How to determine the distance to a star

 

Method 1.  Use parallax angle (only works for nearby stars).

 

Distance to star in light years = 3.26 light years / parallax angle in arcseconds

 

Distance to star in parsecs = 1 / parallax angle in arcseconds

 

Method 2.  Look at spectrum and apparent magnitude of star

 

Spectrum indicates two things:  temperature of the surface of the star (spectrum peak indicates temperature) and luminosity class (spectrum line broadening indicates type of star).

 

Spectrum lines are broad = fairly dense atmosphere = main sequence star

Spectrum lines are thin = thin atmosphere = bloated star = giant or supergiant star

 

Each luminosity class has specific characteristics in the spectrum.  The spectrum of a white dwarf will be different from the spectrum of a white main sequence star.

 

Once you have the temperature and luminosity class of a star, you can determine the absolute magnitude of the star using the H-R diagram.  This is the magnitude of the star IF it was placed 10 parsecs away from the Earth.

 

Now, you compare the absolute magnitude with the apparent magnitude (the observed magnitude) of the star.

 

• If apparent magnitude is brighter than the absolute magnitude, then the star is closer than 10 pc away (32.6 light years away). 

 

• If apparent magnitude is smaller than the absolute magnitude, then the star is farther than 10 pc away.

 

Comparing the two magnitudes is like comparing two 100 watt light bulbs that are at different distances away from you.  See p. 399 for the relation to get the distance to the star.

 

 

Determining the radius of a star

 

The temperature of the star’s surface and the luminosity can also determine the radius of the star.  Using the spectrum of a star, one can get the temperature (which gives information about how much energy is emitted per square meter every second) and the luminosity (which gives information about the total energy emitted per second).  With this information, the surface area of the star can be determined—which leads to the radius of the star.

 

 

Determining the mass of a star

 

The mass of a star can be found using Kepler’s Third Law of Planetary Motion (now applied to stars’ orbits) and Newton’s Law of Universal Gravity IF it belongs to a binary system (two stars orbiting each other via gravity). 

 

Most stars are actually a part of a multiple star system, so finding masses is not difficult.  The hard part is to get information about how the stars orbit each other (like orbital period, etc.). 

 

A visual binary is one that can be resolved as two stars using an optical telescope.  10% of the stars in the sky are visual binaries.  Over time, the positions of the two stars will change (think YEARS or DECADES), and this will give information needed to determine the masses. 

 

An optical double star is NOT a visual binary—they may appear to be close to one another, but in actuality may be very distant from each other. Fig 12-22

 

Some binary systems have stars which are too close to each other to resolve as two stars optically.  These are called spectroscopic binaries—they require the observation of their spectra.  Over time, the overlapping spectral lines of these systems will move against each other, giving clues to their motions.  The Doppler shift affects the spectral line positions, so the stars’ speeds can be determined. 

 

A special type of binary star is the eclipsing binary, where the apparent magnitudes of the system change regularly, like clockwork.  Fig 12-31

 

It turns out that the mass of a star is directly related to the luminosity of a star.  Fig 12-37 indicates that for a number of local binary systems, the mass and luminosity have a simple relationship.

 

 

Cepheid Variable Stars

 

Most stars in the sky have stable sizes—the gravity and thermal pressure have balanced out in a state of hydrostatic equilibrium.  There are a number of stars which have not reached this equilibrium state, and continue to vary in size.  As the size changes, the apparent magnitude wavers periodically.  Henrietta Leavitt used data from many Cepheids to determine statistically the average absolute magnitudes of these stars.  So, from observation, the period can be determined, which can gice information about the average absolute magnitude.  With the apparent magnitude and absolute magnitude known, one can determine the distanced to the Cepheid. 

 

The nice thing about Cepheids, is that these stars can be found in distant objects, like the Large and Small Magellanic Clouds (now known to be 48,000 and 56,000 pc away, respectively), and other galaxies (like the Andromeda Galaxy—2.2 million ly away).