The most basic study of stellar spectra makes it clear that the hydrogen lines are sensitive indicators of the physical conditions in the atmospheres of stars. The Balmer lines, which arise from the n = 2 level of the hydrogen atom, play a key role in spectral classification. Modern classification work is best described as three dimensional, since the classifications may indicate not only the surface temperature and gravity of the star, but also in many cases, abundances.
To cite a simple example, a dead givaway of a subdwarf or metal-poor giant in a low dispersion spectrogram is the appearance of H and K of Ca II at equal strength, along with a relatively clean spectrum showing little other than the Balmer lines and perhaps a G-band (CH molecule).
In spite of the obvious sensitivity of the Balmer lines to the physical and chemical conditions in the atmospheres of stars, they are underutilized in modern spectrographic analyses. We have been attempting to understand why this is so, and to see what can be done to improve the situation.
It is a pleasure for me to acknowledge the collaboration of Dr. Fiorella Castelli, and advice and comments of many colleagues.
Hydrogen is unique among all atoms in that the levels with a given value of the principal quantum number n all have nearly the same energies, independent of the angular quantum number l. This means that the electric fields of nearby electrons or ions can split these close-lying levels, and induce transitions among them. The phenomena is known as the Stark Effect. The net result is that the hydrogen lines are subject to considerably greater broadening than most metallic lines. It is this fact, along with the high abundance of hydrogen in most stars, that gives the Balmer lines their overall strength and great breadth.
The theory that treats the broadening of the hydrogen lines is a very complex one. The electric fields at a radiating hydrogen atom depends not only on the positions of the nearby ions and electrons, but also on their velocities. Calculations of the shapes of hydrogen lines due to Stark broadening are made by specialists, most recently by C. Stehle and her colleagues at the Paris Observatory.
Other sources of broadening are relevant for the Balmer lines. In cooler stars, there are many more neutral atoms than ions. Radiating hydrogen atoms can bump into their neighbors, and this mechanism also broadens the lines. In giant and especially supergiant stars, bulk motions of the gas caused by convection or sound waves can influence the shape of spectral lines. These effects, are often collectively called "astronomical turbulence", and they can influence the cores of the Balmer lines.
In order to predict spectral lines from a star, it is necessary to make a mathematical model of its outer layers, called the photosphere. Various studies have shown that the profiles of the Balmer lines depend on certain assumptions about these models that are only approximately known. Because of this, there is uncertainty in our ability to calculate reliable models of the Balmer lines, and therefore some doubt about their usefulness as analytical tools.
We are attempting to clarify this situation.
If it is possible to make accurate calculations of the Balmer lines in stars, it should be possible to do it for the sun. This is where all studies have started, and ours is no exception. Because the sun is so bright, its spectrum can be obtained with exceptional spectral purity and very low noise levels. The spectra below were made at the Kitt Peak National Observatory. They are for the center of the disk.
The profiles below were all based on the Holweger-Muller empirical model of the solar atmosphere. The Stark profiles of Stehle were used, along with Lorentz broadening by neutral hydrogen, using calculations of Barklem, Piskunov, and O'Mara. No attempt was made to adjust the line broadening theory to fit the observations. We have adjusted our estimate of the level of the continuum for the higher Balmer members.
For the H-beta profile, we have multiplied the continuous opacity by a factor of 1.03 to account for the well-known missing solar ultraviolet opacity. The increase is calculated quantitatively by requiring the calculated specific intensity at disk center match observations tabulated by Heinz Neckel. We also lowered our initial estimate of the position of the continuum by 3%. These two effects, (1) changinging the estimate of the position of the observed continuum, and (2) scaling the continuous opacity in the model, are not independent of one another. For example, we initially scaled the continuous opacity for H-beta by a factor of 1.03 to bring the calculated, disk-center specific intensity into agreement with the observed specific intensity. However, after making the best fit of the calculated profile to the observations, we decided it best to modify our assessment of where the observed continuum was. We actually reduced it a little. This meant that we needed to modify our continuous opacity scaling in order to be consistent.
The "continuum" in the neighborhood of H-beta is a little strange. The thick solid line in the figure below shows high points within 10A bands of the Neckel-Kitt Peak solar spectrum for the disk center(Neckel, H., Solar Phys., 184, 421, 1999; KPN). Filled circles are points chosen to represent a smoothed overall continuum. These filled circles are connected by straight line segments which do not represent our judgment of the continuum in between them. For this purpose, we have adopted a four-point Lagrange interpolation formlua (results not yet shown). The dotted curve represents the continuum chosen by KPN. It is in excellent agreement with the choice of high points made here, with the exception of the region of the H-beta line.
By H-gamma, it is necessary to increase the continuous opacity by a factor of 1.05. A significant portion of the line absorption is molecular.
At H-delta, the contribution of the unknown opacity to the continuum is nearly 17%.
The near wings of H-delta, especially in the region from 4090 to 4098A are depressed by line opacity. This situation is not substantially rectified if we recalculate the region with an additional 698 atomic absorption lines. We still do not account for the depression of the violet wing, though the calculated red wing is in somewhat better agreement with the observations. The missing absorption in this region is probably not due to known molecular absorption, which was not included in the calculation. While there are a few features in the 4090-4098A region due to CN or CH, most of the unaccounted absorption may be due to unidentified lines. This region deserves closer study. In the figure below, the observations are again shown in red.
The calculation shown here is "raw" in the sense that the default oscillator strengths from VALD were combined with broadening parameters from the Michigan synthesis code. Typically, much better-looking fits can be obtained by considering individual lines, and making plausible adjustments to uncertain atomic parameters. However, there would still remain many features that are unidentified.
The figure below shows, in a purely theoretical calculation, the effect of scaling the continuous opacity by factors of 1.00 (no change), 1.20, and 2.00. As far as the hydrogen lines are concerned, scaling the continuous opacity simulates a lower temperature (for late-type stars). In hotter stars, the Balmer lines are sensitive to both temperature and surface gravity. Near spectral type A0, narrow Balmer lines generally means a star has a high luminosity.
Dr. Castelli and I have made a number of calculations to evaluate the accuracy of approximations implemented in a commonly used spectral systhesis routines. These routines make rough interpolations in a tabular grid of Stark profiles, and simplify the convolution of Stark and Lorentz broadening. There is no way (in one version of these codes) to include a possible microturbulence.
We have found that the approximations used give excellent agreement with profiles we have calculated using a full convolution of the relevant broadening profiles, and an improved interpolation in the grid of Stark profiles. We also find no difference in the calculated profiles when the older VCS Stark profiles are replaced by more refined calculations due to Stehle and her colleagues. There are surely differences in the intrinsic Stark profiles, but they seem to average out when integrated through the stellar photospheres.
The only example that we have found (so far) of a difference in the profiles using traditional codes and various improvements and updates is shown in the figure below. Some of the older codes have no provision to take (additional) gaussian broadening by microturbulence (Vt) into account. Such broadening is negligible so long as Vt is significantly lower than the sound speed. When Vt is near the sound speed, the cores of the lines broaden, as illustrated below. iii::
