Chapter 9: Gravity & Planetary Motion

Planetary and Satellite Motion Kepler’s Three Laws: In the early 1600’s, Johannes Kepler proposed three laws of planetary motion. Kepler was able to summarize the carefully collected data of his mentor-Tycho Brahe- with three statements which described the motion of planets in a sun-centered solar system. Mathematician who studied under Tycho Brahe (wanted to test Tycho’s data about planetary motion around sun)

Kepler's 1st Law: "The Law of Ellipses" Kepler’s first law explains that planets orbit the sun in a path described as an ellipse, with the sun being located at one of the foci of that ellipse. An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. Looking at picture, as one moves pencil around circle, the sum of the distances from both foci to a point on circle will be the same

Kepler's 2nd Law: "The Law of Equal Areas" Kepler’s second law describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. This is due to the planets’ elliptical orbit and the fact that the sun is not in the center of the orbital path. If a line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. Over equal periods of time, a planet will sweep out the same area as it orbits around the sun

Kepler's 2nd Law A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun.

Kepler's Third Law: "The Law of Harmonies" Planet Period (sec.) Average Dist. (m) T2/R3 (s2/m3) Earth 3.156 x 107 sec. 1.4957 x 1011 m 2.977 x 10-19 Mars 5.93 x 2.278 x 2.975 x Kepler’s third law compares the orbital period and radius of orbit of a planet to those of other planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every Dividing the square of the period of planets by the cube of its distance from sun will always yield the same number one of the planets. As an illustration, consider the orbital period and average distance from the sun (orbital radius) for earth and mars as given in the table above. Observe that the T2/R3 ratio is the same for the earth as it is for mars.

Newton and Gravity Newton was troubled by the lack of explanation for the planet’s orbits. Newton looked for an explanation for their circular motion. Myth has it that he got an idea while sitting under an apple tree.

Planetary and Satelite Motion The planets in the solar system travel with almost perfect circular motion. This being true, all planets are experiencing a centripetal force. What is the cause of this centripetal force? Answer: We learned earlier in this class that all objects that have mass experience a weak attractive force with each other. This attractive force is called GRAVITY. It is this attractive force between the planets and the sun that causes the centripetal force. Gravity is the centripetal force.

Gravitational Force is the Centripetal Force

Planetary Motion Continued Why are the planets’ orbital paths ellipses and not perfect circles? Answer: Planets are pulled toward the sun with a centripetal force. If this were the only force acting on the planets, their obits would be perfect circles. However, the planets also have weaker gravitational forces pulling on them in other directions. This causes their orbits to be shaped in a manner that is not a perfect circle. Orbits are elliptical in shape for a reason!

Universal Gravitation Newton determined that the gravitational force between two objects is determined by their mass and their distance away from each other.

Universal Gravitation (Cont.)

Centripetal Force is NOT Constant

d2 Fgrav = G m1m2 Universal Gravitation Equation The Value of "G" 6.67259 x 10-11 Nm2/kg2 Universal Gravitation Constant The value of G is an extremely small numerical value. Its smallness accounts for the fact that the force of gravitational attraction is only appreciable for objects with large mass. While two people will indeed exert gravitational forces upon each other, these forces are too small to be noticeable.

Satellite Motion A satellite is an object which is orbiting the earth, sun, or other massive body. Whether a moon, a planet, or some man-made satellite, every satellite’s motion is governed by the same physics principles. Every satellite is a projectile. That is to say, a satellite is an object upon which the only force is gravity. Newton was the first to theorize that a projectile launched with sufficient speed would actually orbit the earth.

So What Launch Speed Does a Satellite Need to Orbit the Earth? For every 8000 meters measured along the horizon of the earth, the earth’s surface curves downward by approximately 5 meters. For a projectile to orbit the Earth, it must travel horizontally a distance of 8000 meters for every 5 meters of vertical fall. dy = ½ at2 Vx = dx A projectile launched at 8000 m/s will orbit the earth. t 5m = ½ -9.8 m/s2 (t)2 Vx = 8000 m Time = 1.0102 sec. 1.0102 sec. Vx = 7919.6 m/s

The Value of "g" Earlier in the year an equation was given for determining the force of gravity (Fgrav) with which an object of mass (m) is attracted to the earth: Fgrav = mg Now, in this chapter, a second equation has been introduced for calculating the force of gravity with which an object is attracted to the earth: Fgrav = G M1M2 d2 The force of gravity is dependent upon location. The further you go away from the center of the earth the lower the force of gravity becomes.