Department of Physics, Middlebury College 1992-93
Modern Physics Laboratory


XVI. Doppler Shift and Zeeman Effect Spectroscopy of the Solar Photosphere


Discussion

In a previous laboratory experiment you were introduced to the solar absorption spectrum, and investigated some of its absorption features. You also gained further experience in configuring our spectrophotometric hardware and software. In this lab you will use this equipment, as well as one or more telescopes, to make physical investigations of the sun.

The heliostat used in Lab XV was not used to form an image of the sun, nor could it, due to the large number of low-quality mirrors used; the light analyzed spectrometrically represented a rough average of the light coming from all parts of the sun's photosphere. However, the sun's photosphere is not uniform in its physical conditions, and an imaging optical system must be used if we wish to study the sun in greater detail. In this experiment you will use a telescope to focus the image of the sun onto the end of an optical fiber; the tiny diameter of the fiber will then admit light coming from only a small solid angular patch of the solar disk. This will allow us to deduce the solar rotation rate by observing the radial velocities of the opposite limbs of the solar disk, and to estimate or measure the strengths of magnetic fields in sunspots (see top panel of Fig. 1).

(1) Solar rotation. Like essentially all celestial objects, the sun exhibits a significant rate of rotation. Consider for the moment the sun and earth rotating around axes parallel to each other (a good first approximation). The various parts of the solar photosphere then exhibit various radial velocities with respect to the earth-based lab. The actual value of the radial velocity will also depend on the earth's rotation and instantaneous orientation, as well as the radial velocity of the sun's center of mass with respect to the earth's. However, the solar rotation rate can be deduced from the difference of the radial velocities at the solar equatorial limb points, as all other factors cancel out in the difference.

The radial velocities of those two diametrically opposite points of the sun can be determined by measuring the Doppler shift of absorption lines in the solar spectrum. The change in the wavelength of light, due to the radial relative velocity of the source and the observer, is given, in the non-relativistic limit, by

obs = rest ( 1 + vrad/c ).        (1)

The size of the shift is large enough to be measurable with our 1 m spectrometer. One major difficulty in this process is the fact that the wavelength calibration of the Spex 1704 is neither perfect, nor even constant over the interval of time typically required to make these measurements. Hence, a reliable wavelength reference is needed. You will use absorption lines of O2, originating in the earth's atmosphere, for this purpose. The four lines near 6300 Å which you studied in Expt. XV (see Fig. 4 of Expt. XV) are nearly ideal for this purpose: they are narrow enough to not be blended, and they provide non-Doppler-shifted reference features immediately adjacent to solar absorption lines.

The actual geometry of the earth-sun system is not quite as simple as that described above, however. The sun's rotation axis is not perpendicular to the earth's orbital plane, so that the observed radial velocities may lead to a slightly lower value of the rotation rate than is correct. This discrepancy is known to be small, however, and we shall ignore it. Furthermore, the fact that the earth's rotation axis is also not perpendicular to the orbital plane makes it difficult to determine which part of the solar limb is on the solar equator. This will be discussed further below.

(2) Zeeman Effect in Sunspots. From the photograph in the top panel of Fig. 1, which was taken in visible light with ordinary color photographic film, it is clear that the solar photosphere is not uniform in brightness. The small dark regions are known as sunspots and were among the first astronomical discoveries made by Galileo, the first scientist to use a telescope to study the heavens. The fundamental cause of sunspots is the existence of localized, high-intensity magnetic fields in the solar photosphere, which otherwise has a very weak magnetic field of at most tens of Gauss. The presence of such strong magnetic fields in the partially ionized gas of the solar photosphere gives rise to a component of gas pressure called "magnetic pressure," whose ultimate source is the Lorentz force on the charged, rapidly-moving (i.e. hot) particles. This magnetic pressure is in addition to the ordinary kinetic gas pressure, which is related to the temperature and density of the gas. As the sunspots are known to be stable for periods of weeks, there must be pressure equilibrium between the gas in the spot and that surrounding it. Since some of the pressure in the spot is due to the high magnetic field strength, there must be a correspondingly smaller contribution from the kinetic pressure, hence a lower temperature; this lower temperature explains, through the Stefan-Boltzmann law, the lower intensity of the emitted light. More detailed discussions are available in the literature.3,6

While the supposition of strong magnetic fields easily explains the darkness of the sunspots, more direct evidence of the magnetic fields would be welcome. We would expect, for example, that the Zeeman effect which you studied in a previous experiment should be visible in at least some of the solar absorption lines. In fact, the Fe lines near 6300 Å which you have already studied show Zeeman splitting into three components in strong magnetic fields. The magnitude of the splitting for such a triplet pattern is related to the magnetic field strength B, the Landé g-factor g, the rest wavelength o, and is given by3

  eB g o2
_________   (cgs units)       (2)
  mec2

and the resultant absorption wavelengths are then o - , o, and o + . For the 6300 Å lines of Fe, g = 2.5, this splitting is sufficient to be resolved with our 1 m spectrometer. The presence of the two O2 lines nearby is, once again, a convenience. The difficulty of the observation requires us to check whether the observed splitting or broadening is physical, or merely an artifact of the observation process. The O2 lines, arising as they do in the Earth's atmosphere, will not be subject to the Zeeman broadening we expect to see in the sunspots.

Procedure

This lab must be performed in the daytime, with clear skies. Even thin cirrus clouds may make it impossible to achieve usable results.

For both parts of this lab you will use a telescope on the roof of the Science Center to provide imaging of the sun. Since the sun moves across the sky at a rapid pace and the solid angle accepted by the optical fiber is extremely small, the telescope must be made to track the Sun continuously, much as the heliostat did in Expt. XV. Your instructor will provide instruction and guidance in the use of the telescope.

The optical fiber is a single filament, 200 Ám in diameter, in a protective plastic sheath. It is coupled to the telescope by a special mount which allows the placement of the end of the fiber to be adjusted with very high precision. The coupler is also fitted with a viewing eyepiece with crosshairs; a flip mirror assembly allows the light from the telescope to alternately fall on the end of the optical fiber or to pass through the eyepiece (lower panel of Fig. 1). Under no circumstances should you look through the viewing eyepiece. The measurements in this experiment are made at a solar intensity which would cause instant and permanent damage to your eyes. Instead, the solar image should be projected through the eyepiece onto a projection screen, which can be simply a white piece of paper attached to the telescope, as seen in the lower panel of Fig. 1. The 3-axis fiber positioner is adjusted so that whatever part of the sun's image falls on the crosshairs (which are visible in the eyepiece projection) will also fall onto the end of the optical fiber when the flip mirror is returned to the normal position. The optical fiber itself is stored in the south side of the well of the pier of the 16 in telescope in the dome on the Science Center roof. Please exercise extreme caution in handling this fragile fiber; do not bend the fiber sharply, and avoid handling the very end of the fiber so as to prevent scratching it.

You may use either the 16 in telescope or the 8 in telescope nearest the dome for the Doppler shift portion of the lab. The 8 in telescope is easier to set up and use, and provides adequate tracking; furthermore, you will probably achieve a higher signal-to-noise (S/N) ratio because you will use the full aperture, so its use is recommended. For a more detailed study of the radial velocity field across the solar disk, the better tracking and easier pointing of the 16 in telescope argues for its use, but you will only use it with a greatly reduced aperture, so that the S/N ratio will also be reduced, making the measurements more time-consuming to complete and to analyze.

For the Zeeman effect measurements, the 16 in telescope must be used. It should only be pointed toward the sun when the 3.5 in aperture mask is in place, to reduce likelihood of damage to the telescope. When the 16 in telescope is used, the finder telescope should be outfitted with its special solar filter to permit safe direct visual use.

In the Laser Spectroscopy Lab (SC 204B), set up the spectrometer hardware and software as for Expt. XV. Position the end of the optical fiber in front of the entrance slits, using a mount that allows vertical adjustment and fine-tune horizontal adjustment. The proper positioning of this end of the fiber is extremely important, as the photon counting rate will be much reduced from the levels of Expt. XV. The lab partner in the dome should point the telescope at the sun while the partner in SC 204B adjusts the lateral position of the fiber mount to achieve the greatest throughput for the system.

Using the experience gained in the previous solar spectroscopy experiment, configure the McSpex software for taking scans in the vicinity of the 6300 Å solar Fe lines, at a scan rate and resolution that will allow you to take spectra which can lead to determination of line centers to a small fraction of a linewidth. The spectrometer slit openings should be chosen wisely; you must make a trade-off between higher signal and optimum resolution.

You are now ready to begin observations.

(1) Solar Rotation. Under the simplified geometrical model described in the discussion section, it would be sufficient to obtain one scan at each of the easternmost and westernmost limbs of the sun, as in Figs. 2(a) and 2(b). The solar limbs can be located using eyepiece projection and moving the telescope until the crosshairs lie just inside the respective solar limbs. Keep in mind that the angular diameter of the optical fiber, projected onto the sun, is the same as the separation of the two parallel lines which make up part of the crosshairs. Due to atmospheric seeing, building vibration, and tracking errors, it is wise to place the crosshairs slightly away from the limb, toward the center of the solar disk. You should, however, obtain at least three scans at each limb. In addition, take three scans with the telescope pointed as close to the center of the solar disk as you can determine.

If time and energy permit, you may take additional scans that will greatly enhance your ability to characterize the solar rotation.

(i) Take a series of scans along the east-west diameter of the solar disk. The radial velocity should be a very specific function of the position along the disk. For the best quantitative results, high-precision pointing is necessary, and therefore use of the 16 in telescope is recommended.
(ii) For the reasons mentioned in the discussion section, scans taken at the east and west limbs may show differential Doppler shifts significantly lower than those that would be obtained at the true equatorial limbs. To ensure that you have observed close to the solar equator, make a series of scans around the limb of the sun, concentrated near the east and west limbs. Again, there should be a specific relationship between the observed radial velocities and the angular position around the limb; and again, use of the 16 in telescope is required.

(2) Zeeman Effect in Sunspots. This part of the lab may not always be possible, especially near times of solar activity minimum. A quick inspection of the sun through an 8 in telescope with the appropriate filter will show whether there is a sufficiently large spot. For the actual measurements, the 16 in telescope is needed.

For the best results, it is probably wise to adjust the tracking rate of the 16 in telescope so that it will follow the sun rather than the stars. Tabular information in the Astronomical Almanac can be used to determine the correct rate.

The measurements themselves are very straightforward, consisting of scans of the sunspot of interest, with occasional scans of the nearby brighter photosphere taken with the same scan rate, RC time constant, and slit widths as those of the sunspot; only the ratemeter scale should be changed. The purpose of these additional scans is to serve as references of the widths of the Fe lines under conditions of low magnetic field; as the observed linewidths depend in part on the instrumentation, it is vital that the slits widths and the RC time constant are not changed.

You will find that obtaining a good scan of the sunspot is much more difficult than obtaining one of the surrounding photosphere, as can be seen from the lower S/N ratio of the spectrum of Fig. 3. There are two reasons for this. The first is simply that the intensity of the light in the sunspot is three to five times weaker. The second is that due to atmospheric seeing (the time-dependent blurring of the images) and building vibration, the optical fiber will occasionally sample part of the surrounding photosphere. Even with a multichannel detector, this would tend to reduce the apparent Zeeman splitting or broadening; since your data acquisition scheme involves scanning through the spectrum, this means your spectra will be subject to a non-uniform spectrum normalization, again because the sunspot light is so much less intense than that of the surrounding region. Therefore you should be prepared to take, and save, several scans of the sunspot. Obtaining twenty or more scans would not be imprudent.

Analysis

(1) Solar Rotation. Make hardcopy plots of each of your east, west, and central scans. For each, determine the centroid wavelengths of each of the four lines, and provide uncertainty estimates for your centroid determinations. For each of your scans, determine the wavelength differences for the pair of Fe lines, and discuss the distribution of values you obtain. Do the same for the wavelength differences for the pair of O2 lines. These results should give you a measure of the quality of the wavelength difference determinations which can be made from your scans. You should also compare these differences to those of Ref. 5.

Show from Eq. (1) that the tangential velocity at the solar limb can be calculated from the measured wavelengths using

vtan = c/2o ( [Fe,West - O2,West] - [Fe,West - O2,East] )        (3)

where o is some suitable average rest wavelength in the region of interest. Note that this method of measurement is insensitive to the state of the earth's motion, as long as that motion does not change during the measurements.

Using the east and west scans, calculate a value for vtan and the corresponding solar rotation period (which may or may not be the equatorial rotation period). Form a histogram of all values so determined. Compare your result to that which can be deduced from tables of astronomical quantities such as the solar diameter and period of rotation.

(2) Zeeman Effect in Sunspots. Make hardcopy plots of all of your scans, including those not of sunspots. How you proceed from this point depends on the quality of your spectra.

If you have one or more scans which show clear broadening (or even splitting!) of the Fe lines, then work directly with those scans (see third scan from top in Fig. 3). Measure the breadth of the Fe lines in both the sunspot and non-sunspot scans, and determine from this the amount of broadening. Keeping in mind that these Zeeman patterns are triplets, calculate the magnetic field strength in the sunspot using Eq. (2).

If many of the scans show hints of broadening but are too noisy to be used as is, you may instead add all of your scans together into one single scan and use it instead, as in Fig. 4. This method is not ideal, because it will tend to show less broadening, due to the contributions to the summed spectrum from the scans which sampled too frequently the brighter photosphere surrounding the sunspot. Furthermore, the wavelength calibration of the SPEX 1704 is unlikely to remain sufficiently constant for the duration of the sequence of scans to avoid artificially broadening all of the features; hence the added importance of the reference photospheric scans.

References
1. E.B. Anthony, The McSpex Project: Laser Spectroscopy Laboratory Interface Program, B. A. thesis, Middlebury College, 1987.
2. J.C. Brandt, The Sun and Stars  (McGraw-Hill, New York, 1966), p. 14.
3. R.J. Bray and R. E. Loughhead, Sunspots  (Chapman and Hall, London, 1964).
4. J.H. Cooley, Four Projects in Solar Spectroscopy, B.A. thesis, Middlebury College, 1991.
5. The Solar Spectrum: 2935 Å to 8770 Å, NBS Monograph No. 64, edited by C. E. Moore, M.G.J. Minnaert, and J. Houtgast (National Bureau of Standards, Washington, DC, 1966).
6. H. Zirin, Astrophysics of the Sun  (Cambridge University Press, Cambridge, 1988).

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