## Hertzsprung Russell diagram

The Herzsprung Russell diagram plots temperature vs. luminosity for stars. Assuming a star lies on the main sequence, the H-R diagram can be used to calculate it distance, using the spectroscopic parallax method. Image source

In 1911,  a Danish astronomer named Ejnar Hertzsprung plotted the absolute magnitude against color for a collection of stars and noticed that there was a strong correlation between these two variables. Independently, an American astronomer named Henry Russell produced a similar plot. Both men were recognized for this discovery, so we call plots like this Hertzsprung-Russell plots, or HR diagrams. Since absolute magnitude can be represented as luminosity and the color of a star can be represented as temperature, we often see these graphs plotted as luminosity vs. temperature. Image source atlasoftheuniverse.com/hr.html

This HR diagram includes data for 23,000 stars from the Hipparcos catalog and the Gliese catalog. As you can see, the vast majority of stars lie along a diagonal strip. Think about what this means.

Suppose you made a plot of people's heights for a large, representative population. You would notice that most people were between 5 and 6 feet tall, with a smaller number of small humans, maybe 3 feet tall. One way to interpret this data would be to guess that humans spent most of their lives at 5-6 feet tall.

This is similar to what the HR diagram tells us. Out of a huge population of stars, most stars lie along the diagonal. We call this collection of stars the "Main Sequence." Most stars spend most of their lives on this line, so it constitutes most of the population. The smaller populations, the upper right giants and lower left dwarfs are stars that are in their final stages of existence.

Animation source

This narrated animation depicts how stars in a globular cluster could be sorted out for color and size to construct a Hertzsrpung Russell diagram of this stellar population.

The diagram tells us more than just the relationship between luminosity and temperature. We can use information to draw other conclusions.

Since the main sequence is diagonal, we can make assumptions about the relationship between temperature and luminosity that tell us more about stars in general. Stars high up on the graph are very luminous. Very luminous main sequence stars also have very high surface temperature. Low luminosity stars have low surface temperature. What does this tell us? Recall that luminosity is proportional to temperature to the fourth power. This means that if two stars have the same temperature, they output the same power per square meter of surface area. If two stars have the same temperature but one is more luminous than the other, it must have a bigger surface area. It must be a bigger star. Few stars are close enough for us to be able to directly measure their sizes. We typically employ an indirect method using the HR diagram to estimate the radius of a star. We can estimate the temperature of a star by using its peak wavelength or spectral class, and then if we assume that it lies on the main sequence, we can guess its luminosity. This allows us to calculate the radius of the star.

## Spectroscopic parallax

We can use the HR diagram to calculate the distance to a star. We begin by finding the temperature, either by using Wien's law analyzing the star's spectrum to find its spectral classification. We then assume the star lies along the main sequence, and estimate its luminosity based on the star's temperature. We measure the star's brightness and use the brightness - luminosity relationship to calculate the distance to the star.

• Find Temperature
• Use Wien’s law
• Max wavelength of blackbody spectrum
• Or spectral type – OBAFGKM
• Assume main sequence
• Use HR diagram to get luminosity L
• Measure apparent brightness B
• Use L and B to find distance r to the star Let's solve a problem using the spectroscopic parallax method.

Suppose the surface temperature of star A is 10,000 K, and the surface temperature of star B is 5000 K. Star A looks as bright as star B. How far away is star A, compared to star B?

Since we know the temperatures, we can use the HR diagram to estimate the luminosities of the stars, assuming they are main sequence stars. We see that 5000 K corresponds to a luminosity of about 1. In these units, that means star B is about as luminous as the sun. Star A, with a temperature of 10,000 K corresponds to a luminosity about 100 times that of the sun. Translating this into math gives us: Also, they have the same apparent brightness. Now we can use the relationship between brightness and luminosity to substitute: Working through the algebra gives us: We find that star A is ten times as far away as star B. © 2005 Pearson Prentice Hall, Inc

The spectroscopic parallax method of finding the distance to a star extends our cosmic distance scale out to a range of about 10,000 parsecs. The Milky Way galaxy is about 30,000 parsecs across, so this means we can estimate the distances to a considerable portion of the stars in our galaxy.

## Stellar masses

Once a star has begun to undergo fusion in its core, it fuses hydrogen into helium for most of its lifetime. The mass and composition of a star determine where it lies along the main sequence, and it stays in that location on the HR diagram until hydrogen fusion in the ceases and it begins to fuse helium. It is important to be able to calculate the mass of a star independently to check this theory.

Stars are born in great clusters. The majority of stars are not singlet stars, like ours; most stars are binary stars. They are paired to another star gravitationally. Binary stars orbit a common center of mass. Some stars are close enough for us to see both of the partners. We call them visual binaries. Binary star systems that are farther away can be monitored by analyzing their Doppler shifts. We call these stars spectroscopic binaries.  © 2005 Pearson Prentice Hall, Inc

We can tell the spectra of the two stars apart, and analyze them separately. The spectrum of an orbiting star is  be blueshifted as it approaches and redshifted as it moves away. This method of calculating mass works for stars near enough to measure semi-major axes. For example, Doppler shift shows that Sirius A moves at half the speed of Sirius B. Using Kepler's laws and Newton's laws, we can calculate that Sirius A must be twice as massive as Sirius B.

## Eclipsing binary star systems © 2005 Pearson Prentice Hall, Inc

If the orbits of the binary stars happen to lie edge-on to us, the stars will completely or partially eclipse each other as one passes between us and its partner. We can make out the dip in the light when this happens, to gain information about how fast the stars are moving and the sizes of their orbits, and use this to calculate the masses of the stars.

Luminosity vs. mass © 2005 Pearson Prentice Hall, Inc

Combining the information gained from all of these methods, we are able to get an understanding of how the radius is related to the mass, as well as how the luminosity is related to the mass. It is not surprising that an increase of stellar mass correlates with an increase in radius as well as luminosity.  © 2005 Pearson Prentice Hall, Inc

We will go into much more detail about the evolution and final stages of stars in future chapters. For now, we will just note that the lifetime of a star is inversely proportional to the cube of its mass. A massive star is much more luminous than a low mass star. It burns hotter and faster, and uses up its fuel much more rapidly than a small low-mass star. © 2005 Pearson Prentice Hall, Inc

This table provides a good summary of stellar characteristics and how they directly measured or indirectly calculated using other measurements.                                    