Astronomy 217E a r t h - M o o n & S u n

Sunlit SystemMany of the most fascinating behaviors of the Earth Moon system are in fact interactions between the Earth-Moon & Sun.

These include

Tides

Lunar Phases

Eclipses

Phases of the MoonAs the most visible change from night to night in the night sky, the cycle of lunar phases held great interest for our ancestors.

The lunar (from Latin) month (from Old English) is the time to complete the cycle of lunar phases (29.53 days).

Sidereal MonthAs with the Earth’s day, the period of the moon’s rotation with respect to the fixed stars is less than the perceived period.

The synodic month is the period from new moon to new moon.

2౫ౡ > 2౫౦ , 2Ӡ238/43E > 23:/64E , 2476/35E

Lunar EclipseEclipses of moons by their much larger planets are very common, as planets cast relatively large shadows.

A lunar eclipse occurs at opposition, meaning that a full moon disappears.

Because of the Sun’s 0.5° angular size, the Earth’s shadow on the moon is slightly smaller (9380 km) than D⨁ and 2.7 DMoon.

With the Moon orbiting at 1.0 km/s, the total lunar eclipse lasts ~ 100 minutes, with 60 minutes of partial shadowing preceding and following.

Solar EclipsesDuring Solar Eclipses, both a dark inner full shadow (Umbra) but also a outer partial shadow (Penumbra) appear. Depending on the orbital orientation and the observers location on Earth, partial, total and annular eclipses are possible. These phenomena occur because of the coincidence of the Sun’s and Moon’s angular size.

Length of the UmbraFor most moons, the umbra falls well short of the planet, but the Moon is a relatively large moon.

a⊕

r0

α

The Moon’s orbital eccentricity is 0.055, making perigee = 0.945 r0 and apogee = 1.055 r0, thus the umbra only reaches the Earth when the Moon is near perigee.

ubo > ӤঃౚӠ ѝ ౪1 , Ψ౭౪ౚ > ౌ౧౧౦Ψ౭౪ౚ

Ψ౭౪ౚ > ౌ౧౧౦Ӥ ѝ ౌ౧౧౦ ঃౚӠ ѝ ౪1 = 3.73 × 105 km = 0.97 r0

ℓumbra

Orbital OrientationThe plane of the Moon’s orbit around the Earth crossed the plane of the Earth’s orbit around the Sun at 5°.

Since solar and lunar eclipses only occur when the Sun, Earth and Moon form a line, eclipses can only occur when the new/full moon lies in the ecliptic.

Crossing EllipsesThe line of nodes is the line formed by intersection of the plane of the Moon’s orbit and the Ecliptic. It connect the ascending node ☊ and descending node ☋.

The line of nodes precesses with an 18.6 year period as a result of the Sun gravitational attraction on the Moon.

Finite SizesBecause of the finite angular sizes of the Sun and Moon,

&

eclipses can occur in a small region around the nodes.

An eclipse can occur if the centers of the Sun and Moon appear within θeclipse ≤ θ☾ + θ☉ = 0.53° to an observer on the Earth’s surface.

But this angle changes over the course of the day because of the Earth’s rotation. To remove the diurnal effects, the angle between the Earth-Moon line and the Earth-Sun line can be calculated relative to the Earth’s center.

Ӥ > ӤౚӠ > 1/38ѣ څ > څ౪1 > 1/37ѣ

θ

Angle from CenterTo derive the angle between the Earth-Moon line and the Earth-Sun line from the Earth’s center.

R⨁

θ⨁ θ

Δ

Law of Sines

But θeclipse ~ 0.5°, so sin (90°+ θ) = 1

⇒ θ⨁ ≈ 0.95°

(90°+ θ) + (90°− Δ) + θ⨁ = 180°

⇒ Δpartial ≈ θ⨁ + θ = θ⨁ + θ☉ + θ☾ = 1.48°Similarly Δtotal ≈ θ⨁ + θ☉ − θ☾= 0.96°

r0Ӡtjo Ӡ > ౪1tjo ):1ѣ , *

tjo Ӡ > Ӡ౪1

Eclipse WindowBecause of the small 5° of the Moon’s orbit relative to the Ecliptic, the Moon spends significant time within 1.5° of the Ecliptic. This time is called the Eclipse window.

ΔΔ

5.1°δθδθ

δθpartial = Δpartial / sin 5.1° = 11 Δpartial = 16.7°δθtotal = Δtotal / sin 5.1° = 11 Δtotal = 10.8°

The Sun moves 360°/year = 0.986°/day = 29.1°/syn month

Comparing this to the eclipse windows, per passage of the nodes, there will be 1-2 partial solar eclipses (2 δθpartial = 33.3° > 29.1°) and 0-1 total solar eclipses (2 δθtotal = 21.6° < 29.1°) and similar for lunar eclipses.

node

Visible EclipsesWithout a doubt, the dramatic darkening of the Sun during a solar eclipse has caught mankind’s attention for millennia.

Such events are quite rare, and historically difficult to predict, because of the 5° mis-alignment between the lunar orbit and ecliptic, the small eccentricity of the lunar orbit, and the fact that the length of the umbra is approximately the Earth-Moon separation.

Hopefully, you witnessed one yourself last year.

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Atomic Physics