Star Properties

Part 1

 

Among the earliest stellar properties to be measured was the brightness.  Hipparchus introduced the system of apparent magnitudes wherein the brightest stars were denoted as first magnitude (1m).  Stars about half as bright as 1m were called 2m.  Another group half as bright as 2m stars were called 3m, and so forth until reaching the naked eye limit at 6m.  This system remained only semi-quantitative until 19th century investigations into sight.  The magnitude system today is scaled so that a ratio of brightness of 100 corresponds to a difference in magnitude of 5m.  A difference in magnitude of 1m corresponds to a brightness ratio of 2.512 (5th root of 100).  The brightest object in the sky is the Sun at -26.8m, several planets and stars have negative magnitudes, the naked eye limit was preserved at 6m, and the largest telescopes in the world can record object of magnitude +28m.  We see that the magnitude scale is inverted - the more prominent (brighter the object) the smaller the number we assign it.

Simply knowing the apparent magnitude of a star does not answer the fundamental question of stellar luminosity.  If I see a bright star in the sky, is it intrinsically bright or simply nearby?  We must measure the distance to the star in order to determine the luminosity.  The only direct method of measuring stellar distance is stellar parallax, the technique that Hipparchus introduced to prove that the Earth orbits the Sun.  As the Earth orbits the Sun, the nearby stars should perform a very tiny mirror of the Earth's motion against the background of much more distant objects.  Measuring the parallax angle allows us to solve for the stellar distance.  The angles here are very small and are usually measured in seconds of arc.  A star with a parallax angle of 1 acrsec is one parsec away from the Sun.  The distance is inversely proportional to the parallax angle.  One parsec is about 3.26 light-year or 206,265 A.U.  The nearest star to the Sun has a parallax angle of 0.7 arcseconds.  All other stars have smaller shifts.  The limit of the technique from ground base is about 200 parsec, requiring the precise measurement of an angle 0.005 arcseconds.  The Hipparcos satellite in Earth orbit can measure angles about five times smaller.

Now we can answer the question concerning the luminosity of stars.  If we measure the apparent magnitude and distance for a star, we can compute how bright the star would be at any other distance.  We choose a standard distance of 10 parsecs.  The brightness in magnitudes that a star would have from ten parsec away is the absolute magnitude (M).  You should see that the apparent magnitude, the absolute magnitude, and the distance are all related.  Find two of them and you can compute the third.  Here we have found apparent magnitude and distance and computed the absolute magnitude.  All other distance measuring techniques rely on a method of obtaining the absolute magnitude.  Measuring the apparent magnitude, then, allows us the compute the distance.  We will look at several techniques like this in the future.

 

Part 2

 

By the end of the first decade of the 20th century, many stellar distances had been determined.  Knowing the distances of the nearby stars allows us to get absolute magnitude.  By 1911-13 two pioneers in astronomy published nearly identical graphs summarizing important properties of the nearby stars.  In Denmark Ejnar Hertzsprung and in the USA Henry Norris Russell presented a graph that now bears their names - the Hertzsprung-Russell diagram, or H-R diagram for short.  It is a plot of absolute magnitude vs. spectral type.  At least these are the measured properties.  But each of these was created to help measure more physical properties of the stars.  The spectral type is a measure of the temperature of the star while the absolute magnitude was a measure of the luminosity.  Notice that we plot the brighter stars higher up on the vertical scale, but the hotter stars appear to the left.  Astronomers are famous for plot at least one scale backwards.  The reference point for measuring star properties is the Sun, so luminosity is measured in solar units.

Stars on the H-R diagram do not fall at random places.  Roughly 75 - 90% of all stars fall along a curve called the Main Sequence.  The Luminosity classes, which are apparent as regions where stars congregate on the diagram, are

 

Luminosity Class       Name           .

V                     Dwarf

IV                     Sub-giant

III                      Giant

II                       Bright Giant

I                       Supergiant

 

Luminosity class implies a relationship to size.  If we choose stars of equal temperature (same spectral type) but different luminosity class, the brighter of the two has more surface area through which to radiate its light.  This points out that only two properties determine the brightness of a star (if we eliminate the obvious effect of distance).  Luminosity is proportional to temperature and radius.  More specifically,

 

L µ T4R2

 

Luminosity class ca be measured in the spectrum.  Recall the argument I have made on pressure broadening of the spectral lines.  Lines in the spectrum that are very thin come from a large star - thicker spectral lines are produced in a smaller star.

We can use the H-R diagram to compare stellar properties.  Among these comparisons are brightness (brighter stars are higher in the diagram), temperature (hotter stars are further left), and radius (luminosity class).  Distance, however, cannot be compared on the diagram since we specifically took out the distance factor when forming the absolute magnitude.

There is a technique called Spectroscopic Parallax that using the H-R diagram to get the distance for a star.  Earlier we noted that the indirect distance determination techniques relied on some means of finding the absolute magnitude.  Then by measuring the apparent magnitude, we can calculate the distance to the star.  In fact, the quantity m-M is referred to as the distance modulus.  Spectroscopic Parallax works like this:

 

·        Measure the apparent magnitude (m) and the spectrum for a star of unknown distance.

·        From the spectrum we can classify the spectral type (which lines are present and in what relative intensity) and the luminosity class (by measuring the width of the spectral lines)

·        Plot the star onto the H-R diagram

·        Read off the absolute magnitude (M)

·        M and m together give you the distance

 

Now we can measure the distance to any star whose spectrum we can take.  We can now reach 5000 pcs with some precision.

 

Part III

 

The H-R diagram is very useful in comparing stellar properties, but the basic data required to plot the graph is extremely difficult to obtain.  The absolute magnitude on the vertical scale requires that we measure distance, one of the most difficult measurements in astronomy.  The horizontal scale of spectral type is simple tedious.  Is there a simpler way of getting the advantages of the H-R diagram without as much work?  The answer is yes, if we make certain restrictions.  We will plot only the stars that fall within star clusters.  An assumption can be made that all of the stars of a particular cluster are the same distance away.  If this is so, the concept of absolute magnitude is not necessary.  Apparent magnitude gives us the same comparison of luminosity that absolute magnitude gives for field stars.  So we will use apparent magnitude as the vertical scale.

The substitute for the horizontal scale (spectral type) is another way of determining the temperature of a star.  The blackbody curve has such a characteristic shape that it is not necessary to measure the intensity at each and every wavelength.  Astronomers have long known that an accurate fit to the blackbody curve (and therefore, an accurate temperature) can be accomplished by measuring the brightness at two wavelengths.  We use a photometer instead of a spectrograph and colored filters in the light path that restrict the wavelengths that get measured to a mere few.  The standard choice for forty years has been a filter that passes blue light only (B) and one that passes yellow-green light only (V).  The job is to measure the brightness of the star through each of these filters and construct the quantity called Color Index, defined as

 

Color Index = B-V = mB - mV  .

 

Notice we are simply subtracting the apparent magnitudes measured through these two filters.  The abbreviation B-V is common in textbooks to denote color index.  As is demonstrated in the videos, color index varies with temperature as can be used as a substitute for temperature.

So our new H-R diagram restricted to star clusters is called a color-magnitude diagram.  The vertical scale is apparent magnitude and the horizontal scale is color index.

The Milky Way galaxy has two types of star clusters.  The open (galactic) clusters has the following properties:

 

·        Irregular in shape

·        Few tens to a few hundred stars

·        Lie in the plane of the galaxy

·        Very young stars

The color-magnitude diagram contains a clearly visible Main Sequence and very few stars off the Main Sequence.

The other type of star cluster in the galaxy are the globular clusters

 

·        Spherical in shape

·        Very rich - 100,000 - 1,000,000 stars

·        Located in a halo distribution about the center of the galaxy

·        Very old stars

 

The color-magnitude diagram for a typical globular has very little left of the Main Sequence.  Most of the stars have advanced to the point in their evolution where they have exhausted the hydrogen in the core region.  The shortness of the Main Sequence for globulars is direct evidence of their great age.