# Star Visual Magnitude Math

## Introduction

I have been reading a number of interesting astronomy articles lately. These articles often refer to the apparent and absolute magnitude of a celestial object or event (example). I thought I would work through a bit of the math associated with these terms.

## Background

### History

Like most tales in science, the story of measuring the visual magnitudes of celestial objects dates back to the ancient Greeks. Most sources say that Hipparchus was the first to develop a system for ranking the brightness of celestial object by their visual magnitudes. He said that the brightest objects were of the "first magnitude" and he ranked less bright objects on a scale from second to sixth magnitude, with sixth magnitude being at the limit of human visibility. This system is the basis for the more formal magnitude system that astronomers use today. I have a number of astronomy-oriented posts planned and I will be using this information in the posts to follow.

### Definitions

The Wikipedia gives the following definitions for apparent and absolute magnitude:

apparent magnitude
The apparent magnitude (m) of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere. The brighter the object appears, the lower the value of its magnitude.
absolute magnitude
Absolute magnitude (M) is the measure of a celestial object's intrinsic brightness. It is the apparent magnitude an object would have if it were at a standard luminosity distance (10 parsecs [pc]) away from the observer, in the absence of astronomical extinction. It allows the true brightnesses of objects to be compared without regard to distance.

## Analysis

### Mathematical Definition

Norman Robert Pogson provided a quantitative framework for the Greek's qualitative approach. He defined a typical first magnitude star as being 100 times brighter as a typical sixth magnitude star. In general, an object with apparent magnitude value less than another object by five will be 100 times brighter. Equation 1 describes this relationship.

 Eq. 1 $\displaystyle \frac{{{L}_{1}}}{{{L}_{2}}}={{100}^{\frac{{{m}_{2}}-{{m}_{1}}}{5}}}={{10}^{\frac{{{m}_{2}}-{{m}_{1}}}{5}\cdot 2}}$

where

• L1 is the luminance of object 1.
• L2 is the luminance of object 2.
• m1 is the apparent magnitude of object 1
• m2 is the apparent magnitude of object 2.

This formula is similar to those listed here.

### Magnitude Variation with Range

The Wikipedia states that the apparent magnitude, absolute magnitudes, and range are related by Equation 2.

 Eq. 2 $\displaystyle 5\cdot \log \left( \frac{R}{10\text{ pc}} \right)=m-M$

where

• R is the distance of the celestial object from the observer
• M is the absolute magnitude of the celestial object when positioned at 10 pc from the observer.
• m is the apparent magnitude of the celestial object at a distance R from the observer.

Figure 1 shows my derivation of this result. The inverse square law figures prominently in this derivation.

### Examples

Figure 2 shows a couple of examples of computing absolute magnitude given apparent magnitude and range. I was able to obtain the same absolute magnitudes as listed in the Wikipedia for the Sun and Eta Carinae.

## Conclusion

I wanted to review the basics of brightness (aka luminance) and magnitude. This is a nice warm-up exercise for some other astronomy posts that I have planned.

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