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Exam 1, etc.● Exam 1 is officially, Monday September 30, here during the class period.

● If you have a conflict please let me know ASAP.

● A document containing guiding “review questions” is available.● A review session is scheduled for Thursday evening at 7:00 in Gilmer

166.● We will distribute a not-a-problem-set with a few quantitative questions

to help you prepare.● Note that end of class science tidbits are for interest but does not

represent exam material.● Problem Set 3 is technically due today, but given the later than planned

distribution let me know if you need more time.

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Time Between Lunar Meridian Transits● The Moon moves in its orbit (significantly) during the course of a

solar day.● The Moon’s motion is about one 27.3th of 360 degrees (the lunar sidereal

orbital period is 27.3 days) – a little more than 10 degrees.

➔ Since the Earth turns 15 degrees per hour this must add about an hour.● Since somebody on the Earth could be considered to “orbit” the center of

the Earth once a day and the Moon orbits this person like a superior planet, the superior planet relation for sidereal vs. synodic periods applies to the time between meridian crossings.

● ``Culmination” is when an astronomical object reaches its highest altitude. For objects tied to the celestial sphere this happens at meridian crossing.

1Pmoonat culmination

= 1

P Earth day−

1P sid Moonorbit

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Time Between Lunar Meridian Transits● The calculation yields a lunar transit period of 24h 50m

● Implying an average time between high tides of 12h 25m

● You can go to the beach and infer the existence of and the orbital period of the Moon

1Pmoonat culmination

= 1

P Earth day−

1P sid Moonorbit

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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 1/2 the Moon's tidal influence on the Earth.

d Fgrav = −2GMm

r3dr

d FSun

d Fmoon

= (M sun

M moon) (rmoonr sun )

3

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The Slowing of Earth Rotation

● Due to viscosity/friction, the Earth's tidal bulges are carried slightly ahead of the Moon (the Earth rotates much more quickly than the Moon orbits it).

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The Slowing of Earth Rotation

● The differing (tidal) forces on the non-spherically-symmetric tidal bulges lead to a slowing of Earth rotation and a corresponding increase in the angular momentum of Moon's orbit.

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The Lengthening Day and Leap Seconds● Due to tidal effects the day gets about 1 second longer every 60,000 years.

● About 900 million years ago the day was only 18 hours long.

● The day will be 25 hours long in another 200 million years.

● Interestingly, the SI second was defined using measurements from 150 years earlier. Using this “stale” second, leap seconds had to be added as soon as the SI second was defined.

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The Lengthening Day and Leap Seconds● Due to tidal effects the day gets about 1 second longer every 60,000 years.

● About 900 million years ago the day was only 18 hours long.

● The day will be 25 hours long in another 200 million years.

● Interestingly, the SI second was defined using measurements from 150 years earlier. Using this “stale” second, leap seconds had to be added as soon as the SI second was defined.

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Consequences for the Moon● Tidal coupling moves the Moon 38 millimeters 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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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 even more effective at slowing the Moon's rotation.

Although it may have originally spun rapidly,was the Moon is now in a state where it turns at the same rate that it orbits the Earth.

d F tidal−moon = −2GMEarth m test

r3diammoon

tidemoon

tideearth

=MEarthdmoonMMoond Earth

The Earth has about 20 times the tidal influence on the Moon compared with the Moon's effect on the Earth.

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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 for the Moon From Earth we can only see one 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.● 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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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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The Dark 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.

X X

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The lunar far side (mostly)

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The Roche Limit● Consider two particles in contact with centers separated by “r”. In

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 R3 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

Planet

mutual gravity

tidal force

Fgrav =Gmm

r2

m m

d Fgrav = −2 GMm

R3Δ r

M

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 R3 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 = 2.44 (ρ planetρ 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.

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The Roche Limit● Planetary rings (Jupiter, Saturn, Uranus, and Neptune) lie inside

the Roche Limit.

r roche = 2.44 (ρplanetρparticles )

1 /3

R planet

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The Roche Limit● Planetary rings (Jupiter, Saturn, Uranus, and Neptune) lie inside

the Roche Limit.

r roche = 2.44 (ρplanetρparticles )

1 /3

R planet

Neptune

Uranus

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The Roche Limit● 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.

r roche = 2.44 (ρplanetρparticles )

1 /3

R planet

Interesting article....

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The Hill Radius● 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 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 of the gravitational potential in a 2 body system

L1 through L5 represent special locations where the gravitational force from the two major bodies balances creating a flat potential.

L2 is the home of many spacecraft that need a stable quiet environment.

The Deep Space Climate Observatory (DSCOVR) lives at L1 to provide early warning of solar particle events and view the daylit hemisphere of the Earth.

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Lagrange Points

● Particles at the Lagrange Points are fixed in the rotating reference frame of the Earth’s orbit.

● L1, L2, and L3 are marginally unstable. L4 and L5 are stable thanks the the Coriolis Force.

● Objects captured at L4 and L5 are known as Trojans.

● Jupiter has both a leading and trailing crowd of Trojan asteroids.

● Neptune has a number of Trojans, the Earth currently has one.

● Saturn’s satellites Tethys and Dione have co-orbital Trojans.

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Helene

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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?

● Using the Earth/Moon system as a concrete example the attractive force between Earth and Moon (left) needs to overwhelm the tidal force on the Moon relative to the Earth (right)

RHill= (M planet

2 Msun)1/3

a planet

Fsun-tide = 2GM sunmm rearth-moon

rsun-earth3F earth-moon=

GM emmrearth-moon

2

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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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Exam Break

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Conservation of Angular Momentum in the Earth-Moon System

LMoon (orbit ) = mv r = mr √GMr = m√GMr

d LMoonorbit

dt= m

2 √GMr drdt

LEarth(rot ) = I ω =25MR2 (

2π

P rot )d LEarth (rot )

dt= − 4π

5MR2

( 1

P rot2

d P rotdt )

m is the Moon's massM is the Earth's massr is the Earth-Moon separationR is the radius of the Earth

dP/dt = 0.0016 sec/century dr/dt = 4 cm/year

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Synodic Lunar Month● The Synodic Lunar Month is the time it takes the Moon to execute

a cycle of phases - Full to Full or New to New.● Since the phases are tied to the Sun and the Earth orbits the Sun

about 1/12 the way around in the course of a lunar sidereal month. The synodic month is about 1/12th a lunar sidereal period longer than a sidereal month.

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Synodic Lunar Month● Mathematically… the Moon orbits the Earth at an angular rate wsid_moon (so units of radians per second, degrees per day….)

● Relative to the Sun the Moon appears to go around the Earth more slowly because the Earth is orbiting the Sun at an angular rate, wsid_earth_orbit

● The synodic orbital rate of the Moon is the difference of these two rates.

ω syn_moon = ω sid_moon_orbit − ω sid_earth_orbit

2π

Psyn_moon

= 2π

Psid_moon_orbit

− 2π

Psid_earth_orbit

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Synodic Lunar Month● Mathematically… the Moon orbits the Earth at an angular rate wsid_moon (so units of radians per second, degrees per day….)

● Relative to the Sun the Moon appears to go around the Earth more slowly because the Earth is orbiting the Sun at an angular rate, wsid_earth_orbit

● The synodic orbital rate of the Moon is the difference of these two rates.

ω syn_moon = ω sid_moon_orbit − ω sid_earth_orbit

129.5

= 1

27.3 −

1365.25

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Eclipses - Understanding Shadows ● An eclipse occurs when one astronomical object casts a shadow

on the other.● Solar Eclipses – The Sun casts a shadow on the Earth

● Lunar Eclipses – The Earth casts a shadow on the Moon

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The Geometry of Shadows● A shadow created from an extended source of light (e.g. The Sun)

has two parts.● A dark umbra (“A”) in which all light from the Sun is blocked

● A less shaded penumbra (“B”, “C”, “D”) where part of the light from the Sun is blocked.

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Moon Phase and Eclipses● Because of the required alignment between Sun, Moon, and Earth

eclipses either happen at Full or New Moon● For a total solar eclipse the Moon is New.

● For a total lunar eclipse the Moon is Full.

Not to Scale!!!!

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Why Eclipses are Rare● The Earth and Moon, when seen to true scale, are tiny compared

to their separation. Alignment must be nearly perfect.● The tilt of the Moon's orbit hinders that alignment.

If this figure were true to scale the Moon would be twice as far from the Earth as shown here!!

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Why Eclipses are Rare● The Earth and Moon, when seen to true scale, are tiny compared

to their separation. Alignment must be nearly perfect.● The tilt of the Moon's orbit hinders that alignment.

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Quantitatively How Limited are the Opportunities?● If we were restricted to observing partial solar eclipses from the

center of the Earth, the Moon could be only a little more than 1/2 degree off of the Ecliptic and it's disk would miss the disk of the Sun.

● The angular diameters of the Earth and Moon are both about 1/2 degree.

● There would be little tolerance for seeing total solar eclipses

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Quantitatively How Limited are the Opportunities?● If we were restricted to observing partial solar eclipses from the

center of the Earth, the Moon could be only a little more than 1/2 degree off of the Ecliptic and it's disk would miss the disk of the Sun.

● The angular diameters of the Earth and Moon are both about 1/2 degree.

● There would be little tolerance for seeing total solar eclipses

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Quantitatively How Limited are the Opportunities?● If we were restricted to observing partial solar eclipses from the

center of the Earth, the Moon could be only a little more than 1/2 degree off of the Ecliptic and it's disk would miss the disk of the Sun.

● The angular diameters of the Earth and Moon are both about 1/2 degree.

● There would be little tolerance for seeing total solar eclipses

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Quantitatively How Limited are the Opportunities?● If we were restricted to observing solar eclipses from the center of

the Earth, the Moon could be only a little more than 1/2 degree off of the Ecliptic and it's disk would miss the disk of the Sun.

● However, the Earth is a large target and it's apparent angular radius as seen from the Moon adds about a degree to the target range – expanding it to 1.5 degrees.

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Quantitatively How Limited are the Opportunities?

● If we were restricted to observing solar eclipses from the center of the Earth, the Moon could be only a little more than 1/2 degree off of the Ecliptic and it's disk would miss the disk of the Sun.

● So how far around the 5-degree inclined orbit of the Moon do you have to go before the separation from the ecliptic reaches 1.5 degrees.

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Why Eclipses are Rare● The Earth and Moon, when seen to true scale, are tiny compared

to their separation. Alignment must be nearly perfect.● There are two times a year, separated by 6 months, when the shadows line

up.

● This “eclipse season” drifts in the calendar due to the precession of the Moon's tilted orbital plane with a period of 18.6 years.

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Solar Eclipses● The tapering umbral shadow of the Moon just barely reaches

Earth (sometimes it doesn't – an annular eclipse)● The Moon's umbral shadow is small – covering a couple of hundred miles

at best.

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Annular vs. Total Solar Eclipse

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Annular vs. Total Solar Eclipse

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Solar Eclipses● The Moon's orbital motion (minus Earth rotation) sweeps the

shadow across the Earth in a matter of hours.● Locations along this eclipse path experience “totality” for a few minutes at

best.

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Solar Eclipses● The Moon's orbital motion and Earth rotation sweep the shadow

across the Earth in a matter of hours.● Locations along this eclipse path experience “totality” for a few minutes at

best.

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Solar Eclipses● Despite the short duration and remote location, people go to

extremes to view eclipses.● Phenomena that are usually washed out by the blue sky become visible –

the solar corona and solar prominences.

http://www.mreclipse.com/

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● Eratosthenes● Lunar Far Side (not quite)