Download Stellar Magnetic Field_1

Stellar Magnetic Field_1
This is a copy of J.D. Landstreet’s &
some copies of UA/NSO Summer
School file
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)
Atom in a magnetic field

For atom in a magnetic field, Hamiltonian is

 
 2
e  
e2
2 2
2
H 
  V (r )  x (r ) L  S  [
B  ( L  2S ) 
B
r
sin
]
2
2m
2mc
8mc


(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
Stellar magnetic regimes
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Because all terms after V(r) are small, may treat effects
(fine structure and magnetic effects) with timeindependent perturbation theory, keeping only important
terms
For most transitions and B < 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
Zeeman effect


In Zeeman limit, atomic structure is only slightly
changed from B = 0 case. Each atomic level is
perturbed by the (e / 2mc ) B  ( L  2S ) 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
Ei  Ei 0  gi (e / 2mc) B(mJ )

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.
Zeeman patterns
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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
Example: Zeeman line
components
Polarisation of Zeeman
components
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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
Linear and circular polarisation
Polarisation effects in line
profiles
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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)
Stokes parameters
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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
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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
Polarisation in stellar line
profiles
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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)
Equations of transfer with
polarisation
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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)
dI

 h I ( I  Bn )  h Q Q  hV V
dt c
dQ

 h Q ( I  Bn )  h I Q  r R Q
dt c
dU

 r R Q  h I U  rW V
dt c
dV

 hV ( I  Bn )  rW U  h I V
dt c
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
h I  (1 / 2)h p sin 2 y  (1 / 4)(h l  h r )(1  cos 2 y )
h Q  [(1 / 2)h p  (1 / 4)(h l  h r )] sin 2 y
hV  (1 / 2)(h r  h l ) cosy
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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
Sample I, Q, U, V calculations
with spectrum synthesis code
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Example of synthesis
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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
Paschen-Back effect
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This regime has few astronomical applications: most fields in nondegenerate 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
(e / 2mc) B(mL  2mS )

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
Zeeman and Paschen-Back
splitting
Quadratic Zeeman effect

When quadratic term in Hamiltonian dominates, line components
shift to shorter wavelengths by about
DQ  (e 2 a0 / 8mc3h)2 n 4 (1  mL ) B 2
2
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2
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
Shift of the centroid,
sigma-component(ML=+_1),
pi-component (ML=0)
The pi-quadratic shift exceed the linear displacement of the sigmacomponents
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
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.
Precise calculations of H for
large B
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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)
Continuum polarisation in MG
fields
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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
Hanle effect
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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
Applications of Hanle effect
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Situations where Hanle effect may be useful arise
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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