# Earth-moon focus

Earth and moon revolve around their common center of gravity - illustration not true to scale

The earth-moon center of gravity is the common center of gravity ( barycenter ) in the earth-moon system . Although the Moon from the Earth seen from ( geocentric ) seemingly around the center of the Earth is rotating , however, both the earth and the moon around their common center of mass and - rotate - in the contemplation of a sun-fixed system. Because of the overwhelming mass of the earth, this is still in the interior of the earth (more precisely: in the earth's mantle ), but does not coincide with the center of the earth .

The lunar orbit is not a circle, but approximately elliptical with the earth-moon center of gravity at a focal point of the ellipse . Due to the eccentricity of the lunar orbit, the earth-moon distance varies by over 13 percent between (on average) 356,410 and 406,740 km in the rhythm of the anomalous month . Both the moon and the center of the earth thus have a varying distance from the earth-moon center of gravity.

The earth-moon system with the earth-moon center of gravity as a common center of gravity orbits the sun as a whole .

## Basics

A body rotating in a weightless state always revolves around its center of mass . In relation to external forces, the center of gravity moves as if the entire mass of the body were united in it. For example, the center of gravity of a hammer thrown upwards moves exactly on a parabola, while the hammer rotates around the center of gravity close to the hammer head.

The same applies to a "double body " ( two-body system ) consisting of two celestial bodies, such as B. Earth and moon: their movement in space can be broken down into

• a movement of the common center of gravity on a Keplerian elliptical orbit around the sun, under the influence of the gravitational force exerted by it ; the " earth orbit " around the sun is actually the orbit of the earth-moon center of gravity .
• a rotation of the earth and moon around the common center of gravity. The rotation around each other results in the earth swinging on its orbit around the sun in radial and vertical directions. This effect is called monthly or lunar orbit disturbances (English lunar perturbation ) of the earth or terrestrial disturbances of the moon. Like the phases of the moon, their period is about 29½ days.

## calculation

In a first approximation it is sufficient to consider the earth and moon as point masses. The position of the common center of gravity on the line connecting the two centers of mass is determined according to the definition of the center of mass as the distance from the center of the earth and as the distance from the center of the moon : ${\ displaystyle r _ {\ mathrm {E}}}$${\ displaystyle r _ {\ mathrm {M}}}$

${\ displaystyle r _ {\ mathrm {E}} = r \, {\ frac {m _ {\ mathrm {M}}} {m _ {\ mathrm {E}} + m _ {\ mathrm {M}}}}}$
${\ displaystyle r _ {\ mathrm {M}} = r \, {\ frac {m _ {\ mathrm {E}}} {m _ {\ mathrm {E}} + m _ {\ mathrm {M}}}}}$

With

• the point mass of the earth${\ displaystyle m _ {\ mathrm {E}}}$
• the punctiform mass of the moon${\ displaystyle m _ {\ mathrm {M}}}$
• their mean mutual distance ${\ displaystyle r = r _ {\ mathrm {E}} + r _ {\ mathrm {M}} \ approx 384,000 \; {\ text {km}}}$

Since the masses of the earth and moon are in a ratio of around 81: 1:

${\ displaystyle m _ {\ mathrm {E}} \ approx 81 \ cdot m _ {\ mathrm {M}}}$

their common center of gravity lies around one eighty-second of the distance between the two centers of mass outside the center of mass of the earth:

${\ displaystyle \ Rightarrow r _ {\ mathrm {E}} \ approx {\ frac {1} {81 + 1}} \ cdot r \ approx {\ frac {384,000 \; {\ text {km}}} {82} }}$

so about 4700 km away from the center of the earth in the direction of the moon or about 1700 km below the earth's surface.