Celestial Mechanics: From Angels to Gravitational Forces

1. "The next question was - what makes planets go around the sun? At the time of Kepler some people answered this problem by saying that there were angels behind them beating their wings and pushing the planets around an orbit. As you will see, the answer is not very far from the truth. The only difference is that the angels sit in a different direction and their wings push inward."       -Richard Feynman

2. Ptolemy 85-165 AD The Almagest

3. 1541: Des Revolutionibus Sun at Center Orbits are Circular

4. Tycho Brahe1546-1601 Tycho was the greatest observational astronomer of his time. Tycho did not believe in the Copernican model because of the lack of observational parallax. He didn’t believe that the Earth Moved.

5. The Rejection of the Copernican Heliocentric Model

6. Tycho Brahe1546-1601 Kepler worked for Tycho as his mathematician. Kepler derived his laws of planetary motion from Tycho’s observational data. Kepler’s Laws are thus empirical - based on observation and not Theory.

7. Kepler’s 3 Laws of Planetary Motion 1: The orbit of each planet about the sun is an ellipse with the sun at one focus. 2. Each planet moves so that it sweeps out equal areas in equal times. 3. The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit.

8. Planet Orbits are Elliptical Why?

9. Man of the MillenniumSir Issac Newton (1642 -1727)

10. Gravitational Force is Universal The same force that makes the apple fall to Earth, causes the moon to fall around the Earth.

11. M d m Universal Law of Gravity 1687 Every particle in the Universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product to their masses and inversely proportional to the square of the distance between them.

12. How Does Newton’s Universal Law of Gravity (ULG) Explain Kepler’s Laws of Planetary Motion? • Kepler’s First Law (Orbits are ellipses) • - Express F = ma as a second order differential equation in polar coordinates, substitute in F as an inverse square law and the radial solutions are ellipses! • (See Wikipedia.com for a simple and elegant solution!) • Kepler’s Second Law (Equal Areas in Equal Time) • - Conservation of Angular Momentum leads to it (See Book): • Kepler’s Third Law (T2 ~ r3) • Direct substitution into ULG of the centripetal acceleration (see book):

13. Measuring G: Cavendish1798 G is the same everywhere in the Universe. G’s small size is a measure of relative strength of gravity. By comparison, the proportionality constant for the electric force is k~109!!

14. Universal Law of Gravity (Minus because of the direction of the unit vector. Attractive Central Forces are negative!)

15. Gravity: Inverse Square Law

16. Gravitational Force INSIDE the Earth How would the force of gravity and the acceleration due to gravity change as you fell through a hole in the Earth? What would your motion be? Assume you jump from rest. Ignore Air Resistance.

17. Gravitational Force INSIDE the Earth Inside the Earth the Gravitational Force is Linear. Acceleration decreases as you fall to the center (where your speed is the greatest) and then the acceleration increases but in the opposite direction, slowing you down to a stop at the other end…but then you would fall back in again, bouncing back and forth forever!

18. Earth-Moon Gravity Calculate the force of gravity between the Earth and the Moon. The distance between the Earth and Moon centers is 3.84x108m

19. Earth-Moon Gravity Calculate the acceleration of the Earth due to the Earth-Moon gravitational interaction.

20. Earth-Moon Gravity Calculate the acceleration of the Moon due to the Earth-Moon gravitational interaction.

21. Earth-Moon Gravity The acceleration of gravity at the Moon due to the Earth is: The acceleration of gravity at the Earth due to the moon is: Why the difference? FORCE is the same. Acceleration is NOT!!! BECAUSE MASSES ARE DIFFERENT!

22. Force is not Acceleration! The forces are equal but the accelerations are not!

23. Source of the Force Reaction to the Force This is your WEIGHT! Independent of your mass! This is why a rock and feather fall with the same acceleration! Finding little g Calculate the acceleration of gravity acting on you at the surface of the Earth. What is g?

24. Source of the Force Reaction to the Force Finding little g Calculate the acceleration of gravity acting on you at the surface of the Earth. What is g? = g!

25. In general, g for any Planet: The gravitational field describes the “effect” that any object has on the empty space around itself in terms of the force that would be present if a second object were somewhere in that space g field

26. Electric FieldTwo flavors of charge (+/-)

27. Problem 13.23Find the Magnitude of the Gravitational Field at O. Since the masses are static, just add the fields due to each mass at O in a vector superposition.. Toward the opposite corner.

28. Before we go on.... Let's go back.....

29. Curvature of Earth Curvature of the Earth: Every 8000 m, the Earth curves by 5 meters! If you threw the ball at 8000 m/s off the surface of the Earth (and there were no buildings or mountains in the way) how far would it travel in the vertical direction in 1 second? The ball will achieve orbit.

30. Orbital Velocity If you can throw a ball at 8000m/s, the Earth curves away from it so that the ball continually falls in free fall around the Earth – it is in orbit around the Earth! Ignoring air resistance. Above the atmosphere

31. Projectile Motion/Orbital Motion Projectile Motion is Orbital motion that hits the Earth!

32. Orbital Motion| & Escape Velocity 8km/s: Circular orbit Between 8 & 11.2 km/s: Elliptical orbit 11.2 km/s: Escape Earth 42.5 km/s: Escape Solar System!

33. Circular Orbits As the ball falls around the Earth in a circular orbit, does the acceleration due to gravity change its orbital speed? NO! It only changes its direction! Ignoring air resistance. Above the atmosphere

34. Circular Orbital Velocity The force of gravity is perpendicular to the velocity of the ball so it doesn’t speed it up – it changes only the direction of the ball. Gravity provides a centripetal acceleration the keeps it in a circle! The PE and KE are the same throughout the orbit. Since F is parallel to r, angular momentum is also conserved. v

35. Elliptical Orbits As the ball falls around the Earth in an elliptical orbit, does the acceleration due to gravity change its orbital speed? YES! There is a component of force (and acceleration) in the direction of motion! Gravity changes the satellite’s speed when in elliptical orbits. Energy is conserved but KE and PE change throughout the orbit. Since F is in the direction of r, Angular Momentum is also conserved. Where is the speed greatest- A or B? A B

36. Orbits Circular Orbit Elliptical Orbit

37. Orbital Speed With Increasing Altitude Why?

38. g and v Above the Earth’s Surface If an object is some distance h above the Earth’s surface, r becomes RE + h The tangential speed of an object is its orbital speed and is given by the centripetal acceleration, g: Orbital speed decreases with increasing altitude!

39. What is this? Notice that this is less 8km/s! Orbit Question Find the orbital speed of a satellite 200 km above the Earth. Assume a circular orbit.

40. Kepler’s 3rd! Orbit Question What is the period of a satellite orbiting 200 km above the Earth? Assume a circular orbit. If you don’t know the velocity: Period increases with r!

41. Orbital Sum….with increasing altitude: g, acceleration decreases v, orbital speed decreases T, orbital period increases

42. Satellite Orbits

43. Global Geostationary Satellite Coverage USA USA Euro Japan China USSR

44. Sun-Synchronous Near Polar Orbits With an orbital period of about 100 minutes, these satellites will complete slightly more than 14 orbits in a single day.

46. Grav. Potential Energy – Work Since the Force is Conservative, the Work is independent of path. • The work done by F along any radial segment is • The work done by a force that is perpendicular to the displacement is 0 • The total work is Recall that the work done by a conservative force on an object is: (As a rock falls, it loses PE but gains KE!!!)

47. Gravitational Potential EnergyNote: We call it U not PE now! • As a particle moves from A to B, its gravitational potential energy changes by • Choose the zero for the gravitational potential energy where the force is zero: Ui = 0 where ri = ∞ • This is valid only for r ≥ RE and not valid for r < RE • U is negative because of the choice of Ui

48. Gravitational Potential Energy for the Earth • Graph of the gravitational potential energy U versus r for an object above the Earth’s surface • The potential energy goes to zero as r approaches infinity. • The potential energy is negative because the force is attractive and we chose the potential energy to be zero at infinite separation • An external agent must do positive work to increase the separation between two objects • The work done by the external agent produces an increase in the gravitational potential energy as the particles are separated • U becomes less negative • The absolute value of the potential energy can be thought of as the binding energy • If an external agent applies a force larger than the binding energy, the excess energy will be in the form of kinetic energy of the particles when they are at infinite separation

49. Problem 13.27 How much energy is required to move a 1 000-kg object from the Earth's surface to an altitude twice the Earth's radius?

50. Problem 13.28 At the Earth's surface a projectile is launched straight up at a speed of 10.0 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth. The height attained is not small compared to the radius of the Earth, so U = mgh does not apply. Use: U = -GmM/r :