Measuring the Temperature of the Photosphere with the H.-Beta Absorption Line

Zachary G. Maas and Michael McLaughlin

University of Minnesota

Methods of Experimental Physics Spring 2013

Introduction

The measurement of the temperature of a star is a complicated process. Common methods are stellar classifi.cation, blackbody .fits, and relating the growths of multiple absorption lines to one another to determine the relative ionization rates of materials. The goal of this experiment was to measure stellar photospheric temperatures while only examining the total intensity removed by one absorption line, the H. line. The equivalent width measured from the absorption line was found and compared to other stars at known temperatures found by spectral type classification, and measured from previous research.

Theory

The H-beta. absorption line will occur due to hydrogen atoms in the second energy state making a transition to the n=4 state when absorbing a photon emitted from a n=4 to n=2 transition, and occurs at 486.1 nm. The strength of an absorbing line is measured by a parameter called equivalent width. Shown in the equation below; W is the equivalent width measured in Angstroms, S is the source continuum intensity, and A is the intensity of the absorption line.

Equivalent width is directly dependent on the number of absorbing atoms. The Boltzmann-Saha equation predicts the relative population of n=2 H atoms to the total number of N atoms. The equation is shown below. A plot of the equation is shown below the equation.

At low temperatures nearly all of the hydrogen atoms are in the ground state as shown in the figure above. If all atoms are in the ground state then there would be no Balmer lines since no atoms would be energetic enough to be in the n=2 state and absorb the H.-beta photon. At higher temperatures all the atoms are ionized so no bound transitions are possible and there will be no H-beta. absorption line in the spectrum. Therefore, the equivalent width measurement for a star must be directly proportional to the number of available absorbing atoms in the n=2 state.

Apparatus

The viewing end of the apparatus consisted of an optical cage containing a neutral density filter. This cage was attached to a motorized telescope. The optical cage was attached to a .ber optic cable which ran to an Ocean Optic PC2000-ISA spectrometer. The spectrometer used a di.raction grating blazed at 400nm and contained 600 grooves.

Results

Below is a diagram of our measured solar spectra before and after calibration, compared to a blackbody at 5700K.

The H-Beta line was isolated from the figure above and its width averaged over multiple data runs. The final results was 2.5 Angstroms with a statistical uncertainty of .1 Angstroms. Our data compared to Gaussian fit curve for other empirical data is shown below.

This .fit recorded a temperature value of 5300K as the temperature of the sun. The uncertainty for this measurement was recorded using the 95% con.fidence bounds for the Gaussian .fit, represented by the dashed lines in figure 4 and 5. Extrapolating the equivalent width value of 2.5A to each con.fidence bound measured a range of 4200K to 6100K. Therefore the our measured temperature of the sun's photosphere is 5300K(-1100K,+800K).

Discussion

The quoted results of 5300K(1100K; +800K) do not match the given value of the sun's temperature of 5780K. Likewise there is a deviation for the equivalent width value. The given equivalent width of the sun is 2.79A compared to the measured value of 2.5. +/-1 Angstrom . The quoted uncertainty is due to the uncertainty on the fitted model. To improve the temperature value, more empirical data is needed. Likewise, the data used for the empirical fi.t has temperatures based purely on spectral type. This means that stars at the same spectral type, but di.fferent equivalent widths would have the same temperature. This makes any .fitted model have large uncertainties on the .fit. This experiment could be improved by using better empirical data with more precise temperature measurements. The lower value on the equivalent width measurement is more puzzling. The reason for this discrepancy is not clear, but mostly likely due to attenuation of intensity due to the atmosphere. Scattering of the light would result in less light seen by the spectrometer, and less apparent strength in the absorption line.

Overall, this experiment found bounds for the solar temperature, but is not useful for precise measurements of the photospheric temperature. Future work on the project would be to use higher resolution equipment to resolve metal lines; they can be better temperature indicators than hydrogen lines in some situation, but are harder to resolve. We had originally planned to use a Digikrom 480 monochromator with 0.6 Angstrom resolution, but experienced difficulty increasing our signal to noise ratio. This problem was likely due to not having our incoming beam correctly collimated.

Acknowledgements

We would like to thank our advisors Dr. Pryke , Dr. Gehrz, Dr. T.J. Jones, and Kurt Wick.

References

Carroll, Bradley W.; Ostlie, Dale A. An Introduction to Modern Astrophysics. Addison Wesley. Boston, Second Edition, 2007

Adams, Russell, 1928, Preliminary Results of a New Method for the Analysis of Stellar Spectra, Astrophysics Journal, vol. 68, p.9

Bowers, Richard; Deeming, Terry. Astrophysics I Stars. Jones and Bartlett. Boston,1984

Zirin, Harold. Astrophysics of the Sun. Cambridge University. Cambridge 1988

Mendoza V., Eugenio E.; Johnson, H.L. Equivalent Widths and Narrow-Band Photometry of Three Stellar Lines, Astronomical Society of the Pacifi.c 1979.

Menzel, 1939, Theoretical Problems of Stellar Absorption Lines, Popular Astronomy, Vol. 47, p.124