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### Usefull Formulas

(Under construction)

Geometrical Formulas

Area of a circle:

A = pi * r^2

Area of a Sphere:

A = 4*pi*r^2

Volume of a Sphere:

V = 4/3 * pi * r^3

The area of a spherical cap:

A = 2*pi*(R^2)*(1-sin(lat))

The area between two bands of latitude in the same hemisphere:

A = |2*pi*(R^2) *(1-sin(lat2)) - 2*pi*(R^2)*(1-sin(lat1))|
= 2*pi*(R^2)*(sin(lat1) - sin(lat2)

Gravitational Formulas:

Orbital Velocity ~= (GM/a)^1/2 in km/s

(For Earth, Orbital Velocity ~= (398,600.4418/a)^1/2)

Escape Velocity = ((2 * G * M) /r) ^ 1/2

(For Earh, the escape velocity is 11.2 km/s at the surface)

Gravitational Potential Energy:

U = -G * (M1 * M2)/R + K

(For Earth, U = m * g * delta-h)

Phanerozoic Insolation:

L(t)/L(c) = 1/(1+2*(1-t/t(c))/5)

Where L(t) is the luminosity at time, t, L(c) is the current luminosity of 3.85 *10^26 Watts,
t is the time in Gigayears from the formation of the Sun, and t(c) is the current time since
the formation of the Sun, or 4.57 Gigayears.

TSI =L(t)/(4*pi*r^2)

Where TSI is the Total Solar Irradiance at the top of the atmosphere, L(t) is the luminosity
at time, t, and r is the distance from the Earth to the Sun (1.496 x 10^11 meters on average).
Hence, on average, at the moment the TSI = 1368.95 W/m^2 by this formula (observed =
1360 W/m^2, which requires an L(c) in the prior formula of 3.825 *10^26 Watts).

Solar forcing = TSI *(1-α)/4

Where TSI is Total Solar Irradiance at the top of the atmosphere, and α is the bond albedo
of the Eath (0.306).