# Orbital elements

This section discusses orbital parameters. We focus on the implementation and conventions in REBOUND.

The following image illustrated the most important angles used. In REBOUND the reference direction is the positive x direction, the reference plane is the xy plane.

## Orbit structure

Variable name Description
d radial distance from reference
v velocity relative to central object's velocity
h specific angular momentum
P orbital period (negative if hyperbolic)
n mean motion (negative if hyperbolic)
a semimajor axis
e eccentricity
inc inclination
Omega longitude of ascending node
omega argument of pericenter
pomega longitude of pericenter
f true anomaly
M mean anomaly
E Eccentric anomaly. Because this requires solving Kepler's equation it is only calculated when needed in python and never calculated in C. To get the eccentric anomaly in C, use the function double reb_tools_M_to_E(double e, double M)
l mean longitude = Omega + omega + M
theta true longitude = Omega + omega + f
T time of pericenter passage
rhill Hill radius, $$r_{\rm hill} =a\sqrt[3]{\frac{m}{3M}}$$

Important

All angles in REBOUND are in radians. Variables which have length, time or velocity units use code units.

### Particle to orbit

The following function allows you to calculate the orbital elements of a particle.

struct reb_simulation* r = create_simulation();
reb_add_fmt(r, "a e", 1., 0.1); // planet
struct reb_orbit o =  reb_tools_particle_to_orbit(r->G, r->particles[1], r->particles[0]);
printf("a=%f e=%f\n", o.a, o.e);

The last argument of the reb_tools_particle_to_orbit function is the primary particle, i.e. the star or the centre of mass.

sim = rebound.Simulation()
o = sim.particles[1].calculate_orbit(primary=sim.particles[0])
print(o.a, o.e)

If primary is not given, Jacobi coordinates are used.

You can also calculate the orbits of all particles in the simulation.

sim = rebound.Simulation()
orbits = sim.calculate_orbits()
for o in orbits:
print(o.a, o.e)


## Conversion functions

### True anomaly

The following function returns the true anomaly $$f$$ for a given eccentricity $$e$$ and mean anomaly $$M$$:

double f = reb_tools_M_to_f(0.1, 1.); // e=0.1, M=1.0

f = rebound.M_to_f(0.1, 1.0) # e=0.1, M=1.0


The following function returns the true anomaly $$f$$ for a given eccentricity $$e$$ and eccentric anomaly $$E$$:

double f = reb_tools_E_to_f(0.1, 1.); // e=0.1, E=1.0

f = rebound.E_to_f(0.1, 1.0) # e=0.1, E=1.0


### Eccentric anomaly

The following function returns the eccentric anomaly $$E$$ for a given eccentricity $$e$$ and mean anomaly $$M$$:

double f = reb_tools_M_to_E(0.1, 1.); // e=0.1, M=1.0

f = rebound.M_to_E(0.1, 1.0) # e=0.1, M=1.0