Download Michael Koohafkan

Michael Koohafkan
What is Tidal Locking?
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Masses exert gravitational pull on other
objects
The Moon and Earth pull each other as the
moon orbits the Earth
Gravitational pull of Moon slows rotational
speed of Earth
Tidal locking occurs when Earth’s rotation
becomes synchronized with Moon’s orbit.
What would tidal locking look like
for us?
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Days would be as long as the orbital
period of the Moon.
The Moon would appear to be ‘locked’ in
place in the sky, and would only ever be
visible on one side of the world.
Does tidal locking only occur
between a mass and a satellite?
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No. Tidal locking can occur between any
two masses that orbit around each other.
Planets can become tidally locked with the
stars they orbit around, and stars in a
binary system can become tidally locked
together.
How can tidal locking be
quantified?
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Gravitational pull between masses can be
measured
Rotational and orbital speed of masses
can be measured
Rate of change of rotational speed can be
calculated. If it can be represented as a
function, then approximate length of time
until tidal locking can be calculated
Why does any of this matter
anyway?
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Tidal locking can help measure the age of
a planet in relation to a satellite. By
measuring the rate at which a planet or
satellite is approaching a tidal lock, we can
extrapolate back and estimate the age of
a satellite or planet.
Tidal locking can have interesting
environmental effects that we can observe
on other planets.
The Math
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Tidal Locking can be represented simply
(and primitively) with a linear function
Further accuracy requires the use of a few
relationships
Important Factors
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Sun’s gravitational pull has effect on
Earth-Moon system
Moon’s gravitational pull is responsible
for tidal locking
Moon’s rotational speed *may* be
changing
Moon’s orbital speed *may* be changing
Moon is drifting away from Earth
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Moon’s gravitational pull is decreasing due
to increase in distance
The Equations
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Depending on what factors we want to
ignore, we can quantize this relationship in
a simple or complex manner
Goal is to determine relationship that
takes as many factors as possible into
account
Linear
Relationship
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Assumptions:
Sun’s gravitational effect is negligible
Moon’s gravitational pull is constant
•Moon’s orbital speed is constant
•Moon’s distance from Earth is constant
Increase in Earth’s day as a function of time currently
has approximate value of 0.0016 seconds/century.
Equation can be modeled linearly:
D(t) = 86400 s + (0.0016 s/century)(t centuries)
D(t) => length of day as function of time centuries
86400 s => Current length of day
0.0016 => Constant increase in day length s/century
Solution: Simply set D(t) = 28.5 days
28.5 days = 2,462,400 s
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2,462,400 s = 86400 s + (0.0016 s/century)(t centuries)
Solve for t
2,462,400 s = 86400 s + (0.0016 s/century)(t centuries)
2,376,000 s = (0.0016 s/century)(t centuries)
1,485,000,000 centuries = t
T = 148,500,000,000 years
148.5 billion years
“To Assume Makes…”
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Linear Equation assumes that force of
gravity is constant
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Untrue; gravitational force varies with
distance as expressed by F = gmM/r2
Moon is drifting away from Earth by 3-4 cm
each year
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3-4 cm adds up over the years
Must the Change in the Moon’s
Gravitational Pull be Included?
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Yes.
Current gravitational Force exerted by Moon:
F = G(7.35x1022 kg)(5.97x1024 kg) / (384x106 m)2
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= 1.98x1020 N
 Linear equation dictates that if gravitational
force is constant, it will take 928.125 billion
years to achieve tidal locking
Check change in r for that length of time at 4
cm/year
r = 384 x 106 m
Δr m = .004 m/year x 148.5 x 109 years
= 5.94 x 108 m
r at tidal locking = r + Δr
= 384 x 106 m + 5.94 x 108 m
= 978 x 106 m
 Moon’s gravitational pull F at tidal locking:
F = GmM/r2
= G(7.35x1022 kg)(5.97x1024 kg) / (978 x 106 m)2
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= 3.06 x 1019 N
Calculating a
Complex Equation
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Assumptions:
Sun’s gravitational effect is negligible
Moon’s gravitational effect is NOT constant
Change in Earth’s rotational speed (and thus increase
in length of day) is dependent on Gravitational force
of Moon.
D(t) is related to strength of Moon’s Gravitational pull
F.
F is related to distance between Earth and Moon by F
= GmM/r2
r is changed by 3-4 cm (.003 m to .004 m) each year
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r = 384 x 106 meters today
Assume maximum change in Moon’s distance
from Earth (.004 meters/year)
r(t) meters = 384 x 106 meters + (.004 meters/year) (t years)
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F = GmM/r2
F = GmM/(r(t))2
F = GmM/(384 x 106 meters + (.004 meters/year) (t years))
2
The Next Step
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Find equation that relates change in
Earth’s rotational speed to gravitational
force exerted by Moon
Another Problem: The Conservation
of Angular Momentum
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Looking at Earth-Moon as isolated system
Gravity is only force acting between
bodies
Gravity is a conservative force
Need to determine where energy from
Earth’s rotation is transferred to (if Earth
slows down, something else must
change/speed up to compensate)
A Case Study?
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Need to determine which factors are in
play and how they effect equations and
results.
Factors: Moon’s drift from Earth, Moon’s
rotational speed, Moon’s orbital speed.
Case 1
Potential energy gained from increase in Moon’s
distance from Earth accounts for all energy
transferred from Earth’s rotation
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If Moon’s 3-4 cm drift per year (and thus
increase in gravitational potential energy)
accounts for change in Earth’s rotational speed,
equation expressing the Moon’s change in orbit
period must be used for solution
Simple solution; change will be a linear equation
(assuming 3-4 cm drift is constant)
Case 2
Potential energy gained from increase in Moon’s
distance from Earth and change in Moon’s
rotational speed accounts for energy transference
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Same solution as Case 1, since change in
rotational speed will not effect
gravitational force of Moon
Case 3
Potential energy gained from increase in Moon’s
distance from Earth and change in Moon’s orbital
speed accounts for all energy transference
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Moon’s orbital period will have to be
expressed as a function
That function will have to be based on
function that expresses the change in both
the orbital distance and the orbital speed
of the Moon
Case Argh
Some variable combination of the previous
three cases
Painful.
Things I Refuse To Look At Right
Now
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Effect that Earth’s change in axial tilt has
on the Moon’s gravitational pull on the
Earth
Non-circular orbits
The gravitational effect of the Sun
My To-Do List
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Determine function that relates change in
Earth’s rotational speed to change in
Moon’s gravitational force
Determine which cases are realistic and
solve for each
Hope that this is actually calculable
Questions/Comments/Insults?
Sources:
Tides
S J Peale
Encyclopedia of Astronomy & Astrophysics
© IOP Publishing Ltd 2006
ISBN 0333750888
Physics For Scientists and Engineers 3rd e
Douglas C Giancoli
© Prentice Hall 2000
ISBN 0-13-021517-1
Universe 7th e
Roger A Freedman
William J Kaufmann III
© W.H. Freeman and company 2005
ISBN 0-7167-9884-0