Natural and artificial perturbations



In [1]:

    
# Temporary hack, see https://github.com/poliastro/poliastro/issues/281
from IPython.display import HTML
HTML('<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>')

import numpy as np

from plotly.offline import init_notebook_mode
init_notebook_mode(connected=True)



In [2]:

    
%matplotlib inline
import matplotlib.pyplot as plt

import functools

import numpy as np
from astropy import units as u
from astropy.time import Time
from astropy.coordinates import solar_system_ephemeris

from poliastro.twobody.propagation import cowell
from poliastro.ephem import build_ephem_interpolant
from poliastro.core.elements import rv2coe

from poliastro.core.util import norm
from poliastro.util import time_range
from poliastro.core.perturbations import (
    atmospheric_drag, third_body, J2_perturbation
)
from poliastro.bodies import Earth, Moon
from poliastro.twobody import Orbit
from poliastro.plotting import OrbitPlotter, plot, OrbitPlotter3D

Atmospheric drag

The poliastro package now has several commonly used natural perturbations. One of them is atmospheric drag! See how one can monitor decay of the near-Earth orbit over time using our new module poliastro.twobody.perturbations!



In [3]:

    
R = Earth.R.to(u.km).value
k = Earth.k.to(u.km**3 / u.s**2).value

orbit = Orbit.circular(Earth, 250 * u.km, epoch=Time(0.0, format='jd', scale='tdb'))

# 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 * u.s).to(u.day).value
tr = time_range(0.0, periods=2000, end=tof, format='jd', scale='tdb')
cowell_with_ad = functools.partial(cowell, ad=atmospheric_drag,
                                   R=R, C_D=C_D, A=A, m=m, H0=H0, rho0=rho0)

rr = orbit.sample(tr, method=cowell_with_ad)



In [4]:

    
plt.ylabel('h(t)')
plt.xlabel('t, days')
plt.plot(tr.value, rr.data.norm() - Earth.R)









    Out[4]:





[<matplotlib.lines.Line2D at 0x7ff1bfb52630>]

Evolution of RAAN due to the J2 perturbation

We can also see how the J2 perturbation changes RAAN over time!



In [5]:

    
r0 = np.array([-2384.46, 5729.01, 3050.46])  # km
v0 = np.array([-7.36138, -2.98997, 1.64354])  # km/s
k = Earth.k.to(u.km**3 / u.s**2).value

orbit = Orbit.from_vectors(Earth, r0 * u.km, v0 * u.km / u.s)

tof = (48.0 * u.h).to(u.s).value
rr, vv = cowell(orbit, np.linspace(0, tof, 2000), ad=J2_perturbation, J2=Earth.J2.value, R=Earth.R.to(u.km).value)
raans = [rv2coe(k, r, v)[3] for r, v in zip(rr, vv)]
plt.ylabel('RAAN(t)')
plt.xlabel('t, s')
plt.plot(np.linspace(0, tof, 2000), raans)









    Out[5]:





[<matplotlib.lines.Line2D at 0x7ff1bfddc400>]

3rd body

Apart from time-independent perturbations such as atmospheric drag, J2/J3, we have time-dependend perturbations. Lets's see how Moon changes the orbit of GEO satellite over time!



In [6]:

    
# database keeping positions of bodies in Solar system over time
solar_system_ephemeris.set('de432s')

j_date = 2454283.0 * u.day  # setting the exact event date is important

tof = (60 * u.day).to(u.s).value

# create interpolant of 3rd body coordinates (calling in on every iteration will be just too slow)
body_r = build_ephem_interpolant(Moon, 28 * u.day, (j_date, j_date + 60 * u.day), rtol=1e-2)

epoch = Time(j_date, format='jd', scale='tdb')
initial = Orbit.from_classical(Earth, 42164.0 * u.km, 0.0001 * u.one, 1 * u.deg, 
                               0.0 * u.deg, 0.0 * u.deg, 0.0 * u.rad, epoch=epoch)

# multiply Moon gravity by 400 so that effect is visible :)
cowell_with_3rdbody = functools.partial(cowell, rtol=1e-6, ad=third_body,
                                        k_third=400 * Moon.k.to(u.km**3 / u.s**2).value, 
                                        third_body=body_r)

tr = time_range(j_date.value, periods=1000, end=j_date.value + 60, format='jd', scale='tdb')
rr = initial.sample(tr, method=cowell_with_3rdbody)



In [7]:

    
frame = OrbitPlotter3D()

frame.set_attractor(Earth)
frame.plot_trajectory(rr, label='orbit influenced by Moon')
frame.show()

Thrusts

Apart from natural perturbations, there are artificial thrusts aimed at intentional change of orbit parameters. One of such changes is simultaineous change of eccenricy and inclination.



In [8]:

    
from poliastro.twobody.thrust import change_inc_ecc

ecc_0, ecc_f = 0.4, 0.0
a = 42164  # km
inc_0 = 0.0  # rad, baseline
inc_f = (20.0 * u.deg).to(u.rad).value  # rad
argp = 0.0  # rad, the method is efficient for 0 and 180
f = 2.4e-6  # km / s2

k = Earth.k.to(u.km**3 / u.s**2).value
s0 = Orbit.from_classical(
    Earth,
    a * u.km, ecc_0 * u.one, inc_0 * u.deg,
    0 * u.deg, argp * u.deg, 0 * u.deg,
    epoch=Time(0, format='jd', scale='tdb')
)
    
a_d, _, _, t_f = change_inc_ecc(s0, ecc_f, inc_f, f)

cowell_with_ad = functools.partial(cowell, rtol=1e-6, ad=a_d)

tr = time_range(0.0, periods=1000, end=(t_f * u.s).to(u.day).value, format='jd', scale='tdb')
rr = s0.sample(tr, method=cowell_with_ad)



In [9]:

    
frame = OrbitPlotter3D()

frame.set_attractor(Earth)
frame.plot_trajectory(rr, label='orbit with artificial thrust')
frame.show()