# How to calculate apparent brightness in relation to the sun

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.

• 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 Feb 27, 2021 at 10:20
• Could you maybe give the source of your formula? Feb 27, 2021 at 10:22
• What do you mean with: I find this isn't helpful in visualizing the brightness in your mind? Feb 27, 2021 at 10:31
• You could simply divide by the apparent brightness of the sun which would give you a result in percent of the sun's brightness. Feb 27, 2021 at 10:40

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