# Cowell’s formulation¶

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:

$\ddot{\mathbb{r}} = -\frac{\mu}{|\mathbb{r}|^3} \mathbb{r} + \mathbb{a}_d$
For an in-depth exploration of this topic, still to be integrated in poliastro, check out https://github.com/Juanlu001/pfc-uc3m

## First example¶

Let’s setup a very simple example with constant acceleration to visualize the effects on the orbit.

In [1]:

%matplotlib inline
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

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]:

array([  8.59072560e+02,  -4.13720368e+03,   5.29556871e+03,
7.37289205e+00,   2.08223573e+00,   4.39999794e-01])

In [4]:

t = np.linspace(0, 10 * iss.period, 500).to(u.s).value
t[:10]

Out[4]:

array([    0.        ,   111.36211826,   222.72423652,   334.08635478,
445.44847304,   556.8105913 ,   668.17270956,   779.53482782,
890.89694608,  1002.25906434])

In [5]:

dt = t[1] - t[0]
dt

Out[5]:

111.36211825977986

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]:

array([  9.60774274e-06,   2.71339728e-06,   5.73371317e-07])

In [8]:

help(func_twobody)

Help on function func_twobody in module poliastro.twobody.propagation:

Differential equation for the initial value two body problem.

This function follows Cowell's formulation.

Parameters
----------
t0 : float
Time.
u_ : ~numpy.ndarray
Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
k : float
Standard gravitational parameter.
Non Keplerian acceleration (km/s2).



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]:

array([[  8.59072560e+02,  -4.13720368e+03,   5.29556871e+03,
7.37289205e+00,   2.08223573e+00,   4.39999794e-01],
[  1.67120051e+03,  -3.87307888e+03,   5.30240756e+03,
7.19314492e+00,   2.65498748e+00,  -3.17310887e-01],
[  2.45692273e+03,  -3.54744387e+03,   5.22509021e+03,
6.89930296e+00,   3.18546088e+00,  -1.06938976e+00],
[  3.20378169e+03,  -3.16548222e+03,   5.06486727e+03,
6.49612475e+00,   3.66524400e+00,  -1.80427142e+00],
[  3.89994802e+03,  -2.73326986e+03,   4.82430776e+03,
5.99011730e+00,   4.08674433e+00,  -2.51027603e+00]])


And we plot the results:

In [10]:

fig = plt.figure(figsize=(10, 10))

ax.plot(*res[:, :3].T)

ax.view_init(14, 70)


## Interactivity¶

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
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.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]:

<function __main__.plot_iss>


## Error checking¶

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]:

(array([  8.59072560e+02,  -4.13720368e+03,   5.29556871e+03,
7.37289205e+00,   2.08223573e+00,   4.39999794e-01]),
13892.424252907538)

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]:

(<Quantity [ -835.92108005, 4151.60692532,-5303.60427969] km>,
<Quantity [ -835.92108005, 4151.60692532,-5303.60427969] km>)

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

/home/juanlu/.miniconda36/envs/poliastro36/lib/python3.6/site-packages/scipy/integrate/_ode.py:1035: UserWarning: dop853: step size becomes too small
self.messages.get(idid, 'Unexpected idid=%s' % idid))

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]:

<matplotlib.legend.Legend at 0x7f9ad3788e80>


## Numerical validation¶

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

r0, v0 = ss.rv()
r, v = cowell(k, r0.to(u.km).value, v0.to(u.km / u.s).value,

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]:

$2.9896209 \times 10^{-6} \; \mathrm{\frac{km}{m}}$
In [26]:

dv_v0 = abs(norm(ss_final.v) - norm(ss.v)) / norm(ss.v)
2 * dv_v0

Out[26]:

$0.0029960537 \; \mathrm{}$
In [27]:

np.allclose(da_a0, 2 * dv_v0, rtol=1e-2)

Out[27]:

True

In [28]:

dv = abs(norm(ss_final.v) - norm(ss.v))
dv

Out[28]:

$0.011403892 \; \mathrm{\frac{km}{s}}$
In [29]:

accel_dt = accel * u.km / u.s**2 * (t[-1] - t[0]) * u.s
accel_dt

Out[29]:

$0.55569697 \; \mathrm{\frac{km}{s}}$
In [30]:

np.allclose(dv, accel_dt, rtol=1e-2, atol=1e-8 * u.km / u.s)

Out[30]:

False


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]:

$6.6621428 \times 10^{-6} \; \mathrm{}$

## References¶

• [Edelbaum, 1961] “Propulsion requirements for controllable satellites”