1. Stellar Magnetic Field_1
This is a copy of J.D. Landstreet’s & some copies of UA/NSO Summer School file

2. Introduction
(Spectro)polarimetry is a major tool for study of stellar magnetic fields
Required to detect, measure, and map most stellar fields
Importance of polarimetry due to fact that magnetic field both splits and polarises spectral lines, but much of information in splitting is lost because of competing line broadening (e.g. rotation)

3. Atom in a magnetic field
For atom in a magnetic field, Hamiltonian is
(NB: cgs Gaussian units)
First 3 terms describe the atom (here in L-S coupling). Final terms are linear and quadratic magnetic terms.
Three regimes: (1) quadratic magnetic term << linear term << fine structure (x(r)L.S): Zeeman effect;
(2) quadratic magnetic term & fine structure term << linear magnetic term: Paschen-Back effect;
(3) quadratic magnetic term >> linear magnetic term & fine structure term: quadratic Zeeman effect

4. Stellar magnetic regimes
Because all terms after V(r) are small, may treat effects (fine structure and magnetic effects) with time-independent perturbation theory, keeping only important terms
For most transitions and B < G (5 T), upper limit for main sequence stars, lines are in Zeeman regime
Fine structure splitting varies a lot in atoms, so a few lines may be in Paschen-Back regime at much smaller B value than others. Paschen-Back splitting of H and Li is easily demonstrated in lab at G
Magnetic white dwarfs, with B of 104 to 108G, are in quadratic Zeeman regime - or even beyond, where perturbation theory is no longer useful

5. Zeeman effect
In Zeeman limit, atomic structure is only slightly changed from B = 0 case. Each atomic level is perturbed by the term
For L-S coupling, J and mJ are good quantum numbers. Magnetic moment of atom is aligned along J, and energy shift depends on dot product of B and J. There are 2J+1 different magnetic sublevels of energies
where gi is the dimensionless Lande factor of the level,given by
gi = 1 + [J(J+1)+S(S+1)-L(L+1)]/[2J(J+1)]
Then wavelengths of spectral line components are computed as (allowed) differences between energy sublevels.

6. Zeeman patterns
Not all transitions are allowed! Allowed transitions have DmJ = 0 (pi), -1 or +1 (sigma). Thus only some combinations of sublevels produce lines
Sometimes spacing of upper and lower sublevels is the same, then only three lines appear (“normal Zeeman effect”). Usually the spacing is not the same and several lines of each of DmJ = -1, 0, 1 occur (“anomalous Zeeman effect”). A few transitions have no splitting at all (“null lines”). Typical line component separation at 1000 G (0.1 T) and 5000 A is about 0.01 A (0.001 nm)
The gi values determine splitting of sublevels. Best values usually from experiment (see Moore’s NBS publications on atomic energy levels) or specific atomic calculations, but L-S coupling values often reasonable

7. Example: Zeeman line components

8. Polarisation of Zeeman components
Typical Zeeman component separation in fields found in MS stars (~0.01 A) is much smaller than normal line width (at least ~0.04 A, usually much more). Thus Zeeman splitting is not usually visible directly
In this situation, we use polarisation properties of Zeeman components to detect, measure, and map fields
For field transverse to line of sight, Zeeman components with DmJ = 0 (pi) are polarised parallel to field (in emission); components with DmJ = -1 and +1 (sigma) are polarised perpendicular to field.
For field parallel to line of sight, DmJ = 0 components vanish, while DmJ = -1 and +1 components are circularly polarised in opposite senses

9. Linear and circular polarisation

15. Polarisation effects in line profiles
Top panels: Zeeman components in longitudinal (left) and transverse (right) field
Panels (b): observed stellar flux line profiles with B = 0 (dotted) and B > 0 (full)
Panels (c): observed line profiles analysed for circular (left) and linear (right) polarisation
Panels (d): circular polarisation (V) signal in line (left) and linear (Q, U) signal (right)

16. Stokes parameters
Describe polarised light using Stokes vector (I, Q, U, V)
Imagine having a set of perfect polarisation analysers and measuring intensity of beam through them
I describes total intensity of light beam (sum of light through two orthogonal polarisers, say I = Ivert + Ihor)
Q describes difference between intensity of vertically and horizontally polarised light, Q = Ivert - Ihor
U is difference between light polarised at 45o and 135o, U = I45 – I135
V is difference between right and left circularly polarised intensities, V = Iright - Ileft
I, Q, U, V are almost always functions of wavelength.
Q, U, V are often normalised to I

17. Polarisation in stellar line profiles
To quantitatively interpret polarisation of Zeeman components in stellar spectrum, we need to examine equation of transfer for Zeeman split lines
Since Zeeman components absorb specific polarisations, we must consider both the direct effects of Zeeman splitting (such as line desaturation and broadening), and effects of radiative transfer of polarised light
In principle these effects influence both model atmosphere and spectrum synthesis, but most attention so far paid to spectrum synthesis (but see Khan & Shulyak 2006, A&A 448, 1153)

18. Equations of transfer with polarisation
Equations are first order linear DE’s, like normal equation of transfer
In LTE, Bn is Planck function, tc is continuum optical depth
h factors are absorption, r factors are anomalous dispersion (retardation)
For line synthesis, solve outwards from unpolarised inner boundary (see e.g. Martin & Wickramasinghe 1979, MN 189, 883)

19. Relation of absorption factors to Zeeman line components
Define hp, hr, hl as ratios of total (line Voigt profiles + continuum) opacity coefficient in pi, right and left sigma Zeeman components to continuum opacity
y is the angle between field and vertical
The hI,Q,V factors are differences between different polarising opacities, much like Stokes polarisation components
So each Zeeman component acts as a polarising Voigt profile which absorbs a specific polarisation, and the coupled equations of transfer follow the resulting polarisation outward to the top of the atmosphere
Result: both absorption and polarisation in emergent stellar spectral lines

20. Sample I, Q, U, V calculations with spectrum synthesis code
Example of synthesis
Cr II 4588 in A0 star
Dipolar field, polar strength = 1000 G
Star not rotating
Viewed from four inclinations from pole: 90, 60, 30, and 0 degrees
Q, U, V all multiplied by 10
Note how much larger V is than Q or U

21. Paschen-Back effect
This regime has few astronomical applications: most fields in non-degenerate stars are too weak to push lines into Paschen-Back regime
A few pairs of levels have very small fine-structure separation and their Zeeman patterns are distorted by “partial” Paschen-Back effect: e.g. Fe II A
In Paschen-Back regime L and S decouple, so J is not good quantum number, but now mL and mS are good quantum numbers, and so perturbation energy becomes
With this perturbation, all lines are split by the same amount, and so only three line components (corresponding to Dm = -1, 0, 1) are seen

22. Zeeman and Paschen-Back splitting

23. Quadratic Zeeman effect
When quadratic term in Hamiltonian dominates, line components shift to shorter wavelengths by about
where wavelengths are in Angstroms, a0 is the Bohr radius and n is the principal quantum number of the upper level.
The quadratic term dominates for hydrogen H10 for B > 104 G
The sigma components shift twice as much as the pi components
At 1 MG, H8 would be shifted by about 350 km/s blueward relative to H8, easily detectable (cf Preston 1970, ApJ 160, L143)
Polarisation effects are similar to those of Zeeman effect

24. Shift of the centroid, sigma-component(ML=+_1), pi-component (ML=0)
The pi-quadratic shift exceed the linear displacement of the sigma-components
When H > 1010 n-4 ;
for Balmer H10 this occurs if H > 106 gauss
If, pi- and sigma-components are weighted as usual, the centroid of their combined pattern will be shifted by

26. Atomic structure in large fields
For fields above 10 MG perturbation theory is no longer adequate. The magnetic terms in the Hamiltonian are comparable to the Coulomb terms, and the combined system must be solved (numerically).
This is very difficult. However, it has been done for H and to some extent for He.
Basically each line component decouples from the others and vary in a dramatic way with field strength.

27. Precise calculations of H for large B
For large B the sigma-like components vary rapidly with field strength and are almost invisible on stars with factor-of-2 field variation over surface
Some pi-like components have little variation over a range of field strength (“stationary components”) and produce visible absorption lines at field strengths of hundreds of MG (e.g. Wunner et al 1985, A&A 149, 102)

28. Continuum polarisation in MG fields
Physically, the fact that free electrons spiral around field lines in a particular sense means that the continuum absorption will be dichroic: right and left circularly polarised light will be absorbed differently, and the continuum radiation will be circularly polarised by a field that is roughly parallel to the line-of-sight
However, emergent continuum polarisation is a complex combination of line and continuum absorption, and one cannot at present compute polarisation spectra that resembles observed spectra (e.g. Koester & Chanmugam 1990, Rep. Prog. Phys. 53, 837, Sec 8)
Significant continuum circular polarisation is found above about 10 MG, linear polarisation above about 100 MG

29. Hanle effect
Hanle effect sometimes allows detection of quite weak fields in situations with large-angle scattering
If unpolarised light is scattered through ~90o, scattered beam is linearly polarised perpendicular to scattering plane
This occurs for resonance scattering as well as continuum scattering
If scattering atom is in magnetic field, J vector precesses about field with period of (4pmc/eB). If this period is comparable to time between atomic absorption and re-emission (decay lifetime of upper state), the polarisation plane of re-emitted photon will be rotated from non-magnetic case

30. Applications of Hanle effect
Situations where Hanle effect may be useful arise
observing scattered radiation at limb of Sun (e.g. Trujillo Bueno 2003, 12th Cambridge Cool Star Workshop)
observing scattered radiation from circumstellar material which is roughly confined to a plane (cf. Ignace et al 2004, ApJ 609, 1018)
The great value of Hanle effect is that with typical upper level lifetimes for scattering, rotation of the plane of linear polarisation is detectable for fields of tens of G
This makes effect valuable for situations where particularly small fields are expected, e.g. in solar chromosphere