The Polarization of Radiation from Stellar Atmospheres
Computer programs used to construct models of stellar atmospheres generally treat the
scattering of the continuum radiation as isotropic. If the dipole angular dependence
and polarization of the scattered radiation are taken into account, the equations
become more complicated. The integral equations for this case can be found in
Harrington (1970) Astrophysics & Space Science, 8, 227-242.
(See also Harrington (1969).)
We solve these equations by approximating the integral transforms, like the $\Lambda$-
transform, by matrices multiplying the source functions at discrete optical depth points.
These matrices are derived by assuming the source functions can be fit by cubic splines.
This reduces the problem to the solution of a set of linear equations. The details are
given here. All we need from the model atmosphere program is the
run of temperature and the ratio kappa/(kappa+sigma) as a function of monochromatic optical
If you wish to experiment with this method, here is a directory with a bit of Fortran code and
some necessary data (i.e., the pre-computed matrix transforms): For_Pol.d.
Cool Stars: The MARCS models.
One extensive compilation of model atmospheres which contain the needed information are the
MARCS models developed at Uppsala University
(see the MARCS website.)
Here are files which contain I(µ), Q(µ) and pol(µ)=Q(µ)/I(µ) extracted
from about 125 of the MARCS model atmospheres in the temperature range
2500K - 8000K and with log(g)= 3, 3.5, 4, 4.5 and 5:
MARCS I(µ) Q(µ). Here is a figure showing the polarization for stars with
effective temperatures of 4000K: Polarization at 3650A
The polarization is higher for metal-poor atmospheres. Here
are results for sets of MARCS models with 1/10 and 1/100 solar metal abundances:
1/10 solar MARCS I(µ) Q(µ) and
1/100 solar MARCS I(µ) Q(µ).
It should be pointed
out, however, that only the continuum absorption has been included in these calculations.
Since the cooler stars are blanketed by line absorption such that there is little or
no free continuum, the polarization given here will be overestimated.
Hot Stars: The TLUSTY models.
For hot stars, there is another compilation of models from which we can extract the necessary
information. These are the NLTE models computed with the code TLUSTY (see the Tlusty webpage).
A new, well-documented version is available on Hubeny's webpage (Hubeny) here: tlusty205 .
For these models, the monochromatic opacities are not available on the website; rather, we need
to run the Tlusty code (just to generate the model tabulated, no iteration needed) with the
keyword IPOPAC=1 set. This generates a file (fort.85) with the needed opacities. From the
opacities we can compute the monochromatic optical depths, while the (electron) scattering
is obtained from the tabulated electron number N_e and the gas density. (There were a few other
other problems which made it necessary to make small changes to the source code of tlusty204.f)
We have computed the polarization and limb darkening of the emergent continuum radiation for
28 wavelengths between 1200A and 8000A for a number of stellar temperatures and gravities:
TLUSTY I(µ) and Q(µ), solar abundances. For the sake of
comparison, here are a few results from models with 1/10 solar abundances:
TLUSTY I(µ) and Q(µ), 1/10 solar.
Here is a figure showing how polarization varies with wavelength:
T=35000K, log g=3.5,4.0,4.5. This is for µ=0.1 (theta = 84 deg). The blue curve is for
log g=3.5, the green for log g=4.5. (Actually, this is a plot of -Q(µ)/I(µ), so where the
curves drop below 0, this indicates polarization perpendicular to the limb.)
Stars of Intermediate Temperatures: The STERNE models.
Neither the MARCS of TLUSTY websites cover the temperature range 8000K to 15,000K. While
this temperature range shows less polarization than the hot or cool stars, we still would like
to have some results available. There is an extensive set of LTE models which cover this intermediate
temperature range (as well as hotter stars):
the STERNE models which can be found
on C.S. Jeffery's webpage. Unfortunately, the models given there contain the monochromatic opacity
for only one wavelength: 4000A. Thus we can only (without getting into the source code) evaluate
the polarization and limb darkening at this wavelength. (Fortunately, this is the blue end of the
visible spectrum, but longward of the Balmer jump, where the polarization is highest.)
Here is a graph of some results for T=12,000K models:
T= 12,000K, log g= 3-4.5.
Here is a comparison (at T=30,000K, log g=4) between the TLUSTY and the STERNE models:
Limb darkening compared, and
The Effects of Absorption Lines: An Example Using the Milne-Eddington Approximation.
The results tabulated here overestimate the polarization to be seen in real stars due to our neglect
of atomic and molecular absorption lines. In fact, in the regions of the spectrum where the polarization
is predicted to be strongest (the blue end of the visible spectrum for cool stars and the far ultraviolet
for hot stars), the continuum may disappear under a multitude of absorption features.
To get some idea of the suppression of polarization by absorption lines, we consider the Milne-Eddington
approximation: we assume that any frequency the ratio of line absorption to continuum absorption is
constant. Further, we assume the line is a pure absorption LTE feature. This may be reasonable for weak
lines, as studies of "The Second Solar Spectrum" show that the great majority of lines are purely
depolarizing and can be explained as pure absorption lines.
Here are the resulting variations in intensity and in
polarization across a Doppler broadened line, where the line center
absorption is 5 times the continuum absorption. If we average the Stokes parameters I and Q over and
interval -4 to +4 Doppler widths, we find the averaged polarization
as a function of µ. The suppression of polarization is mitigated by the fact that where the line absorption
suppresses the polarization, the intensity is also reduced, and so counts less in the average.
While these results for a 4000K star seem straightforward, things get a bit strange when we consider a
hot (30,000K), low-gravity (log g=3.0) star, with our LTE absorption line at 5000A. Here is the
intensity and polarization
across the line. Note that for values of µ≤0.3 the line is in emission! While the polarization is
suppressed for small values of µ (near the limb), at µ=0.1 the continuum is polarized parallel to the limb, while the
core of the line is polarized perpendicular to the limb. At µ=0.2, the continuum is unpolarized but
the line is polarized perpendicular to the limb (positive Q(µ)). For µ>0.2, the continuum is polarized
perpendicular to the limb and the line suppresses the magnitude of this polarization. Here is the
polarization for µ=0.195, 0.200, and 0.205, which bracket the point where the direction of the
continuum polarization changes by 90º: polarization near µ=0.2.
(The appearance of
line emission for small µ is related to "Schuster's emission-line mechanism". While it is difficult
for the emergent flux integrated over all angles to show emission --
see Harrington 1970, ApJ 162, 913 --
apparently emission at small µ is not so hard.) This figure shows the variation of polarization as
a function of µ for the line core (small), the continuum, and for the average over a band-pass of
+/- 4 Doppler widths: Pol(µ).
The bottom line here is that a true calculation of the polarization from a model atmosphere requires
the use of a proper spectrum synthesis code, with the evaluation of the emergent flux F at each
of the closely spaced wavelengths replaced by the evaluation of I(µ) and Q(µ) at each of these wavelengths.
We can then average these Stokes parameters over the desired bandpass.
The Effects of Absorption Lines: A Proper Calculation.
We have modified the Synspec source
code to output the depth-dependent absorption (from which we also compute the monochromatic optical
depth scale) and scattering for a fine wavelength grid. We then can compute the polarization and
limb darkening not only in the continuum, but across the line profiles. (This assumes the lines are
formed by a pure absorption mechanism.) Here is an example of the results for 25,000 wavelengths
between 3900A and 5950A:
Q(µ)/I(µ) vs. wavelength. The green line is the mean polarization found
by integrating the Stokes parameters over the entire band. Here is the polarization over just 42A to
show more detail: Q(µ)/I(µ) over 40A. (The strong lines
depolarize beyond zero, which corresponds to a shift in the plane of polarization by 90 degrees.)
For hot stars, the most heavily line-blanketed part of the spectrum is in the far UV. Here we show a
calculation based of Synspec for a 30,000K star with solar abundances. We have solved for the limb
darkening and polarization at 128,162 wavelengths between 1200A and 8000A. This shows the whole spectrum:
-Q(µ)/I(µ) for 1200-8000A. The red line shows the polarization we find
by averaging the Stokes parameters of each group of 500 wavelengths. We see that shortward of 2000A the
average drops significantly below the upper envelope. To show this in more detail, we plot the results
over just 7A near 1600A: Here is the intensity I(µ=0.1) and this is
the polarization: -Q(µ)/I(µ) near 1600A. This interval was covered
by 400 wavelengths. The red line is the average polarization over this interval. If we go back to
the continuum results for this temperature, gravity, and a wavelength of 1600A and look at the value
of polarization for µ=0.1 as given in the TLUSTY tables presented above, we find a value of -0.033 (shown
in the figure as the green line). Thus the line blanketing reduces the overall polarization by about 30%.
As another example, we consider a star with T=15,000K and log g = 4.0. After computing the polarization
and limb darkening at 168,500 frequencies, we average the Stokes $I$ and $Q$ over groups of 200
frequencies, to obtain the average polarization at 840 points between 1200A and 8000A. Here is a plot
of the results for a viewing angle of µ=0.1. The blue line is the continuum-only results from the
tables above. Continuum vs. Averaged Absorption Line Spectrum
Aside from the hydrogen lines (which are not treated properly anyway), the greatest suppression of
the polarization seems to occur around 2400A. The strongest polarization in these hot stars is in the
far UV, as that is where the outgoing radiation field is most asymmetric. But this is also a region
of strong line blanketing. Here is a 145A segment of the UV spectrum of a T=15,000K, log g=4.5 star
between 1200A and 1345A. The computations for this segment involved evaluation of the Stokes I and Q
parameters at 30,000 wavelengths. We then averaged I and Q over bins of 250 wavelengths, and we here
plot the resulting average polarization as the red line:
Averaged far UV polarization.
It appears that the polarization remains quite high in spite of the absorption lines.
Here is another example: the TLUSTY model of a T=20,000K, log g=3.0, solar abundance star. We show the wavelengh interval 1364A - 1377A where the line blanketing is pretty strong. The plot of the intensity emergent at µ=0.1 shows that the average value is ~20% below
the bits of continuum. The second plot shows that the polarization is suppressed, on
average, by less than 30%. So if we only had the ability to measure UV polarization ...
Far UV emergent intensity. - - - -
Far UV polarization.
Here are the detailed results for stars of a number of surface temperatures and gravities.
We give results at 115,00 to 170,000 wavelength points between 1200A and 8000A.
The first row indicates the T_eff and log g of the model. The 2nd row is the list of 18
µ's (the cosines of the emergent co-latitudes), while the 3rd gives the total nuber of
wavelength points. This is followed by blocks of 5 rows for each wavelelngth:
[row 1] gives the wavelength in A followed by the
total emergent flux at that wavelength,
[rows 2 & 3] give the emergent intensity I(µ) at 18 angles specified by the values of µ = cos(theta), and [rows 4 & 5] give the polarization Q(µ)/I(µ) at these same µ's.
(Note that negative values
of Q(µ)/I(µ) indicate polarization parallel to the stellar surface.)
"2nd Spectrum" for some hot stars..
Synspec and the Kurucz Models: The 2nd Spectrum for Cooler Stars.
Since the above was written, a new version of synspec (synspec51, packaged with tlusty) has appeared:
tlusty205 This program can easily
take the Kurucz models as input. By adding a new file, open(unit=96,status='UNKNOWN'), and the command
after line 2088 of the source code synspec51.f, we can write out the coefficients of monochromatic opacity
(absorption + scattering) and scattering for all depths at each wavelength (big file!). This allows us to
compute the emergent intensity and polarization as a function of µ for each frequency across the lines in
any of the Kurucz models. (We also need to read the density from the synspec output file fort.6). We have
used this method for stars in the 3000K to 20,000K temperature range.
Without showing any detailed results, we can say that for stars of effective temperatures around
4000K, the heavy line blanketing, especially in the blue end of the visual spectrum, depresses the
polarization strongly compared to the continuum-only results given above.
Applications: Rotation, Starspots, Eclipses & Transits.
For an isolated, spherical star, even the modest polarization tabulated here is generally
unobservable because the net polarization cancels out by symmetry. So, unless we can
resolve the stellar disk (as we can for the sun), we need some special circumstances
which break this symmetry.
Long ago we considered the effects of rapid rotation
(Harrington, J.P. and Collins, G.W., 1968).
That study considered the continuum polarization to be that of
the pure scattering solution of Chandrasekhar (1946), which is much larger than the level
of polarization expected from hot stars in the visible. This question could be revisited
using the model atmosphere results presented here. For example, we looked at a B1 V star
(T=25,000K, log g=4.3) rotating at 95/% break-up velocity. Polarization is only significent
in the far UV. Here are the aspect dependent intensity and percentage net polarization for
this case: I_vs_i-UV.pdf and
P_vs_i-UV.pdf. Here is a discussion of these calculations (which are ongoing):
Notes on Rotation.
Another effect of rotation may be revealed at higher spectral resolution.
Since spectral lines will usually supress the continuum polarization, if the rotational
Doppler shift is comparible to the line width, the line may mask the polarization of part
of the star, breaking the symmetry and revealing the continuum polarization of the
unmasked part of the stellar disk, something we may call the Öhman effect.
We can illustrate this effect using the Milne-Eddington line model discussed above. We
put a pure absorption line in the continuum from one of the MARCS models and obtain the
emergent I(µ) and Q(µ) at all wavelengths across the line profile. We then apply the
appropriate Doppler shifts for a grid of points on the visible stellar surface and sum
the Stokes parameters to find the Doppler broadened intensity profile and the residual
polarization Q across the line. The effect is best seen if the rotational velocity and
inclination of the star are such that the Doppler broadening is a few times the Doppler
width of the absorption line (we use 3x in the following examples). Here are a couple
of results for stars of effective temperatures of 4000K and log surface gravities of 3
and 4: T4000-lg3 and
T4000-lg4. We see net polarizations reaching 0.17 and 0.068 percent. (The intensity
scale of the line profiles in the lower part of the figure is arbitrary, and the purple
line is zero intensity.)
Here are some notes with further details:
The next step beyond the Milne-Eddington line model is to use the I(µ) and Q(µ) from an
actual spectral synthesis program. We have provided such results above (under "2nd
Spectrum for some hot stars"). It is simple to integrate the (Doppler shifted) Stokes
parameters over the surface of the rotating star to obtain the net polarization. But
the polarization curve would be choppy since spectral resolution of our input "2nd
spectrum" is barely adaquate for this sort of calculation.
We have thus run the SYNSPEC spectral synthesis code at 0.01A wavelength spacing - the
output tables of I(µ) and Q(µ) are too large to post here. Here, for a T=20,000K, log g
= 3.5 star, is a few Angstroms of spectrum around Si IV 1402.7A for a velocity of 20
km/s: Stokes near 1402.7A.
The scale on the y-axis is the percent polarization (100*Q/I), while below is the Doppler
broadened intensity, and for comparison, the unbroadened emergent flux from the atmosphere
- the scale for the intensity is completely arbitrary. Note that the broad Si IV
line produces little polarization since it is wide compared to Doppler shifts.
This series of plots for a T=30,000K, log g=4.0 star, shows how the polarization varies
with the projected rotational velocity for the NII 4035.08A and Fe III 4035.42A lines:
We see that for this hot star, the polarization in the visible is very small (Q/I
< 6E-5), and the Q profile is inverted: the central peak is negative and the flanking
peaks are positive. This is because for a hot star, the gradient of the Planck
function in the visible is so small that there is actually more radiation traveling
horizontally than vertically near the surface. As a result the plane of the polarization
of the emergent radiation switches - the polarization is perpendicular to the surface
and thus the profile of the Ohman effect is inverted.
A situation of some current interest is the polarization which might result from the
transit of an exoplanet across a stellar disc.
Such polarization was discussed a few years ago (e.g., Davidson et al. 2010, AAS meeting
No. 215 [http://adsabs.harvard.edu/abs/2010AAS...21542303D]; Carciofi & Magalhaes 2005,
ApJ, 635, 570). We have written code that computes the polarization from a transiting
planet, given the limb darkening I(mu) and Stokes Q(mu). (We need only compute the
polarization of the spot occupied by the transiting planet: the observable stellar
polarization is just the compliment of that spot.)
Here are some results:T2500_4600A.jpg.
The scattering and hence polarization will be largest for early type (electron scattering)
and late type (Rayleigh scattering) atmospheres, but not very significant for solar type
stars. Here are the results we obtain for the transit of a Jupiter size planet across the
MARCS solar atmosphere. The light curves are thus:sun_light.
Here is the polarization: sun_pol. The maximum polarization is
1.8e-6 (0.00018 %), largely in agreement with the results of Carciofi & Magalhaes (2005).
The polarization from cool atmospheres increases strongly towards shorter wavelengths.
This is because (1) Rayleigh scattering increases as lambda^(-4), and (2) the Planck
function has a steeper gradient at short wavelengths, resulting in radiation which is
more strongly peaked perpendicular to the atmosphere's surface, which in turn leads to
higher polarization of radiation scattered near the surface. Here are the polarization
curves for the transit of a Jupiter radius planet across a K5 V star with atmosphere
parameters T_eff=4500K and log g=4.5. The curves are for wavelengths of 6000A, 5500A,
5200A, 4800A, 4600A, 4400A, 4200A, 4000A and 3800A; the polarization is increasing with
decreasing wavelength: K5V_Jupiter.
The discovery of the
Kepler-16 system, where a planet orbits a binary system provides a timely example.
We take the larger star "A" (R=0.65 R_sun) to have a
temperature of 4500K and surface gravity of log g=4.5.
Here are the polarization curves for transits of "A" by both star "B" (R=0.22 R_sun)
and by planet "b" (R=0.0775 R_sun): Kepler-16. The transit
by the companion star produces nearly 5 times the polarization as the planet's transit.
(Here are the theoretical light curves: Kepler-16 light
The presence of large starspots will have effects similar to the transit of a planet,
though in this case we must consider the foreshortening of the spot as it approaches
the limb, which will suppress the effect just where the polarization becomes strongest.
As discussed above, we expect some polarization from hot star atmospheres due to the
electron scattering in the atmosphere.
We might be able to observe such polarization during eclipses of hot stars, though
the effects are small. Here are predicted polarization curves for two identical O7
stars as given above: O-star_eclipse.pdf. Three
cases are shown for different impact parameters of the eclipse track. We see here
that the polarization reversal of the emergent radiation leads to complicated
reversals in the eclipse polarization.