Chapter 19 The Stars Distances to stars are measured using parallax.

This is not effective for very distant stars. The angle formed by parallax is measured in arc seconds.

A circle is divided into 360°. One degree is divided into 60 minutes, and one minute is divided into 60 seconds.

Therefore, one arc second is 1/(360 x 60 x 60) of a circle, or 1/ of a circle.

The distance a star must be to have a parallax of one arc second is 20,265 A.U.’s, 3.1 x cm. This distance is called a parsec (parallax in arc seconds).

The farther away a star is the smaller the angle becomes, so: distance (in parsecs) = 1/parallax (in arc seconds)

One parsec is approximately equal to 3.3 light years.

The closest star to Earth is Proxima Centauri. It is a member of a triple star system called the Alpha Centauri System. Proxima Centauri has the largest known stellar parallax at 0.76”.

1/0.76 = 1.3 parsecs; 4.3 light years, or 270,000 A.U.’s. This is a typical interstellar distance in the Milky Way galaxy.

If the Earth were a grain of sand orbiting a golf ball sized Sun at a distance of 1 meter, Proxima Centauri would be another golf ball over 100 km distant.

The next nearest star is Barnard’s Star at 1.8 parsecs (pc), 6.0 light years. There are about 30 stars within 4 pc of Earth.

The annual movement of a star across the sky, relative to other stars, is called proper motion. It is measured by angular displacement.

Barnard’s Star moved 227” over 22 years. This solves to 10.3”/yr. This is the largest known proper motion of any star.

Proper motion is only the transverse velocity (perpendicular to Earth). The other component of motion is radial velocity (found from the Doppler Effect).

True space motion can be found from the Pythagorean Theorem.

Finding Stellar Size – One way is by speckle interferometry. Many short exposure images of a star are pieced together producing a high resolution map of the star.

Another way to find the size of stars is by using the Radius-Luminosity- Temperature Relationship. Energy flux is the energy emitted by a star per unit area per unit time. Energy flux increases proportional to increases in temperature and stellar radius.

_________ √ luminosity radius is proportional to temperature 2 This is used to indirectly determine stellar size.

Example: Omicron Ceti temp: 3000K 1/2 Sun’s Luminosity: 1.6 x erg/sec 400x Sun’s √400 Therefore: radius = = X Sun’s radius

80X Sun’s radius would put the photosphere at Mercury’s orbit. This makes Omicron Ceti a Red Giant. A Giant is 10 to 100x the Sun’s size. A Supergiant is 1000x the Sun’s size.

Example: Sirius B temp: 12,000K 2x Sun’s Luminosity: erg/sec 0.002x Sun’s √0.002 Therefore: radius = = X Sun’s radius

Sirius B is much hotter and much smaller than our Sun. It is roughly the size of Earth. It is a white dwarf star. Any star smaller than our Sun is called a dwarf.

Luminosity is the rate of energy emission by a star. The apparent brightness of a star is how bright it appears from Earth.

A bright star is a powerful emitter, is near Earth, or both. A dim star is a weak emitter, is far from Earth, or both.

The apparent brightness of a star decreases in an inverse square relationship as its distance from the Earth increases.

Doubling the distance from a star makes it appear 2 2, or 4 times dimmer. Tripling the distance makes it appear 3 2, or 9 times dimmer.

The apparent brightness of a star is directly proportional to its luminosity and inversely proportional to the square of its distance.

When comparing the luminosity of stars, astronomers imagine looking at all stars from a standard distance of 10 pc.

The apparent brightness a star would have at 10 pc from Earth is called its absolute brightness.

A star closer than 10 pc from Earth will have an absolute brightness less than its apparent brightness. A star greater than 10 pc will have an absolute brightness greater than its apparent brightness.

The surface temperature of a star can be determined from measurements of its brightness at different frequencies. This is usually measured at a certain frequency of blue light (B) and a certain frequency of visible light (V) to which human vision is most sensitive.

The color index of a luminous object is the ratio of its B to V intensities. It is directly related to the object’s surface temperature and to its color.

Color Index B/V Temp ColorExample ,000K electric blue ,000K blue Rigel ,000K white Vega, Sirius 0.8 8,000K yellow-white Canopus 0.6 6,000K yellow the Sun, Alpha Centauri 0.4 4,000K orange Arcturus, Aldebaran 0.2 3,000K red Betelgeuse

This intensity measurement through a series of filters is called photometry. The UBV system uses Ultraviolet, Blue, and Visible filters to determine a star’s properties.