POLARIZATION DIAGNOSTICS FOR COOL CORE CLUSTER EMISSION LINES

Our targets are well-studied central cluster galaxies with extensive low excitation filament systems, moderate power radio emission, and clear indications of interaction between the X-ray and radio plasma. Observations utilize a 22 × 2 arcsec spectrograph slit, configured as described in Section 2.2.

M87 has a well-known low-ionization optical filament system distributed around the periphery of the inner radio lobes and jet. We recently discovered FUV C iv line emission exactly coincident with this material but arising from gas a factor of 10 higher in temperature, consistent with our prediction for a model in which the filaments are excited by Spitzer thermal conduction from the hot coronal gas (Sparks et al. 2009; Sparks et al. 2012). Spectropolarimetry observations were acquired with three slit locations and orientations, as shown in Figure 1.

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NGC 4696 is the central dominant galaxy in the classical cool-core Centaurus Cluster. This well-studied system formed the basis of the suggestion by Sparks et al. (1989) that mergers in the presence of thermal conduction can be the cause of the X-ray excess and line emission, not cooling flows. The optical filaments are dusty with normal extinction characteristics indicative of a merger origin. The X-ray morphology is similar to that of the optical, though it is extended over a much larger scale, and the radio emission is compact and has a steep spectrum on similar scale to the optical. Figure 2 shows the two slit positions used.

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PKS 0745-19. Heckman et al. (1989), Donahue et al. (2000), and Wilman et al. (2009) show that the optical line emission forms a roughly triangular or conical shape to the west of the nucleus, with two dominant arms of emission. The radio source is irregular and coincident in scale with the optical filaments, which are located primarily to the south. Slits centered on the nucleus with position angles 90° and 45° were used (see Figure 3).

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Hydra A displays the now classical feedback situation (McNamara et al. 2000) with X-ray cavities unarguably at the locations of the outwardly propagating radio jets. The optical line emitting gas spans the region between the radio core and radio knots, shares the S-symmetry of the radio emission, and overlaps with the radio emission only at the edges of the knots (Baum et al. 1988). Our long slit, in p.a.≈20°, runs approximately in the direction of the radio source, orthogonal to an edge-on dust disk (see Figure 4).

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The filaments in all of these targets span a range of brightnesses, and there is a range of line-ratios within the filaments. Although we know that filaments are dusty, we do not expect significant polarization either from scattering or dichroic absorption. In M87, for example, the optical depth τ ≈ 0.01 (Sparks et al. 1993), which would result in a net dichroic polarization for Galactic dust of p_max = 0.09E(B − V), and hence p < 0.03%. Dust characteristics differ from Galactic dust characteristics, we expect to recognize dust through its additional effect on the background galaxy continuum polarization and not just at wavelengths of optical line emission.

We obtained VLT UT1 spectropolarimetry of optical emission filaments using FORS2 in long slit spectropolarimetric mode. In this mode the light entering the spectrograph encounters a polarization slit mask, a rotatable half wave plate, a Wollaston prism to split the polarization o- and e-beams, and a grism to disperse the light. The polarization mask, required so that the dual polarization beams do not overlap on the detector, results in a long slit comprised of 22 arcsec segments. For our analysis, we used only the single 22 arcsec segment centered on the target of interest, as illustrated in Figures 1–4. We used the 300V+10 grism to obtain spectra from ≈450 nm to ≈900 nm and the GG435 order sorting filter. The slit width was 2 arcsec, chosen to be wide in order to maximize the light gathered on the detector.

For a given half wave plate setting, we acquire a single Stokes parameter, and to reduce systematics, the same Stokes parameter is observed with the beams reversed using rotation of the half wave plate. Hence, at least four wave plate rotations are required for a complete set of linear Stokes polarization spectra. Our observations used half wave plate rotation angles of 0, 22.5, 45, and 67.5 deg.

The chosen spectral window encompasses the strong low excitation red emission lines, Hα, [N ii] 6548,6584, [S ii] 6717, 6730, and [O i] 6300 and provides adequate spectral resolution to separate them, though the Hα and [N ii] lines overlap. Weaker lines include [O iii], He i, and [N i]. In principle, the mix of recombination and collisional lines allows us to contrast any polarization found between the different excitation mechanisms. All observations were taken using the ESO VLT service mode and an observing log is presented in Table 1.

Standard star observations were provided to us, of both polarized and unpolarized standards, using the same observing procedures as the target clusters.

A master bias frame was prepared from the median of 55 bias calibration frames. Prescan and overscan rows were used to match the bias level to each data frame and hence to subtract the master bias. A pixel sensitivity flat field (P-flat) was derived separately for the o- and e-beams using white light dome flats, and each flat field section was divided by its row average, where, to a good approximation, the rows correspond to the wavelength direction. The data were all divided by the P-flat after debiassing.

The two-dimensional (2D) subsections of the data arrays corresponding to the two different polarization beams were extracted, and arc line lamp spectra were used to rectify and wavelength calibrate the data. A shear was applied to the data frames to straighten arc lines in the y-direction only, and a second shear was applied to straighten the spectra in the x-direction. Wavelength calibration was assumed to be the same for all observations. We did not resample the data in the wavelength direction, but we did provide an external lookup table of wavelengths for each pixel. The spatial separation of the two polarization beams was measured using standard star observations. Hence, with spline interpolation from each original data frame, we derived two frames, one for each polarization beam, spatially and spectrally registered. The accuracy achieved is much better than a single pixel, so we anticipate no significant error term from these procedures as line polarization measurements integrate over multiple pixels.

A simple cosmic ray rejection algorithm was applied to the data frames by comparing each spectrum of a target to the median of all similar spectra, typically 16. 2D spectra for each retarder configuration were co-added (separately for the o- and e-beams).

To derive the polarization information, there are two methods available. One is the difference method, and the other the flux ratio method (Miller et al. 1987). We processed the data using both techniques but found no significant difference in the results. For the flux ratio (FR) method, the normalized Stokes parameters are given by q = (R_q − 1)/(R_q + 1), where $R_q= \sqrt{(I_0^o/I_0^e)/(I_{45}^o/I_{45}^e)}$, and u = (R_u − 1)/(R_u + 1), where $R_u= \sqrt{(I_{22.5}^o/I_{22.5}^e)/(I_{67.5}^o/I_{67.5}^e)}$. For the O−E difference method (OE), the normalized Stokes parameters are given by $q=0.5((I_0^o- I_0^e)/(I_0^o+I_0^e))-0.5((I_{45}^o-I_{45}^e)/(I_{45}^o+I_{45}^e))$ and $u=0.5((I_{22.5}^o - I_{22.5}^e)/(I_{22.5}^o + I_{22.5}^e)) - 0.5((I_{67.5}^o-I_{67.5}^e)/(I_{67.5}^o + I_{67.5}^e))$.

The retarder offset angles, as provided at the ESO Web site, were also subtracted from the derived position angle data⁷.

Adjustable smoothing parameters in both the y (spatial) and x (wavelength) directions were allowed in the processing, but in the end, we used only data at the highest resolution in order to minimize correlated noise terms. Where appropriate, a throughput correction was applied, derived from the FORS total efficiency as provided by the FORS2 exposure time calculator on the ESO Web site, interpolated and corrected to account for the slowly changing pixel size with wavelength.

We processed the standard star observations in the same fashion as the data, Table 2, and found excellent agreement with the expected values for the polarization and position angle. Figure 5 shows the derived polarization and position angle as a function of wavelength for the polarized standard star Vela 1. The agreement with expectation is on the order of 0.1% in polarization degree and within 1° in position angle over most of the spectrum. There is no significant difference between the two reduction methods (OE or FR) for the standards or any of the targets, and hence throughout we describe only the FR method for convenience. The subsequent analysis steps are illustrated and described within the results Sections 3.1 and 3.2.

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The central region of M87 contains highly polarized synchrotron jet emission and emission from the nucleus, also likely synchrotron in origin (Perlman et al. 2011). This serves as an additional check on our methods while providing astrophysically interesting results. The slit positions for M87 are shown in Figure 1. The slit passes through the nucleus then extends along the bright emission filament north of the famous synchrotron jet, shown in the upper panel. The lower contrast image in the lower panel shows that the compact source within the jet, HST-1, which underwent a massive outburst peaking around 2005 lasting several years (Perlman et al. 2003; Madrid 2009), is also included in our slit. The separation between the nucleus and HST-1 is only 0.8 arcsec, yet they are clearly and cleanly separated in the spectra (see Figure 6 inset), illustrating the excellent seeing conditions for these observations.

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Figure 6 shows the overall data processing approach for the example of M87. The basic CCD reductions, geometric corrections, and wavelength calibrations lead to separate o- and e-beam images, which are combined using the FR method to yield Stokes q and Stokes u images. To derive a Stokes I image, it was necessary to coadd the polarization spectra and remove night sky emission lines. We did this by first deriving a spatially averaged continuum profile (y direction) of the galaxy in a region of the spectrum unaffected by emission lines from the sky or from the activity in the galaxy. We also established a mask indicating the location of emission lines within the objects. At each wavelength, the continuum spatial profile was scaled linearly to the data at that wavelength omitting regions with internal emission lines. The intercept of the linear fit, corresponding to a constant offset, yielded a model of the night sky line distribution that was subtracted from the total intensity image. The resulting image serves as the "Stokes I" total intensity image, I(λ, y). The scaled model continuum and night sky emission line maps were both subtracted to yield images of the galaxy line emission total intensity (see Figure 6).

Given that the spectra all contain a mixture of sources, including continuum stellar emission from the galaxy, line emission from the galaxy associated with its radio source, and, in M87, optical synchrotron continuum emission, it is appropriate to work with the "polarized flux" rather than the polarization degree. This allows us to separate the different constituents of the polarization and study the individual polarization of discrete sources separately. We derive the polarized flux, p_f, and corresponding total intensity Stokes parameters, Q, U, as Q = Iq, U = Iu, and $p_f = \sqrt{Q^2+U^2} = I p_d$ where p_d is the polarization degree.

To examine the M87 synchrotron sources, we extracted spectra three pixels wide spatially, centered on both the nucleus and HST-1. With this width, there is no overlap between the two extractions (HST-1 and the nucleus are measured to be 3.13 pixels apart using quadratic fits to spectrally averaged spatial profiles). We will use the complete polarized flux spectra below when looking at line emission, but to check the nuclear polarization and HST-1 polarization, we used these extracted spectra of the total intensity and polarized flux. We also extracted a galaxy continuum spectrum from a region away from line emission, 3.5–7 arcsec SE of the nucleus and linearly scaled and subtracted this from the total intensity spectra. Comparing the mean values of the polarized flux and the individual Stokes Q, U spectra in the region 550—600 nm to the galaxy-subtracted total intensity levels, we derived a nuclear polarization of 12.0% and a polarization for HST-1 of 23.4%. The formal statistical errors are negligible, and we are likely dominated here by systematic errors from the galaxy subtraction process.

The position angle of the nuclear polarization electric vector around ≈600 nm is 128°  ±  03 (uncertainty from the measured dispersion of the position angle), while the inner jet has position angle ≈290° (Cheung et al. 2007), i.e., a misalignment of ∼18°. If the polarization is synchrotron, it is normal to consider the magnetic field position angle, which is at 90° to the electric vector, hence at p.a. 38°, or ∼18° from perpendicular to the jet. HST-1 is undoubtedly synchrotron, and its magnetic vector position angle is 24° ± 04, which is ≈94° from the jet axis, close to perpendicular to the jet.

For HST-1, Perlman et al. (2011) found a relatively stable magnetic vector position angle of ∼28° for HST-1, with polarization degree ranging 20%—40%, a nuclear position angle (electric vector) varying wildly between 100° and 180°, and polarization of 1–14%. Hence, we are comfortably within this range and in good agreement for HST-1.

Figures 7–10 show images of line emission spectra and polarized flux spectra. As for the case of the M87 nucleus described above, we scale and subtract galaxy continua spectra to derive the line emission spectral images shown. As described above, we also use polarized flux rather than polarization degree to separate different components more easily than polarization degree would allow. For example, again in the case of M87, the polarization degree spectrum dips dramatically at Hα, but the polarized flux spectrum shows a smooth uninterrupted continuum through this region (see Figures 11–13, discussed below).

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To process line emission polarization measurements, we determined a spatial extraction region (range of y values), and then extracted spectra for Stokes I, Q, U, and the line emission image (which is the galaxy continuum subtracted Stokes I image). For each line or group of lines, we fitted a straight-line continuum through emission-free regions of the spectrum on either side of the lines, for both the Q and U spectra. Stokes parameters are linear, and hence if there is emission line polarization, this would be revealed as an additive component to the Stokes parameter spectra in the vicinity of the emission line. Hence, the correct procedure to determine whether line emission polarization is present is to subtract the underlying continuum of the Stokes Q and U spectra, which was done. The estimate of the emission line polarized flux and degree are then:

$$\begin{equation*} p_f = \sqrt{Q_s^2 + U_s^2} - \Delta p \end{equation*}$$

$$\begin{equation*} p_d = p_f/I_e , \end{equation*}$$

where Q_s and U_s are the total values of the continuum subtracted Stokes parameters in the emission line region, I_e is the total value of Stokes I in the emission line region, and Δp is a bias term, described below. As before, the position angle is (1/2)tan ⁻¹(U_s/Q_s) + ϕ, where ϕ is the spectrograph slit position angle on the sky.

For low values of polarization, the positive definite nature of the polarized flux causes a bias towards positive values. If the true polarization is zero, then the expected value of the polarized flux is $\Delta p_c = \sqrt{{{{\sigma _Q^2+\sigma _U^2}}}}$, where σ_Q and σ_U are the uncertainties on Q and U. We estimated σ_Q and σ_U empirically from the root mean square of the residuals from the straight-line continuum fits. Within the line emission region, the count level can be extremely high, and hence we adjusted the derived uncertainties scaled from the empirical continuum level assuming Poisson counting statistics. That is, the assumed bias on the polarized flux is $\Delta p = \sqrt{{{{\sigma _Q^2+\sigma _U^2}}}}$$\sqrt{I_e/I_c}$, where I_c is the continuum Stokes I and I_e is the total flux in the emission line.

Figures 11– 13 illustrate for the example of the M87 nucleus Hα + [N ii] complex. Figure 11 shows the total flux per pixel for the M87 nucleus in the vicinity of the Hα + [N ii] lines, or Stokes I. The solid vertical line shows the center of the complex, and the dotted lines show the bounds used to define the location of the line emission. The (red) crosses indicate the continuum region that was used for fitting purposes. Figure 12 shows the Stokes parameters Q and U. An arbitrarily scaled Stokes I is included for reference, as is the scaled polarization degree showing the strong dip at the locations of the emission lines. For the nucleus of M87, the continuum is highly polarized and well-described as a power-law continuum (see above).

We fitted straight lines using the regions indicated by red crosses to the Stokes Q and U and subtracted the continua as described above. The results of this subtraction are shown in Figure 13, which also illustrates the necessity to remove the bias of the polarized flux due to its positive definite character. The upper red line shows the unbiased polarized flux, i.e., the quadratic sum of the continuum-subtracted Q and U spectra. If the line polarization is zero, then we expect, using the procedure described above to derive a noise model, the dotted red line shown. Clearly, this mimics the behavior of the data quite closely.

To correct the polarized flux spectrum, we therefore subtract this bias term from the polarized flux, producing the solid black line, which effectively removes the apparent polarized flux excess at the location of Hα + [N ii]. The final derived polarization is p = 0.00038 ± 0.00033 for this line complex.

This is the generic procedure followed to populate the primary results presented in Table 3. Table 3 includes polarization measurements for all slit positions in all target objects for the Hα + [N ii] complex, a narrow Hα-only region selected to be of width 2 nm centered on Hα, and the strongest lines, which are [O iii] 5007, [O i] 6300, and [S ii] 6717+6731. Table 3 presents a summary of the results, with polarization upper limits for all strong lines in all objects. Since all measurements are upper limits, we do not present derived position angles as they would be meaningless.

3.2.1. M87

3.2.2. Hydra A

The acquisition image of Hydra A (see Figure 4) shows a compact, edge-on, prominent dust lane through the center of the galaxy running in position angle ≈100°. A lobe of emission at the west end of the dust lane further distorts the galaxy contours. The dust lane also marks the location of a rotating line-emitting disk (Heckman et al. 1989). The long slit runs in position angle 20°, approximately along the axis of the large-scale radio jet. The long slit line emission spectrum, Figure 8, shows strong lines at the location of the nucleus and dust lane, with a patch of emission to the north. The presence of dust distorts the background appearance of the line emission 2D spectrum, though this does not affect our conclusions.

The polarized flux data (see Figure 8) show elevated levels of apparent polarization at the position of the dust lane, and also excess emission at the positions of Hα + [N ii] and [S ii]6717, 6731. However, as shown in Figure 14, this also is a consequence of the positive definite nature of the polarized flux in the presence of higher noise due to Poisson counting statistics. The derived position angle is quite close to that of the dust lane, but again the uncertainties are too large to allow us to conclude we have a definitive measurement of polarization, formally only 1.25σ for Hα on the nucleus. We process the nucleus and the patch of emission offset from the nucleus separately and list both in Table 3.

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3.2.3. NGC 4696

3.2.4. PKS0745-19

Many different sources of energy and excitation mechanisms have been considered to drive the Hα emission filaments in cool core galaxy clusters. For NGC 1275 (Sabra et al. 2000; Conselice et al. 2001) and M87 (Sabra et al. 2003), photoionization by the central AGN, the intracluster medium, young hot stars, and shock heating have all been discussed as the underlying physical mechanisms involved. The conclusion was that "neither shocks nor photoionization alone can reproduce the emission line intensity ratios" (Voit & Donahue 1997) and that some additional source of heating must be present. A study of the optical line ratios in A2597 led Voit & Donahue (1997) to rule out shocks as an excitation mechanism and to conclude that although hot stars might be the best candidate for producing the ionization, even the hottest stars could not power a nebula as hot as observed, so another non-ionizing source of heat must contribute at least a comparable amount of power. Similar conclusions were reached by Hatch et al. (2007) for a number of cool core clusters, and they also note that heating by thermal electrons from the intracluster medium is a plausible mechanism.

4.1.1. Conduction

4.1.2. Saturated Conduction

It has further been noted that in the tenuous intragalactic medium, where there is a large temperature difference between the medium ∼10⁷ K and the filaments ⩽10⁴ K, the electron mean free paths might be sufficiently large so that standard diffusive (Spitzer) conductivity is no longer applicable. Under these circumstances, the conduction becomes "saturated" at a value around the maximum heat flux in a plasma on the order of (Cowie & McKee 1977)

$$\begin{equation} q_{\rm sat} \approx f \frac{3}{2} (n_e k T_e) v_{\rm char}, \end{equation} \tag{ 1 }$$

where n_e and T_e are the electron number density and temperature, respectively, and v_char is a characteristic velocity that one might expect to be on the order of the electron thermal velocity $v_e = \sqrt{3kT_e/m_e}$, where m_e is the electron mass. This is because when conduction reaches its saturated limit, the electrons no longer diffuse (short mean free path) but rather are able to stream freely (mean free path larger than the characteristic local temperature distance scale, T/∣∇T∣). The reduction factor f ≈ 0.4 (Cowie & McKee 1977) accounts for a charge neutrality requirement, which we discuss below. Sparks et al. (2004) have noted that saturated conduction, using the formula (Cowie & McKee 1977)

$$\begin{equation} q_{\rm sat} = 0.4 \left(\frac{2kT_e}{\pi m_e} \right)^{1/2} \: n_ekT_e \end{equation} \tag{ 2 }$$

can provide an adequate heat flux to power the filaments in both M87 and also NGC 1275. Fabian et al. (2011) concur that the surface radiative flux from the outer filaments in NGC 1275 is close to the energy flux impacting on them from particles in the hot gas. They use a different formula for the saturated conduction heat flux, also given by Cowie & McKee (1977),

$$\begin{equation} q_{\rm sat} = 5 \, \phi \, p \, c_s, \end{equation} \tag{ 3 }$$

where ϕ ∼ 1 accounts for uncertain physics, such as f and the average particle mass, p = nkT is the gas pressure (where here n is the total particle density and T the temperature), and c_s is the isothermal sound speed of the gas, $c_s = \sqrt{p/\rho } \sim \sqrt{kT/m_p}$, where m_p is the proton mass.

It is worth remarking that these two formulae (Equations (2) and (3)) emphasize physically different ways of viewing the conduction process, though they are algebraically equivalent. In Equation (2), the characteristic velocity (Equation (1)) is taken to be the free electron speed averaged over direction, on the order of the electron thermal velocity

$$\begin{equation} v_{\rm char} = \left(\frac{8}{9\pi } \right)^{1/2} \left(\frac{kT}{m_e} \right)^{1/2}. \end{equation} \tag{ 4 }$$

In Equation (3), the implied characteristic velocity is the ion (or proton) sound speed v_char ∼ v_i ∼ (m_e/m_p)^1/2v_e.

In fact, the equations are equivalent for the case of equal ion and electron temperature, and the actual characteristic velocity with which electrons and ions cross the boundary is the same. To see this, imagine that initially one sets a hot fully ionized plasma (with large mean free path) next to a cold absorber. Since v_e ≫ v_i, there is a flow of heat from the hot plasma at a rate given by Equation (1). However, this flux of heat, carried by the electrons, leads to a net electric current j into the absorber, and therefore to a build up of (negative) charge on the absorber. What then happens is that "an electrostatic field E will build up to such a value that j vanishes. This field then reduces the flow of heat" (Spitzer & Härm 1953). Eventually, in order to maintain the zero current condition, the net flow speed of both the electrons and the ions when they reach the cold absorber must be the same. Thus, the electrostatic field set up on the absorber is such that it slows the electrons and speeds the ions. An electron loses the same amount of energy when it travels through an electrostatic potential barrier as a proton gains when it falls into an electrostatic potential well. Because m_p ≫ m_e, the net flow speed of both the protons and the electrons must be on the order of v_char ∼ v_i ∼ c_s, in line with the expression for saturated heat flux given by Equation (3).

This now has a further very important implication because, to maintain charge neutrality, the net flow speeds of the electrons and the ions must be the same, so the dominant energy transport is provided by the ions.

We have seen that if the dominant heating mechanism for the filaments is indeed the penetration of the filaments by thermal particles originating in the hot gas (Fabian et al. 2011), then most of the energy is carried by the hot ions. One effect of the excitation of H atoms (and H molecules) by a non-isotropic velocity distribution of electrons or protons is that the resultant emission lines (including Lyα and Hα) can be polarized. This has been discussed in the context of excitation by electron impact in solar flares (Laming 1990a; Aboudarham et al. 1992) and for excitation by both electron and proton impact in non-radiative shock fronts to be found in supernova remnants (Laming 1990b).

To understand how the polarization comes about, consider a collection of H atoms excited by a beam of protons. There is a preferred plane perpendicular to the velocity vector of the beam. As viewed from that plane and for the relevant range of proton energies in the hot gas surrounding the filaments (a few keV). At such energies, the contribution to line polarization for electron impact excitation in Hα is negligible (Aboudarham et al. 1992). Therefore, if heating of the filaments were to proceed via standard diffusive (Spitzer) conductivity, in which most of the heat is carried inwards by electrons, we would expect negligible polarization of the emission lines. For proton energies of a few keV, polarization can arise, but it is not straightforward to calculate the precise level. The difficulty arises mainly because at such energies the proton velocity

$$\begin{equation} v_p = 4.34 \times 10^7 (E/{\rm keV})^{1/2}\ {\rm cm\ s^{-1}} \end{equation} \tag{ 5 }$$

is much less than the electron orbital velocity in the ground state

$$\begin{equation} v_e = \frac{ \hbar c}{4 \pi ^2 e^2} \, c = 2.2 \times 10^8\ {\rm cm\ s^{-1}}, \end{equation} \tag{ 6 }$$

so that the plasma is in the quasi-molecular regime where electronic processes proceed through states that are transiently formed during the collision (Hippler et al. 1988).⁸

Both computational and experimental results for Lyα suggest polarizations in the range of 10%–25% (Kauppila et al. 1970; Hippler et al. 1988; McLaughlin et al. 1997; Keim et al. 2005). Laming (1990b) suggests that at these energies it is appropriate to assume that the Hα polarization is the same as the Lyα polarization for the same energy photons. Computations by Balança & Feautrier (1998) indicate that this is an appropriate assumption and also find that proton impact polarizations for Hα for protons in the range E ≈ 1–5 keV are on the order of 20%–25%.

If the Hα emission lines from the filaments are being induced by hot particles originating in the hot gas, then the observed polarizations are likely to be less than these values. The reduction would come about because the incoming particles do not form an organized beam, because of geometric effects, and the cross-section for excitation to the n = 3 level (in order to excite Hα) can be comparable in this energy range to the cross-section for ionization (Lin et al. 2011). We briefly consider geometry and two mechanisms that have the potential to randomize the proton velocity distribution within the filaments. We conclude that measurable polarization ought to persist for the saturated conduction case.

4.3.1. Geometry

4.3.2. Scattering

If the proton beam is scattered so that the proton velocities become randomized, then this could significantly reduce the degree of polarization. To excite the electron from the n = 1 state to n = 2 or n = 3, the proton needs to come within a few Bohr radii a₀, where

$$\begin{equation} a_0 = \frac{h^2}{4 \pi ^2 m_e e^2} = 0.53 \times 10^{-8} \ {\rm cm}. \end{equation} \tag{ 7 }$$

This value agrees with the typical cross-sections for the interaction given as on the order of ∼2 × 10⁻¹⁷ cm² (e.g., McLaughlin et al. 1997; Balança & Feautrier 1998; Lin et al. 2011), compared to the area of the first Bohr orbit of $\pi a_0^2 = 8.8 \times 10^{-17}$ cm². At the radius of the Bohr orbit, the electrostatic potential energy is, of course, around E₀ ≈ 13.6 eV, which is much less than the typical proton energies E (a few keV) we are interested in. Thus, the angle through which the proton is deflected is on the order of ϕ ∼ E₀/E ≪ 1. We conclude that the act of exciting the Lyman and Balmer lines does not significantly isotropize the directions of the incoming ions.

4.3.3. Magnetic Fields

For typical magnetic field strengths expected within the filaments, the Larmor radius for a few keV proton is of order 10⁹ cm, which is many orders of magnitudes less than the radii of the filaments. Hence the proton velocity distribution could potentially be isotropized if the magnetic field structure within the filaments were strongly randomized.

Fabian et al. (2008) have argued, however, that the filaments in NGC 1275 are "essentially magnetic structures" in which the magnetic pressure dominates the thermal pressure. Werner et al. (2013) came to similar conclusions for the filaments in M87. The suggested value of B ≈ 100 μG would give approximate pressure equilibrium with the external medium (density n ≈ 0.06 cm⁻³ and temperature T ≈ 4 keV) and would imply for their assumed values internal to the filament of density n ≈ 2 cm⁻³ and temperature T ⩽ 10⁴ K that the ratio of thermal to magnetic pressure β is

$$\begin{equation} \beta = \frac{nkT}{B^2/8\pi } \le 7 \times 10^{-3}. \end{equation} \tag{ 8 }$$

Similar arguments are made for the filamentary gas in M87 by Werner et al. (2013), who suggest that the 10⁴ K gas phase, which emits the density sensitive [S ii] λ 6716, λ 6731, requires fields B ≈ 50 μG to maintain pressure balance with the surroundings.

In this picture, it is argued that the magnetic field must lie predominantly along the filaments, in order that they be magnetically dominated structures. Note that Fabian et al. (2008) also deemed it necessary that there is an "unseen" ⁹ component of magnetic field that is perpendicular to the filaments in order to prevent material sliding along the filaments. It is not clear how both of these requirements are to be achieved simultaneously. In addition, it is claimed that the filaments contain Alfvénic turbulence in order to account for the internal velocity dispersion of ∼100 km s⁻¹ (Hatch et al. 2006). The driver for this turbulence remains unspecified.

The idea of turbulence within the filaments is also suggested by Fabian et al. (2011) and Werner et al. (2013). There, in order for external hot plasma to interpenetrate the cold filaments, a process known as "reconnection-diffusion" is introduced. However, these authors agree that (Fabian et al. 2008) "It is natural to assume that the turbulent velocity in filaments is less than the Alfvén speed" because otherwise the turbulence would randomize the field direction and so prevent the existence of long-lived filaments. This is the crux of the matter for our discussion. In order for the filaments to be strongly magnetic, the magnetic field within the filaments must be well-ordered. If the field is well-ordered, magnetic randomization of the velocity distribution of the incoming protons is not going to be effective. Thus, the reduction of polarization caused by randomization of the proton velocity distribution by a chaotic magnetic field configuration is unlikely to be significant.

Our observational limits are very stringent. For individual emission regions, we find polarization levels ≲ 0.1%, with the average polarization degree in Table 3 for the Hα + [N ii] complex <p > ≈ 3 × 10⁻⁴, i.e., a polarization percentage ≈0.03%. By contrast, from theoretical considerations, we have argued that in a saturated conduction regime where the filaments are excited by a highly directional proton beam, polarization levels of plausibly a few percent ought to be present in the emission lines. This arises because the protons do not necessarily ionize the filament of neutral H atoms, and the system retains a degree of the incident anisotropy.

Hence, we conclude that the evidence from these observations and theoretical arguments indicates that if conduction is the dominant process for energy transport into the filament system from the hot ambient coronal X-ray gas, we are unlikely to be in the saturated regime. For the case of M87, Sparks et al. (2012) showed that the line strengths were consistent with a classical non-saturated conduction model, which would not be expected to produce significant polarization. Global energetic considerations do show that saturated conduction can carry the required energy to power the emission filaments, and to an order of magnitude the energy transport is similar for the classical conduction regime, though the details of the interface structure, energy flux, and timescales involved differ.

The use of emission line polarization as a plasma diagnostic is clearly in its infancy for application to galaxy clusters. From other areas of astrophysics, it is apparent that the approach has the potential to provide unique insights into the excitation mechanisms of relevance. Additional theoretical work is needed to determine more accurately the likely levels of polarization, for both the case of conduction and that of anisotropic photon excitation and shocks. Heuristically, one would expect that for photoionization, the resulting polarization distribution will depend primarily on the degree of anisotropy of the photons, both their origin and modification by any dust present, as well as, of course, on the spectral energy distribution of the photons. Plausibly, hot stars and the intracluster medium would result in an approximately isotropic photon excitation and low polarization, while AGN excitation would have much stronger directionality and is therefore more likely to yield polarization. Shock excitation is also highly directional and the consequent polarization depends on the shock speed. Laming (1990b) showed that substantial polarization can arise from fast shocks, on the order of 2000 km s⁻¹, which is much faster than likely shock speeds in these filaments, unless an interaction with, e.g., the relativistic plasma radio lobes is involved.