# Measuring the Stars Continued

### Apparent Magnitude

The naked-eye limit is magnitude 6. This can vary a bit from person to person, but it is a good number. Remember how the magnitude system works: a larger number means a fainter object. Look at Figure 10.6 for examples of the apparent magnitudes of a number of familiar objects.

An astronomer always has access to the apparent magnitude of a star. If you are lucky, someone else has carried out a project of photometry and has measured the magnitude of the star(s) you are interested in. If not, you mount a photometer on a telescope and measure the magnitude(s) yourself.

### Absolute Magnitude

Apparent magnitude tells you how bright a star appears in the night sky but it does not tell you how luminous the star is. That's because distance also affects apparent magnitude. We need a measurement that will tell us the luminosity of a star without distance confusing the issue. That system is called Absolute magnitude. The definition of absolute magnitude is simple: it is the apparent magnitude a star would have IF it were exactly 10 parsecs away. This is a measurement which removes the distance parameter so you can directly compare the stars' luminosity. Absolute magnitude is always written as M.

### The Distance/Magnitude Formula

Distance, apparent magnitude and absolute magnitude are related by a simple formula.

`m-M=5log(d/10)`
If you know any two of the variables (m,M,d), you can quickly calculate the third. The "5" comes from the 5 magnitude steps which constitute a brightness difference of 100 times, which the "10" represents 10 parsecs, the standard distance for absolute magnitude.

### Star Color

There's another property of stars which is sometimes visible to the eye. If the star is bright enough, a hint of color can sometimes be seen. This color is a result of the temperature of the star. The blackbody curves in Figure 10.8 illustrate this effect. A star at 3000 K has its output peak in the infrared; its output in the visible range drops off from red to blue. A 10,000 K star has reasonably flat output over the visible range. A star at 30,000 K has its output peak in the ultraviolet, so its intensity in the visible range rises from red to blue. The first star (3,000 K) will appear reddish-orange, the 10,000 K star will be essentially white (no color) and the 30,000 K star will show a tinge of violet color.

### Spectral Types

There's another way you can take the measure of a star: record its spectrum. You'll get an absorption spectrum as ilustrated in Figure 10.9. Notice that those spectra are not identical. These spectra can be (and were) classified by their line patterns. The work was done about a century ago by one Annie Jump Cannon at Harvard College Observatory. She assigned letters to the types, producing an A,B,C,... sequence. This was before the actual meaning of the difference was known. The assignment was done according to the hydrogen line; type A showed the most hydrogen, B the next, and so on. Later, as knowledge of the physics improved, it became apparent that a temperature sequence would fit better. When the alphabetical types were rearranged in order of temperature, the alpha order got scrambled and a number of redundant types were dropped. What survived is todays list of spectral types.

```O B A F G K M
```
This is the sequence; type O is the hottest and type M is the coolest. How to remember this sequence? Try a nonsense jingle. The classic (and dullest) is the old chestnut "Oh Be A Fine Guy/Gal, Kiss Me." If you want something weirder, can make up one yourself.

In recent years a few more types have been added.

• C - Unusually large carbon abundance. Appears very red. Most are supergiants.
• J - Subset of type C. Contain 13CN.
• L - Cooler than type M - 2000 - 1500K. Prpoperly referenced as L-dwarfs.
• T - Cool brown dwarfs 700-1,300 K. Prominent methane features in spectrum.
• W - Wolf-Rayet star. Very hot like type O but with emission lines.
• Y - Super-cool brown dwarfs. T<600K. Mass 9-25 Jupiters. Few known.
• D - White dwarfs.

### The Sizes of Stars

It is, in fact, possible to determine the radius of a star that is so distant that it cannot be resolved with any telescope. How is this done? It is based upon the knowledge that a star acts like an approximation of a black body. This means that the amount of energy emitted by a square meter of a star's photosphere can be calculated as a function of temperature. Along with this, measurement of the star's absolute magnitude will lead to knowledge of the total amount of energy emitted by the star. When you divide the total energy emitted by the energy per square meter (temperature function), you get a result in square meters, which is an area. Use that area in the formula for the area of a sphere (4 x pi x r2) and very quickly get the radius of the star.

Look at Figure 10.11 to get an idea of how star sizes vary. The giant star Canopus (far south in the winter sky) is shown as 71 times the Sun's radius. The supergiant Betelgeuse, in Orion's right shoulder, is over 1,000 times the Sun's radius; its radius is about 5 AU! Little Proxima Centauri, which is very small, has a radius 1.5 times Earth, or about 12,000 miles.