## Hertzsprung-Russell Diagram

Originally it was proposed, based on the spectral types, that young stars are hot, very bright, and with a large mass, and as they become older, they progressively burn their material, becoming colder, less heavy, and less bright. In other words: there is a linear relationship between the spectral types (i.e. T of the star) and the M (absolute magnitude).

This is a good starting point, but there is much more to it. A plot of the absolute magnitude of stars M vs the color index B-V (which can be related to the temperature of the star) shows groups of stars out of the general linear trend.

This kind of representation is called Hertzprung-Russell diagram (HPD), its “Y-shaped” form is characteristic (see below).

Note: There are different things that one can put in the axis of a HPD.

• An “observational” representation would use properties that can be observed, i.e. absolute magnitude and color index.
• A theorist’s representation would use properties that are calculated, in particular luminosity and effective temperature.

However, the scale is consistant. For magnitude and color index we use a linear scale, for luminosity and temperature we use a logarithmic scale. Why that? Well, because the relation between magnitude/color index and luminosity/temperature is logarithmic.

Hertzsprung-Russell diagram from wikipedia (modified).

The diagram above contains most of the relevant information in this post. A few things to note:

• For one given spectral type. Look in particular at the spectral groups K and G. There are groups of stars with similar temperature but completely different luminosity L. The brighter stars were called giants or supergiants. The dimmer stars were called dwarfs. For a given temperature, larger L means larger radius R.

• There is something called the main sequence stars. These are the ones who follow the linear-ish relationship between temperature and luminosity.
• In fact, the stars of the main sequence don’t line in a line, but in a narrow band. This is caused by variations in temperature, luminosity, and composition.
• Don’t confuse dwarf with white dwarf! White dwarfs are very dim, extremely small, and hot stars, far out of the main sequence.
• Parallel lines with a certain slope in the HRD connect stars with the same radius. These are the lines indicated in the diagram with a label such as 100R⊙, such line means that whatever stars lines on top of it has a radius 100 times larger than the radius of the Sun (the lines I drew are quite approximate, you may see a better approximation here, where the line for the Sun is not so misplaced as in my pic…).
• The position of a star in the main sequence depends only on one factor: the mass.
• Masses of stars of the main sequence are known to range from 60 times the mass of the Sun (top-left) to 0.08 Sun masses (bottom-right).
• The average densities of stars:
– Sun ~1.41 g/cm3
– Sirius ~0.79 g/cm3 (thus stars of the main sequence in the top-left of the HRD are a bit less dense)
– Betelgeuse (a red supergiant) ~10e-8 g/cm3 (extremely low density!)

Two-dimensional Morgan-Keeman system of spectral classification (M-K): based in the differences in strength of spectral lines of giant and main sequence stars of the same spectral type (i.e. similar temperature but different luminosity).

In general, for the same stellar type, more luminous stars give narrower spectral lines. This is related to the atmospheric density of the star.

This classification is also indicated in the HRD above. It consists on classes defined by the Roman numerals I and V:

• The numeral “I” is used for giants and supergiants.
• The numeral “V” denotes a main sequence star.

Thus, the classification:

• I = supergiants, with two subgroups; Ia (really gigantic and very luminous) and Ib (still gigantic but not so much).
• II = bright giants
• III = normal giants
• IV = subgiants
• V = main sequence
• VI, sd = subdwarfs (i.e. smaller than dwarfs, not indicated in the HRD, but they’re right under the V stars toward the bottom-right tail of the main sequence)
• D = white dwarfs.

Spectroscopic parallax: From the information of the spectral lines and the HRD we can calculate how distant a star is by the method of spectroscopic parallax. It contains the word “parallax”, but it has absolutely nothing to do with parallax! That’s how it works:

1. Look at the spectrum, and identify the star according to the M-K classification.
2. Find the approximate position of the star in the HRD diagram.
3. Read from the vertical axis.
4. Measure the apparent magnitude m.
5. Use the formula:

The accuracy of d calculated this way is of a factor of 1.6 (101/5), since the accuracy of our M is of about +/-1.

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