Using Mathematica to model observations of surface magnetic fields

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1 Using Mathematica to model observations of surface magnetic fields version of -Mar-0 (w/ changes marked in red type) Oleg, Thanks for the plots and further info on how Stokes V is computed from B z. While I can appreciate that your code is much more sophisticated, including, for example, nonlinear effects, my interest for now is to generate Stokes V flux profiles from my Mathematica code in the simplest realistic approximation, as described below. ü Basic Milne-Eddington model with linear dependence on strengths of both line and magnetic field From my recent background reading, namely parts of the Mathys (989) review, as well as reviews Mary wrote when she was visiting Gregg in Kingston last fall, it seems the simplest model is based on the Milne-Eddington formalism for line formation. Assuming that the line strength is in the linear part of the curve of growth, then for a moderate strength field, the Stokes V intensity on the stellar surface scales linearly with the line-of-sight field component H z, () V º -g H z Dl z I where the Gaunt factor g and Dl z are just constants given by atomic physics, and the I term reflects the fact that a net StokesV stems from a difference between the Zeeman split components for the right (R) vs. left (L) circular polarizations. For a Planck function that varies linearly with optical depth, B(t) = B o H + b o tl, we find that, for a Milne-Eddington model of a line with strength h and profile function y(l), (a) I = m B h o b y o H+hyL (b) º m B o b o h y, where the latter approximation applies in the case of suitably weak line with h `, so

2 FV-query-v.nb where the latter approximation applies in the case of suitably weak line with `, so that the line depth just scales linearly with the line strength. For simplicity of exposition, I assume this in the analysis below; but it is not too difficult to put back the full /H + hyl factor, as needed. When () is integrated over the whole disk, this results in a measurable net wavelength shift that provides the basic diagnostic of the disk-integrated longitudinal field, viz. () Xl R \ - Xl L \ g Dl z = XH è z\ = p Ÿ p 0 f Ÿ Hz 0 m m = p Ÿ p è 0 f Ÿ 0 H z m m where the wavelength shift is defined in terms of the Stokes V flux profile, (4) Xl R \ - Xl L \= - W l Ÿ - FV HlL l and the last equality introduces the notation, H è z ª m H z. It was only after understanding this that I noted how the integral weighting here has an extra factor m compared to the usual mdm that one uses to compute a surface flux, due essentially to the m scaling of the line intensity derivative in () above. In practice, my calculation of this is done as an integral over the projected surface area of the disk, using the fact that (5) -mdm df = pdp dj = dx dy. Here p is the impact parameter of the line-of-sight to disk center, and x=pcosj and y=psinj are associated Cartesian coordinates on the sky, defined such that y is along the projected rotation axis, and all lengths here are in units of the stellar radius, i.e. Rª. Specifically, the form of the integral in () that I now carry out is (6) XH è z\ = -x x pÿ Ÿ- H è zhx, yl y. - -x My earlier results did not include this extra factor m in the weighting of H z Hx, yl, but now I do. Indeed, in our discussions, you mentioned accounting for limb darkening, but this factor seems to stem from the same physics, namely the B o b o factor that sets the variation of temperature with vertical optical depth.

3 FV-query-v.nb Modeling Stokes V line-flux profiles ü Aside: initial attempts to compare with your plots ü Surface integral to get Stokes V flux To convert to a Stokes V flux, we need again to carry out a surface integral over the Stokes V intensity, viz. (9a) = Ÿ 0 p f Ÿ 0 VHm, f, ll m m (9b) = - g D l 4 p Ÿ yhl-xl - x Ÿ- -x -x H è z y (9c) = - g D l 4 p Ÿ XH è z\ y - yhl-xl x In the second equality, I've used the fact that the profile function y for material at a coordinate position x on the star is centered on a Doppler-shifted line-center given by l o =x, noting also that, since this is independent of y, it can be pulled outside the y-integral. (Recall that x and l are both normalized to vary over the range ±, through the scalings Rª and Vsiniª.) I ve also assumed a standard Eddington gray atmosphere to evaluate the B o and b o (=/) coefficients in the solutions for V, I, and. So the question now is how to carry out the last integral over x. For the usual case that intrinsic profile is symmetric, as, e.g. a Gaussian, we have y(lx)=y(x-l), which further implies yhl-xl = yhx-ll. Using this, we can integrate (9c) x by parts, yielding (9.) = g D l 4 p Ÿ yhx - ll XHè z\ y - x x - AXH è z\ y HxL yhx - llf - +. If we consider the idealized limit that the intrinsic broadening of the line intensity -- e.g. from Stark, natural, thermal, or turbulent broadening -- is small compared to the rotational broadening, then we can approximate y as a Dirac d-function, yielding () = g D l JB XH è z\ y F - AXH è 4 p x z\ y HxL dhx - llf x=l - +. Note that we could also have obtained this directly from (9c) by using the d-function

4 4 FV-query-v.nb Note that we could also have obtained this directly from (9c) by using the -function property, (0) Ÿ - bd x = As a check, we can use () to compute the mean longitudinal field, defined by () XH è z\ = Xl R\ - Xl L \ = - g Dl z g Dl z W l Ÿ l. Integrating again by parts, and using that the equivalent width W l =h/, we find () XH è z\ = pÿ - -x x Ÿ- H è z y -x = p Ÿ 0 p f Ÿ 0 Hz m m, which indeed is the same as given in eqns. () or (6). So at least this is consistent. Thus in the narrow line limit we can use () to compute the Stoke V flux profile. Otherwise, the local contribution of XHè z\ y (at position x on the disk) to the observed x Stokes V at wavelength l has to be weighted by the profile function y(x-l), carrying out the full x-integral given in (9.). ü Mathematica result for Stokes V flux vs. wavelength and rotational phase The above indicates that the Stokes V flux profile can be viewed as scaling with the lateral (x) gradient of the vertically (y) averaged, longitiudinal (z) component of the surface magnetic field, as given in eqn. (). But the further analysis shows that the boundary terms also play a role. But we can also derive this from the direct integration (9c), weighting B z by the local profile-gradient yhl-xl. So here is a comparison of the dynamic spectra for the db z /dx form (left) with profile gradient cases with a narrow (s=0.0; middle plot) and realistically broadened (s=0.8; right plot) Gaussian profile.

5 FV-query-v.nb 5 right plot) Gaussian profile. The agreement from the different models is quite good. ü Line profile stacks ü Conclusion So it seems there are two equivalent ways to cast the integrals for Stokes V profile, either as a profile-weighted field gradient (), or as a profile-gradient-weighted longitudinal field (9c). Both results seem in good agreement with the new profiles you ve sent. Stan