Spectra are plots of the dispersed starlight over all the wavelengths for which there are significant radiant flux. A star’s spectrum is analyzed to determine the physical properties of the star. On a spectre graph, with wavelength along the $x$-axis and flux along the $y$-axis, there are two series of troughs in the curves. The first is called the Balmer break and the second is the Paschen. These series correspond to the energy levels in a hydrogen atom.

The particular wavelengths push the electron to the next energy level, so it is absorbed in the atom. The Paschen is smaller because it is started at the third energy level as opposed to the Balmer, which starts at the second level and jumps up from there. On a blackbody radiation graph of wavelength and flux, a smooth curve means there is no absorption of light caused by hydrogen in the atmosphere of the star. Usually these are observed on a flattened spectrum, as shown below with flattened and the original unflattened versions.

As shown in the sliders, our chosen effective temperature is $9000 K$ and our $\log g$, where $g$ is gravity, is 7. Spectrum are, once found, manipulated in this way to reveal the atomic makeup of the star, or perhaps the chemical makeup of nebulous dust that the starlight is shining through. Spectra of white dwarfs are valuable to many aspects of research.

The big assumption of our research project is that due to convection movement along the surface of white dwarfs, not all of the stars are the same temperature over their surface. Usually astronomers assume stars are one temperature and then use the Koester Grid, which takes $\log g$ and a single temperature to give a spectre graph for that star. However, stars are not a single temperature all over their surface, so there must be some error when constructing their spectra with assuming one temperature, usually called the effective temperature. The overall goal was to find the difference between the spectra produced from a single temperature reading and our model of convection: Gaussian distribution.

We are assuming that due to convection, there are patches across the surface of a star that are warmer and cooler by a significant amount of degrees. These different patches are not random, but in such a way that more of the star are the average temperatures out of the range, while less of the star is at either of the extreme temperatures.

In order to account for this, we created a weighted average, because although we do not exactly know how convection will affect the spread of the temperature across the surface of the star, we are assuming it creates a Gaussian distribution, with the mean temperature being the most common temperature on the surface. This means that the temperature of the surface of the star at any one time is determined by the mean temperature, or the temperature in the center of the distribution, and the spread of the temperature, or the standard deviation of the Gaussian distribution.

Last semester, the researchers working on this project did not use a normal, or Gaussian distribution, to calculate the error, but used a flat distribution across a range of temperatures, with all the temperatures having equal weight. This was a good start to the calculations, but essentially not a realistic model. Our first step was to use $T_c$, the center temperature, and $T_{\text{sig}}$, the standard deviation, along with $\log g$, to create a Gaussian distribution, or a normal distribution. Ideally, the number of spectra used would be along a continuous range, however in order to work with the Koester Grid, which is discrete, a discrete number would be chosen.

In the graph above, there are 100 grey spectra that are combined in a weighted average along a Gaussian distribution to create the blue spectrum.

This next graph has a close up look with only 7 spectrum, in multiple colors, chosen along a range to combine together along the same distribution to create the heavy black spectrum.

The center temperature and $\log g$ for both of these was chosen to be the same as the first graph, where $Tc = 9000 K$ and $\log g = 7$.

Our next function would create a file to begin creating the spectra that would be comparable to a conventionally created spectra graph. We would assume the center temperature, $T_c$, and vary the standard deviation, $T_{\text{sig}}$, in order to find a multitude of spectra and compare them. On the $y$-axis is $T_{\text{fit}}/T_c$; $T_{\text{fit}}$ would be the temperature as conventionally calculated, and $Tc$ would be the center temperature of the gaussian distribution we assumed. Along the $x$-axis is the $T_{\text{sig}}$, the standard deviation of the temperature.

If there were no error in the conventional methods of selecting temperature, the plot would be a straight horizontal line, as $T_{\text{fit}}/T_c$ would be 1. However, what we see is a decreasing curve, suggesting that there are deviations, and that they increase as $T_{\text{sig}}$ increases. So, as the temperatures get more spread out, i.e. a larger range, our ability to create an accurate spectrum for that white dwarf and so to predict the $\log g$, decreases. This would have been our next step, however, our group did not have enough time to complete this.

Had our group finished completely, we would have taken data from stars and used multiple temperatures to find the difference in log g between assuming one temperature instead of a Gaussian spread of temperatures. However, this normal distribution model of convection is not necessarily the most realistic model of convection on a white dwarf’s surface. There could be other models that would work even better, just as a Gaussian distribution was more realistic than a flat average. This new manipulation of spectra and its readings are an interesting addition to the already recognized way of finding spectrum for white dwarfs.

## Works Cited

Koester, D. *White dwarf spectra and atmosphere models. *Memorie della Societa Astronomica Italiana, v.81, p.921-931 (2010). Web. 3 May 2017.

M., Van Horn H. *Unlocking the secrets of white dwarf stars*. Cham: Springer, 2015. Print. 3 May 2017.