Star size distribution

Note: The analysis in this post has some errors which I've attempted to rectify here.

How does the Sun compare to other stars? There are several parameters useful for characterizing stars, including temperature, color, and luminosity, but perhaps the most immediately relatable parameter is physical size. When a friend asked me about this, and after several minutes of googeling failed to provide a clear answer, I decided to do some digging, the results of which I present here.

My starting point for this project was the HYG database, a compilation of data from several star catalogs describing about 120.000 stars, curated by David Nash. I downloaded the HYG 3.0 CSV and loaded it into Python using Pandas, the Python Data Analysis Library. The HYG does not contain data on stellar radii however, so a few assumptions and calculations were needed.

Stars emit energy in the form of electromagnetic radiation,and particles such as neutrinos, protons and electrons. The first simplifying assumption is that the electromagnetic output is that of a black body. This isn't really true, since stellar spectra exhibit emission and absorption lines, but it'll do as a starting approximation.

The emittance of a black body follows the Stefan-Boltzmann law, \begin{align} j^* &= \sigma \cdot T^4\,, \end{align} where the emittance $j^*$ is the power radiated per unit surface area, $\sigma \approx 5.67\cdot 10^{-8}\,\frac{\mathrm{W}}{\mathrm{m}^2\mathrm{K^4}}$ is the Stefan-Boltzmann constant, and $T$ is the temperature measured in kelvin.

Knowing the emmitance, the total power output $L$, also known as the star's luminosity, is \begin{align} L &= A\cdot j^*\,, \end{align} where $A$ is the surface area of the body. Assuming that stars are spherical, the surface area of a star with radius $R$ is $A=4\pi R^2$.

Combining this we get a relation between a star's luminosity, radius, and temperature: \begin{align} L &= 4 \pi \sigma R^2T^4\,. \end{align}

For the Sun we know both the luminosity $L_{\odot}$, the radius $R_{\odot}$ and the surface temperature $T_{\odot}$, and these should of course satisfy the above relation. We recast it using the Sun's parameters as \begin{align} \frac{L}{L_{\odot}} &= \left(\frac{R}{R_{\odot}}\right)^2 \left(\frac{T}{T_{\odot}}\right)^4\,. \end{align}

Isolating $\frac{R}{R_{\odot}}$ gives \begin{align} \frac{R}{R_{\odot}} &= \sqrt{\frac{L}{L_{\odot}}}\cdot \left(\frac{T_{\odot}}{T}\right)^{2}\,. \end{align}

The HYG database contains the luminosity of stars, but not their temperature. What it does have, however, is their color index.

The color index of a star is a number describing its color; the lower the color index, the more blue the star, whereas a higher color index means a redder star. Since the color of a star is determined by its temperature, the temperature can be calculated from the color index.

The color index used is called $B - V$, and the temperature calculation is done using Ballesteros' formula (pdf link): \begin{align} T(B-V) &= 4600\cdot\left(\frac{1}{0.92\cdot(B-V)+1.7} + \frac{1}{0.92\cdot(B-V) + 0.62}\right)\,. \end{align}

Using this obtaining the radius of each star is straightforward: calculate the star's surface temperature from the color index using Ballesteros' formula, and use this temperature and the luminosity to determine the radius using the previous formula.

After performing this calculation on the stars in the database, the radii are plotted in a histogram using a logarithmic primary axis:

The Sun's position in the distribution is shown by the yellow vertical line, and the median, the geometric mean and the arithmetic mean are shown with red, green and black lines, respectively.

The plot shows two prominent peaks around $\sim 2\,R_{\odot}$ and $\sim 10\, R_{\odot}$, and two smaller peaks around $\sim 500 \, R_{\odot}$ and $\sim 3000 \, R_{\odot}$. The peaks represent different star populations, such as main sequence stars and giant stars. It is immediately clear from the plot that most stars are larger than the Sun.

The median radius $R_M = 3.2\, R_{\odot}$ is the middle-most value, meaning that there are equal numbers of stars in the dataset with $R < R_{M}$ and $R > R_{M}$, respectively.

The arithmetic mean $R_A = 203.7 \, R_{\odot}$ is the classic average value. However, since some stars are orders of magnitude larger than the majority of stars the arithmetic mean is not a very good descriptor of the distribution. The geometric mean $R_G = 5.9\, R_{\odot}$, corresponding to the average calculated using $\log(R)$ rather than $R$, is a better descriptor.

There is of course much to be said on why star sizes are distributed the way they are. I will attempt to cover some of this in a later post.