Kepler's Laws Calculator
Please go to the following page and bookmark it.
Note: Please run the orbit simulator for the object of interest and print out the plot of its orbit before using this calculator. Data measured from the print-out will be necessary for the calculations below.
Planets orbit the Sun in ellipses with the Sun at one focus of the ellipse.
The eccentricity of an orbit is a measure of how elongated it is, that is, how non-circular it is. To comprehend this more clearly, students will measure the orbit on the print-out and then calculate the eccentricity from these measurements. These results will then be compared with the original value.
One formula is:
|e2 = 1 - ( b2 / a2 )|
The second formula is:
|e = (distance between the foci) / ( 2 a )|
Compare these two values together and with the eccentricity used to plot the orbit. They should agree to the number of significant digits used to make the appropriate measurements from the print-out.
Planets move in their orbits at a rate such that a line connecting the planet and the Sun will sweep through equal areas in equal time intervals.
Kepler's Second Law only gives a relationship between the velocities of an orbiting object at two points in its orbit and not a way to calculate any particular velocity. Isaac Newton, with his more theoretical understanding of the motion of an object, his formulation of the force of gravity and his concept of the conservation of angular momentum, was able to derive a formula for the velocity of an orbiting object.
The sketch at left shows a visual representation of Kepler's Second Law. Each sector is diagrammed for the same length of time as the object moves in its orbit. Notice how the object is moving slower in its orbit when it is furtherest from the Sun.
Confirm that the information already given at the top of this web page has been correctly entered in the appropriate boxes below, where:
The constants in the following equations have been chosen so as to give the orbital velocity of the object in m/s.
Orbital velocity at aphelion (point in orbit furtherest from the Sun):
|va2 = ( 8.871 X 108 / a ) ( 1 - e ) / ( 1 + e )|
Orbital velocity at perihelion (point in orbit closest to the Sun):
|vp2 = ( 8.871 X 108 / a ) ( 1 + e ) / ( 1 - e )|
If the orbital period is given in years and the semi-major axis of the orbit is given in AU, then the period squared is equal to the semi-major axis cubed.
Kepler formulated his conclusion of how the period of an object orbiting the Sun is related to the average distance the object is from the Sun into his Third Law. The average distance an object is from the Sun is equal to the semi-major axis of its orbit. Again Newton was able to extend Kepler's concept. Newton showed that this relationship also depended upon the sum of mass of the object and the mass of the Sun. However, since Jupiter, the largest planet is only about 0.1% of the mass of the Sun, this additional variable is not significant at the level of three significant digits.
Confirm that the length of the semi-major axis given at the top of this web page has been correctly entered below, where:
With these units, Kepler's Third Law is simply:
|p2 = a3|
Created and maintained by: Richard L. Bowman (1997-2004; last updated: 1-Mar-10)