Stars and Planets

• It can be cast in terms of the relative motion and total gravitational mass
• motion is in a plane

The two-body problem for bodies at a mass ratio . ↗ In the inertial frame both bodies orbit around the center of mass (cross). ↘ relative motion with one of the bodies put at the center. This representation is far more common (and useful) when . However, motion is not in an inertial coordinate frame. The orbit is elliptical, here with an eccentricity of e=0.7.

• It can be cast in terms of the relative motion and total gravitational mass
• motion is in a plane
• conserved quantities:

  Energy
  Angular momentum
  Eccentricity vector

• the orbit

  which describes an ellipse with semi-major axis and semi-minor axis .

• the orbit

  which describes an ellipse with semi-major axis and semi-minor axis .

• energy, angular momentum in terms of orbital elements:
• the orbital period

  where n is the mean motion. This is Kepler's law (not to be confused with Kepler's equation [see below], which describes how is related to time ).

Three anomalies

• Mean anomaly (time),

  where t is time, n the mean motion, and t₀ the time of pericenter passage

• True anomaly

  indicates the position of the particle in the orbit

• Eccentric anomaly E

  another indicator for the position of the orbit (see figure) the eccentric and true anomalies are geometrically related:

Kepler's equation relates the eccentric and mean anomalies

Hence, while we can obtain the mean anomaly (or time) as function of , i.e., , no analytic solutions exist for the reverse problem, . This amounts to solving Kepler's equation — a transcendental equation that must be solved numerically.

There are 6 orbital elements:

1. semi-major axis a
2. eccentricity e
3. argument of pericenter ω.

   this is the angle with a reference direction. In the planar case the reference is fixed (e.g., the x-axis); in the 3D case, it is the angle with respect to the ascending node Ω.

   Sometimes, this angle is replaced by the longitude of pericenter

4. true anomaly

   the position of the particle w.r.t. pericenter. Equivalently, the true longitude Another option is to indicate the positon by the mean anomaly M or by the mean longitude which is not a geometric angle.

for 2D problems, these 4 elements suffice.

There are 6 orbital elements:

1. semi-major axis a
2. eccentricity e
3. argument of pericenter ω.
4. true anomaly
5. inclination i: the angle of the orbit w.r.t. the reference plane.
6. longitude of the ascending node (Ω in the figure →). The angle to the line in the reference plane where the particle intersects the reference planet in the ascending direction

In the guiding center approximation the orbit solution, as function of mean anomaly M reads, to first order in eccentricity:

This is a superposition of:

1. uniform circular motion (M is linear with time); and
2. uniform elliptical motion with the ellipse having a 2:1 aspect ratio.

Note. The 2:1 aspect ratio and the epicycle frequency are valid for a Keplerian potential. This can be generalized to arbitrary potentials.

Note. Historically, pre-Copernican Europeans used circular epicyles to describe the motion of planets under the Ptolemaic model (Earth at center). They did not stop with one epicycle!

↗ Guiding center approximation for a particle in an orbit.
→ motion with respect to the guiding center.

The Galilean satellite system is an example of three-body resonance (Laplace resonance)

The period ratios of the planets obey a conmensurability, where the orbital period ratio is a rational number:

• same for the second and third planet
• o is called the resonance order

Q: Shown are planets at/near a 3:2 mean motion resonance. Inner body is on a circular orbit, outer on an eccentric. Which are in resonance?