# Measuring Sizes and Distances

How do we measure the size of something on the sky? Not with a ruler!

We measure angular sizes, defined in the following way:

• one full circle = 360 degrees (360o)
• 1o = 60 arc minutes (60')
• 1' = 60 arc seconds (60")

 Object Angular Size Night (or Day) Sky 180o Big Dipper 24o Thumb at arm's length ~ 1o Moon 1/2o (=30') Naked eye resolution ~ 1' Jupiter's disk 40" Atmospheric limiting resolution 1" Hubble Space Telescope limiting resolution 0.1" Nearest Star 0.03"

Okay, we can measure apparent sizes (theta). How can we get the true size (r)? Trigonometry! If we know the distance (d), we simply have r=d tan(theta).

So, how do we get a distance?

## Stellar Parallax

Okay, stick your thumb up at arm's length (Aaaayy!). Close one eye, then the other (and open the first one, so you can still see...). See how your thumb's position shifts relative to the background wall? This is called parallax. Now bring your thumb closer to your face, and do it again. The shift is bigger! You can use parallax to measure distance. (Right, Fonzie?)

Okay, so let's do this with stars.

Instead of closing one eye and then the other, we observe a star six months apart, so that we are on opposite sides of the sun for each observation. Watch the star shift against background star field, and measure that shift. Define the parallax angle as half this shift:

Using trigonometry, we can solve for the distance. If we measure p in radians, we have:

(Why?)

But we usually measure angles in arcseconds. Since 1 radian = 57.3o = 206265", we have

Okay, now comes the thinking part. Let's define a unit of measure called the parsec (pc) which is simply 1 pc = 206265 AU.  Then we have

By definition, a parsec is the distance to a star which has a parallax angle of 1"

1 parsec is 3.26 light years. The nearest star has a distance of 4.2 light years, which is 1.3 pc. Its parallax angle is 0.77" -- small!

Stars are so distant that measuring parallax is difficult. The smallest parallax angles which are currently measurable are about 0.001", corresponding to a distance of 1000 parsecs (a kiloparsec). This is much smaller than the size of our galaxy -- we can only use parallax to measure distances of stars in the "solar neighborhood."