def test_coordinates_kepler_equation():
"""Tests the kepler_equation() method of Coordinates module"""
e1, v1 = kepler_equation(0.1, Angle(5.0))
e2, v2 = kepler_equation(0.99, Angle(1.0))
e3, v3 = kepler_equation(0.99, Angle(0.2, radians=True))
assert abs(round(e1(), 6) - 5.554589) < TOL, \
"ERROR: 1st kepler_equation() test, 'e1' value doesn't match"
assert abs(round(v1(), 6) - 6.139762) < TOL, \
"ERROR: 2nd kepler_equation() test, 'v1' value doesn't match"
assert abs(round(e2(), 6) - 24.725822) < TOL, \
"ERROR: 3rd kepler_equation() test, 'e2' value doesn't match"
assert abs(round(v2(), 6) - 144.155952) < TOL, \
"ERROR: 4th kepler_equation() test, 'v2' value doesn't match"
assert abs(round(e3(), 8) - 61.13444578) < TOL, \
"ERROR: 5th kepler_equation() test, 'e3' value doesn't match"
assert abs(round(v3(), 6) - 166.311977) < TOL, \
"ERROR: 6th kepler_equation() test, 'v3' value doesn't match"
def heliocentric_ecliptical_position(self, epoch):
"""This method computes the heliocentric position of a minor celestial
body, providing the result in ecliptical coordinates.
:param epoch: Epoch to compute geocentric position, as an Epoch object
:type epoch: :py:class:`Epoch`
:returns: A tuple containing longitude and latitude, as Angle objects
:rtype: tuple
:raises: TypeError if input value is of wrong type.
>>> a = 2.2091404
>>> e = 0.8502196
>>> q = a * (1.0 - e)
>>> i = Angle(11.94524)
>>> omega = Angle(334.75006)
>>> w = Angle(186.23352)
>>> t = Epoch(1990, 10, 28.54502)
>>> epoch = Epoch(1990, 10, 6.0)
>>> minor = Minor(q, e, i, omega, w, t)
>>> lon, lat = minor.heliocentric_ecliptical_position(epoch)
>>> print(lon.dms_str(n_dec=1))
66d 51' 57.8''
>>> print(lat.dms_str(n_dec=1))
11d 56' 14.4''
"""
# First check that input value is of correct types
if not isinstance(epoch, Epoch):
raise TypeError("Invalid input type")
# Get the mean motion and other orbital parameters
n = self._n
a = self._a
e = self._e
i = self._i
omega = self._omega
w = self._w
t = self._t
# Time since perihelion
t_peri = epoch - t
# Now, compute the mean anomaly, in degrees
m = t_peri * n
m = Angle(m)
# With the mean anomaly, use Kepler's equation to find E and v
ee, v = kepler_equation(e, m)
ee = Angle(ee).to_positive()
# Get r
er = ee.rad()
r = a * (1.0 - e * cos(er))
# Compute the heliocentric rectangular ecliptical coordinates
wr = w.rad()
vr = Angle(v).rad()
ur = wr + vr
omer = omega.rad()
ir = i.rad()
x = r * (cos(omer) * cos(ur) - sin(omer) * sin(ur) * cos(ir))
y = r * (sin(omer) * cos(ur) + cos(omer) * sin(ur) * cos(ir))
z = r * sin(ir) * sin(ur)
lon = atan2(y, x)
lat = atan2(z, sqrt(x * x + y * y))
return Angle(lon, radians=True), Angle(lat, radians=True)
def geocentric_position(self, epoch):
"""This method computes the geocentric position of a minor celestial
body (right ascension and declination) for the given epoch, and
referred to the standard equinox J2000.0. Additionally, it also
computes the elongation angle to the Sun.
:param epoch: Epoch to compute geocentric position, as an Epoch object
:type epoch: :py:class:`Epoch`
:returns: A tuple containing the right ascension, the declination and
the elongation angle to the Sun, as Angle objects
:rtype: tuple
:raises: TypeError if input value is of wrong type.
>>> a = 2.2091404
>>> e = 0.8502196
>>> q = a * (1.0 - e)
>>> i = Angle(11.94524)
>>> omega = Angle(334.75006)
>>> w = Angle(186.23352)
>>> t = Epoch(1990, 10, 28.54502)
>>> minor = Minor(q, e, i, omega, w, t)
>>> epoch = Epoch(1990, 10, 6.0)
>>> ra, dec, p = minor.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
10h 34' 13.7''
>>> print(dec.dms_str(n_dec=0))
19d 9' 32.0''
>>> print(round(p, 2))
40.51
>>> t = Epoch(1998, 4, 14.4358)
>>> q = 1.487469
>>> e = 1.0
>>> i = Angle(0.0)
>>> omega = Angle(0.0)
>>> w = Angle(0.0)
>>> minor = Minor(q, e, i, omega, w, t)
>>> epoch = Epoch(1998, 8, 5.0)
>>> ra, dec, p = minor.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
5h 45' 34.5''
>>> print(dec.dms_str(n_dec=0))
23d 23' 53.0''
>>> print(round(p, 2))
45.73
"""
# First check that input value is of correct types
if not isinstance(epoch, Epoch):
raise TypeError("Invalid input type")
# Get internal parameters
aa, bb, cc = self._aa, self._bb, self._cc
am, bm, cm = self._am, self._bm, self._cm
# Get the mean motion and other orbital parameters
n = self._n
a = self._a
e = self._e
w = self._w
t = self._t
# Time since perihelion
t_peri = epoch - t
# Now, compute the mean anomaly, in degrees
m = t_peri * n
m = Angle(m)
if e < 0.98:
# Elliptic case
# With the mean anomaly, use Kepler's equation to find E and v
ee, v = kepler_equation(e, m)
ee = Angle(ee).to_positive()
# Get r
er = ee.rad()
rr = a * (1.0 - e * cos(er))
elif abs(e - 1.0) < self._tol:
# Parabolic case
q = self._q
ww = (0.03649116245 * (epoch - self._t)) / (q * sqrt(q))
sp = ww / 3.0
iterate = True
while iterate:
s = (2.0 * sp * sp * sp + ww) / (3.0 * (sp * sp + 1.0))
iterate = abs(s - sp) > self._tol
sp = s
v = 2.0 * atan(s)
v = Angle(v, radians=True)
rr = q * (1.0 + s * s)
else:
# We are in the near-parabolic case
v, rr = self._near_parabolic(t_peri)
# Compute the heliocentric rectangular equatorial coordinates
wr = w.rad()
vr = Angle(v).rad()
x = rr * am * sin(aa + wr + vr)
y = rr * bm * sin(bb + wr + vr)
z = rr * cm * sin(cc + wr + vr)
# Now let's compute Sun's rectangular coordinates
xs, ys, zs = Sun.rectangular_coordinates_j2000(epoch)
xi = x + xs
eta = y + ys
zeta = z + zs
delta = sqrt(xi * xi + eta * eta + zeta * zeta)
# We need to correct for the effect of light-time. Compute delay tau
tau = 0.0057755183 * delta
# Recompute some critical parameters
t_peri = epoch - t - tau
# Now, compute the mean anomaly, in degrees
m = t_peri * n
m = Angle(m)
if e < 0.98:
# Elliptic case
# With the mean anomaly, use Kepler's equation to find E and v
ee, v = kepler_equation(e, m)
ee = Angle(ee).to_positive()
# Get r
er = ee.rad()
rr = a * (1.0 - e * cos(er))
elif abs(e - 1.0) < self._tol:
# Parabolic case
q = self._q
ww = (0.03649116245 * (epoch - self._t)) / (q * sqrt(q))
sp = ww / 3.0
iterate = True
while iterate:
s = (2.0 * sp * sp * sp + ww) / (3.0 * (sp * sp + 1.0))
iterate = abs(s - sp) > self._tol
sp = s
v = 2.0 * atan(s)
v = Angle(v, radians=True)
rr = q * (1.0 + s * s)
else:
# We are in the near-parabolic case
v, rr = self._near_parabolic(t_peri)
# Compute the heliocentric rectangular equatorial coordinates
wr = w.rad()
vr = Angle(v).rad()
x = rr * am * sin(aa + wr + vr)
y = rr * bm * sin(bb + wr + vr)
z = rr * cm * sin(cc + wr + vr)
xi = x + xs
eta = y + ys
zeta = z + zs
ra = Angle(atan2(eta, xi), radians=True)
dec = Angle(atan2(zeta, sqrt(xi * xi + eta * eta)), radians=True)
r_sun = sqrt(xs * xs + ys * ys + zs * zs)
psi = acos((xi * xs + eta * ys + zeta * zs) / (r_sun * delta))
psi = Angle(psi, radians=True)
return ra, dec, psi