A&A 398, 41-48 (2003)
DOI: 10.1051/0004-6361:20021648
R. Piffaretti^{1,2} - Ph. Jetzer^{1} - S. Schindler^{3,4}
1 - Institute of Theoretical Physics, University of Zürich, Winterthurerstrasse, 190, 8057 Zürich, Switzerland
2 - Paul Scherrer Institute, Laboratory for Astrophysics, 5232 Villigen,
Switzerland
3 - Institut für Astrophysik, Leopold-Franzens-Universität Innsbruck,
Technikerstrasse 25, 6020 Innsbruck, Austria
4 - Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Birkenhead CH41 1LD, UK
Received 6 May 2002 / Accepted 8 November 2002
Abstract
We present an investigation of the effects of asphericity on the estimates of total mass and gas mass fraction in galaxy clusters from X-ray observations. We model the aspherical shape of galaxy clusters by a triaxial model and compare the true total mass and the true total gas mass fraction with the corresponding quantities obtained with the assumption of spherical symmetry.
In the triaxial model we allow the extent along the line of sight to vary in order to describe elongated and compressed cluster shapes. Using a sample of 10 ROSAT clusters and a recent CHANDRA observation we find the following results. For prolate or oblate shapes the difference between triaxial and spherical model both in the mass and in the gas mass fraction are negligible (less than 3%). For more aspherical shapes the total mass is underestimated (overestimated) in the centre, if the cluster is compressed (elongated). The gas mass fraction is overestimated for compressed clusters and slightly underestimated for elongated clusters. Comparing X-ray masses with gravitational lensing estimates, we find that elongations along the line of sight can resolve discrepancies of masses determined by the two different methods of up to 30%. The combination of Sunyaev-Zel'dovich and X-ray observations is useful to measure the elongation of the cluster along the line of sight. As an application, we estimate the elongation of the cluster CL0016+16 with two different approaches, Sunyaev-Zel'dovich measurements and comparison of weak lensing and X-ray masses, and find reasonable agreement.
Key words: galaxies: clusters: general - X-ray: galaxies: clusters
Clusters of galaxies are the largest gravitationally bound aggregates of matter in the universe and, for many applications, they can be regarded as being representative for the universe as a whole. In particular the ratio of baryonic to total matter in clusters can be assumed to be representative of the universe, because of the large volume and the fact that clusters are closed systems. Combined with big bang nucleosynthesis calculations and observed light-element abundances, this ratio can be used to constrain the cosmological density parameter . Optical velocity dispersion measurements and gravitational lensing are independent methods used to constrain the total mass in galaxy clusters while X-ray observations and recently the Sunyaev-Zel'dovich (SZ) effect (Grego et al. 2001) are used to estimate not only total masses but also gas mass fractions in clusters. The X-ray method is based on the assumptions that the X-ray emitting gas is an ideal gas in hydrostatic equilibrium and the total mass and gas mass fraction are estimated through the X-ray surface brightness distribution and the gas temperature. In this context, the -model (Cavaliere & Fusco-Femiano 1976) is widely used in X-ray astronomy to parametrise the gas density profile in clusters of galaxies by fitting their surface brightness. It is usually used under the assumption of spherical symmetry. If the images show that the cluster emission is smooth and the isophotes show an elliptical shape, the modelling is improved when an elliptical -model is used: Fabricant et al. (1984) studied the non-spherical shape of A2256, McMillan et al. (1989) the morphology of 49 Abell clusters and the elliptical shape of Cl0016+16 was analysed by Neumann & Böhringer (1997) and Hughes & Birkinshaw (1998). Buote & Canizares (1996) studied the elliptical shapes of 5 Abell clusters concluding that, for oblate and prolate shapes, the estimates of the gas masses are insensitive to the ellipticities of the X-ray isophotes.
The aim of this paper is to investigate to which degree the assumption of spherical symmetry in the estimate of total mass and gas mass fraction in galaxy clusters showing elliptical X-ray emission is accurate. We select 10 galaxy clusters showing a smooth, elliptical emission in the ROSAT archive and we model the intracluster medium (ICM) with a triaxial -model and compare the estimates with those of a spherical -model. In particular, a triaxial model allows the description of observations of clusters which may be compressed or elongated along the line of sight, with the latter being probably more frequent, since elongated clusters are selected preferentially due to their higher contrast over the background emission. Since our investigation is focused on a geometrical aspect common to many clusters, we do not present an exhaustive discussion on the results for each single cluster. Instead, we point out the general features common to each sample cluster. In Sect. 2 we describe our model. In Sect. 3 we present the sample and the determination of the morphology and the relevant best fit parameters necessary for the estimate of total masses and gas mass fractions, which are discussed in Sects. 4 and 5, respectively. We discuss the influence of asphericity on the comparison of X-ray and weak lensing mass estimates in Sect. 6 and investigate further possible constrains on the cluster shape by means of the SZ effect in Sect. 7. We present a summary and conclusions in Sect. 8.
If not stated explicitly, and are assumed. uncertainties for the estimated parameters are used throughout this paper.
z | ||
Cl0016+16 | 0.5455 | 7.55^{+0.72}_{-0.58}[1] |
A 478 | 0.0881 | 6.40^{+0.25}_{-0.25}[2] |
A 1795 | 0.0631 | 5.68^{+0.11}_{-0.11}[2] |
A 1068 | 0.1386 | 5.5^{+1.4}_{-0.9}[3] |
A 1413 | 0.1427 | 8.5^{+1.3}_{-0.8}[3] |
A 2390 | 0.231 | 11.1^{+1.0}_{-1.0}[5] |
A 2199 | 0.0298 | 4.22^{+0.06}_{-0.06}[2] |
A 2029 | 0.0765 | 8.47^{+0.41}_{-0.36}[3] |
A 2597 | 0.0852 | 3.6^{+0.2}_{-0.2}[4] |
Hydra A | 0.0522 | 3.71^{+0.14}_{-0.14}[2] |
The clusters are selected from the ROSAT archive. Since our goals require a sample of clusters that are regular and show a smooth, elliptical surface brightness, we exclude all the clusters with bimodal or strongly irregular morphology and those with a high spherical symmetry. In order to have a robust estimate of the parameters that constrain the morphology, we select only those clusters for which both ROSAT High Resolution Imager (HRI) and Position Sensitive Proportional Counter (PSPC) observations are available. Because of the lack of detailed temperature maps and for simplicity, we assume that the gas in each cluster is isothermal. The 10 selected clusters with the additional information needed to deproject the surface brightness are listed in Table 1.
For both HRI and PSPC, we prepare exposure corrected images and remove point sources embedded in the cluster emission to prevent contamination on measurements of the ellipticity of the surface brightness. The determination of the isophotes ellipticity and position angle is performed with the MIDAS routine FIT/ELL3, an iterative least-squares method in which the isophotes are free to translate position, and change ellipticity and orientation (Bender & Möllenhof 1987). From each image processed with this algorithm we obtain a set of ellipses with given position angles, half minor and major axes and centre coordinates. Even though our model implies that the isophotes are similar and concentric, such a set of parameters allows us to identify the isophotes which are suitable to describe the shape of the cluster emission. In the PSPC data we notice that for all the clusters the minor to major axis ratio e tends to unity as we approach the centre. We identify this effect as a distortion due to the point spread function (PSF) of the PSPC and thus discard the isophotes within the central 25 arcsec region.
The X-ray emission can be traced up to
,
which is estimated from the surface brightness profile and listed in Table 5. For each cluster we also can identify a distance from the centre beyond which the isophote parameters jump to arbitrary values. As this distance is approximately equal to
,
we also discard the isophotes beyond it.
We then compute the mean values of the minor to major axis ratio e, position angle and centre position from the parameters of the remaining isophotes. For the HRI images we also notice the same central feature, but due to the smaller field of view, we are not able to estimate the ellipses parameters for the whole cluster emission. In any case we find that the mean values for the minor to major axis ratio e, the position angle and centre computed from HRI and PSPC images agree within the errors when evaluated over the same radial range. For the analysis we use the PSPC images because of the PSPC's superior sensitivity and larger field of view. The relevant morphological parameters (the position angle and the minor to major axis ratio e) from the image analysis are listed in Table 2.
Since we are interested in the differences between spherical and ellipsoidal geometries in the estimate of total masses and gas mass fractions, we extract two surface brightness profiles from the PSPC images, referred to as circular and elliptical profiles. The circular profiles are determined by computing the counts within circular annuli spaced by 5 arcsec, while for the elliptical profiles the bins are similar and concentric ellipses with the minor to major axis ratios e and position angles listed in Table 2 and spaced by 5 arcsec in the direction of the major axis. We parametrise the circular and elliptical profiles with:
e | ||||
Cl0016+16 | 0.64 | 1.80 | ||
A 478 | 0.65 | 1.56 | ||
A 1795 | 0.64 | 1.48 | ||
A 1068 | 0.62 | 1.27 | ||
A 1413 | 0.59 | 1.05 | ||
A 2390 | 0.62 | 1.31 | ||
A 2199 | 0.66 | 1.68 | ||
A 2029 | 0.63 | 1.35 | ||
A 2597 | 0.62 | 1.30 | ||
Hydra A | 0.66 | 1.75 |
Cl0016+16 | |||
A 478 | |||
A 1795 | |||
A 1068 | |||
A 1413 | |||
A 2390 | |||
A 2199 | |||
A 2029 | |||
A 2597 | |||
Hydra A |
Cl0016+16 | |||
A 478 | |||
A 1795 | |||
A 1068 | |||
A 1413 | |||
A 2390 | |||
A 2199 | |||
A 2029 | |||
A 2597 | |||
Hydra A |
Assuming hydrostatic equilibrium and isothermality of the intracluster gas, the total mass density
of a cluster can be computed from the -model with the best fit parameters listed in Tables 3 and 4:
Since most of the mass is contained in the core, we find that at large radii, at for instance, the relative errors are almost independent of i and have values of a few per cent. This means that at large distances from the centre the possible compression or elongation along the line of sight is not important and that the assumption of spherical symmetry yields very reliable results. The relative errors for for the 4 models are listed in Table 6. As R decreases the relative errors in the mass estimations get larger (see Fig. 1): for shapes elongated along the line of sight we always find overestimations which increase with the value of the core radius along the line of sight. The reason is that for strong elongations a large portion of the core is excluded from the sphere within which the mass is estimated. A similar explanation holds for the underestimation found for compressed shapes. In Fig. 2, we show the relative errors at for the 10 sample clusters as a function of i. At we find: , , , , where means that we average over the 10 sample clusters. As the masses are usually estimated at or larger, these errors usually do not affect the mass determination.
We also perform the same analysis on a recent CHANDRA observation of the galaxy cluster RBS797 (Schindler et al. 2001), which shows an elliptical emission with a minor to major axis ratio of 0.77. This cluster is an excellent object for this analysis because the minor to major axis ratio e and the position angle do hardly change with radius. For this cluster we find
and
and we arrive at the same conclusions on the relative errors for the total mass and gas mass fraction obtained from our ROSAT sample. For instance, comparing these values with the estimates from the 4 triaxial models discussed in this section, we find that the relative error for the total mass within
is always less than .
Cl0016+16 | 1.051 | 2.1 | ||
A 478 | 1.492 | 1.5 | ||
A 1795 | 1.456 | 1.2 | ||
A 1068 | 1.384 | 1.2 | ||
A 1413 | 1.677 | 1.9 | ||
A 2390 | 1.731 | 1.9 | ||
A 2199 | 1.288 | 1.3 | ||
A 2029 | 1.713 | 1.1 | ||
A 2597 | 1.172 | 1.4 | ||
Hydra A | 1.170 | 1.3 |
Figure 1: The relative errors for the total mass estimates for A2390 plotted for the 4 triaxial models: compressed along the line of sight ( ) (dotted line), prolate (i=e) (dashed-dotted line), oblate (i=1) (dashed line) and elongated ( ) (solid line) shapes. Positive relative errors imply overestimations of the total mass if spherical symmetry is assumed: this is the case if the cluster is elongated. Underestimations are found for compressed clusters. | |
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Cl0016+16 | -3.44 | -1.36 | -0.08 | 4.72 |
A 478 | 0.84 | 0.89 | 0.93 | 1.04 |
A 1795 | -0.65 | -0.60 | -0.55 | -0.45 |
A 1068 | 0.73 | 0.78 | 0.83 | 0.89 |
A 1413 | 3.73 | 3.84 | 4.06 | 4.09 |
A 2390 | 0.32 | 0.52 | 0.73 | 1.00 |
A 2199 | 0.85 | 0.90 | 0.93 | 1.04 |
A 2029 | 2.05 | 2.08 | 2.10 | 2.15 |
A 2597 | 1.51 | 1.55 | 1.59 | 1.65 |
Hydra A | -0.20 | -0.17 | -0.16 | -0.08 |
We investigate the same 4 triaxial models as in Sect. 4 with respect to the gas mass fraction. For the computation of the gas mass fractions for the 10 sample clusters we deproject the X-ray emission with the -model in order to determine the gas density
and then integrate it within a sphere of radius R. We thus obtain the gas mass fractions
for the spherical geometry, which are listed in Table 5 for
,
and the gas mass fractions
for the triaxial models, where i is defined as in Sect. 2. Similarly, we define the relative error:
Since the gas mass fraction is the ratio of the gas mass to the total mass , the discussion of the relative errors involves many quantities. We notice that, since the central surface brightness is proportional to , the central gas density is larger for the compressed shapes than for elongated. This factor is usually dominant for small R, leading to gas mass overestimations for elongated ellipsoids and underestimations for compressed shapes. In Table 8 we list the gas mass fraction relative errors for the 4 models at . We find small over- and underestimations for oblate and prolate shapes for small R: and at . For elongated and compressed shapes we find larger but in general still negligible errors: at we get and .
At large distances the core radius along the line of sight is dominant: small core radii imply steep profiles and consequently less gas mass. In Fig. 4, we show the relative errors at
for the 10 sample clusters as a function of i. We see that at large distances from the cluster center, at
or larger, where the total masses for a spherical and an elliptical model are almost the same (see Sect. 4), the compressed shapes imply an overestimation of the gas mass fraction for the majority of the clusters. Although this overestimation is small, the more compressed the shape, the larger the overestimation (see Fig. 3).
At
we find the average values:
,
and
at
,
showing small underestimations of the gas mass fractions for elongated shapes (see also Table 9). For oblate (i=1) and prolate (i=e) shapes we find that at
both underestimates and overestimates are present and that they are small (see Table 9).
Averaging the absolute values of the relative errors we find:
and
.
We conclude that these errors usually do not substantially affect the gas mass fraction determination at large radii.
Cl0016+16 | -11.62 | -1.98 | 3.52 | 20.23 |
A 478 | -11.64 | -3.34 | 2.29 | 15.05 |
A 1795 | -9.87 | -0.93 | 5.65 | 17.79 |
A 1068 | -12.19 | -4.65 | 3.16 | 10.28 |
A 1413 | -14.24 | -8.45 | 0.96 | 2.31 |
A 2390 | -7.68 | 1.26 | 9.73 | 18.69 |
A 2199 | -12.18 | -3.83 | 1.03 | 15.40 |
A 2029 | -14.25 | -7.07 | -0.55 | 7.70 |
A 2597 | -13.20 | -5.70 | 1.61 | 9.15 |
Hydra A | -8.13 | -1.67 | 1.77 | 14.08 |
Figure 2: The relative errors for the total mass E_{M}^{i} as a function of i for the 4 models ( , e, 1 and , which are separated by the vertical lines) at for the 10 sample clusters. For clusters elongated along the line of sight we find overestimations, while for compressed clusters the total mass is underestimated. | |
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Since the X-ray method and gravitational lensing are independent methods used to determine the dark matter distribution in cluster of galaxies, their comparison is very useful. First of all, the morphology of the dark matter distribution can be compared: Smail et al. (1995) found good agreement for the position angle and ellipticity of Cl0016+16 determined with weak lensing and X-ray observations. Furthermore, Miralda-Escudé & Babul (1995), presenting a detailed study of three clusters with both arcs and X-ray data available, conclude that the mass estimates from the arc modelling can be nearly a factor 2-3 larger than those from the X-ray observation in the innermost regions. As summarised by Kneib (2000), the discrepancy might be due to different reasons: too simple X-ray modelling, merging and projection effects. Also non-thermal effects can play a role, although magnetic fields can be ruled out as error sources (Dolag & Schindler 2000).
Since gravitational lensing estimates provide projected masses within projected distances from the centre, the X-ray mass must be projected too, in order to achieve the comparison. In Sect. 4 the X-ray mass was calculated within a spherical volume, while for the comparison it has to be computed within a cylindrical volume. We investigate projection effects by considering a triaxial -model and the 4 models used in Sects. 4 and 5. As done in Sect. 4, we compare the estimates from the triaxial modelling with those obtained assuming spherical symmetry, but integrating Eq. (4) within a cylinder of radius R. We thus obtain a projected masses and for the spherical model and a triaxial model with a core radius along the line of sight , respectively. Formally, the integral of the total mass density extends from the observer along the line of sight through the cluster infinitely; in practice, a cutoff is used. In the following, a cutoff equal to is used, as a compromise between to much extrapolation and too short integration length. Although the integration volume is different from the one used in Sect. 4, the ranges of elongations i given in Table 2 apply in this case as well.
Similar to the relative errors defined in Sects. 4 and 5, we define:
In Table 10 we list the relative errors at for and . Within this projected distance we find the average values: , , and . In Table 11 we list the relative errors at for the 4 models. Within we find the average values: , , and . These results show that X-ray estimated projected masses are larger for elongated clusters than for spherical ones and that discrepancies up to 30% between the two methods can be resolved if a triaxial -model with a maximal elongation is used.
For the cluster Cl0016+16, Smail et al. (1995) derive a projected mass of
integrated out to a projected radius of
using X-rays, and
from weak lensing. Using the spherical model, our estimate for the projected mass is
if no cut-off is used and
with a cutoff equal to
.
Instead we find that using a triaxial model the discrepancy disappears if the core radius along the line of sight is
(we use a cutoff equal to
). Cl0016+16 is among the most X-ray luminous clusters known and X-ray selected, thus a shape which is elongated along the line of sight is not surprising. From this estimate we conclude that elongation is one of the factors contributing to the discrepancy between lensing and X-ray mass estimates.
Figure 3: The relative errors for the gas mass fraction estimates for A2390 plotted for the 4 triaxial models: compressed along the line of sight ( ) (dotted line), prolate (i=e) (dashed-dotted line), oblate (i=1) (dashed line) and elongated ( ) (solid line) shapes. For compressed clusters the gas mass fraction is overestimated. | |
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Cl0016+16 | 7.54 | 1.37 | -0.29 | 4.94 |
A 478 | 4.91 | 1.90 | 1.38 | 6.19 |
A 1795 | 3.96 | 0.64 | -0.04 | 3.70 |
A 1068 | 6.78 | 2.94 | 1.57 | 2.94 |
A 1413 | 10.74 | 7.73 | 6.57 | 6.81 |
A 2390 | 5.31 | 1.43 | 0.24 | 2.05 |
A 2199 | 4.53 | 1.83 | 1.52 | 8.06 |
A 2029 | 6.40 | 3.88 | 3.56 | 6.48 |
A 2597 | 7.44 | 3.93 | 2.94 | 4.82 |
Hydra A | 2.30 | 0.40 | 0.40 | 7.92 |
Figure 4: The relative error for the gas mass fraction as a function of i for the 4 models ( , e, 1 and ) at for the 10 sample clusters. The 4 elongations are separated by 3 vertical lines. | |
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Cl0016+16 | 8.57 | 2.52 | 0.30 | -0.13 |
A 478 | 1.92 | -1.10 | -2.32 | -2.60 |
A 1795 | 5.63 | 2.28 | 0.76 | 0.08 |
A 1068 | 5.21 | 0.48 | -3.06 | -5.33 |
A 1413 | 0.65 | -2.42 | -5.41 | -5.66 |
A 2390 | 3.01 | -0.76 | -3.01 | -4.07 |
A 2199 | 1.26 | -1.22 | -1.98 | -0.97 |
A 2029 | -1.15 | -3.48 | -4.54 | -4.45 |
A 2597 | 2.86 | -1.17 | -3.82 | -5.44 |
Hydra A | 4.07 | 1.19 | 0.34 | 1.05 |
From X-ray observations only, it is impossible to have information on the cluster elongation in the direction of the line of sight. The uncertainties on the total mass and gas mass fraction estimates due to this unknown quantity might be reduced by noticing that the core radius along the line of sight can be constrained using a complementary and independent measurement: the SZ effect. Assuming the triaxial -model as in Sect. 1 one gets:
Instead, using (Fukugita & Hogan 2000), we obtain: i=1.24^{+0.54}_{-0.40}. We find: , , and . We conclude that in this case a triaxial -model improves the estimation of the total mass within the central region. We note that the estimated elongation of agrees qualitatively with the conclusion of Sect. 6, but this smaller value suggests that elongations along the line of sight can only partially resolve the discrepancy between lensing and X-ray mass estimates.
Figure 5: The relative errors for the projected mass estimates for A2390 plotted for the 4 triaxial models: compressed along the line of sight ( ) (dotted line), prolate (i=e) (dashed-dotted line), oblate (i=1) (dashed line) and elongated ( ) (solid line) shapes. The cut-off is . Negative relative errors imply underestimations of the total mass if spherical symmetry is assumed: this is the case if the cluster is elongated. Overestimations are found for compressed clusters. Therefore, an elongation along the line of sight can contribute to resolve the discrepancy between X-ray and lensing mass. | |
Open with DEXTER |
Cl0016+16 | 41.54 | 11.03 | -3.55 | -41.07 |
A 478 | 39.43 | 8.29 | -8.61 | -40.80 |
A 1795 | 41.24 | 9.71 | -8.81 | -37.81 |
A 1068 | 40.01 | 11.08 | -12.38 | -30.63 |
A 1413 | 39.10 | 14.26 | -16.44 | -20.27 |
A 2390 | 43.68 | 13.70 | -9.01 | -29.71 |
A 2199 | 38.49 | 6.89 | -7.93 | -44.57 |
A 2029 | 38.75 | 9.11 | -11.98 | -34.46 |
A 2597 | 40.26 | 10.47 | -12.24 | -32.12 |
Hydra A | 39.68 | 6.67 | -7.19 | -46.37 |
Cl0016+16 | 40.70 | 14.35 | 1.65 | -30.89 |
A 478 | 30.77 | 7.31 | -5.70 | -30.72 |
A 1795 | 29.50 | 6.27 | -7.69 | -29.78 |
A 1068 | 31.44 | 9.65 | -8.45 | -22.69 |
A 1413 | 35.65 | 16.04 | -8.78 | -11.90 |
A 2390 | 31.33 | 9.01 | -8.27 | -24.14 |
A 2199 | 30.42 | 6.55 | -4.88 | -33.42 |
A 2029 | 32.91 | 10.20 | -6.35 | -24.18 |
A 2597 | 32.84 | 10.22 | -7.45 | -23.10 |
Hydra A | 29.60 | 5.04 | -5.51 | -35.57 |
The elliptical X-ray emission shown by many galaxy clusters make us ponder about the possible triaxial distribution of the ICM. In this paper we model the X-ray surface brightness by means of a spherical and a triaxial -model and, assuming hydrostatic equilibrium and isothermality of the intracluster gas, we compare the estimates on the total mass and gas mass fraction we obtain from these geometrically different models. Strongly elongated shapes are not allowed in this model, suggesting that if these should be nonetheless observed they may be indicative of unvirialised clusters. Analysing 10 ROSAT clusters we find:
Acknowledgements
We thank D. Puy for useful discussions. We are grateful to the referee for many useful suggestions. This work is partially supported by the Swiss National Science Foundation.