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 Our Our Place Place in in the the Cosmos Cosmos can explain Kepler’s laws of planetary motion in the special case of circular orbits Lecture 8 • A full mathematical derivation of elliptical orbits is beyond the scope of this course Non-Circular Orbits and Tides • We can, however, gain some intuitive understanding of non-circular orbits 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 a bove 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, Captions gaining speed on the way Non-Circular Orbits Escape Velocity • Gravity also predicts unbound orbits • The further a satellite pulls away from the • The greater the speed of a satellite at closest Earth, the more slowly it moves, until it approach, the further it is able to pull away from the reaches a maximum distance Earth and the more eccentric its orbit • It then falls back towards the Earth, gaining • If a satellite is is moving faster than its escape velocity gravity is unable to reverse its outward speed as it does so motion • This is true for any object on an elliptical • The satellite then coasts away from Earth, never to orbit about a more massive body, including a return planet orbiting the Sun • One can show that the escape velocity is a factor � 2 larger than the circular velocity • Gravity thus explains Kepler’s 2nd law, why v esc = � [2 G M / r ] = � 2 v circ - about 11 km/s on Earth planets sweep out equal areas in equal times
Unbound Orbits Bound elliptical orbits v < v esc • If velocity is less than escape velocity, orbit will be elliptical Unbound • If velocity is greater than escape velocity, parabolic orbit orbit will be hyperbolic and will be unbound v = v esc • Parabolic orbits are the limiting case, where Unbound v = v esc (also unbound) hyperbolic orbits v > v esc Mass Estimates Gravity and Extended Objects • The gravitational pull of an extended • Newton’s form of Kepler’s 3rd law can be object (such as the Earth) is equal to rearranged to read M = 4 � 2 / G x ( A 3 / P 2 ) the sum of the gravitational forces of all of the mass elements which comprise • This formula is used throughout astronomy to make mass estimates the object • It still holds when mass of orbiting object is • For a spherically symmetric object, the comparable to central mass net gravitational force is equivalent to a • In this case each object orbits about their common point source located at the centre with centre of mass and M above is the total mass of the the same mass system 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
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 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 Tidal Forces Tidal Forces • We have seen that gravitational forces within • Imagine holding three rocks at different an object (self-gravity) vary with location heights far above the Moon’s surface and let them go at the same time • External gravitational forces will also vary in strength depending on location within and on • The rock closest to the Moon feels the the surface of an object such as the Earth strongest gravitational force and so accelerates fastest towards the Moon • Consider Moon’s gravitational pull on the Earth • The rock furthest from the Moon feels the weakest gravitational force and so • That part of the Earth closer to the Moon feels a stronger force accelerates slowest • That part further away feels a weaker force • As the rocks fall towards the Moon the • Difference is about 7% separation between them increases
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 Captions Moon’s gravitational pull Average force felt by Tidal Forces all parts of Earth is is stronger on the near side of the Earth than on responsible for overall • Gravitational force due to Moon is 300,000 motion towards Moon the far side 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 Difference between actual force • However, the residual acceleration due to the at each point and the average varying strength of gravity with distance from force is the tidal force the Moon is not the same everywhere Tidal Forces Tides • A 1 kg mass on the side of the Earth closer to the • Tidal forces produce an obvious effect on the Moon feels a force towards the Moon of 1.1 x 10 -6 N oceans, the lunar tides relative to the Earth as a whole • There is a tidal bulge in the oceans in • On opposite side, relative force is same but points directions towards and directly away from the away from the Moon Moon • Earth is also squeezed by a net force in direction • As Earth rotates beneath the oceans, the perpendicular to the Moon tides ebb and flow • Earth’s shape is distorted by these residual forces, or tidal stresses, and slightly elongated along direction • In addition, friction between the rotating towards Moon Earth and the ocean drags the tidal bulge in • Note there is no actual force pulling on the far side the direction of rotation, so it does not point of the Earth, the force towards the Moon is just less exactly towards and away from the Moon than average here
Without rotation, Tides tidal bulge would occur along Earth- • High and low tides occur at intervals of about Moon axis 6 1/4 hours rather than exactly 6 hours, since Moon is orbiting Earth as it rotates Rotation drags tidal • It thus takes 25 hours to return to spot that bulge faces the Moon • Tidal range depends on local geology As Earth rotates • Mediterranean is largely enclosed and has beneath the tidal small tidal range bulge, tides rise • Bay of Fundy in Canada experiences 14-16 m and fall tides! Solar Tides Spring and Neap Tides • The side of the Earth closer to the Sun also • If Sun and Moon are aligned their experiences stronger gravitational pull than combined tidal force is greater than far side that of the Moon alone by about 50% • Although Sun’s overall gravitational force is • Strong tides near new or full Moon are 200 times that of the Moon, much larger known as spring tides distance of Sun means only a very small difference in gravitational pull between one • Around 1st and 3rd quarter, solar and side of the Earth and the other lunar tides partially cancel - neap tides • Solar tides are about half the strength of lunar tides 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 Solar and lunar tides Solar and lunar tides speed - tidal locking partially cancel to give add to give large tides • Moon is no longer being continually deformed small tides
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