Formula to calculate the apparent magnitude of Earth from arbitrary distances I would like to calculate the apparent magnitude of Earth from various distances, say from the Voyager probe at this moment or from various stars in the sky. I know there must be a formula that I can enter in the distance from Earth (or the Sun) and get out the apparent magnitude. However, I am failing to find it as everything I can find assumes that I am on Earth and want to calculate the brightness of other things.
The absolute magnitude of the Earth is about $H = -3.99$. This means, from $1$ AU away from the Sun and the Earth, the Earth has an apparent magnitude of $-3.99$.
Voyager 1 is $152.26$ AU from the Earth, and $151.85$ AU from the Sun. Using the formula for apparent magnitude, $${\displaystyle m=H+5\log _{10}{\left({\frac {d_{BS}d_{BO}}{d_{0}^{2}}}\right)}-2.5\log _{10}{q(\alpha )}}$$
With $d_{BO} = 152.26, d_{BS} = 1, d_0=1,$ AU, and $q(\alpha) = 0.306$, we obtain that Earth's apparent magnitude from Voyager 1 would be about $\boxed{8.208}$.