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Code for generating the graphs in https://theconfused.me/blog/numerical-integration-of-light-paths-in-a-schwarzschild-metric/
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# code for generating the graphs in https://theconfused.me/blog/numerical-integration-of-light-paths-in-a-schwarzschild-metric/ | |
import numpy as np | |
import matplotlib.pyplot as plt | |
import scipy.integrate as spi | |
def schw_null_geodesics(w, t, p): | |
r, rdot, phi = w | |
M, L = p | |
phidot = L / r**2 | |
return [ | |
rdot, | |
L**2 * (r - 3*M) / r**4, | |
phidot | |
] | |
def expected_schw_light_bending(r, M): | |
return 4. * M / r | |
def solve(t, initial, p): | |
sol = spi.odeint(schw_null_geodesics, initial, t, args=(p,),) | |
r = sol[:,0] | |
phi = sol[:,2] | |
return r, phi | |
def calculate_deflection(x, y, t): | |
num_entries = len(t) | |
gradient = (y[num_entries*4//5] - y[num_entries-1]) / (x[num_entries*4//5] - x[num_entries-1]) | |
return np.arctan(-gradient) | |
def calculate_initial_conditions(b, initial_x): | |
# calculate initial conditions for each impact parameter | |
initial_r = np.sqrt(b**2 + initial_x**2) | |
initial_phi = np.arccos(initial_x / initial_r) | |
initial_rdot = np.cos(initial_phi) | |
initial_phidot = -np.sqrt((1 - initial_rdot**2) / initial_r**2) | |
L = initial_r**2 * initial_phidot | |
return [initial_r, initial_rdot, initial_phi], L | |
def plot_light_rays(): | |
plt.figure() | |
M = 1. # natural units | |
initial_x = -50 | |
max_t = 10000. | |
# impact parameters to plot | |
bs = np.array([4, 5, 6, 7, 8, 9, 10, 15, 20, 30, 40]) | |
# time points to evaluate | |
t = np.arange(0, max_t, 0.01) | |
for b in bs: | |
initial, L = calculate_initial_conditions(b, initial_x) | |
r, phi = solve(t, initial, [M, L]) | |
x = r * np.cos(phi) | |
y = r * np.sin(phi) | |
plt.plot(x, y) | |
axes = plt.gca() | |
lim = -initial_x | |
axes.set_xlim([-lim, lim]) | |
axes.set_ylim([-lim, lim]) | |
# plot the black hole, schwarzschild radius is 2M | |
circle = plt.Circle((0., 0.), 2*M, color='black', fill=True, zorder=10) | |
plt.legend(bs) | |
axes.add_artist(circle) | |
axes.set_aspect('equal', adjustable='box') | |
def plot_deflection_angles(): | |
plt.figure() | |
M = 1. # natural units | |
initial_x = -50 | |
max_t = 10000. | |
# impact parameters to plot | |
bs = np.linspace(20, 100, 40) | |
# time points to evaluate | |
t = np.arange(0, max_t, 0.01) | |
deflections = [] | |
for b in bs: | |
initial, L = calculate_initial_conditions(b, initial_x) | |
r, phi = solve(t, initial, [M, L]) | |
x = r * np.cos(phi) | |
y = r * np.sin(phi) | |
deflections.append(calculate_deflection(x, y, t)) | |
# plot deflections angles against analytical predicted deflections angles | |
expected = expected_schw_light_bending(bs, M) | |
plt.plot(bs, expected, 'ro') | |
plt.plot(bs, deflections, 'b+') | |
plt.ylabel('Deflection angle') | |
plt.xlabel('Distance of closest approach') | |
plt.legend([r'Theoretical deflection ($\frac{4M}{R}$)', 'Numerical result']) | |
if __name__ == '__main__': | |
plt.style.use('ggplot') | |
plot_light_rays() | |
plot_deflection_angles() | |
plt.show() |
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