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2. 2 Orbits & Gravity

3. 3 Tycho Brahe (1546 – 1601) Tee-ko Bra A Danish nobleman and astronomer His most significant contribution to science: Nearly 30 years of his life was spent accurately documenting the position of the planets against the backdrop of the Celestial Sphere.

4. 4 Tycho’s ‘luxurious’ pretelescope-era observatory on the island of Hven where most of his data was collected.

5. 5 After nearly 30 years of observations, Tycho could not find a satisfactory model that supported the Copernican universe. He eventually settled for a more Aristotelian model.

6. 6 www.mhs.ox.ac.uk/tycho/ Eduard Ender (1855) Rudolph II (sitting) contemplates Tycho’s Universe

7. 7 A Copernican Introversive, referred to himself as ‘the mangy dog’ Brilliantly disciplined in the use of mathematics Developed three laws of planetary motion based on Brahe’s extensive data collection Johanes Kepler (1571 – 1630)

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9. 9 Kepler’s Laws are based on the mathematics/geometry associated with conic sections

10. 10 Note that every ellipse has two focal points

11. 11 Eccentricity, e, is a quantitative measure of the degree to which an object deviates from being circular. Note that for a circle the eccentricity is zero and for an increasingly elliptical shape, the eccentricity is approaching 1. A parabolic path has an e equal to 1 and a hyperbolic path has an e greater than 1 but less than infinity.

12. 12 Kepler’s 1 st Law Law of Orbits

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14. 14 Note that none of the planets within our solar system have orbits that are perfect circles. (the eccentricity is not zero)

15. 15 Planets are not the only objects that follow conic paths around the Sun And the Sun is not the only object that has objects orbiting it. Kepler’s Laws apply equally well to the moon around the Earth and any other object orbiting another!

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17. 17 Kepler’s 2 nd Law Law of Areas Note that equal areas are covered by a line extending from the Sun to the orbiting object in equal time intervals

18. 18 Kepler’s 2 nd Law Law of Areas

19. 19 Kepler’s 3 rd Law Law of Periods (Harmony of Spheres) Note that square of the orbital period (p) of each planet is equal to the cube of the semimajor axis (a).

20. 20 Kepler’s 3 rd Law Law of Periods (Harmony of Spheres)

21. 21 Isaac Newton 1642-1727 1642-1727Achievements:Calculus Laws of Motion Law of Gravitation Nature of Light Adv. Telescopes Just to name a few!!

22. 22 1 st Law of Motion Every body continues in its state of rest, or of uniform motion in a right [straight] line, unless it is compelled to change that state by a force impressed on it. This law is often referred to as ‘The Law of Inertia’ and is credited as an accomplishment of Galileo.

23. 23 1 st Law of Motion This is what Newton would refer to as ‘natural’ motion or inertial motion. Note that ‘Force’ is not required for objects to be in or maintain motion.

24. 24 2 nd Law of Motion The change in motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

25. 25 2 nd Law of Motion The change in motion (acceleration) is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. F = ma

26. 26 2 nd Law of Motion The unit of force is given the name Newton [N]. Based on F=ma, the unit of a newton can be expressed as: 1 Newton = (1kilogram)(1meter) (second) 2

27. 27 3 rd Law of Motion To Every action there is always an equal reaction; or, the mutual actions of two bodies are always equal, and directed to contrary parts [opposite directions].

28. 28 3 rd Law of Motion If motion was bestowed upon one object it must have been taken from another. What if this astronauts jetpack fails? How can he get back to the ship? Knowledge of Newton’s 3 rd will save his life!!

29. 29 3 rd Law of Motion F BC = F CB As the book leans and pushes on the crate, the crate pushes with an equal and oppositely directed force on the book.

30. 30 3 rd Law of Motion F Earth on Ball = F Ball on Earth Each time the ball bounces off the Earth, the Earth and ball exert forces on each other. According to Newton’s 3 rd law, these forces are exactly the same in magnitude and in opposite directions. Why is it that the ball is the only object visible changing direction?

31. 31 Newton’s Universal Law of Gravitation Projectile motion on Earth had been well documented before Newton but the models of motion lacked a mechanism for the movement towards Earth. Newton envisioned a force that gives every object with mass the ability to ‘reach across empty space’ and pull on neighboring bodies of mass. GRAVITY

32. 32 The force that one massive body exerts on another such that the two bodies are drawn together is directly proportional to the masses of each body and inversely proportional to the separation distance of the two bodies. The force that one massive body exerts on another such that the two bodies are drawn together is directly proportional to the masses of each body and inversely proportional to the separation distance of the two bodies. Written in the form of Newton’s 2 nd Law: Written in the form of Newton’s 2 nd Law: F = ma = F g = G(m 1 *m 2 )/r 2 = m 1 (Gm 2 /r 2 ) where G is a constant of proportionality given by the value 6.67x10 -11 Nm 2 /kg 2 The value of G was never known to Newton! It would not be discovered until about 1800 by Henry Cavendish. Newton’s Universal Law of Gravitation

33. 33 The value of G was never known to Newton! It would not be adequately measured experimentally until 1798 by Henry Cavendish. Newton’s Universal Law of Gravitation Henry Cavendish (1731-1810) Cavendish’s Torsion Balance Used to estimate the universal gravitational constant, G

34. 34 How could Newton extend his idea of Earthly gravitation to the motion of the celestial bodies? The moon was the key! Newton’s Universal Law of Gravitation

35. 35 If an apple falls towards the Earth due to gravity, does this same gravitational force extend into the heavens? Is it the same force that keeps the Moon in orbit around the Earth? If an apple falls towards the Earth due to gravity, does this same gravitational force extend into the heavens? Is it the same force that keeps the Moon in orbit around the Earth? Newton’s Universal Law of Gravitation

36. 36 Consideration of Uniform Circular Motion The acceleration of any object undergoing uniform circular motion is: The acceleration of any object undergoing uniform circular motion is: This is based on simple motion experiments performed here on Earth. The acceleration in this type of motion is known as Centripetal Acceleration (or center-seeking acceleration)

37. 37 Acceleration of the Moon Towards the Earth (Approximating it to be Uniform Circular motion) The Moon moves around the Earth in a nearly uniform circular orbit with a speed of ~1016m/s. The distance between the Moon and Earth is known to be ~3.8x10 8 m. Approximating the motion to be uniform and circular yields a centripetal acceleration of: a moon = v 2 /r = (1016m/s) 2 /(380 000 000m) a moon = 0.00272m/s 2

38. 38 Newton then imagined his ‘gravitational force’ extending out into the heavens and pulling on the Moon. Since the distance to the Moon is about 60 times longer than the radius of the Earth, Earth’s gravity should be considerably weaker at the Moon’s location. Based on earthly experimentation, things accelerate downwards at a rate of 9.81m/s 2 when near the surface of the Earth. Assuming an inverse square law relationship between gravity and distance, Newton supposed that the Earth’s gravitational acceleration should be about 60 2 times less (3600 times less) near the Moon than it is on Earth. a moon = (9.81m/s 2 )/3600 a moon = 0.00271m/s 2 Acceleration of the Moon Towards the Earth (Using Newton’s Gravitational Inverse Square Law)

39. 39 Gravity Newton found the connection The forces (gravity) that govern projectile motion here on planet Earth ALSO govern the motion of the moon & the planets. REVOLUTIONARY THINKING!!!

40. 40 Newton’s Gravitation and Kepler’s Laws Kepler’s 3 rd law (p 2 = a 3 ) can be derived (with a little algebra & calculus) from Newton’s Law of Gravitation. G(M 1 + M 2 )P 2 = 4π 2 a 3 Where a is the semimajor axis length, P is the period of orbit, and the M’s are the masses of the orbiting objects

41. 41 Newton’s Gravitation and Kepler’s Laws G(M 1 + M 2 )P 2 = 4π 2 a 3 Newton had given a mechanism for Kepler’s planetary motion laws With this relationship, we can measure distances and periods by observation and then calculate the mass of any object orbiting another!!

42. 42 The masses of the stars and planets G(M 1 + M 2 )*p 2 = 4πa 3 It is from this relationship that we estimate the mass of the Sun, the planets, and any other celestial body orbiting another!! Keypoint: Masses of Celestial objects are derived from observing orbital motion of the objects.

43. 43 History recalls that nearly 20 years passed before Newton was urged by his contemporary, Sir Edmund Halley, to publish his results Newton’s laws of motion and gravity were first published in 1687 as Philosophiae Naturalis Principia Mathematica. Edmund Halley (1656-1742)

44. 44 Fig 3-13, p.72 Phases of the Moon What causes them? Earth/Moon Gravitational Effects The Earth and Moon exert mutual gravitational forces on each other Every particle on the Earth is influenced by the gravitational force of the Moon This has a small effect on the hard surface of the Earth However, the effect is quite noticeable on the Earth’s liquid surface (oceans)

45. 45 Phases of the Moon What causes them? Ocean Tides Two tidal bulges are caused by the pull of the moon There is a tidal bulge on the moon side of the Earth due to strong lunar gravity and another directly on the other side due to weak lunar gravity and the fact that the Earth is accelerating towards the moon (‘essentially leaving the water behind’)

46. 46 Ocean Tides

47. 47 Phases of the Moon What causes them? The Sun & Ocean Tides The Sun can also influence the tides (but to a lesser degree) When the Sun and Moon lie on the a line that passes through Earth (either on different sides, Full Moon or on the same side, New Moon) the tides are higher and are called ‘Spring Tides’ (nothing to do with Spring season) When the Sun and Moon lie on perpendicular lines relative to Earth (either first quarter or third quarter) the tides are lower and are called Neap tides.