Making an HR Diagram

The only things which can be observed for any given star are

• Its apparent flux (measured in ergs cm^-2 sec ^-1)
• Its spectral energy distribution or, more crudely, its color.

From such observations an HR diagram is made:

Which is a plot of the luminosity of a star (y-axis) versus its temperature or color. This diagram is what the entire theory of stellar evolution rests on. The x-axis is reversed in the sense that higher temperaturs, (hotter stars) are on the left and cooler stars are on the right. The Y-axis is luminosity in units of solar luminosities

However, to get the luminosity of the star, one must determine the distance to that star using the method of stellar parallax.

Now when we are making a plot of surface temperature vs energy output (stellar luminosity) we are plotting the two external attributes of stars that we get from observation. We didn't have to go to the star to get this information.

• Stellar temperature comes from measuring the B-V color of the star under the assumption that the star is a blackbody radiator (see more below)

• Stellar luminosity comes from deriving the distance to the star (i.e. from stellar parallax - see below) in combination with the apparent flux of energy received on our digital detector.

Deriving stellar distances using the method of stellar parallax.

Figure 1.3 Schematic Representation of stellar Parallax. Distant stars act as a fixed reference coordinate system. Nearby stars, when observed 6 months apart, will show a small movement with respect to the background of fixed stars. At position 1, the nearby star would be viewed against a background that contained star B while 6 months later, at position 2, the nearby star would be viewed against a background that contained star A.

The angle which we measure with respect to the baseline of the earth's orbit about the sun is called the parallactic or parallax angle.

This angle would have a size of 1 arc second (1/3600 of a degree) for a star that had a distance of 1 parsec from the earth. 1 parsec is equal to 3.26 light years.

The nearest star to us has a distance of 4.1 light years so that all parallactic angles are less than 1 arc second for all stars.

There are three main observational difficulties associated with the accurate determination of stellar parallax. The last one is the most important:

• The distance between the earth provides a relatively small baseline. If we lived on Jupiter, we would have a much easier time measuring stellar parallax.

• Our Solar System is located near the outskirts of our galaxy and the local density of stars is therefore lower. If we lived more near the center of our galaxy, we would have plenty of stars within a distance of 1 parsec.

• Atmospheric motions/smearing make positional measurements of stars, at levels of accuracy less than 1 arc second difficult for individual measurements. Therefore, many measurements of the star are needed to record an accurate parallax. In practice, one usually requires 20 years of measurements of a single star.

If we measure the parallactic angle, then we can directly know the distance to the star. The distance in parsecs is simply

1/p

where p is the angle measured in arcseconds. Thus a star that has p = 0.1 would have distance of 1/p = 10 parseconds = 32.6 light years.

Now let's consider the following scenarios:

Simulation of stellar parallax with no error. This star has a distance of 2 parsecs so its parallax angle would be 0.5 arcseconds.

Noisy measurments: error per measurement is comparable to the actual parallax

Accurate measurments error per measurement s 10 times smaller than the actual parallax (only possible for very nearby stars)

COLOR AND STELLAR TEMPERATURE

The pattern of radiation given off by a perfect radiator is called Blackbody radiation (because the radiation pattern is independent of what the material is made of). This pattern is characterized by a particular function, called the Planck function, which mathematically is expressed as:

The key parameter in this equation is the temperature. Each temperature corresponds to a unique spectrum of emission.

To a high degree of approximation, stars are blackbody radiatiors and hence we can use this ideal to describe the pattern of radiation given off. This pattern of radiation is called the Planck curve

Here are two examples:

Notice in the above sample, the curve with a temperature of 7500 degrees emits the most radiation (e.g. peaks) in the blue portion of the spectrum. Therefore, that object would appear to be blue.

In the example below, the curve with a temprature of 3000 degrees emits very little light in the blue and therefore would appear to be very red.

Examination of these curves shows a fundamental experimental result. As you go to cooler temperatures, the wavelength at which the maximum amount of energy is emitted shifts to longer wavelengths.

In more quantitative terms, we have this relation:

But remember, the optical part of the spectrum

is a very small part of the total spectrum of radiation given off by objects in the Universe:

Furthermore, our atmosphere blocks up much of this spectrum and therefore to observe the total EM spectrum from celestial sources requires Satellites and/or telescopes in space.

In practice, astronomers use various filters and form filter flux ratios to determine the color. This is coded as the B-V color index, which is what we will use as an indicator of temperature.

In the applet below you should check the box labelled draw limits of integration. This will superpose the filter system on the black body curve. As you change the temperature, watch how the B-V index changes in the readout below the graph.

Wavelength in Angstroms
I N T E N S I T Y

In addition to the wavelength dependence on temperature, there is also a strong dependence of the total energy emitted. Let's see if we can ferret this out using this applet: