Transcript ppt

Our Place in the
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 orbits
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 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
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 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
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, orbit
will be hyperbolic and will be unbound
• Parabolic orbits are the limiting case, where
v = vesc (also unbound)
Bound elliptical
orbits
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 = 42/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
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
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
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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
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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
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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
• 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
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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
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
Without rotation,
tidal bulge would
occur along EarthMoon 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
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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
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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
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
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?