**Background: What and Why?**

*Magnitudes* are a system astronomers use to talk about the brightness of objects. They’re often a headache for astronomy students, because some things about them are counterintuitive. They also leave physicists who come to astronomy scratching their heads, because said physicists may not appreciate the historical reason why it actually makes sense to use magnitudes. I go back and forth myself on whether or not I should teach magnitudes in an introductory astronomy class. Sometimes I decide not to, because they really don’t add much to the understanding of the physics of astronomical systems, they’re just one more complication when it comes to dealing with numbers. However, at the moment, my opinion has vacillated towards thinking it *is* worth teaching, for two reasons. First, because they are ubiquitous in astronomy, you will see them referenced a lot. For instance, if you click on an object in Stellarium, you will be told the magnitude of that object. Second, and perhaps more importantly, it’s a worthwhile intellectual exercise. The magnitude scale is a logarithmic scale, and I think it’s good brain exercise to make students wrestle with that.

The two most important things to know about magnitudes— the first of these being one of the things that makes the difficult to work with— are as follows:

- A larger value of the magnitude represents a
*dimmer*object - The magnitude scale is a logarithmic scale, so a given
*difference*of magnitudes represents a given*multiple*of brightness.

I’ll explore both of these in greater depth below as I give the definition of the magnitude system.

Historically, the ancients categorized stars in brightness classes. The brightest stars in the sky (all named) were called “first magnitude” stars. The ones that appeared a step down in brightness were “second magnitude” stars, and so forth. Modern astronomers reverse-engineered this historical listing of stars to give a numerical “magnitude” that corresponds to the measured brightness of the stars such that stars would more or less have the same magnitude as they had when classified by the ancients. Because the response of our eye is much more approximately logarithmic than it is linear, this led to the magnitude scale being a logarithmic scale.

**Flux and Apparent Magnitudes**

When an astronomer talks about “the” magnitude of an object, she is usually referring to the *apparent* magnitude. This corresponds to a measurement of how bright an object is to a given observer (almost always an observer at Earth), *not* to the intrinsic energy output of the object. To quantify brightness, we use the concept of flux or energy flux, which is defined as the rate at which energy is collected by a telescope with a 1 m2 aperture. Flux comes in units of W/m2 (where a W, of course, is a J/s). Notice the m2 in the denominator. This is what makes flux not depend on the telescope that’s looking at the object. If two telescopes look at the same object, the one with the larger aperture will collect more energy from the same object. To figure out the rate of energy collection, you have to multiply the flux by the collecting area of the telescope. (Real telescopes also have an efficiency that combines the reflectively of their mirrors and the intrinsic quantum efficiency of their detector, but we’ll not worry about that here.)

Suppose you’re looking at two stars, one with flux f1, the other with flux f2. The difference between the magnitudes of these two objects is then defined by:

Notice that this is a *relative* definition; it defines the difference in magnitudes in terms of the quotient of the fluxes of the two objects. The logarithm here is a base-10 logarithm. **Warning:** while the button on many calculators that performs this function is often labelled “log”, the log() function in many computer languages and applications actually does a natural logarithm (usually called “ln”) rather than a base-10 logarithm. To find out which you have, try taking the log of 10. If you get 1, then you’re doing a base-10 logarithm. If you’re doing a natural logarithm, you will get 2.3.

**What is a log?**

A logarithm is the inverse operation of exponentiation. This means that if you have

then it’s also true that

You can think of a logarithm as being the operator that returns *whatever you have to raise 10 to in order to get the argument of the logarithm*. The natural logarithm (ln) is the same thing, only it’s what you raise *e* to rather than 10. Because logarithms are exponents, a few interesting properties apply to logs, which are useful when dealing with magnitudes:

**Redux: Flux and Apparent Magnitude**

Returning to our definition of the relative magnitude of two objects:

this also tells us how to get the ratio of fluxes from the magnitudes:

The -2.5 in front of the logarithm is very important. The negative sign gives the feature that *brighter objects have lower magnitudes*. For instance, if object 1 has 10 times the flux of object 2, then:

Because the difference is negative, m1 is *less than* m2, given that object 1 has a higher flux. The 2.5 means tells you how many times one object must be brighter than another to have a certain difference in flux. It’s chosen so that a factor of 100 in flux corresponds to a difference of five magnitudes— it works out, because log(100)=2.

So far, however, I’ve only told you how to compare magnitudes. How, then, can we talk about “the” magnitude of an object? For that to happen, we must have some agreed-upon reference that is the standard of comparison. Here’s where things get sad. There two different standards in widespread usage. A more modern system, known as “AB” magnitudes, defines an object of a given flux as having magnitude 0. The more traditional system, still in widespread use by a lot of astronomers, and the system used by programs such as Stellarium, uses “Vega-based” magnitudes. In this system, the star Vega is defined to have magnitude 0. If you compare the flux of an object to the flux of Vega, you get “the” apparent magnitude for that object. The difference between Vega magnitudes and AB magnitudes matters when you’re talking about magnitudes through different filters. Note, however, that if you are comparing to objects, the *difference* in magnitude will be the same regardless of which system you’re on. The system matters when you cite *the magnitude* of an object in a given bandpass or filter.

As a few examples, on the Vega system the Sun has an apparent magnitude of -26.74. Sirius, the brightest star in the night sky, has a magnitude of -1.46. Vega has a magnitude of 0.03. (*What?* you cry. *Why not exactly 0?* It turns out that “Vega magnitudes” are based on a historical idealized Vega rather than on the real thing.) The North Star, Polaris, has a magnitude of 2.02. The dimmest stars you can see under a good dark sky, on a moonless night and away from city lights, will be magnitude 5 or 6. (Numbers are visual magnitudes, and are from SIMBAD, except for the Sun’s magnitude which is from this page from GSFC.)

**Filters and “Color Index”**

Up to now, we’ve been talking about the flux of an object, implicitly including *all* the flux at all wavelengths. The technical term for this is the *bolometric flux*. In reality, we generally do not collect the flux at all wavelengths. Every detector we use is sensitive only to a finite range of wavelengths. For example, our eyes are only sensitive to wavelengths in about the range 450-650 nm, but stars emit light at wavelengths outside that range as well as inside that range. What’s more, there is a benefit to talking about the flux in even smaller ranges of wavelengths; this allows us to quantify the color of an object, by comparing (say) its flux at red wavelengths to its flux at green wavelengths.

In order to talk about colors and fluxes through different filters, we have to choose a filter set to use. A filter is defined by its transmission function; that is, what fraction of the photons it lets through as a function of wavelength. Two of the common filter systems in use are the Johnson-Cousins filters and the Sloan Digital Sky Survey filters. This image shows the Johnson-Cousins passbands:

UBVRI passbands from Bessel, 1990, PASP, 102, 1181

This system is sometimes called the “UBVRI” system. U stands for Ultraviolet, B for blue, V for ** visual**, R for red, and I for infrared. All of these letters make sense except for V;

**. V is more like green or yellow-green. It corresponds to more or less the peak of the eye’s sensitivity.**

*V is not violet*You can then define the flux through a filter as the flux coming from an object including just the photons transmitted through that filter. This will be less than the flux of the object as a whole (unless the object is very perverse and all of its light is emitted exactly at the wavelength where the transmission of the filter is exactly 1.0). A color, then, corresponds to the ratio of fluxes an object shows through different filters. For a given object, fB/fV would be the ratio of its B-band flux to its V-band flux. This ratio will be higher for bluer objects, because more of the light will be emitted at shorter wavelengths. If f0V and f0B are the fluxes of the reference object (i.e. Vega), then we have:

However, remember that the magnitude of Vega is defined to be 0, so:

If we subtract these two numbers, we get:

That’s a bit of a mouthful, but it corresponds to the “color index” B-V of an object, defined by:

Notice that because a higher magnitude is a dimmer object, as B-V gets larger, it means that the flux in the B filter gets *lower* compared to the flux in the V filter. This is backwards from what might be intuitive.

Astronomers will very frequently cite the color index (something like B-V, B-R, V-I, or similar) as a way of talking about the color of stars or other astronomical objects.

**Absolute Magnitudes and Distance Modulus**

Apparent magnitude (or just “magnitude”) corresponds to flux, or the observed brightness of an object. We also have a magnitude defined that corresponds to the luminosity (intrinsic energy output) of an object, and we call that “absolute magnitude”. The absolute magnitude is defined as the magnitude that would be observed for an object by an observer 10pc away from the object. (pc means parsecs; one parsec is equal to 3.262 light-years, or 3.086×1016 meters.) This may seem arbitrary… and it is. It’s just the reference distance. We had to pick something. So, hey, why not 10pc?

Imagine we’re looking at two stars, the Sun, and another star just like the Sun that is exactly 10pc away. The flux of the Sun is f⊙, and the flux of the other star we shall call fG. Both have the same luminosity (which we’ll just call L). We know that flux depends on distance by:

So, we have:

Simplifying:

From the definition of magnitudes, we also have:

where m is the magnitude of the Sun and M is the magnitude of the Sun as observed 10pc away. Because the magnitude of the Sun as observed 10pc away is the *definition* of absolute magnitude, M is also the absolute magnitude of the Sun. By convention, we use capital letters for the absolute magnitudes of stars. (However, be careful! We’ll often use capital letters for color indexes even when we’re talking about fluxes. It turns out that (discounting redshift effects) the absolute and apparent color index of an object will be exactly the same, so we can afford to be sloppy.)

Putting these two equations together, we have:

This last equation defines the ** distance modulus** for the Sun. To get there from the previous equation, I’ve used a couple of the properties of logarithms given earlier. In general, for an object a distance d away from us, its distance modulus is the difference between its apparent magnitude m and absolute magnitude M, and is:

For the Sun, the distance from the Earth is 1AU. Putting this in to the distance modulus equation, together with a unit conversion from AU to pc, we can figure out the absolute magnitude of the Sun:

As a final note, you can, of course, turn around the distance modulus equation to write the distance in terms of the absolute and observed magnitudes of an object:

**Summary**

The magnitude system is a system astronomers use to talk about the brightnesses of stars. It is a logarithmic system, so a *step* of magnitude corresponds to a *factor* of energy output or energy detected. It is also *backwards*, so that larger magnitudes correspond to dimmer objects. Magnitudes can be “bolometric magnitudes” (taking into account all flux at all wavelengths), but more commonly we talk about magnitudes through a given filter. The color index is the difference of two magnitudes; by convention, we subtract the magnitude through the redder filter from the magnitude through the bluer filter. Absolute magnitudes are defined as the magnitude that would be observed by an observer 10pc away from the object; absolute magnitudes correspond to the luminosity of an object, while apparent (or observed) magnitudes correspond to the flux of an object. Finally, the distance modulus is defined as the difference between the observed and absolute magnitudes of an object, and corresponds directly to the distance between you and that object.

It might be worth noting that there are other logarithmic scales in widespread use, including pH (for acidity/alkalinity) and the Richter scale (for the strength of earthquakes).

Indeed — the decibel system for sound is also a logarithmic scale.