Kepler’s Laws and Newton's Synthesis

1. Kepler’s Laws and Newton's Synthesis • Kepler’s laws describe planetary motion. • The orbit of each planet is an ellipse, with the Sun at one focus.

2. 2. An imaginary line drawn from each planet to the Sun sweeps out equal areas in equal times.

3. As the planet is closest the sun, the planet is moving fastest and as the planet is farthest from the sun,it is moving slowest. Nonetheless, the imaginary line adjoining the center of the planet to the center of the sun sweeps out the same amount of area in each equal interval of time.

4. KEPLER’S THIRD LAW "If T is the period and r is the length of the semi-major axis of a planet’s orbit, then the ratioT2/r3 is the same for all planets."

5. The ratio of the square of a planet’s orbital period is proportional to the cube of its mean distance from the Sun: T2/r3

6. Kepler’s laws can be derived from Newton’s laws. Irregularities in planetary motion led to the discovery of Neptune, and irregularities in stellar motion have led to the discovery of many planets outside our Solar System.

7. 4.18 The mean distance from the Earth to the Sun is 1.496x108 km and the period of its motion about the Sun is one year. The period of Jupiter’s motion around the Sun is 11.86 years. Determine the mean distance from the Sun to Jupiter. KL rE = 1.496x108 km TE = 1 year TJ = 11.86 years = 7.77x108 km

9. Newton’s Law of Universal Gravitation If the force of gravity is being exerted on objects on Earth, what is the origin of that force? Newton’s realization was that the force must come from the Earth. He further realized that this force must be what keeps the Moon in its orbit.

10. Newton’s Law of Universal Gravitation The gravitational force must be proportional to both masses. By observing planetary orbits, Newton concluded that the gravitational force must decrease as the inverse of the square of the distance between the masses. In its final form, the Law of Universal Gravitation reads: Where:

11. The magnitude of the gravitational constant G can be measured . This is the Cavendish experiment.

12. 4.19 Derive Kepler’s Third Law from Newton’s Law of Gravitation. For a Planet 1 of mass m1 and the Sun of mass MS ULG/KL ΣF = ma = mac substituting: rearranging: For a Planet 2 of mass m2 and the Sun of mass MS: therefore:

13. 4.20 What is the force of gravity acting on a 2000 kg spacecraft when it orbits two Earth radii from the Earth’s center above the Earth’s surface? ULG m1 = 2000 kg ME = 5.98x1024 kg rE = 6380x103 (2) =1.276x107 m = 4899 N

14. 4.21 a. Derive the expression for g from the Law of Universal Gravitation. Fg = FUG ULG

15. b. Estimate the value of g on top of the Everest (8848 m) above the Earth’s surface. mE = 5.98x1024 kg RE = 6.38x106 m RT = 8848 + 6.38x106 = 6.388x106 m = 9.77 m/s2

16. The acceleration due to gravity varies over the Earth’s surface due to altitude, local geology, and the shape of the Earth, which is not quite spherical.

17. Satellites Satellites are routinely put into orbit around the Earth. The tangential speed must be high enough so that the satellite does not return to Earth, but not so high that it escapes Earth’s gravity altogether.

19. The satellite is kept in orbit by its speed – it is continually free falling, but the Earth curves from underneath it.

20. 4.22 A geosynchronous satellite is one that stays above the same point on the equator of the Earth. Determine: a. The height above the Earth’s surface such a satellite must orbit UCM mE = 5.98x1024 kg RE = 6.38x106 m T = 1 day = 86400 s FUG = FC GMET2 = 4 π2r3 r = 4.23x107 m from the Earth’s center

21. height = r - rE = 4.23x107 - 6.38x106 m = 3.592x107 m b. The satellite’s speed = 3070 m/s