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orbit_det.py
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orbit_det.py
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import numpy as np
import numpy.polynomial.polynomial as poly
import pandas as pd
from astroquery.jplhorizons import Horizons
from astropy.coordinates import SkyCoord
import astropy.units as u
from pprint import pprint
import math
# https://stackoverflow.com/a/23689767
# helpful class to access a dictionary using dots (useful for reprsenting orbits)
class dotdict(dict):
"""dot.notation access to dictionary attributes"""
__getattr__ = dict.get
__setattr__ = dict.__setitem__
__delattr__ = dict.__delitem__
# get a 3D rotation matrix with angle and axis
def get_rotation_mat(angle, axis="x"):
# create a 2d rotation matrix
mat = np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]])#, dtype=np.float128)
# convert to a 3d rotation matrix by inserting the 0's and 1's
num = ["x", "y", "z"].index(axis)
mat = np.insert(mat, num, [0, 0], axis=1)
mat = np.insert(mat, num, [0, 0, 0], axis=0)
mat[num, num] = 1
return mat
# constants class
class CONSTS:
k = 0.0172020989484
c = 173.144643267 # au / day
obliquity = np.radians(23.4374)
# generate conversions from equatorial to ecliptic & vice-versa
eq2ec_mat = get_rotation_mat(-obliquity, axis="x")
ec2eq_mat = get_rotation_mat(obliquity, axis="x")
# convert degree to sexagesimal
def to_seg(ra, dec):
# convert ra, extract each value
ra_hours = ra // (360/24)
ra %= 360/24
ra_minutes = ra // (360/(24 * 60))
ra %= 360/(24 * 60)
ra_seconds = ra / (360/(24 * 60 * 60))
# convert dec, extract each value
# get the sign of dec so we don't have to deal w/ signage
sign = math.copysign(1, dec)
dec = abs(dec)
dec_deg = dec // 1
dec %= 1
dec_minutes = dec // (1 / 60)
dec %= 1/60
dec_seconds = dec / (1 / (60 * 60))
# return the final result as a string
return f"{int(ra_hours):02d}:{int(ra_minutes):02d}:{ra_seconds:08.5f}", f"{'+' if sign > 0 else '-'}{int(dec_deg):02d}:{int(dec_minutes):02d}:{dec_seconds:08.5f}"
# extract rho_hat using formula
def get_rho_hat(ra, dec):
return np.array(
[
np.cos(ra) * np.cos(dec),
np.sin(ra) * np.cos(dec),
np.sin(dec)
]
)
# taus = [tau1, tau3, tau3 - tau1]
# Ds = [D0, D21, D22, D23]
# scalar equation of lagrange to get inital estimates of the magnitude of r2 (roots) and the length of the rho vectors (rhos)
def SEL(taus, Sun2, rhohat2, Ds, mu=1):
tau1, tau3, tau = taus
# find expressions for variables
A1 = tau3 / tau
B1 = A1 / 6 * (tau ** 2 - tau3 ** 2)
A3 = -tau1 / tau
B3 = A3 / 6 * (tau ** 2 - tau1 ** 2)
A = (A1 * Ds[1] - Ds[2] + A3 * Ds[3]) / (-Ds[0])
B = (B1 * Ds[1] + B3 * Ds[3]) / (-Ds[0])
E = -2 * np.dot(rhohat2, Sun2)
F = np.dot(Sun2, Sun2)
a = -(A ** 2 + A * E + F)
b = -mu * (2 * A * B + B * E)
c = -((mu * B) ** 2)
# create polynomials & solve
coeffs = np.zeros(9)
coeffs[0] = c
coeffs[3] = b
coeffs[6] = a
coeffs[8] = 1
roots = poly.polyroots(coeffs)
# find rhos from roots (which are r2 magnitude)
rhos = A + mu * B / roots.real ** 3
# filter out bad roots
mask = (roots.imag == 0) & (rhos > 0) & (roots.real > 0)
roots = roots[mask].real
rhos = rhos[mask]
return roots, rhos
# find dE with Newton's method for finding f & g
def find_dE(r2, r2dot, tau, n, a, x_guess, eps=1e-12):
x = x_guess
x_prev = 99999
# helpful value to just calculate once, no need to keep calculating every loop
val = np.dot(r2, r2dot) / (n * a ** 2)
r2_mag = np.linalg.norm(r2)
# iterate until change small enough
while abs(x_prev - x) > eps:
f_x = x - (1 - r2_mag / a) * np.sin(x) + val * (1 - np.cos(x)) - n * tau
f_prime = 1 - (1 - r2_mag / a) * np.cos(x) + val * np.sin(x)
x_prev = x
x -= f_x / f_prime
return x
# if flag is 0, then we use func
def fg(tau1, tau3, r2, r2dot, flag=0, mu=1):
r2_mag = np.linalg.norm(r2)
# if flag is > 0, then we assume it's of degree flag
if flag > 0:
u = mu / (r2_mag ** 3)
z = np.dot(r2, r2dot) / (r2_mag ** 2)
q = np.dot(r2dot, r2dot) / (r2_mag ** 2) - u
# generate truncated polynomials
f_series_terms = [
1,
0,
- mu/(2 * (r2_mag ** 3)),
mu * np.dot(r2, r2dot) / (2 * (r2_mag ** 5)),
(3 * u * q - 15 * u * z ** 2 + u ** 2) / 24
][:flag + 1]
g_series_terms = [
0,
1,
0,
-mu / (6 * (r2_mag ** 3)),
(6 * u * z) / 24
][:flag + 1]
# get the powers and compute
powers1 = tau1 ** np.arange(flag + 1)
powers3 = tau3 ** np.arange(flag + 1)
f_series_terms = np.array(f_series_terms)
g_series_terms = np.array(g_series_terms)
return (np.dot(f_series_terms, powers1), np.dot(g_series_terms, powers1)), (np.dot(f_series_terms, powers3), np.dot(g_series_terms, powers3))
elif flag == 0:
# function version
# find the orbital elements needed
a = 1/(2 / r2_mag - np.dot(r2dot, r2dot) / mu)
n = np.sqrt(mu / a ** 3)
e = np.sqrt(1 - np.linalg.norm(np.cross(r2, r2dot)) ** 2 / (mu * a))
# get the init guess
val1 = np.dot(r2, r2dot) / (n * a ** 2)
val2 = n * tau1 - val1
sign = val1 * np.cos(val2) + (1 - r2_mag / a) * np.sin(val2)
guess1 = n * tau1 + np.copysign(0.85 * e, sign) - val1
# calculate dE
dE1 = find_dE(r2, r2dot, tau1, n, a, guess1)
f1 = 1 - a / r2_mag * (1 - np.cos(dE1))
g1 = tau1 + 1 / n * (np.sin(dE1) - dE1)
# same process for obs 3
val1 = np.dot(r2, r2dot) / (n * a ** 2)
val2 = n * tau3 - val1
sign = val1 * np.cos(val2) + (1 - r2_mag / a) * np.sin(val2)
guess3 = n * tau3 + np.copysign(0.85 * e, sign) - val1
dE3 = find_dE(r2, r2dot, tau3, n, a, guess3)
f3 = 1 - a / r2_mag * (1 - np.cos(dE3))
g3 = tau3 + 1 / n * (np.sin(dE3) - dE3)
return (f1, g1), (f3, g3)
else:
raise ValueError("Invalid Flag")
def get_orbit(r, rdot, t, t0, mu=1, k=CONSTS.k):
rmag = np.linalg.norm(r)
### SEMIMAJOR AXIS
a = 1/(2 / rmag - np.dot(rdot, rdot) / mu)
### ECCENTRICITY
h = np.cross(r, rdot)
hmag = np.linalg.norm(h)
e = np.sqrt(1 - hmag ** 2 / (mu * a))
### INCLINATION
# h[2] = h_z
i = np.arccos(h[2] / hmag)
### LONG OF ASC NODE
# h[0] = h_x, h[1] = h_y
omega = np.arctan2(h[0], -h[1])
### ARG OF PERIHELION
## first find U
U = np.arctan2(r[2] / np.sin(i), r[0] * np.cos(omega) + r[1] * np.sin(omega))
## then find v
ecosv = a * (1 - e ** 2) / rmag - 1
esinv = a * (1 - e ** 2) / hmag * np.dot(r, rdot) / rmag
v = np.arctan2(esinv, ecosv) % (2 * np.pi)
## finally find w
w = U - v
### MEAN ANAMOLY AT EPOCH
## first find E
E = np.arccos((1 - rmag/a) / e)
if v > np.pi:
E = 2 * np.pi - E
## then find mean anamoly rn
M = E - e * np.sin(E)
# finally convert to epoch mean anamoly using mean motion
# convert n from per gaussian day to per day using k
n = np.sqrt(mu / a ** 3) * k
M0 = M + n * (t0 - t)
return dotdict(
a=a,
e=e,
i=np.degrees(i) % 360,
omega=np.degrees(omega) % 360,
w=np.degrees(w) % 360,
M0=np.degrees(M0) % 360
)
# approximate E using Newton's method
# inverse of M = E - e*sin(E)
def get_E(M, e, eps=1e-004):
# make guesses
E_guess = M
M_guess = E_guess - e*np.sin(E_guess)
M_true = M
# loop while we are not within epsilon
while np.abs(M_guess - M_true) > eps:
# our approximation for M
M_guess = E_guess - e * np.sin(E_guess)
# update our guess
E_guess = E_guess - (M_true - (E_guess - e * np.sin(E_guess))) / (e * np.cos(E_guess) - 1)
return E_guess
# the Method of Gauss!
def MoG(ras_obs, decs_obs, sun_vecs, times, epoch, debug=False, tolerance=1e-8):
# get rho hats from ras & decs
rho_hats = get_rho_hat(np.radians(ras_obs), np.radians(decs_obs)).T
# find taus by multiplying by k
taus = [(times[i] - times[1]) * CONSTS.k for i in range(3)]
tau = taus[2] - taus[0]
# calculate D values
D0 = np.dot(rho_hats[0], np.cross(rho_hats[1], rho_hats[2]))
# (R_i cross rho2) dot rho3
D1 = [np.dot(np.cross(sun_vecs[i], rho_hats[1]), rho_hats[2]) for i in range(3)]
# (rho1 cross R_i) dot rho3
D2 = [np.dot(np.cross(rho_hats[0], sun_vecs[i]), rho_hats[2]) for i in range(3)]
# rho1 dot (rho2 cross R_i)
D3 = [np.dot(rho_hats[0], np.cross(rho_hats[1], sun_vecs[i])) for i in range(3)]
# get roots/rhos & ask user if more than one solution
roots, rhos = SEL([taus[0], taus[2], tau], sun_vecs[1], rho_hats[1], [D0, *D2])
if len(rhos) > 1:
print("ROOTS:", roots)
print("RHOS:", rhos)
index = int(input("Which root?"))
rho2_mag = rhos[index]
else:
rho2_mag = rhos[0]
# find the vectors
rho2 = rho_hats[1] * rho2_mag
r2 = rho2 - sun_vecs[1]
# we don't have r2dot yet, but we are only going up to the 2nd degree term
(f1, g1), (f3, g3) = fg(taus[0], taus[2], r2, np.array([0., 0., 0.]), flag=3)
# begin to iterate until rho2 magnitude change is small enough
rho2_mag_prev = 9999
i = 0
while abs(rho2_mag_prev - rho2_mag) > tolerance:
if debug:
print("Prev RHO2 Mag", rho2_mag_prev)
print("New RHO2 Mag", rho2_mag)
print("Change", abs(rho2_mag_prev - rho2_mag))
rho2_mag_prev = rho2_mag
if debug:
i += 1
print("ITERATION", i)
# find c's
denom = f1 * g3 - g1 * f3
c1 = g3 / denom
c2 = -1
c3 = -g1 / denom
c = [c1, c2, c3]
# find rho's and r's
rho1 = np.dot(c, D1) / (c1 * D0) * rho_hats[0]
rho2 = np.dot(c, D2) / (c2 * D0) * rho_hats[1]
rho3 = np.dot(c, D3) / (c3 * D0) * rho_hats[2]
r1 = rho1 - sun_vecs[0]
r2 = rho2 - sun_vecs[1]
r3 = rho3 - sun_vecs[2]
rho2_mag = np.linalg.norm(rho2)
# find r2dot from f & g
d1 = -f3 / denom
d3 = f1 / denom
r2dot = d1 * r1 + d3 * r3
# light travel correction
t1 = times[0] - np.linalg.norm(rho1) / CONSTS.c
t2 = times[1] - np.linalg.norm(rho2) / CONSTS.c
t3 = times[2] - np.linalg.norm(rho3) / CONSTS.c
# recalc taus
taus[0] = (t1 - t2) * CONSTS.k
taus[2] = (t3 - t2) * CONSTS.k
tau = t3 - t1
# update f & g
(f1, g1), (f3, g3) = fg(taus[0], taus[2], r2, r2dot, flag=0)
# convert to ecliptic
r2_ecliptic = CONSTS.eq2ec_mat @ r2
r2dot_ecliptic = CONSTS.eq2ec_mat @ r2dot
# get orbital params
o = get_orbit(r2_ecliptic, r2dot_ecliptic, t2, epoch)
return r2_ecliptic, r2dot_ecliptic, o
# read input
obs = []
filename = input("FILENAME: ")
with open(filename) as f:
# ignore commented lines
lines = [line for line in f.readlines() if line[0] != "#"]
tokens = lines[0].split()
epoch = float(tokens[0])
# read in each line
for line in lines[1:]:
tokens = line.split()
jd = float(tokens[0])
# get ra & dec in radians
c = SkyCoord(tokens[1], tokens[2], unit=(u.hourangle, u.deg))
ra = c.ra.rad
dec = c.dec.rad
# 6 token line is test input
if len(tokens) == 6:
sun_vector = np.array([float(tokens[3]), float(tokens[4]), float(tokens[5])])
ra_err = 0
dec_err = 0
obscode="500"
# otherwise get sun vector from horizons
else:
obscode = tokens[3]
query = Horizons(id="10", location=obscode, epochs=jd, id_type="id")
sun_vector = query.vectors(refplane="earth", aberrations="apparent")[0]
sun_vector = np.array([sun_vector["x"], sun_vector["y"], sun_vector["z"]])
obs.append(dict(
ra=ra,
dec=dec,
jd=jd,
sun_vector=sun_vector,
obscode=obscode
))
# pick observations if more than 3
if len(obs) > 3:
i, j, k = input("Which 3 observations (zero-indexing)?\n").replace(",", " ").split()
i, j, k = sorted([int(i), int(j), int(k)])
mog_obs = [obs[i], obs[j], obs[k]]
del obs[k], obs[j], obs[i]
obs = mog_obs + obs
# convert to pandas dataframe to get out the lists
obs = pd.DataFrame(obs)
sun_vecs = np.stack(obs["sun_vector"])
times = np.array(obs["jd"])
ras_obs = np.degrees(np.array(obs["ra"]))
decs_obs = np.degrees(np.array(obs["dec"]))
# get values and output
r2, r2dot, orbit = MoG(ras_obs, decs_obs, sun_vecs, times, epoch, debug=True)
print()
print("RESULTS:")
print("POSITION VECTOR (AU) :", r2)
print("VELOCITY VECTOR (AU/day):", r2dot * CONSTS.k)
print("DIST TO EARTH:", r2 + sun_vecs[1])
print("ORBITAL PARAMS:")
pprint(orbit)