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User guide

Defining the orbit: :py:class:`~poliastro.twobody.orbit.Orbit` objects

The core of poliastro are the :py:class:`~poliastro.twobody.orbit.Orbit` objects inside the :py:mod:`poliastro.twobody` module. They store all the required information to define an orbit:

  • The body acting as the central body of the orbit, for example the Earth.
  • The position and velocity vectors or the orbital elements.
  • The time at which the orbit is defined.

First of all, we have to import the relevant modules and classes:

import numpy as np
import matplotlib.pyplot as plt
plt.ion()  # To immediately show plots

from astropy import units as u

from poliastro.bodies import Earth, Mars, Sun
from poliastro.twobody import Orbit

plt.style.use("seaborn")  # Recommended

From position and velocity

There are several methods available to create :py:class:`~poliastro.twobody.orbit.Orbit` objects. For example, if we have the position and velocity vectors we can use :py:meth:`~poliastro.twobody.orbit.Orbit.from_vectors`:

# Data from Curtis, example 4.3
r = [-6045, -3490, 2500] * u.km
v = [-3.457, 6.618, 2.533] * u.km / u.s

ss = Orbit.from_vectors(Earth, r, v)

And that's it! Notice a couple of things:

  • Defining vectorial physical quantities using Astropy units is very easy. The list is automatically converted to a :py:mod:`astropy.units.Quantity`, which is actually a subclass of NumPy arrays.

  • If we display the orbit we just created, we get a string with the radius of pericenter, radius of apocenter, inclination, reference frame and attractor:

    >>> ss
    7283 x 10293 km x 153.2 deg (GCRS) orbit around Earth (♁)
    
  • If no time is specified, then a default value is assigned:

    >>> ss.epoch
    <Time object: scale='utc' format='jyear_str' value=J2000.000>
    >>> ss.epoch.iso
    '2000-01-01 12:00:00.000'
    
  • The reference frame of the orbit will be one pseudo-inertial frame around the attractor. You can retrieve it using the :py:attr:`~poliastro.twobody.orbit.Orbit.frame` property:

    >>> ss.frame
    <GCRS Frame (obstime=J2000.000, obsgeoloc=(0., 0., 0.) m, obsgeovel=(0., 0., 0.) m / s)>

Note

At the moment, there is no explicit way to set the reference system of an orbit. This is the focus of our next releases, so we will likely introduce changes in the near future. Please subscribe to this issue to receive updates in your inbox.

Intermezzo: quick visualization of the orbit

Plot of the orbit

If we're working on interactive mode (for example, using the wonderful IPython notebook) we can immediately plot the current state:

from poliastro.plotting import plot
plot(ss)

This plot is made in the so called perifocal frame, which means:

  • we're visualizing the plane of the orbit itself,
  • the \(x\) axis points to the pericenter, and
  • the \(y\) axis is turned \(90 \mathrm{^\circ}\) in the direction of the orbit.

The dotted line represents the osculating orbit: the instantaneous Keplerian orbit at that point. This is relevant in the context of perturbations, when the object shall deviate from its Keplerian orbit.

Warning

Be aware that, outside the Jupyter notebook (i.e. a normal Python interpreter or program) you might need to call plt.show() after the plotting commands or plt.ion() before them or they won't show. Check out the Matplotlib FAQ for more information.

From classical orbital elements

We can also define a :py:class:`~poliastro.twobody.orbit.Orbit` using a set of six parameters called orbital elements. Although there are several of these element sets, each one with its advantages and drawbacks, right now poliastro supports the classical orbital elements:

  • Semimajor axis \(a\).
  • Eccentricity \(e\).
  • Inclination \(i\).
  • Right ascension of the ascending node \(\Omega\).
  • Argument of pericenter \(\omega\).
  • True anomaly \(\nu\).

In this case, we'd use the method :py:meth:`~poliastro.twobody.orbit.Orbit.from_classical`:

# Data for Mars at J2000 from JPL HORIZONS
a = 1.523679 * u.AU
ecc = 0.093315 * u.one
inc = 1.85 * u.deg
raan = 49.562 * u.deg
argp = 286.537 * u.deg
nu = 23.33 * u.deg

ss = Orbit.from_classical(Sun, a, ecc, inc, raan, argp, nu)

Notice that whether we create a Orbit from \(r\) and \(v\) or from elements we can access many mathematical properties individually using the :py:attr:`~poliastro.twobody.orbit.Orbit.state` property of :py:class:`~poliastro.twobody.orbit.Orbit` objects:

>>> ss.state.period.to(u.day)
<Quantity 686.9713888628166 d>
>>> ss.state.v
<Quantity [  1.16420211, 26.29603612,  0.52229379] km / s>

To see a complete list of properties, check out the :py:class:`poliastro.twobody.orbit.Orbit` class on the API reference.

Moving forward in time: propagation

Now that we have defined an orbit, we might be interested in computing how is it going to evolve in the future. In the context of orbital mechanics, this process is known as propagation, and can be performed with the propagate method of :py:class:`~poliastro.twobody.orbit.Orbit` objects:

>>> from poliastro.examples import iss
>>> iss
6772 x 6790 km x 51.6 deg (GCRS) orbit around Earth (♁)
>>> iss.epoch
<Time object: scale='utc' format='iso' value=2013-03-18 12:00:00.000>
>>> iss.nu.to(u.deg)
<Quantity 46.595804677061956 deg>
>>> iss.n.to(u.deg / u.min)
<Quantity 3.887010576192155 deg / min>

Using the :py:meth:`~poliastro.twobody.orbit.Orbit.propagate` method we can now retrieve the position of the ISS after some time:

>>> iss_30m = iss.propagate(30 * u.min)
>>> iss_30m.epoch  # Notice we advanced the epoch!
<Time object: scale='utc' format='iso' value=2013-03-18 12:30:00.000>
>>> iss_30m.nu.to(u.deg)
<Quantity 163.1409357544868 deg>

For more advanced propagation options, check out the :py:mod:`poliastro.twobody.propagation` module.

Studying non-keplerian orbits: perturbations

Apart from the Keplerian propagators, poliastro also allows the user to define custom perturbation accelerations to study non Keplerian orbits, thanks to Cowell's method:

>>> from poliastro.twobody.propagation import cowell
>>> from numba import njit
>>> r0 = [-2384.46, 5729.01, 3050.46] * u.km
>>> v0 = [-7.36138, -2.98997, 1.64354] * u.km / u.s
>>> initial = Orbit.from_vectors(Earth, r0, v0)
>>> @njit
... def accel(t0, state, k):
...     """Constant acceleration aligned with the velocity. """
...     v_vec = state[3:]
...     norm_v = (v_vec * v_vec).sum() ** .5
...     return 1e-5 * v_vec / norm_v
...
>>> initial.propagate(3 * u.day, method=cowell, ad=accel)
18255 x 21848 km x 28.0 deg (GCRS) orbit around Earth (♁)

Some natural perturbations are available in poliastro to be used directly in this way. For instance, let us examine the effect of J2 perturbation:

>>> from poliastro.core.perturbations import J2_perturbation
>>> tof = (48.0 * u.h).to(u.s)
>>> final = initial.propagate(tof, method=cowell, ad=J2_perturbation, J2=Earth.J2.value, R=Earth.R.to(u.km).value)

The J2 perturbation changes the orbit parameters (from Curtis example 12.2):

>>> ((final.raan - initial.raan) / tof).to(u.deg / u.h)
<Quantity -0.17232668 deg / h>
>>> ((final.argp - initial.argp) / tof).to(u.deg / u.h)
<Quantity 0.28220397 deg / h>

For more available perturbation options, see the :py:mod:`poliastro.twobody.perturbations` module.

Studying artificial perturbations: thrust

In addition to natural perturbations, poliastro also has built-in artificial perturbations (thrusts) aimed at intentional change of some orbital elements. Let us simultaineously change eccentricy and inclination:

>>> from poliastro.twobody.thrust import change_inc_ecc
>>> from poliastro.twobody import Orbit
>>> from poliastro.bodies import Earth
>>> from poliastro.twobody.propagation import cowell
>>> from astropy import units as u
>>> from astropy.time import Time
>>> ecc_0, ecc_f = 0.4, 0.0
>>> a = 42164
>>> inc_0, inc_f = 0.0, (20.0 * u.deg).to(u.rad).value
>>> argp = 0.0
>>> f = 2.4e-7
>>> 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)
>>> sf = s0.propagate(t_f * u.s, method=cowell, ad=a_d, rtol=1e-8)

The thrust changes orbit parameters as desired (within errors):

>>> sf.inc, sf.ecc
(<Quantity 0.34719734 rad>, <Quantity 0.00894513>)

For more available perturbation options, see the :py:mod:`poliastro.twobody.thrust` module.

Changing the orbit: :py:class:`~poliastro.maneuver.Maneuver` objects

poliastro helps us define several in-plane and general out-of-plane maneuvers with the :py:class:`~poliastro.maneuver.Maneuver` class inside the :py:mod:`poliastro.maneuver` module.

Each Maneuver consists on a list of impulses \(\Delta v_i\) (changes in velocity) each one applied at a certain instant \(t_i\). The simplest maneuver is a single change of velocity without delay: you can recreate it either using the :py:meth:`~poliastro.maneuver.Maneuver.impulse` method or instantiating it directly.

from poliastro.maneuver import Maneuver

dv = [5, 0, 0] * u.m / u.s

man = Maneuver.impulse(dv)
man = Maneuver((0 * u.s, dv))  # Equivalent

There are other useful methods you can use to compute common in-plane maneuvers, notably :py:meth:`~poliastro.maneuver.Maneuver.hohmann` and :py:meth:`~poliastro.maneuver.Maneuver.bielliptic` for Hohmann and bielliptic transfers respectively. Both return the corresponding Maneuver object, which in turn you can use to calculate the total cost in terms of velocity change (\(\sum |\Delta v_i|\)) and the transfer time:

>>> ss_i = Orbit.circular(Earth, alt=700 * u.km)
>>> ss_i
7078 x 7078 km x 0.0 deg (GCRS) orbit around Earth (♁)
>>> hoh = Maneuver.hohmann(ss_i, 36000 * u.km)
>>> hoh.get_total_cost()
<Quantity 3.6173981270031357 km / s>
>>> hoh.get_total_time()
<Quantity 15729.741535747102 s>

You can also retrieve the individual vectorial impulses:

>>> hoh.impulses[0]
(<Quantity 0 s>, <Quantity [ 0.        , 2.19739818, 0.        ] km / s>)
>>> hoh[0]  # Equivalent
(<Quantity 0 s>, <Quantity [ 0.        , 2.19739818, 0.        ] km / s>)
>>> tuple(val.decompose([u.km, u.s]) for val in hoh[1])
(<Quantity 15729.741535747102 s>, <Quantity [ 0.        , 1.41999995, 0.        ] km / s>)

To actually retrieve the resulting Orbit after performing a maneuver, use the method :py:meth:`~poliastro.twobody.orbit.Orbit.apply_maneuver`:

>>> ss_f = ss_i.apply_maneuver(hoh)
>>> ss_f
36000 x 36000 km x 0.0 deg (GCRS) orbit around Earth (♁)

More advanced plotting: :py:class:`~poliastro.plotting.OrbitPlotter` objects

We previously saw the :py:func:`poliastro.plotting.plot` function to easily plot orbits. Now we'd like to plot several orbits in one graph (for example, the maneuver we computed in the previous section). For this purpose, we have :py:class:`~poliastro.plotting.OrbitPlotter` objects in the :py:mod:`~poliastro.plotting` module.

These objects hold the perifocal plane of the first Orbit we plot in them, projecting any further trajectories on this plane. This allows to easily visualize in two dimensions:

from poliastro.plotting import OrbitPlotter

op = OrbitPlotter()
ss_a, ss_f = ss_i.apply_maneuver(hoh, intermediate=True)
op.plot(ss_i, label="Initial orbit")
op.plot(ss_a, label="Transfer orbit")
op.plot(ss_f, label="Final orbit")

Which produces this beautiful plot:

Hohmann transfer

Plot of a Hohmann transfer.

Where are the planets? Computing ephemerides

.. versionadded:: 0.3.0

Thanks to Astropy and jplephem, poliastro can now read Satellite Planet Kernel (SPK) files, part of NASA's SPICE toolkit. This means that we can query the position and velocity of the planets of the Solar System.

The method :py:meth:`~poliastro.twobody.orbit.Orbit.get_body_ephem` will return a planetary orbit using low precision ephemerides available in Astropy and an :py:mod:`astropy.time.Time`:

from astropy import time
epoch = time.Time("2015-05-09 10:43")  # UTC by default

And finally, retrieve the planet orbit:

>>> from poliastro import ephem
>>> Orbit.from_body_ephem(Earth, epoch)
1 x 1 AU x 23.4 deg (ICRS) orbit around Sun (☉)

This does not require any external download. If on the other hand we want to use higher precision ephemerides, we can tell Astropy to do so:

>>> from astropy.coordinates import solar_system_ephemeris
>>> solar_system_ephemeris.set("jpl")
Downloading http://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/de430.bsp
|==========>-------------------------------|  23M/119M (19.54%) ETA    59s22ss23

This in turn will download the ephemerides files from NASA and use them for future computations. For more information, check out Astropy documentation on ephemerides.

Note

The position and velocity vectors are given with respect to the Solar System Barycenter in the International Celestial Reference Frame (ICRF), which means approximately equatorial coordinates.

Traveling through space: solving the Lambert problem

The determination of an orbit given two position vectors and the time of flight is known in celestial mechanics as Lambert's problem, also known as two point boundary value problem. This contrasts with Kepler's problem or propagation, which is rather an initial value problem.

The package :py:obj:`poliastro.iod` allows as to solve Lambert's problem, provided the main attractor's gravitational constant, the two position vectors and the time of flight. As you can imagine, being able to compute the positions of the planets as we saw in the previous section is the perfect complement to this feature!

For instance, this is a simplified version of the example Going to Mars with Python using poliastro, where the orbit of the Mars Science Laboratory mission (rover Curiosity) is determined:

date_launch = time.Time('2011-11-26 15:02', scale='utc')
date_arrival = time.Time('2012-08-06 05:17', scale='utc')
tof = date_arrival - date_launch

ss0 = Orbit.from_body_ephem(Earth, date_launch)
ssf = Orbit.from_body_ephem(Mars, date_arrival)

from poliastro import iod
(v0, v), = iod.lambert(Sun.k, ss0.r, ssf.r, tof)

And these are the results:

>>> v0
<Quantity [-29.29150998, 14.53326521,  5.41691336] km / s>
>>> v
<Quantity [ 17.6154992 ,-10.99830723, -4.20796062] km / s>
MSL orbit

Mars Science Laboratory orbit.

Working with NEOs

NEOs (Near Earth Objects) are asteroids and comets whose orbits are near to earth (obvious, isn't it?). More correctly, their perihelion (closest approach to the Sun) is less than 1.3 astronomical units (≈ 200 * 106 km). Currently, they are being an important subject of study for scientists around the world, due to their status as the relatively unchanged remains from the solar system formation process.

Because of that, a new module related to NEOs has been added to poliastro as part of SOCIS 2017 project.

For the moment, it is possible to search NEOs by name (also using wildcards), and get their orbits straight from NASA APIs, using :py:func:`~poliastro.neos.orbit_from_name`. For example, we can get Apophis asteroid (99942 Apophis) orbit with one command, and plot it:

from poliastro.neos import neows

apophis_orbit = neows.orbit_from_name('apophis')  # Also '99942' or '99942 apophis' works
earth_orbit =  Orbit.from_body_ephem(Earth)

op = OrbitPlotter()
op.plot(earth_orbit, label='Earth')
op.plot(apophis_orbit, label='Apophis')
Apophis asteroid orbit

Apophis asteroid orbit compared to Earth orbit.

Per Python ad astra ;)