Exercise 1.2: Introduction to the H-R Diagram

The Hertzsprung-Russell (H-R) Diagram, pioneered independently by Elnar Hertzsprung and Henry Norris Russell, plots Luminosity as a function of Temperature for stars. In general, it is a graph that looks like this:

In an H-R diagram, each star is represented by a dot. One uses data from lots of stars, so there are lots of dots. The position of each dot on the diagram corresponds to the star's luminosity and its temperature, where the vertical position represents the star's luminosity and the horizontal position represents the star's surface temperature.

In this plot, you can see that the Sun is roughly an average star, lying between the hot and cool stars horizontally and between the dim and bright stars vertically.

So, what kind of trends can we see in a H-R Diagram? When data on the nearest stars to us, or stars in a cluster, are plotted in a H-R diagram, here is what one sees. It is readily apparent that the H-R Diagram is not uniformly populated, but that stars preferentially fall into certain regions of the diagram. The majority of stars fall along a curving diagonal line called the Main Sequence (the point where the stars begin burning hydrogen in their centers), but there are other regions where many stars also fall. These stars are very interesting, but for now, let's concentrate on the main sequence.

The question is, why is there a main sequence? To answer this, we need to look at some physics concepts.

A star's luminosity is the total amount of power (energy per second) it radiates into space and is an intrinsic quantity of the star. A star of a given luminosity will appear to get brighter and brighter as you move closer to it, just as a car's headlights get brighter as it comes closer to you. When we are looking at the stars in the sky we cannot measure this luminosity directly since they are at very different distances. For example, Alpha Centauri and the Sun are about the same luminosity, but certainly the Sun looks brighter. What we can measure is how bright the star appears to be and define this as the amount of light reaching us per unit area, and call it the apparent brightness.

The apparent brightness of any light source obeys an inverse square law with distance. If we viewed the Sun from twice the Earth's distance, it would appear dimmer by a factor of 2² = 4. If we viewed it from 10 times the Earth's distance it would appear 10² = 100 times dimmer. The figure below shows this relationship graphically. Notice that, at a distance of 1 unit, the luminosity covering 1 square area, covers 4 times as much at a distance of 2 units. Similarly, 9 times as much area is covered at a distance of 3 units.

The inverse square law leads to a very simple and important luminosity-distance formula relating the apparent brightness, luminosity, and distance of any light source:

We need another equation, the Stefan-Boltzmann law, to help us measure specific properties of stars. It describes the total amount of energy radiated in each second from any hot surface. The total energy radiated per second by a star is called its luminosity, L.

We know that every warm body radiates. The higher the temperature, the bluer the radiation. About a century ago the Austrian physicists Josef Stefan and Ludwig Boltzmann discovered another characteristic: The higher the temperature, the more energy is radiated each second. Although the subjects of their study were radiating objects in the laboratory, the law applies to stars and all other bodies in the universe. The law gives the total energy, L, radiated per second from a body with temperature T and surface area A. For bodies that radiate efficiently, like stars, L is proportional to the area and to T⁴.

This proportionality shows that if the temperature, or area, of a star increases, the total energy radiated every second increases. Suppose the star has some radius R. Then, its area is A = 4 pi R², and its luminosity is given by:

This variation of the Stefan-Boltzmann law allows us to calculate a star's radius if its luminosity and temperature are known.

This suggests an interesting possibility: maybe all stars are the size of the Sun, but the hotter ones are more luminous just because they are hotter. We can test this with the H-R diagram:

Our hypothesis will be that all main sequence stars fall on the blue line. Here is the comparison:

Our idea was wrong. But we learned something: the coolest main sequence stars are a lot smaller than the Sun and the hottest main sequence stars are a lot bigger than the Sun.

The easiest way to describe apparent brightness is in Watts per square meter, and luminosities are easily described in either Watts or Solar luminosities. However, many amateur and professional astronomers describe stellar brightness using the ancient magnitude scale system devised by the Greek astronomer Hipparchus (c. 190-120 B.C.). The magnitude scale originally classified stars according to how bright they look to our eyes. The brightest stars received the designation "first magnitude", the next brightest "second magnitude", and so on. We call these descriptions apparent magnitudes because they compare how bright different stars appear in the sky.

Absolute magnitude, M, expresses the brightness of a star as it would be ranked on the magnitude scale if it was placed 10 parsecs (32.6 lightyears) from the Earth. Since all stars would be placed at the same distance, absolute magnitudes show differences in actual luminosities.

As this system evolved, some stars were found to be brighter than magnitude 1; for example, Vega is magnitude 0, and Sirius is magnitude -1.4. The first peculiarity to note about the magnitude scale is that the larger negative magnitude a star (or any celestial object) has, the brighter it is; but the larger positive magnitude, the fainter the star.

This is key to note: a brighter star has a more NEGATIVE magnitude value and a dimmer star has a more POSITIVE one.

How the human eye judges light affects the magnitude scale. Suppose you tried to estimate the relative brightness of a 100 Watt and a 200 Watt light bulb. The second bulb does not appear twice as bright as the first. The eye does not sense light in the simple way you might expect; if an object has twice the luminosity, it does not appear twice as bright.

Stars of first magnitude are 100 times brighter than stars of sixth magnitude; so a difference of 5 magnitudes corresponds to a brightness ratio of 100. A difference of 1 magnitude then amounts to a brightness ratio of 2.5118864.. which we will round off to 2.512. This strange number pops up because 2.512 x 2.512 x 2.512 x 2.512 x 2.512 = 2.512⁵ = 100, another way of stating that a difference of five magnitudes equals a ratio of 100 in brightness.

The following table will help keep the magnitude differences and brightness ratios straight.

A Magnitude Difference of:	Equals a Brightness Ratio of:
0.0	1.0
0.2	1.2
1.0	2.5
1.5	4.0
2.0	6.3
2.5	10.0
4.0	40.0
5.0	100.0
7.5	1000.0
10.0	10,000.0

Both the absolute and apparent magnitudes are given on the same scale. For example the apparent magnitude of the Sun is about -26. The brightest star in the sky (other than the Sun), Sirius, has an apparent magnitude of -1.4. The difference in magnitude is about 25; for every five magnitudes the brightness ratio is 100. So Sirius is 10² x 10² x 10² x 10² x 10² = 10¹⁰ times less bright than the Sun in apparent magnitude. This result tells us nothing about the luminosities of the Sun and Sirius, only how bright they appear in the sky. In contrast, the absolute magnitude of the Sun is approximately +4.8, and Sirius is +1.4. The difference in absolute magnitudes is roughly 3.5, corresponding to a brightness ratio of approximately 25. This comparison of absolute magnitudes tells us that Sirius is roughly 25 times as luminous as the Sun. A similar comparison can be made to find the luminosities of other stars.

Using a little algebra, we can convert from magnitude differences to brightness ratios quickly by using the formula:

If we take the above equation and replace the apparent magnitudes with absolute magnitudes, the formula would apply to stellar luminosities:

Notice that on the HR Diagram stars have certain types, corresponding to certain surface temperatures that have certain characteristic luminosities. So, for example, an A star (see below) may have a surface temperature of about 10,000 K and a luminosity of about 80 times the luminosity of the Sun. You could also plot Absolute Magnitude instead of Luminosity. Then you would find that A stars typically have absolute magnitudes of about M=0. So, by knowing the star's type, you know how bright it would be at 10 parsecs. You can measure how bright it looks to you in the sky. Therefore, you should be able to figure out how far away it must be to make it look as faint as it does. Let's figure out a formula that will let us calculate the distance to a star if we know it's absolute and apparent magnitude.

Think about the quantities you want to relate: m, M, and d. What equations do you know that use magnitude and distance?

Now, what if Star1 and Star2 are the same luminosity but different distances? What is their ratio of brightness? Try putting that ratio into the formula for the difference in magnitudes. The final step is to remember that m = M if d=10parsecs. Have fun!

The basic property of stars used for modern stellar classification is surface temperature. Measuring a star's surface temperature is somewhat easier than measuring its luminosity since the measurement is not affected by the star's distance. Instead, we determine surface temperature directly from the star's color or spectrum. A star's surface temperature determines the color of light it emits. A red star is cooler than a yellow star, which is cooler than a blue star. Think of an electric stove: when you turn on the burner it starts to glow a faint red. As it heats up, the red intensifies, and gets brighter. As it gets hotter still, the burner will start to glow orange to yellow.

Another way to determine a star's surface temperature is to look at its spectrum. You have seen a spectrum before if you have ever seen a rainbow. The light passing through the raindrops is being scattered so that you can see the visible light spectrum ranging from red to violet. A star's spectrum depends on what kind of elements are in the atmosphere surrounding the star. These missing colors are called spectral lines and can be seen in the seven colored strips below. Stars displaying spectral lines of elements that are hard to ionize must be very hot, while stars that only have lines of elements that are easy to ionize are cooler. Astronomers thus classify stars according to surface temperature by assigning a Spectral Class determined from the types of spectral lines seen in a star's spectrum.

The first widely used classification system was devised by astronomers at Harvard during the 1890s and 1900s. The spectra were first classified in groups labeled A through Q, but some groups were later dropped, merged with others, or reordered. The final classification includes the seven spectral classes, or types, still used today: O, B, A, F, G, K, M.

This sequence of spectral types is important because it is a temperature sequence. The O stars are the hottest, and the temperature continues to decrease down to the M stars, the coolest of all.

The temperature can be converted to the Spectral Class, for the Main Sequence, by the following table which also shows what color the star appears to be in the sky:

Temperature (Kelvin)	Spectral Class	Color of Star
35,000	O	blue-white
21,000	B	blue-white
10,000	A	white
7,000	F	yellow-white
6,000	G	yellow
4,000	K	orange
3,000	M	red

The hottest stars appear blue (top spectrum), the coolest red (bottom). One can arrange stellar spectra (shown below) by their strength and other characteristics of their lines which gives rise to the stellar spectrum classification scheme.

O
B
A
F
G
K
M

So, when you see an H-R diagram sometimes the axis labels are different. The vertical position represents the star's luminosity which could be the luminosity in Watts, but more commonly it is in units of the Sun's luminosity (as seen in the previous H-R diagrams). The magnitude scale system is also commonly used. The horizontal position represents the star's surface temperature usually labeled by the temperature in Kelvins. It is traditional to have the highest Temperatures go to the left. Instead of temperature sometimes the star's spectral class (OBAFGKM) is used, as seen below, with each letter in the sequence being subdivided into a numbered sequence (0, 1, 2, ... 9).

Click here for an H-R diagram simulation that will draw out a Main Sequence and let you evolve it in time to see how the stars will change over their lifetimes. See if you can find the star most like the Sun and find out how long it will live on the Main Sequence. The Main Sequence, as defined above, is the portion on the H-R diagram where a star is burning hydrogen in its center and making helium. At time goes on (as the star evolves) the amount of hydrogen will dwindle as the amount of helium piles up. As this process continues the star will become dimmer, and therefore move across the H-R diagram towards the right (as seen in the simulation).

References
Davison E. Soper, University of Oregon
The Cosmic Perspective, Bennett, 1999