Engineering Physics

Gravitational Time Dilation for Object (GPS Satellite) in Orbit Equation and Calculator

Schwarzachild Factor Calculate Time Dilation.
​According to the Theory of Relativity also heat, pressure and magnetic fields slow  time.

dt' = sqrt ( 1 - 2 M G / r c² ) dt
or
dt' = ( 1 - 2 M G / r c² )^(1/2) dt

and

dt' = dt / sqrt ( 1 - v² / c² )

Gravitational time dilation is proportional to:
Δ = sqrt ( 1 - 2U )
Where:
U = G M / r c²

Therefore:

Δ = ( 1 - 2 G M / R_p c² )^(1/2) / ( 1 - 2 G M / R_o c² )^(1/2)

Δ_t = Δ · s

Δ_t = Time dilation (difference) between Planet and Orbiting Satellite at specified time duration (sec)
Δ = Time Dilation factor
U = Newtonian gravitational potential in dimensionless form, i.e., divided by the square of the speed of light
c = Speed of light 299792458 m/s
M = Mass of the Earth 5.97219 x 10²⁴ kg (or some other planet given in kg)
G = Gravitational constant 6.67408 x 10^-11 ( m²/ kg / s²)
R_p = Average radius of the Planet (Mass) 6371000 m
R_o = Satellite or Object Orbiting earth or Other Planet/Mass 20180 km
s = Some time duration (seconds) 1 earth day = 86,400 seconds

Contributed by:

Dmitry Vakhtin
Viktor T. Toth