An Ocean Tide Model

CONSTANTS AND EQUATIONS

According to this model the ocean tides are caused by the variations of the sums of the lunar and solar centrifugal and gravitational forces for different points on earth. Those variations are very small differences between large numbers. Therefore, the constants used in calculations must have a sufficient number of meaningful places to accurately reflect those small differences.
Let us first consider the interactions between the oceans of the earth and the sun. The solar centrifugal force
Fsc = w/g · (2π/Ts)2ds, where w is the weight of an earthbound object (say 1 kg of ocean water), g is the acceleration of gravity, Ts is the orbital period (one year), and ds is the distance between the sun and that earthbound object.
The solar gravitational force Fsg = G(w/g · ms)/ds2, where G is Newton's gravitational constant and ms is the mass of the sun. On the line of the orbit Fsc=Fsg and w/g · (2π/Ts)2ds = G(w/g · ms)/ ds2, whence
(2π/Ts)2 = Gms/ds3 .....(1)
Equation (1) expresses a relationship between constants. The constants in equation (1) are Ts, G, ms, and ds. Given values for any three of the constants, the fourth can be calculated. The value of π used in the calculations was 3.1415926536. Here are the results of the calculations:

Table 1 Earth-sun interaction constants
  Literature Calculated Percent
  value value change
Orbital period (days) 365.25 365.0735 -0.0483
Gravitational constant 6.67259x10-11 6.666142x10-11 -0.0966
Mass of sun (kg) 1.991x1030 1.989076x1030 -0.0966
Earth - sun distance(m) 1.496x1011 1.496481x1011 +0.0322


The gravitational constant and orbital period are known with sufficient accuracy to preclude their change. I selected the calculated value of the earth-sun distance, since it required the least percentage change.

In the case of the earth-moon interactions, the centrifugal forces arise from the earth's 'monthly' rotation around the earth-moon barycenter. The distance between the center of the earth and the barycenter
db = dmmm/(me+mm), .....(2)
where dm is the distance between the centers of the earth and moon, mm is the mass of the moon, and me is the mass of the earth. The barycentric centrifugal force Fbc = w/g · (2π/Tm)2db where Tm is the lunar period. The gravitational force of the moon
Fmg = G(w/g · mm)/dm2. On the line of the earth's center's orbit of the barycenter then, Fbc=Fmg and
w/g · (2π/Tm)2db = G(w/g · mm)/dm2. .....(3)
Substituting the right hand side of equation (2) for db in equation (3) yields
me + mm = (2π/Tm)2dm3/G. .....(4)

The constants in equation (4) are me, mm, Tm, dm, and G. Holding four at a time constant and calculating the fifth gives:

Table 2 Earth-moon interaction constants
  Literature Calculated Percent
  value value change
Mass of earth (kg) 5.983x1024 5.957484x1024 +0.426
Mass of moon (kg) 7.347x1022 4.795391x1022 -34.73
Lunar period (days) 27.322 27.26439 -0.211
Moon-earth distance (m) 3.84407x108 3.849482x108 +0.141
Gravitation constant 6.67259x10-11 6.644479x10-11 +0.421


The mass of the earth, the mass of the moon, the lunar period, and the gravitational constant are known to sufficient accuracy to preclude changing them. I selected the calculated value of the moon-earth distance that also required the least percentage change.

The above equations for solar and lunar gravitational force and solar centrifugal force were subsequently used to calculate those forces for a particular location on earth with the distances from sun and moon appropriately changed to reflect distances to that particular location. Lunar (point orbit) centrifugal force per kilogram of ocean water (the same everywhere) was calculated with the equation given above. The appropriate cosine factor was also applied to each calculation in accordance with Figure 4 of the Revised Ocean Tide Model page.