158 lines
5.3 KiB
Python
158 lines
5.3 KiB
Python
"""Formulae for specific earth behaviors and effects."""
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from numpy import (abs, arcsin, arccos, arctan2, array, clip, cos, minimum,
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nan_to_num, pi, sin, sqrt, tan, where, zeros_like)
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from .constants import (AU_M, ANGVEL, DAY_S, DEG2RAD, ERAD,
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IERS_2010_INVERSE_EARTH_FLATTENING, RAD2DEG, T0, tau)
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from .functions import dots
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earth_radius_au = ERAD / AU_M
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one_minus_flattening = 1.0 - 1.0 / IERS_2010_INVERSE_EARTH_FLATTENING
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one_minus_flattening_squared = one_minus_flattening * one_minus_flattening
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def terra(latitude, longitude, elevation, gast):
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"""Deprecated conversion from lat,lon,t -> GCRS; neglects polar motion."""
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zero = zeros_like(gast)
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sinphi = sin(latitude)
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cosphi = cos(latitude)
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c = 1.0 / sqrt(cosphi * cosphi +
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sinphi * sinphi * one_minus_flattening_squared)
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s = one_minus_flattening_squared * c
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ach = earth_radius_au * c + elevation
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ash = earth_radius_au * s + elevation
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# Compute local sidereal time factors at the observer's longitude.
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stlocl = 15.0 * DEG2RAD * gast + longitude
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sinst = sin(stlocl)
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cosst = cos(stlocl)
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# Compute position vector components in au.
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ac = ach * cosphi
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acsst = ac * sinst
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accst = ac * cosst
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pos = array((accst, acsst, zero + ash * sinphi))
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# Compute velocity vector components in au/day.
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vel = ANGVEL * DAY_S * array((-acsst, accst, zero))
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return pos, vel
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def reverse_terra(xyz_au, gast, iterations=3):
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"""Deprecated conversion from GCRS -> lat,lon,t; neglects polar motion."""
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x, y, z = xyz_au
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R = sqrt(x*x + y*y)
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lon = (arctan2(y, x) - 15 * DEG2RAD * gast - pi) % tau - pi
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lat = arctan2(z, R)
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a = ERAD / AU_M
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f = 1.0 / IERS_2010_INVERSE_EARTH_FLATTENING
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e2 = 2.0*f - f*f
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i = 0
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C = 1.0
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while i < iterations:
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i += 1
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C = 1.0 / sqrt(1.0 - e2 * (sin(lat) ** 2.0))
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lat = arctan2(z + a * C * e2 * sin(lat), R)
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elevation_m = ((R / cos(lat)) - a * C) * AU_M
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return lat, lon, elevation_m
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def compute_limb_angle(position_au, observer_au):
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"""Determine the angle of an object above or below the Earth's limb.
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Given an object's GCRS `position_au` |xyz| vector and the position
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of an `observer_au` as a vector in the same coordinate system,
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return a tuple that provides `(limb_ang, nadir_ang)`:
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limb_angle
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Angle of observed object above (+) or below (-) limb in degrees.
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nadir_angle
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Nadir angle of observed object as a fraction of apparent radius
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of limb: <1.0 means below the limb, =1.0 means on the limb, and
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>1.0 means above the limb.
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"""
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# Compute the distance to the object and the distance to the observer.
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disobj = sqrt(dots(position_au, position_au))
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disobs = sqrt(dots(observer_au, observer_au))
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# Compute apparent angular radius of Earth's limb.
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aprad = arcsin(minimum(earth_radius_au / disobs, 1.0))
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# Compute zenith distance of Earth's limb.
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zdlim = pi - aprad
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# Compute zenith distance of observed object.
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coszd = dots(position_au, observer_au) / (disobj * disobs)
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coszd = clip(coszd, -1.0, 1.0)
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zdobj = arccos(coszd)
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# Angle of object wrt limb is difference in zenith distances.
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limb_angle = (zdlim - zdobj) * RAD2DEG
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# Nadir angle of object as a fraction of angular radius of limb.
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nadir_angle = (pi - zdobj) / aprad
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return limb_angle, nadir_angle
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def sidereal_time(t):
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"""Compute Greenwich Mean Sidereal Time (GMST) in hours at time ``t``."""
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theta = earth_rotation_angle(t.whole, t.ut1_fraction)
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# The equinox method. See Circular 179, Section 2.6.2.
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# Precession-in-RA terms in mean sidereal time taken from third
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# reference, eq. (42), with coefficients in arcseconds.
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t = (t.whole - T0 + t.tdb_fraction) / 36525.0
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st = ( 0.014506 +
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(((( - 0.0000000368 * t
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- 0.000029956 ) * t
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- 0.00000044 ) * t
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+ 1.3915817 ) * t
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+ 4612.156534 ) * t)
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return (st / 54000.0 + theta * 24.0) % 24.0
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def earth_rotation_angle(jd_ut1, fraction_ut1=0.0):
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"""Return the value of the Earth Rotation Angle (theta) for a UT1 date.
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Uses the expression from the note to IAU Resolution B1.8 of 2000.
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Returns a fraction between 0.0 and 1.0 whole rotations.
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"""
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th = 0.7790572732640 + 0.00273781191135448 * (jd_ut1 - T0 + fraction_ut1)
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return (th % 1.0 + jd_ut1 % 1.0 + fraction_ut1) % 1.0
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def refraction(alt_degrees, temperature_C, pressure_mbar):
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"""Given an observed altitude, estimate atmospheric refraction, in degrees.
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Zero refraction is returned both for objects very near the zenith,
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as well as for objects more than one degree below the horizon.
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"""
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r = 0.016667 / tan((alt_degrees + 7.31 / (alt_degrees + 4.4)) * DEG2RAD)
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d = r * (0.28 * pressure_mbar / (temperature_C + 273.0))
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return where((-1.0 <= alt_degrees) & (alt_degrees <= 89.9), d, 0.0)
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def refract(alt_degrees, temperature_C, pressure_mbar):
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"""Given an unrefracted `alt` determine where it will appear in the sky."""
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alt = alt_degrees
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while True:
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alt1 = alt
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alt = alt_degrees + refraction(alt, temperature_C, pressure_mbar)
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converged = nan_to_num(abs(alt - alt1)).max() <= 3.0e-5
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if converged:
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break
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return alt
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