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Planetary orbits and Kepler’s laws

Adjust the eccentricity and semi-major radius and compare the orbital area, period, and speed changes at the same time.

    E12 · Planetary orbits and Kepler’s lawsReady
    You can also experiment with the numbers and controls below the scene.

    Earth Science, Astronomy, Space / Principles and Exploration

    Planetary orbits and Kepler’s laws — principles and investigation

    Planets do not always move at the same speed in elliptical orbits. Change the semi-major axis and eccentricity of the orbit and compare the sun's position, area swept in the same time, and orbital period. This is a Kepler experiment that distinguishes between the laws regarding the shape of an ellipse and the laws regarding time and speed.

    How does it work?

    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.

    Key relationship

    E−e sin E=M · T²=a³

    Try this investigation

    1. Set the eccentricity and major radius and execute one revolution. The two colored fans are areas swept for the same T/10.
    2. Record the period, area, and velocity ratio at a=1 AU and 4 AU. Distinguish between the same execution playback time and the actual cycle (years).
    3. Return the semi-major diameter to 1 AU, change only e to 0, and compare it to the constant speed of a circular orbit.

    Interpreting your results

    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.

    Assumptions and limitations

    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.