simply plot r against ∅ on a polar graph to get the shape of the orbit.Code (Text):while rOFtau(1,i) > 2*M && imag(rOFtau(1,i)) == 0 && abs(Dr) > 0

% Iterate over Tau to get a set of Tau for plotting

tau(1,i+1) = tau(1,i) + Dtau;

% Iterate numerical integration over r to get a set of r for plotting

Dr = Dtau * -( e^2 - 1 ...

+ 2*M/rOFtau(1,i) ...

- (l^2 - (a^2)*(e^2 - 1))/(rOFtau(1,i)^2) ...

+ (2*M*(l-a*e)^2)/(rOFtau(1,i)^3) ...

)^(0.5);

rOFtau(1,i+1) = rOFtau(1,i) + Dr;

% Iterate numerical integration over Phi to get a set of Phi for plotting

phi(1,i+1) = phi(1,i) + Dtau * ( (rOFtau(1,i)^2 - 2*M*rOFtau(1,i) + a^2)^(-1) * ...

( (1-(2*M/rOFtau(1,i)))*l + ...

(2*M*a/rOFtau(1,i))*e ...

));

i = i + 1;

end

Now, for the part where I start varying the parameters, I first tried:

...and I found that the orbit was too much of a plunge. (Image: https://scontent.fmnl4-2.fna.fbcdn....=086e26b46f326bb0509950e3f92842e6&oe=58D6F490)Code (Text):% Black Hole Parameters

M = 20;

a = 15;

% Particle Parameters

e = 1;

l = 20;

So at first, I thought acquiring a spiral-like orbit was a matter of increasing ℓ, so I increased it to ℓ = 100, and the program prompted me that it ignored the imaginary parts of the function. The orbit's angle ∅ varied more, but the loop terminated early because I let it stop if an imaginary part of r was found: (Image: https://scontent.fmnl4-2.fna.fbcdn....=82266e02168f425ad10ade24767114e5&oe=58D60994)

I thought that this could be fixed by decreasing e (indicating an initial kinetic energy of the particle), but decreasing it to e = 0.9 gives r an imaginary part immediately after the first iteration. I've also tried altering a and M but the orbit barely changes.

MATLAB is the program I use for coding.

Any suggestions on how to proceed would be greatly appreciated. Or perhaps I misinterpreted a concept or two? Any insights or corrections would be appreciated as well. Thanks!

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# Numerical Plot of Particle Orbits in Kerr Equatorial Plane

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