The Hertzsprung-Russel Diagram

In a recent post (followed by this update) I wrote about how to obtain the radii of stars in the HYG database, and how to visualize the distribution with a simple histogram. Of course, with a large data set such as this there is much more one can do; most importantly, one would want to look for an explanation underlying the particular distribution of star sizes revealed by the histogram.

One the most common ways to visualize the properties of star populations is using the Hertzsprung-Russel diagram (HR-diagram). In this post I present an HR diagram I constructed using the data found in HYG. The HR diagram is shown here:

The lower axis is the color index, as explained in a previous post, and the logarithmic left axis shows luminosity in units of solar luminosity. The right axis gives the bolometric magnitude $M$, which is calculated from the luminosity as \begin{align} M &= 4.74 - 2.5\cdot \log\left(\frac{L}{L_{\odot}}\right). \end{align} where the number 4.74 is the bolometric magnitude of the Sun. The bolometric magnitude, like the luminosity, is a measure of the total brightness of a star. Note that a lower bolometric magnitude corresponds to a greater luminosity, i.e., a brighter star. Since $M$ is a logarithmic function of $L$, the logarithmic $L$-scale corresponds to a linear $M$-scale.

The upper axis in the diagram gives the temperature, calculated with Ballestreros' formula as shown earlier.

Each star is represented by a dot in the diagram, placed according to its color index and its luminosity, and colored based on the color index (I'll discuss the conversion from color index/temperature to rgb color in a later post). One notes how cooler stars are redder, whereas hotter stars are more blue, in agreement with Wien's displacement law, which relates the peak wavelength of a black body spectrum to the temperature: \begin{align} \lambda_{\text{max}} &= \frac{2,898\cdot 10^{-3}\,\mathrm{m}\cdot\mathrm{K}}{T}\,. \end{align}

Lines corresponding to constant radii are traced on the plot.

One notes a number of vertical streaks of points in the diagram; these are likely artefacts in the data, in which stars with undetermined color indices have been assigned specific values. This does not seem to overly disturb the overall pattern, however.

Looking at the plot one sees that the stars are organised into distinct groups, or populations. Below is the HR diagram with the most important of these populations named:

The position of the Sun is marked by the dashed white lines. The Sun's position in the diagram shows that it has a color index of $B - V = 0.676$, luminosity $L = L_{\odot}$ (by definition), a bolometric magnitude $M = 4.74$, and a temperature around $T = 5700\, \mathrm{K}$. Stars located below the Sun in the diagram are fainter, while those above it are brighter. Stars found to the right of the Sun are cooler and redder, and those to the left are hotter and more blue.

The Sun is part of the main sequence, extending along the diagonal of the diagram. The population of subgiants bridge the main sequence to the bulge in the population of giant stars, which are reddish stars with $10\, R_{\odot} < R < 10^2\, R_{\odot}$. Near the bottom of the diagram are the white dwarf stars with $R \sim 10^{-2} \, R_{\odot}$.

The populations revealed the HR diagram can be compared with the histogram from the previous post:

The origins of the peaks identified in the distribution are now clear: the largest peak around $2\, R_{\odot}$ are stars in the white to bluish part of the main sequence, and the peak around $10\, R_{\odot}$ corresponds to the population of giant stars. A very small peak can be made out around $10^{-2} \, R_{\odot}$, corresponding to white dwarfs.

In a later post I will attempt an explanation of the populations revealed in the HR diagram.