def calculate_true_anomaly(epoch, sma, ecc, tau, mtot):
""" Calculate true anomaly
Args:
epoch (float): epoch at which to calculate contrast [mjd]
sma (float): semimajor axis [au]
ecc (float): eccentricity
tau (float): epoch of periastron, in mjd, divided by period
mtot (float): total mass [solar masses]
Returns: true anomaly [float]
History:
2018: shamelessly stolen from orbitize.kepler.calc_orbit() by Sarah Blunt
"""
# compute period (from Kepler's third law) and mean motion
period = np.sqrt((sma**3) / mtot) * 365.25
mean_motion = 2 * np.pi / (period) # in rad/day
# compute mean anomaly
manom = (mean_motion * epoch - 2 * np.pi * tau) % (2.0 * np.pi)
# compute eccentric anomalies
eanom = _calc_ecc_anom(manom, ecc)
# compute the true anomalies
tanom = 2. * np.arctan(
np.sqrt((1.0 + ecc) / (1.0 - ecc)) * np.tan(0.5 * eanom))
return tanom
def profile_mikkola_ecc_anom_solver(n_orbits=1000, use_c=True):
"""
Test orbitize.kepler._calc_ecc_anom() in the iterative solver regime (e < 0.95) by comparing the mean anomaly computed from
_calc_ecc_anom() output vs the input mean anomaly
"""
mean_anoms = np.linspace(0, 2.0 * np.pi, n_orbits)
eccs = np.linspace(.95, 0.999999, n_orbits)
for ee in eccs:
ecc_anoms = kepler._calc_ecc_anom(mean_anoms, ee, use_c=use_c)
def test_iterative_ecc_anom_solver():
"""
Test orbitize.kepler._calc_ecc_anom() in the iterative solver regime (e < 0.95) by comparing the mean anomaly computed from
_calc_ecc_anom() output vs the input mean anomaly
"""
mean_anoms = np.linspace(0, 2.0 * np.pi, 100)
eccs = np.linspace(0, 0.9499999, 100)
for ee in eccs:
ecc_anoms = kepler._calc_ecc_anom(mean_anoms, ee, tolerance=1e-9)
calc_ma = (ecc_anoms - ee * np.sin(ecc_anoms)) % (
2 * np.pi) # plug solutions into Kepler's equation
for meas, truth in zip(calc_ma, mean_anoms):
assert angle_diff(meas, truth) == pytest.approx(0.0, abs=threshold)
def test_analytical_ecc_anom_solver(use_c=False, use_gpu=False):
"""
Test orbitize.kepler._calc_ecc_anom() in the analytical solver regime (e > 0.95) by comparing the mean anomaly computed from
_calc_ecc_anom() output vs the input mean anomaly
"""
mean_anoms = np.linspace(0, 2.0 * np.pi, 1000)
eccs = np.linspace(0.95, 0.999999, 100)
for ee in eccs:
ecc_anoms = kepler._calc_ecc_anom(mean_anoms,
ee,
tolerance=1e-9,
use_c=use_c,
use_gpu=use_gpu)
calc_mm = (ecc_anoms - ee * np.sin(ecc_anoms)) % (
2 * np.pi) # plug solutions into Kepler's equation
for meas, truth in zip(calc_mm, mean_anoms):
assert angle_diff(meas, truth) == pytest.approx(0.0, abs=threshold)
def calc_orbit_3d(epochs,
sma,
ecc,
inc,
argp,
lan,
tau,
plx,
mtot,
mass=None,
tau_ref_epoch=0,
tolerance=1e-9,
max_iter=100):
"""
Returns the separation and radial velocity of the body given array of
orbital parameters (size n_orbs) at given epochs (array of size n_dates)
Based on orbit solvers from James Graham and Rob De Rosa. Adapted by Jason Wang and Henry Ngo.
Args:
epochs (np.array): MJD times for which we want the positions of the planet
sma (np.array): semi-major axis of orbit [au]
ecc (np.array): eccentricity of the orbit [0,1]
inc (np.array): inclination [radians]
argp (np.array): argument of periastron [radians]
lan (np.array): longitude of the ascending node [radians]
tau (np.array): epoch of periastron passage in fraction of orbital period past MJD=0 [0,1]
plx (np.array): parallax [mas]
mtot (np.array): total mass [Solar masses]
mass (np.array, optional): mass of the body [Solar masses]. For planets mass ~ 0 (default)
tau_ref_epoch (float, optional): reference date that tau is defined with respect to (i.e., tau=0)
tolerance (float, optional): absolute tolerance of iterative computation. Defaults to 1e-9.
max_iter (int, optional): maximum number of iterations before switching. Defaults to 100.
Return:
3-tuple:
raoff (np.array): array-like (n_dates x n_orbs) of RA offsets between the bodies
(origin is at the other body) [mas]
deoff (np.array): array-like (n_dates x n_orbs) of Dec offsets between the bodies [mas]
vz (np.array): array-like (n_dates x n_orbs) of radial velocity offset between the bodies [km/s]
Written: Jason Wang, Henry Ngo, 2018
"""
n_orbs = np.size(sma) # num sets of input orbital parameters
n_dates = np.size(epochs) # number of dates to compute offsets and vz
# Necessary for _calc_ecc_anom, for now
if np.isscalar(epochs): # just in case epochs is given as a scalar
epochs = np.array([epochs])
ecc_arr = np.tile(ecc, (n_dates, 1))
# If mass not given, assume test particle case
if mass is None:
mass = np.zeros(n_orbs)
# Compute period (from Kepler's third law) and mean motion
period = np.sqrt(4 * np.pi**2.0 * (sma * u.AU)**3 / (consts.G *
(mtot * u.Msun)))
period = period.to(u.day).value
mean_motion = 2 * np.pi / (period) # in rad/day
# # compute mean anomaly (size: n_orbs x n_dates)
manom = (mean_motion * (epochs[:, None] - tau_ref_epoch) -
2 * np.pi * tau) % (2.0 * np.pi)
# compute eccentric anomalies (size: n_orbs x n_dates)
eanom = _calc_ecc_anom(manom,
ecc_arr,
tolerance=tolerance,
max_iter=max_iter)
# compute the true anomalies (size: n_orbs x n_dates)
# Note: matrix multiplication makes the shapes work out here and below
tanom = 2. * np.arctan(
np.sqrt((1.0 + ecc) / (1.0 - ecc)) * np.tan(0.5 * eanom))
# compute 3-D orbital radius of second body (size: n_orbs x n_dates)
radius = sma * (1.0 - ecc * np.cos(eanom))
# compute ra/dec offsets (size: n_orbs x n_dates)
# math from James Graham. Lots of trig
c2i2 = np.cos(0.5 * inc)**2
s2i2 = np.sin(0.5 * inc)**2
arg1 = tanom + argp + lan
arg2 = tanom + argp - lan
c1 = np.cos(arg1)
c2 = np.cos(arg2)
s1 = np.sin(arg1)
s2 = np.sin(arg2)
# updated sign convention for Green Eq. 19.4-19.7
raoff = radius * (c2i2 * s1 - s2i2 * s2) * plx
deoff = radius * (c2i2 * c1 + s2i2 * c2) * plx
# zoff = np.ones(raoff.shape)
zoff = radius * np.sin(tanom + argp) * np.sin(inc) * plx
# compute the radial velocity (vz) of the body (size: n_orbs x n_dates)
# first comptue the RV semi-amplitude (size: n_orbs x n_dates)
m1 = mtot - mass # mass of the primary star
Kv = np.sqrt(consts.G /
(1.0 - ecc**2)) * (m1 * u.Msun * np.sin(inc)) / np.sqrt(
mtot * u.Msun) / np.sqrt(sma * u.au)
# Convert to km/s
Kv = Kv.to(u.km / u.s)
# compute the vz
vz = Kv.value * (ecc * np.cos(argp) + np.cos(argp + tanom))
# Squeeze out extra dimension (useful if n_orbs = 1, does nothing if n_orbs > 1)
# [()] used to convert 1-element arrays into scalars, has no effect for larger arrays
# raoff = np.transpose(np.squeeze(raoff)[()])
# deoff = np.transpose(np.squeeze(deoff)[()])
# vz = np.transpose(np.squeeze(vz)[()])
vz = np.squeeze(vz)[()]
return raoff, deoff, zoff, vz