def test_atmospheric_drag():
# http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node94.html#sair (10.148)
# given the expression for \dot{r} / r, aproximate \Delta r \approx F_r * \Delta t
R = Earth.R.to(u.km).value
k = Earth.k.to(u.km**3 / u.s**2).value
# parameters of a circular orbit with h = 250 km (any value would do, but not too small)
orbit = Orbit.circular(Earth, 250 * u.km)
r0, _ = orbit.rv()
r0 = r0.to(u.km).value
# parameters of a body
C_D = 2.2 # dimentionless (any value would do)
A = ((np.pi / 4.0) * (u.m**2)).to(u.km**2).value # km^2
m = 100 # kg
B = C_D * A / m
# parameters of the atmosphere
rho0 = Earth.rho0.to(u.kg / u.km**3).value # kg/km^3
H0 = Earth.H0.to(u.km).value
tof = 100000 # s
dr_expected = -B * rho0 * np.exp(-(norm(r0) - R) / H0) * np.sqrt(k * norm(r0)) * tof
# assuming the atmospheric decay during tof is small,
# dr_expected = F_r * tof (Newton's integration formula), where
# F_r = -B rho(r) |r|^2 sqrt(k / |r|^3) = -B rho(r) sqrt(k |r|)
r, v = cowell(orbit, tof, ad=atmospheric_drag, R=R, C_D=C_D, A=A, m=m, H0=H0, rho0=rho0)
assert_quantity_allclose(norm(r) - norm(r0), dr_expected, rtol=1e-2)
def test_atmospheric_drag():
# http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node94.html#sair (10.148)
# given the expression for \dot{r} / r, aproximate \Delta r \approx F_r * \Delta t
R = Earth.R.to(u.km).value
k = Earth.k.to(u.km**3 / u.s**2).value
# parameters of a circular orbit with h = 250 km (any value would do, but not too small)
orbit = Orbit.circular(Earth, 250 * u.km)
r0, _ = orbit.rv()
r0 = r0.to(u.km).value
# parameters of a body
C_D = 2.2 # dimentionless (any value would do)
A = ((np.pi / 4.0) * (u.m**2)).to(u.km**2).value # km^2
m = 100 # kg
B = C_D * A / m
# parameters of the atmosphere
rho0 = Earth.rho0.to(u.kg / u.km**3).value # kg/km^3
H0 = Earth.H0.to(u.km).value
tof = 100000 # s
dr_expected = -B * rho0 * np.exp(-(norm(r0) - R) / H0) * np.sqrt(k * norm(r0)) * tof
# assuming the atmospheric decay during tof is small,
# dr_expected = F_r * tof (Newton's integration formula), where
# F_r = -B rho(r) |r|^2 sqrt(k / |r|^3) = -B rho(r) sqrt(k |r|)
r, v = cowell(orbit, tof, ad=atmospheric_drag, R=R, C_D=C_D, A=A, m=m, H0=H0, rho0=rho0)
assert_quantity_allclose(norm(r) - norm(r0), dr_expected, rtol=1e-2)
def change_ecc_quasioptimal(ss_0, ecc_f, f):
"""Guidance law from the model.
Thrust is aligned with an inertially fixed direction perpendicular to the
semimajor axis of the orbit.
Parameters
----------
ss_0 : Orbit
Initial orbit, containing all the information.
ecc_f : float
Final eccentricity.
f : float
Magnitude of constant acceleration
"""
# We fix the inertial direction at the beginning
k = ss_0.attractor.k.to(u.km**3 / u.s**2).value
a = ss_0.a.to(u.km).value
ecc_0 = ss_0.ecc.value
if ecc_0 > 0.001: # Arbitrary tolerance
ref_vec = ss_0.e_vec / ecc_0
else:
ref_vec = ss_0.r / norm(ss_0.r)
h_unit = ss_0.h_vec / norm(ss_0.h_vec)
thrust_unit = cross(h_unit, ref_vec) * np.sign(ecc_f - ecc_0)
def a_d(t0, u_, k):
accel_v = f * thrust_unit
return accel_v
delta_V, t_f = extra_quantities(k, a, ecc_0, ecc_f, f)
return a_d, delta_V, t_f
def third_body(t0, state, k, k_third, third_body):
r"""Calculates 3rd body acceleration (km/s2)
.. math::
\vec{p} = \mu_{m}\left ( \frac{\vec{r_{m/s}}}{r_{m/s}^3} - \frac{\vec{r_{m}}}{r_{m}^3} \right )
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
third_body: a callable object returning the position of 3rd body
third body that causes the perturbation
Note
----
This formula is taken from Howard Curtis, section 12.10. As an example, a third body could be
the gravity from the Moon acting on a small satellite.
"""
body_r = third_body(t0)
delta_r = body_r - state[:3]
return k_third * delta_r / norm(delta_r)**3 - k_third * body_r / norm(
body_r)**3
def third_body(t0, state, k, k_third, third_body):
r"""Calculates 3rd body acceleration (km/s2)
.. math::
\vec{p} = \mu_{m}\left ( \frac{\vec{r_{m/s}}}{r_{m/s}^3} - \frac{\vec{r_{m}}}{r_{m}^3} \right )
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
third_body: a callable object returning the position of 3rd body
third body that causes the perturbation
Note
----
This formula is taken from Howard Curtis, section 12.10. As an example, a third body could be
the gravity from the Moon acting on a small satellite.
"""
body_r = third_body(t0)
delta_r = body_r - state[:3]
return k_third * delta_r / norm(delta_r) ** 3 - k_third * body_r / norm(body_r) ** 3
def change_inc_ecc(ss_0, ecc_f, inc_f, f):
"""Guidance law from the model.
Thrust is aligned with an inertially fixed direction perpendicular to the
semimajor axis of the orbit.
Parameters
----------
ss_0 : Orbit
Initial orbit, containing all the information.
ecc_f : float
Final eccentricity.
inc_f : float
Final inclination.
f : float
Magnitude of constant acceleration.
"""
# We fix the inertial direction at the beginning
ecc_0 = ss_0.ecc.value
if ecc_0 > 0.001: # Arbitrary tolerance
ref_vec = ss_0.e_vec / ecc_0
else:
ref_vec = ss_0.r / norm(ss_0.r)
h_unit = ss_0.h_vec / norm(ss_0.h_vec)
thrust_unit = cross(h_unit, ref_vec) * np.sign(ecc_f - ecc_0)
inc_0 = ss_0.inc.to(u.rad).value
argp = ss_0.argp.to(u.rad).value
beta_0_ = beta(ecc_0, ecc_f, inc_0, inc_f, argp)
def a_d(t0, u_, k):
r = u_[:3]
v = u_[3:]
nu = rv2coe(k, r, v)[-1]
beta_ = beta_0_ * np.sign(
np.cos(nu)
) # The sign of ß reverses at minor axis crossings
w_ = cross(r, v) / norm(cross(r, v))
accel_v = f * (np.cos(beta_) * thrust_unit + np.sin(beta_) * w_)
return accel_v
delta_V, beta_, t_f = extra_quantities(
ss_0.attractor.k.to(u.km ** 3 / u.s ** 2).value,
ss_0.a.to(u.km).value,
ecc_0,
ecc_f,
inc_0,
inc_f,
argp,
f,
)
return a_d, delta_V, beta_, t_f
def change_inc_ecc(ss_0, ecc_f, inc_f, f):
"""Guidance law from the model.
Thrust is aligned with an inertially fixed direction perpendicular to the
semimajor axis of the orbit.
Parameters
----------
ss_0 : Orbit
Initial orbit, containing all the information.
ecc_f : float
Final eccentricity.
inc_f : float
Final inclination.
f : float
Magnitude of constant acceleration.
"""
# We fix the inertial direction at the beginning
ecc_0 = ss_0.ecc.value
if ecc_0 > 0.001: # Arbitrary tolerance
ref_vec = ss_0.e_vec / ecc_0
else:
ref_vec = ss_0.r / norm(ss_0.r)
h_unit = ss_0.h_vec / norm(ss_0.h_vec)
thrust_unit = cross(h_unit, ref_vec) * np.sign(ecc_f - ecc_0)
inc_0 = ss_0.inc.to(u.rad).value
argp = ss_0.argp.to(u.rad).value
beta_0_ = beta(ecc_0, ecc_f, inc_0, inc_f, argp)
def a_d(t0, u_, k):
r = u_[:3]
v = u_[3:]
nu = rv2coe(k, r, v)[-1]
beta_ = beta_0_ * np.sign(
np.cos(nu)
) # The sign of ß reverses at minor axis crossings
w_ = cross(r, v) / norm(cross(r, v))
accel_v = f * (np.cos(beta_) * thrust_unit + np.sin(beta_) * w_)
return accel_v
delta_V, beta_, t_f = extra_quantities(
ss_0.attractor.k.to(u.km ** 3 / u.s ** 2).value,
ss_0.a.to(u.km).value,
ecc_0,
ecc_f,
inc_0,
inc_f,
argp,
f,
)
return a_d, delta_V, beta_, t_f
def a_d(t0, u_, k):
r = u_[:3]
v = u_[3:]
nu = rv2coe(k, r, v)[-1]
alpha_ = nu - np.pi / 2
r_ = r / norm(r)
w_ = cross(r, v) / norm(cross(r, v))
s_ = cross(w_, r_)
accel_v = f * (np.cos(alpha_) * s_ + np.sin(alpha_) * r_)
return accel_v
def a_d(t0, u_, k):
r = u_[:3]
v = u_[3:]
# Change sign of beta with the out-of-plane velocity
beta_ = beta(t0, V_0=V_0, f=f, beta_0=beta_0_) * np.sign(r[0] * (inc_f - inc_0))
t_ = v / norm(v)
w_ = cross(r, v) / norm(cross(r, v))
# n_ = cross(t_, w_)
accel_v = f * (np.cos(beta_) * t_ + np.sin(beta_) * w_)
return accel_v
def a_d(t0, u_, k):
r = u_[:3]
v = u_[3:]
nu = rv2coe(k, r, v)[-1]
alpha_ = nu - np.pi / 2
r_ = r / norm(r)
w_ = cross(r, v) / norm(cross(r, v))
s_ = cross(w_, r_)
accel_v = f * (np.cos(alpha_) * s_ + np.sin(alpha_) * r_)
return accel_v
def a_d(t0, u_, k):
r = u_[:3]
v = u_[3:]
# Change sign of beta with the out-of-plane velocity
beta_ = beta(t0, V_0=V_0, f=f, beta_0=beta_0_) * np.sign(
r[0] * (inc_f - inc_0))
t_ = v / norm(v)
w_ = cross(r, v) / norm(cross(r, v))
# n_ = cross(t_, w_)
accel_v = f * (np.cos(beta_) * t_ + np.sin(beta_) * w_)
return accel_v
def J2_perturbation(t0, state, k, J2, R):
"""Calculates J2_perturbation acceleration (km/s2)
.. versionadded:: 0.9.0
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
J2: float
oblateness factor
R: float
attractor radius
Notes
-----
The J2 accounts for the oblateness of the attractor. The formula is given in
Howard Curtis, (12.30)
"""
r_vec = state[:3]
r = norm(r_vec)
factor = (3.0 / 2.0) * k * J2 * (R**2) / (r**5)
a_x = 5.0 * r_vec[2]**2 / r**2 - 1
a_y = 5.0 * r_vec[2]**2 / r**2 - 1
a_z = 5.0 * r_vec[2]**2 / r**2 - 3
return np.array([a_x, a_y, a_z]) * r_vec * factor
def J2_perturbation(t0, state, k, J2, R):
"""Calculates J2_perturbation acceleration (km/s2)
.. versionadded:: 0.9.0
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
J2: float
oblateness factor
R: float
attractor radius
Notes
-----
The J2 accounts for the oblateness of the attractor. The formula is given in
Howard Curtis, (12.30)
"""
r_vec = state[:3]
r = norm(r_vec)
factor = (3.0 / 2.0) * k * J2 * (R ** 2) / (r ** 5)
a_x = 5.0 * r_vec[2] ** 2 / r ** 2 - 1
a_y = 5.0 * r_vec[2] ** 2 / r ** 2 - 1
a_z = 5.0 * r_vec[2] ** 2 / r ** 2 - 3
return np.array([a_x, a_y, a_z]) * r_vec * factor
def J3_perturbation(t0, state, k, J3, R):
"""Calculates J3_perturbation acceleration (km/s2)
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
J3: float
oblateness factor
R: float
attractor radius
Notes
-----
The J3 accounts for the oblateness of the attractor. The formula is given in
Howard Curtis, problem 12.8
This perturbation has not been fully validated, see https://github.com/poliastro/poliastro/pull/398
"""
r_vec = state[:3]
r = norm(r_vec)
factor = (1.0 / 2.0) * k * J3 * (R**3) / (r**5)
cos_phi = r_vec[2] / r
a_x = 5.0 * r_vec[0] / r * (7.0 * cos_phi**3 - 3.0 * cos_phi)
a_y = 5.0 * r_vec[1] / r * (7.0 * cos_phi**3 - 3.0 * cos_phi)
a_z = 3.0 * (35.0 / 3.0 * cos_phi**4 - 10.0 * cos_phi**2 + 1)
return np.array([a_x, a_y, a_z]) * factor
def J3_perturbation(t0, state, k, J3, R):
"""Calculates J3_perturbation acceleration (km/s2)
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
J3: float
oblateness factor
R: float
attractor radius
Notes
-----
The J3 accounts for the oblateness of the attractor. The formula is given in
Howard Curtis, problem 12.8
This perturbation has not been fully validated, see https://github.com/poliastro/poliastro/pull/398
"""
r_vec = state[:3]
r = norm(r_vec)
factor = (1.0 / 2.0) * k * J3 * (R ** 3) / (r ** 5)
cos_phi = r_vec[2] / r
a_x = 5.0 * r_vec[0] / r * (7.0 * cos_phi ** 3 - 3.0 * cos_phi)
a_y = 5.0 * r_vec[1] / r * (7.0 * cos_phi ** 3 - 3.0 * cos_phi)
a_z = 3.0 * (35.0 / 3.0 * cos_phi ** 4 - 10.0 * cos_phi ** 2 + 1)
return np.array([a_x, a_y, a_z]) * factor
def atmospheric_drag(t0, state, k, R, C_D, A, m, H0, rho0):
r"""Calculates atmospheric drag acceleration (km/s2)
.. math::
\vec{p} = -\frac{1}{2}\rho v_{rel}\left ( \frac{C_{d}A}{m} \right )\vec{v_{rel}}
.. versionadded:: 0.9.0
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
R : float
radius of the attractor (km)
C_D: float
dimensionless drag coefficient ()
A: float
frontal area of the spacecraft (km^2)
m: float
mass of the spacecraft (kg)
H0 : float
atmospheric scale height, (km)
rho0: float
the exponent density pre-factor, (kg / m^3)
Note
----
This function provides the acceleration due to atmospheric drag. We follow
Howard Curtis, section 12.4
the atmospheric density model is rho(H) = rho0 x exp(-H / H0)
"""
H = norm(state[:3])
v_vec = state[3:]
v = norm(v_vec)
B = C_D * A / m
rho = rho0 * np.exp(-(H - R) / H0)
return -(1.0 / 2.0) * rho * B * v * v_vec
def third_body(t0, state, k, k_third, third_body):
"""Calculates 3rd body acceleration (km/s2)
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
third_body: a callable object returning the position of 3rd body
third body that causes the perturbation
"""
body_r = third_body(t0)
delta_r = body_r - state[:3]
return k_third * delta_r / norm(delta_r) ** 3 - k_third * body_r / norm(body_r) ** 3
def atmospheric_drag(t0, state, k, R, C_D, A, m, H0, rho0):
r"""Calculates atmospheric drag acceleration (km/s2)
.. math::
\vec{p} = -\frac{1}{2}\rho v_{rel}\left ( \frac{C_{d}A}{m} \right )\vec{v_{rel}}
.. versionadded:: 0.9.0
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
C_D: float
dimensionless drag coefficient ()
A: float
frontal area of the spacecraft (km^2)
m: float
mass of the spacecraft (kg)
H0 : float
atmospheric scale height, (km)
rho0: float
the exponent density pre-factor, (kg / m^3)
Note
----
This function provides the acceleration due to atmospheric drag. We follow
Howard Curtis, section 12.4
the atmospheric density model is rho(H) = rho0 x exp(-H / H0)
"""
H = norm(state[:3])
v_vec = state[3:]
v = norm(v_vec)
B = C_D * A / m
rho = rho0 * np.exp(-(H - R) / H0)
return -(1.0 / 2.0) * rho * B * v * v_vec
def izzo(k, r1, r2, tof, M, numiter, rtol):
# Check preconditions
assert tof > 0
assert k > 0
# Chord
c = r2 - r1
c_norm, r1_norm, r2_norm = norm(c), norm(r1), norm(r2)
# Semiperimeter
s = (r1_norm + r2_norm + c_norm) * .5
# Versors
i_r1, i_r2 = r1 / r1_norm, r2 / r2_norm
i_h = cross(i_r1, i_r2)
i_h = i_h / norm(i_h) # Fixed from paper
# Geometry of the problem
ll = np.sqrt(1 - c_norm / s)
if i_h[2] < 0:
ll = -ll
i_h = -i_h
i_t1, i_t2 = cross(i_h, i_r1), cross(i_h, i_r2) # Fixed from paper
# Non dimensional time of flight
T = np.sqrt(2 * k / s ** 3) * tof
# Find solutions
xy = _find_xy(ll, T, M, numiter, rtol)
# Reconstruct
gamma = np.sqrt(k * s / 2)
rho = (r1_norm - r2_norm) / c_norm
sigma = np.sqrt(1 - rho ** 2)
for x, y in xy:
V_r1, V_r2, V_t1, V_t2 = _reconstruct(x, y, r1_norm, r2_norm, ll,
gamma, rho, sigma)
v1 = V_r1 * i_r1 + V_t1 * i_t1
v2 = V_r2 * i_r2 + V_t2 * i_t2
yield v1, v2
def third_body(t0, state, k, k_third, third_body):
"""Calculates 3rd body acceleration (km/s2)
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
third_body: a callable object returning the position of 3rd body
third body that causes the perturbation
"""
body_r = third_body(t0)
delta_r = body_r - state[:3]
return k_third * delta_r / norm(delta_r)**3 - k_third * body_r / norm(
body_r)**3
def izzo(k, r1, r2, tof, M, numiter, rtol):
# Check preconditions
assert tof > 0
assert k > 0
# Chord
c = r2 - r1
c_norm, r1_norm, r2_norm = norm(c), norm(r1), norm(r2)
# Semiperimeter
s = (r1_norm + r2_norm + c_norm) * .5
# Versors
i_r1, i_r2 = r1 / r1_norm, r2 / r2_norm
i_h = cross(i_r1, i_r2)
i_h = i_h / norm(i_h) # Fixed from paper
# Geometry of the problem
ll = np.sqrt(1 - c_norm / s)
if i_h[2] < 0:
ll = -ll
i_h = -i_h
i_t1, i_t2 = cross(i_h, i_r1), cross(i_h, i_r2) # Fixed from paper
# Non dimensional time of flight
T = np.sqrt(2 * k / s**3) * tof
# Find solutions
xy = _find_xy(ll, T, M, numiter, rtol)
# Reconstruct
gamma = np.sqrt(k * s / 2)
rho = (r1_norm - r2_norm) / c_norm
sigma = np.sqrt(1 - rho**2)
for x, y in xy:
V_r1, V_r2, V_t1, V_t2 = _reconstruct(x, y, r1_norm, r2_norm, ll,
gamma, rho, sigma)
v1 = V_r1 * i_r1 + V_t1 * i_t1
v2 = V_r2 * i_r2 + V_t2 * i_t2
yield v1, v2
def a_d(t0, u_, k):
r = u_[:3]
v = u_[3:]
nu = rv2coe(k, r, v)[-1]
beta_ = beta_0_ * np.sign(
np.cos(nu)) # The sign of ß reverses at minor axis crossings
w_ = cross(r, v) / norm(cross(r, v))
accel_v = f * (np.cos(beta_) * thrust_unit + np.sin(beta_) * w_)
return accel_v
def a_d(t0, u_, k):
r = u_[:3]
v = u_[3:]
nu = rv2coe(k, r, v)[-1]
beta_ = beta_0_ * np.sign(
np.cos(nu)
) # The sign of ß reverses at minor axis crossings
w_ = cross(r, v) / norm(cross(r, v))
accel_v = f * (np.cos(beta_) * thrust_unit + np.sin(beta_) * w_)
return accel_v
def radiation_pressure(t0, state, k, R, C_R, A, m, Wdivc_s, star):
r"""Calculates radiation pressure acceleration (km/s2)
.. math::
\vec{p} = -\nu \frac{S}{c} \left ( \frac{C_{r}A}{m} \right )\frac{\vec{r}}{r}
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
R : float
radius of the attractor
C_R: float
dimensionless radiation pressure coefficient, 1 < C_R < 2 ()
A: float
effective spacecraft area (km^2)
m: float
mass of the spacecraft (kg)
Wdivc_s : float
total star emitted power divided by the speed of light (W * s / km)
star: a callable object returning the position of star in attractor frame
star position
Note
----
This function provides the acceleration due to star light pressure. We follow
Howard Curtis, section 12.9
"""
r_star = star(t0)
r_sat = state[:3]
P_s = Wdivc_s / (norm(r_star) ** 2)
nu = float(shadow_function(r_sat, r_star, R))
return -nu * P_s * (C_R * A / m) * r_star / norm(r_star)
def radiation_pressure(t0, state, k, R, C_R, A, m, Wdivc_s, star):
r"""Calculates radiation pressure acceleration (km/s2)
.. math::
\vec{p} = -\nu \frac{S}{c} \left ( \frac{C_{r}A}{m} \right )\frac{\vec{r}}{r}
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
R : float
radius of the attractor
C_R: float
dimensionless radiation pressure coefficient, 1 < C_R < 2 ()
A: float
effective spacecraft area (km^2)
m: float
mass of the spacecraft (kg)
Wdivc_s : float
total star emitted power divided by the speed of light (W * s / km)
star: a callable object returning the position of star in attractor frame
star position
Note
----
This function provides the acceleration due to star light pressure. We follow
Howard Curtis, section 12.9
"""
r_star = star(t0)
r_sat = state[:3]
P_s = Wdivc_s / (norm(r_star)**2)
nu = float(shadow_function(r_sat, r_star, R))
return -nu * P_s * (C_R * A / m) * r_star / norm(r_star)
def rv2coe(k, r, v, tol=1e-8):
"""Converts from vectors to classical orbital elements.
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.
"""
h = cross(r, v)
n = cross([0, 0, 1], h) / norm(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)) %
(2 * np.pi))
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)) %
(2 * np.pi))
elif equatorial and circular:
raan = 0
argp = 0
nu = np.arctan2(r[1], r[0]) % (2 * np.pi) # True longitude
else:
raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
argp = (np.arctan2(e.dot(cross(h, n)) / norm(h), e.dot(n)) %
(2 * np.pi))
nu = (np.arctan2(r.dot(cross(h, e)) / norm(h), r.dot(e))
% (2 * np.pi))
return p, ecc, inc, raan, argp, nu
def rv2coe(k, r, v, tol=1e-8):
"""Converts from vectors to classical orbital elements.
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.
"""
h = cross(r, v)
n = cross([0, 0, 1], h) / norm(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)) % (2 * np.pi))
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)) % (2 * np.pi))
elif equatorial and circular:
raan = 0
argp = 0
nu = np.arctan2(r[1], r[0]) % (2 * np.pi) # True longitude
else:
raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
argp = (np.arctan2(e.dot(cross(h, n)) / norm(h), e.dot(n)) %
(2 * np.pi))
nu = (np.arctan2(r.dot(cross(h, e)) / norm(h), r.dot(e)) % (2 * np.pi))
return p, ecc, inc, raan, argp, nu
def J2_perturbation(t0, state, k, J2, R):
r"""Calculates J2_perturbation acceleration (km/s2)
.. math::
\vec{p} = \frac{3}{2}\frac{J_{2}\mu R^{2}}{r^{4}}\left [\frac{x}{r}\left ( 5\frac{z^{2}}{r^{2}}-1 \right )\vec{i} + \frac{y}{r}\left ( 5\frac{z^{2}}{r^{2}}-1 \right )\vec{j} + \frac{z}{r}\left ( 5\frac{z^{2}}{r^{2}}-3 \right )\vec{k}\right]
.. versionadded:: 0.9.0
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
J2: float
oblateness factor
R: float
attractor radius
Note
----
The J2 accounts for the oblateness of the attractor. The formula is given in
Howard Curtis, (12.30)
"""
r_vec = state[:3]
r = norm(r_vec)
factor = (3.0 / 2.0) * k * J2 * (R ** 2) / (r ** 5)
a_x = 5.0 * r_vec[2] ** 2 / r ** 2 - 1
a_y = 5.0 * r_vec[2] ** 2 / r ** 2 - 1
a_z = 5.0 * r_vec[2] ** 2 / r ** 2 - 3
return np.array([a_x, a_y, a_z]) * r_vec * factor
def J2_perturbation(t0, state, k, J2, R):
r"""Calculates J2_perturbation acceleration (km/s2)
.. math::
\vec{p} = \frac{3}{2}\frac{J_{2}\mu R^{2}}{r^{4}}\left [\frac{x}{r}\left ( 5\frac{z^{2}}{r^{2}}-1 \right )\vec{i} + \frac{y}{r}\left ( 5\frac{z^{2}}{r^{2}}-1 \right )\vec{j} + \frac{z}{r}\left ( 5\frac{z^{2}}{r^{2}}-3 \right )\vec{k}\right]
.. versionadded:: 0.9.0
Parameters
----------
t0 : float
Current time (s)
state : numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
gravitational constant, (km^3/s^2)
J2: float
oblateness factor
R: float
attractor radius
Note
----
The J2 accounts for the oblateness of the attractor. The formula is given in
Howard Curtis, (12.30)
"""
r_vec = state[:3]
r = norm(r_vec)
factor = (3.0 / 2.0) * k * J2 * (R**2) / (r**5)
a_x = 5.0 * r_vec[2]**2 / r**2 - 1
a_y = 5.0 * r_vec[2]**2 / r**2 - 1
a_z = 5.0 * r_vec[2]**2 / r**2 - 3
return np.array([a_x, a_y, a_z]) * r_vec * factor
def normalize_to_Curtis(t0, sun_r):
r = sun_r(t0)
return 149600000 * r / norm(r)
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) * (norm(v) ** 2) / 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 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.core.elements import rv2coe
>>> from poliastro.constants import GM_earth
>>> from astropy import units as u
>>> import numpy as np
>>> k = GM_earth.to(u.km**3 / u.s**2) #Earth gravitational parameter
>>> r = [-6045, -3490, 2500] * u.km
>>> v = [-3.457, 6.618, 2.533] * u.km / u.s
>>> 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: 20.068139973005362 [deg]
>>> print("nu:", np.rad2deg(nu), "[deg]")
nu: 28.445804984192108 [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) / norm(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)) % (2 * np.pi)
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)) % (2 * np.pi)
elif equatorial and circular:
raan = 0
argp = 0
nu = np.arctan2(r[1], r[0]) % (2 * np.pi) # True longitude
else:
raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
argp = np.arctan2(e.dot(cross(h, n)) / norm(h), e.dot(n)) % (2 * np.pi)
nu = np.arctan2(r.dot(cross(h, e)) / norm(h), r.dot(e)) % (2 * np.pi)
return p, ecc, inc, raan, argp, nu
def izzo(k, r1, r2, tof, M, numiter, rtol):
""" Aplies izzo algorithm to solve Lambert's problem.
Parameters
----------
k: float
Gravitational Constant
r1: ~numpy.array
Initial position vector
r2: ~numpy.array
Final position vector
tof: float
Time of flight between both positions
M: int
Number of revolutions
numiter: int
Numbert of iterations
rotl: float
Error tolerance
Returns
-------
v1: ~numpy.array
Initial velocity vector
v2: ~numpy.array
FInal velocity vector
"""
# Check preconditions
assert tof > 0
assert k > 0
# Check collinearity of r1 and r2
if np.all(cross(r1, r2) == 0):
raise ValueError(
"Lambert solution cannot be computed for collinear vectors")
# Chord
c = r2 - r1
c_norm, r1_norm, r2_norm = norm(c), norm(r1), norm(r2)
# Semiperimeter
s = (r1_norm + r2_norm + c_norm) * 0.5
# Versors
i_r1, i_r2 = r1 / r1_norm, r2 / r2_norm
i_h = cross(i_r1, i_r2)
i_h = i_h / norm(i_h) # Fixed from paper
# Geometry of the problem
ll = np.sqrt(1 - min(1.0, c_norm / s))
if i_h[2] < 0:
ll = -ll
i_h = -i_h
i_t1, i_t2 = cross(i_h, i_r1), cross(i_h, i_r2) # Fixed from paper
# Non dimensional time of flight
T = np.sqrt(2 * k / s**3) * tof
# Find solutions
xy = _find_xy(ll, T, M, numiter, rtol)
# Reconstruct
gamma = np.sqrt(k * s / 2)
rho = (r1_norm - r2_norm) / c_norm
sigma = np.sqrt(1 - rho**2)
for x, y in xy:
V_r1, V_r2, V_t1, V_t2 = _reconstruct(x, y, r1_norm, r2_norm, ll,
gamma, rho, sigma)
v1 = V_r1 * i_r1 + V_t1 * i_t1
v2 = V_r2 * i_r2 + V_t2 * i_t2
yield v1, v2
def normalize_to_Curtis(t0, sun_r):
r = sun_r(t0)
return 149600000 * r / norm(r)
def izzo(k, r1, r2, tof, M, numiter, rtol):
""" Aplies izzo algorithm to solve Lambert's problem.
Parameters
----------
k: float
Gravitational Constant
r1: ~numpy.array
Initial position vector
r2: ~numpy.array
Final position vector
tof: float
Time of flight between both positions
M: int
Number of revolutions
numiter: int
Numbert of iterations
rtol: float
Error tolerance
Returns
-------
v1: ~numpy.array
Initial velocity vector
v2: ~numpy.array
Final velocity vector
"""
# Check preconditions
assert tof > 0
assert k > 0
# Check collinearity of r1 and r2
if np.all(cross(r1, r2) == 0):
raise ValueError("Lambert solution cannot be computed for collinear vectors")
# Chord
c = r2 - r1
c_norm, r1_norm, r2_norm = norm(c), norm(r1), norm(r2)
# Semiperimeter
s = (r1_norm + r2_norm + c_norm) * 0.5
# Versors
i_r1, i_r2 = r1 / r1_norm, r2 / r2_norm
i_h = cross(i_r1, i_r2)
i_h = i_h / norm(i_h) # Fixed from paper
# Geometry of the problem
ll = np.sqrt(1 - min(1.0, c_norm / s))
if i_h[2] < 0:
ll = -ll
i_h = -i_h
i_t1, i_t2 = cross(i_h, i_r1), cross(i_h, i_r2) # Fixed from paper
# Non dimensional time of flight
T = np.sqrt(2 * k / s ** 3) * tof
# Find solutions
xy = _find_xy(ll, T, M, numiter, rtol)
# Reconstruct
gamma = np.sqrt(k * s / 2)
rho = (r1_norm - r2_norm) / c_norm
sigma = np.sqrt(1 - rho ** 2)
for x, y in xy:
V_r1, V_r2, V_t1, V_t2 = _reconstruct(
x, y, r1_norm, r2_norm, ll, gamma, rho, sigma
)
v1 = V_r1 * i_r1 + V_t1 * i_t1
v2 = V_r2 * i_r2 + V_t2 * i_t2
yield v1, v2
