# Measuring the Stars

## Finding Out About the Stars

This chapter will give you the sense of how we know so much about stars. They are very distant, but we can know a lot about them.

### Parallax

Parallax is an apparent back-and-forth motion of something against its distant background as you change your viewing location. You can demonstrate this by holding your arm out straight with your thumb up. Look with first one eye, then the other, and you will see your thumb jumping back and forth. The thumb isn't moving - your viewing location is.

This will work for stars - if you can get two viewing locations that are far enough apart. Opposite sides of Earth are 8,000 miles apart, but that's not enough spacing. Actually. it's easy. Simply look at the star, wait six months, then do it again. In that six months the Earth has carried you (and everybody else) halfway around its orbit, to the other side of the Sun. Those two viewing locations are 186,000,000 miles, or 300,000,000 km, apart. That spacing is enough. Two observations at 6-month intervals define the triangle.

The method is based on a "skinny" triangle; that's an isosceles triangle (two sides are equal) with a base that is VERY small compared to the two equal sides. That small base is 2 AU long (Earth's orbit diameter).

Along with parallax measurement comes a new distance unit - the parsec. That's a made-up word; it means PARallax SECond, or the distance to an object which has a parallax angle of 1 second of arc from Earth. One parsec turns about to equal 206,265 AU, which is also about 3.26 light years (LY). That triangle is so skinny that its height is 206,265 times its base. This illustration shows the whole thing.

There's another nice benefit of this skinny triangle - distance and parallax relate by an EXTREMELY simple formula.

```      distance = 1/parallax
```
The parallax is expressed in seconds of arc.

Note: the nearest star is more than 1 parsec away.

Another note: the distance to a star is not an intrinsic property of the star. How far away a star is not related to what the star is.

### Motion

The stars are actually moving through space. You don't notice this because the motion is, at most, a few arcseconds per year. That motion on the sky is known as proper motion. It is measured in arcseconds per year. Notice that the distance to the star not a part of the measurement. It is important, however, because a more distant star will appear to move more slowly than a nearby one. Measuring proper motion is often done by comparing photographs of a region of sky taken decades apart; over 20 or 30 years the proper motion will be large enough to see. At least two large photographic sky surveys have been rephotographed after several decades.

This animation shows what happens to the stars of the Big Dipper over a period of about 200000 years.

The actual motion of a star is called the space velocity; it describes which way in space the star is moving and how fast. This motion is expressed relative to the Sun (has to be with respect to something). This works better than Earth because you don't have to look out for Earth's orbital motion. The velocity has two parts - radial velocity and transverse velocity. Radial velocity is the star's motion along our line of sight and transverse velocity is motion across (perpendicular to) our line of sight.

Radial velocity is measured by looking measuring the Doppler shift in the star's spectrum. This does not require knowing the distance to the star. Transverse velocity requires knowing both the distance to the star and its proper motion. Using a triangle, these two will yield the transverse velocity.

### Brightness

The brightness of a star is one of two properties that you can see without optical aid (the other is color in bright stars). You can find very bright and very dim stars in the night sky.

What you see in the night sky is the apparent brightness - how it appears to you. It doesn't tell you anything about how luminous the star is. Luminosity is the amount of energy radiated by the star. Some stars are enormously luminous, radiating huge amounts of energy, while others are small and dim, shining faintly even if nearby.

Understanding what distance does to the brightness of a star requires understanding the inverse square law. This law governs the way light spreads out from a source into the universe. It is a result of a three-dimensional space. We used two demonstrations to illustrate the effect. If you double the distance from a light source to a detector, the light intensity drops by a factor of 4! See page 282 or look at this graphic.

We need a formal system of measuring star brightness. The concept of this dates to ancient times and the Greek astronomer Hipparchos. The basic system assigns a value of 6, or 6th magnitude, to the faintest stars one can see with the naked eye. The brightest stars are labeled 1st magnitude. Later astronomers, using telescopes and measuring instruments, found that the brightest stars were about 100 times brighter than the faintest. This was the basis for the formal system we use. Magnitude 6 is assigned to the faintest star you can see without optical aid (that's magnitude 6.0). A magnitude 1 star is defined as being exactly 100 times brighter. Five magnitude steps (1 to 2, 2 to 3, 3 to 4, 4 to 5, 5 to 6) add up to 100 times brightness difference. The non-linear response of the human eye produces an interesting property of this system; one magnitude step is not 20 (100/5), but rather the fifth root of 100, which we round to 2.512 (try this with your calculator).

```Magnitude  1 ---|
2    |
3    | = 100 times brightness
4    |
5    |
6 ---|
```

It turns out that this range is not sufficient to describe all visible objects. As a result, the magnitude scale continues to 0, then to -1, -2, -3 and so on to magnitude -26.8, which is the Sun's apparent magnitude. See Figure 10.6. The negative magnitudes may look weird, but they work fine mathematically.

If one star is 2 magnitudes brighter than another, the difference is not 2.512+2.512, but rather 2.512*2.512 (product, not sum). Actually, 2.512 multiplied by itself 5 times will yield 100 (actually only very close to 100 because of rounding). The actual value of the fifth root of 100 is 2.5118864. Try taking this to the 5th power.