The comet 67P/Churyumov–Gerasimenko (iPython)¶
will download the data from NASA Horizons and visualize the orbit using matplotlib.
This tutorial assumes that you have already installed REBOUND.
If you’re interested in Solar System dynamics, you have probably heard of NASA Horizons. It’s a large database of Solar System objects, their orbits and physical properties. It includes planets, moons, satellites, asteroids, comets and spacecrafts. With REBOUND, you can easily import data from NASA Horizons. As an example, let’s pull in the present day positions of Jupiter, Saturn and the Sun:
import rebound sim = rebound.Simulation() sim.add("Sun") sim.add("Jupiter") sim.add("Saturn")
Searching NASA Horizons for 'Sun'... Found: Sun (10). Searching NASA Horizons for 'Jupiter'... Found: Jupiter Barycenter (5). Searching NASA Horizons for 'Saturn'... Found: Saturn Barycenter (6).
Now all the data is in REBOUND! Let’s have a look at the orbits of the two planets.
for orbit in sim.calculate_orbits(): print(orbit)
<rebound.Orbit instance, a=5.203404756087003 e=0.0487339750460116 inc=0.022753035945471716 Omega=1.7543452919187963 omega=-1.509668088895102 f=4.4722512023910825> <rebound.Orbit instance, a=9.541682795513731 e=0.05457050713906437 inc=0.0434177408803769 Omega=1.9827272207965274 omega=-0.36202170650080945 f=3.4443492206301825>
Although there are three bodies, the
only returns two objects as the orbit for the Sun would be a little
boring. The function returns the orbits in Jacobi coordinates. Since we
didn’t specify a falue for \(G\), REBOUND assumes that \(G=1\).
The unit of length is one astronomical unit, the unit of time is one
Let’s add something more interesting to our simulation: the comet 67P/Churyumov-Gerasimenko.
Searching NASA Horizons for 'Churyumov-Gerasimenko'... Found: 67P/Churyumov-Gerasimenko.
/Users/rein/git/rebound/rebound/horizons.py:137: RuntimeWarning: Warning: Mass cannot be retrieved from NASA HORIZONS. Set to 0. warnings.warn("Warning: Mass cannot be retrieved from NASA HORIZONS. Set to 0.", RuntimeWarning)
When searching for a body by name, REBOUND takes the first dataset that Horizons offers. In this case, it’s a set of parameters from 1962. You probably want to go to the Horizons website and check that the values you are using are up-to-date and appropriate for what you want to do. You can also use more complicted Horizons queries, for example, to get the most recent apparition solution for the comet, use
You can also use the IAU asteroid number for numbered asteroids, or the
database record numbers from Horizons for objects not yet numbered by
the IAU (but note that database record numbers can change as the
database gets rearranged with new discoveries, see
http://ssd.jpl.nasa.gov/?horizons_doc#sb for details). In our case the
current database record number is 900647, so you could use
sim.add("900647") to get the newest set of orbital parameters for
NASA Horizons doesn’t have masses for all bodies. If REBOUND doesn’t
find a mass, you get a warning message (see above). In our case, we
don’t need the mass of the comet (it’s really smal). However, it you
want, you can add it manually using the syntax
Before we integrate the orbits, let’s plot the instantaneous orbits
using the built-in REBOUND function
%matplotlib inline fig = rebound.OrbitPlot(sim, unitlabel="[AU]")
Integration with IAS15
We will integrate backwards in time for 70 years. Because we don’t know what will happen yet (hint: a close encounter) we will use the IAS15 integrator. It is fast, accurate and has adaptive timesteps to capture any potential close encounters.
To integrate backwards, we could set a negative timestep or multiply all velocities with \(-1\). We’ll choose the first option:
sim.dt = -0.01
While we’re integrating, let’s store the positions of Jupiter and the comet at 1000 times during the interval. We’ll need to prepare a few variables to do that:
import numpy as np Noutputs = 1000 year = 2.*np.pi # One year in units where G=1 times = np.linspace(0.,-70.*year, Noutputs) x = np.zeros((2,Noutputs)) y = np.zeros((2,Noutputs))
Now we’re ready to start the integration:
sim.integrator = "ias15" # IAS15 is the default integrator, so we actually don't need this line sim.move_to_com() # We always move to the center of momentum frame before an integration ps = sim.particles # ps is now an array of pointers and will change as the simulation runs for i,time in enumerate(times): sim.integrate(time) x[i] = ps.x # This stores the data which allows us to plot it later y[i] = ps.y x[i] = ps.x y[i] = ps.y
Visualization with matplotlib
Let’s plot the orbits of Jupiter (blue) and the comet (green) to get an idea of what was going on during our integration.
import matplotlib.pyplot as plt fig = plt.figure(figsize=(5,5)) ax = plt.subplot(111) ax.set_xlim([-6,6]) ax.set_ylim([-6,6]) plt.plot(x, y); plt.plot(x, y);
As you can see in the above image, the comet 67P had a rather strong encounter with Jupiter a few years ago. Of course, if you wanted to do a realistic simulation of that encounter, you’d need to include all the other planets and maybe even some non-gravitational effects for the comet. However, let’s stick with our simplistic model and try to find out when exactly the two bodies had a close encouter. We already stored the data, so we can just plot their distance as a function of time.
fig = plt.figure(figsize=(12,5)) ax = plt.subplot(111) ax.set_xlabel("time [yrs]") ax.set_ylabel("distance [AU]") distance = np.sqrt(np.square(x-x)+np.square(y-y)) plt.plot(times/year, distance); closeencountertime = times[np.argmin(distance)]/year print("Minimum distance (%f AU) occured at time: %f years." % (np.min(distance),closeencountertime))
Minimum distance (0.046555 AU) occured at time: -60.680681 years.
We can see that the minimum distance occured approximately 56 years ago (as of writing this tutorial). Let’s see what date that was using some python magic and the datetime module:
import datetime encounterdate = datetime.datetime.today() + datetime.timedelta(days=365.25*closeencountertime) encounterdate.strftime("%Y-%m-%d %H:%M")
If you check wikipedia, the encounter happened on Feb. 4th, 1959, so we are off by a few days. It turns out that’s because of jets from the comet!