9712

The Celestial Two-Body Problem

Two celestial bodies interacting gravitationally can establish a stable system in which both trace out elliptical (or possibly circular) orbits about their mutual barycenter (center of mass), marked with a red dot. Each orbit individually follows Kepler's three laws of planetary motion.
Kepler's first law specifies that each orbit is an ellipse with one focus at the barycenter.
According to Kepler's second law, each orbit sweeps out equal areas in equal times. Thus for an eccentric orbit, the speed increases closer to the barycenter.
Kepler's third law states that the square of the period of an orbit is proportional to the cube of its semimajor axis. Accordingly, the outer planets of the Solar System have much longer "years" than Earth.
When one of the interacting bodies is much more massive than the other, as in the typical case of a planet orbiting a star, the more elementary form of Kepler's laws pertains, with the star itself at the focus of an elliptical orbit. Actually, the barycenter is then located within the body of the star, but not at its exact center. This can cause the star to "wobble" when a Jupiter-sized planet orbits around it. In recent years, the existence of extrasolar planets has been detected in this way.
In this Demonstration, you can vary the mass ratio and the eccentricity ϵ of the orbit. For a circular orbit,, while for an ellipse . You can adjust the apoapses, the vectors from the barycenter to the farthest point of each orbit, also known as Laplace–Runge–Lenz vectors. (Apogee, aphelion, and apoastron are more familiar synonyms for apoapsis pertaining to specific celestial bodies.) Rotating the three-dimensional figure can show the orbits from different angles. From an edge-on perspective, the motions of the orbiting bodies appear to follow irregular linear trajectories.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Snapshot 1: the binary or double star 61 Cygni, also known as Bessel's star; the period of rotation is approximately 659 years
Snapshot 2: the dwarf planet Pluto with its companion Charon, with about a seventh of the mass; the barycenter of the system orbits the Sun with a period of 248.5 years
Snapshot 3: a star with a planet 1/50 of its mass; the star's wobbling motion is apparent
Snapshot 4: approximate representation of a comet in orbit around the Sun; the eccentricity approaches 1
Snapshot 5: the lowest Bohr orbit for a muonic atom: a negative muon orbiting a proton; the Bohr radius is about 1/207 that of a hydrogen atom
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+