import numpy as np
import keplertools.CyKeplerSTM
[docs]
class planSys:
"""
Kepler State Transition Matrix
Class container for defining a planetary system (or group of planets in multiple
systems) via their gravitational parameters and state vectors. Contains methods
for propagating state vectors forward in time via the Kepler state transition
matrix.
Args:
x0 (numpy.ndarray):
6n vector of stacked positions and velocities for n planets
mu (numpy.ndarray):
n vector of standard gravitational parameters mu = G(m+m_s) where m is
the planet mass, m_s is the star mass and G is the gravitational
constant
epsmult (float):
default multiplier on floating point precision, used as convergence
metric. Higher values mean faster convergence, but sacrifice precision.
prefVallado (bool):
If True, always try the Vallado algorithm first, otherwise try Shepherd
first. Defaults False.
noc (bool):
Do not attempt to use cythonized code. Defaults False.
.. note::
All units must be complementary (i.e., if position is AU and velocity
is AU/day, mu must be in AU^3/day^2.
.. note::
Two algorithms are implemented, both using Batting/Goodyear universal variables.
The first is from Shepperd (1984), using continued fraction to solve the Kepler
equation. The second is from Vallado (2004), using Newton iteration to solve the
time equation. One algorithm is used preferentially, and the other is called
only in the case of convergence failure on the first. All convergence is
calculated to machine precision of the data type and variable size, scaled by a
user-selected multiple.
"""
def __init__(self, x0, mu, epsmult=4.0, prefVallado=False, noc=False):
# determine number of planets and validate input
nplanets = x0.size / 6.0
if nplanets - np.floor(nplanets) > 0:
raise Exception("The length of x0 must be a multiple of 6.")
if mu.size != nplanets:
raise Exception("The length of mu must be the length of x0 divided by 6")
self.nplanets = int(nplanets)
self.mu = np.squeeze(mu)
if self.mu.size == 1:
self.mu = np.array(mu)
self.epsmult = epsmult
if prefVallado:
self.algOrder = [self.calcSTM_vallado, self.calcSTM]
else:
self.algOrder = [self.calcSTM, self.calcSTM_vallado]
# create position and velocity index matrices
tmp = np.reshape(np.arange(self.nplanets * 6), (self.nplanets, 6)).T
self.rinds = tmp[0:3]
self.vinds = tmp[3:6]
if noc:
self.havec = False
else:
self.havec = True
self.updateState(np.squeeze(x0))
[docs]
def updateState(self, x0):
"""Update internal state variable and associated constants
Args:
x0 (~numpy.ndarray(float)):
6n vector of stacked positions and velocities for n planets
"""
self.x0 = x0
r0 = self.x0[self.rinds]
v0 = self.x0[self.vinds]
# constants
self.r0norm = np.sqrt(np.sum(r0 ** 2.0, 0)) # ||r0||
self.v0norm2 = np.sum(v0 * v0, 0) # ||v0||^2
self.nu0 = np.sum(r0 * v0, 0) # r0 \cdot v0
self.beta = 2 * self.mu / self.r0norm - self.v0norm2 # -2E
self.alpha = self.beta / self.mu
self.nu0osmu = self.nu0 / np.sqrt(self.mu)
[docs]
def takeStep(self, dt):
"""Propagate state by input time
Args:
dt (float):
Time step
"""
if self.havec:
try:
tmp = keplertools.CyKeplerSTM.CyKeplerSTM(
self.x0, dt, self.mu, self.epsmult
)
self.updateState(tmp)
return
except: # noqa: E722
print("Cython propagation failed. Falling back to python.")
try:
Phi = self.algOrder[0](dt)
except ValueError as detail:
print("First algorithm error: %s\n Trying second algorithm." % (detail))
Phi = self.algOrder[1](dt)
self.updateState(np.dot(Phi, self.x0))
[docs]
def calcSTM(self, dt):
"""Compute STM for input time
Args:
dt (float):
Time step
Returns:
~numpy.ndarray(float):
6x6 STM
"""
# allocate
u = np.zeros(self.nplanets)
deltaU = np.zeros(self.beta.size)
t = np.zeros(self.nplanets)
counter = 0
# For elliptic orbits, calculate period effects
eorbs = self.beta > 0
if any(eorbs):
P = 2 * np.pi * self.mu[eorbs] * self.beta[eorbs] ** (-3.0 / 2.0)
n = np.floor((dt + P / 2 - 2 * self.nu0[eorbs] / self.beta[eorbs]) / P)
deltaU[eorbs] = 2 * np.pi * n * self.beta[eorbs] ** (-5.0 / 2.0)
# loop until convergence of the time array to the time step
while (np.max(np.abs(t - dt)) > self.epsmult * np.spacing(dt)) and (
counter < 1000
):
q = self.beta * u ** 2.0 / (1 + self.beta * u ** 2.0)
U0w2 = 1.0 - 2.0 * q
U1w2 = 2.0 * (1.0 - q) * u
temp = self.contFrac(q)
U = 16.0 / 15.0 * U1w2 ** 5.0 * temp + deltaU
U0 = 2.0 * U0w2 ** 2.0 - 1.0
U1 = 2.0 * U0w2 * U1w2
U2 = 2.0 * U1w2 ** 2.0
U3 = self.beta * U + U1 * U2 / 3.0
r = self.r0norm * U0 + self.nu0 * U1 + self.mu * U2
t = self.r0norm * U1 + self.nu0 * U2 + self.mu * U3
u = u - (t - dt) / (4.0 * (1.0 - q) * r)
counter += 1
if counter == 1000:
raise ValueError(
"Failed to converge on t: %e/%e"
% (np.max(np.abs(t - dt)), self.epsmult * np.spacing(dt))
)
# Kepler solution
f = 1 - self.mu / self.r0norm * U2
g = self.r0norm * U1 + self.nu0 * U2
F = -self.mu * U1 / r / self.r0norm
G = 1 - self.mu / r * U2
Phi = np.zeros([6 * self.nplanets] * 2)
for j in np.arange(self.nplanets):
st = j * 6
Phi[st : st + 6, st : st + 6] = np.vstack(
(
np.hstack((np.eye(3) * f[j], np.eye(3) * g[j])),
np.hstack((np.eye(3) * F[j], np.eye(3) * G[j])),
)
)
return Phi
[docs]
def contFrac(self, x, a=5.0, b=0.0, c=5.0 / 2.0):
"""Compute continued fraction
Args:
x (~numpy.ndarray(float)):
iterant
a (float):
a parameter
b (float):
b parameter
c (float):
c parameter
Returns:
~numpy.ndarray(float):
converged iterant
"""
# initialize
k = 1 - 2 * (a - b)
l = 2 * (c - 1)
d = 4 * c * (c - 1)
n = 4 * b * (c - a)
A = np.ones(x.size)
B = np.ones(x.size)
G = np.ones(x.size)
Gprev = np.zeros(x.size) + 2
counter = 0
# loop until convergence of continued fraction
while (np.max(np.abs(G - Gprev)) > self.epsmult * np.max(np.spacing(G))) and (
counter < 1000
):
k = -k
l = l + 2.0
d = d + 4.0 * l
n = n + (1.0 + k) * l
A = d / (d - n * A * x)
B = (A - 1.0) * B
Gprev = G
G = G + B
counter += 1
if counter == 1000:
raise ValueError(
"Failed to converge on G, most likely due to divergence in continued fractions."
)
return G
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def calcSTM_vallado(self, dt):
"""Compute STM for input time
Args:
dt (float):
Time step
Returns:
~numpy.ndarray(float):
6x6 STM
"""
# classify orbits
epsval = 1e-12
eorbs = self.alpha >= epsval
porbs = np.abs(self.alpha) < epsval
horbs = self.alpha <= -epsval
xi = np.zeros(self.nplanets)
if np.any(eorbs):
atmp = self.alpha[eorbs]
tmp = np.sqrt(self.mu[eorbs]) * dt * atmp
circinds = np.abs(atmp - 1) > epsval
if any(circinds):
tmp[circinds] *= 0.97
xi[eorbs] = tmp
if np.any(porbs):
r0 = self.x0[self.rinds]
v0 = self.x0[self.vinds]
h = np.cross(r0[:, porbs].T, v0[:, porbs].T).T
p = np.sum(h * h, 0) / self.mu[porbs]
s = np.arctan2(1.0, (3.0 * np.sqrt(self.mu[porbs] / p ** 3.0) * dt)) / 2.0
w = np.arctan((np.tan(s)) ** (1.0 / 3.0))
xi[porbs] = np.sqrt(p) * 2.0 / np.tan(2 * w)
self.alpha[porbs] = 0
if np.any(horbs):
a = 1.0 / (self.alpha[horbs])
xi[horbs] = (
np.sign(dt)
* np.sqrt(-a)
* np.log(
-2
* self.mu[horbs]
* self.alpha[horbs]
* dt
/ (
self.nu0[horbs]
+ np.sign(dt)
* np.sqrt(-self.mu[horbs] * self.alpha[horbs])
* (1.0 - self.r0norm[horbs] * self.alpha[horbs])
)
)
)
# loop
counter = 0
r = self.r0norm
xiup = 10.0 * np.max(np.abs(np.hstack((xi, r))))
while (
np.max(np.abs(xiup))
> self.epsmult * np.spacing(np.max(np.abs(np.hstack((xi, r)))))
) and (counter < 1000):
ps = xi ** 2.0 * self.alpha
c2, c3 = self.psi2c2c3(ps)
r = (
xi ** 2.0 * c2
+ self.nu0osmu * xi * (1 - ps * c3)
+ self.r0norm * (1 - ps * c2)
)
xiup = (
np.sqrt(self.mu) * dt
- xi ** 3.0 * c3
- self.nu0osmu * xi ** 2.0 * c2
- self.r0norm * xi * (1 - ps * c3)
) / r
xi += xiup
counter += 1
if counter == 1000:
raise ValueError(
"Failed to converge on xi: %e/%e"
% (
np.max(np.abs(xiup)),
self.epsmult * np.spacing(np.max(np.abs(np.hstack((xi, r))))),
)
)
# kepler solution
f = 1.0 - xi ** 2.0 / self.r0norm * c2
g = dt - xi ** 3.0 / np.sqrt(self.mu) * c3
F = np.sqrt(self.mu) / r / self.r0norm * xi * (ps * c3 - 1.0)
G = 1.0 - xi ** 2.0 / r * c2
Phi = np.zeros([6 * self.nplanets] * 2)
for j in np.arange(self.nplanets):
st = j * 6
Phi[st : st + 6, st : st + 6] = np.vstack(
(
np.hstack((np.eye(3) * f[j], np.eye(3) * g[j])),
np.hstack((np.eye(3) * F[j], np.eye(3) * G[j])),
)
)
return Phi
[docs]
def psi2c2c3(self, psi0):
"""Compute c_2 and c_3 values given psi
Args:
psi0 (float):
Input psi value
Returns:
tuple:
c2 (float):
c2 value
c3 (float):
c3 value
"""
c2 = np.zeros(len(psi0))
c3 = np.zeros(len(psi0))
psi12 = np.sqrt(np.abs(psi0))
pos = psi0 >= 0
neg = psi0 < 0
if np.any(pos):
c2[pos] = (1 - np.cos(psi12[pos])) / psi0[pos]
c3[pos] = (psi12[pos] - np.sin(psi12[pos])) / psi12[pos] ** 3.0
if any(neg):
c2[neg] = (1 - np.cosh(psi12[neg])) / psi0[neg]
c3[neg] = (np.sinh(psi12[neg]) - psi12[neg]) / psi12[neg] ** 3.0
tmp = c2 + c3 == 0
if any(tmp):
c2[tmp] = 1.0 / 2.0
c3[tmp] = 1.0 / 6.0
return c2, c3