def MST(jdnumber):
assert isinstance(jdnumber, JulianDayNumber), 'Invalid Julian Day Number'
# Number of Julian centuries since J2000
T = (jdnumber.jdn - epoch_j2000.jdn) / julian_century
ret_deg = 280.46061837 \
+ 360.98564736629*(jdnumber.jdn - epoch_j2000.jdn) \
+ (0.000387933 - T/38710000.0)*T*T
return Longitude(radians(ret_deg))
def to_altazimuthal(self, lati_, hangle_):
assert isinstance(lati_, Latitude), 'lati_ must be a Latitude'
assert isinstance(hangle_, Longitude), 'hangle_ must be a Longitude'
lati_rad, decl_rad, hangle_rad = lati_.rads, self.b.rads, hangle_.rads
azimuth_rad = atan2(sin(hangle_rad), cos(hangle_rad)*sin(lati_rad) \
- tan(decl_rad)*cos(lati_rad))
altitude_rad = asin(sin(lati_rad)*sin(decl_rad) \
+ cos(lati_rad)*cos(decl_rad)*cos(hangle_rad))
return AltAzimuthal(Longitude(azimuth_rad), Latitude(altitude_rad))
def to_ecliptical(self, epsilon_):
assert isinstance(epsilon_, Angle), 'epsilon_ must be Angle'
alpha_rad, delta_rad = self.a.rads, self.b.rads
epsilon_rad = epsilon_.rads
lambda_rad = atan2(
sin(alpha_rad) * cos(epsilon_rad) +
tan(delta_rad) * sin(epsilon_rad), cos(alpha_rad))
beta_rad = asin(
sin(delta_rad) * cos(epsilon_rad) -
cos(delta_rad) * sin(epsilon_rad) * sin(alpha_rad))
return Ecliptical(Longitude(lambda_rad), Latitude(beta_rad))
def to_equatorial(self, epsilon_):
assert isinstance(epsilon_, Angle), 'epsilon_ must be Angle'
lambda_rad, beta_rad = self.a.rads, self.b.rads
epsilon_rad = epsilon_.rads
alpha_rad = atan2(
sin(lambda_rad) * cos(epsilon_rad) -
tan(beta_rad) * sin(epsilon_rad), cos(lambda_rad))
delta_rad = asin(
sin(beta_rad) * cos(epsilon_rad) +
cos(beta_rad) * sin(epsilon_rad) * sin(lambda_rad))
return Equatorial(Longitude(alpha_rad), Latitude(delta_rad))
def deprecated_precession(coord, epoch_start, epoch_end):
"""Calculate the precession in the RA and Declination at the ending epoch
of a body given the mean coordinates at a starting epoch.
@param coord The mean coordinates referred to epoch_start as a Equatorial object
@param epoch_start The epoch to which the coord is referred as a
Julian Day Number in TD
@param epoch_end The epoch for which the new coord is to be computed as a
Julian Day Number in TD
@return The RA and Declination for epoch_end as a SphCoord object
@caution We assume that the coord is already corrected for the proper
motion of the body over the epoch interval in question
"""
assert isinstance(coord, Equatorial), 'coord must be a Equatorial'
assert isinstance(epoch_start, JulianDayNumber) \
and isinstance(epoch_end, JulianDayNumber), \
'epochs much be JulianDayNumbers'
T = (epoch_start.jdn - epoch_j2000.jdn) / julian_century
t = (epoch_end.jdn - epoch_start.jdn) / julian_century
K = (2306.2181 + (1.39656 - 0.000139 * T) * T) * t
t_square = t**2
zeta = K + (
(0.30188 - 0.000344 * T) + 0.017998 * t) * t_square # arc-seconds
zeta = radians(zeta / 3600.0)
zappa = K + (
(1.09468 + 0.000066 * T) + 0.018203 * t) * t_square # arc-seconds
zappa = radians(zappa / 3600.0)
theta = (2004.3109 - (0.85330 + 0.000217 * T) * T) * t # arc-seconds
theta -= (
(0.42665 + 0.000217 * T) + 0.041833 * t) * t_square # arc-seconds
theta = radians(theta / 3600.0)
alpha = coord.a.rads + zeta
delta0 = coord.b.rads
A = cos(delta0) * sin(alpha)
B = cos(theta) * cos(delta0) * cos(alpha) - sin(theta) * sin(delta0)
C = sin(theta) * cos(delta0) * cos(alpha) + cos(theta) * sin(delta0)
alpha = zappa + atan2(A, B)
# If the object is close to the celestial pole
if fabs(delta0 - pi / 2) < radians(0.5):
delta = acos(sqrt(A**2 + B**2))
else:
delta = asin(C)
return Equatorial(Longitude(alpha), Latitude(delta))
def equation_of_kepler_iterative(M_, e_, prec_):
"""Calculate the Eccentric anomaly from the Mean Anomaly using iteration.
@param M_ The Mean anomaly as a Longitude object
@param e_ The eccentricity of the orbit
@param prec_ The precision at which to stop the iteration
@return The Eccentric anomaly as a Longitude object
"""
assert isinstance(M_, Longitude), 'M_ should be a Longitude'
assert isinstance(e_, Number), 'e_ should be a Number'
assert 0 <= e_ < 1, 'e_ should lie between 0 and 1'
assert isinstance(prec_, Number), 'prec_ should be a Number'
E = M_.rads
while True:
dE = (M_.rads + e_ * sin(E) - E) / (1 - e_ * cos(E))
E += dE
if fabs(dE) <= prec_: break
return Longitude(E)
def apply_correction(self, coord):
assert isinstance(coord, Equatorial), 'coord must be a Equatorial'
alpha = coord.a.rads + self.zeta
delta0 = coord.b.rads
A = cos(delta0)*sin(alpha)
B = cos(self.theta)*cos(delta0)*cos(alpha) - sin(self.theta)*sin(delta0)
C = sin(self.theta)*cos(delta0)*cos(alpha) + cos(self.theta)*sin(delta0)
alpha = self.zappa + atan2(A,B)
# If the object is close to the celestial pole
if fabs(delta0 - pi/2) < radians(0.5):
delta = acos(sqrt(A**2+B**2))
else:
delta = asin(C)
return Equatorial(Longitude(alpha), Latitude(delta))
def proper_motion_classical(coord, annual_pm, epoch_yrs):
"""Compute the proper motion using the classical method of uniform changes
in RA and declination.
@param coord The coordinates of the star as a Equatorial object at the epoch.
@param annual_pm The annual proper motion of the star as a tuple of Angles.
@param epoch_yrs The number of years from the starting epoch.
@return The updated coordinates for the star as an Equatorial object.
"""
assert isinstance(coord, Equatorial), 'coord should be a Equatorial'
assert isinstance(annual_pm, tuple), 'annual_pm should be a tuple'
assert len(annual_pm) == 2, 'annual_pm should be a 2-element tuple'
assert isinstance(annual_pm[0], Angle) and isinstance(annual_pm[1],Angle), \
'annual_pm elements should be Angles'
alpha_0, delta_0 = coord.a.rads, coord.b.rads
dalpha, ddelta = annual_pm[0].rads, annual_pm[1].rads
alpha_new = alpha_0 + dalpha * epoch_yrs
delta_new = delta_0 + ddelta * epoch_yrs
return Equatorial(Longitude(alpha_new), Latitude(delta_new))
def proper_motion(coord, r_parsecs, v_parsecs_per_year, annual_pm, epoch_yrs):
"""Compute the proper motion of a star over the given number of years.
@param coord The coordinates of the star as a Equatorial object at the epoch.
@param r_parsecs The radial distance to the star in parsecs.
@param v_parsecs_per_year The radial velocity of the star in parsecs/year.
@param annual_pm The annual proper motion of the star as a tuple of Angles.
@param epoch_yrs The number of years from the starting epoch.
@return The updated coordinates for the star as an Equatorial object.
"""
assert isinstance(coord, Equatorial), 'coord should be a Equatorial'
assert isinstance(annual_pm, tuple), 'annual_pm should be a tuple'
assert len(annual_pm) == 2, 'annual_pm should be a 2-element tuple'
assert isinstance(annual_pm[0], Angle) and isinstance(annual_pm[1],Angle), \
'annual_pm elements should be Angles'
alpha_0, delta_0 = coord.a.rads, coord.b.rads
dalpha, ddelta = annual_pm[0].rads, annual_pm[1].rads
x = r_parsecs * cos(delta_0) * cos(alpha_0) # parsecs
y = r_parsecs * cos(delta_0) * sin(alpha_0) # parsecs
z = r_parsecs * sin(delta_0) # parsecs
dx = (x / r_parsecs
) * v_parsecs_per_year - z * ddelta * cos(alpha_0) - y * dalpha
dy = (y / r_parsecs
) * v_parsecs_per_year - z * ddelta * sin(alpha_0) + x * dalpha
dz = (z /
r_parsecs) * v_parsecs_per_year + r_parsecs * ddelta * cos(delta_0)
# dx,dy,dz are in parsecs/year
x_new = x + epoch_yrs * dx # parsecs
y_new = y + epoch_yrs * dy # parsecs
z_new = z + epoch_yrs * dz # parsecs
alpha_new = atan2(y_new, x_new)
delta_new = atan2(z_new, sqrt(x_new**2 + y_new**2))
return Equatorial(Longitude(alpha_new), Latitude(delta_new))
def equation_of_kepler_binarysearch(M_, e_, loop_=33):
"""Calculate the Eccentric anomaly from the Mean Anomaly using binary search.
@param M_ The Mean anomaly as a Longitude object
@param e_ The eccentricity of the orbit
@param loop_ The number of times to
@return The Eccentric anomaly as a Longitude object
"""
assert isinstance(M_, Longitude), 'M_ should be a Longitude'
assert isinstance(e_, Number), 'e_ should be a Number'
assert 0 <= e_ < 1, 'e_ should lie between 0 and 1'
assert type(loop_) is IntType, 'loop_ should be an integer'
M_ = pfmod(M_.rads, 2 * pi)
if M_ > pi:
sign = -1
M_ = 2 * pi - M_
else:
sign = 1
E0, D = pi / 2, pi / 4
for idx in range(loop_):
M1 = E0 - e * sin(E0)
E0 += copysign(D, M_ - M1)
D /= 2
E0 *= sign
return Longitude(E0)
def get_ecliptical_apparent(self):
return Ecliptical(Longitude(self.lambda_apparent), Latitude(0))
def get_ecliptical(self):
return Ecliptical(Longitude(self.L), Latitude(0))
def apply_correction(self, coord):
assert isinstance(coord, Equatorial), 'coord must be a Equatorial'
alpha = coord.a.rads + self.zeta
delta0 = coord.b.rads
A = cos(delta0)*sin(alpha)
B = cos(self.theta)*cos(delta0)*cos(alpha) - sin(self.theta)*sin(delta0)
C = sin(self.theta)*cos(delta0)*cos(alpha) + cos(self.theta)*sin(delta0)
alpha = self.zappa + atan2(A,B)
# If the object is close to the celestial pole
if fabs(delta0 - pi/2) < radians(0.5):
delta = acos(sqrt(A**2+B**2))
else:
delta = asin(C)
return Equatorial(Longitude(alpha), Latitude(delta))
if __name__ == "__main__":
epoch_start = epoch_j2000
epoch_end = JulianDayNumber(Date(2028,11,13),Time(4,28,0))
prec = Precession(epoch_start, epoch_end)
# theta Persei (proper motion corrected)
theta_persei_epoch_start = Equatorial(
Longitude(radians(41.054063)), Latitude(radians(49.227750))
)
print prec.apply_correction(theta_persei_epoch_start) # 2h46m11.331s, +49deg20'54.54"
star_coords = [copy.copy(star_) for date in dates]
return conjunction(dates_, star_coords, coords_, prec_)
if __name__ == "__main__":
dates = [
JulianDayNumber(Date(1991, 8, 5), Time(0, 0, 0)),
JulianDayNumber(Date(1991, 8, 6), Time(0, 0, 0)),
JulianDayNumber(Date(1991, 8, 7), Time(0, 0, 0)),
JulianDayNumber(Date(1991, 8, 8), Time(0, 0, 0)),
JulianDayNumber(Date(1991, 8, 9), Time(0, 0, 0)),
]
# Mercury
coords1 = [
SphCoord(
Longitude(radians((10.0 + 24.0 / 60.0 + 30.125 / 3600.0) * 15)),
Latitude(radians(6 + 26.0 / 60.0 + 32.05 / 3600.0))),
SphCoord(
Longitude(radians((10.0 + 25.0 / 60.0 + 0.342 / 3600.0) * 15)),
Latitude(radians(6 + 10.0 / 60.0 + 57.72 / 3600.0))),
SphCoord(
Longitude(radians((10.0 + 25.0 / 60.0 + 12.515 / 3600.0) * 15)),
Latitude(radians(5 + 57.0 / 60.0 + 33.08 / 3600.0))),
SphCoord(
Longitude(radians((10.0 + 25.0 / 60.0 + 6.235 / 3600.0) * 15)),
Latitude(radians(5 + 46.0 / 60.0 + 27.07 / 3600.0))),
SphCoord(
Longitude(radians((10.0 + 24.0 / 60.0 + 41.185 / 3600.0) * 15)),
Latitude(radians(5 + 37.0 / 60.0 + 48.45 / 3600.0)))
]
# Venus
D /= 2
E0 *= sign
return Longitude(E0)
if __name__ == "__main__":
## Refraction tests
print refraction(Angle(0))
print refraction(Angle(pi / 4))
print refraction(Angle(pi / 2))
## Angular Separation Tests
# List of (alpha1, delta1, alpha2, delta2) tuples
data = [
# sph1 a number
(pi / 2, SphCoord(Longitude(pi / 6), Latitude(pi / 3))),
# sph2 a tuple
(SphCoord(Longitude(pi / 6), Latitude(pi / 3)), (0, pi / 2)),
# (0,45), (30,60). Should be just over 23 degrees
(SphCoord(Longitude(0), Latitude(pi / 4)),
SphCoord(Longitude(pi / 6), Latitude(pi / 3))),
# Example 17.a, Arcturus and Spica. (32.7930 degrees)
(SphCoord(Longitude(radians(213.9154)), Latitude(radians(19.1825))),
SphCoord(Longitude(radians(201.2983)), Latitude(radians(-11.1614)))),
]
for item in data:
try:
sph1, sph2 = item
print angular_separation(sph1, sph2)
except Exception as e:
print 'Error:', e
def get_equatorial(self):
return Equatorial(Longitude(self.alpha), Latitude(self.delta))
def get_equatorial_apparent(self):
return Equatorial(Longitude(self.alpha_apparent),
Latitude(self.delta_apparent))
class AltAzimuthal(SphCoord):
def __unicode__(self):
return u'(A:' + self.a.dms() + u', h:' + self.b.dms() + u')'
def __str__(self):
return unicode(self).encode(sys.stdout.encoding or DEFAULT_ENCODING,
'replace')
if __name__ == "__main__":
## Spherical Coordinate Tests
data = [
(0, pi / 2), # Not Longitudes or Latitudes
(Latitude(0), Latitude(pi / 2)), # alpha is not a Longitude
(Longitude(0), Longitude(pi / 2)), # delta is not a Latitude
(Longitude(0), Angle(pi / 2)) # delta is not a Latitude
]
for tup in data:
try:
sph = SphCoord(tup[0], tup[1])
except AssertionError as e:
print 'Error:', e
print
equa = Equatorial(Longitude(0), Latitude(0))
print equa
ecli = equa.to_ecliptical(epsilon_j2000)
print ecli
print ecli.to_equatorial(epsilon_j2000)
print