Deriving the Exoplanet Brightness Equation.

The equation below shows the transformation of the brightness of a planet (B_planet) as viewed from earth here in our own solar system to how it would appear if the same planet were placed in the same orbit around another star at greatest elongation (B). Each of the terms on the right show the step-by-step transformation of its brightness.

where

B_planet = brightness of planet as viewed from earth on given date.

R_planet = distance of planet on given date.

Ø_planet = phase of planet on given date.

R_star = distance from earth to star.

L_star = absolute luminosity of star.

L_sun = absolute luminosity of sun.

1. moved from the solar system to the other star (2nd term),
2. adjusted to a phase of 0.5--the phase at greatest elongation (3rd term), and
3. then adjusted to the different brightness of other star (4th term).

Brightnesses are usually quoted in the literature in a logarithmic unit of astronomical magnitudes.

Applying this transformation to the previous equation yields:

or,

where

m = the apparent magnitude of the planet.

m_planet = magnitude of planet viewed within our solar system at given phase and distance.

m_star = apparent magnitude of the planet.

m_sun = apparent magnitude of the sun.