astrometry.net-0.85/util/celestial_mechanics.py

# celestial_mechanics.py
#   celestial mechanics utilities for exoplanet ephemerides
#
# intellectual property:
#   Copyright 2009 David W. Hogg.  All rights reserved.
#   Licensed under a 3-clause BSD style license - see LICENSE
#
# comments:
#   - Written for clarity, not speed.  The code is intended to be human-
#     readable.
#
# bugs:
#   - Need to make Fourier expansion functions.
#    - What to do if e is close to 1.0 in eccentric_anomaly
#
from __future__ import print_function

from math import pi
import unittest
import sys

import numpy
import numpy as np
from numpy import *
from scipy.special import jn
import matplotlib.pyplot as plt

from astrometry.util.starutil_numpy import *

ihat = array([1.,0.,0.])
jhat = array([0.,1.,0.])
khat = array([0.,0.,1.])
default_tolerance = 1e-15 # (radians) don't set to zero or less
default_maximum_iteration = 1000 # should never hit this limit!
default_order = 32
default_K = 1.0

(Equinox, Solstice, EclipticPole) = ecliptic_basis()

c_au_per_yr = 63239.6717 # google says

#GM_sun = 2.9591310798672560E-04 #AU^3/d^2
# Google says mass of the sun * G = 39.4775743 (au^3) / (yr^2)
GM_sun = 39.4775743 # AU^3 / yr^2

def norm1d(x):
    assert(len(x.shape) == 1)
    return np.sqrt(np.sum(x**2))

def deg2rad(x):
    return x * pi/180.
    #return radians(x)

def orbital_elements_to_ss_xyz(E, observer=None, light_travel=True):
    (a,e,i,Omega,pomega,M,GM) = E
    # ugh, it's hard to be units-agnostic.
    # we just assert here so we have to think about this!
    # This means:
    #  distances in AU
    #  angles in radians
    #  times in years
    assert(GM == GM_sun)
    if light_travel:
        assert(observer is not None)
        # orbital angular velocity  [radians/yr]
        meanfrequency = np.sqrt(GM / a**3)
    # Correct for light-time delay.
    # dM: [radians]
    dM = 0.
    lastdM = dM
    dx = None
    for ii in range(100):
        (x,v) = phase_space_coordinates_from_orbital_elements(
            a,e,i,Omega,pomega,M-dM,GM)
        if not light_travel:
            if observer is not None:
                dx = x - observer
            break
        dx = x - observer
        # light-travel distance [AU]
        r = norm1d(dx)
        # light-travel time [yr]
        travel = r / c_au_per_yr
        #print 'light-travel time:', travel, 'yr, or', travel*365.25, 'd'
        # light travel in angular periods [radians]
        dM = travel * meanfrequency
        if abs(lastdM - dM) < 1e-12:
            break
        lastdM = dM
    if ii == 99:
        print('Warning: orbital_elements_to_ss_xyz: niters', ii)
    return x,dx

def orbital_elements_to_xyz(E, observer, light_travel=True, normalize=True):
    (x,dx) = orbital_elements_to_ss_xyz(E, observer, light_travel)
    if normalize:
        dx /= norm1d(dx)
    edx = dx[0] * Equinox + dx[1] * Solstice + dx[2] * EclipticPole
    return edx

# E = (a,e,i,Omega,pomega,M, GM)
# observer = 3-vector
# light_travel: correct for light travel time?
# Returns RA,Dec in degrees.
def orbital_elements_to_radec(E, observer, light_travel=True):
    xyz = orbital_elements_to_xyz(E, observer, light_travel)
    return xyztoradec(xyz)

# convert orbital elements into vectors in the plane of the orbit.
def orbital_vectors_from_orbital_elements(i, Omega, pomega):
    ascendingnodevector = np.cos(Omega) * ihat + np.sin(Omega) * jhat
    tmpydir= np.cross(khat, ascendingnodevector)
    zhat= np.cos(i) * khat - np.sin(i) * tmpydir
    tmpydir= np.cross(zhat, ascendingnodevector)
    xhat= np.cos(pomega) * ascendingnodevector + np.sin(pomega) * tmpydir
    yhat = np.cross(zhat, xhat)
    return (xhat, yhat, zhat)

def position_from_orbital_vectors(xhat, yhat, a, e, M):
    E = eccentric_anomaly_from_mean_anomaly(M, e)
    cosE = np.cos(E)
    sinE = np.sin(E)
    b = a*np.sqrt(1. - e**2)
    x =  a * (cosE - e)  * xhat + b * sinE        * yhat
    return x

# convert orbital elements to phase-space coordinates
#  a       - semi-major axis (length units)
#  e       - eccentricity
#  i       - inclination (rad)
#  Omega   - longitude of ascending node (rad)
#  pomega  - argument of periapsis (rad)
#  M       - mean anomaly (rad)
#  GM      - Newton's constant times central mass (length units cubed over time units squared)
#  return  - (x,v)
#            position, velocity (length units, length units per time unit)
def phase_space_coordinates_from_orbital_elements(a, e, i, Omega, pomega, M, GM):
    (xhat, yhat, zhat) = orbital_vectors_from_orbital_elements(i, Omega, pomega)
    # [radians/yr]
    dMdt = np.sqrt(GM / a**3)
    E = eccentric_anomaly_from_mean_anomaly(M, e)
    cosE = np.cos(E)
    sinE = np.sin(E)
    # [radians/yr]
    dEdt = 1.0 / (1.0 - e * cosE) * dMdt
    # [AU]
    b = a*np.sqrt(1. - e**2)
    x =  a * (cosE - e)  * xhat + b * sinE        * yhat
    # [AU/yr]
    v = -a * sinE * dEdt * xhat + b * cosE * dEdt * yhat
    return (x, v)

class UnboundOrbitError(ValueError):
    pass

def potential_energy_from_position(x, GM):
    return -1. * GM / norm1d(x)

def energy_from_phase_space_coordinates(x, v, GM):
    return 0.5 * np.dot(v, v) + potential_energy_from_position(x, GM)

# convert phase-space coordinates to orbital elements
#  x       - position (3-vector, length units)
#  v       - velocity (3-vector, length units per time unit)
#  GM      - Newton's constant times central mass (length units cubed over time units squared)
#  return  - (a, e, i, Omega, pomega, M)
#          - see "phase_space_coordinates" for definitions
def orbital_elements_from_phase_space_coordinates(x, v, GM):
    energy = energy_from_phase_space_coordinates(x, v, GM)
    if energy > 0:
        raise UnboundOrbitError('orbital_elements_from_phase_space_coordinates: Unbound orbit')

    angmom = np.cross(x, v)
    zhat = angmom / norm1d(angmom)
    evec = np.cross(v, angmom) / GM - x / norm1d(x)
    e = norm1d(evec)
    if e == 0:
        # by convention:
        xhat = np.cross(jhat, zhat)
        xhat /= norm1d(xhat)
    else:
        xhat = evec / e
    yhat = np.cross(zhat, xhat)
    a = -0.5 * GM / energy
    i = np.arccos(angmom[2] / norm1d(angmom))
    if i == 0:
        Omega = 0.0
    else:
        Omega = np.arctan2(angmom[1], angmom[0]) + 0.5 * pi
        if Omega < 0:
            Omega += 2.*pi
        if i < 0:
            i *= -1.
            Omega += pi
    cosOmega = cos(Omega)
    sinOmega = sin(Omega)
    if e == 0:
        pomega = 0. - Omega
    else:
        pomega = np.arccos(min(1.0, (evec[0] * cosOmega + evec[1] * sinOmega) / e))
    horriblescalar = ( sinOmega * evec[2] * angmom[0]
             - cosOmega * evec[2] * angmom[1]
             + cosOmega * evec[1] * angmom[2]
             - sinOmega * evec[0] * angmom[2])
    if horriblescalar < 0.:
        pomega = 2.0 * pi - pomega
    if pomega < 0.0:
        pomega += 2.0 * pi
    if pomega > 2.0 * pi:
        pomega -= 2.0 * pi
    f = np.arctan2(np.dot(yhat, x), np.dot(xhat, x))
    M = mean_anomaly_from_true_anomaly(f, e)
    if M < 0:
        M += 2.*pi
    return (a, e, i, Omega, pomega, M)

# convert eccentric anomaly to mean anomaly
#  E       - eccentric anomaly (radians)
#  e       - eccentricity
#  return  - mean anomaly (radians)
def mean_anomaly_from_eccentric_anomaly(E, e):
    return (E - e * np.sin(E))

def mean_anomaly_from_true_anomaly(f, e):
    return mean_anomaly_from_eccentric_anomaly(eccentric_anomaly_from_true_anomaly(f, e), e)

# convert mean anomaly to eccentric anomaly
#  M       - [array of] mean anomaly (radians)
#  e       - eccentricity
#  [tolerance - read the source]
#  [maximum_iteration - read the source]
#  return  - eccentric anomaly (radians)
def eccentric_anomaly_from_mean_anomaly(M, e, tolerance = default_tolerance,
              maximum_iteration = default_maximum_iteration, verbose=False):
    E = M + e * np.sin(M)
    iteration = 0
    deltaM = 100.0
    while (iteration < maximum_iteration) and (abs(deltaM) > tolerance):
        deltaM = (M - mean_anomaly_from_eccentric_anomaly(E, e))
        E = E + deltaM / (1. - e * cos(E))
        iteration += 1
    if verbose: print('eccentric anomaly iterations:',iteration)
    return E

def eccentric_anomaly_from_true_anomaly(f, e):
    E = np.arccos((np.cos(f) + e) / (1.0 + e * np.cos(f)))
    E *= (np.sign(np.sin(f)) * np.sign(np.sin(E)))
    return E

# convert eccentric anomaly to true anomaly
#  E       - eccentric anomaly (radians)
#  e       - eccentricity
#  return  - true anomaly (radians)
def true_anomaly_from_eccentric_anomaly(E, e):
    f = np.arccos((np.cos(E) - e) / (1.0 - e * np.cos(E)))
    f *= (np.sign(np.sin(f)) * np.sign(np.sin(E)))
    return f

# compute radial velocity
#  K       - radial velocity amplitude
#  f       - true anomaly (radians)
#  e       - eccentricity
#  pomega  - eccentric longitude (radians)
#  return  - radial velocity (same units as K)
def radial_velocity(K, f, e, pomega):
    return K * (np.sin(f + pomega) + e * np.sin(pomega))

# compute radial velocity
#  K       - radial velocity amplitude
#  M       - mean anomaly (radians)
#  e       - eccentricity
#  pomega  - eccentric longitude (radians)
#  return  - radial velocity (same units as K)
def radial_velocity_from_M(K, M, e, pomega):
    E = M + e*np.sin(M)
    term1 = np.cos(pomega) * np.sqrt(1 - e**2) * np.sin(E) / (1 - e*np.cos(E))
    term2 = np.sin(pomega) * (np.cos(E) - e) / (1 - e*np.cos(E))
    term3 = e*np.sin(pomega)
    return K * (term1 + term2 + term3)

# compute radial velocity using a truncated Fourier series
#  K       - radial velocity amplitude
#  M       - mean anomaly (radians) APW: you may want to change this input
#  e       - eccentricity
#  pomega  - eccentric longitude (radians)
#  phi       - phase
#  [order  - read the source]
#  return  - radial velocity (same units as K)
def radial_velocity_fourier_series(K, M, e, pomega, phi, order=default_order):
    vr = 0.0
    for n in arange(0, order+1, 1):
        vr += K*(fourier_coeff_A(n, pomega, phi, e) * np.cos(n*(M-phi)) \
            + fourier_coeff_B(n, pomega, phi, e) * np.sin(n*(M-phi)))
    return vr

# the following is based on the naming convention in Itay's notes on Fourier analysis
#  - fourier_coeff_A and fourier_coeff_B are the actual coefficients in the series
#  - aprime and bprime are just used to simplify the code, and break it up to make it more readable

#  n       - order of the coefficient
#  e       - eccentricity
def aprime(n,e):
    return np.sqrt(1. - e**2)*( ((np.sqrt(1. - e**2) - 1)/e)*jn(n,n*e) + jn(n-1,n*e))

def bprime(n,e):
    return np.sqrt(1. - e**2)*( ((np.sqrt(1. - e**2) - 1)/e)*jn(n,n*e) - jn(n-1,n*e))

def fourier_coeff_A(n, pomega, phi, e):
    return 0.5 * (aprime(n,e) * np.sin(pomega + n * phi) + bprime(n,e) * np.sin(pomega - n * phi))

def fourier_coeff_B(n, pomega, phi, e):
    return 0.5 * (aprime(n,e) * np.cos(pomega + n * phi) - bprime(n,e) * np.cos(pomega - n * phi))

# APW: adjust function call as necessary
#  return  - amplitudes as a list of tuples (An,Bn)
def radial_velocity_fourier_amplitudes(K, phi, e, pomega, order=default_order):
    amplitudes = []
    for n in range(order):
        amplitudes.append((fourier_coeff_A(n, pomega, phi, e), fourier_coeff_B(n, pomega, phi, e)))
    return amplitudes

def eccentricity_from_fourier_amplitudes(amplitudes):
    K = np.sqrt(amplitude[0]**2 + amplitude[1]**2)
    phi = np.arctan(amplitude[1] / amplitude[0]) # WRONG?
    e = 0 # WRONG
    pomega = 0 # WRONG
    return (K, phi, e, pomega)


# some functional testing
if __name__ == '__main__':
    from test_celestial_mechanics import *
    #suite = unittest.TestLoader().loadTestsFromTestCase(TestOrbitalElements)

    suite = unittest.TestSuite()
    #suite.addTest(TestOrbitalElements('testEdgeCases'))
    suite.addTest(TestOrbitalElements('testAgainstJPL_2'))

    unittest.TextTestRunner(verbosity=2).run(suite)
    import sys
    sys.exit(0)

    # -- test Earth ephemeris against JPL at Holmes' closest approach
    # -- test Holmes against JPL         ---------''------------
    # -- test direction2radec()

    try:
        arg1 = sys.argv[1]
    except IndexError:
        arg1 = 0

    for e in [0.01, 0.1, 0.9, 0.99, 0.999]:
        print('eccentricity:', e)
        M = arange(-3.16,-3.14,0.001) # easy
        #M = arange(-10., 10., 2.1)    # easy
        #M = arange(-0.01,0.01,0.001)  # hard
        print('mean anomaly input:', M)
        E = eccentric_anomaly_from_mean_anomaly(M, e, verbose=True)
        print('eccentric anomaly output:', E)
        f = true_anomaly_from_eccentric_anomaly(E, e)
        print('true anomaly output:', f)
        M2 = mean_anomaly_from_eccentric_anomaly(E, e)
        print('round-trip error:', M2 - M)

    if arg1 == "plot":
        # This code will do the plotting:
        range_min = 0.0
        range_max = 20.0
        step_size = 0.01
        phase = 0.0
        pomegas = [0.0, pi/5., 3.*pi/5., 5.*pi/5., 7.*pi/5., 9.*pi/5.]
        pomegas_str = ["pomega = $0.0$", "pomega = $\pi/5$", "pomega = $3\pi/5$", "pomega = $\pi$", \
            "pomega = $7\pi/5$", "pomega = $9\pi/5$"]
        eccens = [0.01, 0.1, 0.5, 0.9, 0.99]
        orders = [2, 4, 8, 16, 32]
        M = arange(range_min,range_max,step_size)

        for n in orders:
            for e in eccens:
                i = 1
                for pomega in pomegas:
                    plt.clf()
                    plt.suptitle(pomegas_str[i-1] + ", e = %.2f, n = %i" % (e,n))
                    plt.subplot(211)
                    plt.plot(M, radial_velocity_fourier_series(default_K, M, e, pomega, phase, order=n), 'r')
                    plt.plot(M, radial_velocity_from_M(default_K, M, e, pomega), 'k--')
                    plt.axis([range_min,range_max,-(default_K+1.),default_K+1.])
                    plt.subplot(212)
                    plt.plot(M, radial_velocity_from_M(default_K, M, e, pomega) \
                        - radial_velocity_fourier_series(default_K, M, e, pomega, phase, order=n), 'k')
                    plt.savefig("celestial_mechanics_plots/pomega_%i_e_%.2f_n_%i.png" % (i,e,n))
                    i += 1