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autograd.py
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autograd.py
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# Experimental JAX based autograd approach to orbit fitting
# Result: Not nearly as fast as the physics-based approach
import os
os.environ["JAX_ENABLE_X64"] = "True"
import numba
import jax.numpy as np
import jax
# use JIT to precompile functions
from jax import grad, jit
from jax.experimental import optimizers
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
from tqdm import tqdm
import matplotlib.pyplot as plt
# 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}"
# get a 3D rotation matrix with angle and axis
def get_rotation_mat(angle, axis="x"):
if axis == "x":
mat = np.array([
[1, 0, 0],
[0, np.cos(angle), -np.sin(angle)],
[0, np.sin(angle), np.cos(angle)]
], dtype=np.float64)
elif axis == "y":
mat = np.array([
[np.cos(angle), 0, np.sin(angle)],
[0, 1, 0],
[-np.sin(angle), 0, np.cos(angle)]
], dtype=np.float64)
elif axis == "z":
mat = np.array([
[np.cos(angle), -np.sin(angle), 0],
[np.sin(angle), np.cos(angle), 0],
[0, 0, 1]
], dtype=np.float64)
return mat
class CONSTS:
k = 0.0172020989484
c = 173.144643267 # au / day
obliquity = np.radians(23.4374)
eq2ec_mat = get_rotation_mat(-obliquity, axis="x")
ec2eq_mat = get_rotation_mat(obliquity, axis="x")
# find dE with Newton's method for finding f & g
@jit
def find_dE(r2, r2dot, tau, n, a, x_guess, eps=1e-12, iter=5):
x = x_guess
# 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)
# use fixed interation count so that autograd can work properly
for i in range(iter):
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 -= f_x / f_prime
return x
@jit
def fg_single(tau, r2, r2dot, flag=0, mu=1):
r2_mag = np.linalg.norm(r2)
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
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]
powers = tau ** np.arange(flag)
f_series_terms = np.array(f_series_terms)
g_series_terms = np.array(g_series_terms)
return np.dot(f_series_terms, powers), np.dot(g_series_terms, powers)
elif flag == 0:
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))
val1 = np.dot(r2, r2dot) / (n * a ** 2)
val2 = n * tau - val1
sign = val1 * np.cos(val2) + (1 - r2_mag / a) * np.sin(val2)
guess1 = n * tau + np.copysign(0.85 * e, sign) - val1
dE1 = find_dE(r2, r2dot, tau, n, a, guess1)
f1 = 1 - a / r2_mag * (1 - np.cos(dE1))
g1 = tau + 1 / n * (np.sin(dE1) - dE1)
return (f1, g1)
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 dict(
a=a,
e=e,
i=np.degrees(i) % 360,
omega=np.degrees(omega) % 360,
w=np.degrees(w) % 360,
M0=np.degrees(M0) % 360
)
@jit
# with light correction, num_iter is number times of iteration to acc for light correction
def ephem_r(r, rdot, t_list, t2, R_list, num_iter=3, c=CONSTS.c, k=CONSTS.k):
t_list_orig = np.array(t_list)
for _ in range(num_iter):
fs = []
gs = []
taus = []
for t in t_list:
tau = k * (t - t2)
f, g = fg_single(tau, r, rdot)
fs.append([f])
gs.append([g])
taus.append(t)
fs = np.array(fs)
gs = np.array(gs)
r_pred = fs * r + gs * rdot
rho_pred = R_list + r_pred
rho_pred_mags = np.sum(rho_pred ** 2, axis=1) ** 0.5
t_list = t_list_orig - rho_pred_mags / c
normed = rho_pred / np.expand_dims(rho_pred_mags, axis=1)
ra = np.arctan2(normed[:, 1], normed[:, 0])
dec = np.arcsin(normed[:, 2])
return np.degrees(ra) % 360, np.degrees(dec)
obs = []
with open("obs.txt") as f:
lines = [line for line in f.readlines() if line[0] != "#"]
tokens = lines[0].split()
epoch = float(tokens[0])
name = " ".join(tokens[1:])
for line in lines[1:]:
tokens = line.split()
jd = float(tokens[0])
c = SkyCoord(tokens[1], tokens[2], unit=(u.hourangle, u.deg))
ra = c.ra.rad
dec = c.dec.rad
if len(tokens) == 7:
sun_vector = np.array([float(tokens[3]), float(tokens[4]), float(tokens[5])], dtype=np.float64)
ra_err = 0
dec_err = 0
obscode="500"
else:
ra_err = float(tokens[3]) / 3600
dec_err = float(tokens[4]) / 3600
obscode = tokens[5]
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"]], dtype=np.float64)
obs.append(dict(
ra=ra,
dec=dec,
ra_err=ra_err,
dec_err=dec_err,
jd=jd,
sun_vector=sun_vector,
obscode=obscode
))
obs = pd.DataFrame(obs)
sun_vecs = np.stack(obs["sun_vector"])
times = np.array(obs["jd"], dtype=np.float64)
ras_obs = np.degrees(np.array(obs["ra"], dtype=np.float64))
decs_obs = np.degrees(np.array(obs["dec"], dtype=np.float64))
ras_obs_err = np.array(obs["ra_err"], dtype=np.float64)
decs_obs_err = np.array(obs["dec_err"], dtype=np.float64)
r = np.array([ 0.23026246, -1.50169513, -0.18074596] , dtype=np.float64)
rdot = np.array([0.66525549, 0.56062365, 0.32885496], dtype=np.float64)
t2 = 2459398.85665902
def rms(r, rdot, times, t2, sun_vecs, ras_obs, decs_obs, debug=False):
ra, dec = ephem_r(r, rdot, times, t2, sun_vecs)
delta_ras = ras_obs - ra
delta_dec = decs_obs - dec
deltas = np.concatenate((delta_ras, delta_dec))
rms = np.sqrt(np.sum(deltas ** 2) / (len(deltas) - 6))
if debug:
print(deltas)
return rms
grad_rms = grad(rms, argnums=[0, 1])
lr = 0.00000001
rmss = []
for i in tqdm(range(1000)):
rms_val = rms(r, rdot, times, t2, sun_vecs, ras_obs, decs_obs, debug=False)
print(rms_val)
rmss.append(rms_val)
grad_r, grad_rdot = grad_rms(r, rdot, times, t2, sun_vecs, ras_obs, decs_obs)
r -= grad_r * lr
rdot -= grad_rdot * lr
print(rmss)
ra, dec = ephem_r(r, rdot, times, t2, sun_vecs)
ras_fit, decs_fit = ephem_r(r, rdot, times, t2, sun_vecs)
delta_ras = ras_obs - ras_fit
delta_dec = decs_obs - decs_fit
for i in range(len(times)):
print()
print(f"OBS{i+1}: ", *to_seg(ras_obs[i], decs_obs[i]))
print("PRED: ", *to_seg(ras_fit[i], decs_fit[i]))
print(f"RESD: {delta_ras[i] * 3600:0.3f} {delta_dec[i] * 3600:0.3f}")
pprint(get_orbit(r, rdot, t2, epoch))