Astronomy 162: Introduction to Stars, Galaxies, & the Universe Prof. Richard Pogge, MTWThF 9:30

Lecture 10: Synthesis:
The Hertzsprung-Russell Diagram

Key Ideas

Luminosity-Radius-Temperature Relation for stars.

The Hertzsprung-Russell (H-R) Diagram of stars

A plot of Stellar Luminosity vs. Effective Temperature

H-R Diagram Features:

Main Sequence (most stars)

Giant & Supergiant Branches

White Dwarfs

Luminosity Classes

Summary of Stellar Properties

• 10^-4 to 10⁶ L_sun

Large range of Stellar Radii:

• 10^-2 to 10³ R_sun

Modest range of Stellar Temperatures:

• 3000 to >50,000 K

Wide Range of Stellar Masses:

• 0.1 to ~50 M_sun

Luminosity-Radius-Temperature Relation

1. The "Effective Temperature" of the stellar photosphere, T_*
2. The total surface area of the star, which depends on its radius, R_*

Because stars are hot dense balls of gas, they obey the Stefan-Boltzmann Law that relates the amount of energy emitted per second from each square meter of its surface to the temperature of the star:

Stefan-Boltzmann Law:

The total surface area of a spherical star is:

Combining these two, the total Stellar Luminosity (energy emitted per second) is therefore:

This is the Luminosity-Radius-Temperature Relation for stars.

In words:

"The Luminosity of a star is proportional to its Effective Temperature to the 4^th power and its Radius squared."

Example 1:

Two stars are the same size, (R_A=R_B), but star A is 2x hotter than star B (T_A=2T_B):

Therefore: Star A is 2⁴ or 16x brighter than Star B.

In words: "If two stars are the same size, the hotter star is brighter."

Example 2:

Two stars have the same effective temperature, (T_A=T_B), but star A is 2x bigger than star B (R_A=2R_B):

Therefore, Star A is 2² or 4x brighter than Star B.

In words: "If two stars have the same effective temperature, the larger star is brighter."

Hertzsprung-Russell Diagram

• estimate the effective temperature, T, from its Spectral Type
• estimate the luminosity, L, from its Apparent Brightness & distance

Plot Temperature from hottest to coolest along the horizontal axis, and Luminosity from faintest to brightest along the vertical axis:

This diagram was first made independently in 1912 by Danish astronomer Eljnar Hertzsprung (who plotted the diagram for nearby clusters of stars), and American astronomer Henry Norris Russell (who plotted the diagram for nearby stars).

As you can see, stars don't land just anywhere on the H-R Diagram, but fall in one of a number of specific regions, as follows:

Main Sequence

Ranges of stellar properties:

• L=10^-2 to 10⁶ L_sun
• T=3000 to >50,0000 K
• R=0.1 to 10 R_sun

Giants & Supergiants

The Luminosity-Radius-Temperature relation tells us that the stars in these bands must therefore be larger in radius than Main Sequence stars.

There are two groups of giant stars:

Giants

Large but cool stars with a wide range of luminosities:

• R = 10 to 100 R_sun
• L = 10³ to 10⁵ L_sun

Supergiants

The very largest stars, arranged along the top of the H-R diagram with a wide range of effective temperatures but relatively narrow range of luminosities:

• R > 10³ R_sun
• L = 10⁵ to 10⁶ L_sun

White Dwarfs

The Luminosity-Radius-Temperature relation tells us that these stars must therefore be smaller in radius than Main Sequence stars.

How small? Using the Luminosity-Radius-Temperature relation, we can make a prediction:

• R ~ 0.01 R_sun

These stars are called White Dwarfs. "White" because they tend to be very hot ("white hot"), and "Dwarfs" because they are so tiny.

Luminosity Classification

Luminosity classification is based upon the widths of the absorption lines in the star's spectrum.

It works because absorption lines are Pressure-sensitive:

• Lines get broader as the pressure increases.
• Big stars are puffier, which means the pressure in their atmosphere is lower.

Implications:

• Larger stars have narrower absorption lines.
• Larger stars are brighter at the same temperature.

This gives us a way to assign a relative Luminosity to stars based upon their spectral line properties!

Locating stars on the H-R Diagram

The Sun: G2v (G2 Main-Sequence star)

Betelgeuse: M2Ib (M2 Supergiant star)

Rigel: B8Ia (B8 Bright Supergiant star)

Sirius: A1v (A1 Main-Sequence star)

Aldebaran: K5III (K5 Giant star)

The full spectral classification of a star tells us the approximate location of the star on the H-R Diagram:

• The spectral class gives the star's relative temperature, hence its horizontal location in the H-R Diagram.
• The luminosity class gives the star's relative luminosity at a given temperature, hence its vertical location in the H-R diagram.

Questions:

Why don't stars have just any Luminosity or Temperature?

Why is there such a distinct Main Sequence of stars? What makes one Main Sequence star different from another?

Were Giant, Supergiant, and White Dwarf stars born that way, or is something else going on?

The answers to these questions forms the basis of the next unit of this course, namely the internal structure and evolution of the stars.

The H-R diagram will prove to be one of our most powerful tools for unravelling the mystery of the stars.

Supplement:
The Hipparcos H-R Diagram of the Solar Neighborhood

The Hipparcos satellite has made the most precise measurements of stellar parallaxes to date for nearby stars. Using data for 4907 stars with distances measured to better than 5% accuracy, this H-R Diagram for the Solar Neighborhood has been made. Because the points for many stars will overlap each other in the plot, the Hipparcos team uses colors to show how many stars sit under a single point. Red means more than 10 stars at the place on the plot.

This remarkable H-R diagram has the following notable features:

• Most nearby stars (~85%) lie along the Main Sequence, as advertised. Why this is true is an important clue to the nature of stellar evolution.

• There are few if any Supergiants in the Solar Neighborhood. In later lectures we'll learn why Supergiant stars are so rare.

• There are also very few White Dwarfs on this diagram. This time, however, it is not a profound new result, but rather it is an artifact of how the data were collected! White dwarf stars are intrinsically very faint, but Hipparcos could only measure good quality parallaxes for brighter stars. This particular H-R diagram uses only the 4907 best parallaxes measured by Hipparcos, and so it excludes the many faint white dwarf stars with poorer-quality parallaxes; only a handful of white dwarfs are close enough, and hence bright enough, to yield 5% accuracy parallaxes. This is an example of what we call a "selection effect". We'll be meeting selection effects in various places during the quarter.