Module 5—Circular Motion

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Kepler's Three Laws of Planetary Motion

 

Despite the difficulties in his personal life, Kepler (1571–1630) formulated the three laws of planetary motion. These laws describe the manner in which the planets move around the Sun, but they do not explain why the planets move in this fashion. This explanation was provided later by Isaac Newton (1642–1727).

 

elliptical: having the shape of an ellipse or oval

 

epicycle: a circle that rolls along the circumference of another circle

Kepler's three laws are based on observations by Tycho Brahe (1546–1601). They allowed a great simplification and increased accuracy in the calculation of the planets' motions compared to the earlier Copernican model. This advance was brought about when Kepler realized that planets were moving in elliptical orbits rather than on epicycles of circles as in the Copernican model.

 

Kepler's laws are now known to be only an approximation, but a very good one. The observed deviations of the planets' motions from Kepler's laws can mostly be explained in terms of Newton's laws of motion applied to the planets in combination with his universal law of gravitation.

 

The Newtonian analysis shows that Kepler's laws would be true if the gravitational force exerted on the planets by the Sun was the only force acting on the planets. However, the planets also exert gravitational forces on each other; and it is these additional small forces that cause the planets' motions to be slightly different from those predicted by Kepler's laws.

 

Newton's theory can explain these deviations from Kepler's laws except for one remaining small discrepancy, which was accounted for early in the twentieth century by Albert Einstein's theory of general relativity.

 

Watch and Listen

 

Watch this video on Kepler and Tycho Brahe.




Data in Astronomical Units

 

The study of planetary motion, by necessity, involves distances in space that are incredibly large. For this reason, distance measurements in space are not always made using conventional distance units, such as metres. The AU is the astronomical unit. It is a unit of distance that is equal in length to the semi-major axis (shown in red in the ellipse at the right) of Earth's orbit around the Sun (the mean or average distance between Earth and Sun).

 

1 AU = 1.496 × 1011 m

 

astronomical unit: the average distance between Earth and the Sun

 

A simulation that uses astronomical units will be used to explore Kepler’s three laws and show how they accurately describe planetary motion.

 

The applet used in this simulation allows you to

sidereal year: the orbital period of Earth

 

calendar year: 365 days or 366 days (leap year)

The year used in the simulation is the sidereal year, which is the time it takes Earth to make one revolution around the Sun relative to the fixed stars.

 

1 Year = 365.26 mean solar days

 

The sidereal year must be distinguished from the calendar year, which is equal to 365 or 366 mean solar days, depending on the particular year. The abbreviation for the sidereal year is either the letter y or the letter a (as in annual). 

 

Open the Planetary Motion silmulation. You can learn more about the simulation and how to use it by reading Show Me found at the top of the simulation screen. To view the astronomical data on the simulation, click the “Data” button (). Observe that various quantities are displayed in units of AU and Year. On the simulation, some of the quantities are set by sliders that also use the astronomical and year units. They are convenient units in an astronomical context.

 

energy/mass (E/m):

angualr momentum/mass (L/m):

areal velocity (dA/dt):

angular velocity (ω):

 

time (t):

distance (r):

speed (v):

acceleration (a):

−19.74 AU2/Year2

5.51 AU2/Year

2.75 AU2/Year

4.52 rad/Year

 

0.21 Year

1.10 AU

5.68 AU/Year

32.40 AU/Year2


 

Self-Check

 

SC 1. If time (t) is given in years and distance (r) is given in astronomical units (AU), substitute these units into the formula below to determine the units of speed for this simulation.

 

 

The units for change in velocity are the same as for speed. Use this fact to determine the units of acceleration in the formula below.

 

 

Verify that your units for speed and acceleration are consistent with those displayed in the table above or click the “Data” button in the simulation.

 

Check your work.
Self-Check Answers

 

SC 1.