Satellites and Kepler’s Laws: An Argument for Simplicity Quiz

PDF page 283: The conditions are satisfied, to good approximation, by Earth’s satellites (including the Moon), by objects orbiting the Sun, and by the satellites of other planets. Historically, planets were studied first, and there is a classical set of three laws, called Kepler’s laws of planetary motion, that describe the orbits of all bodies satisfying the two previous conditions (not just planets in our solar system). These descriptive laws are named for the German astronomer Johannes Kepler (1571–1630), who devised them after careful study (over some 20 years) of a large amount of meticulously recorded observations of planetary motion done by Tycho Brahe (1546–1601). Such careful collection and detailed recording of methods and data are hallmarks of good science. Data constitute the evidence from which new interpretations and meanings can be constructed.

PDF page 283: The conditions are satisfied, to good approximation, by Earth’s satellites (including the Moon), by objects orbiting the Sun, and by the satellites of other planets. Historically, planets were studied first, and there is a classical set of three laws, called Kepler’s laws of planetary motion, that describe the orbits of all bodies satisfying the two previous conditions (not just planets in our solar system). These descriptive laws are named for the German astronomer Johannes Kepler (1571–1630), who devised them after careful study (over some 20 years) of a large amount of meticulously recorded observations of planetary motion done by Tycho Brahe (1546–1601). Such careful collection and detailed recording of methods and data are hallmarks of good science. Data constitute the evidence from which new interpretations and meanings can be constructed.

PDF page 284: FIGURE 6.26 (a) An ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci ( and ) is a

PDF page 284: constant. You can draw an ellipse as shown by putting a pin at each focus, and then placing a string around a pencil and the pins and tracing a line on paper. A circle is a special case of an ellipse in which the two foci coincide (thus any point on the circle is the same distance from

PDF page 285: FIGURE 6.27 The shaded regions have equal areas. It takes equal times for to go from A to B, from C to D, and from E to F. The mass

PDF page 285: moves fastest when it is closest to . Kepler’s second law was originally devised for planets orbiting the Sun, but it has broader validity. Note again that while, for historical reasons, Kepler’s laws are stated for planets orbiting the Sun, they are actually valid for all bodies satisfying the two previously stated conditions.

PDF page 285: Given that the Moon orbits Earth each 27.3 d and that it is an average distance of from the center of Earth, calculate the period of an artificial satellite orbiting at an average altitude of 1500 km above Earth’s surface. Strategy The period, or time for one orbit, is related to the radius of the orbit by Kepler’s third law, given in mathematical

PDF page 285: Given that the Moon orbits Earth each 27.3 d and that it is an average distance of from the center of Earth, calculate the period of an artificial satellite orbiting at an average altitude of 1500 km above Earth’s surface. Strategy The period, or time for one orbit, is related to the radius of the orbit by Kepler’s third law, given in mathematical

PDF page 285: . The given information tells us that the orbital radius of the Moon is , and that the period of the Moon is . The height of the artificial satellite above Earth’s surface is given, and so we must add the radius of Earth (6380 km) to get . Now all quantities are known, and so can be found. Solution Kepler’s third law is

PDF page 286: Discussion This is a reasonable period for a satellite in a fairly low orbit. It is interesting that any satellite at this altitude will orbit in the same amount of time. This fact is related to the condition that the satellite’s mass is small compared with that of Earth.

PDF page 286: People immediately search for deeper meaning when broadly applicable laws, like Kepler’s, are discovered. It was Newton who took the next giant step when he proposed the law of universal gravitation. While Kepler was able to discover what was happening, Newton discovered that gravitational force was the cause.

PDF page 286: beginning of this section. Gravity supplies the centripetal force to mass . Starting with Newton’s second law applied to circular motion,

PDF page 286: The fact that cancels out is another aspect of the oft-noted fact that at a given location all masses fall with the same acceleration. Here we see that at a given orbital radius , all masses orbit at the same speed. (This was implied by the result of the preceding worked example.) Now, to get at Kepler’s third law, we must get the period into the equation. By definition, period is the time for one complete orbit. Now the average speed is the circumference divided by the period—that is,