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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.

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    $\begingroup$ Hi! As you have seen, I have edited your question using MathJax. For future questions, you might find this helpful: MathJax basic tutorial and quick reference $\endgroup$
    – Jonas
    Feb 27, 2021 at 10:20
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    $\begingroup$ Could you maybe give the source of your formula? $\endgroup$
    – User123
    Feb 27, 2021 at 10:22
  • $\begingroup$ What do you mean with: I find this isn't helpful in visualizing the brightness in your mind? $\endgroup$
    – User123
    Feb 27, 2021 at 10:31
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    $\begingroup$ You could simply divide by the apparent brightness of the sun which would give you a result in percent of the sun's brightness. $\endgroup$
    – Jonas
    Feb 27, 2021 at 10:40

1 Answer 1

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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: $$\Delta M = -2.5\log_{10}(B^*)$$

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