For cases where we only study the gravitational forces, solving the Kepler's equation is enough to propagate the orbit forward in time. However, when we want to take perturbations that deviate from Keplerian forces into account, we need a more complex method to solve our initial value problem: one of them is Cowell's formulation.
In this formulation we write the two body differential equation separating the Keplerian and the perturbation accelerations:
$$\ddot{\mathbb{r}} = -\frac{\mu}{|\mathbb{r}|^3} \mathbb{r} + \mathbb{a}_d$$
In [1]:
import numpy as np
from astropy import units as u
from matplotlib import ticker
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
plt.ion()
from scipy.integrate import ode
from poliastro.bodies import Earth
from poliastro.twobody import Orbit
from poliastro.examples import iss
from poliastro.twobody.propagation import func_twobody
from poliastro.util import norm
from ipywidgets.widgets import interact, fixed
In [2]:
def state_to_vector(ss):
r, v = ss.rv()
x, y, z = r.to(u.km).value
vx, vy, vz = v.to(u.km / u.s).value
return np.array([x, y, z, vx, vy, vz])
In [3]:
u0 = state_to_vector(iss)
u0
Out[3]:
In [4]:
t = np.linspace(0, 10 * iss.period, 500).to(u.s).value
t[:10]
Out[4]:
In [5]:
dt = t[1] - t[0]
dt
Out[5]:
In [6]:
k = Earth.k.to(u.km**3 / u.s**2).value
To provide an acceleration depending on an extra parameter, we can use closures like this one:
In [7]:
def constant_accel_factory(accel):
def constant_accel(t0, u, k):
v = u[3:]
norm_v = (v[0]**2 + v[1]**2 + v[2]**2)**.5
return accel * v / norm_v
return constant_accel
constant_accel_factory(accel=1e-5)(t[0], u0, k)
Out[7]:
In [8]:
help(func_twobody)
Now we setup the integrator manually using scipy.integrate.ode. We cannot provide the Jacobian since we don't know the form of the acceleration in advance.
In [9]:
res = np.zeros((t.size, 6))
res[0] = u0
ii = 1
accel = 1e-5
rr = ode(func_twobody).set_integrator('dop853') # All parameters by default
rr.set_initial_value(u0, t[0])
rr.set_f_params(k, constant_accel_factory(accel))
while rr.successful() and rr.t + dt < t[-1]:
rr.integrate(rr.t + dt)
res[ii] = rr.y
ii += 1
res[:5]
Out[9]:
And we plot the results:
In [10]:
fig = plt.figure(figsize=(10, 10))
ax = fig.add_subplot(111, projection='3d')
ax.plot(*res[:, :3].T)
ax.view_init(14, 70)
This is the last time we used scipy.integrate.ode directly. Instead, we can now import a convenient function from poliastro:
In [11]:
from poliastro.twobody.propagation import cowell
In [12]:
def plot_iss(thrust=0.1, mass=2000.):
r0, v0 = iss.rv()
k = iss.attractor.k
t = np.linspace(0, 10 * iss.period, 500).to(u.s).value
u0 = state_to_vector(iss)
res = np.zeros((t.size, 6))
res[0] = u0
accel = thrust / mass
# Perform the whole integration
r0 = r0.to(u.km).value
v0 = v0.to(u.km / u.s).value
k = k.to(u.km**3 / u.s**2).value
ad = constant_accel_factory(accel)
r, v = r0, v0
for ii in range(1, len(t)):
r, v = cowell(k, r, v, t[ii] - t[ii - 1], ad=ad)
x, y, z = r
vx, vy, vz = v
res[ii] = [x, y, z, vx, vy, vz]
fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111, projection='3d')
ax.set_xlim(-20e3, 20e3)
ax.set_ylim(-20e3, 20e3)
ax.set_zlim(-20e3, 20e3)
ax.view_init(14, 70)
return ax.plot(*res[:, :3].T)
In [13]:
interact(plot_iss, thrust=(0.0, 0.2, 0.001), mass=fixed(2000.))
Out[13]:
In [14]:
rtol = 1e-13
full_periods = 2
In [15]:
u0 = state_to_vector(iss)
tf = ((2 * full_periods + 1) * iss.period / 2).to(u.s).value
u0, tf
Out[15]:
In [16]:
iss_f_kep = iss.propagate(tf * u.s, rtol=1e-18)
In [17]:
r0, v0 = iss.rv()
r, v = cowell(k, r0.to(u.km).value, v0.to(u.km / u.s).value, tf, rtol=rtol)
iss_f_num = Orbit.from_vectors(Earth, r * u.km, v * u.km / u.s, iss.epoch + tf * u.s)
In [18]:
iss_f_num.r, iss_f_kep.r
Out[18]:
In [19]:
assert np.allclose(iss_f_num.r, iss_f_kep.r, rtol=rtol, atol=1e-08 * u.km)
assert np.allclose(iss_f_num.v, iss_f_kep.v, rtol=rtol, atol=1e-08 * u.km / u.s)
In [20]:
#assert np.allclose(iss_f_num.a, iss_f_kep.a, rtol=rtol, atol=1e-08 * u.km)
#assert np.allclose(iss_f_num.ecc, iss_f_kep.ecc, rtol=rtol)
#assert np.allclose(iss_f_num.inc, iss_f_kep.inc, rtol=rtol, atol=1e-08 * u.rad)
#assert np.allclose(iss_f_num.raan, iss_f_kep.raan, rtol=rtol, atol=1e-08 * u.rad)
#assert np.allclose(iss_f_num.argp, iss_f_kep.argp, rtol=rtol, atol=1e-08 * u.rad)
#assert np.allclose(iss_f_num.nu, iss_f_kep.nu, rtol=rtol, atol=1e-08 * u.rad)
Too bad I cannot access the internal state of the solver. I will have to do it in a blackbox way.
In [21]:
u0 = state_to_vector(iss)
full_periods = 4
tof_vector = np.linspace(0, ((2 * full_periods + 1) * iss.period / 2).to(u.s).value, num=100)
rtol_vector = np.logspace(-3, -12, num=30)
res_array = np.zeros((rtol_vector.size, tof_vector.size))
for jj, tof in enumerate(tof_vector):
rf, vf = iss.propagate(tof * u.s, rtol=1e-12).rv()
for ii, rtol in enumerate(rtol_vector):
rr = ode(func_twobody).set_integrator('dop853', rtol=rtol, nsteps=1000)
rr.set_initial_value(u0, 0.0)
rr.set_f_params(k, constant_accel_factory(0.0)) # Zero acceleration
rr.integrate(rr.t + tof)
if rr.successful():
uf = rr.y
r, v = uf[:3] * u.km, uf[3:] * u.km / u.s
res = max(norm((r - rf) / rf), norm((v - vf) / vf))
else:
res = np.nan
res_array[ii, jj] = res
In [22]:
fig, ax = plt.subplots(figsize=(16, 6))
xx, yy = np.meshgrid(tof_vector, rtol_vector)
cs = ax.contourf(xx, yy, res_array, levels=np.logspace(-12, -1, num=12),
locator=ticker.LogLocator(), cmap=plt.cm.Spectral_r)
fig.colorbar(cs)
for nn in range(full_periods + 1):
lf = ax.axvline(nn * iss.period.to(u.s).value, color='k', ls='-')
lh = ax.axvline((2 * nn + 1) * iss.period.to(u.s).value / 2, color='k', ls='--')
ax.set_yscale('log')
ax.set_xlabel("Time of flight (s)")
ax.set_ylabel("Relative tolerance")
ax.set_title("Maximum relative difference")
ax.legend((lf, lh), ("Full period", "Half period"))
Out[22]:
According to [Edelbaum, 1961], a coplanar, semimajor axis change with tangent thrust is defined by:
$$\frac{\operatorname{d}\!a}{a_0} = 2 \frac{F}{m V_0}\operatorname{d}\!t, \qquad \frac{\Delta{V}}{V_0} = \frac{1}{2} \frac{\Delta{a}}{a_0}$$So let's create a new circular orbit and perform the necessary checks, assuming constant mass and thrust (i.e. constant acceleration):
In [24]:
ss = Orbit.circular(Earth, 500 * u.km)
tof = 20 * ss.period
ad = constant_accel_factory(1e-7)
r0, v0 = ss.rv()
r, v = cowell(k, r0.to(u.km).value, v0.to(u.km / u.s).value,
tof.to(u.s).value, ad=ad)
ss_final = Orbit.from_vectors(Earth, r * u.km, v * u.km / u.s, ss.epoch + rr.t * u.s)
In [25]:
da_a0 = (ss_final.a - ss.a) / ss.a
da_a0
Out[25]:
In [26]:
dv_v0 = abs(norm(ss_final.v) - norm(ss.v)) / norm(ss.v)
2 * dv_v0
Out[26]:
In [27]:
np.allclose(da_a0, 2 * dv_v0, rtol=1e-2)
Out[27]:
In [28]:
dv = abs(norm(ss_final.v) - norm(ss.v))
dv
Out[28]:
In [29]:
accel_dt = accel * u.km / u.s**2 * (t[-1] - t[0]) * u.s
accel_dt
Out[29]:
In [30]:
np.allclose(dv, accel_dt, rtol=1e-2, atol=1e-8 * u.km / u.s)
Out[30]:
This means we successfully validated the model against an extremely simple orbit transfer with approximate analytical solution. Notice that the final eccentricity, as originally noticed by Edelbaum, is nonzero:
In [31]:
ss_final.ecc
Out[31]: