def frozen(
cls,
attractor,
alt,
inc=None,
argp=None,
raan=0 * u.deg,
arglat=0 * u.deg,
ecc=None,
epoch=J2000,
plane=Planes.EARTH_EQUATOR,
):
r"""Return frozen Orbit. If any of the given arguments results in an impossibility, some values will be overwritten
To achieve frozen orbit these two equations have to be set to zero.
.. math::
\dfrac {d\overline {e}}{dt}=\dfrac {-3\overline {n}J_{3}R^{3}_{E}\sin \left( \overline {i}\right) }{2a^{3}\left( 1-\overline {e}^{2}\right) ^{2}}\left( 1-\dfrac {5}{4}\sin ^{2}\overline {i}\right) \cos \overline {w}
.. math::
\dfrac {d\overline {\omega }}{dt}=\dfrac {3\overline {n}J_{2}R^{2}_{E}}{a^{2}\left( 1-\overline {e}^{2}\right) ^{2}}\left( 1-\dfrac {5}{4}\sin ^{2}\overline {i}\right) \left[ 1+\dfrac {J_{3}R_{E}}{2J_{2}\overline {a}\left( 1-\overline {e}^{2}\right) }\left( \dfrac {\sin ^{2}\overline {i}-\overline {e}\cos ^{2}\overline {i}}{\sin \overline {i}}\right) \dfrac {\sin \overline {w}}{\overline {e}}\right]
The first approach would be to nullify following term to zero:
.. math::
( 1-\dfrac {5}{4}\sin ^{2})
For which one obtains the so-called critical inclinations: i = 63.4349 or 116.5651 degrees. To escape the inclination requirement, the argument of periapsis can be set to w = 90 or 270 degrees to nullify the second equation. Then, one should nullify the right-hand side of the first equation, which yields an expression that correlates the inclination of the object and the eccentricity of the orbit:
.. math::
\overline {e}=-\dfrac {J_{3}R_{E}}{2J_{2}\overline {a}\left( 1-\overline {e}^{2}\right) }\left( \dfrac {\sin ^{2}\overline {i}-\overline {e}\cos ^{2} \overline {i}}{\sin \overline {i}}\right)
Assuming that e is negligible compared to J2, it can be shown that:
.. math::
\overline {e}\approx -\dfrac {J_{3}R_{E}}{2J_{2}\overline {a}}\sin \overline {i}
The implementation is divided in the following cases:
1. When the user gives a negative altitude, the method will raise a ValueError
2. When the attractor has not defined J2 or J3, the method will raise an AttributeError
3. When the attractor has J2/J3 outside of range 1 to 10 , the method will raise an NotImplementedError. Special case for Venus.See "Extension of the critical inclination" by Xiaodong Liu, Hexi Baoyin, and Xingrui Ma
4. If argp is not given or the given argp is a critical value:
* if eccentricity is none and inclination is none, the inclination is set with a critical value and the eccentricity is obtained from the last formula mentioned
* if only eccentricity is none, we calculate this value with the last formula mentioned
* if only inclination is none ,we calculate this value with the formula for eccentricity with critical argp.
5. If inc is not given or the given inc is critical:
* if the argp and the eccentricity is given we keep these values to create the orbit
* if the eccentricity is given we keep this value, if not, default to the eccentricity of the Moon's orbit around the Earth
6. if it's not possible to create an orbit with the the argp and the inclination given, both of them are set to the critical values and the eccentricity is calculate with the last formula
Parameters
----------
attractor : Body
Main attractor.
alt : ~astropy.units.Quantity
Altitude over surface.
inc : ~astropy.units.Quantity, optional
Inclination, default to critical value.
argp : ~astropy.units.Quantity, optional
Argument of the pericenter, default to critical value.
raan : ~astropy.units.Quantity, optional
Right ascension of the ascending node, default to 0 deg.
arglat : ~astropy.units.Quantity, optional
Argument of latitude, default to 0 deg.
ecc : ~astropy.units.Quantity
Eccentricity, default to the eccentricity of the Moon's orbit around the Earth
epoch: ~astropy.time.Time, optional
Epoch, default to J2000.
plane : ~poliastro.frames.Planes
Fundamental plane of the frame.
"""
critical_argps = [np.pi / 2, 3 * np.pi / 2] * u.rad
critical_inclinations = [
63.4349 * np.pi / 180, 116.5651 * np.pi / 180
] * u.rad
try:
if 1 <= np.abs(attractor.J2 / attractor.J3) <= 10:
raise NotImplementedError(
"This has not been implemented for {}".format(
attractor.name))
assert alt > 0
argp = critical_argps[0] if argp is None else argp
a = attractor.R + alt
critical_argp = find_closest_value(argp, critical_argps)
if np.isclose(argp, critical_argp, 1e-8, 1e-5 * u.rad):
if inc is None and ecc is None:
inc = critical_inclinations[0]
ecc = get_eccentricity_critical_argp(attractor, a, inc)
elif ecc is None:
ecc = get_eccentricity_critical_argp(attractor, a, inc)
else:
inc = get_inclination_critical_argp(attractor, a, ecc)
return cls.from_classical(attractor, a, ecc, inc, raan, argp,
arglat, epoch, plane)
inc = critical_inclinations[0] if inc is None else inc
critical_inclination = find_closest_value(inc,
critical_inclinations)
if np.isclose(inc, critical_inclination, 1e-8, 1e-5 * u.rad):
ecc = get_eccentricity_critical_inc(ecc)
return cls.from_classical(attractor, a, ecc, inc, raan, argp,
arglat, epoch, plane)
argp = critical_argps[0]
inc = critical_inclinations[0]
ecc = get_eccentricity_critical_argp(attractor, a, inc)
return cls.from_classical(attractor, a, ecc, inc, raan, argp,
arglat, epoch, plane)
except AttributeError:
raise AttributeError(
"Attractor {} has not spherical harmonics implemented".format(
attractor.name))
except AssertionError:
raise ValueError(
"The semimajor axis may not be smaller that {}'s radius".
format(attractor.name))
def frozen(
cls,
attractor,
alt,
inc=None,
argp=None,
raan=0 * u.deg,
arglat=0 * u.deg,
ecc=None,
epoch=J2000,
plane=Planes.EARTH_EQUATOR,
):
r"""Return frozen Orbit. If any of the given arguments results in an impossibility, some values will be overwritten
To achieve frozen orbit these two equations have to be set to zero.
.. math::
\dfrac {d\overline {e}}{dt}=\dfrac {-3\overline {n}J_{3}R^{3}_{E}\sin \left( \overline {i}\right) }{2a^{3}\left( 1-\overline {e}^{2}\right) ^{2}}\left( 1-\dfrac {5}{4}\sin ^{2}\overline {i}\right) \cos \overline {w}
.. math::
\dfrac {d\overline {\omega }}{dt}=\dfrac {3\overline {n}J_{2}R^{2}_{E}}{a^{2}\left( 1-\overline {e}^{2}\right) ^{2}}\left( 1-\dfrac {5}{4}\sin ^{2}\overline {i}\right) \left[ 1+\dfrac {J_{3}R_{E}}{2J_{2}\overline {a}\left( 1-\overline {e}^{2}\right) }\left( \dfrac {\sin ^{2}\overline {i}-\overline {e}\cos ^{2}\overline {i}}{\sin \overline {i}}\right) \dfrac {\sin \overline {w}}{\overline {e}}\right]
The first approach would be to nullify following term to zero:
.. math::
( 1-\dfrac {5}{4}\sin ^{2})
For which one obtains the so-called critical inclinations: i = 63.4349 or 116.5651 degrees. To escape the inclination requirement, the argument of periapsis can be set to w = 90 or 270 degrees to nullify the second equation. Then, one should nullify the right-hand side of the first equation, which yields an expression that correlates the inclination of the object and the eccentricity of the orbit:
.. math::
\overline {e}=-\dfrac {J_{3}R_{E}}{2J_{2}\overline {a}\left( 1-\overline {e}^{2}\right) }\left( \dfrac {\sin ^{2}\overline {i}-\overline {e}\cos ^{2} \overline {i}}{\sin \overline {i}}\right)
Assuming that e is negligible compared to J2, it can be shown that:
.. math::
\overline {e}\approx -\dfrac {J_{3}R_{E}}{2J_{2}\overline {a}}\sin \overline {i}
The implementation is divided in the following cases:
1. When the user gives a negative altitude, the method will raise a ValueError
2. When the attractor has not defined J2 or J3, the method will raise an AttributeError
3. When the attractor has J2/J3 outside of range 1 to 10 , the method will raise an NotImplementedError. Special case for Venus.See "Extension of the critical inclination" by Xiaodong Liu, Hexi Baoyin, and Xingrui Ma
4. If argp is not given or the given argp is a critical value:
* if eccentricity is none and inclination is none, the inclination is set with a critical value and the eccentricity is obtained from the last formula mentioned
* if only eccentricity is none, we calculate this value with the last formula mentioned
* if only inclination is none ,we calculate this value with the formula for eccentricity with critical argp.
5. If inc is not given or the given inc is critical:
* if the argp and the eccentricity is given we keep these values to create the orbit
* if the eccentricity is given we keep this value, if not, default to the eccentricity of the Moon's orbit around the Earth
6. if it's not possible to create an orbit with the the argp and the inclination given, both of them are set to the critical values and the eccentricity is calculate with the last formula
Parameters
----------
attractor : Body
Main attractor.
alt : ~astropy.units.Quantity
Altitude over surface.
inc : ~astropy.units.Quantity, optional
Inclination, default to critical value.
argp : ~astropy.units.Quantity, optional
Argument of the pericenter, default to critical value.
raan : ~astropy.units.Quantity, optional
Right ascension of the ascending node, default to 0 deg.
arglat : ~astropy.units.Quantity, optional
Argument of latitude, default to 0 deg.
ecc : ~astropy.units.Quantity
Eccentricity, default to the eccentricity of the Moon's orbit around the Earth
epoch: ~astropy.time.Time, optional
Epoch, default to J2000.
plane : ~poliastro.frames.Planes
Fundamental plane of the frame.
"""
critical_argps = [np.pi / 2, 3 * np.pi / 2] * u.rad
critical_inclinations = [63.4349 * np.pi / 180, 116.5651 * np.pi / 180] * u.rad
try:
if 1 <= np.abs(attractor.J2 / attractor.J3) <= 10:
raise NotImplementedError(
"This has not been implemented for {}".format(attractor.name)
)
assert alt > 0
argp = critical_argps[0] if argp is None else argp
a = attractor.R + alt
critical_argp = find_closest_value(argp, critical_argps)
if np.isclose(argp, critical_argp, 1e-8, 1e-5 * u.rad):
if inc is None and ecc is None:
inc = critical_inclinations[0]
ecc = get_eccentricity_critical_argp(attractor, a, inc)
elif ecc is None:
ecc = get_eccentricity_critical_argp(attractor, a, inc)
else:
inc = get_inclination_critical_argp(attractor, a, ecc)
return cls.from_classical(
attractor, a, ecc, inc, raan, argp, arglat, epoch, plane
)
inc = critical_inclinations[0] if inc is None else inc
critical_inclination = find_closest_value(inc, critical_inclinations)
if np.isclose(inc, critical_inclination, 1e-8, 1e-5 * u.rad):
ecc = get_eccentricity_critical_inc(ecc)
return cls.from_classical(
attractor, a, ecc, inc, raan, argp, arglat, epoch, plane
)
argp = critical_argps[0]
inc = critical_inclinations[0]
ecc = get_eccentricity_critical_argp(attractor, a, inc)
return cls.from_classical(
attractor, a, ecc, inc, raan, argp, arglat, epoch, plane
)
except AttributeError:
raise AttributeError(
"Attractor {} has not spherical harmonics implemented".format(
attractor.name
)
)
except AssertionError:
raise ValueError(
"The semimajor axis may not be smaller that {}'s radius".format(
attractor.name
)
)