Editing Gravitational redshift (section)

==== Newtonian limit ====
In the Newtonian limit, i.e. when <math>R_\text{e}</math> is sufficiently large compared to the Schwarzschild radius <math>r_\text{S}</math>, the redshift can be approximated as
: <math>z = \frac{\Delta\lambda}{\lambda} \approx \frac{1}{2}\frac{r_\text{S}}{R_\text{e}} = \frac{GM}{R_\text{e} c^2} = \frac{g R_\text{e}}{c^2}</math>
where <math>g</math> is the [[gravitational acceleration]] at <math>R_\text{e}</math>. For Earth's surface with respect to infinity, ''z'' is approximately {{val|7|e=−10}} (the equivalent of a 0.2&nbsp;m/s radial Doppler shift); for the Moon it is approximately {{val|3|e=−11}} (about 1&nbsp;cm/s). The value for the surface of the Sun is about {{val|2|e=−6}}, corresponding to 0.64&nbsp;km/s. (For non-relativistic velocities, the radial [[Relativistic Doppler effect|Doppler equivalent velocity]] can be approximated by multiplying ''z'' with the speed of light.)

The z-value can be expressed succinctly in terms of the [[escape velocity]] at <math>R_\text{e}</math>, since the [[gravitational potential]] is equal to half the square of the [[escape velocity]], thus:
: <math>z \approx \frac{1}{2}\left( \frac{v_\text{e}}{c} \right)^2</math>
where <math>v_\text{e}</math> is the escape velocity at <math>R_\text{e}</math>.

It can also be related to the circular orbit velocity <math>v_\text{o}</math> at <math>R_\text{e}</math>, which equals <math>v_\text{e}/\sqrt{2}</math>, thus
: <math>z \approx \left( \frac{v_\text{o}}{c} \right)^2</math>.

For example, the gravitational blueshift of distant starlight due to the Sun's gravity, which the Earth is orbiting at about 30&nbsp;km/s, would be approximately 1 × 10<sup>−8</sup> or the equivalent of a 3&nbsp;m/s radial Doppler shift.

For an object in a (circular) orbit, the gravitational redshift is of comparable magnitude as the [[transverse Doppler effect]], <math>z \approx \tfrac{1}{2} \beta^2</math> where {{nowrap|1=''β'' = ''v''/''c''}}, while both are much smaller than the [[Relativistic Doppler effect|radial Doppler effect]], for which <math>z \approx \beta</math>.