__Angular Acceleration of the Earth__

__Constants__

Garvitational Constant:

[1]

Radius of the Earth:

[2]

Angle that the tidal bulge leads the moon by (given):

[3]

Height of the tidal bulge (given):

[4]

Mass of the Earth:

[5]

Density of water:

[6]

__Calculations__

Tidal acceleration

[7]

Density of the bulges:

[8]

Volume in the bulges:

[9]

Mass in the bulges:

[10]

Torque of the bulges:

[11]

Rotational inertia of the earth:

[12]

Angular acceleration of the Earth:

[13]

[14]

__Radial Velocity of the Moon__

__Constants__

Mass of the Moon:

[15]

Mean distance between the Earth and Moon now:

[16]

Rotational period of the Earth:

[17]

Radius of the moon:

[18]

Orbital period of the moon:

[19]

__Calculations__

Orbital velocity of the moon:

[20]

Angular velocity of the moon:

[21]

Angular velocity of the Earth:

[22]

Rotational velocity of the Earth:

[23]

Rotational inertia of the moon:

[24]

Ratio of the moon's roatational inertia to the Earth's rotational inertia:

[25]

Period of the moon's orbit:

[26]

Derivative of the total angular

momentum w/ respect to time:

[27a]

Solving for the radial velocity:

[27b]

[28]

__Extrapolating Forward (and backward) in Time__

__Calculations__

Orbital kinetic energy of the moon:

[29]

Gravitational potential energy of the moon:

[30]

Rotational kinetic energy of the moon:

[31]

Rotational kinetic energy of the Earth:

[32]

Total energy of the system today:

[33]

Total angular momentum today:

[34]

__Difference in total angular energy from today as a function of mean distance__

(I switch from w to W in this section to avoid confusion since I am now talking about angular velocities that are changing with distance)

[35]

Solving for Wearth:

[36]

[37]

Solving for E_{diff}

[38]

__Figure 2__

Visual guess of the equilibrium distance:

[39]

Solving for the angular velocity of the moon at equilibrium:

[40]

guess value for solve block:

__Solve Block:__

[41]

__Equilibrium:__

Length of a day at equilibrium:

[42]

Mean distance of moon at equilibrium:

[43]

__Roche Limit:__

[44]

[45]

[46]

The Roche limit for the moon in Earth radii:

[47]

The Roche limit for the moon in current Earth-moon distances:

__Rate of energy loss:__

Rate of energy loss today (given):

[48]

Combined angular velocity of the system:

[49]

Experimentally determining a function for the rate of energy loss:

[50]

Guess function for rate of energy loss:

[51]

__Figure 3__

o - Today

__Time between two distances:__

[52]

[53]

__Figure 4__