Meaning and mechanism
The major axis length, denoted as ‘a’, represents the orbital size, while the eccentricity ‘e’ indicates how much it deviates from a circle. The Sun is not at the center of the ellipse; rather, it lies at one of its foci.
Near the sun, planets move faster, and farther away, they move slower. This is because the same area is swept over in the same amount of time when a line connecting the sun and a planet is drawn.
Relationships and conditions
A small-mass planet orbiting the same central body follows the relationship T² ∝ a³, where T is the period and a is the semi-major axis. In the Solar System, using AU·year units, this is approximately T² = a³, so if a=4 AU is one AU, then T=8 years.
In other central celestial bodies, the proportionality constant differs. The simple two-body model neglects perturbations from other planets or relativistic effects.
What to observe in the simulation
The sun is at one focus of the ellipse and covers the same area at the same time. If a=4 AU, T=8 years.
These are models of two celestial bodies with fixed solar masses. In units of AU·year, the gravitational constant is set at 4π² and planetary masses, perturbations of other planets, and relativity are omitted. Kepler's equations are solved using Newton's method. The scene is zoomed in to match the current a, and the object size and velocity arrows are for illustrative purposes only.
Key relationships and symbols
E−e sin E=M · T²=a³
| Symbol | Meaning | Unit |
|---|---|---|
| a | Semimajor axis of the ellipse | AU or m |
| e | Eccentricity | 0 ≤ e < 1 |
| T | Orbital period | years or s |
Worked example
What is the perihelion/aphelion velocity ratio at e=0.5?
(1+e)/(1−e)=1.5/0.5=3. It's fast when it's close and slow when it's far away, but the area is the same for the same amount of time.
Common misconceptions
- The planets do not always move the same length of orbital range for the same period of time.