## The Earth Moon System

The size of the Earth was first determined by the ‘Greek’ astronomer Eratosthenes who noted that at the summer solstice the noon altitude in the city of Syrene differed by 7° 12′ from the altitude in Alexandria. Assuming the Earth was spherical and knowing that Alexandria was due north and 5,000 stadia from Syrene, he found the Earth’s circumference to be (360°/7.2°) 5,000 ≈ 40,000km.

The Moon is one of the largest and most massive satellites in the solar system, relative to its primary planet. Its radius is 1738km (0.272R_{⊕}) and its mass is 7.35 x 10^{22}kg (0.0123R_{⊕}). The mean distance between the centres of the Earth and Moon is 384,405km (60.3R_{⊕}). The barycentre (centre of mass) of the system is *M*_{m}*a*_{m}/(*M*_{⊕} + *M*_{m}) ≈ 4,671km from the Earth’s centre. The Earth and Moon orbit this point, which is buried 1,707km below the Earth’s surface.

Using Kepler’s third law: *M*_{⊕} + *M*_{m} ≈ *M*_{⊕} = 4π^{2}*a*_{s}^{3}/*GP*_{s}^{2}

⇒ *M*_{⊕} ≈ 5.98 x 10^{24}kg

From the Moon’s apparent diameter, which averages ½ ° , and the average Earth-Moon distance, we can calculate its diameter to be 3,476km. Because of its eccentricity (e = 0.055), its orbital distance ranges from 363,263km at perigee, to 405,547km at apogee. The angular diameter varies between 32.9′ and 29.5′.

With respect to the stars, the Moon orbits the barycentre in one sidereal month (27.322^{d}); with respect to the Sun, this orbit takes one synodic month (the month of phases, 29.531^{d}). With respect to the Moon’s line of nods, the orbital period is one nodical or draconic month (27.212^{d}) and with respect to its perigee, the period is one anomalistic month (27.555^{d}). The Moon’s synchronous rotation period is one sidereal month.

Differential solar gravitational forces on the Earth-Moon system, tend to

- elongate the Moon’s orbit at quadrature
- cause the perigee of the moons orbit to precess eastwards with a period of 8.85 years
- produce torque on the inclined orbit which causes the line of nodes (intersection of the Moon’s orbital plane and the ecliptic plane) to regress westwards along the ecliptic with a period of 18.6 years.

The Sun always illuminates one hemisphere of the Moon, but from Erath we see varying fractions of the sunlit hemisphere depending on the Moon’s elongation. The cycle of geocentric phases lasts synodic month of; New Moon (inferior conjunction), waxing crescent, first quarter (quadrature), waxing gibbous, full (opposition), waning gibbous, third quarter (quadrature), waxing crescent.

An eclipse occurs when the shadow of one celestial body falls on another.

The Sun’s apparent diameter is 32′, almost the same as the Moon. The line of nodes of the moon’s orbit point toward the Sun twice each year, so that the new Moon can occur close enough to the ecliptic for the Moon to cover the Sun – a Solar Eclipse. When the Moon passes to one side of the centre of the Sun, we see a partial eclipse. If the Moon is not near perigee in its orbit, its angular size is slightly smaller and we see an annular eclipse. Near its perigee, when the Moon’s centre crosses the Sun’s centre, we have a total eclipse.

At the Moon’s orbit, the Earth’s shadow has an angular diameter of 1° 22.4′. When the line of nodes points down this shadow, a lunar eclipse can take place. Both partial and total lunar eclipses are possible, with the Moon remaining dark for up to 1^{h}40^{m}.

The Greek astronomer Meton saw that the Moon exhibits the same phase on the same day of the month at intervals of 18.6 years – the Metonic cycle. The regression of the nodes of the Moon’s orbit is known as the Saros cycle: Similar Solar and Lunar eclipses take place at intervals of 223 synodic months (18^{y}10^{d}) because the Saros cycle interval is 6,585.32^{d}. We must wait 3 Saros cycles for an eclipse to repeat at the same place on the Earth.

The mean or bulk density of the Earth is,

<*ρ*> = 3*M*_{⊕}/4π*R*_{⊕}^{3} ≈ 5,520kgm^{-3}

The density of rocks at the surface is ~ 2,800kgm^{-3} which suggests the interior of the Earth must be *very* dense. The Earth’s interior is stratified:

- a 35km thick crust of 3,300kgm
^{-3} - a solid mantle with densities ranging from 3,400 to 5,500kgm
^{-3}down to a depth of 2,900km - a 2,200km deep liquid outer core (
*ρ*≈ 9,900 to 12,000kgm^{-3}) - a solid inner core of radius 1,300km and
*ρ*≈ 13,000gm^{-3}

This picture has been deduced from studies of the propagation of earthquake waves (seismology). Earthquakes generate longitudinal compression waves (P-waves) and transverse distortion waves (S-waves). Both are refracted by the changing density and composition of the Earth’s internal material. Only the P-waves can propagate through the liquid outer core and no S-waves are seen through the core. The inner solid core is metallic, probably nickel-iron.

The solid inner core rotates faster than the rest of the planet. The core rotates about 2/3 second faster per day, about 1° per year, so that over a century the core gains about 90°.

The Earth is neither expanding nor contracting, the downward force of gravity is balanced by the upward pressure (Hydrostatic equilibrium) i.e.

*dP*/*dr* = – *ρ*(*r*)(*GM*/*r*^{2})

(The equation of hydrostatic equilibrium)

This suggests that the central pressure of the planet is;

∫_{Pc}^{0 }*dP* = -<*ρ*>^{2}(4/3)π*G* ∫_{0}^{R}*r **dr*

⇒ *P*_{c} = (2/3)π*G*<*ρ*>^{2}*R*^{2}

*P*_{c} ≈ 1.7 x 10^{11}Pa = 1.7 x 10^{6}atm

From its mass and radius, the Moon’s average density is 3,370kgm^{-3}, similar to the Earth’s crust. Returned lunar surface samples have a density of 3,000kgm^{-3}, so the density cannot increase much towards the Moon’s centre.