Catch that asteroid!¶

In [1]:

import matplotlib.pyplot as plt
plt.ion()

from astropy import units as u
from astropy.time import Time

In [2]:

from astropy.utils.data import conf
conf.dataurl

Out[2]:

'http://data.astropy.org/'

In [3]:

conf.remote_timeout

Out[3]:

10.0


First, we need to increase the timeout time to allow the download of data occur properly

In [4]:

conf.remote_timeout = 10000


Then, we do the rest of the imports and create our initial orbits.

In [5]:

from astropy.coordinates import solar_system_ephemeris
solar_system_ephemeris.set("jpl")

from poliastro.bodies import *
from poliastro.twobody import Orbit
from poliastro.plotting import OrbitPlotter, plot

EPOCH = Time("2017-09-01 12:05:50", scale="tdb")

In [6]:

earth = Orbit.from_body_ephem(Earth, EPOCH)
earth

Out[6]:

1 x 1 AU x 23.4 deg orbit around Sun (☉)

In [7]:

plot(earth, label=Earth);

In [8]:

from poliastro.neos import neows

In [9]:

florence = neows.orbit_from_name("Florence")
florence

Out[9]:

1 x 3 AU x 22.1 deg orbit around Sun (☉)


Two problems: the epoch is not the one we desire, and the inclination is with respect to the ecliptic!

In [10]:

florence.epoch

Out[10]:

<Time object: scale='tdb' format='jd' value=2458200.5>

In [11]:

florence.epoch.iso

Out[11]:

'2018-03-23 00:00:00.000'

In [12]:

florence.inc

Out[12]:

$22.144811 \; \mathrm{{}^{\circ}}$

We first propagate:

In [13]:

florence = florence.propagate(EPOCH)
florence.epoch.tdb.iso

Out[13]:

'2017-09-01 12:05:50.000'


And now we have to convert to another reference frame, using http://docs.astropy.org/en/stable/coordinates/.

In [14]:

from astropy.coordinates import (
ICRS, GCRS,
CartesianRepresentation, CartesianDifferential
)
from poliastro.frames import HeliocentricEclipticJ2000


The NASA servers give the orbital elements of the asteroids in an Heliocentric Ecliptic frame. Fortunately, it is already defined in Astropy:

In [15]:

florence_heclip = HeliocentricEclipticJ2000(
x=florence.r[0], y=florence.r[1], z=florence.r[2],
v_x=florence.v[0], v_y=florence.v[1], v_z=florence.v[2],
representation=CartesianRepresentation,
differential_type=CartesianDifferential,
obstime=EPOCH
)
florence_heclip

Out[15]:

<HeliocentricEclipticJ2000 Coordinate (obstime=2017-09-01 12:05): (x, y, z) in km
(1.45898575e+08, -58567565.51964308, 2279107.73676029)
(v_x, v_y, v_z) in km / s
(7.40829065, 31.11151452, 12.79669448)>


Now we just have to convert to ICRS, which is the “standard” reference in which poliastro works:

In [16]:

florence_icrs_trans = florence_heclip.transform_to(ICRS)
florence_icrs_trans.representation = CartesianRepresentation
florence_icrs_trans

Out[16]:

<ICRS Coordinate: (x, y, z) in km
(1.46265478e+08, -53881737.41800184, -20898600.46334482)
(v_x, v_y, v_z) in km / s
(7.3998822, 23.46299461, 24.12028277)>

In [17]:

florence_icrs = Orbit.from_vectors(
Sun,
r=[florence_icrs_trans.x, florence_icrs_trans.y, florence_icrs_trans.z] * u.km,
v=[florence_icrs_trans.v_x, florence_icrs_trans.v_y, florence_icrs_trans.v_z] * (u.km / u.s),
epoch=florence.epoch
)
florence_icrs

Out[17]:

1 x 3 AU x 44.5 deg orbit around Sun (☉)

In [18]:

florence_icrs.rv()

Out[18]:

(<Quantity [ 1.46265478e+08, -5.38817374e+07, -2.08986005e+07] km>,
<Quantity [ 7.3998822 , 23.46299461, 24.12028277] km / s>)


Let us compute the distance between Florence and the Earth:

In [19]:

from poliastro.util import norm

In [20]:

norm(florence_icrs.r - earth.r) - Earth.R

Out[20]:

$7057830.8 \; \mathrm{km}$

This value is consistent with what ESA says! $$7\,060\,160$$ km

In [21]:

from IPython.display import HTML

HTML(
)

Out[21]:


And now we can plot!

In [22]:

frame = OrbitPlotter()

frame.plot(earth, label="Earth")

frame.plot(Orbit.from_body_ephem(Mars, EPOCH))
frame.plot(Orbit.from_body_ephem(Venus, EPOCH))
frame.plot(Orbit.from_body_ephem(Mercury, EPOCH))

frame.plot(florence_icrs, label="Florence");


The difference between doing it well and doing it wrong is clearly visible:

In [23]:

frame = OrbitPlotter()

frame.plot(earth, label="Earth")

frame.plot(florence, label="Florence (Ecliptic)")
frame.plot(florence_icrs, label="Florence (ICRS)");


And now let’s do something more complicated: express our orbit with respect to the Earth! For that, we will use GCRS, with care of setting the correct observation time:

In [24]:

florence_gcrs_trans = florence_heclip.transform_to(GCRS(obstime=EPOCH))
florence_gcrs_trans.representation = CartesianRepresentation
florence_gcrs_trans

Out[24]:

<GCRS Coordinate (obstime=2017-09-01 12:05, obsgeoloc=(0., 0., 0.) m, obsgeovel=(0., 0., 0.) m / s): (x, y, z) in km
(4960528.40227817, -5020204.24301458, 306195.40673516)
(v_x, v_y, v_z) in km / s
(-2.76863621, -1.95773248, 13.09966915)>

In [25]:

florence_hyper = Orbit.from_vectors(
Earth,
r=[florence_gcrs_trans.x, florence_gcrs_trans.y, florence_gcrs_trans.z] * u.km,
v=[florence_gcrs_trans.v_x, florence_gcrs_trans.v_y, florence_gcrs_trans.v_z] * (u.km / u.s),
epoch=EPOCH
)
florence_hyper

Out[25]:

7064205 x -7068561 km x 104.3 deg orbit around Earth (♁)


Notice that the ephemerides of the Moon is also given in ICRS, and therefore yields a weird hyperbolic orbit!

In [26]:

moon = Orbit.from_body_ephem(Moon, EPOCH)
moon

Out[26]:

151218466 x -151219347 km x 23.3 deg orbit around Earth (♁)

In [27]:

moon.a

Out[27]:

$-440.42131 \; \mathrm{km}$
In [28]:

moon.ecc

Out[28]:

$343350.57 \; \mathrm{}$

So we have to convert again.

In [29]:

moon_icrs = ICRS(
x=moon.r[0], y=moon.r[1], z=moon.r[2],
v_x=moon.v[0], v_y=moon.v[1], v_z=moon.v[2],
representation=CartesianRepresentation,
differential_type=CartesianDifferential
)
moon_icrs

Out[29]:

<ICRS Coordinate: (x, y, z) in km
(1.41399531e+08, -49228391.42507221, -21337616.62766309)
(v_x, v_y, v_z) in km / s
(11.10890252, 25.6785744, 11.0567569)>

In [30]:

moon_gcrs = moon_icrs.transform_to(GCRS(obstime=EPOCH))
moon_gcrs.representation = CartesianRepresentation
moon_gcrs

Out[30]:

<GCRS Coordinate (obstime=2017-09-01 12:05, obsgeoloc=(0., 0., 0.) m, obsgeovel=(0., 0., 0.) m / s): (x, y, z) in km
(94189.90120828, -367278.24304992, -133087.21297573)
(v_x, v_y, v_z) in km / s
(0.94073662, 0.25786326, 0.03569047)>

In [31]:

moon = Orbit.from_vectors(
Earth,
[moon_gcrs.x, moon_gcrs.y, moon_gcrs.z] * u.km,
[moon_gcrs.v_x, moon_gcrs.v_y, moon_gcrs.v_z] * (u.km / u.s),
epoch=EPOCH
)
moon

Out[31]:

367937 x 405209 km x 19.4 deg orbit around Earth (♁)


And finally, we plot the Moon:

In [32]:

plot(moon, label=Moon)
plt.gcf().autofmt_xdate()


And now for the final plot:

In [33]:

frame = OrbitPlotter()

# This first plot sets the frame
frame.plot(florence_hyper, label="Florence")

# And then we add the Moon
frame.plot(moon, label=Moon)

plt.xlim(-1000000, 8000000)
plt.ylim(-5000000, 5000000)

plt.gcf().autofmt_xdate()