How does planet mass affect orbit velocity?

The Earth takes a bit more than 365 days to complete one orbit around the Sun. Mercury orbits the Sun in only 88 days, but Pluto takes more than 90,000 days. Why are the periods of planetary orbits so different?

One factor is the distance each planet must travel in its circuit. The Earth is about 2.5 times more distant from the Sun than Mercury, so the circumference of its orbit is about 2.5 times larger. Does that mean that Earth's orbital period is 2.5 times longer? Let's see; the actual ratio is (365 days / 88 days) = 4.1 times longer. Hmmm. And Pluto's journey around the Sun is about 40 times farther than the Earth's, yet it takes Pluto roughly 247 times longer to make the trip. It seems that the size of the orbit alone can't explain the period. What else could be happening?

Kepler developed his 3^rd Law based on solar system observations and he had no way to know what the effects of having different solar masses might be. Newton solved the general problem, and took mass into account and came up with a modified 3^rd law: p² = a³/(M+m), where M and m are the star and planet masses respectively. In choosing the Earth orbit to set the scales for the Solar System, Kepler was using M = 1 without realizing it, and since planets are so small when compared to their stars, the ‘M + m’ factor was essentially 1.

Therefore, for different central stars, planetary systems will give a straight line when plotting p² vs a³, but the slope of the line is 1/M; the bigger the star the smaller the slope. If p² vs a³ is plotted for 3 different planetary systems with 3 different mass stars, 3 straight lines would result with 3 different slopes. However, if p² is plotted against a³/M, all objects for all systems would appear on the one line!

Your goal is to explore the differences between Kepler’s original equation for orbital velocity and Newton’s modification that included mass.

There are several control options:

• Adjust the Star’s Mass.
• Change the Orbit Radius of Earth.
• Choose between graphing Kepler’s original 3rd Law (using a³) or Newton’s Modified version (using a³/M).
• Clear the graph of all the points you have plotted so far. Screen reader users: The clear graph and play/pause buttons are for the visuals only.

The color of each dot represents stellar mass, so that you can isolate the effects of a change in orbital size (look at all dots of the same color) from a change in stellar mass (look at all the dots in the same vertical position.)

If sound is on, higher tones indicate faster planetary movement. You can turn the sound off in the top of the window in the banner region.

Notes: The Sun and Earth are drawn much larger than their true sizes, relative to the size of the orbit. Because the interactive measures orbital radius in Astronomical Units, mass in solar masses, and velocity in kilometers per second, the gravitational constant "G" has a value of about 890; this is very different from the usual textbook value (based on units of meters, kilograms, and meters per second, respectively.)