def mikkola_coe(k, p, ecc, inc, raan, argp, nu, tof):
a = p / (1 - ecc**2)
n = np.sqrt(k / np.abs(a)**3)
# Solve for specific geometrical case
if ecc < 1.0:
# Equation (9a)
alpha = (1 - ecc) / (4 * ecc + 1 / 2)
M0 = E_to_M(nu_to_E(nu, ecc), ecc)
else:
alpha = (ecc - 1) / (4 * ecc + 1 / 2)
M0 = F_to_M(nu_to_F(nu, ecc), ecc)
M = M0 + n * tof
beta = M / 2 / (4 * ecc + 1 / 2)
# Equation (9b)
if beta >= 0:
z = (beta + np.sqrt(beta**2 + alpha**3))**(1 / 3)
else:
z = (beta - np.sqrt(beta**2 + alpha**3))**(1 / 3)
s = z - alpha / z
# Apply initial correction
if ecc < 1.0:
ds = -0.078 * s**5 / (1 + ecc)
else:
ds = 0.071 * s**5 / (1 + 0.45 * s**2) / (1 + 4 * s**2) / ecc
s += ds
# Solving for the true anomaly
if ecc < 1.0:
E = M + ecc * (3 * s - 4 * s**3)
f = E - ecc * np.sin(E) - M
f1 = 1.0 - ecc * np.cos(E)
f2 = ecc * np.sin(E)
f3 = ecc * np.cos(E)
f4 = -f2
f5 = -f3
else:
E = 3 * np.log(s + np.sqrt(1 + s**2))
f = -E + ecc * np.sinh(E) - M
f1 = -1.0 + ecc * np.cosh(E)
f2 = ecc * np.sinh(E)
f3 = ecc * np.cosh(E)
f4 = f2
f5 = f3
# Apply Taylor expansion
u1 = -f / f1
u2 = -f / (f1 + 0.5 * f2 * u1)
u3 = -f / (f1 + 0.5 * f2 * u2 + (1.0 / 6.0) * f3 * u2**2)
u4 = -f / (f1 + 0.5 * f2 * u3 + (1.0 / 6.0) * f3 * u3**2 +
(1.0 / 24.0) * f4 * (u3**3))
u5 = -f / (f1 + f2 * u4 / 2 + f3 * (u4 * u4) / 6.0 + f4 *
(u4 * u4 * u4) / 24.0 + f5 * (u4 * u4 * u4 * u4) / 120.0)
E += u5
if ecc < 1.0:
nu = E_to_nu(E, ecc)
else:
if ecc == 1.0:
# Parabolic
nu = D_to_nu(E)
else:
# Hyperbolic
nu = F_to_nu(E, ecc)
return nu
def nu_from_delta_t(delta_t, ecc, k=1.0, q=1.0, delta=1e-2):
"""True anomaly for given elapsed time since periapsis.
Parameters
----------
delta_t : float
Time elapsed since periapsis.
ecc : float
Eccentricity.
k : float
Gravitational parameter.
q : float
Periapsis distance.
delta : float
Parameter that controls the size of the near parabolic region.
Returns
-------
nu : float
True anomaly.
"""
if ecc < 1 - delta:
# Strong elliptic
n = np.sqrt(k * (1 - ecc)**3 / q**3)
M = n * delta_t
# This might represent several revolutions,
# so we wrap the true anomaly
E = M_to_E((M + np.pi) % (2 * np.pi) - np.pi, ecc)
nu = E_to_nu(E, ecc)
elif 1 - delta <= ecc < 1:
E_delta = np.arccos((1 - delta) / ecc)
# We compute M assuming we are in the strong elliptic case
# and verify later
n = np.sqrt(k * (1 - ecc)**3 / q**3)
M = n * delta_t
# We check against abs(M) because E_delta could also be negative
if E_to_M(E_delta, ecc) <= abs(M):
# Strong elliptic, proceed
# This might represent several revolutions,
# so we wrap the true anomaly
E = M_to_E((M + np.pi) % (2 * np.pi) - np.pi, ecc)
nu = E_to_nu(E, ecc)
else:
# Near parabolic, recompute M
n = np.sqrt(k / (2 * q**3))
M = n * delta_t
D = M_to_D_near_parabolic(M, ecc)
nu = D_to_nu(D)
elif ecc == 1:
# Parabolic
n = np.sqrt(k / (2 * q**3))
M = n * delta_t
D = M_to_D(M)
nu = D_to_nu(D)
elif 1 < ecc <= 1 + delta:
F_delta = np.arccosh((1 + delta) / ecc)
# We compute M assuming we are in the strong hyperbolic case
# and verify later
n = np.sqrt(k * (ecc - 1)**3 / q**3)
M = n * delta_t
# We check against abs(M) because F_delta could also be negative
if F_to_M(F_delta, ecc) <= abs(M):
# Strong hyperbolic, proceed
F = M_to_F(M, ecc)
nu = F_to_nu(F, ecc)
else:
# Near parabolic, recompute M
n = np.sqrt(k / (2 * q**3))
M = n * delta_t
D = M_to_D_near_parabolic(M, ecc)
nu = D_to_nu(D)
# elif 1 + delta < ecc:
else:
# Strong hyperbolic
n = np.sqrt(k * (ecc - 1)**3 / q**3)
M = n * delta_t
F = M_to_F(M, ecc)
nu = F_to_nu(F, ecc)
return nu
def rv2coe(k, r, v, tol=1e-8):
r"""Converts from vectors to classical orbital elements.
1. First the angular momentum is computed:
.. math::
\vec{h} = \vec{r} \times \vec{v}
2. With it the eccentricity can be solved:
.. math::
\begin{align}
\vec{e} &= \frac{1}{\mu}\left [ \left ( v^{2} - \frac{\mu}{r}\right ) \vec{r} - (\vec{r} \cdot \vec{v})\vec{v} \right ] \\
e &= \sqrt{\vec{e}\cdot\vec{e}} \\
\end{align}
3. The node vector line is solved:
.. math::
\begin{align}
\vec{N} &= \vec{k} \times \vec{h} \\
N &= \sqrt{\vec{N}\cdot\vec{N}}
\end{align}
4. The rigth ascension node is computed:
.. math::
\Omega = \left\{ \begin{array}{lcc}
cos^{-1}{\left ( \frac{N_{x}}{N} \right )} & if & N_{y} \geq 0 \\
\\ 360^{o} -cos^{-1}{\left ( \frac{N_{x}}{N} \right )} & if & N_{y} < 0 \\
\end{array}
\right.
5. The argument of perigee:
.. math::
\omega = \left\{ \begin{array}{lcc}
cos^{-1}{\left ( \frac{\vec{N}\vec{e}}{Ne} \right )} & if & e_{z} \geq 0 \\
\\ 360^{o} -cos^{-1}{\left ( \frac{\vec{N}\vec{e}}{Ne} \right )} & if & e_{z} < 0 \\
\end{array}
\right.
6. And finally the true anomaly:
.. math::
\nu = \left\{ \begin{array}{lcc}
cos^{-1}{\left ( \frac{\vec{e}\vec{r}}{er} \right )} & if & v_{r} \geq 0 \\
\\ 360^{o} -cos^{-1}{\left ( \frac{\vec{e}\vec{r}}{er} \right )} & if & v_{r} < 0 \\
\end{array}
\right.
Parameters
----------
k : float
Standard gravitational parameter (km^3 / s^2)
r : array
Position vector (km)
v : array
Velocity vector (km / s)
tol : float, optional
Tolerance for eccentricity and inclination checks, default to 1e-8
Returns
-------
p : float
Semi-latus rectum of parameter (km)
ecc: float
Eccentricity
inc: float
Inclination (rad)
raan: float
Right ascension of the ascending nod (rad)
argp: float
Argument of Perigee (rad)
nu: float
True Anomaly (rad)
Examples
--------
>>> from poliastro.constants import GM_earth
>>> from astropy import units as u
>>> k = GM_earth.to(u.km ** 3 / u.s ** 2).value # Earth gravitational parameter
>>> r = np.array([-6045., -3490., 2500.])
>>> v = np.array([-3.457, 6.618, 2.533])
>>> p, ecc, inc, raan, argp, nu = rv2coe(k, r, v)
>>> print("p:", p, "[km]")
p: 8530.47436396927 [km]
>>> print("ecc:", ecc)
ecc: 0.17121118195416898
>>> print("inc:", np.rad2deg(inc), "[deg]")
inc: 153.2492285182475 [deg]
>>> print("raan:", np.rad2deg(raan), "[deg]")
raan: 255.27928533439618 [deg]
>>> print("argp:", np.rad2deg(argp), "[deg]")
argp: 20.068139973005366 [deg]
>>> print("nu:", np.rad2deg(nu), "[deg]")
nu: 28.445804984192122 [deg]
Note
----
This example is a real exercise from Orbital Mechanics for Engineering
students by Howard D.Curtis. This exercise is 4.3 of 3rd. Edition, page 200.
"""
h = cross(r, v)
n = cross([0, 0, 1], h)
e = ((v.dot(v) - k / (norm(r))) * r - r.dot(v) * v) / k
ecc = norm(e)
p = h.dot(h) / k
inc = np.arccos(h[2] / norm(h))
circular = ecc < tol
equatorial = abs(inc) < tol
if equatorial and not circular:
raan = 0
argp = np.arctan2(e[1], e[0]) % (2 * np.pi) # Longitude of periapsis
nu = np.arctan2(h.dot(cross(e, r)) / norm(h), r.dot(e))
elif not equatorial and circular:
raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
argp = 0
# Argument of latitude
nu = np.arctan2(r.dot(cross(h, n)) / norm(h), r.dot(n))
elif equatorial and circular:
raan = 0
argp = 0
nu = np.arctan2(r[1], r[0]) % (2 * np.pi) # True longitude
else:
a = p / (1 - (ecc**2))
ka = k * a
if a > 0:
e_se = r.dot(v) / sqrt(ka)
e_ce = norm(r) * v.dot(v) / k - 1
nu = E_to_nu(np.arctan2(e_se, e_ce), ecc)
else:
e_sh = r.dot(v) / sqrt(-ka)
e_ch = norm(r) * (norm(v)**2) / k - 1
nu = F_to_nu(np.log((e_ch + e_sh) / (e_ch - e_sh)) / 2, ecc)
raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
px = r.dot(n)
py = r.dot(cross(h, n)) / norm(h)
argp = (np.arctan2(py, px) - nu) % (2 * np.pi)
nu = (nu + np.pi) % (2 * np.pi) - np.pi
return p, ecc, inc, raan, argp, nu
def mikkola(k, r0, v0, tof, rtol=None):
""" Raw algorithm for Mikkola's Kepler solver.
Parameters
----------
k : ~astropy.units.Quantity
Standard gravitational parameter of the attractor.
r : ~astropy.units.Quantity
Position vector.
v : ~astropy.units.Quantity
Velocity vector.
tofs : ~astropy.units.Quantity
Array of times to propagate.
rtol: float
This method does not require of tolerance since it is non iterative.
Returns
-------
rr : ~astropy.units.Quantity
Propagated position vectors.
vv : ~astropy.units.Quantity
Note
----
Original paper: https://doi.org/10.1007/BF01235850
"""
# Solving for the classical elements
p, ecc, inc, raan, argp, nu = rv2coe(k, r0, v0)
M0 = nu_to_M(nu, ecc, delta=0)
a = p / (1 - ecc**2)
n = np.sqrt(k / np.abs(a)**3)
M = M0 + n * tof
# Solve for specific geometrical case
if ecc < 1.0:
# Equation (9a)
alpha = (1 - ecc) / (4 * ecc + 1 / 2)
else:
alpha = (ecc - 1) / (4 * ecc + 1 / 2)
beta = M / 2 / (4 * ecc + 1 / 2)
# Equation (9b)
if beta >= 0:
z = (beta + np.sqrt(beta**2 + alpha**3))**(1 / 3)
else:
z = (beta - np.sqrt(beta**2 + alpha**3))**(1 / 3)
s = z - alpha / z
# Apply initial correction
if ecc < 1.0:
ds = -0.078 * s**5 / (1 + ecc)
else:
ds = 0.071 * s**5 / (1 + 0.45 * s**2) / (1 + 4 * s**2) / ecc
s += ds
# Solving for the true anomaly
if ecc < 1.0:
E = M + ecc * (3 * s - 4 * s**3)
f = E - ecc * np.sin(E) - M
f1 = 1.0 - ecc * np.cos(E)
f2 = ecc * np.sin(E)
f3 = ecc * np.cos(E)
f4 = -f2
f5 = -f3
else:
E = 3 * np.log(s + np.sqrt(1 + s**2))
f = -E + ecc * np.sinh(E) - M
f1 = -1.0 + ecc * np.cosh(E)
f2 = ecc * np.sinh(E)
f3 = ecc * np.cosh(E)
f4 = f2
f5 = f3
# Apply Taylor expansion
u1 = -f / f1
u2 = -f / (f1 + 0.5 * f2 * u1)
u3 = -f / (f1 + 0.5 * f2 * u2 + (1.0 / 6.0) * f3 * u2**2)
u4 = -f / (f1 + 0.5 * f2 * u3 + (1.0 / 6.0) * f3 * u3**2 +
(1.0 / 24.0) * f4 * (u3**3))
u5 = -f / (f1 + f2 * u4 / 2 + f3 * (u4 * u4) / 6.0 + f4 *
(u4 * u4 * u4) / 24.0 + f5 * (u4 * u4 * u4 * u4) / 120.0)
E += u5
if ecc < 1.0:
nu = E_to_nu(E, ecc)
else:
if ecc == 1.0:
# Parabolic
nu = D_to_nu(E)
else:
# Hyperbolic
nu = F_to_nu(E, ecc)
return coe2rv(k, p, ecc, inc, raan, argp, nu)