Apparent Visual Magnitude of Binary Stars

Quote of the Day

Once you know one programming language, you pretty much know them all.

— George Winsky. I often refer to this quote as 'Winsky's Law'. I used to work with George at Hewlett-Packard. He said this after I mentioned how learning Basic, Pascal, C, etc. was all easy after having FORTRAN in school.

Introduction

Figure 1: Sirius and its Companion (Source).
Binary stars are very common in the universe.

I was reading a Wikipedia article on the star Iota Apodis (Figure 1), which is a binary star, and noticed that three apparent visual magnitudes were listed for the two stars:  5.41 (5.90/6.46). The visual magnitudes listed represented the combined and individual brightness of the two components (in parentheses). I became curious as to how the magnitudes were summed.

In this post, I will look at how to determine the combined visual magnitudes of two stars. I will also discuss the limitations of this sort of calculation.

This calculation is almost identical to the calculation that electrical engineers do when they must sum powers that are expressed in dB. We first must "un-dB" the values, sum them, then "re-dB" the sum. As with dB, I do not consider visual magnitudes to be a dimensional unit – they really are a scaling.

For those who are interested, I include my Mathcad source and a PDF here.

Background

Definitions

Apparent Magnitude (m)
The apparent magnitude of a celestial object is a number that is a measure of its brightness as seen by an observer on Earth. The brighter an object appears, the lower its magnitude value (i.e. inverse relation). The Sun, at apparent magnitude of −27, is the brightest object in the sky. It is adjusted to the value it would have in the absence of the atmosphere.

The magnitude scale is logarithmic: a difference of one in magnitude corresponds to a change in brightness by a factor of $\sqrt[5]{100}$ or about 2.512.(Source)

Absolute Magnitude (M)
Absolute magnitude is the measure of intrinsic brightness of a celestial object. It is the hypothetical apparent magnitude of an object at a standard distance of exactly 10 parsecs (32.6 light years) from the observer, assuming no astronomical extinction of starlight. This places the objects on a common basis and allows the true energy output of astronomical objects to be compared without the distortion introduced by distance.
As with all astronomical magnitudes, the absolute magnitude can be specified for different wavelength intervals; for stars the most commonly quoted absolute magnitude is the absolute visual magnitude, which uses only the visual (V) band of the spectrum (UBV system). Also commonly used is the absolute bolometric magnitude, which is the total luminosity expressed in magnitude units that takes into account energy radiated at all wavelengths, whether visible or not. (Source)

Apparent Magnitude and Luminosity

Equation 1 shows the key relationship between luminosity and visual magnitude (Source).

 Eq. 1 $\displaystyle {{m}_{{\text{star}}}}={{m}_{\odot }}-2.5{{\log }_{{10}}}\left[ {\frac{{{{L}_{{\text{star}}}}}}{{{{L}_{\odot }}}}{{{\left( {\frac{{{{d}_{\odot }}}}{{{{d}_{{\text{star}}}}}}} \right)}}^{2}}} \right]$

where

• mStar is the apparent magnitude of a star.
• m is the apparent magnitude of the Sun (it could be that of any reference star).
• LStar is the visual luminosity of a star.
• L is the visual luminosity of the Sun (it could be that of any reference star).
• dStar is the distance to the star.
• d is the distance to the Sun.

Analysis

Derivation

Figure 2 shows my derivation for the magnitude of the sum of magnitudes. The process is straightforward:

• convert each magnitude to a luminosity (i.e. optical power).
• sum the luminosities
• convert the luminosities back to a magnitude.

Figure 2: Derivation of the Apparent Magnitude of Two Stars.

Test Cases

Figure 3 shows how I tested this formula. The process I used was to:

• randomly grabbed  eight test cases from the Wikipedia's list of binary stars.
• apply the formula
• check the formula against the Wikipedia's value for the combined magnitude.

I should note that the agreement is not perfect because the Wikipedia shows measured values, not computed values. This means that there will be some random variation.

Figure 3: Check the Formula Against the Wikipedia Values.

Conclusion

The agreement seems reasonable. One issue that I see with the calculation is that it is not always clear whether total luminosity or visual luminosity is being used. For example, Spica emits most of its energy outside of the visual band. As shown in Figure 4, the total luminosity and visual luminosity can quite different (Figure 4).

Figure 4: Quick Look at Spica.

After I finished writing this post, I found a Wikipedia paragraph that calls out the magnitude summation formula derived here.

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One Response to Apparent Visual Magnitude of Binary Stars

1. Dulli Chandra Agrawal says:

A very good attempt for those who want to learn such a technique. Please read mine paper
Apparent magnitude scale: incandescent lamp
available at
https://www.researchgate.net/publication/325247291_Apparent_magnitude_scale_incandescent_lamps