# User guide¶

## Defining the orbit: `Orbit`

objects¶

The core of poliastro are the `Orbit`

objects
inside the `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:

```
from astropy import units as u
from poliastro.bodies import Earth, Mars, Sun
from poliastro.twobody import Orbit
```

## From position and velocity¶

There are several methods available to create
`Orbit`

objects. For example, if we have the
position and velocity vectors we can use
`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
orb = 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

`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:

>>> orb 7283 x 10293 km x 153.2 deg (GCRS) orbit around Earth (♁) at epoch J2000.000 (TT)

If no time is specified, then a default value is assigned:

>>> orb.epoch <Time object: scale='tt' format='jyear_str' value=J2000.000> >>> orb.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

`frame`

property:>>> orb.get_frame() <GCRS Frame (obstime=J2000.000, obsgeoloc=(0., 0., 0.) m, obsgeovel=(0., 0., 0.) m / s)>

### Intermezzo: quick visualization of the orbit¶

If we’re working on interactive mode (for example, using the wonderful Jupyter notebook) we can immediately plot the current orbit:

```
orb.plot()
```

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.

Note

This visualization uses Plotly under the hood and works best in a Jupyter notebook.
To use the static interface based on matplotlib,
which might be more useful for batch jobs and publication-quality plots,
check out the `poliastro.plotting.static.StaticOrbitPlotter`

.

## From classical orbital elements¶

We can also define an `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
`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
orb = Orbit.from_classical(Sun, a, ecc, inc, raan, argp, nu)
```

Notice that whether we create an `Orbit`

from \((r)\) and \((v)\) or from
elements we can access many mathematical properties of the orbit:

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

To see a complete list of properties, check out the
`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
`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 `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
`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
`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 simultaneously 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
`poliastro.twobody.thrust`

module.

### Changing the orbit: `Maneuver`

objects¶

poliastro helps us define several in-plane and general out-of-plane
maneuvers with the `Maneuver`

class inside the
`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 `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 `hohmann()`

and
`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.get_total_cost()
<Quantity 3.6173981270031357 km / s>
>>> hoh.get_total_time()
<Quantity 15729.741535747102 s>
>>> hoh = Maneuver.hohmann(ss_i, 36000 * u.km)
```

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 `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: `OrbitPlotter`

objects¶

We previously saw the `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 `OrbitPlotter`

objects in the
`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 OrbitPlotter2D
op = OrbitPlotter2D()
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:

### Where are the planets? Computing ephemerides¶

New in version 0.14.0.

Thanks to Astropy and jplephem, poliastro can 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 `poliastro.ephem.Ephem`

class allows us to retrieve
a planetary orbit using low precision ephemerides available in
Astropy and an `astropy.time.Time`

:

```
from astropy import time
epoch = time.Time("2020-04-29 10:43") # UTC by default
```

And finally, retrieve the planet ephemerides:

```
>>> from poliastro.ephem import Ephem
>>> earth = Ephem.from_body(Earth, epoch.tdb)
>>> earth
Ephemerides at 1 epochs from 2020-04-29 10:44:09.186 (TDB) to 2020-04-29 10:44:09.186 (TDB)
```

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.

Warning

This is the preferred method over

`poliastro.twobody.orbit.Orbit.from_body_ephem()`

, which is now deprecated and will be removed in the next release.

If we want to retrieve the **osculating orbit** at a given epoch,
we can do so using `from_ephem()`

:

```
>>> Orbit.from_ephem(Sun, earth, epoch)
1 x 1 AU x 23.4 deg (HCRS) orbit around Sun (☉) at epoch 2020-04-29 10:43:00.000 (UTC)
```

Note

Notice that the position and velocity vectors are given with respect to the
**Heliocentric Celestial Reference System** (HCRS)
which means equatorial coordinates centered on the Sun.

## 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 `poliastro.iod`

holds the different raw algorithms 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!

Since poliastro version 0.13 it is possible to solve Lambert’s problem by making
use of the `poliastro.maneuver`

module. We just need to pass as
arguments the initial and final orbits and the output will be a
`poliastro.maneuver.Maneuver`

instance. Time of flight is computed
internally since orbits epochs are known.

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')
ss0 = Orbit.from_ephem(Sun, Ephem.from_body(Earth, date_launch), date_launch)
ssf = Orbit.from_ephem(Sun, Ephem.from_body(Mars, date_arrival), date_arrival)
man_lambert = Maneuver.lambert(ss0, ssf)
dv_a, dv_b = man_lambert.impulses
```

And these are the results:

```
>>> dv_a
(<Quantity 0. s>, <Quantity [-2.06420561, 2.58796837, 0.23911543] km / s>)
>>> dv_b
(<Quantity 21910501.00019529 s>, <Quantity [287832.91384349, 58935.96079319, -94156.93383463] km / d>)
```

## Fetching Orbits from external sources¶

As of now, poliastro supports fetching orbits from 2 online databases from Jet Propulsion Laboratory: SBDB and Horizons.

JPL Horizons can be used to generate ephemerides for solar-system bodies, while JPL SBDB (Small-Body Database Browser) provides model orbits for all known asteroids and many comets.

The data is fetched using the wrappers to these services provided by astroquery.

```
Ephem.from_horizons("Ceres")
Orbit.from_sbdb("Apophis")
```

## Creating a CZML document¶

We can create CZML documents which can then be visualized with the help of Cesium.

First we load the orbital data and the CZML Extractor:

```
from poliastro.examples import molniya, iss
from poliastro.czml.extract_czml import CZMLExtractor
```

Then we specify the starting and ending epoch, as well as the number of sample points (the higher the number, the more accurate the trajectory):

```
start_epoch = iss.epoch
end_epoch = iss.epoch + molniya.period
sample_points = 10
extractor = CZMLExtractor(start_epoch, end_epoch, sample_points)
extractor.add_orbit(molniya, label_text="Molniya")
extractor.add_orbit(iss, label_text="ISS")
```

We can find the generated CZML file by calling `extractor.packets`

.
There is more information in this sample Cesium
application.

*Per Python ad astra* ;)