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
= 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).
