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Gravity and Tides
Tides result from the differential gravitational force across a finite
sized object.
●
●
simply put, the near side of the object feels a stronger tug of gravity than the far
side.
Horribly
not to
scale!!!
2
Gravity and Tides
●
Tides result from the differential gravitational force across a finite
sized object.
–
consider the gravitational force acting on a parcel of mass at the center of
the object.
Fgrav
GMm
=
2
r
different “r” for
different parts of
the Earth.
Gravity and Tides
●
Tides result from the differential gravitational force across a finite
sized object.
–
consider the gravitational force acting on a parcel of mass at the center of
the object.
Fgrav
GMm
=
2
r
subtract the force at the center from all the other forces....
3
Gravity and Tides
●
Tides result from the differential gravitational force across a finite
sized object.
–
subtract the force at the center of the object from all of the other force
vectors and the resultant forces represent the tidal stresses on the object.
4
Gravity and Tides
●
Tides result from the differential gravitational force across a finite
sized object.
–
Quantify tidal forces by differentiating the gravitational force law with
respect to “r”
Fgrav
GMm
=
2
r
d Fgrav = − 2GMm
dr
3
r
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Consequences of Solid Body Tides
Europa's Ocean
Volcanoes on Jupiter's Io
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How Much Do Things Stretch in Response
●
●
A completely rigid object will not change shape in response to tidal
forces.
A fluid will adjust to fill a surface of constant potential energy.
–
a fluid on a more rigid sphere (Earth's oceans) will flex differentially to the
solid surface.
●
solid body lunar tides in the Earth are about 0.3 meter on average
●
ocean tides are about 1 meter on average
–
this 1 meter is in excess of the solid body tides
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Ocean Tides
●
●
High ocean ties occur approximately every 12.5 hours as the
Earth rotates beneath the slowly orbiting Moon.
A given location experiences two high tides a day as Earth rotation
carries the observer through each of the tidal bulges.
Direction of Earth Rotation
The Moon orbits
slowly enough
relative to Earth's
rotation rate (29.5
day synodic month
vs. 1 day-long day)
that we can consider
it occupying a fixed
location in space
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Ocean Tides
●
●
High ocean ties occur approximately every 12.5 hours as the
Earth rotates beneath the slowly orbiting Moon.
A given location experiences two high tides a day as Earth rotation
carries the observer through each of the tidal bulges.
Direction of Earth Rotation
The Moon orbits
slowly enough
relative to Earth's
rotation rate (29.5
day synodic month
vs. 1 day-long day)
that we can consider
it occupying a fixed
location in space
Ocean Tides vs. Latitude
●
●
●
The Moon's orbit lies close to
the ecliptic plane and thus is
significantly inclined to the
Earth's equator.
Add to that the 5-degree
inclination of the Moon's orbit
to the Ecliptic and the Moon
can reach a declination of
nearly 30 degrees.
A location's two high tides can
be quite different in a given
day depending on the Moon's
declination.
The image above shows the
significant difference between the
two daily tides when the Moon is
at high declination
(if the Moon is on the celestial
equator the simple tides would
be uniform at a given latitude).
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Real Ocean Tides
●
Landmasses substantially complicate the simple situation.
–
The image below shows the tidal range at different ocean locations.
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12
Bay of Fundy, Canada
Solar vs. Lunar Tides
●
●
●
In the force equations M is the mass of the tide-causing object, r is
the separation between the two objects. dr is the size of the
object on which the tides are being raised.
The Sun is 30 million times the mass of the Moon, but the Moon is
400 times closer than the Sun.
The Sun has about 1/3 the Moon's tidal influence on the Earth.
d FSun
M sun r moon
=
d F moon
M moon r sun
(
)( )
3
d Fgrav = − 2GMm
dr
3
r
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Superposition of the Solar and Lunar Tides

When the Sun and Moon align (New and Full Moon) tides are
higher than when they raise tides in different directions (First and
Last Quarter Moon).
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Superposition of the Solar and Lunar Tides

When the Sun and Moon align (New and Full Moon) tides are
higher than when they raise tides in different directions (First and
Last Quarter Moon).
Note that planetary tides, often
invoked by nutcase theories of
global doom from planetary
alignment, are vanishingly
insignificant compared to the Sun
and Moon.
Jupiter's tidal force on Earth is
1/100,000th that of the Sun's.
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The Sun's Role in Tides


The Sun also has significant gravitational influence on the Earth.

It is much further away than the Moon, but also much more massive.

Solar tides are about 1/3 the strength of Lunar tides.
When the Sun and Moon align (New and Full Moon) tides are
higher than when the raise tides in different directions (First and
Last Quarter Moon).
Note that planetary tides, often
invoked by nutcase theories of
global doom from planetary
alignment, are vanishingly
insignificant compared to the Sun
and Moon.
Jupiter's tidal force on Earth is
1/100,000th that of the Sun's.
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The Slowing of Earth Rotation
●
Due to friction, the Earth's tidal bulges are carried slightly ahead of
the Moon.
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The Slowing of Earth Rotation
●
The net forces lead to a slowing of Earth rotation and a
corresponding increase in the angular momentum of Moon's orbit.
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The Changing Day


Due to tidal effects the day gets about 1 second longer every
60,000 years.

About 400 million years ago the day was only 22 hours long.

The day will be 25 hours long in another 200 million years.
Interestingly, the second was defined using measurements from
more than 200 years earlier.

Using this “stale” second, The Earth runs slow enough that we have to
add a leap second into timekeeping every couple of years.
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Consequences for the Moon

Tidal coupling moves the Moon a few centimeters further from
the Earth each year.

Although small, this effect is measured to great accuracy with pulses of
laser light bounced off of retro-reflectors on the Moon.
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Conservation of Angular Momentum in the
Earth-Moon System
√
GM
L Moon (orbit) = m v r = mr
= m √ GMr
r
d L Moon
= m GM dr
dt
2
r dt
√
m is the Moon's mass
M is the Earth's mass
r is the Earth-Moon separation
R is the radius of the Earth
2
2 2π
L Earth (rot ) = I ω = MR
5
P rot
( )
d L Earth (rot )
d P rot
2
4
π
1
=−
MR
2
dt
5
P rot dt
(
dP/dt = 0.0016 sec/century
dr/dt = 4 cm/year
)
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Consequences for the Moon

Tidal coupling moves the Moon a few centimeters further from
the Earth each year.

The Moon was once much closer – maybe 1/20th it's current distance.

We live in the last era where total solar eclipses are possible.
−
Total eclipses are becoming increasingly less frequent.
−
soon (in about 100 million years) all central eclipses will be annular.
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Consequences for the Moon

Tidal coupling moves the Moon a few centimeters further from
the Earth each year.

The Moon was once much closer – maybe 1/20th it's current distance.

We live in the last era where total solar eclipses are possible.
−
Total eclipses are becoming increasingly less frequent.
−
soon (in about 100 million years) all central eclipses will be annular.
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Consequences for the Moon

The Moon's rotation has “stopped” relative to the Earth

The Earth was even more effective at slowing the Moon's rotation.
−
Although it may have originally spun rapidly, the Moon is now in a
state where it turns at the same rate that it orbits the Earth.
d Fgrav−moon = −
2GM Earth m test
r
3
tide moon
M Earth d moon
=
tide earth
M Moon d Earth
The Earth has about 20 times
the tidal influence on the Moon
compared with the Moon's
effect on the Earth.
diam moon
Consequences for the Moon

The Moon's rotation has “stopped” relative to the Earth



This “tidal locking” is the natural end state of a planet/moon system.
Even now, the Moon is slowing the Earth's rotation toward the goal of the
Earth always keeping the same face toward the Moon.
Once an object is in “synchronous rotation” it's tidal bulges remain aligned
and there is no more tidal friction.
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Consequences Throughout the Solar System

Nearly all major satellites are synchronously locked to their
planets (certainly all the “close in” ones).

Pluto and Charon are in synchronous lock with each other.

Mercury spins twice for every three orbits around the Sun
– - this funky synchronous lock results from Mercury's significantly elliptical
orbit. As its rotation slowed it found the 3:2 spin/orbit stable before slowing
down to reach 1:1
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Consequences for the Moon

From Earth we can only see one side of the Moon.
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The X
Dark
X Far Side of the Moon

From Earth we can only see one side of the Moon.

The other side of the Moon (which has 2-week long days just like the near
side) was not observed until the Space Age.
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The lunar far
side (mostly)
Lunar Libration
●
We actually see about 59% of the Moon from the Earth due to
three effects.
–
Diurnal libration – peaking around the edge as Earth rotation changes your
perspective
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Lunar Libration
●
We actually see about 59% of the Moon from the Earth due to
three effects.
–
Libration in Longitude – due to the changing speed of the Moon along its
elliptical (e=0.055) orbit.
–
At the extreme this effect results in seeing about 6 degrees in longitude
around either side of the Moon.
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Lunar Libration
●
We actually see about 59% of the Moon from the Earth due to
three effects.
–
Libration in latitude – the Moon's rotation axis is tipped about 6 degrees to
its orbital plane.
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Lunar Libration
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Lunar Phases
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Lunar Phases
Keep in mind:
●
●
●
Earth rotates “counterclockwise”
looking down on the North Pole
Moon revolves counterclockwise.
First quarter Moon is “ahead” of the
Sun along the Ecliptic.
●
At the Spring Equinox the first
quarter Moon will be at the
Summer Solstice location (high
positive declination) “3-months”
ahead of the Sun.
Last quarter Moon lags 3-months
“behind” (or is 9 months ahead...)
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The Roche Limit
●
Consider two particles in contact with centers separated by “r”. It
the tug of war between tidal forces trying to separate the objects
and mutual gravity trying to hold them together, who wins?
–
Tidal effects fall off as R 3 whereas the mutual gravity of the two particles is
always the same.
–
There must be a distance at which there is a transition from tidal dominance
(particles get torn apart close to a planet) to mutual gravitational dominance
(particles stick and grow far away) – the Roche Limit
tidal
force
Planet
m
d F grav = −
2GMm
Δr
3
R
m
M
R
mutual
gravity
F grav =
Gmm
2
(r )
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The Roche Limit
●
Consider two particles in contact. It the tug of war between tidal
forces trying to separate the objects and mutual gravity trying to
hold them together, who wins?
–
Tidal effects fall off as R 3 whereas the mutual gravity of the two particles is
always the same.
–
There must be a distance at which there is a transition from tidal dominance
(particles get torn apart close to a planet) to mutual gravitational dominance
(particles stick and grow far away) – the Roche Limit
r roche
ρ planet
= 2.44 ρ
particles
(
)
1/3
R planet
The coefficient 2.44 above derives from a formal treatment of a liquid
droplet being sheared apart by tidal forces.
The Roche Limit
●
r roche
ρ planet
= 2.44 ρ
particles
(
38
1 /3
)
R planet
Phobos orbits inside Mars' Roche Limit, and although it is
expected to be a “rubble pile” given its density (1.8 g/cc), there are
enough cohesive forces to hold it together.
–
Tidal coupling is moving Phobos closer to Mars. Likely in <100 million
years the remains of Phobos will become a (temporary) ring around Mars.
Interesting article....
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The Roche Limit
●
r roche
ρ planet
= 2.44 ρ
particles
(
40
1 /3
)
R planet
Planetary rings (Jupiter, Saturn, Uranus, and Neptune) lie inside
the Roche Limit.
The Roche Limit
●
r roche
ρ planet
= 2.44 ρ
particles
(
41
1 /3
)
R planet
Planetary rings (Jupiter, Saturn, Uranus, and Neptune) lie inside
the Roche Limit.
Uranus
Neptune
The Hill Radius
●
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●
The gravitational force exerted on the Moon by the Sun is twice
the gravitational force exerted by Earth on the Moon???
Just what determines if a planet can hold on to a satellite?
The answer is not as simple as tracking forces – both the Earth
and Moon are “falling” around the Sun in their mutual orbit – to first
order not knowing that the Sun is even there.
Presuming the Earth stays at constant distance from the Sun, the
Moon is sometimes closer to and sometimes farther than the Earth
is from the Sun
–
If the difference in solar gravitational acceleration at these extremes relative
to the Earth exceeds the gravitational acceleration of the Moon by the Earth
the Earth will likely lose the Moon.
–
This boundary, known as the Hill Radius or Hill Sphere, is somewhat fuzzy
because weak long-term disturbances have more significant effects on
stability than simple instantaneous conditions, especially in multi-planet
situations.
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The Hill Radius
●
Illustration gravitational potential in a 2 body system
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The Hill Radius
●
Consider a particle in orbit around a planet.
–
How does the gravitational acceleration of the planet on the particle
compare with the difference in acceleration between the planet and the Sun
vs. the particle and the Sun.
M planet
R Hill =
2 M sun
(
)
1/ 3
a planet
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The Hill Radius
●
Consider a particle in orbit around a planet.
–
How does the gravitational acceleration of the planet on the particle
compare with the difference in acceleration between the planet and the Sun
vs. the particle and the Sun
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