# 3.1: Types of Stars


## Spectral Types

The mass of a star determines it's temperature, how long it will live, and what atomic elements can be manufactured through fusion reactions. Stellar mass is one of the most fundamental attributes of a star. Unfortunately, stellar masses are notoriously difficult to measure.

Instead of stellar mass, brightness and color are the first two measurements that astronomers made of stars. The color of a star can be determined by measuring the brightness through two filters, typically, a blue and a red filter. Color is closely correlated with the temperature of the star. Red stars may be a relatively "cool" 3000K, while blue stars are much hotter: 10,000K or more.

In 1850, Fraunhofer dispersed light from the Sun and observed characteristic absorption lines. A decade later, Kirchoff and Bunsen (famous for the Bunsen burner) realized that the dark lines in the spectrum were caused by absorption of light during specific energy transitions of atoms in the atmosphere of the Sun. When other stars were observed, they sometimes had different intensities or completely different absorption lines. The absorption lines in spectra provided a quantitative tool for classifying stars.

The Women Computers

In 1890, the Directory of the Harvard College Observatory, Edward Pickering, needed a workforce to analyze and catalog the extensive photographic data that had been collected. He employed dozens of women "computers" as his assistants. He believed that these women (unappealingly referred to as "Pickering's Harem") were ideal for the job because they did excellent work and they could be paid half the salary of men. Henrietta Swan Leavitt, who classified Cepheid variable stars, and Annie Jump Cannon were two of Pickering's women computers.

The Director of the Harvard Observatory, Edward Pickering, obtained spectra for 10,000 stars. He and his staff grouped the spectra according to complexity with letters from A through Q (from the simplest looking to the most complex spectra). Director Pickering enlisted the help of Annie Jump Cannon, one of his "women computers." Cannon and Antonia Maury found a more natural ordering that merged the stellar types into the smooth continuous sequence shown in the spectral sequence above. But it was not clear what this classification meant - what physical processes were driving the differences in the spectra?

Annie Jump Cannon expanded the catalog to include more than a million stars and she developed a new system for spectral classification with seven main groups: O, B, A, F, G, K, M and three classes of peculiar stars. As you might imagine, the original classification of stars was in the form A, B, C... but it was later realized that the bins of stars needed to be shuffled to sort them by temperature, or equivalently color, to have a measurable parameter. Each spectral type is further subdivided from 0 to 9 (increasing numbers moving toward the next cooler spectral type). Our Sun is a G2 star.

Characteristics of different spectral types for normal (hydrogen-burning) main sequence stars. Which stars have the shortest lifetimes? Which stars are most common? What is the spectral type of the Sun? How does the brightness of an O-type star compare to the brightness of an M-dwarf?

High mass stars are hotter and more luminous than low mass stars. You can see this from the data above: the temperature of stars is directly tied to their color: higher mass, hot stars are blue while lower mass, cooler stars are red. The color differences are subtle, but can still be distinguished by eye when looking at stars in the night sky. The use of filters or spectroscopy allows the color differences to be easily quantified.

The lifetime of the hydrogen-burning phase for a star depends on two factors: how much hydrogen exists and the rate of hydrogen fusion. For massive O-type stars, there is 100 times more hydrogen fuel than for stars like the Sun. However, the rate of fusion is so high that the O-type stars burn through this fuel in a million years. In comparison, stars like the Sun will burn hydrogen for about 10 billion years. And the lowest mass Mdwarf stars will burn hydrogen for more than 100 billion years (since the galaxy is only 13.7 Gyr old, there are no Mdwarf stars that have depleted all of their hydrogen).

Sometimes, stars form in pairs or small groups instead of single stars. If we can map out the orbits of the stars, we can use the equation for gravitational force to derive their masses. For main sequence stars with measured masses, astronomers were able to derive a correlation between color (or equivalently temperature or spectral type) and stellar mass. Piecing together all of this information was a triumph that launched the field of astrophysics -- a study of the physical processes of stars.

## Luminosity Class

Stars have both a spectral type (OBAFGKM) and a luminosity class (I, II, III, IV, V). Spectral type is directly tied to the temperature (or equivalently, the color) of a star. The energy output of a star (luminosity) is directly tied to both temperature and size. Therefore, two stars can have the same temperature, but they can have different luminosities if their radii are different. The radius of a star is constant while it is burning hydrogen - this is the "main sequence" or luminosity class V for a star. When hydrogen is depleted from the core in a star that is similar in mass to the Sun, the core of the star collapses, but the outer envelope expands so that overall, the radius of the star increases. Now, the star evolves across the subgiant branch (luminosity class IV) and the surface temperature cools as the star expands. Even as the outer envelope of the star is increasing, the core of the star is collapsing; when the pressure is high enough in the core, helium fusion begins and the star enters the giant phase (luminosity class III). Stars that are more massive than the Sun will evolve from class V to giants (class III) or supergiants (class II or class I) depending on their mass.

There are also subdwarfs (luminosity class VI) and white dwarfs (luminosity class VII) but by convention, those luminosity classes are not often used. Instead, those stars are usually just called "subdwarfs" or "white dwarfs."

Watch the video below to review star formation, spectral types, luminosity classes and stellar evolution. What is the difference between stellar classes and luminosity classes (I, II, III, IV and V)? What is the spectral type and luminosity class of our Sun?

## Distances to the stars

In Chapter 1.1 "Pointing to the stars" we discussed the celestial coordinates of right ascension and declination. This coordinate system treated the sky as a two-dimensional projection on the celestial sphere. However, there is an additional coordinate: the distances to stars (very few stars are Cepheid variables).

We need to know distances to stars to be able to derive the intrinsic luminosity of the star that is listed in Table above. Two stars with equal luminosity can have different apparent brightnesses because they are at different distances. Conversely, if we know the distance to a star, we can work backwards and calculate the intrinsic luminosity. Trigonometry is used to measure the distances to nearby stars. To understand how this works, hold up something like a pencil at a distance of just a few inches from your nose. Now, hold up another pencil at the greatest distance that you can reach (as demonstrated by Tyler, below). If you wink your eyes (left then right), you'll see that the pencil that is closest to your face appears to have a much larger shift (right to left) than the distant pencil.

We use the same principle to measure distances to stars. Stars that are close to us will have larger parallaxes than more distant stars. Winking our eyes won't work for this measurement because the baseline (the distance between our eyes) is too small. Instead, we use the baseline of the Earth's orbit to see apparent position of the foreground stars jump relative to the background stars.

The parallax sketch illustrates the (very exaggerated) geometry for measuring the parallax (trigonometric distances) to stars. Notice the triangle subtended by the parallax angle (with vertices at the Sun, the Earth in July and a nearby star). Because the distance to even nearby stars is so enormous, the parallax angle is tiny and the hypotenuse of this angle is essentially the same as the distance from the Sun to the star. This condition is called the small angle approximation. Sine and tangent functions can also be represented by Taylor-Maclaurin expansions where the higher order terms drop out for small angles. When θ is a small angle:

sinθ∼tanθ∼θ

The baseline for this measurement is always 1 AU. However, the parallax angle measured by our telescope and the distance to various stars will be different.

$\theta\,=\,\frac{1AU}{d}$

θ is the parallax angle in units of arcseconds (identified as "p'' in the parallax sketch above) and d is the distance from the Earth Sun to the star in units of parsecs.

## What is a parsec?

By definition:

360=2$$\pi$$radians.

So 1 radian = 57.2959 degrees = 3437.75 arcminutes = 206265 arcseconds.

For convenience, a new unit called a "parsec'' was defined such that 1 parsec = 206265 AU and subtends the physical length of 1 AU with an angle of one arcsecond (1") as shown in the Figure below.

$\theta\,=\frac{1\,AU}{206265\,AU}\,=\frac{1\,AU}{1\,parsec}$

Parallaxes have been measured for millions of stars using space missions like Hipparcos. If you need to know the distance to a star and the parallax for that star has been measured, then equation 2 is what you need. What is a parsec in other units?

$1\,parsec\,=\,206265\,AU\,=\,3\times 10^{13}\,kilometers\,=\,3.26ly$

In equation 3, 3.26 light years is the distance that light travels in 3.26 years. A light year may sound like a measure of time, but it is a distance. To instill a little awe, light could travel a distance equivalent to 7 times around the equator of the Earth in just one second. A light second is an incredible distance. A light year is conceptually almost impossible to grasp.

## Apparent and Absolute Magnitudes

Astronomers classify the brightness of stars with a magnitude system. This system dates back to the Greek scientist, Hipparchus, who called the brightest stars "first magnitude" stars. The faintest type of stars that Hipparchus could see were "sixth magnitude" stars. Unfortunately for astronomy students today, this scale was based on the sensitivity of the human eye. It turns out that a 6th magnitude star is 100 times fainter than a 1st magnitude star. Another way of saying this is that a Δm of 5 magnitudes corresponds to a brightness difference of 100 (or Δm of 1 corresponds to a brightness difference of 2.512). You also need to remember that brighter stars have smaller magnitudes.

The apparent magnitude, m, tells us how bright an object appears to be. The apparent magnitude is the quantity that an astronomer can easily measure with a telescope. However, two objects may have the same intrinsic brightness (luminosity), but one star may appear to be fainter than the other because it is farther away.

It is the absolute magnitude, M, that can compare the true brightness of stars. The absolute magnitude is the brightness that a star would be if it were located at a distance of 10 parsecs.

$M\,=\,m-5\log(\frac{d}{10})$

where d is in parsecs. As a reality check, plug d=10 pc into equation 4. This says that the apparent magnitude is equal to the absolute magnitude at d=10 pc, since $$\log_{10}(1)\,=\,0$$.

If d < 10 pc, then the absolute brightness must be dimmer than the apparent brightness (the star appears brighter because it is closer than 10 pc). If d > 10 pc then the absolute magnitude is brighter (smaller) than the apparent magnitude.

Like luminosity (the energy that a star emits), the absolute magnitude is a measure of the true intrinsic brightness and can be used to compare one star to another. By definition, the effect of distance has been eliminated (all stars are conceptually at 10 parsecs). The trick is to figure out what the luminosity or the absolute magnitude is... because all stars are not lined up at a distance of 10 parsecs. Thanks to trigonometric parallaxes, the distances to nearby stars is the easier parameter to measure. We can then solve for the absolute magnitude (or luminosity).

## Sorting the Stars: H-R Diagrams

Humans like to sort and compare things because this helps us to make sense of our environment. After astronomers were able to determine distances and calculate the intrinsic brightness (absolute magnitudes) of stars, they were able to sort them in a meaningful way. The Hertzprung-Russell (H-R) diagram organizes different groups of stars. Cutting diagonally across the schematic H-R diagram are the "main sequence" stars. To anthropomorphize a bit, all of these stars are in their normal adult years, steadily burning hydrogen in their cores.

The H-R diagram

Locate the Sun on the H-R diagram. What is the absolute magnitude of the Sun? What is the spectral type and temperature?

All of the known stars closer than 200 parsecs are shown in the Figure below. The data for this plot were drawn from data cataloged from the Hipparcos satellite, which measured parallaxes (equivalently, distances) for 118,000 stars. Stellar distances and the measured apparent brightness at visible wavelengths, mV, were used to calculate the absolute magnitude of each star:

$d\,=\frac{1}{parallax}\quad\qquad\quad\,M_v\,=\,m_v-5\log_{10}(d)\,=\,5$

The added constant of +5 in Eqn 5 comes from applying one of the rules of logarithms to Eqn 4:

$$\log(\frac{a}{b})\,=\log (a) - \log (b)$$

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