# this is a fork of jaxoplanet/src/jaxoplanet/core/kepler.py, many thanks to the original authors
import jax
jax.config.update("jax_enable_x64", True)
import jax.numpy as jnp
from jax.interpreters import ad
[docs]
@jax.jit
def kepler(M, ecc):
"""Solve Kepler's equation to compute the true anomaly.
This implementation is based on that within `jaxoplanet <https://github.com/exoplanet-dev/jaxoplanet/>`_, many thanks to the authors.
Args:
M (Array [Radian]): Mean anomaly
ecc (Array [Dimensionless]): Eccentricity
Returns:
Array: True anomaly in radians
"""
sinf, cosf = _kepler(M, ecc)
# this is the only bit that's different from jaxoplanet-
# puts true anomalies into the range [0, 2*pi)
f = jnp.arctan2(sinf, cosf)
return jnp.where(f < 0, f + 2 * jnp.pi, f)
@jax.custom_jvp
def _kepler(M, ecc):
# Wrap into the right range
M = M % (2 * jnp.pi)
# We can restrict to the range [0, pi)
high = jnp.pi < M
M = jnp.where(high, 2 * jnp.pi - M, M)
# Solve
ome = 1 - ecc
E = _starter(M, ecc, ome)
E = _refine(M, ecc, ome, E)
# Re-wrap back into the full range
E = jnp.where(high, 2 * jnp.pi - E, E)
# Convert to true anomaly; tan(0.5 * f)
tan_half_f = jnp.sqrt((1 + ecc) / (1 - ecc)) * jnp.tan(0.5 * E)
tan2_half_f = jnp.square(tan_half_f)
# Then we compute sin(f) and cos(f) using:
# sin(f) = 2*tan(0.5*f)/(1 + tan(0.5*f)^2), and
# cos(f) = (1 - tan(0.5*f)^2)/(1 + tan(0.5*f)^2)
denom = 1 / (1 + tan2_half_f)
sinf = 2 * tan_half_f * denom
cosf = (1 - tan2_half_f) * denom
return sinf, cosf
@_kepler.defjvp
def _(primals, tangents):
M, e = primals
M_dot, e_dot = tangents
sinf, cosf = _kepler(M, e)
# Pre-compute some things
ecosf = e * cosf
ome2 = 1 - e**2
def make_zero(tan):
if type(tan) is ad.Zero:
return ad.zeros_like_aval(tan.aval)
else:
return tan
# Propagate the derivatives
f_dot = make_zero(M_dot) * (1 + ecosf) ** 2 / ome2**1.5
f_dot += make_zero(e_dot) * (2 + ecosf) * sinf / ome2
return (sinf, cosf), (cosf * f_dot, -sinf * f_dot)
def _starter(M, ecc, ome):
M2 = jnp.square(M)
alpha = 3 * jnp.pi / (jnp.pi - 6 / jnp.pi)
alpha += 1.6 / (jnp.pi - 6 / jnp.pi) * (jnp.pi - M) / (1 + ecc)
d = 3 * ome + alpha * ecc
alphad = alpha * d
r = (3 * alphad * (d - ome) + M2) * M
q = 2 * alphad * ome - M2
q2 = jnp.square(q)
w = jnp.square(jnp.cbrt(jnp.abs(r) + jnp.sqrt(q2 * q + r * r)))
return (2 * r * w / (jnp.square(w) + w * q + q2) + M) / d
def _refine(M, ecc, ome, E):
sE = E - jnp.sin(E)
cE = 1 - jnp.cos(E)
f_0 = ecc * sE + E * ome - M
f_1 = ecc * cE + ome
f_2 = ecc * (E - sE)
f_3 = 1 - f_1
d_3 = -f_0 / (f_1 - 0.5 * f_0 * f_2 / f_1)
d_4 = -f_0 / (f_1 + 0.5 * d_3 * f_2 + (d_3 * d_3) * f_3 / 6)
d_42 = d_4 * d_4
dE = -f_0 / (f_1 + 0.5 * d_4 * f_2 + d_4 * d_4 * f_3 / 6 - d_42 * d_4 * f_2 / 24)
return E + dE
def _x(a, e, f, Omega, i, omega):
return (
a
* (-1 + e**2)
* (
jnp.sin(f)
* (
jnp.cos(Omega) * jnp.sin(omega)
+ jnp.cos(i) * jnp.cos(omega) * jnp.sin(Omega)
)
+ jnp.cos(f)
* (
-(jnp.cos(omega) * jnp.cos(Omega))
+ jnp.cos(i) * jnp.sin(omega) * jnp.sin(Omega)
)
)
) / (1 + e * jnp.cos(f))
def _y(a, e, f, Omega, i, omega):
return -(
(
a
* (-1 + e**2)
* (
jnp.cos(i) * jnp.cos(Omega) * jnp.sin(f + omega)
+ jnp.cos(f + omega) * jnp.sin(Omega)
)
)
/ (1 + e * jnp.cos(f))
)
def _z(a, e, f, Omega, i, omega):
return -((a * (-1 + e**2) * jnp.sin(i) * jnp.sin(f + omega)) / (1 + e * jnp.cos(f)))
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@jax.jit
def skypos(a, e, f, Omega, i, omega, **kwargs):
"""Compute the cartesian coordinates of the center of the planet in the sky frame
given its orbital elements.
Args:
a (Array [Rstar]): Semi-major axis of the orbit in units of R_star
e (Array [Dimensionless]): Eccentricity of the orbit
f (Array [Radian]): True anomaly, the angle between the direction of periapsis
and the current position of the planet as seen from
the star.
Omega (Array [Radian]): Longitude of the ascending node
i (Array [Radian]): Orbital inclination
omega (Array [Radian]): Argument of periapsis
**kwargs: Unused additional keyword arguments. These are included so that we
can can take in a larger state dictionary that includes all of the
required parameters along with other unnecessary ones.
Returns:
Array: The cartesian coordinates of the planet in the sky frame. Shape [3, N].
"""
return jnp.array(
[
_x(a, e, f, Omega, i, omega),
_y(a, e, f, Omega, i, omega),
_z(a, e, f, Omega, i, omega),
]
)
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def true_anomaly_at_transit_center(e, i, omega):
"""Computes the true anomaly at the instant of minimum star/planet separation.
Uses equations 4.12-4.18 of
`Kipping 2011 <https://ui.adsabs.harvard.edu/abs/2011PhDT.......294K/abstract>`_
to compute the true anomaly at the instant of minimum star/planet separation.
Args:
e (Array [Dimensionless]): Eccentricity of the orbit
i (Array [Radian]): Orbital inclination
omega (Array [Radian]): Argument of periapsis
Returns:
Array: True anomaly at the instant of minimum star/planet separation in radians.
"""
hp = e * jnp.sin(omega)
kp = e * jnp.cos(omega)
eta_1 = (kp / (1 + hp)) * (jnp.cos(i) ** 2)
eta_2 = (kp / (1 + hp)) * (1 / (1 + hp)) * (jnp.cos(i) ** 2) ** 2
eta_3 = (
-(kp / (1 + hp))
* ((-6 * (1 + hp) + kp**2 * (-1 + 2 * hp)) / (6 * (1 + hp) ** 3))
* (jnp.cos(i) ** 2) ** 3
)
eta_4 = (
-(kp / (1 + hp))
* ((-2 * (1 + hp) + kp**2 * (-1 + 3 * hp)) / (2 * (1 + hp) ** 4))
* (jnp.cos(i) ** 2) ** 4
)
eta_5 = (
(kp / (1 + hp))
* (
(
40 * (1 + hp) ** 2
- 40 * kp**2 * (-1 + 3 * hp + 4 * hp**2)
+ kp**4 * (3 - 19 * hp + 8 * hp**2)
)
/ (40 * (1 + hp) ** 6)
)
* (jnp.cos(i) ** 2) ** 5
)
eta_6 = (
(kp / (1 + hp))
* (
(
24 * (1 + hp) ** 2
- 40 * kp**2 * (-1 + 4 * hp + 5 * hp**2)
+ 9 * kp**4 * (1 - 8 * hp + 5 * hp**2)
)
/ (24 * (1 + hp) ** 7)
)
* (jnp.cos(i) ** 2) ** 6
)
return jnp.pi / 2 - omega - eta_1 - eta_2 - eta_3 - eta_4 - eta_5 - eta_6
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def t0_to_t_peri(e, i, omega, period, t0, **kwargs):
f = true_anomaly_at_transit_center(e, i, omega)
eccentric_anomaly = jnp.arctan2(jnp.sqrt(1 - e**2) * jnp.sin(f), e + jnp.cos(f))
mean_anomaly = eccentric_anomaly - e * jnp.sin(eccentric_anomaly)
return t0 - period / (2 * jnp.pi) * mean_anomaly
