616 lines
20 KiB
Python
616 lines
20 KiB
Python
from __future__ import division
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import sys
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import math
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from numpy import (
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abs, amax, amin, arange, arccos, arctan, array, atleast_1d,
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clip, copy, copyto, cos, cosh, exp, full_like, log, ndarray, newaxis,
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pi, power, repeat, sign, sin, sinh, sqrt, sum, tan, tanh, zeros_like,
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)
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from skyfield.constants import AU_KM, DAY_S, DEG2RAD
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from skyfield.functions import dots, length_of, mxv
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from skyfield.descriptorlib import reify
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from skyfield.elementslib import OsculatingElements, normpi
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from skyfield.units import Distance, Velocity
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from skyfield.vectorlib import VectorFunction
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from skyfield.sgp4lib import _cross
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_CONVERT_GM = DAY_S * DAY_S / AU_KM / AU_KM / AU_KM
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class _KeplerOrbit(VectorFunction):
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def __init__(self,
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position,
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velocity,
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epoch,
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mu_au3_d2,
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center=None,
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target_name=None,
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):
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""" Calculates the position of an object using 2 body propagation
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Parameters
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----------
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position : Distance
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Position vector at epoch with shape (3,)
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velocity : Velocity
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Velocity vector at epoch with shape (3,)
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epoch : Time
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Time corresponding to `position` and `velocity`
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mu_au_d : float
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Value of mu (G * M) in au^3/d^2
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center : int
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NAIF ID of the primary body, 399 for geocentric orbits, 10 for
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heliocentric orbits
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target : int
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NAIF ID of the secondary body
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"""
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self.position_at_epoch = position
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self.velocity_at_epoch = velocity
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self.epoch = epoch
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self.mu_au3_d2 = mu_au3_d2
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self.center = center
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self.target_name = target_name
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self._rotation = None # TODO: make argument?
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@property
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def target(self):
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return self # this instance itself represents the target object
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@classmethod
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def _from_periapsis(
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cls,
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semilatus_rectum_au,
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eccentricity,
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inclination_degrees,
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longitude_of_ascending_node_degrees,
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argument_of_perihelion_degrees,
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t_periapsis,
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gm_km3_s2,
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center=None,
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target_name=None,
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):
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"""Build a `KeplerOrbit` given its parameters and date of periapsis."""
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gm_au3_d2 = gm_km3_s2 * _CONVERT_GM
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pos, vel = ele_to_vec(
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semilatus_rectum_au,
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eccentricity,
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DEG2RAD * inclination_degrees,
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DEG2RAD * longitude_of_ascending_node_degrees,
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DEG2RAD * argument_of_perihelion_degrees,
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0.0,
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gm_au3_d2,
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)
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return cls(
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Distance(pos),
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Velocity(vel),
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t_periapsis,
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gm_au3_d2,
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center,
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target_name,
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)
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@classmethod
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def _from_true_anomaly(cls, p, e, i, Om, w, v,
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epoch,
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mu_km_s=None,
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mu_au3_d2=None,
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center=None,
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target_name=None,
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):
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""" Creates a `KeplerOrbit` object from elements using true anomaly
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Parameters
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----------
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p : Distance
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Semi-Latus Rectum
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e : float
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Eccentricity
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i : Angle
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Inclination
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Om : Angle
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Longitude of Ascending Node
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w : Angle
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Argument of periapsis
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v : Angle
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True anomaly
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epoch : Time
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Time corresponding to `position` and `velocity`
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mu_km_s : float
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Value of mu (G * M) in km^3/s^2
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mu_au3_d2 : float
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Value of mu (G * M) in au^3/d^2
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center : int
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NAIF ID of the primary body, 399 for geocentric orbits, 10 for
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heliocentric orbits
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target : int
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NAIF ID of the secondary body
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"""
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if (mu_km_s and mu_au3_d2) or (not mu_km_s and not mu_au3_d2):
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raise ValueError('Either mu_km_s or mu_au3_d2 should be used, but not both')
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if mu_au3_d2:
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mu_km_s = mu_au3_d2 * AU_KM**3 / DAY_S**2
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position, velocity = ele_to_vec(p.km,
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e,
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i.radians,
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Om.radians,
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w.radians,
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v.radians,
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mu_km_s,
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)
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return cls(Distance(km=position),
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Velocity(km_per_s=velocity),
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epoch,
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mu_km_s,
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center=center,
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target_name=target_name,
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)
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@classmethod
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def _from_mean_anomaly(
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cls,
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semilatus_rectum_au,
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eccentricity,
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inclination_degrees,
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longitude_of_ascending_node_degrees,
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argument_of_perihelion_degrees,
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mean_anomaly_degrees,
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epoch,
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gm_km3_s2,
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center=None,
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target_name=None,
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):
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""" Creates a `KeplerOrbit` object from elements using mean anomaly
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Parameters
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----------
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p : Distance
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Semi-Latus Rectum
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e : float
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Eccentricity
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i : Angle
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Inclination
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Om : Angle
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Longitude of Ascending Node
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w : Angle
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Argument of periapsis
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M : Angle
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Mean anomaly
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epoch : Time
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Time corresponding to `position` and `velocity`
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mu_km_s : float
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Value of mu (G * M) in km^3/s^2
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mu_au3_d2 : float
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Value of mu (G * M) in au^3/d^2
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center : int
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NAIF ID of the primary body, 399 for geocentric orbits, 10 for
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heliocentric orbits
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target : int
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NAIF ID of the secondary body
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"""
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M = DEG2RAD * mean_anomaly_degrees
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gm_au3_d2 = gm_km3_s2 * _CONVERT_GM
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if eccentricity < 1.0:
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E = eccentric_anomaly(eccentricity, M)
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v = true_anomaly_closed(eccentricity, E)
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elif eccentricity > 1.0:
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E = eccentric_anomaly(eccentricity, M)
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v = true_anomaly_hyperbolic(eccentricity, E)
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else:
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v = true_anomaly_parabolic(semilatus_rectum_au, gm_au3_d2, M)
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pos, vel = ele_to_vec(
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semilatus_rectum_au,
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eccentricity,
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DEG2RAD * inclination_degrees,
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DEG2RAD * longitude_of_ascending_node_degrees,
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DEG2RAD * argument_of_perihelion_degrees,
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v,
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gm_au3_d2,
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)
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return cls(
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Distance(pos),
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Velocity(vel),
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epoch,
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gm_au3_d2,
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center,
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target_name,
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)
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def _at(self, time):
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"""Propagate the KeplerOrbit to the given Time object
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The Time object can contain one time, or an array of times
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"""
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pos, vel = propagate(
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self.position_at_epoch.au,
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self.velocity_at_epoch.au_per_d,
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self.epoch.tt,
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time.tt,
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self.mu_au3_d2,
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)
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if self._rotation is not None:
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pos = mxv(self._rotation, pos)
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vel = mxv(self._rotation, vel)
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return pos, vel, None, None
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@reify
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def elements_at_epoch(self):
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return OsculatingElements(self.position_at_epoch,
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self.velocity_at_epoch,
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self.epoch,
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mu_km_s = self.mu_au3_d2 / _CONVERT_GM,
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)
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def __str__(self):
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ele = self.elements_at_epoch
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if self.target_name:
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return 'KeplerOrbit {0} -> {1}'.format(
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self.center_name,
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self.target_name,
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)
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ele = self.elements_at_epoch
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string = ('KeplerOrbit {} ->'
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' q={:.2}au e={:.3f} i={:.1f} Om={:.1f} w={:.1f}')
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return string.format(
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self.center_name,
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ele.periapsis_distance.au,
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ele.eccentricity,
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ele.inclination.degrees,
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ele.longitude_of_ascending_node.degrees,
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ele.argument_of_periapsis.degrees,
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)
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def __repr__(self):
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return '<{0}>'.format(str(self))
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_ten_iterations = tuple([None] * 10)
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def eccentric_anomaly(e, M):
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"""Iterate to solve Kepler's equation to find the eccentric anomaly.
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See arXiv:2108.03215.
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"""
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M = normpi(M)
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sign_M = sign(M)
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M *= sign_M
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ebar = 0.25 * pi/e - 1.0
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E = 0.5 * pi * ebar * (sign(ebar) * sqrt(1 + M/(e*ebar*ebar)) - 1.0)
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for _ in _ten_iterations:
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f1 = 1.0 - e*cos(E)
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f2 = e*sin(E)
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f = E - f2 - M
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dE = f*f1 / (f1*f1 - 0.5*f*f2)
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E -= dE
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if abs(dE) < 1e-14:
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return E * sign_M
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raise ValueError('eccentric anomaly failed to converge')
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def true_anomaly_hyperbolic(e, E):
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"""Calculates true anomaly from eccentricity and eccentric anomaly.
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Valid for hyperbolic orbits. Equations from the relevant Wikipedia entries.
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"""
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return 2.0 * arctan(sqrt((e + 1.0) / (e - 1.0)) * tanh(E/2))
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def true_anomaly_closed(e, E):
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"""Calculates true anomaly from eccentricity and eccentric anomaly.
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Valid for closed orbits. Equations from the relevant Wikipedia entries.
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"""
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return 2.0 * arctan(sqrt((1.0 + e) / (1.0 - e)) * tan(E/2))
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def true_anomaly_parabolic(p, gm, M):
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"""Calculates true anomaly from semi-latus rectum, gm, and mean anomaly.
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Valid for parabolic orbits. Equations from
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https://en.wikipedia.org/wiki/Parabolic_trajectory.
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"""
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delta_t = sqrt(2 * p**3 / gm) * M # from http://www.bogan.ca/orbits/kepler/orbteqtn.html
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periapsis_distance = p / 2
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A = 3 / 2 * sqrt(gm / (2 * periapsis_distance**3)) * delta_t
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B = (A + (A*A + 1))**(1/3)
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return 2 * arctan(B - 1/B)
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def ele_to_vec(p, e, i, Om, w, v, mu):
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"""Calculates state vectors from orbital elements. Also checks for invalid
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sets of elements.
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Based on equations from this document:
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https://web.archive.org/web/*/http://ccar.colorado.edu/asen5070/handouts/kep2cart_2002.doc
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"""
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# Checks that true anomaly is less than arccos(-1/e) for hyperbolic orbits
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if isinstance(e, ndarray) and isinstance(v, ndarray):
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inds = (e>1)
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if (v[inds]>arccos(-1/e[inds])).any():
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raise ValueError('If eccentricity is >1, abs(true anomaly) cannot be more than arccos(-1/e)')
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elif isinstance(e, ndarray) and not isinstance(v, ndarray):
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inds = (e>1)
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if (v>arccos(-1/e[inds])).any():
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raise ValueError('If eccentricity is >1, abs(true anomaly) cannot be more than arccos(-1/e)')
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elif isinstance(v, ndarray) and not isinstance(e, ndarray):
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if e>1 and (v>arccos(-1/e)).any():
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raise ValueError('If eccentricity is >1, abs(true anomaly) cannot be more than arccos(-1/e)')
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else:
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if e>1 and v>arccos(-1/e):
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raise ValueError('If eccentricity is >1, abs(true anomaly) cannot be more than arccos(-1/e)')
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# Checks that inclination is in the range [0, pi]
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if isinstance(i, ndarray):
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if not ((i>=0) * (i <= pi)).all():
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raise ValueError('Inclination outside the range [0, pi] radians')
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else:
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if not 0 <= i <= pi:
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raise ValueError('Inclination outside the range [0, pi] radians')
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r = p/(1 + e*cos(v))
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h = sqrt(p*mu)
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u = v+w
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X = r*(cos(Om)*cos(u) - sin(Om)*sin(u)*cos(i))
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Y = r*(sin(Om)*cos(u) + cos(Om)*sin(u)*cos(i))
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Z = r*(sin(i)*sin(u))
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X_dot = X*h*e/(r*p)*sin(v) - h/r*(cos(Om)*sin(u) + sin(Om)*cos(u)*cos(i))
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Y_dot = Y*h*e/(r*p)*sin(v) - h/r*(sin(Om)*sin(u) - cos(Om)*cos(u)*cos(i))
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Z_dot = Z*h*e/(r*p)*sin(v) + h/r*sin(i)*cos(u)
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# z and z_dot are independent of Om, so if Om is an array and the other
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# elements are scalars, z and z_dot need to be repeated
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if Z.size!=X.size:
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Z = repeat(Z, X.size)
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Z_dot = repeat(Z_dot, X.size)
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return array([X, Y, Z]), array([X_dot, Y_dot, Z_dot])
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dpmax = sys.float_info.max
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def find_trunc():
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denom = 2
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factr = 2
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trunc = 1
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x = 1 / denom
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while 1+x > 1:
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denom = denom * (2+factr) * (1+factr)
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factr = factr + 2
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trunc = trunc + 1
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x = 1 / denom
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return trunc
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trunc = find_trunc()
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odd_factorials = array([math.factorial(i) for i in range(3, trunc*2, 2)])
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even_factorials = array([math.factorial(i) for i in range(2, trunc*2, 2)])
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exponents = arange(0, trunc-1)
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stumpff_bound = -(log(2) + log(dpmax))**2
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def stumpff(x):
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"""Calculates Stumpff functions
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Based on the function toolkit/src/spicelib/stmp03.f from the SPICE toolkit,
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which can be downloaded from naif.jpl.nasa.gov/naif/toolkit_FORTRAN.html
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"""
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if x.min() < stumpff_bound:
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raise ValueError('Argument below lower bound')
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z = sqrt(abs(x))
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c0 = zeros_like(x)
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c1 = zeros_like(x)
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c2 = zeros_like(x)
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c3 = zeros_like(x)
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low = x < -1
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c0[low] = cosh(z[low])
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c1[low] = sinh(z[low])/z[low]
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high = x > 1
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c0[high] = cos(z[high])
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c1[high] = sin(z[high])/z[high]
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mid = ~(low|high)
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if sum(mid):
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numerators = repeat(x[mid][:, newaxis], trunc-1, axis=1)
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numerators[:, 1::2] *= -1
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c3[mid] = sum(power(numerators, exponents)/odd_factorials, axis=1)
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c2[mid] = sum(power(numerators, exponents)/even_factorials, axis=1)
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c1[mid] = 1 - x[mid]*c3[mid]
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c0[mid] = 1 - x[mid]*c2[mid]
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not_mid = ~mid
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c2[not_mid] = (1 - c0[not_mid])/x[not_mid]
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c3[not_mid] = (1 - c1[not_mid])/x[not_mid]
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return c0, c1, c2, c3
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def propagate(position, velocity, t0, t1, gm):
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"""Propagates a position and velocity vector with an array of times.
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Based on the function toolkit/src/spicelib/prop2b.f from the SPICE toolkit,
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which can be downloaded from naif.jpl.nasa.gov/naif/toolkit_FORTRAN.html
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Parameters
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----------
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position : ndarray
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Position vector with shape (3,)
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velocity : ndarray
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Velocity vector with shape (3,)
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t0 : float
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Time corresponding to `position` and `velocity`
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t1 : float or ndarray
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Time or times to propagate to
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gm : float
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Gravitational parameter in units that match the other arguments
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"""
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output_shape = (3,) + t1.shape
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gm = atleast_1d(gm)
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if (gm <= 0).any():
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raise ValueError("'gm' should be positive")
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if (length_of(velocity)).any() == 0:
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raise ValueError('Velocity vector has zero magnitude')
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if (length_of(position)).any() == 0:
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raise ValueError('Position vector has zero magnitude')
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if position.ndim == 1:
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position = position[:, newaxis]
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if velocity.ndim == 1:
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velocity = velocity[:, newaxis]
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r0 = length_of(position)
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rv = dots(position, velocity)
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hvec = _cross(position, velocity)
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h2 = dots(hvec, hvec)
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if (h2 == 0).any():
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raise ValueError('Motion is not conical')
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eqvec = _cross(velocity, hvec)/gm + -position/r0
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e = length_of(eqvec)
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q = h2 / (gm * (1+e))
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f = 1 - e
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b = sqrt(q/gm)
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br0 = b * r0
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b2rv = b * b * rv
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bq = b * q
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qovr0 = q / r0
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maxc = amax(array([abs(br0),
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abs(b2rv),
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abs(bq),
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abs(qovr0/bq)]), axis=0)
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hyperbolic = (f<0)
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bound = zeros_like(f)
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fixed = log(dpmax/2) - log(maxc[hyperbolic])
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rootf = sqrt(-f[hyperbolic])
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logf = log(-f[hyperbolic])
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bound[hyperbolic] = amin(array([fixed/rootf, (fixed + 1.5*logf)/rootf]), axis=0)
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logbound = (log(1.5) + log(dpmax) - log(maxc[~hyperbolic])) / 3
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bound[~hyperbolic] = exp(logbound)
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|
|
|
# each of these arrays has 1 entry per orbit, so its shape is (#orbits, 1)
|
|
f = f[:, newaxis]
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bq = bq[:, newaxis]
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b2rv = b2rv[:, newaxis]
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br0 = br0[:, newaxis]
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qovr0 = qovr0[:, newaxis]
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|
bound = bound[:, newaxis]
|
|
|
|
def kepler(x):
|
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_, c1, c2, c3 = stumpff(f*x*x)
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return x*(br0*c1 + x*(b2rv*c2 + x*bq*c3))
|
|
|
|
def kepler_1d(x, orb_inds):
|
|
_, c1, c2, c3 = stumpff(x*x*repeat(f, orb_inds))
|
|
return x*(c1*repeat(br0, orb_inds) + x*(c2*repeat(b2rv, orb_inds) + x*(c3*repeat(bq, orb_inds))))
|
|
|
|
t1 = atleast_1d(t1)
|
|
t0 = atleast_1d(t0)
|
|
if len(t0) == 1:
|
|
t0 = repeat(t0, position.shape[1])
|
|
|
|
# shape of 2 dimensional arrays from here on out should be (#orbits, len(t1))
|
|
dt = t1 - t0[:, newaxis]
|
|
|
|
x = dt/bq
|
|
copyto(x, -bound, where=(x<-bound))
|
|
copyto(x, bound, where=(x>bound))
|
|
|
|
kfun = kepler(x)
|
|
|
|
past = dt < 0
|
|
future = dt > 0
|
|
|
|
upper = zeros_like(dt, dtype='float64')
|
|
lower = zeros_like(dt, dtype='float64')
|
|
oldx = zeros_like(dt, dtype='float64')
|
|
|
|
copyto(lower, x, where=past)
|
|
copyto(upper, x, where=future)
|
|
|
|
while (kfun[past] > dt[past]).any():
|
|
copyto(upper, lower, where=past)
|
|
lower[past] *= 2
|
|
copyto(oldx, x, where=past)
|
|
orb_ind = sum(past, axis=1)
|
|
x[past] = clip(lower[past], repeat(-bound, orb_ind), repeat(bound, orb_ind))
|
|
if (x[past] == oldx[past]).any():
|
|
raise ValueError('The input delta time (dt) has a value of {0}.'
|
|
'This is beyond the range of DT for which we '
|
|
'can reliably propagate states. The limits for '
|
|
'this GM and initial state are from {1}'
|
|
'to {2}.'.format(dt, kepler(-bound), kepler(bound)))
|
|
kfun[past] = kepler_1d(x[past], orb_ind)
|
|
|
|
while (kfun[future] < dt[future]).any():
|
|
copyto(lower, upper, where=future)
|
|
upper[future] *= 2
|
|
copyto(oldx, x, where=future)
|
|
orb_ind = sum(future, axis=1)
|
|
x[future] = clip(upper[future], repeat(-bound, orb_ind), repeat(bound, orb_ind))
|
|
if (x[future] == oldx[future]).any():
|
|
raise ValueError('The input delta time (dt) has a value of {0}.'
|
|
'This is beyond the range of DT for which we '
|
|
'can reliably propagate states. The limits for '
|
|
'this GM and initial state are from {1} '
|
|
'to {2}.'.format(dt, kepler(-bound), kepler(bound)))
|
|
kfun[future] = kepler_1d(x[future], orb_ind)
|
|
|
|
x = copy(upper)
|
|
copyto(x, (upper+lower)/2, where=(lower<=upper))
|
|
|
|
lcount = zeros_like(dt)
|
|
mostc = full_like(dt, 1000)
|
|
not_done = (lower < x) & (x < upper)
|
|
|
|
while not_done.any():
|
|
orb_inds = sum(not_done, axis=1)
|
|
kfun[not_done] = kepler_1d(x[not_done], orb_inds)
|
|
|
|
high = (kfun > dt) & not_done
|
|
low = (kfun < dt) & not_done
|
|
same = (~high & ~low) & not_done
|
|
|
|
copyto(upper, x, where=(high|same))
|
|
copyto(lower, x, where=(low|same))
|
|
|
|
condition = not_done & (mostc > 64) & (upper != 0) & (lower != 0)
|
|
mostc[condition] = 64
|
|
lcount[condition] = 0
|
|
|
|
copyto(x, upper, where=(not_done & (lower>upper)))
|
|
copyto(x, (upper+lower)/2, where=(not_done & (lower<=upper)))
|
|
|
|
lcount += 1
|
|
not_done = (lower < x) & (x < upper) & (lcount < mostc)
|
|
|
|
c0, c1, c2, c3 = stumpff(f*x*x)
|
|
br = br0*c0 + x*(b2rv*c1 + x*bq*c2)
|
|
|
|
pc = 1 - qovr0 * x * x * c2
|
|
vc = dt - bq * x**3 * c3
|
|
pcdot = -qovr0 / br * x * c1
|
|
vcdot = 1 - bq / br * x * x * c2
|
|
|
|
position_prop = pc[newaxis, :, :]*position[:, :, newaxis] + vc[newaxis, :, :]*velocity[:, :, newaxis]
|
|
velocity_prop = pcdot[newaxis, :, :]*position[:, :, newaxis] + vcdot[newaxis, :, :]*velocity[:, :, newaxis]
|
|
|
|
position_prop.shape = output_shape
|
|
velocity_prop.shape = output_shape
|
|
return position_prop, velocity_prop
|