periastron_optimization_function

luminet.black_hole_math.periastron_optimization_function(p: float, ir_radius: float, ir_angle: float, bh_mass: float, incl: float, order: int = 0) → float

Cost function for the optimization of the periastron value.

This function is optimized to find the periastron value that solves Equation 13 in cite:t:Luminet1979:

\[4 M P - r (Q - P + 2 M) + r (Q - P + 6 M) \text{sn}^2 \left( \frac{\gamma}{2 \sqrt{P/Q}} + F(\zeta_{\infty}, k) \right) = 0\]

When the above equation is zero, the photon periastron value \(P\) is correct.

See also

solve_for_periastron() to calculate the periastron of a photon orbit, given an accretion disk radius of origin \(R\).

Parameters:

• periastron (float) – Periastron distance

• ir_radius (float) – Radius in the black hole frame

• ir_angle (float) – Angle in the black hole frame

• bh_mass (float) – Black hole mass

• incl (float) – Inclination of the black hole

• order (int) – Order of the image. Default is \(0\) (direct image).

Returns:

float – Cost function value. Should be zero when the photon periastron value is correct.