How to calculate apparent brightness in relation to the sun

Question

I know that we can determine the apparent brightness of an object in our solar system using the following formula:

$$B=\frac{A\cdot L_S\cdot R^2}{D^2\cdot d^2}$$

$B$ = Brightness of observed object in watts
$A$ = Albedo of observed object
$L_S$ = Luminosity of the Sun in watts
$R$ = Radius of the observed object in meters
$D$ = Distance between the Sun and the observed object in meters
$d$ = Distance between the Earth and the observed object in meters

However, I find this isn't helpful in visualizing the brightness in your mind. I was wondering what modifications would be needed to this formula to make the result relative to the sun.

Answer 1

If you're just looking for an equivalent expression, but relative to the sun, the modification is pretty simple. Let's start with the original expression: $$B=\frac{A \cdot L_S \cdot R^2}{D^2 \cdot d^2}$$

where

$B$ = Brightness of observed object in watts
$A$ = Albedo of observed object
$L_S$ = Luminosity of the Sun in watts
$R$ = Radius of the observed object in meters
$D$ = Distance between the Sun and the observed object in meters
$d$ = Distance between the Earth and the observed object in meters

To make this relative to the sun we're like an expression for $B^* = B / L_S$ so we divide both side of the original expression by $L_S$: $$B^* = \frac{B}{L_S} = \frac{A \cdot R^2}{D^2 \cdot d^2}$$

where

$B^*$ = Brightness of the observed object relative to the brightness of the sun

This is of course in linear units. To convert to relative magnitude, which might be more helpful: $$\Delta M = -2.5\log_{10}(B^*)$$