# Magnitudes

#### Why use magnitudes?

One of the most fundamental properties of a star is its brightness. Astronomers measure stellar brightness in units called magnitudes, which seem at first counterintuitive and confusing. That's because they ARE counterintuitive and confusing -- they are in large part a legacy of ancient times. However, once you get used to them, they turn out to have a few redeeming qualities.

Why do we continue to use this system? There are several of reasons:

1. The objects we study cover a huge range in apparent brightness: the brightest star visible with the naked eye is over 600 times brighter than the faintest one. The Sun is over 6 TRILLION times brighter than the faintest star visible to the naked eye. It's awkward to deal with numbers this large.
2. As you will see below, the magnitude system is logarithmic, which turns the huge range in brightness ratios into a much smaller range in magnitude differences: the difference between the Sun and the faintest star visible to the naked eye is only 32 magnitudes.
3. Over the past few hundred years, astronomers have built up a vast literature of catalogs and measurements in the magnitude system.
4. Astronomers have figured out how to use magnitudes in some practical ways which turn out to be easier to compute than the corresponding brightness ratios.

Astronomers who study objects outside the optical wavelengths -- in the radio, ultraviolet, or X-ray regimes -- do not have any historical measurements to incorporate into their work: these fields are all very recent, dating to the 1930s or later. In those regimes, measurements are almost always quoted in "more rational" systems: units which are linear with intensity (rather than logarithmic) and which become larger for brighter objects. In the radio, for example, sources are typically measured in janskys, where

```     1 Jansky  =  10^(-26)  watts / square meter / Hertz
```

A source of strength 5 Janskys is 5 times brighter than a source of 1 Jansky, just as one would expect.

#### History of stellar measurements

People in all corners of the world have looked up at the stars (after all, without television, what else could they do at night?). We have detailed records from several cultures in the Middle East; in some, priests spent years studying the motions of the stars and planets, often trying to predict events in the future. Their motives may have been ill-founded, but in some cases, they did make very good measurements of what they could see.

Hipparchus of Rhodes compiled a catalog of about 850 stars. He described the brightness of each star by placing it in one of six categories, which one could call "brighest, bright, not so bright, not so faint, faint, faintest." Later scientists used the word "magnitude" to describe these categories.

```mag-ni-tude  n.

1. a.  Greatness of rank or position: "such duties as were expected
of a landowner of his magnitude" (Anthony Powell).
b.  Greatness in size or extent: The magnitude of the flood
was impossible to comprehend.
c.  Greatness in significance or influence:
was shocked by the magnitude of the crisis.
```
From the American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2000 by Houghton Mifflin Company.

The brightest stars were assigned to "the first magnitude", just as we would call the best movies or restaurants "first rate." The next-brightest stars were called "second magnitude", and so on down to the faintest stars visible to the unaided eye, which were called "sixth magnitude."

This is the origin of the peculiar convention that

• bright stars are denoted by small magnitudes
• faint stars are denoted by large magnitudes

#### The quantitative basis for magnitudes

In the nineteenth century, astronomers devised a number of tools which allowed them for the first time to make accurate, quantitative measurements of stellar brightness. They discovered two properties of the traditional magnitude classifications:

1. each step in the magnitude scale corresponded to roughly the same ratio in stellar brightness
2. the stars traditionally called "first magnitude" were roughly one hundred times brighter than those called "sixth magnitude".

An astronomer named N. R. Pogson came up with a system which would roughly preserve the ancient magnitude values, while allowing modern scientists to extend it to more precise measurements. He proposed that the magnitude system be defined as follows: given two stars with intensity of brightness I(1), I(2), define a magnitude difference which is based on their ratio of intensities:

```
I(2)
(m1 - m2)    =   2.5 * log   [ ---- ]
10   I(1)
```

So, for example,

```
intensity ratio I(1)/I(2)            magnitude difference  (m1 - m2)
----------------------------------------------------------------------
0.01                                  +5.00
0.1                                   +2.50
0.5                                   +0.75
1.0                                    0.0
2                                     -0.75
10                                     -2.50
100                                     -5.00
```

Note again the counterintuitive sign of magnitude differences: the brighter star has a smaller magnitude.

Note also that this definition says nothing about the zero-point of a magnitude: it provides only the DIFFERENCE between two stars. Exactly where to set the zero-point of the magnitude scale is a matter of some debate, and eventually comes down to an arbitrary choice. We'll deal with it later.

Exercises:
1. One of the stars in the handle of the Big Dipper is really a pair of stars, Alcor and Mizar, just far enough apart for people with good eyes to distinguish. Given magnitudes of 4.00 for Alcor and 2.06 for Mizar, how many times brighter is Mizar than Alcor?

2. What is the ratio of intensities for a pair of stars 5 magnitudes apart? 10 magnitudes? 15 magnitudes? 20 magnitudes? Is there a simple rule for calculating these intensity ratios quickly?

3. The average diameter of the dark-adapted pupil in a human eye is about 6 millimeters; the average person can see a star of magnitude 6 on a clear, dark night. If the same person were to look through typical 7x35 binoculars, how faint a star might he be able to detect?

#### A very handy coincidence

Despite their arcane definition, magnitudes can occasionally be very quick and handy to use in practice. For example, one often looks for small changes in the brightness of a star or asteroid. A common way to describe small changes is in terms of percentages: "alpha Orionis faded by 3 percent over the past week." It turns out that there is a simple relationship between small percentage changes in intensity and the corresponding changes in magnitude:

```    if
a star changes its intensity by   N     percent,

then
its magnitude changes by about  0.01*N  mag.
```

For example, if alpha Orionis fades by 3 percent, then its magnitude increases by about 0.03 mag.

This rule is accurate to about ten percent -- the real change corresponding to fading by 3 percent is about 0.033 mag, not 0.030 mag. But under most circumstances, it's good enough. It works best for very small changes; applying it to changes greater than 15 or 20 percent yields results which are increasingly incorrect.

Exercise:
1. Using the definition of magnitudes given above, derive this relationship. Hint: look up the series expansion for e-to-the-x, when x is much less than 1.

#### The painful side of magnitudes: adding and subtracting

There are drawbacks to the magnitude system. One of the big ones is the work one must do when trying to figure out the result of adding or subtracting two stellar sources, rather than multiplying or dividing them. Suppose there are two stars, A and B, with magnitudes m(A) and m(B), which appear so close together that their light blends into a single source. What is the magnitude of the resulting blend?

```      m(A + B)  =?    m(A) + m(B)               NO!
```

The proper way to do this calculation is to convert the magnitudes back into intensities, add together the intensities, and then convert back into magnitudes. There's no way around it.

Exercise:
1. My eyesight is so poor that I can't distinguish Alcor from Mizar without my eyeglasses. Given magnitudes of 4.00 for Alcor and 2.06 for Mizar, what is the magnitude of the single blurry object I see?