Lecture 3

HOW FAR IS A STAR?

Key Concepts

• Distances in the universe are important to know, but difficult to measure.
• The distance to a nearby star can be found from its PARALLAX.
• On average, a nearby star will have a larger PROPER MOTION than a distant star.

(1a) Distances in the universe are important to know.

Distances are required to compute:

• The luminosity of an object (a dim nearby star looks just like a bright distant star)
• The mass of an object (this can be computed from Kepler's Third Law if the object is part of an orbiting system).
• The physical size of an object (a large distant object [e.g. the Sun] can be the same angular size as a small nearby object [e.g. the Moon])

(Besides, knowing the distance to a star is interesting in its own right.)

(1b) Distances in the universe are difficult to measure.

It took a long time to measure distances to the nearest stars.

• Ptolemy [2nd century AD]: thought that all stars were at the same (relatively short) distance, attached to a crystalline sphere
• Copernicus [16th century AD]: knew that stars were at a large (but unknown) distance
• Bessel [1837]: first to measure the distance to a star other than the Sun - used the method of parallax (see below)

(2) The distance to a nearby star can be found from its parallax.

Definition: Parallax is the change in the apparent position of an object which results from a change in the observer's position.

Textbook example: Hold thumb at arm's length. Look first through one eye, then through other. Thumb's position changes relative to background. The closer your thumb to your eyes, the larger the jump in position.

Similar idea: Look at star, first in April, then in October. Your location changes by 2 AU between these times. Therefore, star's position changes changes relative to background. The angle p (see diagram below) is the star's parallax.

After measuring p, and knowing the size of the Earth's orbit, we can measure the distance to the star using trigonometry.

Note: The distance to even the nearest stars is much larger than 1 AU. Therefore the angle p is small.

Measuring small angles:

• 360 degrees = full circle
• 60 arcminutes = 1 degree
• 60 arcseconds = 1 arcminute

Full circle = 360 X 60 X 60 = 1,296,000 arcseconds

1 arcsecond = angular size of dime 2 kilometers away

Important Equation: Computing a Distance from a Parallax

d = 1/p

d = distance to star, measured in parsecs

p = parallax, measured in arcseconds

Note: 1 parsec is defined as the distance at which a star has a parallax of 1 arcsecond.

1 parsec = 3.26 light years = 206,000 AU

Example:

The star Proxima Centauri has a parallax p = 0.77 arcsecond.

d = 1/0.77 = 1.30 parsec = 4.23 light years

Parallax can only be computed for nearby stars.

• From the ground, smallest measurable parallax is p = 0.01 arcsec (d = 100 parsecs = 326 light years).
• From a satellite, smallest measurable parallax is p = 0.002 arcsec (d = 500 parsecs).

For comparison, distance to center of galaxy = 8500 parsecs.

(3) On average, a nearby star will have a large proper motion.

Definition: Proper motion is the steady change in a star's apparent position on the sky, resulting from its motion through space.

Proper motion is measured in arcseconds per year.

On average, close stars have faster proper motions than distant stars. (This is only an average statement: a star which happens to be moving directly toward or away from the Sun, for instance, will have no proper motion.)

Examples:

• Proxima Centauri: proper motion = 3.85 arcseconds/year, distance = 1.3 parsecs
• Betelgeuse: proper motion = 0.03 arcseconds/year, distance = 150 parsecs

Searching for stars likely to be nearby? Look for stars with large proper motion.

Amusing sidelight: proper motions cause distortions in the observed shapes of constellations, on time scales of thousands of years.

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