Measuring and modelling stellar magnetic fields

1
Measuring and modelling stellar magnetic fields

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

2
Introduction

• We now turn to three major questions connecting
  the atomic physics of magnetic fields with
  astronomy
• Which of the various atomic effects of a magnetic
  field are actually observable in stars
• How do we use these effects to obtain empirical
  information about stellar magnetic fields
• To what extend do observations allow us to
  actually model the field structure of a star

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Basic observation methods

• To search for magnetic fields in stars, we could
  use either the splitting properties of Zeeman
  effect, or polarising properties
• Direct line splitting can be detected in some
  stars, but only under somewhat exceptional
  circumstances typical field (in kG) gt v sin i
  (in km/s)
• When we do detect splitting, measurement of
  (mean) displacement of sigma components from pi
  components gives direct measurement of average
  magnitude of field over visible hemisphere, often
  called mean field modulus, Bs or ltBgt

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Mean field modulus

• Segment of spectrum of magnetic Ap star HD 94660
  (Mathys) shows visible Zeeman splitting in a
  field of about 6 kG
• Laboratory and theoretical Zeeman patterns are
  shown beneath stellar spectrum

5
Field detection from spectropolarimetry

• For most non-degenerate stars, field detection is
  achieved by spectropolarimetry, using
  polarisation properties of Zeeman effect
• As we saw in first lecture, longitudinal field
  component produces first-order separation of line
  profiles in right and left circularly polarised
  light.
• Measuring this separation yields an average of
  the line of sight component of the field over the
  visible stellar hemisphere (the linearly
  polarised transverse field component contributes
  the same profile to both right and left
  circularly analysed line profiles)
• The quantity is usually called the mean
  longitudinal field, ltBzgt or sometimes Be

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Useful data analysis methods from equations of
polarised transfer

• If field is so weak that Zeeman splitting is
  small compared to natural width of local line
  profile, mean longitudinal field ltBzgt may be
  found from
• V(l) 4.67 10-13gl2 (dI/dl) ltBzgt (l in
  A)
• (e.g. Landstreet 1982, ApJ 258, 639 Bagnulo
  et al 2002, AA 389, 191)
• If the lines are fairly weak (often used on
  strong lines too)
• where B is in G, l is in A, g is the mean
  Lande factor of the line, and I and V are
  functions of velocity shift v from line centre
  (e.g. Donati et al 1997, MN 291, 658). This
  result is widely used to interpret magnetic
  circular polarisation spectra, including LSD
  spectra (see below)

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The oblique rotator model of magnetic A and B
(Ap) stars

• First non-zero (longitudinal) magnetic fields
  found in magnetic Ap stars (A and B stars of very
  peculiar atmospheric chemistry) by Babcock in
  1947
• When ltBzgt is measured repeatedly for one Ap
  star, found to vary periodically with period
  that varies inversely with v sin i. Many such
  stars are also periodically variable in light
  and/or spectrum, with same period as field.
• Clearly the period is the rotation period of the
  star, and the field varies because it is not
  axisymmetric. As the star rotates we see
  different field configurations on the visible
  hemisphere the oblique (dipole) rotator
• Because ltBzgt usually varies almost sinusoidally,
  we deduce that the field is simple, roughly
  dipolar. Ratio of largest value of ltBzgt to
  typical ltBgt is consistent with this idea

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Example of observed Ap star variations

• HD 184927 shows periodic variations in
  longitudinal field, He line strength, and
  photometric brightness and colour with a period
  of 9.53 d (Wade et al 1997, AA 320, 172)

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The oblique rotator

• Sketch of the oblique rotator model of a star
  showing magnetic field lines (black vectors
  projecting from the surface) and the variation of
  abundance of a variable chemical element over the
  surface (colour coding)
• As the star rotates about an axis other than the
  field axis, both field and spectrum vary.

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Recent advances

• Spectropolarimetry allows detection of remarkably
  small magnetic fields. Polarisation is measured
  differentially, so values as low as a few times
  10-4 of I can be measured reliably. This allows
  detection of magnetic fields as small as a few
  10s of G (in bright stars! e.g. Shorlin et al
  2002, AA 392, 637)
• Linear polarisation, variable with the rotation
  period (hence intrinsic, not interstellar) has
  been detected in broad bands (Leroy 1995, AAS
  114, 79). This is apparently due to Zeeman linear
  polarisation in individual saturated spectral
  lines, and corresponds very well to line linear
  polarisation, now observed with Musicos and
  Espadons (Wade et al 2000, MN 313, 823)

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Least squares deconvolution

• Polarisation spectra of fainter stars often lack
  sufficient S/N to detect the very small signals
  due to the Zeeman effect. Donati has introduced
  and implemented a very valuable tool for
  co-addition of these small signals LSD
• This technique allows one to recover very small
  polarisation signals. In the case of V signals
  the individual polarisation profiles are
  sufficiently similar that the mean V signal may
  be treated as the signal of a single line for Q
  and U (linear polarisation) signals, the
  dispersion of individual signals is much larger
  it is not clear what the meaning of the mean
  signal is (except for providing detection and
  estimate of amplitude).

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Detection of a weak field in e UMa using LSD
(with TBL-Musicos)
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Broad-band linear polarisation in 78 Vir

• Upper panel longitudinal magnetic field of Ap
  star 78 Vir
• Lower panel broad-band linear polarisation
  observed by Leroy (1995, AAS 114, 79) compared
  with value synthesized from LSD line profile
  linear polarisation in same bandpass (Wade et al
  2000, MN 313, 851)

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Moment techniques

• The longitudinal field measurement can be viewed
  as the first order (wavelength or velocity)
  moment of the circular polarisation spectrum (and
  the equivalent width is the zero-order moment of
  the intensity spectrum) recall
• Mathys (e.g. in Astrophysical Spectropolarimetry,
  eds Trujillo-Bueno et al, 2000, p 101) has shown
  that higher moments of the I and V spectra
  contain useful information about the
  configuration of the magnetic field under
  observation
• Simplest modelling strategy assume simple field
  structure, fit one or more field moments

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Combined variations of several field moments

• Typical variations of mean longitudinal field
  (upper left of each group) and mean field modulus
  (lower right) for two stars for which both fields
  are measured (models Landstreet Mathys 2000
  AA)
• Also shown are two other field moments. Lower
  left panel shows mean quadratic field (similar
  to mean field modulus) upper right is
  crossover field which detects reversals of
  field polarity (cf Mathys 1995, AA 293, 733
  746)

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Modelling with synthesis codes

• Most physically correct method of deducing field
  structure from observed intensity and
  polarisation spectra is by computing spectra that
  match observations rather than fitting simple
  models to field moments.
• Generally this requires model atmosphere and line
  synthesis programmes. Both may take magnetic
  effects into account at various levels
• Zeeman splitting of lines,
• polarised radiative transfer,
• magnetic field effects on hydrostatic
  equilibrium,
• effects on convection and transport of chemical
  elements horizontally and vertically

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Physics and strategy of a simple code (Zeeman)

• Zeeman (Landstreet 1988, ApJ 326, 967) is simple
  magnetic line synthesis programme
• Reads in specifications of star, spectral window
• Computes one or more (I, Q, U, V) spectra
  including magnetic line splitting and polarised
  radiative transfer, using line list and line
  parameters from VALD and precomputed Atlas
  atmosphere
• Compares computed spectrum(a) with observed
  one(s), determines vrad, v sin i, and c2 of fit
• If desired, iterates abundance of one element to
  fit one or many spectral lines chosen by
  programme
• Contains simple parametrised models of magnetic
  field, abundance distribution over several
  co-axial rings. With several spectra, can
  optimise model

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Fit to a spectrum of Sirius

• Synthesis fits to non-magnetic stars can be very
  accurate
• Example Sirius

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Simple parametrised model of 53 Cam

• However, parametrised models which fit moment
  data usually do not describe more detailed
  polarisation spectra accurately (Bagnulo et al
  2001, AA 369, 889)

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More powerful mapping methods do fit 53 Cam

• Solution is to develop powerful mapping code
  which can fit field and abundance at many points
  on stellar surface by iterative adjustment
  (Kochukhov et al 2004, AA 414, 613)

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53 Cam Fe abundance and field maps from 3 lines