# 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.

### The H-R Diagram

Early in the 20th century, Danish astronomer Einar Hertzsprung and American astronomer Henry Norris Russell independently looked at a possible relation between luminosity and temperature. That was a very significant research question. The result is shown in figures 10.12, 10.13, 10.14 and 10.15. Figure 10.15 shows the obvious relationship that ultimately appeared. The band running from upper left to lower right is the Main Sequence; it clearly reveals the relationship. The hottest (type O) stars are also the most luminous while the coolest (type M) are the least luminous (dimmest). This graphic shows the H-R Diagram quite nicely.

Main sequence stars are "normal" stars which are burning hydrogen as their fuel.

There are two other regions of the diagram to note: the red giant and white dwarf regions. The red giants are found in the upper right. These are very cool (3,000 K) stars which are VERY luminous. Their excuse for this is that they are VERY large. Each square meter doesn't emit a lot of energy as stars go, but the huge star has a LOT of surface area, so it is quite bright. The white dwarfs are found in the lower left. These objects are quite hot and very dim. Each square meter emits a lot of energy but the star is small and has a small surface area to radiate, so it is dim. See More Precisely 10-2. Here's a H-R plot of nearby stars using data from the Hipparcos spacecraft.

### Spectroscopic Parallax

Spectroscopic parallax isn't exactly a parallax (angle) measurement, but it does present a means of getting a distance estimate for a star that is too far away for actual parallax measurement. You do it by examining the spectrum of a star and determining its spectral type. Looking at the Main Sequence band at that type allows you to look to the left of the diagram and get an estimate of the absolute magnitude. Since you always can find m (apparent magnitude) and you have an estimate of M (absolute magnitude), the formula will yield a distance. The Main Sequence does have some width (is not a thin line), so the estimate of M, and therefore the distance, is somewhat uncertain (has an error bar). That is, however, a LOT better than knowing nothing about the distance.

### Finding Stellar Masses

There's only one way to determine the mass of a star directly - measure its gravitational influence on something else. For the Sun it's easy; use the orbits of its planets to find its gravity. For other stars that do not have planets we can see it is a problem. The only thing we are likely to be able to use in finding the star's gravitational influence is another star, and that requires having two stars orbiting each other in a binary star system. Fortunately, over half the stars in the Galaxy are in binary systems, so there are a lot of opportunities.

The masses are determined by studying the orbit of the binary system. Remember that the two stars orbit around the common center of mass (CM). Astronomers will determine the following.

• Orbital period - by observation over time.
• Distance from Earth - by parallax or spectroscopic parallax.
• Size of orbit - by apparent size and measured distance or doppler measurement of orbital velocity.
• Ratio of distances from CM - by measuring angle.
Kepler's 3rd law will give the sum of the two masses. The ratio of distances from the CM will give the mass ratio. This gives two simple equations in two unknowns, which can be quickly solved for the masses.

Once masses have been determined for a significant number of stars, we find that mass and luminosity are related. See Figure 10.21 for an indication of the Mass-luminosity relationship (or use this graphic); type O stars are the most massive and type M stars (red dwarfs) are the least massive.

Also note that stellar radius and spectral type are related. The more massive stars are the largest, while the least massive stars are smallest.

Be sure you understand this chapter. Use the questions at the end of the chapter for practice.