8 Total mass profile

8.1 Calculation of the mass profile

The mass profile is calculated under the usual assumptions of hydrostatic equilibrium and spherical symmetry. The integrated mass profile can be calculated from the gas density, $n_{\rm g}$, and temperature profiles:

$$M(r) = - \frac{ kT r}{ G \mu m_{\rm p}} \left[ \frac{{\rm d} ... ...\rm d} \ln{r}} + \frac{{\rm d} \ln{T}}{{\rm d} \ln{r}} \right]$$ (12)

where G and $m_{\rm p}$ are the gravitational constant and proton mass and $\mu = 0.597$.

If the gas density profile is described by the KBB model (Eq. (5)), then the mass profile is described by:

$$r < R_{\rm cut}~~~M(r) - \frac{{k} r^2}{{G} \mu m_{\rm p}} \left[- \frac{3 \beta_{\rm in... ...)} T(r)}{r^{2 \xi} + r_{\rm c,in}^{2 \xi}} + \frac{{\rm d} T}{{\rm d} r}\right]$$ =

$$r > R_{\rm cut}~~~M(r) - \frac{k r^2}{{\rm G} \mu m_{\rm p}} \left[- \frac{3 \beta r T(r)}{r^2 + r_{\rm c}^2} + \frac{{\rm d} T}{{\rm d} r}\right]$$ = (13)

where $r_{\rm c,in}$, $\xi$, $r_{\rm c}$ and ${\beta }$ are the parameters of the KBB model, $\beta_{\rm in}$ being linked to them by Eq. (8).

The mass profile is calculated using the Monte Carlo method of Neumann & Böhringer (1995). The gas density profile parameters are fixed to their best fit values. This method calculates random temperature profiles within the bounds of the observed profile. We calculated 10 000 such random temperature profiles, using a window size of 150 kpc and a smoothing parameter of $0.1~\rm keV$. The final output is the mean mass profile and the corresponding errors for each data point in the input temperature profile. The errors are calculated using the $90\%$ errors in the temperature profile and the standard deviation of the mass at any given radius. The resulting errors are calculated to correspond to $1\sigma$ errors in the mass profile.

The errors on the mass profile due to the error on the density gradient, ${\rm d} \ln{n_{\rm g}}/{\rm d} \ln{r}$, are then calculated. As the parameters $r_{\rm c,in}$, $\xi$, $r_{\rm c}$ and ${\beta }$ are correlated, this error cannot be deduced directly from the errors on these parameters. We used a method similar to the one described in Elbaz et al. (1995). For each considered radius, the surface brightness profile is fitted, considering ${\rm d} \ln{n_{\rm g}}/{\rm d} \ln{r}$ estimated at this radius as a free parameter, instead of ${\beta }$. The $1\sigma$ error on this parameter is then classically derived from the $\chi^2$ variation, the other parameters (normalisation, $r_{\rm c,in}$, $r_{\rm c}$ and $\xi$) being optimised. Finally the errors due to the density and temperature profiles (derived from the Monte-Carlo method) are added quadratically.

The resulting mass profile, with $1\sigma$ error bars, is plotted in Fig. 10.

Figure 10: The mass profile of A1413 derived from XMM-Newton surface brightness and temperature profiles. Data points: mass derived from the HE equation, using the Monte-Carlo method with the best fit KBB model for the gas density (Eq. (5)) and the observed temperature profile. Errors bars are $1\sigma$ and take into account both errors on the temperature and gas density profiles. Full line: mass profile derived from the HE equation and the best fit polytropic model with the KBB model for the gas density. Dashed line: same assuming isothermality. Dotted line: mass profile derived from the HE equation and an isothermal ${\beta }$-model, fitting the outer gas density profile.

8.2 Factors influencing the mass profile

The temperature profile of A1413 is well determined out to $\sim$ 0.7 r₂₀₀, and shows a gradual decline which is well described by a polytrope of index $\gamma =1.07$ beyond the CF region. Assuming such a polytropic description, the mass profile can be calculated analytically from the best fit gas density KBB model. The polytropic mass profile lies well within the errors of the Monte Carlo profile, except for the central point due to the temperature drop observed in the center (full line in Fig. 10). We also note that the derived mass at large radii (r > 1.3 Mpc) lies at the lower range of the Monte Carlo mass. This is due to the drop of temperature in the last radial bin, the best fit temperature being below the polytropic value.

In the classic approach, the gas density is described with an isothermal ${\beta }$-model, in which the temperature profile is assumed to be isothermal and the gas density distribution is parameterised by a ${\beta }$-model. In Fig. 10 (dotted line) we show the mass profile obtained using this approach, with the ${\beta }$-model best fitting the outer part of the cluster and the average cluster temperature outside the CF region, $kT_{\rm X}$. Not surprisingly, the mass is greatly underestimated in the centre ( $r < R_{\rm cut}$), where the gas density profile is more concentrated (higher gradient) than the extrapolated ${\beta }$-model. If we instead parameterise the gas density using the best fit KBB model, the mass distribution towards the centre is recovered (Fig. 10, dashed line). Beyond  $R_{\rm cut}$the mass profile is slightly steeper than that derived from the true temperature profile, as expected from the observed $\gamma$ value, slightly larger than 1. This comparison shows that the temperature gradient has a small but systematic effect on the derived mass profile.

The mass profile is remarkably well constrained: the $1\sigma$ error is less than $\pm$$5\%$ below 1.4 Mpc and rises to $\sim$$\pm 18\%$ at 1.8 Mpc. The temperature logarithmic gradient is much smaller than the density logarithmic gradient ($7\%$ for  $\gamma =1.07$), except in the very outer part, where the temperature gradient is both larger and the constraints are poorer. As a consequence (see Eq. (12)), except in this outer region, the error on the mass profile is dominated by the error on the gas density gradient (in the range $0.5\%{-}3\%$) and on the average temperature ($2.3\%$). For the same reason the mass profile is very robust versus possible systematic errors on the temperature profile. We have shown in Sect. 5.2 that spectral deprojection or PSF correction do not have a significant effect on the form of the profile. One might also ask what effect the ellipticity of the cluster might have on the derived radial quantities. We extracted spectra in elliptical annuli and compared the projected temperature profile with that produced using circular annuli. All temperatures agree within the respective errors, and so we conclude that the cluster ellipticity is also a minor source of error.

   
8.3 Modelling of the mass profile

Navarro et al. (1997, NFW) performed high resolution N-body simulations which showed that the density profiles of dark matter halos have a universal shape, regardless of halo mass and values of cosmological parameters. The NFW profile is given by:

$$\rho(r) = \frac{\rho_{\rm c}(z) \delta_{\rm c}}{(r/r_{\rm s}) (1+ r/r_{\rm s})^2}$$ (14)

where $\rho(r)$ is the mass density and $\rho_{\rm c} (z)$is the critical density at the observed redshift, which, for a matter dominated $\Omega = 1, \Lambda = 0$ Universe is:

$$\rho_{\rm c}(z) = \frac{3 H_0^2}{ 8 \pi G} (1 + z)^3.$$ (15)

The parameters of the model are $r_{\rm s}$, a scale length and $\delta_{\rm c}$, a characteristic dimensionless density dependent on the formation epoch of the dark matter halo. $\delta_{\rm c}$ can be expressed in term of the equivalent concentration parameter, c:

$$\delta_{\rm c}= \frac{200}{3} \frac{c^3}{[{\rm ln} (1+c) - c/(1+c)]}\cdot$$ (16)

The radius corresponding to a density contrast of 200 is  $r_{200}= cr_{\rm s}$.

Figure 11: The mass profile of A1413 fitted with the NFW profile (dotted line) and Moore et al. (1999) profile (full line). Bottom panels: residual between the data and the model. The best fit concentration parameters are c=5.4 and c=2.5 respectively (see Table 5). The radius corresponding to a density contrast of 500 is indicated by an arrow.

 

 

Table 5: NFW and Moore et al. (1999) fits to the mass profile of A1413. Errors are $1\sigma$.

Parameter	NFW model	MQGSL model
c	$5.4 \pm 0.2$	$2.6 \pm 0.1$
$r_{\rm s}$ (kpc)	$401 \pm 17$	$845 \pm 43$
r200 (kpc)	2169	2221
$M_{200} (M_{\odot})$	$8.9 \times 10^{14}$	$9.5 \times 10^{14}$
$\chi^2 / \nu$	8.76/9	6.44/9

The NFW density profile varies from $\rho_{\rm NFW} \propto r^{-1}$at small radii to  $\rho_{\rm NFW} \propto r^{-3}$ at large radii. As we are fitting the mass profile M(r), we use the integrated mass of the NFW profile for the fit (e.g. Suto et al. 1998):

$$4\pi \rho_{\rm c}(z) \delta_{\rm c} r_{\rm s}^3 m(r/r_{\rm s})$$                  M(r) = (17)

$${\rm ln} (1+ x) - \frac{x}{1 + x}\cdot$$ m(x) = (18)

More recent, higher resolution simulations by Moore et al. (1999, hereafter MQGSL) suggest a profile described by:

$$\rho(r) = \frac{\rho_{\rm c}(z) \delta_{\rm c}}{(r / r_{\rm s})^{3/2} \left[1+\left(r/r_{\rm s}\right)^{3/2}\right]}~,$$ (19)

where

$$\delta_{\rm c}= \frac{100 c^{3} }{\ln {(1 + c^{3/2})}}\cdot$$ (20)

This is essentially identical to the NFW profile at large radii but is steeper near the centre ( $\rho_{m} \propto r^{-1.5}$). Again, as we are fitting the mass profile, we use the integrated mass of the MQGSL profile, given by (Suto et al. 1998):

$$4\pi \rho_{\rm c}(z) \delta_{\rm c} r_{\rm s}^3 m(r/r_{\rm s})$$                 M(r) = (21)

$$\frac{2}{3} \ln (1+ x^{3/2}).$$ m(x) = (22)

The derived parameters from each fit are given in Table 5 and the best fit models are compared to the data in Fig. 11. We find that the data are extremely well described by the MQGSL profile across the entire radial range. The NFW profile can also be used to describe the data, but shows a small divergence at small radii ( $\chi^{2}=4.54$ for the first 3 points). The radius corresponding to a density contrast of 500, computed from the data, is indicated by an arrow. There is a slight hint that the measured mass, $M_{500} = 7.7^{+1.2}_{-0.8}~\times~ 10^{14}~M_{\odot}$, is higher than the MQGSL and NFW models ( $6.8\times 10^{14}~M_{\odot}$ and $6.5\times 10^{14}~M_{\odot}$, respectively) around that radius (see Sect. 9.3 for further discussion).