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 depth.

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.

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.

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.)

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

Polarization compared.

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.

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.)

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

write(96,*) id,wlam(ij),abso(ij),emis(ij),scat(ij),sc(ij,id)

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.

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.

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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: Doppler Polarization.

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: v=5 km/s, v=7.5 km/s, v=10 km/s, v=12.5 km/s, v=15 km/s, v=17.5 km/s, v=20 km/s, v=25 km/s, v=30 km/s. 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.

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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 curves.)

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.