def perihelion():
"""
An example to showcase the usage of the various modules in ``einsteinpy.metric``. \
Here, we assume a Schwarzschild spacetime and obtain & plot the apsidal precession of \
test particle orbit in it.
Returns
-------
geod: ~einsteinpy.geodesic.Geodesic
Geodesic defining test particle trajectory
"""
M = 6e24 # Mass
t = 10000 # Coordinate Time (has no effect in this case - Schwarzschild)
x_vec = np.array([130.0, np.pi / 2, -np.pi / 8]) # 3-Pos
v_vec = np.array([0.0, 0.0, 1900.0]) # 3-Vel
# Schwarzschild Metric Object
ms_cov = Schwarzschild(M=M)
# Getting Position 4-Vector
x_4vec = four_position(t, x_vec)
# Calculating Schwarzschild Metric at x_4vec
ms_cov_mat = ms_cov.metric_covariant(x_4vec)
# Getting stacked (Length-8) initial vector, containing 4-Pos and 4-Vel
init_vec = stacked_vec(ms_cov_mat, t, x_vec, v_vec, time_like=True)
# Calculating Geodesic
geod = Geodesic(metric=ms_cov, init_vec=init_vec, end_lambda=0.002, step_size=5e-8)
return geod
def test_compare_calculate_trajectory_iterator_cartesian_kerrnewman(
test_input):
t, a, Q, q, end_lambda, step_size = test_input
M = 2e24
x_vec = np.array([1e6, 1e6, 20.5])
v_vec = np.array([1e4, 1e4, -30.])
mkn_cov = KerrNewman(coords="BL", M=M, a=a, Q=Q, q=q)
x_4vec = four_position(t, x_vec)
mkn_cov_mat = mkn_cov.metric_covariant(x_4vec)
init_vec = stacked_vec(mkn_cov_mat, t, x_vec, v_vec, time_like=True)
OdeMethodKwargs = {"stepsize": step_size}
return_cartesian = True
geod = Geodesic(metric=mkn_cov,
init_vec=init_vec,
end_lambda=end_lambda,
step_size=step_size,
return_cartesian=return_cartesian)
traj = geod.trajectory
traj_iter = geod.calculate_trajectory_iterator(
OdeMethodKwargs=OdeMethodKwargs, return_cartesian=return_cartesian)
traj_iter_list = list()
for _, val in zip(range(20), traj_iter):
traj_iter_list.append(val[1])
traj_iter_arr = np.array(traj_iter_list)
assert_allclose(traj[:20], traj_iter_arr)
def test_calculate_trajectory_iterator_RuntimeWarning_kerrnewman():
M = 0.5 * 5.972e24
a = 0.
Q = 0.
q = 0.
t = 0.
x_vec = np.array([306, np.pi / 2, np.pi / 2])
v_vec = np.array([0., 0.01, 10.])
mkn_cov = KerrNewman(coords="BL", M=M, a=a, Q=Q, q=q)
x_4vec = four_position(t, x_vec)
mkn_cov_mat = mkn_cov.metric_covariant(x_4vec)
init_vec = stacked_vec(mkn_cov_mat, t, x_vec, v_vec, time_like=True)
end_lambda = 200.
step_size = 0.4e-6
OdeMethodKwargs = {"stepsize": step_size}
geod = Geodesic(metric=mkn_cov,
init_vec=init_vec,
end_lambda=end_lambda,
step_size=step_size)
with warnings.catch_warnings(record=True) as w:
it = geod.calculate_trajectory_iterator(
OdeMethodKwargs=OdeMethodKwargs, )
for _, _ in zip(range(1000), it):
pass
assert len(w) >= 1
def test_calculate_trajectory_iterator_RuntimeWarning_kerr():
t = 0.
M = 1e25
a = 0.
x_vec = np.array([306., np.pi / 2, np.pi / 2])
v_vec = np.array([0., 0.01, 10.])
mk_cov = Kerr(coords="BL", M=M, a=a)
x_4vec = four_position(t, x_vec)
mk_cov_mat = mk_cov.metric_covariant(x_4vec)
init_vec = stacked_vec(mk_cov_mat, t, x_vec, v_vec, time_like=True)
end_lambda = 1.
stepsize = 0.4e-6
OdeMethodKwargs = {"stepsize": stepsize}
geod = Geodesic(metric=mk_cov,
init_vec=init_vec,
end_lambda=end_lambda,
step_size=stepsize,
return_cartesian=False)
with warnings.catch_warnings(record=True) as w:
it = geod.calculate_trajectory_iterator(
OdeMethodKwargs=OdeMethodKwargs, )
for _, _ in zip(range(1000), it):
pass
assert len(w) >= 1
def test_calculate_trajectory1_kerrnewman():
# This needs more investigation
# the test particle should not move as gravitational & electromagnetic forces are balanced
t = 0.
M = 0.5 * 5.972e24
a = 0.
Q = 11604461683.91822052001953125
q = _G * M / _Cc
r = 1e6
end_lambda = 1000.0
step_size = 0.5
x_vec = np.array([r, 0.5 * np.pi, 0.])
v_vec = np.array([0., 0., 0.])
mkn_cov = KerrNewman(coords="BL", M=M, a=a, Q=Q, q=q)
x_4vec = four_position(t, x_vec)
mkn_cov_mat = mkn_cov.metric_covariant(x_4vec)
init_vec = stacked_vec(mkn_cov_mat, t, x_vec, v_vec, time_like=True)
geod = Geodesic(metric=mkn_cov,
init_vec=init_vec,
end_lambda=end_lambda,
step_size=step_size,
return_cartesian=False)
ans = geod.trajectory
assert_allclose(ans[0][1], ans[-1][1], 1e-2)
def test_calculate_trajectory_iterator_kerr(x_vec, v_vec, t, M, a, end_lambda,
step_size, OdeMethodKwargs,
return_cartesian):
mk_cov = Kerr(coords="BL", M=M, a=a)
x_4vec = four_position(t, x_vec)
mk_cov_mat = mk_cov.metric_covariant(x_4vec)
init_vec = stacked_vec(mk_cov_mat, t, x_vec, v_vec, time_like=True)
geod = Geodesic(metric=mk_cov,
init_vec=init_vec,
end_lambda=end_lambda,
step_size=step_size,
return_cartesian=return_cartesian)
traj = geod.trajectory
traj_iter = geod.calculate_trajectory_iterator(
OdeMethodKwargs=OdeMethodKwargs, return_cartesian=return_cartesian)
traj_iter_list = list()
for _, val in zip(range(50), traj_iter):
traj_iter_list.append(val[1])
traj_iter_arr = np.array(traj_iter_list)
assert_allclose(traj[:50, :], traj_iter_arr, rtol=1e-10)
def test_calculate_trajectory_kerr(x_vec, v_vec, t, M, a, end_lambda,
step_size):
mk_cov = Kerr(coords="BL", M=M, a=a)
x_4vec = four_position(t, x_vec)
mk_cov_mat = mk_cov.metric_covariant(x_4vec)
init_vec = stacked_vec(mk_cov_mat, t, x_vec, v_vec, time_like=True)
geod = Geodesic(metric=mk_cov,
init_vec=init_vec,
end_lambda=end_lambda,
step_size=step_size,
return_cartesian=False)
ans = geod.trajectory
testarray = list()
for i in ans:
x = i[:4]
g = mk_cov.metric_covariant(x)
testarray.append(g[0][0] * (i[4]**2) + g[1][1] * (i[5]**2) + g[2][2] *
(i[6]**2) + g[3][3] * (i[7]**2) +
2 * g[0][3] * i[4] * i[7])
testarray = np.array(testarray, dtype=float)
assert_allclose(testarray, 1., 1e-4)
def test_str_repr():
"""
Tests, if the ``__str__`` and ``__repr__`` messages match
"""
t = 0.
M = 1e25
x_vec = np.array([306., np.pi / 2, np.pi / 2])
v_vec = np.array([0., 0.01, 10.])
ms_cov = Schwarzschild(M=M)
x_4vec = four_position(t, x_vec)
ms_cov_mat = ms_cov.metric_covariant(x_4vec)
init_vec = stacked_vec(ms_cov_mat, t, x_vec, v_vec, time_like=True)
end_lambda = 1.
step_size = 0.4e-6
geod = Geodesic(metric=ms_cov,
init_vec=init_vec,
end_lambda=end_lambda,
step_size=step_size)
assert str(geod) == repr(geod)
def test_calculate_trajectory2_schwarzschild():
# based on the revolution of earth around sun
# data from https://en.wikipedia.org/wiki/Earth%27s_orbit
t = 0.
M = 1.989e30
distance_at_perihelion = 147.10e9
speed_at_perihelion = 30290
angular_vel = (speed_at_perihelion / distance_at_perihelion)
x_vec = np.array([distance_at_perihelion, np.pi / 2, 0])
v_vec = np.array([0.0, 0.0, angular_vel])
ms_cov = Schwarzschild(M=M)
x_4vec = four_position(t, x_vec)
ms_cov_mat = ms_cov.metric_covariant(x_4vec)
init_vec = stacked_vec(ms_cov_mat, t, x_vec, v_vec, time_like=True)
end_lambda = 3.154e7
geod = Geodesic(metric=ms_cov,
init_vec=init_vec,
end_lambda=end_lambda,
step_size=end_lambda / 2e3,
return_cartesian=False)
ans = geod.trajectory
# velocity should be 29.29 km/s at aphelion(where r is max)
i = np.argmax(ans[:, 1]) # index where radial distance is max
v_aphelion = (((ans[i][1] * ans[i][7]) * (u.m / u.s)).to(u.km / u.s)).value
assert_allclose(v_aphelion, 29.29, rtol=0.01)
def dummy_data():
M = 6e24
t = 0.
x_vec = np.array([130.0, np.pi / 2, -np.pi / 8])
v_vec = np.array([0.0, 0.0, 1900.0])
metric = Schwarzschild(M=M)
x_4vec = four_position(t, x_vec)
metric_mat = metric.metric_covariant(x_4vec)
init_vec = stacked_vec(metric_mat, t, x_vec, v_vec, time_like=True)
end_lambda = 0.002
step_size = 5e-8
return metric, init_vec, end_lambda, step_size
def test_calculate_trajectory0_kerrnewman():
# Based on the revolution of earth around sun
# Data from https://en.wikipedia.org/wiki/Earth%27s_orbit
# Initialized with cartesian coordinates
# Function returning cartesian coordinates
t = 0.
M = 1.989e30
a = 0.
Q = 0.
q = 0.
distance_at_perihelion = 147.10e9
speed_at_perihelion = 30290
x_sph = CartesianConversion(distance_at_perihelion / np.sqrt(2),
distance_at_perihelion / np.sqrt(2), 0.,
-speed_at_perihelion / np.sqrt(2),
speed_at_perihelion / np.sqrt(2),
0.).convert_spherical()
x_vec = x_sph[:3]
v_vec = x_sph[3:]
mkn_cov = KerrNewman(coords="BL", M=M, a=a, Q=Q, q=q)
x_4vec = four_position(t, x_vec)
mkn_cov_mat = mkn_cov.metric_covariant(x_4vec)
init_vec = stacked_vec(mkn_cov_mat, t, x_vec, v_vec, time_like=True)
end_lambda = 3.154e7
geod = Geodesic(
metric=mkn_cov,
init_vec=init_vec,
end_lambda=end_lambda,
step_size=end_lambda / 1.5e3,
)
ans = geod.trajectory
# velocity should be 29.29 km/s at aphelion(where r is max)
R = np.sqrt(ans[:, 1]**2 + ans[:, 2]**2 + ans[:, 3]**2)
i = np.argmax(R) # index where radial distance is max
v_aphelion = ((np.sqrt(ans[i, 5]**2 + ans[i, 6]**2 + ans[i, 7]**2) *
(u.m / u.s)).to(u.km / u.s)).value
assert_allclose(v_aphelion, 29.29, rtol=0.01)
def test_calculate_trajectory3_schwarzschild():
# same test as with test_calculate_trajectory2_schwarzschild(),
# but initialized with cartesian coordinates
# and function returning cartesian coordinates
t = 0.
M = 1.989e30
distance_at_perihelion = 147.10e9
speed_at_perihelion = 30290
x_sph = CartesianConversion(distance_at_perihelion / np.sqrt(2),
distance_at_perihelion / np.sqrt(2), 0.,
-speed_at_perihelion / np.sqrt(2),
speed_at_perihelion / np.sqrt(2),
0.).convert_spherical()
x_vec = x_sph[:3]
v_vec = x_sph[3:]
ms_cov = Schwarzschild(M=M)
x_4vec = four_position(t, x_vec)
ms_cov_mat = ms_cov.metric_covariant(x_4vec)
init_vec = stacked_vec(ms_cov_mat, t, x_vec, v_vec, time_like=True)
end_lambda = 3.154e7
geod = Geodesic(
metric=ms_cov,
init_vec=init_vec,
end_lambda=end_lambda,
step_size=end_lambda / 2e3,
)
ans = geod.trajectory
# velocity should be 29.29 km/s at aphelion(where r is max)
R = np.sqrt(ans[:, 1]**2 + ans[:, 2]**2 + ans[:, 3]**2)
i = np.argmax(R) # index where radial distance is max
v_aphelion = ((np.sqrt(ans[i, 5]**2 + ans[i, 6]**2 + ans[i, 7]**2) *
(u.m / u.s)).to(u.km / u.s)).value
assert_allclose(v_aphelion, 29.29, rtol=0.01)
