Stability map with MEGNO and WHFast (iPython)

Orbits) and the symplectic integrator WHFast (Rein and Tamayo 2015).

We will integrate a two planet system with massive planets. We vary two orbital parameters, the semi-major axis \(a\) and the eccentricity \(e\). Let us first define a function that runs one simulation for a given set of initial conditions \((a, e)\).

def simulation(par):
    a, e = par # unpack parameters
    sim = rebound.Simulation()
    sim.integrator = "whfast"
    sim.ri_whfast.safe_mode = 0
    sim.dt = 5.
    sim.add(m=1.) # Star
    sim.add(m=0.000954, a=5.204, M=0.600, omega=0.257, e=0.048)
    sim.add(m=0.000285, a=a, M=0.871, omega=1.616, e=e)

    sim.exit_max_distance = 20.
        sim.integrate(5e2*2.*np.pi, exact_finish_time=0) # integrate for 500 years, integrating to the nearest
        #timestep for each output to keep the timestep constant and preserve WHFast's symplectic nature
        megno = sim.calculate_megno()
        return megno
    except rebound.Escape:
        return 10. # At least one particle got ejected, returning large MEGNO.

Let’s try this out and run one simulation

import rebound
import numpy as np

The return value is the MEGNO. It is about 2, thus the system is regular for these initial conditions. Let’s run a whole array of simulations.

Ngrid = 80
par_a = np.linspace(7.,10.,Ngrid)
par_e = np.linspace(0.,0.5,Ngrid)
parameters = []
for e in par_e:
    for a in par_a:
from rebound.interruptible_pool import InterruptiblePool
pool = InterruptiblePool()
results =,parameters)

On my laptop (dual core CPU), this takes only 3 seconds!

Let’s plot it!

results2d = np.array(results).reshape(Ngrid,Ngrid)
%matplotlib inline
import matplotlib.pyplot as plt
fig = plt.figure(figsize=(7,5))
ax = plt.subplot(111)
extent = [min(par_a),max(par_a),min(par_e),max(par_e)]
ax.set_xlabel("semi-major axis $a$")
ax.set_ylabel("eccentricity $e$")
im = ax.imshow(results2d, interpolation="none", vmin=1.9, vmax=4, cmap="RdYlGn_r", origin="lower", aspect='auto', extent=extent)
cb = plt.colorbar(im, ax=ax)
cb.set_label("MEGNO $\\langle Y \\rangle$")