THE STARS

What properties of the stars can we measure? How are they related, and what can we learn from them?

• distance
• velocity
• temperature
• size
• luminosity
• chemical composition
• mass

DISTANCES TO THE STARS

can be measured by parallax for nearby stars (say, out to 100 ly).

• Even nearby stars are so distant one needs the whole orbit of the Earth as a baseline; even with this baseline, the nearest stars have parallactic angle <1".
• First measured on 61 Cygni by Bessel in 1838
• proves that the Earth moves about the Sun: answers objection of ancients to heliocentric hypothesis
• stars were further away than anyone expected

define a unit of distance, parsec: distance (in parsecs)=1/parallax (in arc seconds)

1 parsec is about 3.3 ly, 2x10⁵ A.U., 3.1x10¹³ km: a huge distance

• orbit of Pluto=0.0002 pc
• The nearest star (Alpha Centauri) is 1.3 pc (4.3 ly) away. At 65 MPH, it would take 45 million years to drive there!
• Pioneer 10, travelling at 28,000 MPH, will reach vicinity of Aldebaran (68 ly) in 2 million years.

VELOCITIES OF STARS

In addition to their apparent motion due to the rotation and revolution of the Earth, stars have real velocities. The velocity can be large, but it looks small from Earth because they are so far away.

• Radial velocity can be determined from Doppler shift
• Transverse velocity can be measured from motion of star with respect to background (proper motion) over a period of years, after correcting for parallax. The more distant the star, the less apparent the proper motion.

TEMPERATURE OF STARS

Can be determined from the continuous (black body) part of their spectrum (Wien's law), as well as from line spectra of elements that have lost their electrons because of high temperature.

From distance and temperature, we can determine the size of stars, as we shall see.

LUMINOSITY OF STARS

• The luminosity L of a star is how much energy it gives off.
• The apparent brightness B of a star is how bright it appears from Earth. This is determined by the amount of light per unit area reaching the Earth. That governs how much light enters a fixed aperture, such as the pupil of the eye, or the aperture of a telescope. Since the total power output of a star is spread out over a sphere, after travelling a distance d it is spread out over the surface of a sphere, 4d², so it falls off as the inverse square of the distance.

  BL/d²

  Just from looking at a star, you can't tell if it's close and dim or bright and far away.

Luminosity scale

Hipparchus, in his star catalogue, invented a scale of brightness or apparent magnitude from 1 to 6. The brightest visible stars he called first magnitude stars, the dimmest, 6th magnitude stars, and the brightness levels between were divided up evenly. Note well: large magnitude indicates a dim star

The luminosity of a star is sometimes expressed as an absolute magnitude, which is the apparent magnitude it would appear to have from a distance of 10 pc.

SIZE OF STARS

• Direct measurement is possible for a few dozen relatively close, very large stars: from angular size of disk and known distance, diameter can be deduced
• Indirect measurement. From the distance of a star and its apparent brightness, we can calculate its luminosity. But we can calculate the absolute luminosity another way: Stefan's law (chapter 2) tells us that a hot body gives off an amount of energy that depends on the size and temperature of the body. Since we know the temperature of the star, comparing the two values of the absolute luminosity enables us to determine the size of the star.

Stars come in a wide variety of sizes from the dwarf star Sirius B, 100 times smaller (about the size of the Earth), to giant stars such as Betelgeuse in Orion's shoulder (10-100 times the size of the Sun) to supergiants (1000 times size of Sun)

CLASSIFICATION OF THE STARS, COMPOSITION

Between 1880 and 1920, Annie Jump Cannon attempted to classify the stars by their chemical composition (determined from discrete dark lines). Later it was realized that most stars have approximately the same chemical composition (mostly hydrogen, some helium) although they do differ in the amount of trace elements. The differences in line spectra was mostly due to the temperature of the star (whether or not the elements have lost their electrons due to high temperature, etc.)

We retain the original letters of the spectral classification, but the order has been changed to order by temperature. The sequence: OBAFGKM ("Oh, Be A Fine Girl/Guy, Kiss Me")

Class	Temperature	Color	Examples
O	30,000 K	blue
B	20,000 K		Rigel
A	10,000 K		Vega, Sirius
F	8,000 K	white	Canopus
G	6,000 K		Sun, Centauri
K	4,000 K		Arcturus
M	3,000 K	red	Betelgeuse

THE HERTZSPRUNG-RUSSELL DIAGRAM

Plot luminosity (vertical scale) against spectral type or temperature (horizontal scale)

Most stars (90%) lie in a band across the diagonal called the main sequence (from red dwarfs to blue giants). These are ordinary stars in the prime of life. They're all pretty much alike, except for size: small stars are cool and dim (red dwarfs) and large stars are hot and bright (blue giants) but normal stars are not, for instance, cool and bright.

The exceptions are stars in their death throes:

• red giants, large, cool and bright
• white dwarfs, small, hot and dim

THE COSMIC DISTANCE LADDER

Spectroscopic parallax

If we assume that distant stars are like nearby stars, then from the spectrum of a main sequence star we can determine (approximately) its absolute luminosity; then from comparing its apparent brightness with the absolute luminosity, we can determine its distance. This enables us to determine distances out to 1000 pc (with an uncertainty of 25%).

The cosmic distance ladder

• This assumes that we've got the luminosities in the HR diagram correctly
• that requires that we know the distances of nearby stars from parallax
• that requires that we know the size of the Earth's orbit (from radar)

STELLAR MASSES

For an isolated star, impossible to determine. However, some stars form binary systems. These are stars that orbit one another. From the period of the orbit, and the size of the orbit, we can determine the mass of the stars (using Kepler's laws and Newton's law of gravitation)

Types of binaries

• visual binaries, can be resolved in a telescope
• spectroscopic binaries, too close together or too far away to be distinguished in a telescope, can be analyzed by studying their Doppler shift
• Eclipsing binaries: if Earth lies in the orbital plane of the binaries, we can observe the stars eclipse one another

Beware of optical doubles, which are stars that just happen to lie along the same line of sight from Earth (e.g., Albireo)

Mass and the main sequence

The position of a star in the main sequence is found to be determined by its mass, which varies from 0.2 solar masses (red dwarfs) to 10-20 solar masses (blue giants). i.e. the mass of a star in the main sequence determines (approximately) its absolute luminosity and its spectral class (temperature).

Stellar lifetimes

From the luminosity, we can determine the rate of energy release, and thus the rate of fuel consumption. Given the mass (the amount of fuel the star has to burn), we can estimate lifetimes:

• Large hot blue stars live 20 million years (more mass to burn, but they burn it even faster)
• Mid-sized stars like the Sun have a lifetime of 10 billion years
• Small cool red dwarfs live trillions of years (small amount of fuel, but they burn it slowly)

STAR CLUSTERS

Open (or galactic) clusters Example: the Pleiades.

• Typically young, contain blue stars;
• size 10-100 stars, few parsecs across;
• contain traces of heavy elements

Globular clusters

• tight, spherical cluster;
• size, hundred thousand-million stars, 50 pc across;
• older (deficient in hot blue stars; sometimes include red giants); age estimated to be about 10 billion years.
• lie outside galactic plane;
• deficient in heavy elements.

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