Observable effects of a stellar magnetic field

1
Observable effects of a stellar magnetic field

• John D Landstreet
• Department of Physics Astronomy
• University of Western Ontario
• London, Canada West

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 ltlt
  linear term ltlt fine structure (x(r)L.S) Zeeman
  effect
• (2) quadratic magnetic term fine structure
  term ltlt linear magnetic term Paschen-Back
  effect
• (3) quadratic magnetic term gtgt linear
  magnetic term fine structure term quadratic
  Zeeman effect

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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 lt 50000 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 30000 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

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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 2J1 different magnetic
  sublevels of energies
• where gi is the dimensionless Lande factor
  of the level,given by
• gi 1 J(J1)S(S1)-L(L1)/2J(J1)
• Then wavelengths of spectral line components are
  computed as (allowed) differences between energy
  sublevels.

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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 Moores
  NBS publications on atomic energy levels) or
  specific atomic calculations, but L-S coupling
  values often reasonable

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Example Zeeman line components
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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

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Linear and circular polarisation
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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 gt 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)

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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

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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, AA 448, 1153)

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Equations of transfer with polarisation

• Equations are first order linear DEs, 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)

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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

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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

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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 6147-49 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

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Zeeman and Paschen-Back splitting
18
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 gt 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 Ha, easily detectable (cf
  Preston 1970, ApJ 160, L143)
• Polarisation effects are similar to those of
  Zeeman effect

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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.

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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, AA 149, 102)

21
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 paralel 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

22
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

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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