Our Our Place Place in in the the Cosmos Cosmos can explain - - PDF document

Our Our Place Place in in the the Cosmos Cosmos

Lecture 8 Non-Circular Orbits and Tides

Non-Circular Orbits

• In the previous lecture we saw that Newton’s

universal law of gravitation Fg = G x m1 x m2 / r2 can explain Kepler’s laws of planetary motion in the special case of circular orbits

• A full mathematical derivation of elliptical
• rbits is beyond the scope of this course
• 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 above a circular orbit and hence its distance from Earth (“vertical distance”) increases

• Exactly like a ball thrown in the air, the pull
• f gravity slows vertical motion until vertical

motion stops and is then reversed as ball/satellite falls back towards Earth, gaining speed on the way Captions

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
• rbit 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

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 vesc = [2G M/r] = 2 vcirc - about 11 km/s on Earth

Unbound Orbits

• If velocity is less than escape velocity, orbit

will be elliptical

• If velocity is greater than escape velocity,
• rbit will be hyperbolic and will be unbound
• Parabolic orbits are the limiting case, where

v = vesc (also unbound)

Bound elliptical

• rbits

v < vesc Unbound parabolic orbit v = vesc Unbound hyperbolic orbits v > vesc

Mass Estimates

• Newton’s form of Kepler’s 3rd law can be

rearranged to read M = 42/G x (A3/P2)

• 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

Gravity and Extended Objects

• The gravitational pull of an extended
• bject (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

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

• f 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
• bject such as the Earth or Sun

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%

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

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

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 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

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
• f the Earth, the force towards the Moon is just less

than average here

Tides

• Tidal forces produce an obvious effect on the
• ceans, 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

Without rotation, tidal bulge would

• ccur along Earth-

Moon axis Rotation drags tidal bulge As Earth rotates beneath the tidal bulge, tides rise and fall

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!

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

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

Solar and lunar tides add to give large tides Solar and lunar tides partially cancel to give small 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 speed - tidal locking

• Moon is no longer being continually deformed

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

Summary

• Newton’s theory of gravity predicts elliptical
• rbits 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

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?

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?