Newton and Kepler`s Third Law

Newton and Kepler's Third Law
•  Newton's laws of gravity and motion showed that the
relationship between the orbital period and average
orbital distance of a system tells us the total mass of the
system.
•  Examples:
–  Earth's orbital period (1 year) and average distance
(1 AU) tell us the Sun's mass.
–  Orbital period and distance of a satellite from Earth
tell us Earth's mass.
–  Orbital period and distance of a moon of Jupiter tell us
Jupiter's mass.
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Newton's Version of Kepler's Third Law
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G = Newton’s gravitational constant
p = orbital period
a = average orbital distance (between centers)
(M1 + M2) = sum of object masses
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Newton's Version of Kepler's Third Law
3
a
2
P =
M1 + M 2
p = orbital period in years
a = average orbital distance in AU (between centers)
(M1 + M2) = sum of object masses in Sun masses
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What have we learned?
•  What determines the strength of gravity?
–  Directly proportional to the product of the
masses (M × m)
–  Inversely proportional to the square of the
separation
•  How does Newton's law of gravity allow us to
extend Kepler's laws?
–  Applies to other objects, not just planets
–  Includes unbound orbit shapes: parabola,
hyperbola
–  Can be used to measure mass of orbiting
systems
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4.5 Orbits, Tides, and the Acceleration of
Gravity
•  Our goals for learning:
–  How do gravity and energy together allow
us to understand orbits?
–  How does gravity cause tides?
–  Why do all objects fall at the same rate?
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How do gravity and energy together allow
us to understand orbits?
•  Total orbital energy
(gravitational +
kinetic) stays
constant if there is
no external force.
•  Orbits cannot
change
spontaneously.
Total orbital energy stays constant.
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Changing an Orbit
•  So what can make an
object gain or lose orbital
energy?
•  Friction or atmospheric
drag
•  A gravitational encounter
(an external force)
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Escape Velocity
•  If an object gains orbital
energy, it moves to a
more distant orbit.
•  If an object gains enough
energy, it may escape
(change from a bound to
unbound orbit).
•  Escape velocity from
Earth ≈ 11 km/s from sea
level (about 40,000 km/hr)
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How does gravity cause tides?
•  Moon's gravity pulls harder on near side of Earth
than on far side.
•  Difference in Moon's gravitational pull from one
side to the other “stretches” the Earth.
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Tides and Phases
•  Since the tidal
stretching is twosided, as the Earth
rotates, there are two
high tides and two
low tides a day.
•  Strength of tides
depends on phase of
Moon.
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Tidal Friction
•  The Earth’s rotation drags the tidal bulge forward
from the Earth-Moon line.
•  Since the bulge leads the Moon, the Moon is
pulled a little bit forward in its orbit. It gains orbital
energy.
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Tidal Friction
•  If the Moon is gaining energy, it must be moving
away from the Earth!
•  How fast? About 38 millimeters per year!
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Tidal Friction
Tidal Friction
•  If the moon is moving away, it must be gaining
angular momentum:
Angular momentum = mass x distance x speed
•  Angular momentum is conserved, so something
must also be losing angular momentum. What?
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Tidal Friction
•  The Earth must be slowing its rotation – days are
getting longer!
•  How fast? About one second longer every 50,000
years!
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Tidal Friction
Tidal rhythmite
Layers of sediment
laid down on
regular basis.
Shows that ~ 600
million years ago,
the day was only
about 21.9 hours
long!
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Tidal Friction
•  When will it all stop? When the Earth’s rotation slows
enough so that it no longer drags the bulges forward.
In about 50 billion years, we would achieve synchronous
rotation: 1 day = 1 month = 47 current days!
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Tidal Friction
Synchronous Rotation
Earth rotation period = Moon orbital period
•  What would the Moon look like from Earth?
•  What would the Earth look like from the Moon?
What other object have we talked about that shows
synchronous rotation?
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What have we learned?
•  How do gravity and energy together allow us
to understand orbits?
–  Change in total energy is needed to change
orbit
–  Add enough energy (escape velocity) and
object leaves.
•  How does gravity cause tides?
–  The Moon's gravity stretches Earth and its
oceans.
–  Tidal friction causing Moon to move further
away from Earth.
–  Earth’s day is getting longer.
© 2014 Pearson Education, Inc.