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#!/usr/bin/env python3
#
# MIT License
#
# Copyright (c) 2019 Don Cross <cosinekitty@gmail.com>
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
#
# The above copyright notice and this permission notice shall be included in all
# copies or substantial portions of the Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
# SOFTWARE.
#
"""Astronomy Engine by Don Cross
See the GitHub project page for full documentation, examples,
and other information:
https://github.com/cosinekitty/astronomy
"""
import math
import datetime
import enum
_PI2 = 2.0 * math.pi
_EPOCH = datetime.datetime(2000, 1, 1, 12)
_T0 = 2451545.0
_MJD_BASIS = 2400000.5
_Y2000_IN_MJD = _T0 - _MJD_BASIS
_ASEC360 = 1296000.0
_ASEC2RAD = 4.848136811095359935899141e-6
_ARC = 3600.0 * 180.0 / math.pi # arcseconds per radian
_C_AUDAY = 173.1446326846693 # speed of light in AU/day
_ERAD = 6378136.6 # mean earth radius in meters
_AU = 1.4959787069098932e+11 # astronomical unit in meters
_KM_PER_AU = 1.4959787069098932e+8
_ANGVEL = 7.2921150e-5
_SECONDS_PER_DAY = 24.0 * 3600.0
_SOLAR_DAYS_PER_SIDEREAL_DAY = 0.9972695717592592
_MEAN_SYNODIC_MONTH = 29.530588
_EARTH_ORBITAL_PERIOD = 365.256
_REFRACTION_NEAR_HORIZON = 34.0 / 60.0
_SUN_RADIUS_AU = 4.6505e-3
_MOON_RADIUS_AU = 1.15717e-5
_ASEC180 = 180.0 * 60.0 * 60.0
_AU_PER_PARSEC = _ASEC180 / math.pi
def _LongitudeOffset(diff):
offset = diff
while offset <= -180.0:
offset += 360.0
while offset > 180.0:
offset -= 360.0
return offset
def _NormalizeLongitude(lon):
while lon < 0.0:
lon += 360.0
while lon >= 360.0:
lon -= 360.0
return lon
class Vector:
def __init__(self, x, y, z, t):
self.x = x
self.y = y
self.z = z
self.t = t
def Length(self):
return math.sqrt(self.x**2 + self.y**2 + self.z**2)
@enum.unique
class Body(enum.IntEnum):
"""The celestial bodies supported by Astronomy Engine calculations.
Values
------
Invalid: An unknown, invalid, or undefined celestial body.
Mercury: The planet Mercury.
Venus: The planet Venus.
Earth: The planet Earth.
Mars: The planet Mars.
Jupiter: The planet Jupiter.
Saturn: The planet Saturn.
Uranus: The planet Uranus.
Neptune: The planet Neptune.
Pluto: The planet Pluto.
Sun: The Sun.
Moon: The Earth's moon.
"""
Invalid = -1
Mercury = 0
Venus = 1
Earth = 2
Mars = 3
Jupiter = 4
Saturn = 5
Uranus = 6
Neptune = 7
Pluto = 8
Sun = 9
Moon = 10
def BodyCode(name):
"""Finds the Body enumeration value, given the name of a body.
Parameters
----------
name: str
The common English name of a supported celestial body.
Returns
-------
Body
If `name` is a valid body name, returns the enumeration
value associated with that body.
Otherwise, returns `Body.Invalid`.
Example
-------
>>> astronomy.BodyCode('Mars')
<Body.Mars: 3>
"""
if name not in Body.__members__:
return Body.Invalid
return Body[name]
def _IsSuperiorPlanet(body):
return body in [Body.Mars, Body.Jupiter, Body.Saturn, Body.Uranus, Body.Neptune, Body.Pluto]
_PlanetOrbitalPeriod = [
87.969,
224.701,
_EARTH_ORBITAL_PERIOD,
686.980,
4332.589,
10759.22,
30685.4,
60189.0,
90560.0
]
class Error(Exception):
def __init__(self, message):
Exception.__init__(self, message)
class EarthNotAllowedError(Error):
def __init__(self):
Error.__init__(self, 'The Earth is not allowed as the body.')
class InvalidBodyError(Error):
def __init__(self):
Error.__init__(self, 'Invalid astronomical body.')
class BadVectorError(Error):
def __init__(self):
Error.__init__(self, 'Vector is too small to have a direction.')
class InternalError(Error):
def __init__(self):
Error.__init__(self, 'Internal error - please report issue at https://github.com/cosinekitty/astronomy/issues')
class NoConvergeError(Error):
def __init__(self):
Error.__init__(self, 'Numeric solver did not converge - please report issue at https://github.com/cosinekitty/astronomy/issues')
def _SynodicPeriod(body):
if body == Body.Earth:
raise EarthNotAllowedError()
if body < 0 or body >= len(_PlanetOrbitalPeriod):
raise InvalidBodyError()
if body == Body.Moon:
return _MEAN_SYNODIC_MONTH
return abs(_EARTH_ORBITAL_PERIOD / (_EARTH_ORBITAL_PERIOD/_PlanetOrbitalPeriod[body] - 1.0))
def _AngleBetween(a, b):
r = a.Length() * b.Length()
if r < 1.0e-8:
return BadVectorError()
dot = (a.x*b.x + a.y*b.y + a.z*b.z) / r
if dot <= -1.0:
return 180.0
if dot >= +1.0:
return 0.0
return math.degrees(math.acos(dot))
class _delta_t_entry_t:
def __init__(self, mjd, dt):
self.mjd = mjd
self.dt = dt
_DT = [
_delta_t_entry_t(-72638.0, 38),
_delta_t_entry_t(-65333.0, 26),
_delta_t_entry_t(-58028.0, 21),
_delta_t_entry_t(-50724.0, 21.1),
_delta_t_entry_t(-43419.0, 13.5),
_delta_t_entry_t(-39766.0, 13.7),
_delta_t_entry_t(-36114.0, 14.8),
_delta_t_entry_t(-32461.0, 15.7),
_delta_t_entry_t(-28809.0, 15.6),
_delta_t_entry_t(-25156.0, 13.3),
_delta_t_entry_t(-21504.0, 12.6),
_delta_t_entry_t(-17852.0, 11.2),
_delta_t_entry_t(-14200.0, 11.13),
_delta_t_entry_t(-10547.0, 7.95),
_delta_t_entry_t(-6895.0, 6.22),
_delta_t_entry_t(-3242.0, 6.55),
_delta_t_entry_t(-1416.0, 7.26),
_delta_t_entry_t(410.0, 7.35),
_delta_t_entry_t(2237.0, 5.92),
_delta_t_entry_t(4063.0, 1.04),
_delta_t_entry_t(5889.0, -3.19),
_delta_t_entry_t(7715.0, -5.36),
_delta_t_entry_t(9542.0, -5.74),
_delta_t_entry_t(11368.0, -5.86),
_delta_t_entry_t(13194.0, -6.41),
_delta_t_entry_t(15020.0, -2.70),
_delta_t_entry_t(16846.0, 3.92),
_delta_t_entry_t(18672.0, 10.38),
_delta_t_entry_t(20498.0, 17.19),
_delta_t_entry_t(22324.0, 21.41),
_delta_t_entry_t(24151.0, 23.63),
_delta_t_entry_t(25977.0, 24.02),
_delta_t_entry_t(27803.0, 23.91),
_delta_t_entry_t(29629.0, 24.35),
_delta_t_entry_t(31456.0, 26.76),
_delta_t_entry_t(33282.0, 29.15),
_delta_t_entry_t(35108.0, 31.07),
_delta_t_entry_t(36934.0, 33.150),
_delta_t_entry_t(38761.0, 35.738),
_delta_t_entry_t(40587.0, 40.182),
_delta_t_entry_t(42413.0, 45.477),
_delta_t_entry_t(44239.0, 50.540),
_delta_t_entry_t(44605.0, 51.3808),
_delta_t_entry_t(44970.0, 52.1668),
_delta_t_entry_t(45335.0, 52.9565),
_delta_t_entry_t(45700.0, 53.7882),
_delta_t_entry_t(46066.0, 54.3427),
_delta_t_entry_t(46431.0, 54.8712),
_delta_t_entry_t(46796.0, 55.3222),
_delta_t_entry_t(47161.0, 55.8197),
_delta_t_entry_t(47527.0, 56.3000),
_delta_t_entry_t(47892.0, 56.8553),
_delta_t_entry_t(48257.0, 57.5653),
_delta_t_entry_t(48622.0, 58.3092),
_delta_t_entry_t(48988.0, 59.1218),
_delta_t_entry_t(49353.0, 59.9845),
_delta_t_entry_t(49718.0, 60.7853),
_delta_t_entry_t(50083.0, 61.6287),
_delta_t_entry_t(50449.0, 62.2950),
_delta_t_entry_t(50814.0, 62.9659),
_delta_t_entry_t(51179.0, 63.4673),
_delta_t_entry_t(51544.0, 63.8285),
_delta_t_entry_t(51910.0, 64.0908),
_delta_t_entry_t(52275.0, 64.2998),
_delta_t_entry_t(52640.0, 64.4734),
_delta_t_entry_t(53005.0, 64.5736),
_delta_t_entry_t(53371.0, 64.6876),
_delta_t_entry_t(53736.0, 64.8452),
_delta_t_entry_t(54101.0, 65.1464),
_delta_t_entry_t(54466.0, 65.4573),
_delta_t_entry_t(54832.0, 65.7768),
_delta_t_entry_t(55197.0, 66.0699),
_delta_t_entry_t(55562.0, 66.3246),
_delta_t_entry_t(55927.0, 66.6030),
_delta_t_entry_t(56293.0, 66.9069),
_delta_t_entry_t(56658.0, 67.2810),
_delta_t_entry_t(57023.0, 67.6439),
_delta_t_entry_t(57388.0, 68.1024),
_delta_t_entry_t(57754.0, 68.5927),
_delta_t_entry_t(58119.0, 68.9676),
_delta_t_entry_t(58484.0, 69.2201),
_delta_t_entry_t(58849.0, 69.87),
_delta_t_entry_t(59214.0, 70.39),
_delta_t_entry_t(59580.0, 70.91),
_delta_t_entry_t(59945.0, 71.40),
_delta_t_entry_t(60310.0, 71.88),
_delta_t_entry_t(60675.0, 72.36),
_delta_t_entry_t(61041.0, 72.83),
_delta_t_entry_t(61406.0, 73.32),
_delta_t_entry_t(61680.0, 73.66)
]
def _DeltaT(mjd):
if mjd <= _DT[0].mjd:
return _DT[0].dt
if mjd >= _DT[-1].mjd:
return _DT[-1].dt
# Do a binary search to find the pair of indexes this mjd lies between.
lo = 0
hi = len(_DT) - 2 # Make sure there is always an array element after the one we are looking at.
while True:
if lo > hi:
# This should never happen unless there is a bug in the binary search.
raise Error('Could not find delta-t value.')
c = (lo + hi) // 2
if mjd < _DT[c].mjd:
hi = c-1
elif mjd > _DT[c+1].mjd:
lo = c+1
else:
frac = (mjd - _DT[c].mjd) / (_DT[c+1].mjd - _DT[c].mjd)
return _DT[c].dt + frac*(_DT[c+1].dt - _DT[c].dt)
def _TerrestrialTime(ut):
return ut + _DeltaT(ut + _Y2000_IN_MJD) / 86400.0
class Time:
"""Represents a date and time used for performing astronomy calculations.
All calculations performed by Astronomy Engine are based on
dates and times represented by `Time` objects.
Parameters
----------
ut : float
UT1/UTC number of days since noon on January 1, 2000.
See the `ut` attribute of this class for more details.
Attributes
----------
ut : float
The floating point number of days of Universal Time since noon UTC January 1, 2000.
Astronomy Engine approximates UTC and UT1 as being the same thing, although they are
not exactly equivalent; UTC and UT1 can disagree by up to 0.9 seconds.
This approximation is sufficient for the accuracy requirements of Astronomy Engine.
Universal Time Coordinate (UTC) is the international standard for legal and civil
timekeeping and replaces the older Greenwich Mean Time (GMT) standard.
UTC is kept in sync with unpredictable observed changes in the Earth's rotation
by occasionally adding leap seconds as needed.
UT1 is an idealized time scale based on observed rotation of the Earth, which
gradually slows down in an unpredictable way over time, due to tidal drag by the Moon and Sun,
large scale weather events like hurricanes, and internal seismic and convection effects.
Conceptually, UT1 drifts from atomic time continuously and erratically, whereas UTC
is adjusted by a scheduled whole number of leap seconds as needed.
The value in `ut` is appropriate for any calculation involving the Earth's rotation,
such as calculating rise/set times, culumination, and anything involving apparent
sidereal time.
Before the era of atomic timekeeping, days based on the Earth's rotation
were often known as *mean solar days*.
tt : float
Terrestrial Time days since noon on January 1, 2000.
Terrestrial Time is an atomic time scale defined as a number of days since noon on January 1, 2000.
In this system, days are not based on Earth rotations, but instead by
the number of elapsed [SI seconds](https://physics.nist.gov/cuu/Units/second.html)
divided by 86400. Unlike `ut`, `tt` increases uniformly without adjustments
for changes in the Earth's rotation.
The value in `tt` is used for calculations of movements not involving the Earth's rotation,
such as the orbits of planets around the Sun, or the Moon around the Earth.
Historically, Terrestrial Time has also been known by the term *Ephemeris Time* (ET).
"""
def __init__(self, ut):
self.ut = ut
self.tt = _TerrestrialTime(ut)
self.etilt = None
@staticmethod
def Make(year, month, day, hour, minute, second):
"""Creates a #Time object from a UTC calendar date and time.
Parameters
----------
year : int
The UTC 4-digit year value, e.g. 2019.
month : int
The UTC month in the range 1..12.
day : int
The UTC day of the month, in the range 1..31.
hour : int
The UTC hour, in the range 0..23.
minute : int
The UTC minute, in the range 0..59.
second : float
The real-valued UTC second, in the range [0, 60).
Returns
-------
Time
"""
micro = round(math.fmod(second, 1.0) * 1000000)
second = math.floor(second - micro/1000000)
d = datetime.datetime(year, month, day, hour, minute, second, micro)
ut = (d - _EPOCH).total_seconds() / 86400
return Time(ut)
@staticmethod
def Now():
ut = (datetime.datetime.utcnow() - _EPOCH).total_seconds() / 86400.0
return Time(ut)
def AddDays(self, days):
return Time(self.ut + days)
def __str__(self):
millis = round(self.ut * 86400000.0)
n = _EPOCH + datetime.timedelta(milliseconds=millis)
return '{:04d}-{:02d}-{:02d}T{:02d}:{:02d}:{:02d}.{:03d}Z'.format(n.year, n.month, n.day, n.hour, n.minute, n.second, math.floor(n.microsecond / 1000))
def Utc(self):
return _EPOCH + datetime.timedelta(days=self.ut)
def _etilt(self):
# Calculates precession and nutation of the Earth's axis.
# The calculations are very expensive, so lazy-evaluate and cache
# the result inside this Time object.
if self.etilt is None:
self.etilt = _e_tilt(self)
return self.etilt
class Observer:
"""Represents the geographic location of an observer on the surface of the Earth.
Parameters
----------
latitude : float
Geographic latitude in degrees north of the equator.
longitude : float
Geographic longitude in degrees east of the prime meridian at Greenwich, England.
height : float
Elevation above sea level in meters.
"""
def __init__(self, latitude, longitude, height=0):
self.latitude = latitude
self.longitude = longitude
self.height = height
class _iau2000b:
def __init__(self, time):
t = time.tt / 36525.0
el = math.fmod((485868.249036 + t*1717915923.2178), _ASEC360) * _ASEC2RAD
elp = math.fmod((1287104.79305 + t*129596581.0481), _ASEC360) * _ASEC2RAD
f = math.fmod((335779.526232 + t*1739527262.8478), _ASEC360) * _ASEC2RAD
d = math.fmod((1072260.70369 + t*1602961601.2090), _ASEC360) * _ASEC2RAD
om = math.fmod((450160.398036 - t*6962890.5431), _ASEC360) * _ASEC2RAD
dp = 0
de = 0
sarg = math.sin(om)
carg = math.cos(om)
dp += (-172064161.0 - 174666.0*t)*sarg + 33386.0*carg
de += (92052331.0 + 9086.0*t)*carg + 15377.0*sarg
arg = 2.0*f - 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-13170906.0 - 1675.0*t)*sarg - 13696.0*carg
de += (5730336.0 - 3015.0*t)*carg - 4587.0*sarg
arg = 2.0*f + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-2276413.0 - 234.0*t)*sarg + 2796.0*carg
de += (978459.0 - 485.0*t)*carg + 1374.0*sarg
arg = 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (2074554.0 + 207.0*t)*sarg - 698.0*carg
de += (-897492.0 + 470.0*t)*carg - 291.0*sarg
sarg = math.sin(elp)
carg = math.cos(elp)
dp += (1475877.0 - 3633.0*t)*sarg + 11817.0*carg
de += (73871.0 - 184.0*t)*carg - 1924.0*sarg
arg = elp + 2.0*f - 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-516821.0 + 1226.0*t)*sarg - 524.0*carg
de += (224386.0 - 677.0*t)*carg - 174.0*sarg
sarg = math.sin(el)
carg = math.cos(el)
dp += (711159.0 + 73.0*t)*sarg - 872.0*carg
de += (-6750.0)*carg + 358.0*sarg
arg = 2.0*f + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-387298.0 - 367.0*t)*sarg + 380.0*carg
de += (200728.0 + 18.0*t)*carg + 318.0*sarg
arg = el + 2.0*f + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-301461.0 - 36.0*t)*sarg + 816.0*carg
de += (129025.0 - 63.0*t)*carg + 367.0*sarg
arg = -elp + 2.0*f - 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (215829.0 - 494.0*t)*sarg + 111.0*carg
de += (-95929.0 + 299.0*t)*carg + 132.0*sarg
arg = 2.0*f - 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (128227.0 + 137.0*t)*sarg + 181.0*carg
de += (-68982.0 - 9.0*t)*carg + 39.0*sarg
arg = -el + 2.0*f + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (123457.0 + 11.0*t)*sarg + 19.0*carg
de += (-53311.0 + 32.0*t)*carg - 4.0*sarg
arg = -el + 2.0*d
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (156994.0 + 10.0*t)*sarg - 168.0*carg
de += (-1235.0)*carg + 82.0*sarg
arg = el + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (63110.0 + 63.0*t)*sarg + 27.0*carg
de += (-33228.0)*carg - 9.0*sarg
arg = -el + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-57976.0 - 63.0*t)*sarg - 189.0*carg
de += (31429.0)*carg - 75.0*sarg
arg = -el + 2.0*f + 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-59641.0 - 11.0*t)*sarg + 149.0*carg
de += (25543.0 - 11.0*t)*carg + 66.0*sarg
arg = el + 2.0*f + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-51613.0 - 42.0*t)*sarg + 129.0*carg
de += (26366.0)*carg + 78.0*sarg
arg = -2.0*el + 2.0*f + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (45893.0 + 50.0*t)*sarg + 31.0*carg
de += (-24236.0 - 10.0*t)*carg + 20.0*sarg
arg = 2.0*d
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (63384.0 + 11.0*t)*sarg - 150.0*carg
de += (-1220.0)*carg + 29.0*sarg
arg = 2.0*f + 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-38571.0 - 1.0*t)*sarg + 158.0*carg
de += (16452.0 - 11.0*t)*carg + 68.0*sarg
arg = -2.0*elp + 2.0*f - 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (32481.0)*sarg
de += (-13870.0)*carg
arg = -2.0*el + 2.0*d
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-47722.0)*sarg - 18.0*carg
de += (477.0)*carg - 25.0*sarg
arg = 2.0*el + 2.0*f + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-31046.0 - 1.0*t)*sarg + 131.0*carg
de += (13238.0 - 11.0*t)*carg + 59.0*sarg
arg = el + 2.0*f - 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (28593.0)*sarg - carg
de += (-12338.0 + 10.0*t)*carg - 3.0*sarg
arg = -el + 2.0*f + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (20441.0 + 21.0*t)*sarg + 10.0*carg
de += (-10758.0)*carg - 3.0*sarg
arg = 2.0*el
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (29243.0)*sarg - 74.0*carg
de += (-609.0)*carg + 13.0*sarg
arg = 2.0*f
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (25887.0)*sarg - 66.0*carg
de += (-550.0)*carg + 11.0*sarg
arg = elp + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-14053.0 - 25.0*t)*sarg + 79.0*carg
de += (8551.0 - 2.0*t)*carg - 45.0*sarg
arg = -el + 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (15164.0 + 10.0*t)*sarg + 11.0*carg
de += (-8001.0)*carg - sarg
arg = 2.0*elp + 2.0*f - 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-15794.0 + 72.0*t)*sarg - 16.0*carg
de += (6850.0 - 42.0*t)*carg - 5.0*sarg
arg = -2.0*f + 2.0*d
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (21783.0)*sarg + 13.0*carg
de += (-167.0)*carg + 13.0*sarg
arg = el - 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-12873.0 - 10.0*t)*sarg - 37.0*carg
de += (6953.0)*carg - 14.0*sarg
arg = -elp + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-12654.0 + 11.0*t)*sarg + 63.0*carg
de += (6415.0)*carg + 26.0*sarg
arg = -el + 2.0*f + 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-10204.0)*sarg + 25.0*carg
de += (5222.0)*carg + 15.0*sarg
arg = 2.0*elp
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (16707.0 - 85.0*t)*sarg - 10.0*carg
de += (168.0 - 1.0*t)*carg + 10.0*sarg
arg = el + 2.0*f + 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-7691.0)*sarg + 44.0*carg
de += (3268.0)*carg + 19.0*sarg
arg = -2.0*el + 2.0*f
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-11024.0)*sarg - 14.0*carg
de += (104.0)*carg + 2.0*sarg
arg = elp + 2.0*f + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (7566.0 - 21.0*t)*sarg - 11.0*carg
de += (-3250.0)*carg - 5.0*sarg
arg = 2.0*f + 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-6637.0 - 11.0*t)*sarg + 25.0*carg
de += (3353.0)*carg + 14.0*sarg
arg = -elp + 2.0*f + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-7141.0 + 21.0*t)*sarg + 8.0*carg
de += (3070.0)*carg + 4.0*sarg
arg = 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-6302.0 - 11.0*t)*sarg + 2.0*carg
de += (3272.0)*carg + 4.0*sarg
arg = el + 2.0*f - 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (5800.0 + 10.0*t)*sarg + 2.0*carg
de += (-3045.0)*carg - sarg
arg = 2.0*el + 2.0*f - 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (6443.0)*sarg - 7.0*carg
de += (-2768.0)*carg - 4.0*sarg
arg = -2.0*el + 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-5774.0 - 11.0*t)*sarg - 15.0*carg
de += (3041.0)*carg - 5.0*sarg
arg = 2.0*el + 2.0*f + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-5350.0)*sarg + 21.0*carg
de += (2695.0)*carg + 12.0*sarg
arg = -elp + 2.0*f - 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-4752.0 - 11.0*t)*sarg - 3.0*carg
de += (2719.0)*carg - 3.0*sarg
arg = -2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-4940.0 - 11.0*t)*sarg - 21.0*carg
de += (2720.0)*carg - 9.0*sarg
arg = -el - elp + 2.0*d
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (7350.0)*sarg - 8.0*carg
de += (-51.0)*carg + 4.0*sarg
arg = 2.0*el - 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (4065.0)*sarg + 6.0*carg
de += (-2206.0)*carg + sarg
arg = el + 2.0*d
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (6579.0)*sarg - 24.0*carg
de += (-199.0)*carg + 2.0*sarg
arg = elp + 2.0*f - 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (3579.0)*sarg + 5.0*carg
de += (-1900.0)*carg + sarg
arg = el - elp
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (4725.0)*sarg - 6.0*carg
de += (-41.0)*carg + 3.0*sarg
arg = -2.0*el + 2.0*f + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-3075.0)*sarg - 2.0*carg
de += (1313.0)*carg - sarg
arg = 3.0*el + 2.0*f + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-2904.0)*sarg + 15.0*carg
de += (1233.0)*carg + 7.0*sarg
arg = -elp + 2.0*d
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (4348.0)*sarg - 10.0*carg
de += (-81.0)*carg + 2.0*sarg
arg = el - elp + 2.0*f + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-2878.0)*sarg + 8.0*carg
de += (1232.0)*carg + 4.0*sarg
sarg = math.sin(d)
carg = math.cos(d)
dp += (-4230.0)*sarg + 5.0*carg
de += (-20.0)*carg - 2.0*sarg
arg = -el - elp + 2.0*f + 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-2819.0)*sarg + 7.0*carg
de += (1207.0)*carg + 3.0*sarg
arg = -el + 2.0*f
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-4056.0)*sarg + 5.0*carg
de += (40.0)*carg - 2.0*sarg
arg = -elp + 2.0*f + 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-2647.0)*sarg + 11.0*carg
de += (1129.0)*carg + 5.0*sarg
arg = -2.0*el + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-2294.0)*sarg - 10.0*carg
de += (1266.0)*carg - 4.0*sarg
arg = el + elp + 2.0*f + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (2481.0)*sarg - 7.0*carg
de += (-1062.0)*carg - 3.0*sarg
arg = 2.0*el + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (2179.0)*sarg - 2.0*carg
de += (-1129.0)*carg - 2.0*sarg
arg = -el + elp + d
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (3276.0)*sarg + carg
de += (-9.0)*carg
arg = el + elp
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-3389.0)*sarg + 5.0*carg
de += (35.0)*carg - 2.0*sarg
arg = el + 2.0*f
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (3339.0)*sarg - 13.0*carg
de += (-107.0)*carg + sarg
arg = -el + 2.0*f - 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-1987.0)*sarg - 6.0*carg
de += (1073.0)*carg - 2.0*sarg
arg = el + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-1981.0)*sarg
de += (854.0)*carg
arg = -el + d
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (4026.0)*sarg - 353.0*carg
de += (-553.0)*carg - 139.0*sarg
arg = 2.0*f + d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (1660.0)*sarg - 5.0*carg
de += (-710.0)*carg - 2.0*sarg
arg = -el + 2.0*f + 4.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-1521.0)*sarg + 9.0*carg
de += (647.0)*carg + 4.0*sarg
arg = -el + elp + d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (1314.0)*sarg
de += (-700.0)*carg
arg = -2.0*elp + 2.0*f - 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-1283.0)*sarg
de += (672.0)*carg
arg = el + 2.0*f + 2.0*d + om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (-1331.0)*sarg + 8.0*carg
de += (663.0)*carg + 4.0*sarg
arg = -2.0*el + 2.0*f + 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (1383.0)*sarg - 2.0*carg
de += (-594.0)*carg - 2.0*sarg
arg = -el + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (1405.0)*sarg + 4.0*carg
de += (-610.0)*carg + 2.0*sarg
arg = el + elp + 2.0*f - 2.0*d + 2.0*om
sarg = math.sin(arg)
carg = math.cos(arg)
dp += (1290.0)*sarg
de += (-556.0)*carg
self.dpsi = -0.000135 + (dp * 1.0e-7)
self.deps = +0.000388 + (de * 1.0e-7)
def _mean_obliq(tt):
t = tt / 36525
asec = (
(((( - 0.0000000434 * t
- 0.000000576 ) * t
+ 0.00200340 ) * t
- 0.0001831 ) * t
- 46.836769 ) * t + 84381.406
)
return asec / 3600.0
class _e_tilt:
def __init__(self, time):
e = _iau2000b(time)
self.dpsi = e.dpsi
self.deps = e.deps
self.mobl = _mean_obliq(time.tt)
self.tobl = self.mobl + (e.deps / 3600.0)
self.tt = time.tt
self.ee = e.dpsi * math.cos(math.radians(self.mobl)) / 15.0
def _ecl2equ_vec(time, ecl):
obl = math.radians(_mean_obliq(time.tt))
cos_obl = math.cos(obl)
sin_obl = math.sin(obl)
return [
ecl[0],
ecl[1]*cos_obl - ecl[2]*sin_obl,
ecl[1]*sin_obl + ecl[2]*cos_obl
]
def _precession(tt1, pos1, tt2):
eps0 = 84381.406
if tt1 != 0 and tt2 != 0:
raise Error('One of (tt1, tt2) must be zero.')
t = (tt2 - tt1) / 36525
if tt2 == 0:
t = -t
psia = (((((- 0.0000000951 * t
+ 0.000132851 ) * t
- 0.00114045 ) * t
- 1.0790069 ) * t
+ 5038.481507 ) * t)
omegaa = (((((+ 0.0000003337 * t
- 0.000000467 ) * t
- 0.00772503 ) * t
+ 0.0512623 ) * t
- 0.025754 ) * t + eps0)
chia = (((((- 0.0000000560 * t
+ 0.000170663 ) * t
- 0.00121197 ) * t
- 2.3814292 ) * t
+ 10.556403 ) * t)
eps0 *= _ASEC2RAD
psia *= _ASEC2RAD
omegaa *= _ASEC2RAD
chia *= _ASEC2RAD
sa = math.sin(eps0)
ca = math.cos(eps0)
sb = math.sin(-psia)
cb = math.cos(-psia)
sc = math.sin(-omegaa)
cc = math.cos(-omegaa)
sd = math.sin(chia)
cd = math.cos(chia)
xx = cd * cb - sb * sd * cc
yx = cd * sb * ca + sd * cc * cb * ca - sa * sd * sc
zx = cd * sb * sa + sd * cc * cb * sa + ca * sd * sc
xy = -sd * cb - sb * cd * cc
yy = -sd * sb * ca + cd * cc * cb * ca - sa * cd * sc
zy = -sd * sb * sa + cd * cc * cb * sa + ca * cd * sc
xz = sb * sc
yz = -sc * cb * ca - sa * cc
zz = -sc * cb * sa + cc * ca
if tt2 == 0.0:
# Perform rotation from other epoch to J2000.0.
return [
xx * pos1[0] + xy * pos1[1] + xz * pos1[2],
yx * pos1[0] + yy * pos1[1] + yz * pos1[2],
zx * pos1[0] + zy * pos1[1] + zz * pos1[2]
]
# Perform rotation from J2000.0 to other epoch.
return [
xx * pos1[0] + yx * pos1[1] + zx * pos1[2],
xy * pos1[0] + yy * pos1[1] + zy * pos1[2],
xz * pos1[0] + yz * pos1[1] + zz * pos1[2]
]
class Equatorial:
def __init__(self, ra, dec, dist):
self.ra = ra
self.dec = dec
self.dist = dist
def _vector2radec(pos):
xyproj = pos[0]*pos[0] + pos[1]*pos[1]
dist = math.sqrt(xyproj + pos[2]*pos[2])
if xyproj == 0.0:
if pos[2] == 0.0:
# Indeterminate coordinates: pos vector has zero length.
raise Error('Cannot convert vector to polar coordinates')
ra = 0.0
if pos[2] < 0.0:
dec = -90.0
else:
dec = +90.0
else:
ra = math.degrees(math.atan2(pos[1], pos[0])) / 15.0
if ra < 0:
ra += 24
dec = math.degrees(math.atan2(pos[2], math.sqrt(xyproj)))
return Equatorial(ra, dec, dist)
def _nutation(time, direction, inpos):
tilt = time._etilt()
oblm = math.radians(tilt.mobl)
oblt = math.radians(tilt.tobl)
psi = tilt.dpsi * _ASEC2RAD
cobm = math.cos(oblm)
sobm = math.sin(oblm)
cobt = math.cos(oblt)
sobt = math.sin(oblt)
cpsi = math.cos(psi)
spsi = math.sin(psi)
xx = cpsi
yx = -spsi * cobm
zx = -spsi * sobm
xy = spsi * cobt
yy = cpsi * cobm * cobt + sobm * sobt
zy = cpsi * sobm * cobt - cobm * sobt
xz = spsi * sobt
yz = cpsi * cobm * sobt - sobm * cobt
zz = cpsi * sobm * sobt + cobm * cobt
if direction == 0:
# forward rotation
return [
xx * inpos[0] + yx * inpos[1] + zx * inpos[2],
xy * inpos[0] + yy * inpos[1] + zy * inpos[2],
xz * inpos[0] + yz * inpos[1] + zz * inpos[2]
]
# inverse rotation
return [
xx * inpos[0] + xy * inpos[1] + xz * inpos[2],
yx * inpos[0] + yy * inpos[1] + yz * inpos[2],
zx * inpos[0] + zy * inpos[1] + zz * inpos[2]
]
def _era(time): # Earth Rotation Angle
thet1 = 0.7790572732640 + 0.00273781191135448 * time.ut
thet3 = math.fmod(time.ut, 1.0)
theta = 360.0 * math.fmod((thet1 + thet3), 1.0)
if theta < 0.0:
theta += 360.0
return theta
def _sidereal_time(time):
t = time.tt / 36525.0
eqeq = 15.0 * time._etilt().ee # Replace with eqeq=0 to get GMST instead of GAST (if we ever need it)
theta = _era(time)
st = (eqeq + 0.014506 +
(((( - 0.0000000368 * t
- 0.000029956 ) * t
- 0.00000044 ) * t
+ 1.3915817 ) * t
+ 4612.156534 ) * t)
gst = math.fmod((st/3600.0 + theta), 360.0) / 15.0
if gst < 0.0:
gst += 24.0
return gst
def _terra(observer, st):
erad_km = _ERAD / 1000.0
df = 1.0 - 0.003352819697896 # flattening of the Earth
df2 = df * df
phi = math.radians(observer.latitude)
sinphi = math.sin(phi)
cosphi = math.cos(phi)
c = 1.0 / math.sqrt(cosphi*cosphi + df2*sinphi*sinphi)
s = df2 * c
ht_km = observer.height / 1000.0
ach = erad_km*c + ht_km
ash = erad_km*s + ht_km
stlocl = math.radians(15.0*st + observer.longitude)
sinst = math.sin(stlocl)
cosst = math.cos(stlocl)
return [
ach * cosphi * cosst / _KM_PER_AU,
ach * cosphi * sinst / _KM_PER_AU,
ash * sinphi / _KM_PER_AU
]
def _geo_pos(time, observer):
gast = _sidereal_time(time)
pos1 = _terra(observer, gast)
pos2 = _nutation(time, -1, pos1)
outpos = _precession(time.tt, pos2, 0.0)
return outpos
def _spin(angle, pos1):
angr = math.radians(angle)
cosang = math.cos(angr)
sinang = math.sin(angr)
return [
+cosang*pos1[0] + sinang*pos1[1],
-sinang*pos1[0] + cosang*pos1[1],
pos1[2]
]
#----------------------------------------------------------------------------
# BEGIN CalcMoon
def _Array1(xmin, xmax):
return dict((key, 0j) for key in range(xmin, 1+xmax))
def _Array2(xmin, xmax, ymin, ymax):
return dict((key, _Array1(ymin, ymax)) for key in range(xmin, 1+xmax))
class _moonpos:
def __init__(self, lon, lat, dist):
self.geo_eclip_lon = lon
self.geo_eclip_lat = lat
self.distance_au = dist
def _CalcMoon(time):
T = time.tt / 36525
ex = _Array2(-6, 6, 1, 4)
def Sine(phi):
return math.sin(_PI2 * phi)
def Frac(x):
return x - math.floor(x)
T2 = T*T
DLAM = 0
DS = 0
GAM1C = 0
SINPI = 3422.7000
S1 = Sine(0.19833+0.05611*T)
S2 = Sine(0.27869+0.04508*T)
S3 = Sine(0.16827-0.36903*T)
S4 = Sine(0.34734-5.37261*T)
S5 = Sine(0.10498-5.37899*T)
S6 = Sine(0.42681-0.41855*T)
S7 = Sine(0.14943-5.37511*T)
DL0 = 0.84*S1+0.31*S2+14.27*S3+ 7.26*S4+ 0.28*S5+0.24*S6
DL = 2.94*S1+0.31*S2+14.27*S3+ 9.34*S4+ 1.12*S5+0.83*S6
DLS =-6.40*S1 -1.89*S6
DF = 0.21*S1+0.31*S2+14.27*S3-88.70*S4-15.30*S5+0.24*S6-1.86*S7
DD = DL0-DLS
DGAM = ((-3332E-9 * Sine(0.59734-5.37261*T)
-539E-9 * Sine(0.35498-5.37899*T)
-64E-9 * Sine(0.39943-5.37511*T)))
L0 = _PI2*Frac(0.60643382+1336.85522467*T-0.00000313*T2) + DL0/_ARC
L = _PI2*Frac(0.37489701+1325.55240982*T+0.00002565*T2) + DL /_ARC
LS = _PI2*Frac(0.99312619+ 99.99735956*T-0.00000044*T2) + DLS/_ARC
F = _PI2*Frac(0.25909118+1342.22782980*T-0.00000892*T2) + DF /_ARC
D = _PI2*Frac(0.82736186+1236.85308708*T-0.00000397*T2) + DD /_ARC
I = 1
while I <= 4:
if I == 1:
ARG=L; MAX=4; FAC=1.000002208
elif I == 2:
ARG=LS; MAX=3; FAC=0.997504612-0.002495388*T
elif I == 3:
ARG=F; MAX=4; FAC=1.000002708+139.978*DGAM
else:
ARG=D; MAX=6; FAC=1.0
ex[0][I] = complex(1, 0)
ex[1][I] = complex(FAC * math.cos(ARG), FAC * math.sin(ARG))
J = 2
while J <= MAX:
ex[J][I] = ex[J-1][I] * ex[1][I]
J += 1
J = 1
while J <= MAX:
ex[-J][I] = ex[J][I].conjugate()
J += 1
I += 1
# AddSol(13.902000, 14.060000, -0.001000, 0.260700, 0.000000, 0.000000, 0.000000, 4.000000)
z = ex[4][4]
DLAM += 13.902 * z.imag
DS += 14.06 * z.imag
GAM1C += -0.001 * z.real
SINPI += 0.2607 * z.real
# AddSol(0.403000, -4.010000, 0.394000, 0.002300, 0.000000, 0.000000, 0.000000, 3.000000)
z = ex[3][4]
DLAM += 0.403 * z.imag
DS += -4.01 * z.imag
GAM1C += 0.394 * z.real
SINPI += 0.0023 * z.real
# AddSol(2369.912000, 2373.360000, 0.601000, 28.233300, 0.000000, 0.000000, 0.000000, 2.000000)
z = ex[2][4]
DLAM += 2369.912 * z.imag
DS += 2373.36 * z.imag
GAM1C += 0.601 * z.real
SINPI += 28.2333 * z.real
# AddSol(-125.154000, -112.790000, -0.725000, -0.978100, 0.000000, 0.000000, 0.000000, 1.000000)
z = ex[1][4]
DLAM += -125.154 * z.imag
DS += -112.79 * z.imag
GAM1C += -0.725 * z.real
SINPI += -0.9781 * z.real
# AddSol(1.979000, 6.980000, -0.445000, 0.043300, 1.000000, 0.000000, 0.000000, 4.000000)
z = ex[1][1] * ex[4][4]
DLAM += 1.979 * z.imag
DS += 6.98 * z.imag
GAM1C += -0.445 * z.real
SINPI += 0.0433 * z.real
# AddSol(191.953000, 192.720000, 0.029000, 3.086100, 1.000000, 0.000000, 0.000000, 2.000000)
z = ex[1][1] * ex[2][4]
DLAM += 191.953 * z.imag
DS += 192.72 * z.imag
GAM1C += 0.029 * z.real
SINPI += 3.0861 * z.real
# AddSol(-8.466000, -13.510000, 0.455000, -0.109300, 1.000000, 0.000000, 0.000000, 1.000000)
z = ex[1][1] * ex[1][4]
DLAM += -8.466 * z.imag
DS += -13.51 * z.imag
GAM1C += 0.455 * z.real
SINPI += -0.1093 * z.real
# AddSol(22639.500000, 22609.070000, 0.079000, 186.539800, 1.000000, 0.000000, 0.000000, 0.000000)
z = ex[1][1]
DLAM += 22639.500 * z.imag
DS += 22609.07 * z.imag
GAM1C += 0.079 * z.real
SINPI += 186.5398 * z.real
# AddSol(18.609000, 3.590000, -0.094000, 0.011800, 1.000000, 0.000000, 0.000000, -1.000000)
z = ex[1][1] * ex[-1][4]
DLAM += 18.609 * z.imag
DS += 3.59 * z.imag
GAM1C += -0.094 * z.real
SINPI += 0.0118 * z.real
# AddSol(-4586.465000, -4578.130000, -0.077000, 34.311700, 1.000000, 0.000000, 0.000000, -2.000000)
z = ex[1][1] * ex[-2][4]
DLAM += -4586.465 * z.imag
DS += -4578.13 * z.imag
GAM1C += -0.077 * z.real
SINPI += 34.3117 * z.real
# AddSol(3.215000, 5.440000, 0.192000, -0.038600, 1.000000, 0.000000, 0.000000, -3.000000)
z = ex[1][1] * ex[-3][4]
DLAM += 3.215 * z.imag
DS += 5.44 * z.imag
GAM1C += 0.192 * z.real
SINPI += -0.0386 * z.real
# AddSol(-38.428000, -38.640000, 0.001000, 0.600800, 1.000000, 0.000000, 0.000000, -4.000000)
z = ex[1][1] * ex[-4][4]
DLAM += -38.428 * z.imag
DS += -38.64 * z.imag
GAM1C += 0.001 * z.real
SINPI += 0.6008 * z.real
# AddSol(-0.393000, -1.430000, -0.092000, 0.008600, 1.000000, 0.000000, 0.000000, -6.000000)
z = ex[1][1] * ex[-6][4]
DLAM += -0.393 * z.imag
DS += -1.43 * z.imag
GAM1C += -0.092 * z.real
SINPI += 0.0086 * z.real
# AddSol(-0.289000, -1.590000, 0.123000, -0.005300, 0.000000, 1.000000, 0.000000, 4.000000)
z = ex[1][2] * ex[4][4]
DLAM += -0.289 * z.imag
DS += -1.59 * z.imag
GAM1C += 0.123 * z.real
SINPI += -0.0053 * z.real
# AddSol(-24.420000, -25.100000, 0.040000, -0.300000, 0.000000, 1.000000, 0.000000, 2.000000)
z = ex[1][2] * ex[2][4]
DLAM += -24.420 * z.imag
DS += -25.10 * z.imag
GAM1C += 0.040 * z.real
SINPI += -0.3000 * z.real
# AddSol(18.023000, 17.930000, 0.007000, 0.149400, 0.000000, 1.000000, 0.000000, 1.000000)
z = ex[1][2] * ex[1][4]
DLAM += 18.023 * z.imag
DS += 17.93 * z.imag
GAM1C += 0.007 * z.real
SINPI += 0.1494 * z.real
# AddSol(-668.146000, -126.980000, -1.302000, -0.399700, 0.000000, 1.000000, 0.000000, 0.000000)
z = ex[1][2]
DLAM += -668.146 * z.imag
DS += -126.98 * z.imag
GAM1C += -1.302 * z.real
SINPI += -0.3997 * z.real
# AddSol(0.560000, 0.320000, -0.001000, -0.003700, 0.000000, 1.000000, 0.000000, -1.000000)
z = ex[1][2] * ex[-1][4]
DLAM += 0.560 * z.imag
DS += 0.32 * z.imag
GAM1C += -0.001 * z.real
SINPI += -0.0037 * z.real
# AddSol(-165.145000, -165.060000, 0.054000, 1.917800, 0.000000, 1.000000, 0.000000, -2.000000)
z = ex[1][2] * ex[-2][4]
DLAM += -165.145 * z.imag
DS += -165.06 * z.imag
GAM1C += 0.054 * z.real
SINPI += 1.9178 * z.real
# AddSol(-1.877000, -6.460000, -0.416000, 0.033900, 0.000000, 1.000000, 0.000000, -4.000000)
z = ex[1][2] * ex[-4][4]
DLAM += -1.877 * z.imag
DS += -6.46 * z.imag
GAM1C += -0.416 * z.real
SINPI += 0.0339 * z.real
# AddSol(0.213000, 1.020000, -0.074000, 0.005400, 2.000000, 0.000000, 0.000000, 4.000000)
z = ex[2][1] * ex[4][4]
DLAM += 0.213 * z.imag
DS += 1.02 * z.imag
GAM1C += -0.074 * z.real
SINPI += 0.0054 * z.real
# AddSol(14.387000, 14.780000, -0.017000, 0.283300, 2.000000, 0.000000, 0.000000, 2.000000)
z = ex[2][1] * ex[2][4]
DLAM += 14.387 * z.imag
DS += 14.78 * z.imag
GAM1C += -0.017 * z.real
SINPI += 0.2833 * z.real
# AddSol(-0.586000, -1.200000, 0.054000, -0.010000, 2.000000, 0.000000, 0.000000, 1.000000)
z = ex[2][1] * ex[1][4]
DLAM += -0.586 * z.imag
DS += -1.20 * z.imag
GAM1C += 0.054 * z.real
SINPI += -0.0100 * z.real
# AddSol(769.016000, 767.960000, 0.107000, 10.165700, 2.000000, 0.000000, 0.000000, 0.000000)
z = ex[2][1]
DLAM += 769.016 * z.imag
DS += 767.96 * z.imag
GAM1C += 0.107 * z.real
SINPI += 10.1657 * z.real
# AddSol(1.750000, 2.010000, -0.018000, 0.015500, 2.000000, 0.000000, 0.000000, -1.000000)
z = ex[2][1] * ex[-1][4]
DLAM += 1.750 * z.imag
DS += 2.01 * z.imag
GAM1C += -0.018 * z.real
SINPI += 0.0155 * z.real
# AddSol(-211.656000, -152.530000, 5.679000, -0.303900, 2.000000, 0.000000, 0.000000, -2.000000)
z = ex[2][1] * ex[-2][4]
DLAM += -211.656 * z.imag
DS += -152.53 * z.imag
GAM1C += 5.679 * z.real
SINPI += -0.3039 * z.real
# AddSol(1.225000, 0.910000, -0.030000, -0.008800, 2.000000, 0.000000, 0.000000, -3.000000)
z = ex[2][1] * ex[-3][4]
DLAM += 1.225 * z.imag
DS += 0.91 * z.imag
GAM1C += -0.030 * z.real
SINPI += -0.0088 * z.real
# AddSol(-30.773000, -34.070000, -0.308000, 0.372200, 2.000000, 0.000000, 0.000000, -4.000000)
z = ex[2][1] * ex[-4][4]
DLAM += -30.773 * z.imag
DS += -34.07 * z.imag
GAM1C += -0.308 * z.real
SINPI += 0.3722 * z.real
# AddSol(-0.570000, -1.400000, -0.074000, 0.010900, 2.000000, 0.000000, 0.000000, -6.000000)
z = ex[2][1] * ex[-6][4]
DLAM += -0.570 * z.imag
DS += -1.40 * z.imag
GAM1C += -0.074 * z.real
SINPI += 0.0109 * z.real
# AddSol(-2.921000, -11.750000, 0.787000, -0.048400, 1.000000, 1.000000, 0.000000, 2.000000)
z = ex[1][1] * ex[1][2] * ex[2][4]
DLAM += -2.921 * z.imag
DS += -11.75 * z.imag
GAM1C += 0.787 * z.real
SINPI += -0.0484 * z.real
# AddSol(1.267000, 1.520000, -0.022000, 0.016400, 1.000000, 1.000000, 0.000000, 1.000000)
z = ex[1][1] * ex[1][2] * ex[1][4]
DLAM += 1.267 * z.imag
DS += 1.52 * z.imag
GAM1C += -0.022 * z.real
SINPI += 0.0164 * z.real
# AddSol(-109.673000, -115.180000, 0.461000, -0.949000, 1.000000, 1.000000, 0.000000, 0.000000)
z = ex[1][1] * ex[1][2]
DLAM += -109.673 * z.imag
DS += -115.18 * z.imag
GAM1C += 0.461 * z.real
SINPI += -0.9490 * z.real
# AddSol(-205.962000, -182.360000, 2.056000, 1.443700, 1.000000, 1.000000, 0.000000, -2.000000)
z = ex[1][1] * ex[1][2] * ex[-2][4]
DLAM += -205.962 * z.imag
DS += -182.36 * z.imag
GAM1C += 2.056 * z.real
SINPI += 1.4437 * z.real
# AddSol(0.233000, 0.360000, 0.012000, -0.002500, 1.000000, 1.000000, 0.000000, -3.000000)
z = ex[1][1] * ex[1][2] * ex[-3][4]
DLAM += 0.233 * z.imag
DS += 0.36 * z.imag
GAM1C += 0.012 * z.real
SINPI += -0.0025 * z.real
# AddSol(-4.391000, -9.660000, -0.471000, 0.067300, 1.000000, 1.000000, 0.000000, -4.000000)
z = ex[1][1] * ex[1][2] * ex[-4][4]
DLAM += -4.391 * z.imag
DS += -9.66 * z.imag
GAM1C += -0.471 * z.real
SINPI += 0.0673 * z.real
# AddSol(0.283000, 1.530000, -0.111000, 0.006000, 1.000000, -1.000000, 0.000000, 4.000000)
z = ex[1][1] * ex[-1][2] * ex[4][4]
DLAM += 0.283 * z.imag
DS += 1.53 * z.imag
GAM1C += -0.111 * z.real
SINPI += 0.0060 * z.real
# AddSol(14.577000, 31.700000, -1.540000, 0.230200, 1.000000, -1.000000, 0.000000, 2.000000)
z = ex[1][1] * ex[-1][2] * ex[2][4]
DLAM += 14.577 * z.imag
DS += 31.70 * z.imag
GAM1C += -1.540 * z.real
SINPI += 0.2302 * z.real
# AddSol(147.687000, 138.760000, 0.679000, 1.152800, 1.000000, -1.000000, 0.000000, 0.000000)
z = ex[1][1] * ex[-1][2]
DLAM += 147.687 * z.imag
DS += 138.76 * z.imag
GAM1C += 0.679 * z.real
SINPI += 1.1528 * z.real
# AddSol(-1.089000, 0.550000, 0.021000, 0.000000, 1.000000, -1.000000, 0.000000, -1.000000)
z = ex[1][1] * ex[-1][2] * ex[-1][4]
DLAM += -1.089 * z.imag
DS += 0.55 * z.imag
GAM1C += 0.021 * z.real
# AddSol(28.475000, 23.590000, -0.443000, -0.225700, 1.000000, -1.000000, 0.000000, -2.000000)
z = ex[1][1] * ex[-1][2] * ex[-2][4]
DLAM += 28.475 * z.imag
DS += 23.59 * z.imag
GAM1C += -0.443 * z.real
SINPI += -0.2257 * z.real
# AddSol(-0.276000, -0.380000, -0.006000, -0.003600, 1.000000, -1.000000, 0.000000, -3.000000)
z = ex[1][1] * ex[-1][2] * ex[-3][4]
DLAM += -0.276 * z.imag
DS += -0.38 * z.imag
GAM1C += -0.006 * z.real
SINPI += -0.0036 * z.real
# AddSol(0.636000, 2.270000, 0.146000, -0.010200, 1.000000, -1.000000, 0.000000, -4.000000)
z = ex[1][1] * ex[-1][2] * ex[-4][4]
DLAM += 0.636 * z.imag
DS += 2.27 * z.imag
GAM1C += 0.146 * z.real
SINPI += -0.0102 * z.real
# AddSol(-0.189000, -1.680000, 0.131000, -0.002800, 0.000000, 2.000000, 0.000000, 2.000000)
z = ex[2][2] * ex[2][4]
DLAM += -0.189 * z.imag
DS += -1.68 * z.imag
GAM1C += 0.131 * z.real
SINPI += -0.0028 * z.real
# AddSol(-7.486000, -0.660000, -0.037000, -0.008600, 0.000000, 2.000000, 0.000000, 0.000000)
z = ex[2][2]
DLAM += -7.486 * z.imag
DS += -0.66 * z.imag
GAM1C += -0.037 * z.real
SINPI += -0.0086 * z.real
# AddSol(-8.096000, -16.350000, -0.740000, 0.091800, 0.000000, 2.000000, 0.000000, -2.000000)
z = ex[2][2] * ex[-2][4]
DLAM += -8.096 * z.imag
DS += -16.35 * z.imag
GAM1C += -0.740 * z.real
SINPI += 0.0918 * z.real
# AddSol(-5.741000, -0.040000, 0.000000, -0.000900, 0.000000, 0.000000, 2.000000, 2.000000)
z = ex[2][3] * ex[2][4]
DLAM += -5.741 * z.imag
DS += -0.04 * z.imag
SINPI += -0.0009 * z.real
# AddSol(0.255000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 2.000000, 1.000000)
z = ex[2][3] * ex[1][4]
DLAM += 0.255 * z.imag
# AddSol(-411.608000, -0.200000, 0.000000, -0.012400, 0.000000, 0.000000, 2.000000, 0.000000)
z = ex[2][3]
DLAM += -411.608 * z.imag
DS += -0.20 * z.imag
SINPI += -0.0124 * z.real
# AddSol(0.584000, 0.840000, 0.000000, 0.007100, 0.000000, 0.000000, 2.000000, -1.000000)
z = ex[2][3] * ex[-1][4]
DLAM += 0.584 * z.imag
DS += 0.84 * z.imag
SINPI += 0.0071 * z.real
# AddSol(-55.173000, -52.140000, 0.000000, -0.105200, 0.000000, 0.000000, 2.000000, -2.000000)
z = ex[2][3] * ex[-2][4]
DLAM += -55.173 * z.imag
DS += -52.14 * z.imag
SINPI += -0.1052 * z.real
# AddSol(0.254000, 0.250000, 0.000000, -0.001700, 0.000000, 0.000000, 2.000000, -3.000000)
z = ex[2][3] * ex[-3][4]
DLAM += 0.254 * z.imag
DS += 0.25 * z.imag
SINPI += -0.0017 * z.real
# AddSol(0.025000, -1.670000, 0.000000, 0.003100, 0.000000, 0.000000, 2.000000, -4.000000)
z = ex[2][3] * ex[-4][4]
DLAM += 0.025 * z.imag
DS += -1.67 * z.imag
SINPI += 0.0031 * z.real
# AddSol(1.060000, 2.960000, -0.166000, 0.024300, 3.000000, 0.000000, 0.000000, 2.000000)
z = ex[3][1] * ex[2][4]
DLAM += 1.060 * z.imag
DS += 2.96 * z.imag
GAM1C += -0.166 * z.real
SINPI += 0.0243 * z.real
# AddSol(36.124000, 50.640000, -1.300000, 0.621500, 3.000000, 0.000000, 0.000000, 0.000000)
z = ex[3][1]
DLAM += 36.124 * z.imag
DS += 50.64 * z.imag
GAM1C += -1.300 * z.real
SINPI += 0.6215 * z.real
# AddSol(-13.193000, -16.400000, 0.258000, -0.118700, 3.000000, 0.000000, 0.000000, -2.000000)
z = ex[3][1] * ex[-2][4]
DLAM += -13.193 * z.imag
DS += -16.40 * z.imag
GAM1C += 0.258 * z.real
SINPI += -0.1187 * z.real
# AddSol(-1.187000, -0.740000, 0.042000, 0.007400, 3.000000, 0.000000, 0.000000, -4.000000)
z = ex[3][1] * ex[-4][4]
DLAM += -1.187 * z.imag
DS += -0.74 * z.imag
GAM1C += 0.042 * z.real
SINPI += 0.0074 * z.real
# AddSol(-0.293000, -0.310000, -0.002000, 0.004600, 3.000000, 0.000000, 0.000000, -6.000000)
z = ex[3][1] * ex[-6][4]
DLAM += -0.293 * z.imag
DS += -0.31 * z.imag
GAM1C += -0.002 * z.real
SINPI += 0.0046 * z.real
# AddSol(-0.290000, -1.450000, 0.116000, -0.005100, 2.000000, 1.000000, 0.000000, 2.000000)
z = ex[2][1] * ex[1][2] * ex[2][4]
DLAM += -0.290 * z.imag
DS += -1.45 * z.imag
GAM1C += 0.116 * z.real
SINPI += -0.0051 * z.real
# AddSol(-7.649000, -10.560000, 0.259000, -0.103800, 2.000000, 1.000000, 0.000000, 0.000000)
z = ex[2][1] * ex[1][2]
DLAM += -7.649 * z.imag
DS += -10.56 * z.imag
GAM1C += 0.259 * z.real
SINPI += -0.1038 * z.real
# AddSol(-8.627000, -7.590000, 0.078000, -0.019200, 2.000000, 1.000000, 0.000000, -2.000000)
z = ex[2][1] * ex[1][2] * ex[-2][4]
DLAM += -8.627 * z.imag
DS += -7.59 * z.imag
GAM1C += 0.078 * z.real
SINPI += -0.0192 * z.real
# AddSol(-2.740000, -2.540000, 0.022000, 0.032400, 2.000000, 1.000000, 0.000000, -4.000000)
z = ex[2][1] * ex[1][2] * ex[-4][4]
DLAM += -2.740 * z.imag
DS += -2.54 * z.imag
GAM1C += 0.022 * z.real
SINPI += 0.0324 * z.real
# AddSol(1.181000, 3.320000, -0.212000, 0.021300, 2.000000, -1.000000, 0.000000, 2.000000)
z = ex[2][1] * ex[-1][2] * ex[2][4]
DLAM += 1.181 * z.imag
DS += 3.32 * z.imag
GAM1C += -0.212 * z.real
SINPI += 0.0213 * z.real
# AddSol(9.703000, 11.670000, -0.151000, 0.126800, 2.000000, -1.000000, 0.000000, 0.000000)
z = ex[2][1] * ex[-1][2]
DLAM += 9.703 * z.imag
DS += 11.67 * z.imag
GAM1C += -0.151 * z.real
SINPI += 0.1268 * z.real
# AddSol(-0.352000, -0.370000, 0.001000, -0.002800, 2.000000, -1.000000, 0.000000, -1.000000)
z = ex[2][1] * ex[-1][2] * ex[-1][4]
DLAM += -0.352 * z.imag
DS += -0.37 * z.imag
GAM1C += 0.001 * z.real
SINPI += -0.0028 * z.real
# AddSol(-2.494000, -1.170000, -0.003000, -0.001700, 2.000000, -1.000000, 0.000000, -2.000000)
z = ex[2][1] * ex[-1][2] * ex[-2][4]
DLAM += -2.494 * z.imag
DS += -1.17 * z.imag
GAM1C += -0.003 * z.real
SINPI += -0.0017 * z.real
# AddSol(0.360000, 0.200000, -0.012000, -0.004300, 2.000000, -1.000000, 0.000000, -4.000000)
z = ex[2][1] * ex[-1][2] * ex[-4][4]
DLAM += 0.360 * z.imag
DS += 0.20 * z.imag
GAM1C += -0.012 * z.real
SINPI += -0.0043 * z.real
# AddSol(-1.167000, -1.250000, 0.008000, -0.010600, 1.000000, 2.000000, 0.000000, 0.000000)
z = ex[1][1] * ex[2][2]
DLAM += -1.167 * z.imag
DS += -1.25 * z.imag
GAM1C += 0.008 * z.real
SINPI += -0.0106 * z.real
# AddSol(-7.412000, -6.120000, 0.117000, 0.048400, 1.000000, 2.000000, 0.000000, -2.000000)
z = ex[1][1] * ex[2][2] * ex[-2][4]
DLAM += -7.412 * z.imag
DS += -6.12 * z.imag
GAM1C += 0.117 * z.real
SINPI += 0.0484 * z.real
# AddSol(-0.311000, -0.650000, -0.032000, 0.004400, 1.000000, 2.000000, 0.000000, -4.000000)
z = ex[1][1] * ex[2][2] * ex[-4][4]
DLAM += -0.311 * z.imag
DS += -0.65 * z.imag
GAM1C += -0.032 * z.real
SINPI += 0.0044 * z.real
# AddSol(0.757000, 1.820000, -0.105000, 0.011200, 1.000000, -2.000000, 0.000000, 2.000000)
z = ex[1][1] * ex[-2][2] * ex[2][4]
DLAM += 0.757 * z.imag
DS += 1.82 * z.imag
GAM1C += -0.105 * z.real
SINPI += 0.0112 * z.real
# AddSol(2.580000, 2.320000, 0.027000, 0.019600, 1.000000, -2.000000, 0.000000, 0.000000)
z = ex[1][1] * ex[-2][2]
DLAM += 2.580 * z.imag
DS += 2.32 * z.imag
GAM1C += 0.027 * z.real
SINPI += 0.0196 * z.real
# AddSol(2.533000, 2.400000, -0.014000, -0.021200, 1.000000, -2.000000, 0.000000, -2.000000)
z = ex[1][1] * ex[-2][2] * ex[-2][4]
DLAM += 2.533 * z.imag
DS += 2.40 * z.imag
GAM1C += -0.014 * z.real
SINPI += -0.0212 * z.real
# AddSol(-0.344000, -0.570000, -0.025000, 0.003600, 0.000000, 3.000000, 0.000000, -2.000000)
z = ex[3][2] * ex[-2][4]
DLAM += -0.344 * z.imag
DS += -0.57 * z.imag
GAM1C += -0.025 * z.real
SINPI += 0.0036 * z.real
# AddSol(-0.992000, -0.020000, 0.000000, 0.000000, 1.000000, 0.000000, 2.000000, 2.000000)
z = ex[1][1] * ex[2][3] * ex[2][4]
DLAM += -0.992 * z.imag
DS += -0.02 * z.imag
# AddSol(-45.099000, -0.020000, 0.000000, -0.001000, 1.000000, 0.000000, 2.000000, 0.000000)
z = ex[1][1] * ex[2][3]
DLAM += -45.099 * z.imag
DS += -0.02 * z.imag
SINPI += -0.0010 * z.real
# AddSol(-0.179000, -9.520000, 0.000000, -0.083300, 1.000000, 0.000000, 2.000000, -2.000000)
z = ex[1][1] * ex[2][3] * ex[-2][4]
DLAM += -0.179 * z.imag
DS += -9.52 * z.imag
SINPI += -0.0833 * z.real
# AddSol(-0.301000, -0.330000, 0.000000, 0.001400, 1.000000, 0.000000, 2.000000, -4.000000)
z = ex[1][1] * ex[2][3] * ex[-4][4]
DLAM += -0.301 * z.imag
DS += -0.33 * z.imag
SINPI += 0.0014 * z.real
# AddSol(-6.382000, -3.370000, 0.000000, -0.048100, 1.000000, 0.000000, -2.000000, 2.000000)
z = ex[1][1] * ex[-2][3] * ex[2][4]
DLAM += -6.382 * z.imag
DS += -3.37 * z.imag
SINPI += -0.0481 * z.real
# AddSol(39.528000, 85.130000, 0.000000, -0.713600, 1.000000, 0.000000, -2.000000, 0.000000)
z = ex[1][1] * ex[-2][3]
DLAM += 39.528 * z.imag
DS += 85.13 * z.imag
SINPI += -0.7136 * z.real
# AddSol(9.366000, 0.710000, 0.000000, -0.011200, 1.000000, 0.000000, -2.000000, -2.000000)
z = ex[1][1] * ex[-2][3] * ex[-2][4]
DLAM += 9.366 * z.imag
DS += 0.71 * z.imag
SINPI += -0.0112 * z.real
# AddSol(0.202000, 0.020000, 0.000000, 0.000000, 1.000000, 0.000000, -2.000000, -4.000000)
z = ex[1][1] * ex[-2][3] * ex[-4][4]
DLAM += 0.202 * z.imag
DS += 0.02 * z.imag
# AddSol(0.415000, 0.100000, 0.000000, 0.001300, 0.000000, 1.000000, 2.000000, 0.000000)
z = ex[1][2] * ex[2][3]
DLAM += 0.415 * z.imag
DS += 0.10 * z.imag
SINPI += 0.0013 * z.real
# AddSol(-2.152000, -2.260000, 0.000000, -0.006600, 0.000000, 1.000000, 2.000000, -2.000000)
z = ex[1][2] * ex[2][3] * ex[-2][4]
DLAM += -2.152 * z.imag
DS += -2.26 * z.imag
SINPI += -0.0066 * z.real
# AddSol(-1.440000, -1.300000, 0.000000, 0.001400, 0.000000, 1.000000, -2.000000, 2.000000)
z = ex[1][2] * ex[-2][3] * ex[2][4]
DLAM += -1.440 * z.imag
DS += -1.30 * z.imag
SINPI += 0.0014 * z.real
# AddSol(0.384000, -0.040000, 0.000000, 0.000000, 0.000000, 1.000000, -2.000000, -2.000000)
z = ex[1][2] * ex[-2][3] * ex[-2][4]
DLAM += 0.384 * z.imag
DS += -0.04 * z.imag
# AddSol(1.938000, 3.600000, -0.145000, 0.040100, 4.000000, 0.000000, 0.000000, 0.000000)
z = ex[4][1]
DLAM += 1.938 * z.imag
DS += 3.60 * z.imag
GAM1C += -0.145 * z.real
SINPI += 0.0401 * z.real
# AddSol(-0.952000, -1.580000, 0.052000, -0.013000, 4.000000, 0.000000, 0.000000, -2.000000)
z = ex[4][1] * ex[-2][4]
DLAM += -0.952 * z.imag
DS += -1.58 * z.imag
GAM1C += 0.052 * z.real
SINPI += -0.0130 * z.real
# AddSol(-0.551000, -0.940000, 0.032000, -0.009700, 3.000000, 1.000000, 0.000000, 0.000000)
z = ex[3][1] * ex[1][2]
DLAM += -0.551 * z.imag
DS += -0.94 * z.imag
GAM1C += 0.032 * z.real
SINPI += -0.0097 * z.real
# AddSol(-0.482000, -0.570000, 0.005000, -0.004500, 3.000000, 1.000000, 0.000000, -2.000000)
z = ex[3][1] * ex[1][2] * ex[-2][4]
DLAM += -0.482 * z.imag
DS += -0.57 * z.imag
GAM1C += 0.005 * z.real
SINPI += -0.0045 * z.real
# AddSol(0.681000, 0.960000, -0.026000, 0.011500, 3.000000, -1.000000, 0.000000, 0.000000)
z = ex[3][1] * ex[-1][2]
DLAM += 0.681 * z.imag
DS += 0.96 * z.imag
GAM1C += -0.026 * z.real
SINPI += 0.0115 * z.real
# AddSol(-0.297000, -0.270000, 0.002000, -0.000900, 2.000000, 2.000000, 0.000000, -2.000000)
z = ex[2][1] * ex[2][2] * ex[-2][4]
DLAM += -0.297 * z.imag
DS += -0.27 * z.imag
GAM1C += 0.002 * z.real
SINPI += -0.0009 * z.real
# AddSol(0.254000, 0.210000, -0.003000, 0.000000, 2.000000, -2.000000, 0.000000, -2.000000)
z = ex[2][1] * ex[-2][2] * ex[-2][4]
DLAM += 0.254 * z.imag
DS += 0.21 * z.imag
GAM1C += -0.003 * z.real
# AddSol(-0.250000, -0.220000, 0.004000, 0.001400, 1.000000, 3.000000, 0.000000, -2.000000)
z = ex[1][1] * ex[3][2] * ex[-2][4]
DLAM += -0.250 * z.imag
DS += -0.22 * z.imag
GAM1C += 0.004 * z.real
SINPI += 0.0014 * z.real
# AddSol(-3.996000, 0.000000, 0.000000, 0.000400, 2.000000, 0.000000, 2.000000, 0.000000)
z = ex[2][1] * ex[2][3]
DLAM += -3.996 * z.imag
SINPI += 0.0004 * z.real
# AddSol(0.557000, -0.750000, 0.000000, -0.009000, 2.000000, 0.000000, 2.000000, -2.000000)
z = ex[2][1] * ex[2][3] * ex[-2][4]
DLAM += 0.557 * z.imag
DS += -0.75 * z.imag
SINPI += -0.0090 * z.real
# AddSol(-0.459000, -0.380000, 0.000000, -0.005300, 2.000000, 0.000000, -2.000000, 2.000000)
z = ex[2][1] * ex[-2][3] * ex[2][4]
DLAM += -0.459 * z.imag
DS += -0.38 * z.imag
SINPI += -0.0053 * z.real
# AddSol(-1.298000, 0.740000, 0.000000, 0.000400, 2.000000, 0.000000, -2.000000, 0.000000)
z = ex[2][1] * ex[-2][3]
DLAM += -1.298 * z.imag
DS += 0.74 * z.imag
SINPI += 0.0004 * z.real
# AddSol(0.538000, 1.140000, 0.000000, -0.014100, 2.000000, 0.000000, -2.000000, -2.000000)
z = ex[2][1] * ex[-2][3] * ex[-2][4]
DLAM += 0.538 * z.imag
DS += 1.14 * z.imag
SINPI += -0.0141 * z.real
# AddSol(0.263000, 0.020000, 0.000000, 0.000000, 1.000000, 1.000000, 2.000000, 0.000000)
z = ex[1][1] * ex[1][2] * ex[2][3]
DLAM += 0.263 * z.imag
DS += 0.02 * z.imag
# AddSol(0.426000, 0.070000, 0.000000, -0.000600, 1.000000, 1.000000, -2.000000, -2.000000)
z = ex[1][1] * ex[1][2] * ex[-2][3] * ex[-2][4]
DLAM += 0.426 * z.imag
DS += 0.07 * z.imag
SINPI += -0.0006 * z.real
# AddSol(-0.304000, 0.030000, 0.000000, 0.000300, 1.000000, -1.000000, 2.000000, 0.000000)
z = ex[1][1] * ex[-1][2] * ex[2][3]
DLAM += -0.304 * z.imag
DS += 0.03 * z.imag
SINPI += 0.0003 * z.real
# AddSol(-0.372000, -0.190000, 0.000000, -0.002700, 1.000000, -1.000000, -2.000000, 2.000000)
z = ex[1][1] * ex[-1][2] * ex[-2][3] * ex[2][4]
DLAM += -0.372 * z.imag
DS += -0.19 * z.imag
SINPI += -0.0027 * z.real
# AddSol(0.418000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 4.000000, 0.000000)
z = ex[4][3]
DLAM += 0.418 * z.imag
# AddSol(-0.330000, -0.040000, 0.000000, 0.000000, 3.000000, 0.000000, 2.000000, 0.000000)
z = ex[3][1] * ex[2][3]
DLAM += -0.330 * z.imag
DS += -0.04 * z.imag
def ADDN(coeffn, p, q, r, s):
return coeffn * (ex[p][1] * ex[q][2] * ex[r][3] * ex[s][4]).imag
N = 0
N += ADDN(-526.069, 0, 0,1,-2)
N += ADDN( -3.352, 0, 0,1,-4)
N += ADDN( +44.297,+1, 0,1,-2)
N += ADDN( -6.000,+1, 0,1,-4)
N += ADDN( +20.599,-1, 0,1, 0)
N += ADDN( -30.598,-1, 0,1,-2)
N += ADDN( -24.649,-2, 0,1, 0)
N += ADDN( -2.000,-2, 0,1,-2)
N += ADDN( -22.571, 0,+1,1,-2)
N += ADDN( +10.985, 0,-1,1,-2)
DLAM += (
+0.82*Sine(0.7736 -62.5512*T)+0.31*Sine(0.0466 -125.1025*T)
+0.35*Sine(0.5785 -25.1042*T)+0.66*Sine(0.4591+1335.8075*T)
+0.64*Sine(0.3130 -91.5680*T)+1.14*Sine(0.1480+1331.2898*T)
+0.21*Sine(0.5918+1056.5859*T)+0.44*Sine(0.5784+1322.8595*T)
+0.24*Sine(0.2275 -5.7374*T)+0.28*Sine(0.2965 +2.6929*T)
+0.33*Sine(0.3132 +6.3368*T)
)
S = F + DS/_ARC
lat_seconds = (1.000002708 + 139.978*DGAM)*(18518.511+1.189+GAM1C)*math.sin(S) - 6.24*math.sin(3*S) + N
return _moonpos(
_PI2 * Frac((L0+DLAM/_ARC) / _PI2),
(math.pi / (180 * 3600)) * lat_seconds,
(_ARC * (_ERAD / _AU)) / (0.999953253 * SINPI)
)
def GeoMoon(time):
"""Calculates the geocentric position of the Moon at a given time.
Given a time of observation, calculates the Moon's position as a vector.
The vector gives the location of the Moon's center relative to the Earth's center
with x-, y-, and z-components measured in astronomical units.
This algorithm is based on Nautical Almanac Office's *Improved Lunar Ephemeris* of 1954,
which in turn derives from E. W. Brown's lunar theories from the early twentieth century.
It is adapted from Turbo Pascal code from the book
[Astronomy on the Personal Computer](https://www.springer.com/us/book/9783540672210)
by Montenbruck and Pfleger.
Parameters
----------
time : Time
The date and time for which to calculate the Moon's position.
Returns
-------
Vector
The Moon's position as a vector in J2000 Cartesian equatorial coordinates.
"""
m = _CalcMoon(time)
# Convert geocentric ecliptic spherical coordinates to Cartesian coordinates.
dist_cos_lat = m.distance_au * math.cos(m.geo_eclip_lat)
gepos = [
dist_cos_lat * math.cos(m.geo_eclip_lon),
dist_cos_lat * math.sin(m.geo_eclip_lon),
m.distance_au * math.sin(m.geo_eclip_lat)
]
# Convert ecliptic coordinates to equatorial coordinates, both in mean equinox of date.
mpos1 = _ecl2equ_vec(time, gepos)
# Convert from mean equinox of date to J2000.
mpos2 = _precession(time.tt, mpos1, 0)
return Vector(mpos2[0], mpos2[1], mpos2[2], time)
# END CalcMoon
#----------------------------------------------------------------------------
# BEGIN VSOP
_vsop = [
# Mercury
[
[
[
[4.40250710144, 0.00000000000, 0.00000000000],
[0.40989414977, 1.48302034195, 26087.90314157420],
[0.05046294200, 4.47785489551, 52175.80628314840],
[0.00855346844, 1.16520322459, 78263.70942472259],
[0.00165590362, 4.11969163423, 104351.61256629678],
[0.00034561897, 0.77930768443, 130439.51570787099],
[0.00007583476, 3.71348404924, 156527.41884944518]
],
[
[26087.90313685529, 0.00000000000, 0.00000000000],
[0.01131199811, 6.21874197797, 26087.90314157420],
[0.00292242298, 3.04449355541, 52175.80628314840],
[0.00075775081, 6.08568821653, 78263.70942472259],
[0.00019676525, 2.80965111777, 104351.61256629678]
]
],
[
[
[0.11737528961, 1.98357498767, 26087.90314157420],
[0.02388076996, 5.03738959686, 52175.80628314840],
[0.01222839532, 3.14159265359, 0.00000000000],
[0.00543251810, 1.79644363964, 78263.70942472259],
[0.00129778770, 4.83232503958, 104351.61256629678],
[0.00031866927, 1.58088495658, 130439.51570787099],
[0.00007963301, 4.60972126127, 156527.41884944518]
],
[
[0.00274646065, 3.95008450011, 26087.90314157420],
[0.00099737713, 3.14159265359, 0.00000000000]
]
],
[
[
[0.39528271651, 0.00000000000, 0.00000000000],
[0.07834131818, 6.19233722598, 26087.90314157420],
[0.00795525558, 2.95989690104, 52175.80628314840],
[0.00121281764, 6.01064153797, 78263.70942472259],
[0.00021921969, 2.77820093972, 104351.61256629678],
[0.00004354065, 5.82894543774, 130439.51570787099]
],
[
[0.00217347740, 4.65617158665, 26087.90314157420],
[0.00044141826, 1.42385544001, 52175.80628314840]
]
]
],
# Venus
[
[
[
[3.17614666774, 0.00000000000, 0.00000000000],
[0.01353968419, 5.59313319619, 10213.28554621100],
[0.00089891645, 5.30650047764, 20426.57109242200],
[0.00005477194, 4.41630661466, 7860.41939243920],
[0.00003455741, 2.69964447820, 11790.62908865880],
[0.00002372061, 2.99377542079, 3930.20969621960],
[0.00001317168, 5.18668228402, 26.29831979980],
[0.00001664146, 4.25018630147, 1577.34354244780],
[0.00001438387, 4.15745084182, 9683.59458111640],
[0.00001200521, 6.15357116043, 30639.85663863300]
],
[
[10213.28554621638, 0.00000000000, 0.00000000000],
[0.00095617813, 2.46406511110, 10213.28554621100],
[0.00007787201, 0.62478482220, 20426.57109242200]
]
],
[
[
[0.05923638472, 0.26702775812, 10213.28554621100],
[0.00040107978, 1.14737178112, 20426.57109242200],
[0.00032814918, 3.14159265359, 0.00000000000]
],
[
[0.00287821243, 1.88964962838, 10213.28554621100]
]
],
[
[
[0.72334820891, 0.00000000000, 0.00000000000],
[0.00489824182, 4.02151831717, 10213.28554621100],
[0.00001658058, 4.90206728031, 20426.57109242200]
],
[
[0.00034551041, 0.89198706276, 10213.28554621100]
]
]
],
# Earth
[
[
[
[1.75347045673, 0.00000000000, 0.00000000000],
[0.03341656453, 4.66925680415, 6283.07584999140],
[0.00034894275, 4.62610242189, 12566.15169998280],
[0.00003417572, 2.82886579754, 3.52311834900],
[0.00003497056, 2.74411783405, 5753.38488489680],
[0.00003135899, 3.62767041756, 77713.77146812050],
[0.00002676218, 4.41808345438, 7860.41939243920],
[0.00002342691, 6.13516214446, 3930.20969621960],
[0.00001273165, 2.03709657878, 529.69096509460],
[0.00001324294, 0.74246341673, 11506.76976979360],
[0.00000901854, 2.04505446477, 26.29831979980],
[0.00001199167, 1.10962946234, 1577.34354244780],
[0.00000857223, 3.50849152283, 398.14900340820],
[0.00000779786, 1.17882681962, 5223.69391980220],
[0.00000990250, 5.23268072088, 5884.92684658320],
[0.00000753141, 2.53339052847, 5507.55323866740],
[0.00000505267, 4.58292599973, 18849.22754997420],
[0.00000492392, 4.20505711826, 775.52261132400],
[0.00000356672, 2.91954114478, 0.06731030280],
[0.00000284125, 1.89869240932, 796.29800681640],
[0.00000242879, 0.34481445893, 5486.77784317500],
[0.00000317087, 5.84901948512, 11790.62908865880],
[0.00000271112, 0.31486255375, 10977.07880469900],
[0.00000206217, 4.80646631478, 2544.31441988340],
[0.00000205478, 1.86953770281, 5573.14280143310],
[0.00000202318, 2.45767790232, 6069.77675455340],
[0.00000126225, 1.08295459501, 20.77539549240],
[0.00000155516, 0.83306084617, 213.29909543800]
],
[
[6283.07584999140, 0.00000000000, 0.00000000000],
[0.00206058863, 2.67823455808, 6283.07584999140],
[0.00004303419, 2.63512233481, 12566.15169998280]
],
[
[0.00008721859, 1.07253635559, 6283.07584999140]
]
],
[
[
],
[
[0.00227777722, 3.41376620530, 6283.07584999140],
[0.00003805678, 3.37063423795, 12566.15169998280]
]
],
[
[
[1.00013988784, 0.00000000000, 0.00000000000],
[0.01670699632, 3.09846350258, 6283.07584999140],
[0.00013956024, 3.05524609456, 12566.15169998280],
[0.00003083720, 5.19846674381, 77713.77146812050],
[0.00001628463, 1.17387558054, 5753.38488489680],
[0.00001575572, 2.84685214877, 7860.41939243920],
[0.00000924799, 5.45292236722, 11506.76976979360],
[0.00000542439, 4.56409151453, 3930.20969621960],
[0.00000472110, 3.66100022149, 5884.92684658320]
],
[
[0.00103018607, 1.10748968172, 6283.07584999140],
[0.00001721238, 1.06442300386, 12566.15169998280]
],
[
[0.00004359385, 5.78455133808, 6283.07584999140]
]
]
],
# Mars
[
[
[
[6.20347711581, 0.00000000000, 0.00000000000],
[0.18656368093, 5.05037100270, 3340.61242669980],
[0.01108216816, 5.40099836344, 6681.22485339960],
[0.00091798406, 5.75478744667, 10021.83728009940],
[0.00027744987, 5.97049513147, 3.52311834900],
[0.00010610235, 2.93958560338, 2281.23049651060],
[0.00012315897, 0.84956094002, 2810.92146160520],
[0.00008926784, 4.15697846427, 0.01725365220],
[0.00008715691, 6.11005153139, 13362.44970679920],
[0.00006797556, 0.36462229657, 398.14900340820],
[0.00007774872, 3.33968761376, 5621.84292321040],
[0.00003575078, 1.66186505710, 2544.31441988340],
[0.00004161108, 0.22814971327, 2942.46342329160],
[0.00003075252, 0.85696614132, 191.44826611160],
[0.00002628117, 0.64806124465, 3337.08930835080],
[0.00002937546, 6.07893711402, 0.06731030280],
[0.00002389414, 5.03896442664, 796.29800681640],
[0.00002579844, 0.02996736156, 3344.13554504880],
[0.00001528141, 1.14979301996, 6151.53388830500],
[0.00001798806, 0.65634057445, 529.69096509460],
[0.00001264357, 3.62275122593, 5092.15195811580],
[0.00001286228, 3.06796065034, 2146.16541647520],
[0.00001546404, 2.91579701718, 1751.53953141600],
[0.00001024902, 3.69334099279, 8962.45534991020],
[0.00000891566, 0.18293837498, 16703.06213349900],
[0.00000858759, 2.40093811940, 2914.01423582380],
[0.00000832715, 2.46418619474, 3340.59517304760],
[0.00000832720, 4.49495782139, 3340.62968035200],
[0.00000712902, 3.66335473479, 1059.38193018920],
[0.00000748723, 3.82248614017, 155.42039943420],
[0.00000723861, 0.67497311481, 3738.76143010800],
[0.00000635548, 2.92182225127, 8432.76438481560],
[0.00000655162, 0.48864064125, 3127.31333126180],
[0.00000550474, 3.81001042328, 0.98032106820],
[0.00000552750, 4.47479317037, 1748.01641306700],
[0.00000425966, 0.55364317304, 6283.07584999140],
[0.00000415131, 0.49662285038, 213.29909543800],
[0.00000472167, 3.62547124025, 1194.44701022460],
[0.00000306551, 0.38052848348, 6684.74797174860],
[0.00000312141, 0.99853944405, 6677.70173505060],
[0.00000293198, 4.22131299634, 20.77539549240],
[0.00000302375, 4.48618007156, 3532.06069281140],
[0.00000274027, 0.54222167059, 3340.54511639700],
[0.00000281079, 5.88163521788, 1349.86740965880],
[0.00000231183, 1.28242156993, 3870.30339179440],
[0.00000283602, 5.76885434940, 3149.16416058820],
[0.00000236117, 5.75503217933, 3333.49887969900],
[0.00000274033, 0.13372524985, 3340.67973700260],
[0.00000299395, 2.78323740866, 6254.62666252360]
],
[
[3340.61242700512, 0.00000000000, 0.00000000000],
[0.01457554523, 3.60433733236, 3340.61242669980],
[0.00168414711, 3.92318567804, 6681.22485339960],
[0.00020622975, 4.26108844583, 10021.83728009940],
[0.00003452392, 4.73210393190, 3.52311834900],
[0.00002586332, 4.60670058555, 13362.44970679920],
[0.00000841535, 4.45864030426, 2281.23049651060]
],
[
[0.00058152577, 2.04961712429, 3340.61242669980],
[0.00013459579, 2.45738706163, 6681.22485339960]
]
],
[
[
[0.03197134986, 3.76832042431, 3340.61242669980],
[0.00298033234, 4.10616996305, 6681.22485339960],
[0.00289104742, 0.00000000000, 0.00000000000],
[0.00031365539, 4.44651053090, 10021.83728009940],
[0.00003484100, 4.78812549260, 13362.44970679920]
],
[
[0.00217310991, 6.04472194776, 3340.61242669980],
[0.00020976948, 3.14159265359, 0.00000000000],
[0.00012834709, 1.60810667915, 6681.22485339960]
]
],
[
[
[1.53033488271, 0.00000000000, 0.00000000000],
[0.14184953160, 3.47971283528, 3340.61242669980],
[0.00660776362, 3.81783443019, 6681.22485339960],
[0.00046179117, 4.15595316782, 10021.83728009940],
[0.00008109733, 5.55958416318, 2810.92146160520],
[0.00007485318, 1.77239078402, 5621.84292321040],
[0.00005523191, 1.36436303770, 2281.23049651060],
[0.00003825160, 4.49407183687, 13362.44970679920],
[0.00002306537, 0.09081579001, 2544.31441988340],
[0.00001999396, 5.36059617709, 3337.08930835080],
[0.00002484394, 4.92545639920, 2942.46342329160],
[0.00001960195, 4.74249437639, 3344.13554504880],
[0.00001167119, 2.11260868341, 5092.15195811580],
[0.00001102816, 5.00908403998, 398.14900340820],
[0.00000899066, 4.40791133207, 529.69096509460],
[0.00000992252, 5.83861961952, 6151.53388830500],
[0.00000807354, 2.10217065501, 1059.38193018920],
[0.00000797915, 3.44839203899, 796.29800681640],
[0.00000740975, 1.49906336885, 2146.16541647520]
],
[
[0.01107433345, 2.03250524857, 3340.61242669980],
[0.00103175887, 2.37071847807, 6681.22485339960],
[0.00012877200, 0.00000000000, 0.00000000000],
[0.00010815880, 2.70888095665, 10021.83728009940]
],
[
[0.00044242249, 0.47930604954, 3340.61242669980],
[0.00008138042, 0.86998389204, 6681.22485339960]
]
]
],
# Jupiter
[
[
[
[0.59954691494, 0.00000000000, 0.00000000000],
[0.09695898719, 5.06191793158, 529.69096509460],
[0.00573610142, 1.44406205629, 7.11354700080],
[0.00306389205, 5.41734730184, 1059.38193018920],
[0.00097178296, 4.14264726552, 632.78373931320],
[0.00072903078, 3.64042916389, 522.57741809380],
[0.00064263975, 3.41145165351, 103.09277421860],
[0.00039806064, 2.29376740788, 419.48464387520],
[0.00038857767, 1.27231755835, 316.39186965660],
[0.00027964629, 1.78454591820, 536.80451209540],
[0.00013589730, 5.77481040790, 1589.07289528380],
[0.00008246349, 3.58227925840, 206.18554843720],
[0.00008768704, 3.63000308199, 949.17560896980],
[0.00007368042, 5.08101194270, 735.87651353180],
[0.00006263150, 0.02497628807, 213.29909543800],
[0.00006114062, 4.51319998626, 1162.47470440780],
[0.00004905396, 1.32084470588, 110.20632121940],
[0.00005305285, 1.30671216791, 14.22709400160],
[0.00005305441, 4.18625634012, 1052.26838318840],
[0.00004647248, 4.69958103684, 3.93215326310],
[0.00003045023, 4.31676431084, 426.59819087600],
[0.00002609999, 1.56667394063, 846.08283475120],
[0.00002028191, 1.06376530715, 3.18139373770],
[0.00001764763, 2.14148655117, 1066.49547719000],
[0.00001722972, 3.88036268267, 1265.56747862640],
[0.00001920945, 0.97168196472, 639.89728631400],
[0.00001633223, 3.58201833555, 515.46387109300],
[0.00001431999, 4.29685556046, 625.67019231240],
[0.00000973272, 4.09764549134, 95.97922721780]
],
[
[529.69096508814, 0.00000000000, 0.00000000000],
[0.00489503243, 4.22082939470, 529.69096509460],
[0.00228917222, 6.02646855621, 7.11354700080],
[0.00030099479, 4.54540782858, 1059.38193018920],
[0.00020720920, 5.45943156902, 522.57741809380],
[0.00012103653, 0.16994816098, 536.80451209540],
[0.00006067987, 4.42422292017, 103.09277421860],
[0.00005433968, 3.98480737746, 419.48464387520],
[0.00004237744, 5.89008707199, 14.22709400160]
],
[
[0.00047233601, 4.32148536482, 7.11354700080],
[0.00030649436, 2.92977788700, 529.69096509460],
[0.00014837605, 3.14159265359, 0.00000000000]
]
],
[
[
[0.02268615702, 3.55852606721, 529.69096509460],
[0.00109971634, 3.90809347197, 1059.38193018920],
[0.00110090358, 0.00000000000, 0.00000000000],
[0.00008101428, 3.60509572885, 522.57741809380],
[0.00006043996, 4.25883108339, 1589.07289528380],
[0.00006437782, 0.30627119215, 536.80451209540]
],
[
[0.00078203446, 1.52377859742, 529.69096509460]
]
],
[
[
[5.20887429326, 0.00000000000, 0.00000000000],
[0.25209327119, 3.49108639871, 529.69096509460],
[0.00610599976, 3.84115365948, 1059.38193018920],
[0.00282029458, 2.57419881293, 632.78373931320],
[0.00187647346, 2.07590383214, 522.57741809380],
[0.00086792905, 0.71001145545, 419.48464387520],
[0.00072062974, 0.21465724607, 536.80451209540],
[0.00065517248, 5.97995884790, 316.39186965660],
[0.00029134542, 1.67759379655, 103.09277421860],
[0.00030135335, 2.16132003734, 949.17560896980],
[0.00023453271, 3.54023522184, 735.87651353180],
[0.00022283743, 4.19362594399, 1589.07289528380],
[0.00023947298, 0.27458037480, 7.11354700080],
[0.00013032614, 2.96042965363, 1162.47470440780],
[0.00009703360, 1.90669633585, 206.18554843720],
[0.00012749023, 2.71550286592, 1052.26838318840]
],
[
[0.01271801520, 2.64937512894, 529.69096509460],
[0.00061661816, 3.00076460387, 1059.38193018920],
[0.00053443713, 3.89717383175, 522.57741809380],
[0.00031185171, 4.88276958012, 536.80451209540],
[0.00041390269, 0.00000000000, 0.00000000000]
]
]
],
# Saturn
[
[
[
[0.87401354025, 0.00000000000, 0.00000000000],
[0.11107659762, 3.96205090159, 213.29909543800],
[0.01414150957, 4.58581516874, 7.11354700080],
[0.00398379389, 0.52112032699, 206.18554843720],
[0.00350769243, 3.30329907896, 426.59819087600],
[0.00206816305, 0.24658372002, 103.09277421860],
[0.00079271300, 3.84007056878, 220.41264243880],
[0.00023990355, 4.66976924553, 110.20632121940],
[0.00016573588, 0.43719228296, 419.48464387520],
[0.00014906995, 5.76903183869, 316.39186965660],
[0.00015820290, 0.93809155235, 632.78373931320],
[0.00014609559, 1.56518472000, 3.93215326310],
[0.00013160301, 4.44891291899, 14.22709400160],
[0.00015053543, 2.71669915667, 639.89728631400],
[0.00013005299, 5.98119023644, 11.04570026390],
[0.00010725067, 3.12939523827, 202.25339517410],
[0.00005863206, 0.23656938524, 529.69096509460],
[0.00005227757, 4.20783365759, 3.18139373770],
[0.00006126317, 1.76328667907, 277.03499374140],
[0.00005019687, 3.17787728405, 433.71173787680],
[0.00004592550, 0.61977744975, 199.07200143640],
[0.00004005867, 2.24479718502, 63.73589830340],
[0.00002953796, 0.98280366998, 95.97922721780],
[0.00003873670, 3.22283226966, 138.51749687070],
[0.00002461186, 2.03163875071, 735.87651353180],
[0.00003269484, 0.77492638211, 949.17560896980],
[0.00001758145, 3.26580109940, 522.57741809380],
[0.00001640172, 5.50504453050, 846.08283475120],
[0.00001391327, 4.02333150505, 323.50541665740],
[0.00001580648, 4.37265307169, 309.27832265580],
[0.00001123498, 2.83726798446, 415.55249061210],
[0.00001017275, 3.71700135395, 227.52618943960],
[0.00000848642, 3.19150170830, 209.36694217490]
],
[
[213.29909521690, 0.00000000000, 0.00000000000],
[0.01297370862, 1.82834923978, 213.29909543800],
[0.00564345393, 2.88499717272, 7.11354700080],
[0.00093734369, 1.06311793502, 426.59819087600],
[0.00107674962, 2.27769131009, 206.18554843720],
[0.00040244455, 2.04108104671, 220.41264243880],
[0.00019941774, 1.27954390470, 103.09277421860],
[0.00010511678, 2.74880342130, 14.22709400160],
[0.00006416106, 0.38238295041, 639.89728631400],
[0.00004848994, 2.43037610229, 419.48464387520],
[0.00004056892, 2.92133209468, 110.20632121940],
[0.00003768635, 3.64965330780, 3.93215326310]
],
[
[0.00116441330, 1.17988132879, 7.11354700080],
[0.00091841837, 0.07325195840, 213.29909543800],
[0.00036661728, 0.00000000000, 0.00000000000],
[0.00015274496, 4.06493179167, 206.18554843720]
]
],
[
[
[0.04330678039, 3.60284428399, 213.29909543800],
[0.00240348302, 2.85238489373, 426.59819087600],
[0.00084745939, 0.00000000000, 0.00000000000],
[0.00030863357, 3.48441504555, 220.41264243880],
[0.00034116062, 0.57297307557, 206.18554843720],
[0.00014734070, 2.11846596715, 639.89728631400],
[0.00009916667, 5.79003188904, 419.48464387520],
[0.00006993564, 4.73604689720, 7.11354700080],
[0.00004807588, 5.43305312061, 316.39186965660]
],
[
[0.00198927992, 4.93901017903, 213.29909543800],
[0.00036947916, 3.14159265359, 0.00000000000],
[0.00017966989, 0.51979431110, 426.59819087600]
]
],
[
[
[9.55758135486, 0.00000000000, 0.00000000000],
[0.52921382865, 2.39226219573, 213.29909543800],
[0.01873679867, 5.23549604660, 206.18554843720],
[0.01464663929, 1.64763042902, 426.59819087600],
[0.00821891141, 5.93520042303, 316.39186965660],
[0.00547506923, 5.01532618980, 103.09277421860],
[0.00371684650, 2.27114821115, 220.41264243880],
[0.00361778765, 3.13904301847, 7.11354700080],
[0.00140617506, 5.70406606781, 632.78373931320],
[0.00108974848, 3.29313390175, 110.20632121940],
[0.00069006962, 5.94099540992, 419.48464387520],
[0.00061053367, 0.94037691801, 639.89728631400],
[0.00048913294, 1.55733638681, 202.25339517410],
[0.00034143772, 0.19519102597, 277.03499374140],
[0.00032401773, 5.47084567016, 949.17560896980],
[0.00020936596, 0.46349251129, 735.87651353180]
],
[
[0.06182981340, 0.25843511480, 213.29909543800],
[0.00506577242, 0.71114625261, 206.18554843720],
[0.00341394029, 5.79635741658, 426.59819087600],
[0.00188491195, 0.47215589652, 220.41264243880],
[0.00186261486, 3.14159265359, 0.00000000000],
[0.00143891146, 1.40744822888, 7.11354700080]
],
[
[0.00436902572, 4.78671677509, 213.29909543800]
]
]
],
# Uranus
[
[
[
[5.48129294297, 0.00000000000, 0.00000000000],
[0.09260408234, 0.89106421507, 74.78159856730],
[0.01504247898, 3.62719260920, 1.48447270830],
[0.00365981674, 1.89962179044, 73.29712585900],
[0.00272328168, 3.35823706307, 149.56319713460],
[0.00070328461, 5.39254450063, 63.73589830340],
[0.00068892678, 6.09292483287, 76.26607127560],
[0.00061998615, 2.26952066061, 2.96894541660],
[0.00061950719, 2.85098872691, 11.04570026390],
[0.00026468770, 3.14152083966, 71.81265315070],
[0.00025710476, 6.11379840493, 454.90936652730],
[0.00021078850, 4.36059339067, 148.07872442630],
[0.00017818647, 1.74436930289, 36.64856292950],
[0.00014613507, 4.73732166022, 3.93215326310],
[0.00011162509, 5.82681796350, 224.34479570190],
[0.00010997910, 0.48865004018, 138.51749687070],
[0.00009527478, 2.95516862826, 35.16409022120],
[0.00007545601, 5.23626582400, 109.94568878850],
[0.00004220241, 3.23328220918, 70.84944530420],
[0.00004051900, 2.27755017300, 151.04766984290],
[0.00003354596, 1.06549007380, 4.45341812490],
[0.00002926718, 4.62903718891, 9.56122755560],
[0.00003490340, 5.48306144511, 146.59425171800],
[0.00003144069, 4.75199570434, 77.75054398390],
[0.00002922333, 5.35235361027, 85.82729883120],
[0.00002272788, 4.36600400036, 70.32818044240],
[0.00002051219, 1.51773566586, 0.11187458460],
[0.00002148602, 0.60745949945, 38.13303563780],
[0.00001991643, 4.92437588682, 277.03499374140],
[0.00001376226, 2.04283539351, 65.22037101170],
[0.00001666902, 3.62744066769, 380.12776796000],
[0.00001284107, 3.11347961505, 202.25339517410],
[0.00001150429, 0.93343589092, 3.18139373770],
[0.00001533221, 2.58594681212, 52.69019803950],
[0.00001281604, 0.54271272721, 222.86032299360],
[0.00001372139, 4.19641530878, 111.43016149680],
[0.00001221029, 0.19900650030, 108.46121608020],
[0.00000946181, 1.19253165736, 127.47179660680],
[0.00001150989, 4.17898916639, 33.67961751290]
],
[
[74.78159860910, 0.00000000000, 0.00000000000],
[0.00154332863, 5.24158770553, 74.78159856730],
[0.00024456474, 1.71260334156, 1.48447270830],
[0.00009258442, 0.42829732350, 11.04570026390],
[0.00008265977, 1.50218091379, 63.73589830340],
[0.00009150160, 1.41213765216, 149.56319713460]
]
],
[
[
[0.01346277648, 2.61877810547, 74.78159856730],
[0.00062341400, 5.08111189648, 149.56319713460],
[0.00061601196, 3.14159265359, 0.00000000000],
[0.00009963722, 1.61603805646, 76.26607127560],
[0.00009926160, 0.57630380333, 73.29712585900]
],
[
[0.00034101978, 0.01321929936, 74.78159856730]
]
],
[
[
[19.21264847206, 0.00000000000, 0.00000000000],
[0.88784984413, 5.60377527014, 74.78159856730],
[0.03440836062, 0.32836099706, 73.29712585900],
[0.02055653860, 1.78295159330, 149.56319713460],
[0.00649322410, 4.52247285911, 76.26607127560],
[0.00602247865, 3.86003823674, 63.73589830340],
[0.00496404167, 1.40139935333, 454.90936652730],
[0.00338525369, 1.58002770318, 138.51749687070],
[0.00243509114, 1.57086606044, 71.81265315070],
[0.00190522303, 1.99809394714, 1.48447270830],
[0.00161858838, 2.79137786799, 148.07872442630],
[0.00143706183, 1.38368544947, 11.04570026390],
[0.00093192405, 0.17437220467, 36.64856292950],
[0.00071424548, 4.24509236074, 224.34479570190],
[0.00089806014, 3.66105364565, 109.94568878850],
[0.00039009723, 1.66971401684, 70.84944530420],
[0.00046677296, 1.39976401694, 35.16409022120],
[0.00039025624, 3.36234773834, 277.03499374140],
[0.00036755274, 3.88649278513, 146.59425171800],
[0.00030348723, 0.70100838798, 151.04766984290],
[0.00029156413, 3.18056336700, 77.75054398390]
],
[
[0.01479896629, 3.67205697578, 74.78159856730]
]
]
],
# Neptune
[
[
[
[5.31188633046, 0.00000000000, 0.00000000000],
[0.01798475530, 2.90101273890, 38.13303563780],
[0.01019727652, 0.48580922867, 1.48447270830],
[0.00124531845, 4.83008090676, 36.64856292950],
[0.00042064466, 5.41054993053, 2.96894541660],
[0.00037714584, 6.09221808686, 35.16409022120],
[0.00033784738, 1.24488874087, 76.26607127560],
[0.00016482741, 0.00007727998, 491.55792945680],
[0.00009198584, 4.93747051954, 39.61750834610],
[0.00008994250, 0.27462171806, 175.16605980020]
],
[
[38.13303563957, 0.00000000000, 0.00000000000],
[0.00016604172, 4.86323329249, 1.48447270830],
[0.00015744045, 2.27887427527, 38.13303563780]
]
],
[
[
[0.03088622933, 1.44104372644, 38.13303563780],
[0.00027780087, 5.91271884599, 76.26607127560],
[0.00027623609, 0.00000000000, 0.00000000000],
[0.00015355489, 2.52123799551, 36.64856292950],
[0.00015448133, 3.50877079215, 39.61750834610]
]
],
[
[
[30.07013205828, 0.00000000000, 0.00000000000],
[0.27062259632, 1.32999459377, 38.13303563780],
[0.01691764014, 3.25186135653, 36.64856292950],
[0.00807830553, 5.18592878704, 1.48447270830],
[0.00537760510, 4.52113935896, 35.16409022120],
[0.00495725141, 1.57105641650, 491.55792945680],
[0.00274571975, 1.84552258866, 175.16605980020]
]
]
],
]
def _CalcVsop(model, time):
spher = []
t = time.tt / 365250.0
for formula in model:
tpower = 1.0
coord = 0.0
for series in formula:
coord += tpower * sum(A * math.cos(B + C*t) for (A, B, C) in series)
tpower *= t
spher.append(coord)
# Convert spherical coordinates to ecliptic cartesian coordinates.
r_coslat = spher[2] * math.cos(spher[1])
ex = r_coslat * math.cos(spher[0])
ey = r_coslat * math.sin(spher[0])
ez = spher[2] * math.sin(spher[1])
# Convert ecliptic cartesian coordinates to equatorial cartesian coordinates.
vx = ex + 0.000000440360*ey - 0.000000190919*ez
vy = -0.000000479966*ex + 0.917482137087*ey - 0.397776982902*ez
vz = 0.397776982902*ey + 0.917482137087*ez
return Vector(vx, vy, vz, time)
def _CalcEarth(time):
return _CalcVsop(_vsop[Body.Earth], time)
# END VSOP
#----------------------------------------------------------------------------
# BEGIN CHEBYSHEV
_pluto = [
{ 'tt':-109573.500000, 'ndays':26141.000000, 'coeff':[
[-30.303124711144, -18.980368465705, 3.206649343866],
[20.092745278347, -27.533908687219, -14.641121965990],
[9.137264744925, 6.513103657467, -0.720732357468],
[-1.201554708717, 2.149917852301, 1.032022293526],
[-0.566068170022, -0.285737361191, 0.081379987808],
[0.041678527795, -0.143363105040, -0.057534475984],
[0.041087908142, 0.007911321580, -0.010270655537],
[0.001611769878, 0.011409821837, 0.003679980733],
[-0.002536458296, -0.000145632543, 0.000949924030],
[0.001167651969, -0.000049912680, 0.000115867710],
[-0.000196953286, 0.000420406270, 0.000110147171],
[0.001073825784, 0.000442658285, 0.000146985332],
[-0.000906160087, 0.001702360394, 0.000758987924],
[-0.001467464335, -0.000622191266, -0.000231866243],
[-0.000008986691, 0.000004086384, 0.000001442956],
[-0.001099078039, -0.000544633529, -0.000205534708],
[0.001259974751, -0.002178533187, -0.000965315934],
[0.001695288316, 0.000768480768, 0.000287916141],
[-0.001428026702, 0.002707551594, 0.001195955756]]
},
{ 'tt':-83432.500000, 'ndays':26141.000000, 'coeff':[
[67.049456204563, -9.279626603192, -23.091941092128],
[14.860676672314, 26.594121136143, 3.819668867047],
[-6.254409044120, 1.408757903538, 2.323726101433],
[0.114416381092, -0.942273228585, -0.328566335886],
[0.074973631246, 0.106749156044, 0.010806547171],
[-0.018627741964, -0.009983491157, 0.002589955906],
[0.006167206174, -0.001042430439, -0.001521881831],
[-0.000471293617, 0.002337935239, 0.001060879763],
[-0.000240627462, -0.001380351742, -0.000546042590],
[0.001872140444, 0.000679876620, 0.000240384842],
[-0.000334705177, 0.000693528330, 0.000301138309],
[0.000796124758, 0.000653183163, 0.000259527079],
[-0.001276116664, 0.001393959948, 0.000629574865],
[-0.001235158458, -0.000889985319, -0.000351392687],
[-0.000019881944, 0.000048339979, 0.000021342186],
[-0.000987113745, -0.000748420747, -0.000296503569],
[0.001721891782, -0.001893675502, -0.000854270937],
[0.001505145187, 0.001081653337, 0.000426723640],
[-0.002019479384, 0.002375617497, 0.001068258925]]
},
{ 'tt':-57291.500000, 'ndays':26141.000000, 'coeff':[
[46.038290912405, 73.773759757856, 9.148670950706],
[-22.354364534703, 10.217143138926, 9.921247676076],
[-2.696282001399, -4.440843715929, -0.572373037840],
[0.385475818800, -0.287872688575, -0.205914693555],
[0.020994433095, 0.004256602589, -0.004817361041],
[0.003212255378, 0.000574875698, -0.000764464370],
[-0.000158619286, -0.001035559544, -0.000535612316],
[0.000967952107, -0.000653111849, -0.000292019750],
[0.001763494906, -0.000370815938, -0.000224698363],
[0.001157990330, 0.001849810828, 0.000759641577],
[-0.000883535516, 0.000384038162, 0.000191242192],
[0.000709486562, 0.000655810827, 0.000265431131],
[-0.001525810419, 0.001126870468, 0.000520202001],
[-0.000983210860, -0.001116073455, -0.000456026382],
[-0.000015655450, 0.000069184008, 0.000029796623],
[-0.000815102021, -0.000900597010, -0.000365274209],
[0.002090300438, -0.001536778673, -0.000709827438],
[0.001234661297, 0.001342978436, 0.000545313112],
[-0.002517963678, 0.001941826791, 0.000893859860]]
},
{ 'tt':-31150.500000, 'ndays':26141.000000, 'coeff':[
[-39.074661990988, 30.963513412373, 21.431709298065],
[-12.033639281924, -31.693679132310, -6.263961539568],
[7.233936758611, -3.979157072767, -3.421027935569],
[1.383182539917, 1.090729793400, -0.076771771448],
[-0.009894394996, 0.313614402007, 0.101180677344],
[-0.055459383449, 0.031782406403, 0.026374448864],
[-0.011074105991, -0.007176759494, 0.001896208351],
[-0.000263363398, -0.001145329444, 0.000215471838],
[0.000405700185, -0.000839229891, -0.000418571366],
[0.001004921401, 0.001135118493, 0.000406734549],
[-0.000473938695, 0.000282751002, 0.000114911593],
[0.000528685886, 0.000966635293, 0.000401955197],
[-0.001838869845, 0.000806432189, 0.000394594478],
[-0.000713122169, -0.001334810971, -0.000554511235],
[0.000006449359, 0.000060730000, 0.000024513230],
[-0.000596025142, -0.000999492770, -0.000413930406],
[0.002364904429, -0.001099236865, -0.000528480902],
[0.000907458104, 0.001537243912, 0.000637001965],
[-0.002909908764, 0.001413648354, 0.000677030924]]
},
{ 'tt':-5009.500000, 'ndays':26141.000000, 'coeff':[
[23.380075041204, -38.969338804442, -19.204762094135],
[33.437140696536, 8.735194448531, -7.348352917314],
[-3.127251304544, 8.324311848708, 3.540122328502],
[-1.491354030154, -1.350371407475, 0.028214278544],
[0.361398480996, -0.118420687058, -0.145375605480],
[-0.011771350229, 0.085880588309, 0.030665997197],
[-0.015839541688, -0.014165128211, 0.000523465951],
[0.004213218926, -0.001426373728, -0.001906412496],
[0.001465150002, 0.000451513538, 0.000081936194],
[0.000640069511, 0.001886692235, 0.000884675556],
[-0.000883554940, 0.000301907356, 0.000127310183],
[0.000245524038, 0.000910362686, 0.000385555148],
[-0.001942010476, 0.000438682280, 0.000237124027],
[-0.000425455660, -0.001442138768, -0.000607751390],
[0.000004168433, 0.000033856562, 0.000013881811],
[-0.000337920193, -0.001074290356, -0.000452503056],
[0.002544755354, -0.000620356219, -0.000327246228],
[0.000534534110, 0.001670320887, 0.000702775941],
[-0.003169380270, 0.000816186705, 0.000427213817]]
},
{ 'tt':21131.500000, 'ndays':26141.000000, 'coeff':[
[74.130449310804, 43.372111541004, -8.799489207171],
[-8.705941488523, 23.344631690845, 9.908006472122],
[-4.614752911564, -2.587334376729, 0.583321715294],
[0.316219286624, -0.395448970181, -0.219217574801],
[0.004593734664, 0.027528474371, 0.007736197280],
[-0.001192268851, -0.004987723997, -0.001599399192],
[0.003051998429, -0.001287028653, -0.000780744058],
[0.001482572043, 0.001613554244, 0.000635747068],
[0.000581965277, 0.000788286674, 0.000315285159],
[-0.000311830730, 0.001622369930, 0.000714817617],
[-0.000711275723, -0.000160014561, -0.000050445901],
[0.000177159088, 0.001032713853, 0.000435835541],
[-0.002032280820, 0.000144281331, 0.000111910344],
[-0.000148463759, -0.001495212309, -0.000635892081],
[-0.000009629403, -0.000013678407, -0.000006187457],
[-0.000061196084, -0.001119783520, -0.000479221572],
[0.002630993795, -0.000113042927, -0.000112115452],
[0.000132867113, 0.001741417484, 0.000743224630],
[-0.003293498893, 0.000182437998, 0.000158073228]]
},
{ 'tt':47272.500000, 'ndays':26141.000000, 'coeff':[
[-5.727994625506, 71.194823351703, 23.946198176031],
[-26.767323214686, -12.264949302780, 4.238297122007],
[0.890596204250, -5.970227904551, -2.131444078785],
[0.808383708156, -0.143104108476, -0.288102517987],
[0.089303327519, 0.049290470655, -0.010970501667],
[0.010197195705, 0.012879721400, 0.001317586740],
[0.001795282629, 0.004482403780, 0.001563326157],
[-0.001974716105, 0.001278073933, 0.000652735133],
[0.000906544715, -0.000805502229, -0.000336200833],
[0.000283816745, 0.001799099064, 0.000756827653],
[-0.000784971304, 0.000123081220, 0.000068812133],
[-0.000237033406, 0.000980100466, 0.000427758498],
[-0.001976846386, -0.000280421081, -0.000072417045],
[0.000195628511, -0.001446079585, -0.000624011074],
[-0.000044622337, -0.000035865046, -0.000013581236],
[0.000204397832, -0.001127474894, -0.000488668673],
[0.002625373003, 0.000389300123, 0.000102756139],
[-0.000277321614, 0.001732818354, 0.000749576471],
[-0.003280537764, -0.000457571669, -0.000116383655]]
}]
def _ChebScale(t_min, t_max, t):
return (2*t - (t_max + t_min)) / (t_max - t_min)
def _CalcChebyshev(model, time):
# Search for a record that overlaps the given time value.
for record in model:
x = _ChebScale(record['tt'], record['tt'] + record['ndays'], time.tt)
if -1 <= x <= +1:
coeff = record['coeff']
pos = []
for d in range(3):
p0 = 1
sum = coeff[0][d]
p1 = x
sum += coeff[1][d] * p1
for k in range(2, len(coeff)):
p2 = (2 * x * p1) - p0
sum += coeff[k][d] * p2
p0 = p1
p1 = p2
pos.append(sum - coeff[0][d]/2)
return Vector(pos[0], pos[1], pos[2], time)
raise Error('Cannot extrapolate Chebyshev model for given Terrestrial Time: {}'.format(time.tt))
# END CHEBYSHEV
#----------------------------------------------------------------------------
# BEGIN Search
def _QuadInterp(tm, dt, fa, fm, fb):
Q = (fb + fa)/2 - fm
R = (fb - fa)/2
S = fm
if Q == 0:
# This is a line, not a parabola.
if R == 0:
# This is a HORIZONTAL line... can't make progress!
return None
x = -S / R
if not (-1 <= x <= +1):
return None # out of bounds
else:
# It really is a parabola. Find roots x1, x2.
u = R*R - 4*Q*S
if u <= 0:
return None
ru = math.sqrt(u)
x1 = (-R + ru) / (2 * Q)
x2 = (-R - ru) / (2 * Q)
if -1 <= x1 <= +1:
if -1 <= x2 <= +1:
# Two solutions... so parabola intersects twice.
return None
x = x1
elif -1 <= x2 <= +1:
x = x2
else:
return None
t = tm + x*dt
df_dt = (2*Q*x + R) / dt
return (x, t, df_dt)
def Search(func, context, t1, t2, dt_tolerance_seconds):
dt_days = abs(dt_tolerance_seconds / _SECONDS_PER_DAY)
f1 = func(context, t1)
f2 = func(context, t2)
iter = 0
iter_limit = 20
calc_fmid = True
while True:
iter += 1
if iter > iter_limit:
raise Error('Excessive iteration in Search')
dt = (t2.tt - t1.tt) / 2.0
tmid = t1.AddDays(dt)
if abs(dt) < dt_days:
# We are close enough to the event to stop the search.
return tmid
if calc_fmid:
fmid = func(context, tmid)
else:
# We already have the correct value of fmid from the previous loop.
calc_fmid = True
# Quadratic interpolation:
# Try to find a parabola that passes through the 3 points we have sampled:
# (t1,f1), (tmid,fmid), (t2,f2).
q = _QuadInterp(tmid.ut, t2.ut - tmid.ut, f1, fmid, f2)
if q:
(q_x, q_ut, q_df_dt) = q
tq = Time(q_ut)
fq = func(context, tq)
if q_df_dt != 0.0:
dt_guess = abs(fq / q_df_dt)
if dt_guess < dt_days:
# The estimated time error is small enough that we can quit now.
return tq
# Try guessing a tighter boundary with the interpolated root at the center.
dt_guess *= 1.2
if dt_guess < dt / 10.0:
tleft = tq.AddDays(-dt_guess)
tright = tq.AddDays(+dt_guess)
if (tleft.ut - t1.ut)*(tleft.ut - t2.ut) < 0.0:
if (tright.ut - t1.ut)*(tright.ut - t2.ut) < 0.0:
fleft = func(context, tleft)
fright = func(context, tright)
if fleft < 0.0 and fright >= 0.0:
f1 = fleft
f2 = fright
t1 = tleft
t2 = tright
fmid = fq
calc_fmid = False
continue
# Quadratic interpolation attempt did not work out.
# Just divide the region in two parts and pick whichever one appears to contain a root.
if f1 < 0.0 and fmid >= 0.0:
t2 = tmid
f2 = fmid
continue
if fmid < 0.0 and f2 >= 0.0:
t1 = tmid
f1 = fmid
continue
# Either there is no ascending zero-crossing in this range
# or the search window is too wide (more than one zero-crossing).
return None
# END Search
#----------------------------------------------------------------------------
def HelioVector(body, time):
if body == Body.Pluto:
return _CalcChebyshev(_pluto, time)
if 0 <= body <= len(_vsop):
return _CalcVsop(_vsop[body], time)
if body == Body.Sun:
return Vector(0.0, 0.0, 0.0, time)
if body == Body.Moon:
e = _CalcEarth(time)
m = GeoMoon(time)
return Vector(e.x+m.x, e.y+m.y, e.z+m.z, time)
raise InvalidBodyError()
def GeoVector(body, time, aberration):
if body == Body.Moon:
return GeoMoon(time)
if body == Body.Earth:
return Vector(0.0, 0.0, 0.0, time)
if not aberration:
# No aberration, so calculate Earth's position once, at the time of observation.
earth = _CalcEarth(time)
# Correct for light-travel time, to get position of body as seen from Earth's center.
ltime = time
for iter in range(10):
h = HelioVector(body, ltime)
if aberration:
# Include aberration, so make a good first-order approximation
# by backdating the Earth's position also.
# This is confusing, but it works for objects within the Solar System
# because the distance the Earth moves in that small amount of light
# travel time (a few minutes to a few hours) is well approximated
# by a line segment that substends the angle seen from the remote
# body viewing Earth. That angle is pretty close to the aberration
# angle of the moving Earth viewing the remote body.
# In other words, both of the following approximate the aberration angle:
# (transverse distance Earth moves) / (distance to body)
# (transverse speed of Earth) / (speed of light).
earth = _CalcEarth(ltime)
geo = Vector(h.x-earth.x, h.y-earth.y, h.z-earth.z, time)
if body == Body.Sun:
# The Sun's heliocentric coordinates are always (0,0,0). No need to correct.
return geo
ltime2 = time.AddDays(-geo.Length() / _C_AUDAY)
dt = abs(ltime2.tt - ltime.tt)
if dt < 1.0e-9:
return geo
ltime = ltime2
raise Error('Light-travel time solver did not converge: dt={}'.format(dt))
def Equator(body, time, observer, ofdate, aberration):
gc_observer = _geo_pos(time, observer)
gc = GeoVector(body, time, aberration)
j2000 = [
gc.x - gc_observer[0],
gc.y - gc_observer[1],
gc.z - gc_observer[2]
]
if not ofdate:
return _vector2radec(j2000)
temp = _precession(0, j2000, time.tt)
datevect = _nutation(time, 0, temp)
return _vector2radec(datevect)
@enum.unique
class Refraction(enum.IntEnum):
"""Selects if/how to correct for atmospheric refraction.
Some functions allow enabling or disabling atmospheric refraction
for the calculated apparent position of a celestial body
as seen by an observer on the surface of the Earth.
Values
------
Airless: No atmospheric refraction correction.
Normal: Recommended correction for standard atmospheric refraction.
JplHorizons: Used only for compatibility testing with JPL Horizons online tool.
"""
Airless = 0
Normal = 1
JplHorizons = 2
class HorizontalCoordinates:
def __init__(self, azimuth, altitude, ra, dec):
self.azimuth = azimuth
self.altitude = altitude
self.ra = ra
self.dec = dec
def Horizon(time, observer, ra, dec, refraction):
if not (Refraction.Airless <= refraction <= Refraction.JplHorizons):
raise Error('Invalid refraction type: ' + str(refraction))
latrad = math.radians(observer.latitude)
lonrad = math.radians(observer.longitude)
decrad = math.radians(dec)
rarad = math.radians(ra * 15.0)
sinlat = math.sin(latrad)
coslat = math.cos(latrad)
sinlon = math.sin(lonrad)
coslon = math.cos(lonrad)
sindc = math.sin(decrad)
cosdc = math.cos(decrad)
sinra = math.sin(rarad)
cosra = math.cos(rarad)
uze = [coslat*coslon, coslat*sinlon, sinlat]
une = [-sinlat*coslon, -sinlat*sinlon, coslat]
uwe = [sinlon, -coslon, 0.0]
angle = -15.0 * _sidereal_time(time)
uz = _spin(angle, uze)
un = _spin(angle, une)
uw = _spin(angle, uwe)
p = [cosdc*cosra, cosdc*sinra, sindc]
pz = p[0]*uz[0] + p[1]*uz[1] + p[2]*uz[2]
pn = p[0]*un[0] + p[1]*un[1] + p[2]*un[2]
pw = p[0]*uw[0] + p[1]*uw[1] + p[2]*uw[2]
proj = math.sqrt(pn*pn + pw*pw)
az = 0.0
if proj > 0.0:
az = math.degrees(-math.atan2(pw, pn))
if az < 0:
az += 360
if az >= 360:
az -= 360
zd = math.degrees(math.atan2(proj, pz))
hor_ra = ra
hor_dec = dec
if refraction != Refraction.Airless:
zd0 = zd
# http://extras.springer.com/1999/978-1-4471-0555-8/chap4/horizons/horizons.pdf
# JPL Horizons says it uses refraction algorithm from
# Meeus "Astronomical Algorithms", 1991, p. 101-102.
# I found the following Go implementation:
# https://github.com/soniakeys/meeus/blob/master/v3/refraction/refract.go
# This is a translation from the function "Saemundsson" there.
# I found experimentally that JPL Horizons clamps the angle to 1 degree below the horizon.
# This is important because the 'refr' formula below goes crazy near hd = -5.11.
hd = 90.0 - zd
if hd < -1.0:
hd = -1.0
refr = (1.02 / math.tan(math.radians(hd+10.3/(hd+5.11)))) / 60.0
if refraction == Refraction.Normal and zd > 91.0:
# In "normal" mode we gradually reduce refraction toward the nadir
# so that we never get an altitude angle less than -90 degrees.
# When horizon angle is -1 degrees, zd = 91, and the factor is exactly 1.
# As zd approaches 180 (the nadir), the fraction approaches 0 linearly.
refr *= (180.0 - zd) / 89.0
zd -= refr
if refr > 0.0 and zd > 3.0e-4:
zdrad = math.radians(zd)
sinzd = math.sin(zdrad)
coszd = math.cos(zdrad)
zd0rad = math.radians(zd0)
sinzd0 = math.sin(zd0rad)
coszd0 = math.cos(zd0rad)
pr = [(((p[j] - coszd0 * uz[j]) / sinzd0)*sinzd + uz[j]*coszd) for j in range(3)]
proj = math.sqrt(pr[0]*pr[0] + pr[1]*pr[1])
if proj > 0:
hor_ra = math.degrees(math.atan2(pr[1], pr[0])) / 15
if hor_ra < 0:
hor_ra += 24
if hor_ra >= 24:
hor_ra -= 24
else:
hor_ra = 0
hor_dec = math.degrees(math.atan2(pr[2], proj))
return HorizontalCoordinates(az, 90.0 - zd, hor_ra, hor_dec)
class EclipticCoordinates:
def __init__(self, ex, ey, ez, elat, elon):
self.ex = ex
self.ey = ey
self.ez = ez
self.elat = elat
self.elon = elon
def _RotateEquatorialToEcliptic(pos, obliq_radians):
cos_ob = math.cos(obliq_radians)
sin_ob = math.sin(obliq_radians)
ex = +pos[0]
ey = +pos[1]*cos_ob + pos[2]*sin_ob
ez = -pos[1]*sin_ob + pos[2]*cos_ob
xyproj = math.sqrt(ex*ex + ey*ey)
if xyproj > 0.0:
elon = math.degrees(math.atan2(ey, ex))
if elon < 0.0:
elon += 360.0
else:
elon = 0.0
elat = math.degrees(math.atan2(ez, xyproj))
return EclipticCoordinates(ex, ey, ez, elat, elon)
def SunPosition(time):
# Correct for light travel time from the Sun.
# Otherwise season calculations (equinox, solstice) will all be early by about 8 minutes!
adjusted_time = time.AddDays(-1.0 / _C_AUDAY)
earth2000 = _CalcEarth(adjusted_time)
sun2000 = [-earth2000.x, -earth2000.y, -earth2000.z]
# Convert to equatorial Cartesian coordinates of date.
stemp = _precession(0.0, sun2000, adjusted_time.tt)
sun_ofdate = _nutation(adjusted_time, 0, stemp)
# Convert equatorial coordinates to ecliptic coordinates.
true_obliq = math.radians(adjusted_time._etilt().tobl)
return _RotateEquatorialToEcliptic(sun_ofdate, true_obliq)
def Ecliptic(equ):
# Based on NOVAS functions equ2ecl() and equ2ecl_vec().
ob2000 = 0.40909260059599012 # mean obliquity of the J2000 ecliptic in radians
return _RotateEquatorialToEcliptic([equ.x, equ.y, equ.z], ob2000)
def EclipticLongitude(body, time):
if body == Body.Sun:
raise InvalidBodyError()
hv = HelioVector(body, time)
eclip = Ecliptic(hv)
return eclip.elon
def AngleFromSun(body, time):
if body == Body.Earth:
raise EarthNotAllowedError()
sv = GeoVector(Body.Sun, time, True)
bv = GeoVector(body, time, True)
return _AngleBetween(sv, bv)
def LongitudeFromSun(body, time):
if body == Body.Earth:
raise EarthNotAllowedError()
sv = GeoVector(Body.Sun, time, True)
se = Ecliptic(sv)
bv = GeoVector(body, time, True)
be = Ecliptic(bv)
return _NormalizeLongitude(be.elon - se.elon)
class ElongationEvent:
def __init__(self, time, visibility, elongation, ecliptic_separation):
self.time = time
self.visibility = visibility
self.elongation = elongation
self.ecliptic_separation = ecliptic_separation
def Elongation(body, time):
angle = LongitudeFromSun(body, time)
if angle > 180.0:
visibility = 'morning'
esep = 360.0 - angle
else:
visibility = 'evening'
esep = angle
angle = AngleFromSun(body, time)
return ElongationEvent(time, visibility, angle, esep)
def _rlon_offset(body, time, direction, targetRelLon):
plon = EclipticLongitude(body, time)
elon = EclipticLongitude(Body.Earth, time)
diff = direction * (elon - plon)
return _LongitudeOffset(diff - targetRelLon)
def SearchRelativeLongitude(body, targetRelLon, startTime):
if body == Body.Earth:
raise EarthNotAllowedError()
if body == Body.Moon or body == Body.Sun:
raise InvalidBodyError()
syn = _SynodicPeriod(body)
direction = +1 if _IsSuperiorPlanet(body) else -1
# Iterate until we converge on the desired event.
# Calculate the error angle, which will be a negative number of degrees,
# meaning we are "behind" the target relative longitude.
error_angle = _rlon_offset(body, startTime, direction, targetRelLon)
if error_angle > 0.0:
error_angle -= 360.0 # force searching forward in time
time = startTime
iter = 0
while iter < 100:
# Estimate how many days in the future (positive) or past (negative)
# we have to go to get closer to the target relative longitude.
day_adjust = (-error_angle/360.0) * syn
time = time.AddDays(day_adjust)
if abs(day_adjust) * _SECONDS_PER_DAY < 1.0:
return time
prev_angle = error_angle
error_angle = _rlon_offset(body, time, direction, targetRelLon)
if abs(prev_angle) < 30.0 and prev_angle != error_angle:
# Improve convergence for Mercury/Mars (eccentric orbits)
# by adjusting the synodic period to more closely match the
# variable speed of both planets in this part of their respective orbits.
ratio = prev_angle / (prev_angle - error_angle)
if 0.5 < ratio < 2.0:
syn *= ratio
iter += 1
raise NoConvergeError()
def _neg_elong_slope(body, time):
dt = 0.1
t1 = time.AddDays(-dt/2.0)
t2 = time.AddDays(+dt/2.0)
e1 = AngleFromSun(body, t1)
e2 = AngleFromSun(body, t2)
return (e1 - e2)/dt
def SearchMaxElongation(body, startTime):
if body == Body.Mercury:
s1 = 50.0
s2 = 85.0
elif body == Body.Venus:
s1 = 40.0
s2 = 50.0
else:
raise InvalidBodyError()
syn = _SynodicPeriod(body)
iter = 1
while iter <= 2:
plon = EclipticLongitude(body, startTime)
elon = EclipticLongitude(Body.Earth, startTime)
rlon = _LongitudeOffset(plon - elon) # clamp to (-180, +180]
# The slope function is not well-behaved when rlon is near 0 degrees or 180 degrees
# because there is a cusp there that causes a discontinuity in the derivative.
# So we need to guard against searching near such times.
if rlon >= -s1 and rlon < +s1:
# Seek to the window [+s1, +s2].
adjust_days = 0.0
# Search forward for the time t1 when rel lon = +s1.
rlon_lo = +s1
# Search forward for the time t2 when rel lon = +s2.
rlon_hi = +s2
elif rlon > +s2 or rlon < -s2:
# Seek to the next search window at [-s2, -s1].
adjust_days = 0.0
# Search forward for the time t1 when rel lon = -s2.
rlon_lo = -s2
# Search forward for the time t2 when rel lon = -s1.
rlon_hi = -s1
elif rlon >= 0.0:
# rlon must be in the middle of the window [+s1, +s2].
# Search BACKWARD for the time t1 when rel lon = +s1.
adjust_days = -syn / 4.0
rlon_lo = +s1
rlon_hi = +s2
# Search forward from t1 to find t2 such that rel lon = +s2.
else:
# rlon must be in the middle of the window [-s2, -s1].
# Search BACKWARD for the time t1 when rel lon = -s2.
adjust_days = -syn / 4.0
rlon_lo = -s2
# Search forward from t1 to find t2 such that rel lon = -s1.
rlon_hi = -s1
t_start = startTime.AddDays(adjust_days)
t1 = SearchRelativeLongitude(body, rlon_lo, t_start)
if t1 is None:
return None
t2 = SearchRelativeLongitude(body, rlon_hi, t1)
if t2 is None:
return None
# Now we have a time range [t1,t2] that brackets a maximum elongation event.
# Confirm the bracketing.
m1 = _neg_elong_slope(body, t1)
if m1 >= 0.0:
raise InternalError() # there is a bug in the bracketing algorithm!
m2 = _neg_elong_slope(body, t2)
if m2 <= 0.0:
raise InternalError() # there is a bug in the bracketing algorithm!
# Use the generic search algorithm to home in on where the slope crosses from negative to positive.
tx = Search(_neg_elong_slope, body, t1, t2, 10.0)
if tx is None:
return None
if tx.tt >= startTime.tt:
return Elongation(body, tx)
# This event is in the past (earlier than startTime).
# We need to search forward from t2 to find the next possible window.
# We never need to search more than twice.
startTime = t2.AddDays(1.0)
iter += 1
def _sun_offset(targetLon, time):
ecl = SunPosition(time)
return _LongitudeOffset(ecl.elon - targetLon)
def SearchSunLongitude(targetLon, startTime, limitDays):
t2 = startTime.AddDays(limitDays)
return Search(_sun_offset, targetLon, startTime, t2, 1.0)
def MoonPhase(time):
return LongitudeFromSun(Body.Moon, time)
def _moon_offset(targetLon, time):
angle = MoonPhase(time)
return _LongitudeOffset(angle - targetLon)
def SearchMoonPhase(targetLon, startTime, limitDays):
# To avoid discontinuities in the _moon_offset function causing problems,
# we need to approximate when that function will next return 0.
# We probe it with the start time and take advantage of the fact
# that every lunar phase repeats roughly every 29.5 days.
# There is a surprising uncertainty in the quarter timing,
# due to the eccentricity of the moon's orbit.
# I have seen up to 0.826 days away from the simple prediction.
# To be safe, we take the predicted time of the event and search
# +/-0.9 days around it (a 1.8-day wide window).
# But we must return None if the final result goes beyond limitDays after startTime.
uncertainty = 0.9
ya = _moon_offset(targetLon, startTime)
if ya > 0.0:
ya -= 360.0 # force searching forward in time, not backward
est_dt = -(_MEAN_SYNODIC_MONTH * ya) / 360.0
dt1 = est_dt - uncertainty
if dt1 > limitDays:
return None # not possible for moon phase to occur within the specified window
dt2 = min(limitDays, est_dt + uncertainty)
t1 = startTime.AddDays(dt1)
t2 = startTime.AddDays(dt2)
return Search(_moon_offset, targetLon, t1, t2, 1.0)
class MoonQuarter:
def __init__(self, quarter, time):
self.quarter = quarter
self.time = time
def SearchMoonQuarter(startTime):
angle = MoonPhase(startTime)
quarter = int(1 + math.floor(angle / 90.0)) % 4
time = SearchMoonPhase(90.0 * quarter, startTime, 10.0)
if time is None:
# The search should never fail. We should always find another lunar quarter.
raise InternalError()
return MoonQuarter(quarter, time)
def NextMoonQuarter(mq):
# Skip 6 days past the previous found moon quarter to find the next one.
# This is less than the minimum possible increment.
# So far I have seen the interval well contained by the range (6.5, 8.3) days.
time = mq.time.AddDays(6.0)
next_mq = SearchMoonQuarter(time)
# Verify that we found the expected moon quarter.
if next_mq.quarter != (1 + mq.quarter) % 4:
raise InternalError()
return next_mq
class IlluminationInfo:
def __init__(self, time, mag, phase, helio_dist, geo_dist, gc, hc, ring_tilt):
self.time = time
self.mag = mag
self.phase_angle = phase
self.phase_fraction = (1.0 + math.cos(math.radians(phase))) / 2.0
self.helio_dist = helio_dist
self.geo_dist = geo_dist
self.gc = gc
self.hc = hc
self.ring_tilt = ring_tilt
def _MoonMagnitude(phase, helio_dist, geo_dist):
# https://astronomy.stackexchange.com/questions/10246/is-there-a-simple-analytical-formula-for-the-lunar-phase-brightness-curve
rad = math.radians(phase)
mag = -12.717 + 1.49*abs(rad) + 0.0431*(rad**4)
moon_mean_distance_au = 385000.6 / _KM_PER_AU
geo_au = geo_dist / moon_mean_distance_au
mag += 5.0 * math.log10(helio_dist * geo_au)
return mag
def _SaturnMagnitude(phase, helio_dist, geo_dist, gc, time):
# Based on formulas by Paul Schlyter found here:
# http://www.stjarnhimlen.se/comp/ppcomp.html#15
# We must handle Saturn's rings as a major component of its visual magnitude.
# Find geocentric ecliptic coordinates of Saturn.
eclip = Ecliptic(gc)
ir = math.radians(28.06) # tilt of Saturn's rings to the ecliptic, in radians
Nr = math.radians(169.51 + (3.82e-5 * time.tt)) # ascending node of Saturn's rings, in radians
# Find tilt of Saturn's rings, as seen from Earth.
lat = math.radians(eclip.elat)
lon = math.radians(eclip.elon)
tilt = math.asin(math.sin(lat)*math.cos(ir) - math.cos(lat)*math.sin(ir)*math.sin(lon-Nr))
sin_tilt = math.sin(abs(tilt))
mag = -9.0 + 0.044*phase
mag += sin_tilt*(-2.6 + 1.2*sin_tilt)
mag += 5.0 * math.log10(helio_dist * geo_dist)
ring_tilt = math.degrees(tilt)
return (mag, ring_tilt)
def _VisualMagnitude(body, phase, helio_dist, geo_dist):
# For Mercury and Venus, see: https://iopscience.iop.org/article/10.1086/430212
c0 = c1 = c2 = c3 = 0
if body == Body.Mercury:
c0 = -0.60; c1 = +4.98; c2 = -4.88; c3 = +3.02
elif body == Body.Venus:
if phase < 163.6:
c0 = -4.47; c1 = +1.03; c2 = +0.57; c3 = +0.13
else:
c0 = +0.98; c1 = -1.02
elif body == Body.Mars:
c0 = -1.52; c1 = +1.60
elif body == Body.Jupiter:
c0 = -9.40; c1 = +0.50
elif body == Body.Uranus:
c0 = -7.19; c1 = +0.25
elif body == Body.Neptune:
c0 = -6.87
elif body == Body.Pluto:
c0 = -1.00; c1 = +4.00
else:
raise InvalidBodyError()
x = phase / 100.0
mag = c0 + x*(c1 + x*(c2 + x*c3))
mag += 5.0 * math.log10(helio_dist * geo_dist)
return mag
def Illumination(body, time):
if body == Body.Earth:
raise EarthNotAllowedError()
earth = _CalcEarth(time)
if body == Body.Sun:
gc = Vector(-earth.x, -earth.y, -earth.z, time)
hc = Vector(0.0, 0.0, 0.0, time)
phase = 0.0 # placeholder value; the Sun does not have a phase angle.
else:
if body == Body.Moon:
# For extra numeric precision, use geocentric moon formula directly.
gc = GeoMoon(time)
hc = Vector(earth.x + gc.x, earth.y + gc.y, earth.z + gc.z, time)
else:
# For planets, heliocentric vector is most direct to calculate.
hc = HelioVector(body, time)
gc = Vector(hc.x - earth.x, hc.y - earth.y, hc.z - earth.z, time)
phase = _AngleBetween(gc, hc)
geo_dist = gc.Length() # distance from body to center of Earth
helio_dist = hc.Length() # distance from body to center of Sun
ring_tilt = None # only reported for Saturn
if body == Body.Sun:
mag = -0.17 + 5.0*math.log10(geo_dist / _AU_PER_PARSEC)
elif body == Body.Moon:
mag = _MoonMagnitude(phase, helio_dist, geo_dist)
elif body == Body.Saturn:
mag, ring_tilt = _SaturnMagnitude(phase, helio_dist, geo_dist, gc, time)
else:
mag = _VisualMagnitude(body, phase, helio_dist, geo_dist)
return IlluminationInfo(time, mag, phase, helio_dist, geo_dist, gc, hc, ring_tilt)
def _mag_slope(body, time):
# The Search() function finds a transition from negative to positive values.
# The derivative of magnitude y with respect to time t (dy/dt)
# is negative as an object gets brighter, because the magnitude numbers
# get smaller. At peak magnitude dy/dt = 0, then as the object gets dimmer,
# dy/dt > 0.
dt = 0.01
t1 = time.AddDays(-dt/2)
t2 = time.AddDays(+dt/2)
y1 = Illumination(body, t1)
y2 = Illumination(body, t2)
return (y2.mag - y1.mag) / dt
def SearchPeakMagnitude(body, startTime):
# s1 and s2 are relative longitudes within which peak magnitude of Venus can occur.
s1 = 10.0
s2 = 30.0
if body != Body.Venus:
raise InvalidBodyError()
iter = 1
while iter <= 2:
# Find current heliocentric relative longitude between the
# inferior planet and the Earth.
plon = EclipticLongitude(body, startTime)
elon = EclipticLongitude(Body.Earth, startTime)
rlon = _LongitudeOffset(plon - elon)
# The slope function is not well-behaved when rlon is near 0 degrees or 180 degrees
# because there is a cusp there that causes a discontinuity in the derivative.
# So we need to guard against searching near such times.
if -s1 <= rlon < +s1:
# Seek to the window [+s1, +s2].
adjust_days = 0.0
# Search forward for the time t1 when rel lon = +s1.
rlon_lo = +s1
# Search forward for the time t2 when rel lon = +s2.
rlon_hi = +s2
elif rlon >= +s2 or rlon < -s2:
# Seek to the next search window at [-s2, -s1].
adjust_days = 0.0
# Search forward for the time t1 when rel lon = -s2.
rlon_lo = -s2
# Search forward for the time t2 when rel lon = -s1.
rlon_hi = -s1
elif rlon >= 0:
# rlon must be in the middle of the window [+s1, +s2].
# Search BACKWARD for the time t1 when rel lon = +s1.
syn = _SynodicPeriod(body)
adjust_days = -syn / 4
rlon_lo = +s1
# Search forward from t1 to find t2 such that rel lon = +s2.
rlon_hi = +s2
else:
# rlon must be in the middle of the window [-s2, -s1].
# Search BACKWARD for the time t1 when rel lon = -s2.
syn = _SynodicPeriod(body)
adjust_days = -syn / 4
rlon_lo = -s2
# Search forward from t1 to find t2 such that rel lon = -s1.
rlon_hi = -s1
t_start = startTime.AddDays(adjust_days)
t1 = SearchRelativeLongitude(body, rlon_lo, t_start)
t2 = SearchRelativeLongitude(body, rlon_hi, t1)
# Now we have a time range [t1,t2] that brackets a maximum magnitude event.
# Confirm the bracketing.
m1 = _mag_slope(body, t1)
if m1 >= 0.0:
raise InternalError()
m2 = _mag_slope(body, t2)
if m2 <= 0.0:
raise InternalError()
# Use the generic search algorithm to home in on where the slope crosses from negative to positive.
tx = Search(_mag_slope, body, t1, t2, 10.0)
if tx is None:
# The search should have found the ascending root in the interval [t1, t2].
raise InternalError()
if tx.tt >= startTime.tt:
return Illumination(body, tx)
# This event is in the past (earlier than startTime).
# We need to search forward from t2 to find the next possible window.
# We never need to search more than twice.
startTime = t2.AddDays(1.0)
iter += 1
# We should have found the peak magnitude in at most 2 iterations.
raise InternalError()
class HourAngleEvent:
def __init__(self, time, hor):
self.time = time
self.hor = hor
def SearchHourAngle(body, observer, hourAngle, startTime):
if body == Body.Earth:
raise EarthNotAllowedError()
if hourAngle < 0.0 or hourAngle >= 24.0:
raise Error('Invalid hour angle.')
iter = 0
time = startTime
while True:
iter += 1
# Calculate Greenwich Apparent Sidereal Time (GAST) at the given time.
gast = _sidereal_time(time)
ofdate = Equator(body, time, observer, True, True)
# Calculate the adjustment needed in sidereal time to bring
# the hour angle to the desired value.
delta_sidereal_hours = math.fmod(((hourAngle + ofdate.ra - observer.longitude/15) - gast), 24.0)
if iter == 1:
# On the first iteration, always search forward in time.
if delta_sidereal_hours < 0.0:
delta_sidereal_hours += 24.0
else:
# On subsequent iterations, we make the smallest possible adjustment,
# either forward or backward in time.
if delta_sidereal_hours < -12.0:
delta_sidereal_hours += 24.0
elif delta_sidereal_hours > +12.0:
delta_sidereal_hours -= 24.0
# If the error is tolerable (less than 0.1 seconds), stop searching.
if abs(delta_sidereal_hours) * 3600.0 < 0.1:
hor = Horizon(time, observer, ofdate.ra, ofdate.dec, Refraction.Normal)
return HourAngleEvent(time, hor)
# We need to loop another time to get more accuracy.
# Update the terrestrial time (in solar days) adjusting by sidereal time.
delta_days = (delta_sidereal_hours / 24.0) * _SOLAR_DAYS_PER_SIDEREAL_DAY
time = time.AddDays(delta_days)
@enum.unique
class Direction(enum.IntEnum):
"""Indicates whether a body is rising above or setting below the horizon.
Specifies the direction of a rising or setting event for a body.
For example, `Direction.Rise` is used to find sunrise times,
and `Direction.Set` is used to find sunset times.
Values
------
Rise: First appearance of a body as it rises above the horizon.
Set: Last appearance of a body as it sinks below the horizon.
"""
Rise = +1
Set = -1
class _peak_altitude_context:
def __init__(self, body, direction, observer, body_radius_au):
self.body = body
self.direction = direction
self.observer = observer
self.body_radius_au = body_radius_au
def _peak_altitude(context, time):
# Return the angular altitude above or below the horizon
# of the highest part (the peak) of the given object.
# This is defined as the apparent altitude of the center of the body plus
# the body's angular radius.
# The 'direction' parameter controls whether the angle is measured
# positive above the horizon or positive below the horizon,
# depending on whether the caller wants rise times or set times, respectively.
ofdate = Equator(context.body, time, context.observer, True, True)
# We calculate altitude without refraction, then add fixed refraction near the horizon.
# This gives us the time of rise/set without the extra work.
hor = Horizon(time, context.observer, ofdate.ra, ofdate.dec, Refraction.Airless)
alt = hor.altitude + math.degrees(context.body_radius_au / ofdate.dist)
return context.direction * (alt + _REFRACTION_NEAR_HORIZON)
def SearchRiseSet(body, observer, direction, startTime, limitDays):
if body == Body.Earth:
raise EarthNotAllowedError()
elif body == Body.Sun:
body_radius = _SUN_RADIUS_AU
elif body == Body.Moon:
body_radius = _MOON_RADIUS_AU
else:
body_radius = 0.0
if direction == Direction.Rise:
ha_before = 12.0 # minimum altitude (bottom) happens BEFORE the body rises.
ha_after = 0.0 # maximum altitude (culmination) happens AFTER the body rises.
elif direction == Direction.Set:
ha_before = 0.0 # culmination happens BEFORE the body sets.
ha_after = 12.0 # bottom happens AFTER the body sets.
else:
raise Error('Invalid value for direction parameter')
context = _peak_altitude_context(body, direction, observer, body_radius)
# See if the body is currently above/below the horizon.
# If we are looking for next rise time and the body is below the horizon,
# we use the current time as the lower time bound and the next culmination
# as the upper bound.
# If the body is above the horizon, we search for the next bottom and use it
# as the lower bound and the next culmination after that bottom as the upper bound.
# The same logic applies for finding set times, only we swap the hour angles.
time_start = startTime
alt_before = _peak_altitude(context, time_start)
if alt_before > 0.0:
# We are past the sought event, so we have to wait for the next "before" event (culm/bottom).
evt_before = SearchHourAngle(body, observer, ha_before, time_start)
time_before = evt_before.time
alt_before = _peak_altitude(context, time_before)
else:
# We are before or at the sought ebvent, so we find the next "after" event (bottom/culm),
# and use the current time as the "before" event.
time_before = time_start
evt_after = SearchHourAngle(body, observer, ha_after, time_before)
alt_after = _peak_altitude(context, evt_after.time)
while True:
if alt_before <= 0.0 and alt_after > 0.0:
# Search between the "before time" and the "after time" for the desired event.
event_time = Search(_peak_altitude, context, time_before, evt_after.time, 1.0)
if event_time is not None:
return event_time
# We didn't find the desired event, so use the "after" time to find the next "before" event.
evt_before = SearchHourAngle(body, observer, ha_before, evt_after.time)
evt_after = SearchHourAngle(body, observer, ha_after, evt_before.time)
if evt_before.time.ut >= time_start.ut + limitDays:
return None
time_before = evt_before.time
alt_before = _peak_altitude(context, evt_before.time)
alt_after = _peak_altitude(context, evt_after.time)
class SeasonInfo:
def __init__(self, mar_equinox, jun_solstice, sep_equinox, dec_solstice):
self.mar_equinox = mar_equinox
self.jun_solstice = jun_solstice
self.sep_equinox = sep_equinox
self.dec_solstice = dec_solstice
def _FindSeasonChange(targetLon, year, month, day):
startTime = Time.Make(year, month, day, 0, 0, 0)
time = SearchSunLongitude(targetLon, startTime, 4.0)
if time is None:
# We should always be able to find a season change.
raise InternalError()
return time
def Seasons(year):
mar_equinox = _FindSeasonChange(0, year, 3, 19)
jun_solstice = _FindSeasonChange(90, year, 6, 19)
sep_equinox = _FindSeasonChange(180, year, 9, 21)
dec_solstice = _FindSeasonChange(270, year, 12, 20)
return SeasonInfo(mar_equinox, jun_solstice, sep_equinox, dec_solstice)
def _MoonDistance(time):
return _CalcMoon(time).distance_au
def _distance_slope(direction, time):
dt = 0.001
t1 = time.AddDays(-dt/2.0)
t2 = time.AddDays(+dt/2.0)
dist1 = _MoonDistance(t1)
dist2 = _MoonDistance(t2)
return direction * (dist2 - dist1) / dt
@enum.unique
class ApsisKind(enum.IntEnum):
"""Represents whether a satellite is at a closest or farthest point in its orbit.
An apsis is a point in a satellite's orbit that is closest to,
or farthest from, the body it orbits (its primary).
`ApsisKind` is an enumerated type that indicates which of these
two cases applies to a particular apsis event.
Values
------
Pericenter: The satellite is at its closest point to its primary.
Apocenter: The satellite is at its farthest point from its primary.
Invalid: A placeholder for an undefined, unknown, or invalid apsis.
"""
Pericenter = 0
Apocenter = 1
Invalid = 2
class Apsis:
def __init__(self, time, kind, dist_au):
self.time = time
self.kind = kind
self.dist_au = dist_au
self.dist_km = dist_au * _KM_PER_AU
def SearchLunarApsis(startTime):
increment = 5.0 # number of days to skip on each iteration
t1 = startTime
m1 = _distance_slope(+1, t1)
iter = 0
while iter * increment < 2.0 * _MEAN_SYNODIC_MONTH:
t2 = t1.AddDays(increment)
m2 = _distance_slope(+1, t2)
if m1 * m2 <= 0.0:
# There is a change of slope polarity within the time range [t1, t2].
# Therefore this time range contains an apsis.
# Figure out whether it is perigee or apogee.
if m1 < 0.0 or m2 > 0.0:
# We found a minimum-distance event: perigee.
# Search the time range for the time when the slope goes from negative to positive.
apsis_time = Search(_distance_slope, +1, t1, t2, 1.0)
kind = ApsisKind.Pericenter
elif m1 > 0.0 or m2 < 0.0:
# We found a maximum-distance event: apogee.
# Search the time range for the time when the slope goes from positive to negative.
apsis_time = Search(_distance_slope, -1, t1, t2, 1.0)
kind = ApsisKind.Apocenter
else:
# This should never happen. It should not be possible for both slopes to be zero.
raise InternalError()
if apsis_time is None:
# We should always be able to find a lunar apsis!
raise InternalError()
dist = _MoonDistance(apsis_time)
return Apsis(apsis_time, kind, dist)
# We have not yet found a slope polarity change. Keep searching.
t1 = t2
m1 = m2
iter += 1
# It should not be possible to fail to find an apsis within 2 synodic months.
raise InternalError()
def NextLunarApsis(apsis):
skip = 11.0 # number of days to skip to start looking for next apsis event
time = apsis.time.AddDays(skip)
next = SearchLunarApsis(time)
# Verify that we found the opposite apsis from the previous one.
if apsis.kind not in [ApsisKind.Apocenter, ApsisKind.Pericenter]:
raise Error('Parameter "apsis" contains an invalid "kind" value.')
if next.kind + apsis.kind != 1:
raise InternalError()
return next
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