OUTLINE
IMPORTANT PAPERS ON STELLAR MAGNETIC FILEDS
- K. R. Lang Astrophysical Formulae (Springer 1999) Section 2.17 (theory of the Zeeman effect).
- G. Mathys, S.K. Solanki, and D.T. Wickramasinghe (Eds) Magnetic Fields Across the Hertzsprung-Russell Diagram (ASP Conference Series Vol.248) (2001).
- Valenti and Johns-Krull in Mathys et al ASP Conference Series Vol 248, p. 179 (2001).
- Schrijver and Zwaan Solar and Stellar Magnetic activity, Cambridge Univ. Press (2000).
ROLES THAT MAGNETIC FIELDS PLAY IN SOLAR AND STELLAR ATMOSPHERES
HOW TO MEASURE STELLAR MAGNETIC FIELDS
- Zeeman polarization technique
- A magnetic field of strength B (Gauss) will split the terms of singlet lines into 2J=1 equally spaced levels (J=angular momentum quantum number).
- Energy of the splitting is E = ehBM/4Pimc = 9.3 10^{-21} erg/Gauss.
- Larmor frequency is eB/4Pimc = 1.4 10^6 B.
- Selection rule is Delta M = 0, +1, or -1.
- If the magnetic field is along the observer's line of sight (longitudinal), one sees the two Sigma components separated by +/-E from the rest wavelength, with opposite circular polarizations.
- Separation in wavelength units is proportional to B (Lambda)^2.
- If the magnetic field is perpendicular to the line of sight (transverse), then one sees only the Pi component at the rest wavelength, which is linearly polarized.
- Other geometries yield a more complex signal, but solar photospheric magnetic fields are generally longitudinal, so one sees mostly the Sigma components.
- For other atomic and molecular transitions the splitting is more complex with a Lande g factor between 0 and 2.5. There also can be more than 3 components.
- For high spatial resolution solar observations, magnetographs that measure the wavelength splitting between opposite circular polarizations measure magnetic fluxes (not magnetic field strengths).
- Observations using this technique for the Sun as a star and for cool stars generally show almost zero net signal because of cancellation by oppositely directed magnetic fields.
- Observations of chemically peculiar B and A-type stars generally show strong magnetic fluxes because the fields are simple (nearly dipoles). There is more to the story according to Piskunov.
- Zeeman line broadening technique
- Since the magnetic field geometry for cool stars is likely complex, as is seen in the Sun, one can search for Zeeman broadening of spectral lines in unpolarized light.
- Compare pairs of unblended spectral lines that are as similar as possible (optical depths, excitation energy, formation at similar levels and temperatures in an atmosphere) but differ in their Lande g factors. Observe many line pairs or model many lines to increase S/N. Note: this is a challenge.
- High resolution spectra with high S/N should show that the high Lande g lines are broader than the low Lande g lines.
- This technique was first shown to work by Robinson, Worden, and Harvey ApJ 236, L155 (1980)).
- Works best for infrared lines because the splitting is proportional to (Lambda)^2 whereas thermal broadening is proportional to Lambda.
- Review by Saar on magnetic field measurements using Zeeman broadening (1996)
- Good examples of applications of the Zeeman broadening technique are in Valenti and Johns-Krull in ASP Conf. Series Vol. 248, p. 179.
- Review by Saar in Cool Stars 11 (2001).
- Hanle effect
- A good reference is Stenflo (Solar Physics 80, 209 (1982)).
- Coherent scattering produces linear polarization of spectral lines. The Hanle effect depolarizes the signal.
- Radiative transfer explanation: magnetic fields rotate the plane of linear polarization along the line of sight, thereby decreasing the net linear polarization for light travelling along the line of sight.
- Quantum mechanical explanation: magnetic fields relax the quantum mechanical interferences between the different magnetic sublevels of an atomic level excited by the incident radiation field.
- Useful for measuring weak magnetic fields (10-100 G) in the Sun and maybe stars.
- Gyrosynchrotron spectrum
- In principle, the radio frequency at which gyrosynchrotron has a peak can provide information on the local magnetic field, but other parameters enter.
- For mildly relativistic electrons, the peak frequency is proportional to B^{0.68 +0.03Delta}, where Delta is the index of the power law electron energy distribution.
- For ultrarelativistic electrons, the peak frequency is proportional to B^{(Delta+2)/(Delta+4)}.
- Doppler imaging
- A good technique for studying the sizes, locations, and evolution of starspots, which are presumably magnetic. Also useful for studying differential rotation and magnetic cycles.
- Vogt and Penrod (PASP 95, 565 (1983)) is (to my knowledge) the first paper that lays out the technique.
- Many subsequent papers by Vogt, Collier-Cameron, Piskunov, and Strassmeier that have developed the technique much further.
- An 11 year program of monitoring the spots on the RS CVn system HR 1099 (Vogt et al ApJS 121, 546 (1999)). An excellent study of starspot evolution and migration.
- Strassmeier review paper on Doppler imaging results in Proceedings of Cool Stars 11 (2002).
- Chromospheric imaging of the HR 1099 system using the Mg II h and k lines in the UV to study active regions (Busa et al A+A 350, 571 (1999).
- Zeeman Doppler imaging
- Extension of the Doppler imaging technique to include circular polarization in the Stokes vector using high resolution spectropolarimetry.
- Components of the Stokes vector for polarized light:
- I=intensity of the wave
- V=circular polarization amplitude (right handed is positive)
- Q=linear polarization amplitude along one axis
- U=linear polarization amplitude along the other axis
- I^2 = Q^2 + U^2 + V^2.
- Reconstruct stellar images of the radial, azimuthal, and meridianal magnetic field.
- Magnetic topology and surface differential rotation on the K1 subgiant of the RS CVn system HR 1099 (Petit et al MN 348, (1175)).
- Use maximum entropy image reconstruction, but there are other techniques for regularization of the data and selecting the image with the most credible magnetic topology consistent with the quality and completeness of the data.
- Large, axisymmetric regions where the magnetic field is mainly azimuthal (e.g. strong fields near the pole usually centered near 60 degrees).
- Large scale features persist for at least 13 years.
- Small scale structures change on time scales of 4-6 weeks.
- Equator rotates faster than the pole (laps in 480 days which is 4 times slower than for the Sun). Slow differential rotation may be due to tidal forces of the nearby secobdary star.
- Another example of applications of the Zeeman Doppler imaging technique is in Valenti and Johns-Krull in ASP Conf. Series Vol. 248, p. 179.
A TOUR THROUGH THE HR DIAGRAM IN SEARCH OF MAGNETIC FIELDS
IMPORTANT RESULTS
- B increases to cool stars and is equipartition, f and fB increase with rotation. But for the most active stars B is much larger than the equipartition value. See Valenti and Johns-Krull (2001).
- The magnetic flux (fB) correlates well with the Rossby number (ratio of rotational period to convective turnover time) showing that magnetic flux in the photosphere is generated by dynamo activity.
- The relationship between X-ray radiance and magnetic flux (Pevtsov et al ApJ 598, 1387 (2003)).
- Magnetic flux is the integral of the (signed) magnetic field strength over the spatial resolution element. The same as magnetic field strength only when the field is uniform.
- Same relationship fits the solar and stellar data over 12 orders of magnitude, even though stars are unresolved and the Sun is partly resolved.
- This suggests a universal relationship between magnetic flux and power dissipated through coronal heating.
- Two-component theoretical chromospheres models for K dwarfs of different magnetic activity (Cuntz et al ApJ 522, 1053 (1999)).
- The nonmagnetic component is heated by acoustic waves.
- The magnetic component is heated by longitudinal tube waves.
- The filling factor for the magnetic component is determined empirically from the statistical relation between the measured magnetic coverage of stars and rotation period.
- Heating is strongest when the magnetic flux tubes spread least with height, which occurs for rapidly-rotating stars that have large magnetic filling factors.
- Reason: Rapid spread of flux tubes with height means that the cross-sectional area grows rapidly with height so that wave amplitudes increase slowly with height and shocks are weak and occur at high altitude where densities are low.
- When flux tubes spread slowly with height, the decrease in density with height in a nearly constant cross-sectional area means that shocks occur with strong amplitde at low heights leading to strong emission.
- Slow rotators have weak heating and Ca II fluxes predicted to be at the basal flux lower limit.
- [Saturated heating may just be a consequence of a very high magnetic filling factor and thus little flux tube spreading with height.]
TASKS FOR THE FUTURE
- Need more multicomponent theoretical models of stellar atmospheres (nonmagnetic, magnetic, starspot) that bring together energy balance arguments with observational data on magnetic fields and radiation from stellar chromospheres and coronae.
- We need more measurements of magnetic fields in stars with different ages, masses, orientation, and binarity.
- We need to monitor the magnetic fields of stars over long time periods to study stellar cycles.
- We need to measure magnetic fields in starspots of active stars.
PROBLEMS FOR THE STUDENTS