6 The spatial distribution of the total SZ effect

In this section we want to study the effect of the superposition of two distinct electron populations on the spatial distribution of the total SZ effect in galaxy clusters. In fact, the thermal and the non-thermal populations have, in general, a different spatial distribution as indicated by the spatial extension of the X-ray emission and of the radio halo emission in many clusters (see Giovannini et al. 2001). Also, there are indications that the EUV emission in some nearby clusters is more extended than the thermal X-ray emission suggesting that there is a warm gas and/or a population of relativistic electrons which are spatially more extended than the X-ray emitting IC gas (Lieu et al. 1999).

Let us first consider a galaxy cluster in which two different electron populations are present: a thermal population, described by a relativistic Maxwellian velocity distribution, and a non-thermal population, with an energy spectrum described by a double power-law (see Eq. (34)). We consider these two populations as independent and spatially superposed, consistently with our analysis of Sect. 4. These assumptions are indeed reasonable since in the cluster A2163, for instance, the radio halo emitting electrons occupy approximately the same volume of the X-ray emitting, thermal electrons (see Feretti et al. 2001) and, moreover, the relativistic electrons with $E_{\rm e} \sim$ a few GeV (which produce the radio-halo synchrotron emission) do not sensitively affect the IC thermal electrons.

Figure 21: The behaviour of the zero of the total SZ effect produced by a combination of a thermal and non-thermal population given in Eq. (34) with p1=0.5 is shown as a function of $n_{\rm e,rel}( \tilde{p}_1=100)$.

Figure 22: The function P1(s) computed for a combination of two thermal populations with $k_{\rm B} T_{\rm e,1}=8.5$ and $k_{\rm B} T_{\rm e,2}=0.5$ keV (solid line) and with $k_{\rm B} T_{\rm e,1}=8.5$ and $k_{\rm B} T_{\rm e,2}=1$ keV (dashed line) is compared with that of the single thermal population with $k_{\rm B} T_{\rm e}= 8.5$ keV (dotted line).

Figure 23: The spectral distortion in units of $2(k_{\rm B} T_0)^3/(hc)^2$ produced by a single thermal population with $k_{\rm B} T_{\rm e,1}=8.5$ keV and $n_{\rm e,1} = 3 \times 10^{-3}$ cm-3 (solid line) is compared with that produced by a combination of two thermal populations with $k_{\rm B} T_{\rm e,2}=1$ keV and $n_{\rm e,2}=10^{-3}$ (dashed line), $3\times 10^{-3}$ (dotted), $5\times 10^{-3}$ (dot-dashed), $7\times 10^{-3}$ (long dashes) and 10-2 (dot-dot-dashes) cm-3.

As for the spatial distribution of the thermal electrons we use here a isothermal $\beta$-model (see, e.g., Cavaliere & Fusco-Femiano 1976), according to which the temperature $T_{\rm e}$ of the ICM is constant and the electron density is described by a spherical distribution

$$n_{\rm e}(r)=n_{\rm e0} \left( 1+ \frac{r^2}{r_{\rm c}^2} \right)^{-3\beta/2} ,$$ (69)

where $n_{\rm e0}$ is the central IC gas density, $r_{\rm c}$ is the core radius of the cluster and $\beta = \mu m_{\rm p} v^2/k_{\rm B} T_{\rm e}$ is observed in the range 0.6-1 (see Sarazin 1988 for a review). Under this assumption the angular dependence of the optical depth of the thermal population is given by

 

$$% \tau(\theta) \sigma_{\rm T} \int n_{\rm e}(r) {\rm d}\ell= \sigma_{\rm T} n_{\... ... \left[ 1+ \left( \frac{r}{r_{\rm c}} \right)^2 \right]^{-3\beta/2} {\rm d}\ell$$ =

$$\equiv n_{\rm e0} \sigma_{\rm T} r_{\rm c} Y(\theta),$$ (70)

where $\theta$ is the angle between the centre of cluster and the direction of observation. All the angular dependence is contained in the function

$$Y(\theta)=\sqrt{\pi} \frac{\Gamma\left( \frac{3}{2}\beta-\fra... ...\rm c}} \right)^2 \right]^{ \frac{1}{2} - \frac{3}{2} \beta} ,$$ (71)

where $\theta_{\rm c}=r_{\rm c}/D_{\rm A}$, in terms of the angular diameter distance $D_{\rm A}$ (see Weinberg 1972).

Figure 24: The behaviour of the zero of the total SZ effect for a combination of two thermal populations with $k_{\rm B} T_{\rm e,1}=8.5$ keV and $k_{\rm B} T_{\rm e,2}=1$ keV (solid line) and $k_{\rm B} T_{\rm e,1}=8.5$ keV and $k_{\rm B} T_{\rm e,2}=0.5$ keV (dashed line) is shown as a function of the density $n_{\rm e,2}$ (left panel) and of the pressure ratio P2/P1 (right panel). A value $n_{\rm e,1} = 3 \times 10^{-3}$ cm-3 is adopted here.

Figure 25: The behaviour of the zero of the total SZ effect for a combination of two thermal populations with $k_{\rm B} T_{\rm e,1}=8.5$ keV and $n_{\rm e,2}=1\times 10^{-3}$ cm-3 is shown as a function of the temperature $k_{\rm B} T_{\rm e,2}$(left panel) and of the pressure ratio P2/P1 (right panel). A value $n_{\rm e,1} = 3 \times 10^{-3}$ cm-3 is adopted here.

In the following we will describe in detail, for the sake of clarity, the spatial distribution of the approximated (to first and second order in $\tau$) expression of the total SZ effect and we will give also the general expression for the exact calculations.

The angular dependence of the SZ-induced spectral distortion evaluated at first order in $\tau$ writes as

$$\Delta i(x,\theta)=\tau(\theta) [j_1(x)-j_0(x)] ,$$ (72)

where $\Delta i$ is in units of $2(k_{\rm B} T_0)^3/(hc)^2$ and $\tau(\theta)$ is given by Eq. (70). Since the spectral distortion can be written in the general form $\Delta i(x)=y \tilde{g}(x)$, the Comptonization parameter writes as

 

$$% y(\theta) \frac{\sigma_{\rm T}}{m_{\rm e} c^2}\int P {\rm d}\ell=\frac{\sig... ... d}\ell= \frac{\langle k_{\rm B} T_{\rm e} \rangle}{m_{\rm e} c^2} \tau(\theta)$$ =

$$\frac{\sigma_{\rm T}}{m_{\rm e} c^2} P_0 r_{\rm c} Y(\theta) ,$$ = (73)

where the quantity $\langle k_{\rm B} T_{\rm e} \rangle$ is defined in Eq. (46) and depends only on the parameters of the electron distribution (here the central pressure is $P_0=n_{\rm e0} \langle k_{\rm B} T_{\rm e} \rangle$). The spectral shape of the SZ effect at an angular distance $\theta$ from the cluster center can be written, at first order in $\tau$, as

$$% \tilde{g}(x) \frac{\Delta i(x,\theta)}{y(\theta)}= \frac{\sigma_{\rm T} n_{\rm... ...{ \frac{\sigma_{\rm T}}{m_{\rm e} c^2} P_0 r_{\rm c} Y(\theta)} [j_1(x)-j_0(x)]$$ =

$$\frac{m_{\rm e} c^2}{\langle k_{\rm B} T_{\rm e} \rangle} [j_1(x)-j_0(x)],$$ = (74)

and it does not contain the angular dependence of the effect. However, at second order in $\tau$ the spectral distortion writes as

$$\Delta i(x,\theta)=\frac{1}{2} \tau^2(\theta) [j_2(x)-2j_1(x)+j_0(x)],$$ (75)

and, as a consequence, the second order approximation of the function $\tilde{g}(x)$ is

 

$$% \tilde{g}(x,\theta) \frac{1}{2} \frac{\sigma_{\rm T}^2 n_{{\rm e}o}^2 r_{\rm c}^2 Y^2... ...\sigma_{\rm T}}{m_{\rm e} c^2} P_0 r_{\rm c} Y(\theta)} [j_2(x)-2j_1(x)+j_0(x)]$$ =

$$\frac{1}{2} \frac{\sigma_{\rm T} m_{\rm e} c^2 n_{\rm e0}^2 r_{\rm c} Y(\theta)}{P_0} [j_2(x)-2j_1(x)+j_0(x)] ,$$ = (76)

which depends explicitly on $\theta$. It is important to notice that the angular dependence of the spectral distortion is present both in the Comptonization parameter y and in the spectral shape $\tilde{g}(x)$. The dependence of the SZ effect from $\theta$ is present at all orders n>1 of the approximations and hence in the full, exact expression of the SZ effect. Thus, also for the study of the spatial distribution of the total SZ effect, considering the first order approximation is neither appropriate nor consistent.

For a combination of two electron populations, the total spectral distortion is given, at first order approximation in $\tau$, by the sum of the single spectral distortions of each population

$$\Delta i(x,\theta)=\Delta i_{\rm th}(x,\theta) +\Delta i_{\rm nonth} (x,\theta).$$ (77)

This expression can be written more explicitly as

 

$$% \Delta i(x,\theta) y_{\rm th}(\theta) \tilde{g}_{\rm th}(x)+y_{\rm nonth} (\theta) \tilde{g}_{\rm nonth}(x)$$ =

$$\frac{\sigma_{\rm T}}{m_{\rm e} c^2}P_{\rm0,th} r_{\rm c,X} Y_{\rm th}(\theta) \tilde{g}_{\rm th}(x)$$ =

$$+\frac{\sigma_{\rm T}}{m_{\rm e} c^2}P_{\rm0,nonth} r_{\rm c,rad} Y_{\rm nonth}(\theta) \tilde{g}_{\rm nonth}(x)$$

$$\frac{\sigma_{\rm T}}{m_{\rm e} c^2}P_{\rm0,th}[r_{\rm c,X}Y_{\rm th}(\theta) \tilde{g}_{\rm th}(x)$$ =

$$+ r_{\rm c,rad} \bar{P} Y_{\rm nonth}(\theta) \tilde{g}_{\rm nonth}(x)]$$ (78)

where $\bar{P}= P_{\rm0,nonth}/P_{\rm0,th}$ is the pressure ratio at the cluster center. We assume here that the spatial distribution of the thermal and of the non-thermal electron populations are given by a $\beta$-model with core radii $r_{\rm c,X}$ and $r_{\rm c,rad}$ and parameters $\beta_{\rm X}$ and $\beta_{\rm rad}$, respectively. Such an expression is also valid in the case of the combination of two thermal populations. At higher approximation orders in $\tau$, the total spectral distortion contains cross-correlation terms of the parameters of each population and thus it is not possible to write it as a linear combination of the spectral distortions of the separate electron populations. An exact derivation (or at least the 3rd order approximation in $\tau$) is required to describe correctly the spatial dependence of the total SZ effect in galaxy clusters. The general calculation of the spatial distribution of the total SZ effect can be made using the exact expressions derived in Eqs. (21-23, 62, 63) in which the spatial dependence of the thermal and non-thermal electron populations are considered.

In Fig. 26 we show the spatial behaviour of the total SZ effect in the case of a Coma-like cluster.

Figure 26: The spatial dependence of the total SZ effect produced by the combination of the thermal and relativistic electrons in a Coma-like cluster. We show the thermal SZ effect (dots) and the total SZ effect with $\bar{P} = 0.05$ (solid), 0.49 (short dashes) and 1.48 (long dashes). Calculations are done for an a-dimensional frequency of x=2.3 where the total SZ effect has its minimum value (upper panel) and for x=6.5 where the total SZ effect has its maximum value (lower panel).

We use specifically the following parameters: $r_{\rm c,X} = 0.42 ~h_{50}^{-1}$Mpc, $r_{\rm c,rad} =0.4 ~h_{50}^{-1}$ Mpc, $R_{\rm X} = 4.2 ~h_{50}^{-1}$ Mpc, $R_{\rm halo} = 1.25 ~h_{50}^{-1}$ Mpc, $\beta _{\rm X} =0.75$, $\beta _{\rm rad} = 0.8$, $n_{\rm e0,X} = 2.89 \times 10^{-3} \;h_{50}^{2}$ cm^-3, $k_{\rm b}T_{\rm e} = 8.21$ keV. The non-thermal electron population fitting the radio halo spectrum is taken with spectral parameters p₁ = 1000 and $\alpha =2.5$. The total SZ effect shows a peculiar spatial behaviour which is particularly evident at the frequencies $x \sim 2.3$ where it attains its minimum value and at frequencies $x \sim 6.5$ close to its maximum. At $x \sim 2.3$ the spatial profile of the total SZ effect is predicted to be higher than that due only to the thermal population because the non-thermal SZ effect sums up in the region of the radio halo which extends up to $\sim1.3~ h_{50}^{-1}$ Mpc (see Fig. 26). At larger distances from the cluster center, the SZ effect is dominated by the thermal component and resemble its spatial behaviour. An opposite behaviour is expected at frequencies higher than that of the zero of the total SZ effect. For instance at $x \sim 6.5$the spatial profile of the total SZ effect is lower than that of the single thermal effect because the spectrum of the total SZ effect is lower than that of the thermal case. So, a depression in the spatial profile of the total SZ effect is expected at radii smaller than the radio halo extension $R_{\rm halo}$. At distances larger than the radio halo extension, the total SZ effect is again dominated by the thermal component. The amplitude of the non-thermal SZ contribution depends from the pressure ratio $\bar{P} = P_{\rm rel}/P_{\rm th}$ as shown in Fig. 26. Such spatial features of the total SZ effect should be detectable with sensitive instruments with good spatial resolution and narrow spectral frequency band. Such instrumental capabilities can also be able to detect the transition between the ambient thermal SZ effect and the non-thermal SZ effect expected in radio relics and in the radio lobes of active galaxies found in the cluster environment.