# The Distance to the Stars

## Outline

1. How far are the stars, and how do we know?
2. Brightness of stars: distance vs. luminosity

# Terms to Know

parallax
parsec (pc)

## 1. How far are the stars, and how do we know?

The ancient Greeks (as well as more "recent" astronomers, such as Tycho Brahe) based major theories on the lack of observed parallax of the "fixed" stars; i.e., the stars didn't appear to wiggle back and forth as the Earth rotates on its axis and revolves in its orbit around the Sun. But as we now know, some stars do show that wiggle, or parallax, every year. You can use that parallax to measure the distance to the nearest stars:

Use the Earth's orbit around the sun as baseline for triangulation:

Baseline = 2 A.U. (Earth's orbit)

Formula (just simple geometry using the small-angle approximation -- see "By the Numbers" 3.1, p. 31):

distance (parsecs) = 1 / angle (arcseconds)

the 'angle' is referred to as the 'parallax'

1 parsec ("PARallax SECond") = 3.26 light-years

1 arcminute (') = 1/60th of one degree

1 arcsecond (") = 1/60th of one arcmin

Nearest star: Proxima Centauri 1.3 parsecs

- actually a 3-star system, one like the Sun

- visible to the eye

Next closest: Barnard's star 1.8 parsecs

- invisible to the naked eye Inference:
Stars must come in a wide range of luminosity

Could space be filled with dim stars that we just can't see?

Limitation of parallax:

- difficult to measure parallax angle for objects more than a few parsecs away -- most stars in the Milky Way Galaxy (and all galaxies outside the Milky Way) are too far to use parallax for measuring distances.

But remember importance of parallax:

- primary measurement of distance in astronomy.

- the 'bottom rung' of the cosmic distance ladder.

- with parallax, we calibrate all other methods of determining distances to objects that are farther away.

## 2. Brightness of stars: distance vs. luminosity

The luminosity of a star is its intrinsic, or physical brightness, independent of its distance. Even if you can't see the star, its luminosity remains the same.

Recall the Inverse Square law for light: I 1/r2

# Terms to Know

Hertzsprung-Russell diagram
the Main Sequence
dwarf, giant, and supergiant stars

Temperatures and luminosities of stars are observed to be highly correlated (like weight and height for humans)

"HR diagram"

The Hertzsprung-Russell diagram (1911-1913) is a plot of temperature (or color -- basically the same thing as color, for black bodies) vs. luminosity. Every star can be plotted somewhere on the H-R diagram. It is one of the most important graphs in astronomy, and shows the complete life cycles of stars of all different masses, ages, temperatures, colors, and luminosities.

90% of all stars have luminosities (or absolute magnitudes) and temperatures (or colors) that place them in a narrow diagonal band in the HR diagram, called the Main Sequence. This makes sense, since stars are almost perfect black bodies, and we know that cold black bodies are dim and red, while hot black bodies are luminous and blue.

But there are some stars that are dim and blue, and others that are brilliant and red! What's going on?

There must be a third characteristic coming into play: the radius of the star! Bigger stars have more surface area, so they appear brighter, while smaller stars have less surface area and appear dimmer, even if they are very hot.

Which is more important: T or r?

Recall: The area of a sphere is given by A = 4 r2.

Now remember the Stefan-Boltzmann Law? To be more accurate, we should actually write it:

 L R2T4

So they're both important.

Here are all the parts of the H-R diagram:

The Main Sequence

temperature and luminosity are correlated

Giants

more luminous (and larger size) than Main Sequence stars of same temperature

Super Giants

even more extreme than giants

White Dwarfs

very small, hot, and faint

How are they all related?

An evolutionary sequence? Sequences?

Note 1:

Don't confuse white dwarfs with luminosity class V ``dwarfs.''

They are NOT the same!

Red dwarfs are the cool end of the Main Sequence, i.e. luminosity class V, and are NOT the same as white dwarfs.

Note 2:

Don't confuse "motion" in the HR diagram with true motion in space! A star that stays in one place all its life but changes temperature and luminosity will move around in the HR diagram. Meanwhile, a star that is zooming around in the Milky Way Galaxy could stay perfectly still in the HR diagram, if its luminosity and temperature don't change.