SCATTERING POLARIZATION IN THE Ca ii INFRARED TRIPLET WITH VELOCITY GRADIENTS

We assume an atomic model consisting of the five lowest energetic, fine structure levels of Ca ii (see Figure 1). The excitation state of the atomic system is given by the populations of its 18 magnetic sublevels and the coherences among them. We neglect coherences between different energy levels of the same term (multilevel approximation). Moreover, since the problem we consider here (plane-parallel, non-magnetic atmosphere with vertical velocity fields) is axially symmetric around the vertical direction, no coherences between different magnetic sublevels exist when the symmetry axis is taken for quantizing the angular momentum. We use the multipolar components of each J-level,

$$\begin{equation} \rho ^K_0=\sum _{M=-J}^{+J}(-1)^{J-M}\sqrt{2K+1}\left(\begin{array}{@{}ccc@{}}J & J & K \\ M & -M & 0 \end{array} \right)N_{M}, \end{equation} \tag{ 1 }$$

where K = 0,..., 2J, N_M is the population of the sublevel with magnetic quantum number M, and the symbol between brackets is the Wigner 3j-symbol (e.g., Brink & Satchler 1968). Due to the symmetry of the scattering process (no magnetic field, no polarized incident radiation in the atmosphere's boundaries) in a given level N_+M = N_−M, and the excitation state of the system is described by just nine independent sublevel populations. Consequently, odd-K elements (orientation components) in Equation (1) vanish for all five levels, and the only independent variables of the problem in the spherical components formalism are the total populations of the five levels ($\sqrt{2J+1}\rho ^0_0$); the alignment components (ρ²₀) of levels 2, 3, and 5; and ρ⁴₀ of level 3, whose role is negligible for our problem.

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The statistical equilibrium equations (SEEs) accounting for the radiative and collisional excitations and deexcitations in the five-level system of Figure 1 are given explicitly in Manso Sainz & Trujillo Bueno (2010; hereafter MSTB2010). We have particularized them to the no-coherence case (only ρ^K₀ elements) in Appendix B. The SEEs for the ρ⁰₀ components contain terms that are equal to those appearing in the SEEs for the populations in a standard (no polarization) NLTE problem (e.g., Mihalas 1978), plus higher order terms ∼J²₀ρ²₀ (see Equations (B1)–(B5)). The SEEs for the alignment (ρ²₀ components) are formally similar to the ones for the populations with additional terms ∼J²₀ρ⁰₀ accounting for the generation of alignment from the anisotropy of the radiation field, and (negligible) higher order terms ∼J²₀ρ²₀ and J²₀ρ⁴₀ (see Equations (B6)–(B8)). These equations are expressed in the atom reference frame (comoving system).

Since the radiation field is axially symmetric, just two radiation field tensor elements (J⁰₀ and J²₀) are necessary to describe the symmetry properties of the spectral line radiation. Let I(ν, μ) and Q(ν, μ) be the Stokes parameters expressed in the observer's frame at a given height z, where ν is the frequency, μ = cos θ and θ is the angle that the ray forms with the vertical direction. Then, the corresponding values seen by a comoving frame with vertical velocity v_z with respect to the observer's frame are I'(ν', μ) = I(ν, μ) and Q'(ν', μ) = Q(ν, μ), where ν' = ν(1 − v_zμ/c) and ν = ν'(1 + v_zμ/c) (to first order in v_z/c). Therefore, the mean intensity at the considered height can be expressed from one or another reference frame as

$$\begin{eqnarray} \bar{J}^0_0 &=& \frac{1}{2}\int d\nu \int _{-1}^1 d\mu \phi ^{\prime }_{\ell u}(\nu,\mu) I(\nu,\mu) \nonumber \\ &=&\frac{1}{2}\int d\nu ^{\prime } \int _{-1}^1 d\mu \phi _{\ell u}(\nu ^{\prime }) I(\nu ^{\prime }(1+v_z\mu /c), \mu), \end{eqnarray} \tag{ 2 }$$

where ϕ_ℓu(ν) is the absorption profile (e.g., for a Gaussian profile, we would have ϕ_ℓu(ν) = π^−1/2Δν_D⁻¹exp (− (ν − ν₀)²/Δν_D²), with ν₀ the central line frequency and Δν_D the Doppler width) and ϕ'_ℓu(ν, μ) = ϕ_ℓu(ν(1 − v_zμ/c)), with v_z > 0 for upflowing material. Analogously, the anisotropy in the observer's frame is

$$\begin{eqnarray} \bar{J}^2_0 &=& \frac{1}{4\sqrt{2}}\int d\nu \int _{-1}^1 d\mu \phi _{\ell u}^{\prime }(\mu, \nu) \nonumber \\ && \times [(3\mu ^2-1)I(\nu, \mu)+3(1-\mu ^2)Q(\nu, \mu)], \end{eqnarray} \tag{ 3 }$$

The important quantity that controls the ability of an anisotropic radiation field to generate atomic polarization is the line anisotropy factor for each transition, which can be calculated as

$$\begin{equation} {\sc }w_{\mathrm{line}}={\sqrt{2}} \frac{\bar{J}^2_0}{\bar{J}^0_0}. \end{equation} \tag{ 4 }$$

Its range goes from $\sc w_{\mathrm{line}}=-0.5$ (for a radiation field coming entirely from the horizontal plane) to $\sc w_{\mathrm{line}}=1$ (for a collimated vertical beam).

Due to symmetry, in a non-magnetized plane-parallel medium with a vertical velocity field, light can only be linearly polarized parallel or perpendicularly to the stellar limb. Therefore, choosing the reference direction for positive Q parallel to the limb, the only non-vanishing Stokes parameters are I and Q, and they satisfy the following radiative transfer equations (RTEs):

$$\begin{equation} \frac{d}{{d}s}I = \epsilon _I-\eta _II-\eta _QQ, \end{equation} \tag{ 5a }$$

$$\begin{equation} \frac{d}{{d}s}Q = \epsilon _Q-\eta _QI-\eta _IQ, \end{equation} \tag{ 5b }$$

where s is the distance along the ray. The absorption and emission coefficients are (MSTB2010)

$$\begin{eqnarray} \epsilon _I &=& \,\epsilon _I^{\rm cont}+\epsilon _I^{\rm line} = \, {\eta _{I}}^{\rm cont}B_{\nu }+ \epsilon _0 \nonumber \\ && \times \left[\rho ^0_0(u)+w_{J_uJ_{\ell }}^{(2)} \frac{1}{2\sqrt{2}} (3\mu ^2-1)\rho ^2_0(u)\right], \end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray} \eta _I &=& \,\eta _I^{\rm cont}+\eta _I^{\rm line} = \,{\eta _{I}}^{\rm cont}+\eta _0\nonumber\\ && \times \left[\rho ^0_0(\ell)+w_{J_{\ell }J_u}^{(2)} \frac{1}{2\sqrt{2}} (3\mu ^2-1)\rho ^2_0(\ell)\right], \end{eqnarray} \tag{ 6b }$$

$$\begin{equation} \epsilon _Q\, = \,\epsilon ^{\rm line}_Q\,=\,\epsilon _0 w_{J_uJ_{\ell }}^{(2)} \frac{3}{2\sqrt{2}}(1-\mu ^2)\rho ^2_0(u), \end{equation} \tag{ 7a }$$

$$\begin{equation} \eta _Q\,=\,\eta ^{\rm line}_Q\,=\,\eta _0 w_{J_{\ell }J_u}^{(2)} \frac{3}{2\sqrt{2}}(1-\mu ^2)\rho ^2_0(\ell), \end{equation} \tag{ 7b }$$

where η_I^cont and _I^cont are the continuum absorption and emission coefficients for intensity, respectively. Likewise, η^line_I and η^line_Q are the line absorption coefficients for Stokes I and Q, respectively, while ^line_I and ^line_Q are the line emission coefficients for Stokes I and Q, respectively. The coefficients $w_{J_{\ell }J_u}^{(2)}$ and $w_{J_{u}J_{\ell }}^{(2)}$ depend only on the transition and are detailed in Table (1). The subscripts u and ℓ refer to the upper and lower level of the transition considered, respectively, and B_ν is the Planck function at the central frequency ν₀ of the transition. Note also that

$$\begin{equation} \epsilon _0=\frac{h\nu }{4\pi }A_{u{\ell }}{\phi ^{\prime }_{\ell u}(\mu,\nu)}{\cal N}\sqrt{2J_u+1}, \end{equation} \tag{ 8a }$$

$$\begin{equation} \eta _0=\frac{h\nu }{4\pi }B_{{\ell }u}{\phi ^{\prime }_{\ell u}(\mu,\nu)}{\cal N}\sqrt{2J_{\ell }+1}, \end{equation} \tag{ 8b }$$

where $\cal N$ is the total number of atoms per unit volume.

Table 1. Short List of Ca ii Atomic Data Parameters

λ	u	ℓ	Auℓ	$w_{J_{\ell }J_u}^{(2)}$	$w_{J_uJ_{\ell }}^{(2)}$
(Å)			(s−1)
Allowed transitions
3934 (K)	5	1	1.4 × 108	0	$\sqrt{2}/2$
3969 (H)	4	1	1.4 × 108	0	0
8498	5	2	1.11 × 106	$-2\sqrt{2}/5$	$-2\sqrt{2}/5$
8542	5	3	9.6 × 106	$\sqrt{7}/5$	$\sqrt{2}/10$
8662	4	2	1.06 × 107	$\sqrt{2}/2$	0

Notes. From left to right: the central wavelength, the upper and the lower level of each transition, the radiative rates from NIST atomic spectra database (http://www.nist.gov/physlab/data/asd.cfm), and the atomic polarizability coefficients introduced by Landi Degl'Innocenti (1984).

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With the total absorption coefficient for the intensity, the line of sight (los) optical depth for each frequency is calculated by the following integral along the ray:

$$\begin{equation} \tau ^{\rm los}_{\nu }=- \int \eta _I(\mu ^{\rm los},\nu) \frac{dz}{\mu ^{\rm los}}. \end{equation} \tag{ 9 }$$

The solution to the non-LTE problem of the second kind considered here (the self-consistent solution of the SEEs for the density matrix elements together with the RTEs for the Stokes parameters) is carried out by generalizing the computer program developed by Manso Sainz & Trujillo Bueno (2003a), to allow for radial macroscopic velocity fields. For integrating the RTE, a parabolic short-characteristics scheme (Kunasz & Auer 1988) is used. At each iterative step, the RTE is solved, and $\bar{J}^0_0$ and $\bar{J}^2_0$ are computed and used to solve the SEE following the accelerated Lambda iteration method outlined in the Appendix of MSTB2010. Once the solution for the multipolar components of the density matrix is consistently reached, the emergent Stokes parameters are calculated for the desired los, which in all the figures of this paper is μ = 0.1. This final step is done increasing the frequency grid resolution to a large value in order to correctly sample the small features and peaks of the emergent profiles.

Some technical considerations have to be kept in mind for the treatment of velocity fields. Due to the presence of Doppler shifts, the frequency axis used to compute $\bar{J}^0_0$ and $\bar{J}^2_0$ must include the required extension and resolution, because the spectral line radiation may be now shifted and asymmetric. In our strategy for the frequency grid, the resolution is larger in the core than in the wings, keeping the same frequency grid for all heights.

The cutoff frequency for the core (where resolution is appreciably higher) is dictated by the maximum expected Doppler shift. Thus, the core bandwidth is estimated allowing for a range of 2V_max around the zero-velocity central frequency of the spectral lines, with V_max the maximum velocity found in the atmosphere (in Doppler units). Apart from that, a minimum typical resolution for the core is set to two points per Doppler width (Δν_D). Then, the height with the smallest Doppler width determines the core resolution, and the height with the maximum macro-velocity states its bandwidth.

Furthermore, as frequencies and angles are inextricably entangled (through terms ν − v_zμν/c appearing in the absorption/emission profiles due to the Doppler effect, like in Equation (2)), the maximum angular increment (Δμ_max) is restricted by the maximum frequency increment (Δx_max ≈ 1/2, in Doppler units). Thus, it must occur that Δμ_max · V_max ⩽ 1/2. In the worst case, the maximum allowed angular increment will be smaller (more angular resolution needed) when the maximum vertical velocity increases. Besides this consideration, the maximum angular increment could be even more demanding because of the high sensitivity of the polarization profiles to the angular discretization.

Finally, the depth grid must be fine enough, in such a way that the maximum difference in velocity between consecutive points is not too large, the typical difference being equal to half the Doppler width (|V(z_i) − V(z_{i − 1})| ⩽ 1/2). If the difference is larger, the absorption/emission profiles would change abruptly with height, producing imprecisions in the optical depth increments (Mihalas 1978).

Consider an absorption spectral line with a Gaussian profile emerging from a static atmosphere with a linear limb darkening law,

$$\begin{equation} I(\nu, \mu)=I^{(0)}(1-u+u\mu)\left[1-a\exp \left(-\left(\frac{\nu -\nu _0}{w} \right)^2 \right)\right], \end{equation} \tag{ 10 }$$

where I⁽⁰⁾ is the continuum intensity at disk center, u is the limb darkening coefficient, a < 1 measures the intensity depression of the line, and w its width. In this approximation, we assume that all the parameters are constant. Now imagine that, at the top of the atmosphere, there is a thin cloud scattering the incident light given by Equation (10) and moving radially at velocity v_z with respect to the bottom layers of the atmosphere, supposed static. We will assume that the absorption profile is Gaussian (dominated by Doppler broadening), with width Δν_D. When Δν_D ≪ w, the incident spectral line radiation is much broader than the absorption profile (in fact, for α = Δν_D/w = 0 the absorption profile is formally a Dirac-δ function). Then, from Equations (2) and (3), we can derive explicit expressions for the mean intensity and anisotropy of the radiation field as a function of the scatterer velocity (see Appendix A): $\bar{J}^0_0={\cal I}_0(\alpha;\xi)/2$, $\bar{J}^2_0={\cal I}_2(\alpha;\xi)/4\sqrt{2}$, where the ${\cal I}_{0, 2}$ functions are defined by Equations (A4) and (A5). The behavior of $\bar{J}^0_0$ and $\bar{J}^2_0$ with the adimensional velocity ξ = v_zν₀/(cw) (Figure 2) is most clearly illustrated in their asymptotic limits at low velocities:

$$\begin{equation} \bar{J}^0_0 =\frac{1}{4}\left(1-\frac{a}{\sqrt{1+\alpha ^2}} \right)(2-u) + \frac{a(4-u)}{ 12(1+\alpha ^2)^{3/2}}\xi ^2 + {\rm O}(\xi ^3), \end{equation} \tag{ 11 }$$

$$\begin{eqnarray} \sqrt{2}\frac{\bar{J}^2_0}{\bar{J}^0_0} &=&\frac{u}{4(2-u)}\nonumber\\ && +\frac{a(64-56u+7u^2)}{ 120(2-u)^2(1+\alpha ^2)(\sqrt{1+\alpha ^2}-a)}\xi ^2 + {\rm O}(\xi ^3).\nonumber\\ \end{eqnarray} \tag{ 12 }$$

Equation (11) shows that, for an absorption line (a > 0), $\bar{J}^0_0$ is always increasing with the velocity since the coefficient of ξ² is positive, and this, regardless of the sign of v_z (i.e., regardless of whether the scatterers move upward or downward); if the line is in emission (a < 0), $\bar{J}^0_0$ monotonically decreases. These are the Doppler brightening and Doppler dimming effects (e.g., Landi Degl'Innocenti & Landolfi 2004). An analogous analysis applies to $\bar{J}^2_0$ (Equation (12)). Note that, in the absence of limb darkening (u = 0), the anisotropy vanishes in a static atmosphere, while the mere presence of a relative velocity between the scatterers and the underlying static atmosphere induces anisotropy in the radiation field—hence, a polarization signal. A real atmosphere could then be understood as a superposition of scatterers that modify the anisotropy depending on the local velocity gradient and the illumination received from lower shells. An interesting discussion on the effect of velocities with directions other than radial can be found in Landi Degl'Innocenti & Landolfi (2004, Section 12.4).

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Clearly, all the above discussion depends on the Doppler shift induced by the velocity v_z normalized to the width of the spectral line, i.e., on ξ. A large velocity gradient on a broad line has the same effect as that of a smaller velocity gradient on a narrow line. This is important to keep in mind since the response of different spectral lines to the same velocity gradient will be different, which can help us decipher the velocity stratification. Even different spectral lines belonging to the same atomic species may have very different widths, as for example, the Ca ii IR triplet and the UV doublet studied in the next section.

The discussion above explains the basic mechanism by means of which a velocity gradient enhances the anisotropy of the radiation field. Now we can get further insight into the structure of the radiation field within an atmosphere with velocity gradients from just the formal solution of the RT equation for the intensity (e.g., Mihalas 1978). As before, we neglect effects due to polarization and J⁰₀ and J²₀ are calculated from Stokes I alone. We consider a semi-infinite, plane-parallel atmosphere with a source function S = S₀(1 + βτ_l), where τ_l is the integrated line optical depth in the static limit (hence, the element of optical depth dτ_ν = (r + ϕ[ν(1 − v_z(τ_l)μ/c)])dτ_l, where r = κ_c/κ_l is the ratio of continuum to line opacity). We begin by considering a vertical velocity field v_z(τ_l) = v₀/[1 + (τ_l/τ₀)] (positive outward the star), shown in Figure 3. Equivalently, we may express the velocity in adimensional terms by using ξ = (ν₀/c)v_z/Δν_D (the width Δν_D of the Gaussian absorption profile is assumed to be constant with depth). The parameter τ₀ fixes the position of the maximum velocity gradient. Note that the frequency dependence of the Doppler effect (δν = ν₀v_z/c) is canceled in the adimensional problem, where velocities are measured in Doppler units. It is easy to calculate I_z(ν, μ) numerically at every point in the atmosphere and thus, the mean intensity and anisotropy of the radiation field (Figure 4).

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The rise in $\bar{J}^0_0$ in higher layers with respect to the static case corresponds to the Doppler brightening discussed above. Thanks to the Doppler shifts, the atoms see more and more of the brighter continuum below, which enhances $\bar{J}^0_0$. When the maximum velocity gradient takes place at optically thick enough layers (τ₀ ≳ 1), $\bar{J^0_0}$ is also larger than for the static case, but it decreases monotonically with height in the atmosphere (τ₀ = 10², dotted lines in upper panel of Figure 4). Note that the important quantity that modulates the increase in $\bar{J^0_0}$ is not the maximum velocity but the velocity gradient (difference in velocity between optically thick and optically thin parts of the atmosphere). The larger the gradient, the more pronounced the radiative decoupling is between different heights. An extreme example of such radiative decoupling could be found in supernovae explosions, where the vertical velocity gradients are huge.

In our case (vertical motions), the Doppler brightening implies an enhancement of the contribution of vertical radiation to Equation (3) with respect to the horizontal radiation, with the latter remaining almost equal to the static case (no horizontal motions, no horizontal Doppler brightening). This velocity-induced limb darkening is the origin of the anisotropy enhancement.

However, note that the maximum anisotropy does not rise indefinitely when increasing the maximum velocity. If the velocity gradient in units of the Doppler width is larger than ∼3 (see curves for ξ_max = 5 in Figure 4), the anisotropy at the surface saturates and decreases (even below the curves corresponding to shorter velocity gradients). It is accompanied by a bump around τ_l = 1 when the maximum velocity gradient is taking place at low-density layers (τ₀ ≲ 1). This behavior can be understood using Equation (3). When ξ_max ≲ 3, an increment in ξ_max entails a rise in $\bar{J}^0_0$, $\bar{J}^2_0$, and $\bar{J}^2_0/\bar{J}^0_0$ (w_line) in the upper atmosphere, which means that the velocity gradients enhance the imbalance between vertical and horizontal radiation. However, if ξ_max is above that threshold, $\bar{J}^0_0$ and $\bar{J}^2_0$ rise, but the ratio $\bar{J}^2_0/\bar{J}^0_0$ saturates and diminishes. The reason is that a large velocity gradient makes the absorption profiles associated to almost horizontal outgoing rays ($0<\mu <1/\sqrt{3}$) shifted so much that they also capture the background continuum radiation. Their contributions are negative to the angular integral of $\bar{J}^2_0$ but positive for $\bar{J}^0_0$.

Separating the contributions of rays with angles in the range $1/\sqrt{3}<|\mu |<1$ (which we refer to with the label +) and angles in the range $0<|\mu |<1/\sqrt{3}$ (which we refer to with the label −), the line anisotropy can be written as $w_{\rm {line}}=w^+_{\rm {line}}+w^-_{\rm {line}}=\bar{J}^{2+}_0/\bar{J}^0_0-|\bar{ J}^{2-}_0|/\bar{J}^0_0$. Here, $\bar{J}^0_0$ and $\bar{J}^{2+}_0$ always grow with ξ_max, but $|\bar{J}^{2-}_0|$ only grows appreciably when ξ_max ≳ 3. Therefore, although w_line increases for all velocity gradients, its enhancement is smaller for large velocity gradients than for smaller ones. This effect occurs as well when motions take place deeper (τ₀ ≳ 1) but it is less important and the anisotropy bump and saturation are reduced.

For the considered velocity fields (with a negligible gradient in the upper atmosphere), $\bar{J}^0_0$ and $\bar{J}^2_0$ reach an asymptotic value in optically thin regions. We have verified that this effect does not occur if the velocity gradient is not zero in those layers. In any case, the presence of a large anisotropy in optically thin heights barely affects the emergent linear polarization profiles.

Before considering a more realistic case, a final illustrative example is considered. In this case, we assume the same parameterization of the velocities than in the previous example, but now we solve the complete iterative RT problem with a two-level atom model and a specific temperature stratification. Consequently, the source function and the anisotropy are consistently obtained in a moving atmosphere. The intensity source function is S_I = r_νμS^line_I + (1 − r_νμ)B (e.g., Rybicki & Hummer 1992), with r_νμ = ϕ'_ℓu(ν, μ)/(r_c + ϕ'_ℓu(ν, μ)) and the expression for the line source function remains formally equal to that of the static case, being $S^{\rm {line}}_I=(1-\epsilon)\bar{J}^0_0+\epsilon B$, where B is the imposed Planck function, is the inelastic collisional parameter, and $\bar{J}^0_0$ is calculated with Equation (2).

The qualitative behavior explained in the previous subsection is maintained in these two-level atom calculations. For small values (large NLTE effects), the source function $S_I \approx \bar{J}^0_0$ shows Doppler brightening effects and its surface value depends on the maximum velocity gradient and on the maximum background continuum set by the photospheric conditions (upper panel in Figure 5). The anisotropy rises proportionally to the velocity gradient until a saturation occurs (lower panel in Figure 5). A similar behavior is found when the maximum velocity gradient occurs higher in the atmosphere (see Figure 10).

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In a static atmosphere, the radiation field anisotropy is dominated by the presence of gradients in the intensity source function (Trujillo Bueno 2001; Landi Degl'Innocenti & Landolfi 2004), which can be modified via the Planck function (equivalently, the temperature). In the dynamical case that we are dealing with, the slope of the source function is also modified due to the existence of velocity gradients thanks to the frequency decoupling caused by relative motions between absorption profiles (Doppler brightening). In general, both mechanisms act together (velocity-induced and temperature-induced modification of the source function gradient) and the ensuing anisotropy and the emergent linear polarization profiles are modified accordingly.

It is important to note that the adimensional velocity ξ depends both on the velocity and also on the line Doppler width because $\xi (\tau)= \delta \nu /\Delta \nu _D= v_z/\sqrt{2k_BT/m}$, with k_B the Boltzmann's constant, T the temperature, and m the mass of the atom. In the photosphere, where velocities are much lower than in the chromosphere, ξ is expected to be negligible. In the chromosphere, plasma motions are important and the temperature is still comparable to that of the photosphere, inducing ξ to be controlled by the velocity field. However, for layers in the transition region and above, the high temperatures reduce the value of ξ. In any case, at these heights, the density is so low that, although ξ (and consequently the anisotropy) could have a highly variable behavior, the emergent polarization profiles of chromospheric lines will not be sensitive to them.

For simplicity, we set linear velocity fields (constant velocity gradient along z) between z = −100 and 2150 km (see Figure 6). Consequently, the adimensional velocity field ξ_z has a non-monotonic behavior due to its dependence on the temperature (upper right panel in Figure 6). In the chromosphere, where the Ca ii triplet lines form, ξ_z is dominated by the macroscopic motions. Here, the velocity gradients produce variations in the anisotropy of the triplet lines that agree with the behavior outlined in the previous sections. Namely, an amplification and a subsequent saturation of the anisotropy factor due to the significant velocity gradient at those heights (see the lower panels and middle right panel in Figure 6). Above the chromosphere, on the contrary, the temperature dominates (ξ_z stabilizes and diminishes) and the anisotropy slightly decreases with height.

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If the (adimensional) velocity gradient is negligible where the line forms (around $\tau ^\mathrm{los}_{\nu _0}\sim 1$), the anisotropy remains unaffected. Otherwise, if a spectral line forms at very hot layers, where the absorption profiles are wider and their sensitivity to the velocity gradients is lower, the Doppler brightening will not be so efficient amplifying the anisotropy. This is the case of the anisotropy of the Ca ii K line (middle left panel in Figure 6). Compare how the slope of ξ_z is smaller where the Ca ii K line forms (black line on Figure 6) than where the triplet lines do. Consequently, the enhancement of the line anisotropy through the presence of velocity gradients in this line is reduced.

All our calculations demonstrate that the anisotropy in the Ca ii IR triplet can be amplified through chromospheric vertical velocity gradients. This results suggest that the same occurs with the ensuing linear polarization profiles.

To investigate the effect of vertical velocity fields on the emergent fractional polarization profiles we perform the following numerical experiments. First, we impose velocity gradients with the same absolute value but opposite signs (top left panel of Figure 7). The resulting emergent Q/I profiles (remaining left panels of Figure 7) are magnified by a significant factor (>2 for all the transitions) with respect to the static case (black dotted line). The linear polarization profiles have the same amplitude, independently of the sign of the velocity gradient. Another remarkable feature is the asymmetry of the profiles, having a higher blue wing in those cases in which the velocity gradient is positive and a higher red wing when the velocity gradient is negative, independently of the velocity sign. Note also that the Q/I profile is shifted in frequency due to the relative velocity between the plasma and the observer.

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As a second experiment, we consider different velocity fields with increasing gradients (right upper panel in Figure 7). In the ensuing Q/I profiles we see that the larger the velocity gradient, the larger the frequency shift of the emergent profiles and the larger the amplitude. In all transitions, one of the lateral lobes of the signal remains almost constant. Thus, what really changes is the central part of the profiles, being a "valley" in the λ8498 line and a "peak" in the other two transitions. To quantify these variations, we define (Q/I)_pp (peak-to-peak amplitude of Q/I) as the difference between the lowest and the highest value of the emergent Q/I signal, which is also a measure of its contrast. Note that, as expected from the first experiment, (Q/I)_pp depends only on the absolute value of the gradient. Figure 8 summarizes these results.

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The sensitivity of the linear polarization to the velocity gradient can be measured approximately as commented in Section 3.1, using a parameter α = Δν_D/w that accounts for the difference in width of the absorption profile with respect to the emergent intensity profile. If α ∼ 1, small adimensional velocities will produce large changes in shape; if α ≪ 1, much larger ξ_z values are needed for the same effect. In the case of the IR triplet lines, α ∼ 0.355 in the formation region of λ8498 and around 0.29 and 0.285 in the formation region of λ8542 and λ8662 (having wider profiles), respectively. Then, the former is more sensitive to velocity variations in its formation region (Figure 8). Finally, the K line has $\alpha \,(\tau ^{\rm {los}}_{\nu _0}=1) \sim 0.015$, a low value due to its wider spectral wings.

The enhancement of the polarization signals is a consequence of the increase in the anisotropy. Therefore, since this increase is produced by the presence of velocity fields, the polarization signals of the Ca ii IR triplet are sensitive also to the dynamic state of the chromosphere.

In order to get physical insight on the formation of the emergent polarization profiles, we use an analytical approximation. Following Trujillo Bueno (2003), the emergent fractional linear polarization for a strong line at the central wavelength can be approximated with

$$\begin{equation} \frac{Q}{I} \approx \frac{3}{2\sqrt{2}} (1-\mu ^2) \left[ {\sc }w^{(2)}_{J_u J_{\ell }} \cdot \sigma ^2_0(J_u) - {\sc }w^{(2)}_{J_{\ell } J_u} \cdot \sigma ^2_0(J_{\ell }) \right]. \end{equation} \tag{ 13 }$$

The symbols ${\sc }w^{(2)}_{J J^{\prime }}$ are numerical coefficients already introduced in Section 2. The quantities σ²₀(J_u) and σ²₀(J_ℓ) are the fractional alignment coefficients (σ²₀ = ρ²₀/ρ⁰₀) evaluated at τ^los_ν = 1 for the upper and lower level of the transition, respectively. This is the generalization of the Eddington–Barbier approximation to the scattering polarization and establishes that changes in linear polarization (for a static case) are induced by changes in the atomic alignment of the energy levels.

Our calculations show that vertical velocity fields with moderate gradients (≲ 10 m s⁻¹ km⁻¹ in a linear velocity field, as the ones shown in the figures) do indeed produce variations in the fractional alignment, which are small for |σ²₀(J_u)| and significant for |σ²₀(J_ℓ)| (see Figure 9). The lower level alignment is the main driver of the changes produced in the Q/I signals. This is strictly true for the λ8662 line, whose upper level with J = 1/2 cannot be aligned (zero-field dichroism polarization; Manso Sainz & Trujillo Bueno 2003b). In the other transitions, a certain influence of the upper level alignment becomes important only for large gradients. The reason is that the K transition is so strong in comparison with the IR triplet that it is dictating the common upper level 5 alignment (Figure 1). In fact, σ²₀(J₅) is driven by the K-line anisotropy which, at chromospheric heights, is almost unaffected for the considered velocity gradients, as we discussed in Section 4.1 (Figure 6). Thus, the strong H and K lines feed population to the upper levels and the K line controls the alignment of the ²P_3/2 level (see Figure 1), while the polarization signals of the IR triplet change with velocity fields affecting σ²₀(J_ℓ) (through the anisotropy enhancement).

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To illustrate the well-known link between the alignment and the anisotropy we can follow the next reasoning. For the Ca ii model atom that we deal with in this work, it is possible to derive a simple analytic expression that relates the anisotropy and the alignments for the λ8542 transition. Making use of Equations (B3) and (B7) and neglecting second-order terms and collisions, we find that

$$\begin{equation} {2}\sigma ^2_0(J_5)-{\sqrt{7}}\sigma ^2_0(J_3) \simeq {\sc w}_{{\rm line}}(3\rightarrow 5). \end{equation} \tag{ 14 }$$

As before, we can roughly assume that σ²₀(J₅) ∼ constant at the chromosphere because it is controlled by the K line. Then, Equation (14) suggests that, if the radiation anisotropy increases at that heights, an amplification of |σ²₀(J₃)| occurs (note that σ²₀(J₃) is negative for these lines). A more aligned atomic population produces a more intense scattering polarization signal.