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The Shape of the
Hydrosphere of Earth

The Analysis

The purpose of this analysis is to derive an equation for the shape of the hydrosphere of Earth and find an approximate solution. The hydrosphere here refers to the oceans and the analysis pertains to the tidal bulge in the oceans that occurs as a result of the attraction of the moon.

The origin of the coordinate system is at the center of the Earth with the x-axis projecting outward along the line connecting the centers of the Earth and moon. The y-axix coincides with the rotation axis of the Earth.

The distance between the centers of the Earth and moon is denoted as R. The Earth and moon rotate about their common center of mass. If M and M1 are the masses of the Earth and moon, respectively, then distance z from the center of the Earth to the common center of mass of the Earth-moon system is:

z = M1R/(M-M1).

The mass of the Earth is 5.97x1024 kg and that of the moon is 7.35x1022 kg/The ratio is 0.01231. The mean distance from the center of the Earth to the center of the moon is 384,400 km. Therefore the distance from the center of the Earth to the center of mass of the Earth-moon system is 4791.6 km. Since the radius of the Earth is 6370 km the center of mass of the Earth-moon system is about 1600 km below the surface of the Earth.

The rate of rotation ω about the center of mass is determined so as to balance the gravitational attraction of the Earth to the moon with the centrifugal force experienced by the Earth due to its rotation about the center of mass of the Earth-moon system; i.e.,

GMM1/R2 = Mω2z = Mω2M1R/(M-M1)

where G is the gravitational constant. This reduces to

G/R2 - ω2R/(M-M1) = 0
or
ω2 = G(M-M1)/R3

For a point (x,y) the components of the force per unit mass are:

Fx = -(x/r)(GM/r2) + ω2(z+x) - GM1/(R+x)2
and
Fy = -(y/r)(GM/r2)

where r = (x2+y2)1/2 and z is the distance from the center of the Earth to the center of mass of the Earth-moon system. In the above equations it is assumed that the gravitational attraction to the moon is parallel to the x-axis. This is of course not precisely true but since the angle subtended by the radius of the Earth from the center of the moon is about 1° it is a reasonable approximation. The centrifugal force due to the rotation of the Earth is left out of the equations because the hydrosphere does not rotate with the Earth but instead remains aligned with the Earth-moon axis.

Since z = M1R/(M-M1) the above equations reduce to:

Fx = -x(GM/r3) + ω2(M1R/(M-M1 + x) - GM1/(R+x)2
and
Fy = -y(GM/r3)

Since (R+x)2 = R2(1+(x/R))2 and x/R is small

1/(R+x)2 = (1/R2)(1-2x/R)

With the above approximation the equations for the components of the force per unit mass are:

Fx =
-x(GM/r3) + ω2x + GM12x/R3 +
ω2M1R/(M-M1) - GM1/R2
and
Fy = -y(GM/r3)

Because of the condition for the determination of ω, previously cited, the last two terms in the equation for Fx cancel out leaving:

Fx = -x(GM/r3) + ωx + GM12x/R3
and
Fy = -y(GM/r3)

The differential equation for the shape of the cross section curve is:

dy/dx = - (Fx/Fy)

With previous expressions for Fx and Fy and multiplying the numerator and denominator by r3/GM the differential equation becomes:

dy/dx = - (x/y)[1 - ω2r3/GM + 2(M1/M)(r/R)3]

Since r is approximately constant and the shape of the cross section curve is approximately an ellipse the relations for an ellipse can be used to determine properties of the cross section. For an ellipse

dy/dx = - (x/y)(b/a)2

where b/a is the ratio of the semi-major axes of the ellipse. This means that

(b/a)2 = 1 - ω2r3/GM + 2(M1/M)(r/R)3.

The Hydrosphere Due to Lunar Attraction

For the Earth-moon system the RHS of the above equation evaluates to 1-4.485x10-6 so

(b/a) = 1 - 2.2424x10-6
and thus
(b-a) = 0.0143 km = 14.3 m = 39 feet

Thus the shape of the hydrosphere is approximately an ellipse in which the difference between the semi-major axes is 14.3 meters or 39 feet. This means the maximum difference between high tide and low tide at the equator is predicted to be 39 feet. This is the proper order of magnitude.

The Hydrosphere Due to Solar Attraction

In the sun-Earth system the mass of the sun is 1.99x1030 kg compared to 5.97x1024 kg for the Earth. The mass ratio is 3x10-6and thus the center of mass of the sun-Earth system is only 449 km from the center of the sun.

Despite the immensely larger mass of the sun the even more immensely larger distance of 1.496x108 km results in the effect of the attraction of the sun on Earth's hydrosphere being very much smaller than that due to the attraction of the moon. The difference in high and low solar tides based upon the previous formula and a rate of rotation of 2π radians per 365.25 days is only 0.0822 meters or 2.7 inches.



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