1. Our Place in the Cosmos Lecture 8 Non-Circular Orbits and Tides

2. Non-Circular Orbits In the previous lecture we saw that Newton’s universal law of gravitation F g = G x m 1 x m 2 / r 2 can explain Kepler’s laws of planetary motion in the special case of circular orbits A full mathematical derivation of elliptical orbits is beyond the scope of this course We can, however, gain some intuitive understanding of non-circular orbits

3. Non-Circular Orbits Consider a satellite in a circular orbit and imagine giving a boost to its orbital velocity Earth’s gravitational pull is unchanged but the greater speed of the satellite causes it to climb above a circular orbit and hence its distance from Earth (“vertical distance”) increases Exactly like a ball thrown in the air, the pull of gravity slows vertical motion until vertical motion stops and is then reversed as ball/satellite falls back towards Earth, gaining speed on the way

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5. Non-Circular Orbits The further a satellite pulls away from the Earth, the more slowly it moves, until it reaches a maximum distance It then falls back towards the Earth, gaining speed as it does so This is true for any object on an elliptical orbit about a more massive body, including a planet orbiting the Sun Gravity thus explains Kepler’s 2nd law, why planets sweep out equal areas in equal times

6. Escape Velocity Gravity also predicts unbound orbits The greater the speed of a satellite at closest approach, the further it is able to pull away from the Earth and the more eccentric its orbit If a satellite is is moving faster than its escape velocity gravity is unable to reverse its outward motion The satellite then coasts away from Earth, never to return One can show that the escape velocity is a factor  2 larger than the circular velocity v esc =  [2G M/r] =  2 v circ - about 11 km/s on Earth

7. Unbound Orbits If velocity is less than escape velocity, orbit will be elliptical If velocity is greater than escape velocity, orbit will be hyperbolic and will be unbound Parabolic orbits are the limiting case, where v = v esc (also unbound)

8. Bound elliptical orbits v < v esc Unbound parabolic orbit v = v esc Unbound hyperbolic orbits v > v esc

9. Mass Estimates Newton’s form of Kepler’s 3rd law can be rearranged to read M = 4  2 /G x (A 3 /P 2 ) This formula is used throughout astronomy to make mass estimates It still holds when mass of orbiting object is comparable to central mass In this case each object orbits about their common centre of mass and M above is the total mass of the system

10. Gravity and Extended Objects The gravitational pull of an extended object (such as the Earth) is equal to the sum of the gravitational forces of all of the mass elements which comprise the object For a spherically symmetric object, the net gravitational force is equivalent to a point source located at the centre with the same mass

12. Gravity Within the Earth Consider a hypothetical observer at the centre of the Earth They would feel an equal gravitational pull in all directions and so the net gravitational force would be zero - they would be truly weightless

14. Gravity Within the Earth Now consider an observer part way out from the centre of the Earth Think of the Earth as made up of two pieces 1. a sphere containing those parts of the Earth closer to the centre 2. a shell comprising the rest of the Earth The gravitational force from the first part is the same as that as a point with the mass of the smaller sphere located at the centre The outer shell provides zero net gravitational force

16. Gravity Within a Sphere Only mass closer to the centre exerts a net gravitational pull This mass acts as a point mass located at the centre Gravity thus gets weaker as we get closer to the centre This is true within any spherically-symmetric object such as the Earth or Sun

17. Tidal Forces We have seen that gravitational forces within an object (self-gravity) vary with location External gravitational forces will also vary in strength depending on location within and on the surface of an object such as the Earth Consider Moon’s gravitational pull on the Earth That part of the Earth closer to the Moon feels a stronger force That part further away feels a weaker force Difference is about 7%

18. Tidal Forces Imagine holding three rocks at different heights far above the Moon’s surface and let them go at the same time The rock closest to the Moon feels the strongest gravitational force and so accelerates fastest towards the Moon The rock furthest from the Moon feels the weakest gravitational force and so accelerates slowest As the rocks fall towards the Moon the separation between them increases

19. Tidal Forces If the rocks were connected by springs, the springs would stretch - an observer on the middle rock would perceive forces pulling on the other rocks in opposite directions The same thing happens if we replace the three rocks with different parts of the Earth Differences in the Moon’s gravitational pull try to stretch the Earth out along a line pointing towards the Moon

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21. Tidal Forces Gravitational force due to Moon is 300,000 times weaker than that due to the Earth Nevertheless, Moon’s pull causes Earth to wobble by more than 9000 km back and forth during a month We do not perceive this motion since everything on Earth falls together towards the Moon However, the residual acceleration due to the varying strength of gravity with distance from the Moon is not the same everywhere

22. Moon’s gravitational pull is stronger on the near side of the Earth than on the far side Average force felt by all parts of Earth is responsible for overall motion towards Moon Difference between actual force at each point and the average force is the tidal force

23. Tidal Forces A 1 kg mass on the side of the Earth closer to the Moon feels a force towards the Moon of 1.1 x 10 -6 N relative to the Earth as a whole On opposite side, relative force is same but points away from the Moon Earth is also squeezed by a net force in direction perpendicular to the Moon Earth’s shape is distorted by these residual forces, or tidal stresses, and slightly elongated along direction towards Moon Note there is no actual force pulling on the far side of the Earth, the force towards the Moon is just less than average here

24. Tides Tidal forces produce an obvious effect on the oceans, the lunar tides There is a tidal bulge in the oceans in directions towards and directly away from the Moon As Earth rotates beneath the oceans, the tides ebb and flow In addition, friction between the rotating Earth and the ocean drags the tidal bulge in the direction of rotation, so it does not point exactly towards and away from the Moon

25. Without rotation, tidal bulge would occur along Earth- Moon axis Rotation drags tidal bulge As Earth rotates beneath the tidal bulge, tides rise and fall

26. Tides High and low tides occur at intervals of about 6 1/4 hours rather than exactly 6 hours, since Moon is orbiting Earth as it rotates It thus takes 25 hours to return to spot that faces the Moon Tidal range depends on local geology Mediterranean is largely enclosed and has small tidal range Bay of Fundy in Canada experiences 14-16 m tides!

27. Solar Tides The side of the Earth closer to the Sun also experiences stronger gravitational pull than far side Although Sun’s overall gravitational force is 200 times that of the Moon, much larger distance of Sun means only a very small difference in gravitational pull between one side of the Earth and the other Solar tides are about half the strength of lunar tides

28. Spring and Neap Tides If Sun and Moon are aligned their combined tidal force is greater than that of the Moon alone by about 50% Strong tides near new or full Moon are known as spring tides Around 1st and 3rd quarter, solar and lunar tides partially cancel - neap tides

29. Solar and lunar tides add to give large tides Solar and lunar tides partially cancel to give small tides

30. Tidal Locking Earth itself is distorted by about 30cm between high and low tide Energy taken to deform planet causes Earth’s rotation to gradually slow - day length is getting longer by 0.0015 seconds each century Moon is also distorted by Earth’s tidal force - by about 20 metres! Early deformation of Moon’s shape slowed its rotation until rotation speed matched orbital speed - tidal locking Moon is no longer being continually deformed

31. Moon is permanently elongated along the direction towards the Earth so it does not rotate through its tidal bulge The same side always faces the Earth

32. Summary Newton’s theory of gravity predicts elliptical orbits if a satellite is not moving at exactly the circular velocity If a satellite exceeds the escape velocity it will be on an unbound orbit Tides are due to the diminishing gravitational pull from the Moon from the side of the Earth facing the Moon to the opposite side Spring tides occur when Sun and Moon pull in same or opposite directions

33. Discussion Topics Orbits of planets around the Sun are ellipses rather than perfect circles - Why? Would a pendulum swing if it were in orbit? If the Sun were dark and invisible, explain how we could still tell that we are in an elliptical orbit about a large mass and at which focus the mass was located The escape velocity at the Earth’s surface is 11.2 km/s. What would be the escape velocity on the surface of an asteroid with radius 10 -4 and mass 10 -12 that of the Earth?

34. Discussion Topics Is an astronaut in an orbiting shuttle weightless? What about somebody at the centre of the Earth? If Earth had constant density and was exactly spherical, what would be your weight at the bottom of a deep well reaching halfway to the Earth’s centre compared to your surface weight? During which phases of the Moon and at what times of day do the lowest tides occur?