Chandra measurements of the distribution of mass in the luminous lensing cluster Abell 2390

Abstract

We present spatially resolved X-ray spectroscopy of the luminous lensing cluster Abell 2390, using observations made with the Chandra observatory. The temperature of the X-ray gas rises with increasing radius within the central ∼ 200 kpc of the cluster, and then remains approximately isothermal, with kT = 11.5_−1.6^+1.5 keV, out to the limits of the observations at r∼1.0 Mpc. The total mass profile determined from the Chandra data has a form in good agreement with the predictions from numerical simulations. Using the parametrization of Navarro, Frenk and White, we measure a scale radius r_s∼0.8 Mpc and a concentration parameter c∼3. The best-fitting X-ray mass model is in good agreement with independent gravitational lensing results and optical measurements of the galaxy velocity dispersion in the cluster. The X-ray gas to total mass ratio rises with increasing radius with f_gas∼21 per cent at r = 0.9 Mpc. The azimuthally averaged 0.3–7.0 keV surface brightness profile exhibits a small core radius and a clear ‘break’ at r∼500 kpc, where the slope changes from S_X∼^∝ r^−1.5 to S_X∼^∝ r^−3.6. The data for the central region of the cluster indicate the presence of a cooling flow with a mass deposition rate of 200–300 M_⊙ yr⁻¹ and an effective age of 2–3 Gyr.

gravitational lensing, galaxies: clusters: individual: Abell 2390, cooling flows, intergalactic medium, X-rays: galaxies: clusters

1 Introduction

Accurate measurements of the masses of clusters of galaxies are of profound importance to cosmological studies. Originally, most measurements of cluster masses were based on optical studies of their galaxy dynamics, wherein the motions of individual galaxies were used to trace the cluster potentials. Although such studies were shown to be sensitive to systematic uncertainties due to velocity anisotropies, substructure and projection effects (e.g. Lucey 1983; Frenk et al. 1990; van Haarlem, Frenk & White 1997), more recent work based on large galaxy samples and employing careful selection techniques has led to significant progress (e.g. Carlberg et al. 1996; den Hartog & Katgert 1996; Fadda et al. 1996; Mazure et al. 1996; Borgani et al. 1999; Geller, Diaferio & Kurtz 1999; Koranyi & Geller 2000).

Recent years have also seen the development of two further techniques for measuring the masses of clusters, based on X-ray observations and studies of gravitational lensing by clusters, respectively. X-ray mass measurements use the assumption that the X-ray-emitting gas that pervades clusters is in hydrostatic equilibrium; the total mass distribution is determined once the radial distributions of the X-ray gas density and temperature are known (see e.g. Sarazin 1988). Since the X-ray emissivity is proportional to the square of the gas density, and the relaxation time-scale for the X-ray gas is relatively short (of the order of a few sound crossing times), the X-ray method is relatively free from the projection and substructure effects that hamper the aforementioned optical studies.

In contrast to the X-ray and optical dynamical techniques, gravitational lensing offers a method for measuring the projected masses through clusters that is essentially free from assumptions about the dynamical state of the gravitating matter (for reviews see Fort & Mellier 1994; Bartelmann & Schneider 1999; Mellier 1999). The primary observational challenges of requiring deep exposures, excellent seeing conditions, wide-field imaging and accurate point spread function models have now been mostly overcome with improved instrumentation (e.g. Bacon et al. 2000 and references therein), although the recovery of the three-dimensional mass distributions in clusters can be complicated by projection effects and uncertainties in the redshift distributions of the lensed sources.

Clearly, the best approach when attempting to measure the masses of clusters reliably is to combine these three methods. The first combined X-ray and lensing studies of galaxy clusters (Miralda-Escudé & Babul 1995) suggested that strong lensing masses, measured within r ≲ 200 kpc of the cluster centre, might typically overestimate X-ray-determined masses by a factor of ∼2–3. This sparked debate into the possible effects of oblate/prolate cluster geometries, projection effects, complex temperature structures and pressure support from bulk and/or turbulent motions and magnetic fields in the X-ray gas (e.g. Loeb & Mao 1994; Miralda-Escudé & Babul 1995; Waxman & Miralda-Escudé 1995; Kneib et al. 1995; Bartelmann & Steinmetz 1996). Later work (Allen, Fabian & Kneib 1996; Allen 1998; Böhringer et al. 1998; Wu 2000) highlighted a clear difference between the results obtained for cooling-flow (CF) and non-cooling-flow (NCF) clusters. For CF clusters, the X-ray and strong lensing mass measurements generally show good agreement, once the effects of their cool spectral components are accounted for in the X-ray analysis. For NCF systems, however, the masses inferred from the strong lensing data invariably exceed the X-ray values, determined under the hydrostatic assumption, by a factor of 2–4.

The origin of the different results obtained for CF and NCF clusters is thought to lie in the different dynamical states of these systems: whereas X-ray and optical imaging and detailed lensing analyses of CF clusters show them to be relatively regular and dynamically relaxed systems, NCF clusters generally appear to be undergoing major subcluster merger events (e.g. Edge, Stewart & Fabian 1992; Kneib et al. 1995; Buote & Tsai 1996; Smail et al. 1995, 1997; Squires et al. 1997). Significant offsets between the X-ray and lensing centroids are observed in NCF clusters, demonstrating a loss of hydrostatic equilibrium in their central regions. The X-ray core radii for NCF systems also appear to have been inflated by the dynamical activity, in agreement with the predictions from numerical simulations (e.g. Roetigger, Burns & Loken 1996); this inflation of the X-ray core radii can account for the bulk of the X-ray/strong lensing mass discrepancy in most NCF systems (Allen 1998). On larger spatial scales (r ≳ 0.5 Mpc), comparisons between weak lensing, X-ray and optical dynamical mass measurements generally provide a consistent picture, with excellent agreement between the results obtained for both CF and NCF clusters (e.g. Squires et al. 1996; Smail et al. 1997; Wu & Fang 1997; Allen 1998; Lewis et al. 1999). This suggests that the loss of hydrostatic equilibrium in NCF clusters is primarily restricted to their inner regions.

The most significant uncertainty associated with the X-ray mass measurements in previous joint X-ray/lensing studies has been the absence of any direct measurements of the X-ray temperature profiles in the clusters, which impacts directly on the mass measurements through the hydrostatic equation. Although limited spatially resolved spectroscopy for bright, nearby clusters was possible using ASCA and BeppoSAX observations (e.g. Markevitch et al. 1998; Kikuchi et al. 1999; de Grandi & Molendi 1999; White 2000; Irwin & Bregman 2000), for the more distant lensing clusters, typically observed at redshifts z ≲ 0.2, only a single, integrated cluster spectrum was normally available. In most cases, only a mean emission-weighted X-ray temperature was therefore determined, although more sophisticated studies also accounted for the effects of cooling flows on the X-ray data (e.g. Allen 1998; Böhringer et al. 1998). For the mass analyses, it was then necessarily assumed that the mass-weighted temperature profile followed some particular form, and usually that it remained approximately isothermal with radius. However, the validity of this assumption remains uncertain, especially within the strong lensing regime.

The launch of the Chandra observatory (Weisskopf et al. 2000) in 1999 July provides the first opportunity for detailed, spatially resolved X-ray spectroscopy of clusters of galaxies at moderate redshifts. The Advanced CCD Imaging Spectrometer (ACIS) on Chandra permits the first direct, simultaneous measurements of the X-ray temperature and density profiles and, via the hydrostatic assumption, the mass distribution in luminous lensing clusters, spanning both the weak and strong lensing regimes. In this paper we present the first results from Chandra observations of the massive CF lensing cluster Abell 2390, which has been the subject of several previous combined X-ray/optical/lensing studies (e.g. Pierre et al. 1996; Squires et al. 1996; Allen 1998; Böhringer et al. 1998; Lewis et al. 1999). We present detailed results on the mass distribution in the cluster determined from the Chandra data and compare our results with those from strong and weak lensing analyses and optical dynamical studies. We also re-examine the properties of the cooling flow in the cluster and discuss their relation to the dynamical history of the system.

The cosmological parameters , and are assumed throughout. At the redshift of Abell 2390 an angular scale of 1 arcsec corresponds to a physical length of 4.652 kpc.

2 Observations

The Chandra observations of Abell 2390 were carried out using the ACIS on 1999 November 7. The target was observed in the back-illuminated charge-coupled device (CCD) detectors, close to the nominal aim point for the ACIS S3 detector. (The source was positioned near the centre of node 1 on the chip.) The focal plane temperature at the time of the observations was −110°C.

We have used the ciao software (version 1.1.3) and the level-2 events file provided by the standard Chandra pipeline processing for our analysis. The light curve for the observation was of high quality with no strong background flaring. Only those X-ray events with grade classifications of 0, 2, 3, 4 and 6 were included in our final cleaned data set, for which the net exposure time was 9.13 ks.

3 X-ray imaging analysis

3.1 X-ray morphology

The raw image of the central region of Abell 2390 is shown in Fig. 1(a). The pixel size is , corresponding to raw detector pixels. Fig. 1(b) shows an adaptively smoothed version of the same image, using the smoothing algorithm of Ebeling, Rangarajan & White (2001). The position (RA Dec.) of the peak of the X-ray emission from the cluster, (J2000), is in excellent agreement with the optical centroid for the dominant cluster galaxy of (J2000; Pierre et al. 1996). The X-ray image is elongated along an approximately north-west–south-east direction, in a similar manner to the optical isophotes of the dominant galaxy (Pierre et al. 1996).

Figure 1.

(Left panel) The raw Chandra image of Abell 2390. The pixel size is four detector pixels (1.97 arcsec). (Right panel) Contour plot of the same region, adaptively smoothed using the code of Ebeling et al. (2000), with a threshold value of 3.5σ. The contours have equal logarithmic spacing.

Some substructure is apparent in the Chandra image. In the central region, a relatively bright ridge of enhanced emission extends ∼ to the north-west of the X-ray peak (Fig. 2). Enhanced emission in the same direction is also observed in the optical blue continuum and optical/UV line emission from the dominant cluster galaxy (Lémonon et al. 1998; Edge et al. 1999; Hutchings & Balogh 2000). A correspondence between the brightest X-ray emission with excess blue continuum and optical/UV line emission (probably associated with the formation of young, massive stars) is also observed in the nearby CF clusters Hydra A (McNamara et al. 2000) and Abell 1795 (Fabian et al. 2001). A second, fainter region of enhanced X-ray emission is located ∼ 6 arcsec to the south-south-east.

Figure 2.

An adaptively smoothed image of the central regions of Abell 2390 on a finer spatial scale (pixel size , which is equivalent to raw detector pixels). Note the ridge of enhanced emission extending ∼ to the north-west of the X-ray peak which is coincident with the excess blue continuum and optical/UV line emission reported by Lémonon et al. (1998) and Hutchings & Balogh (2000).

On medium scales , the X-ray emission is extended to the north-west, in a similar manner to the optical luminosity distribution and lensing mass models of Pierre et al. (1996)– these latter authors also note the presence of an X-ray extension in the same direction using ROSAT High Resolution Imager data. On large scales (≳2 arcmin), the X-ray emission is extended towards the east. The presence of such substructure in the Chandra image suggests that the cluster has not fully relaxed following its most recent merger activity. However, the agreement between the X-ray, optical and gravitational lensing mass measurements discussed in Section 5.2 argues that overall the assumption of hydrostatic equilibrium in the cluster is a reasonable one.

Three point sources are detected to the west of the X-ray peak, in the central regions of the cluster, at positions , and . The X-ray and submillimetre properties of these sources are discussed by Fabian et al. (2000). Four more point sources are also visible at larger radii in Fig. 1(b).

3.2 The surface brightness profile

The azimuthally averaged, X-ray surface brightness profile for Abell 2390 is shown in Fig. 3. The profile has been flat-fielded and background-subtracted using a rectangular background region of size , located ∼ 5 arcmin from the cluster centre. All obvious point sources were excluded from the analysis.

Figure 3.

(a) The background-subtracted, flat-fielded, azimuthally averaged radial surface brightness profile for Abell 2390 in the band. The bin size is 0.984 arcsec (4.58 kpc). (b) The same profile, rebinned by a factor of 4, in the range , with the best-fitting broken power-law model overlaid. At the break radius of , the slope of the surface brightness profile changes from to .

The X-ray emission from the cluster extends beyond the 5 arcmin (∼1.4 Mpc) radius associated with our on-chip background region (see also Pierre et al. 1996; Böhringer et al. 1998). However, beyond this radius, background counts dominate the flux in the ACIS-S3 detector. Fig. 3 shows the data for the central 900 kpc (193 arcsec), for which systematic errors associated with the background subtraction and flat-fielding are negligible. The bin size in Fig. 3(a) is two detector pixels (0.984 arcsec). Fig. 3(b) shows the data for the outer regions of the cluster with a larger bin size of eight detector pixels (3.94 arcsec).

Within a radius of 500 kpc (107 arcsec), the X-ray surface brightness profile can be parametrized for 106 degrees of freedom) by a standard β model (e.g. Jones & Forman 1984) of the form , with a core radius and a slope parameter (1σ errors; . On larger scales, however, the β model does not provide an acceptable fit: examining the data for the central 900-kpc radius, we obtain for 194 degrees of freedom, with best-fitting parameter values of and . Ignoring the central (18 arcsec) region, associated with the possible cooling flow (Section 6), the fit is improved for 177 degrees of freedom, with best-fitting parameter values and , although it is still formally unacceptable.

The main reason for the poor fit obtained with the β model at larger radii is the presence of a ‘break’ in the surface brightness profile at (Fig. 3b). This break is not obviously due to substructure in any particular direction in the cluster: Fig. 4 shows the surface brightness profile measured in the four quadrants covering position angles , , and degrees. We see that the profile appears remarkably similar in three of the four directions, although the emission is slightly more extended towards the east (as is also evident in the images presented in Fig. 1).

Figure 4.

The surface brightness profile in quadrants covering position angles degrees (filled circles), degrees (open circles), degrees (triangles) and degrees (squares). The extension towards the east is also evident in Fig. 1. The results for the other quadrants are in good agreement with each other.

A good fit to the surface brightness profile beyond the central cooling region can be obtained using a simple broken power-law model. Fitting the data from , we obtain for 176 degrees of freedom, with a break at a radius of , and slopes in the regions internal and external to the break radius of and , respectively. Interestingly, these slopes are similar to the values expected at small and large radii for the dark matter in a Navarro, Frenk & White (1997; hereafter NFW) potential in which . However, isothermal gas in an NFW-like potential should not exhibit a sharp break at the scale radius r_s, but rather a slow rollover [although individual clusters in the simulations presented by Thomas et al. (2000) do exhibit sharp breaks in their dark matter distributions]. Fitting the surface brightness profile external to the cooling flow with the prescription for isothermal gas in an NFW potential described by Ettori & Fabian (1999), we measure (formal 1σ errors), with for 177 degrees of freedom. Thus, the fit with the NFW mass model assuming strict isothermality in the X-ray gas is formally unacceptable, although it provides a better description of the data in the region than the β model. As discussed in Section 4, the Chandra data show that the X-ray gas in Abell 2390 is not isothermal and that the temperature rises with increasing radius within the central . In Section 5 we show that an NFW mass model can provide a good description of the Chandra data, once the assumption of isothermality is relaxed.

4 Spatially resolved spectroscopy

4.1 Method of analysis

For our spectral analysis, we divided the cluster into annular regions, as detailed in Table 1. A spectrum was extracted from each region in 1024 pulse height analyser (PHA) channels. The spectra were regrouped to contain a minimum of 20 counts per PHA channel, thereby allowing χ² statistics to be used. (For the two outer annuli, a larger grouping of 40 counts per PHA channel was used because of the increased background contribution.) Background spectra appropriate for the regions studied were extracted from the ACIS-S3 blank-field data sets provided by M. Markevitch and made available from the Chandra X-ray Center. All obvious point sources were masked out and excluded from the analysis. Separate photon-weighted response matrices and effective area files were constructed for each region using the calibration and response files appropriate for the focal plane temperature, available from the Chandra X-ray Center.

Table 1.

The results from the analysis of the annular spectra using spectral model A in the 0.5–7.0 keV band. Temperatures (kT) are in keV, metallicities (Z) in solar units and normalizations (K) in units of {10⁻¹⁷/4γ[D_A(1 + z)]²} n_en_H dV, where D_A is the angular size distance to the source (in cm), n_e is the electron density (in cm⁻³) and n_H is the hydrogen ion density (in cm⁻³). The absorbing column density has been fixed at the nominal Galactic value of 6.8 × 10²⁰ atom cm⁻². The total η² values and number of degrees of freedom (DOF) in the fits are listed in column 5. Error bars at both the 1σ (Δη ² = 1.0) and 90 per cent (Δη ² = 2.71; in parentheses) confidence limits on each single parameter of interest are listed. For the outermost 0.7–1.0 Mpc annulus, where the instrumental background is most significant, only data in the energy range 0.5–6.0 keV range were used.

Two separate energy ranges were examined. First, a conservative band was used over which the calibration of the back-illuminated CCD detectors is currently best understood. Secondly, for the central 100 kpc region, where a cooling flow is thought to exist (Section 6), we have also examined a more extended energy range, which provides extra constraints on the presence of cool emission components and/or intrinsic absorption in the cluster.

4.2 The spectral models

The analysis of the spectral data has been carried out using the xspec software package (version 11.01; Arnaud 1996). The spectra were modelled using the plasma emission code of Kaastra & Mewe (1993), incorporating the Fe L calculations of Liedhal, Osterheld & Goldstein 1995), and the photoelectric absorption models of Balucinska-Church & McCammon (1992). We first examined each annular spectrum using a simple, single-temperature model with the absorbing column density fixed at the nominal Galactic value Dickey & Lockman 1990). This model is hereafter referred to as model A. The free parameters in model A were the temperature (kT) and metallicity (Z) of the plasma [measured relative to the solar photospheric values of Anders & Grevesse (1989), with the various elements assumed to be present in their solar ratios] and the emission measure (K). We also examined a second single-temperature model (model B), which was identical to model A but with the absorbing column density (N_H) also included as a free parameter in the fits.

The image deprojection and X-ray colour profile analyses discussed in Section 6 indicate the presence of a strong cooling flow in the central region of the cluster. We have therefore also examined the spectral data for this region using a series of more sophisticated, multiphase models in which the emission properties of the cooling flow were explicitly accounted for. The first such model (model C1) introduced an extra emission component into model A, with a spectrum appropriate for gas cooling at constant pressure from the ambient cluster temperature (following the prescription of Johnstone et al. 1992). The normalization of this component was parametrized in terms of a mass deposition rate, M˙, which was a free parameter in the fits. In the second case, model C2, the cooling gas was modelled as an isothermal cooling flow, following Nulsen (1998); we assume a value for , where the integrated mass deposition rate within radius r, . The mean gas temperature and metallicity in both the cooling flow and isothermal emission component (which accounts for the emission from gas at larger radii viewed in projection) were assumed to be equal. Finally, we also examined a more general emission model, model D, in which the cooling gas was modelled by a second, cooler isothermal emission component, with the temperature and normalization of this component included as free fitting parameters. Model D provides a more flexible parametrization, with an additional degree of freedom over models C1 and C2, and invariably provides a good match to the more specific cooling-flow models at the spectral resolution and signal-to-noise ratios typical of ACIS cluster observations. However, the parameter values determined with model D were not well constrained for Abell 2390, and thus we do not quote explicit results for this model here.

With each of the cooling-flow emission models, we have also examined the effects of including extra absorption, using a variety of different absorption models. In the first case [absorption model (i)], the only absorption included was that due to cold gas in our Galaxy, with the equivalent column density fixed to the nominal Galactic value (Dickey & Lockman 1990); for a single-temperature emission model, this is identical to spectral model A. In the second model [absorption model (ii)], the absorption was again assumed to be due to Galactic (zero-redshift) cold gas, but with the column density, N_H, included as a free parameter in the fits. (For a single-temperature emission model, this would be equivalent to spectral model B.) In the third case [absorption model (iii)], an intrinsic absorption component with column density ΔN_H due to cold gas at the redshift of the cluster was introduced. The absorber was assumed to lie in a uniform screen in front of the cooling flow, with the column density included as a free fitting parameter. In the fourth case [absorption model (iv)], the intrinsic absorption was assumed to cover only a fraction, ƒ, of the emission from the cooling flow. The fifth and final case [absorption model (v)] was similar to model (iii) but with the gaseous absorber replaced by an intrinsic absorption edge, with the edge depth, τ, and energy, E_edge, free parameters in the fits. This more general absorption model may be used to approximate the effects of a dusty and/or ionized absorber.

In those cases where the absorption has been quantified in terms of an equivalent hydrogen column density, solar metallicity in the absorbing gas is assumed. We note that in absorption models (iii)–(v), the absorption acting on the ambient cluster emission was fixed at the nominal Galactic value. However, allowing the Galactic absorption to vary from this value did not significantly improve the fits.

4.3 Results from the single-phase analysis

The best-fitting parameter values and 1σ and 90 per cent confidence limits determined from the fits in the band with the single-temperature models are summarized in Tables 1 and 2. The temperature profile determined with spectral model A is shown in Fig. 5(a). The measured temperature is approximately isothermal beyond a radius of 200 kpc, out to the limits of the data at . A combined fit to the data in the range with model A gives a mean temperature of . Fitting the results with a simple power-law model of the form , we measure [1σ bootstrap errors obtained using the Akritas & Bershady (1996) modification of the ordinary least-squares statistic]. We observe a clear drop in the emission-weighted temperature within the central 100 kpc, with a mean value for the central of .

Table 2.

The results from the analysis of the annular spectra in the 0.5–7.0 keV band using spectral model B. The absorbing column density (N_H) is in units of 10²⁰ atom cm^‒2. Other details as in Table 1.

Figure 5.

(a) The projected X-ray gas temperature profile (and 1σ errors) measured with spectral model A in the energy range. The dotted lines mark the 90 per cent confidence limits on the mean ambient cluster temperature determined from a joint analysis of the data in the region. (b) The ambient X-ray gas temperature as a function of radius, corrected for the effects of the cooling flow in the central 100 kpc using spectral model C2(iii) (see Section 4.4,). For , the values are determined in an identical manner to those in (a).

We detect marginal evidence for a metallicity gradient in the cluster, with a mean, best-fitting value for the central 100 kpc of , which compares to a mean value of for the outer region (Fig. 6a; 1σ errors). The results determined with spectral model B in the band also provide marginal evidence for increased absorption towards the cluster core (Fig. 6b).

Figure 6.

(a) The (projected) variation of metallicity in the cluster measured with spectral model A in the energy range. (b) The absorbing column density measured with spectral model B. The dashed line shows the Galactic column density determined from H i studies (Dickey & Lockman 1990).

In all cases, the χ² values obtained from the fits to the data with the single-temperature models are acceptable, the only marginal case being the data for the central 50 kpc region, where the spectrum is complicated by the effects of the cooling flow.

4.4 Multiphase analysis of the cooling core

The results determined from the more detailed, multiphase analysis of the central 100 kpc radius in the extended energy band are summarized in Table 3. The results demonstrate a clear requirement for excess absorption in this region, over and above the nominal Galactic value (Dickey & Lockman 1990), using each of the different emission models. The systematic variations between the column density measurements obtained with the different emission and absorption models are similar to those determined from previous ASCA studies (e.g. Allen 2000). Unfortunately, the present Chandra data for Abell 2390 cannot statistically discriminate between the single-phase and multiphase emission models for the central 100 kpc region, which provide comparable χ² values with each absorption model. [We note, however, that at some level the spectrum for the central 100 kpc must be multiphase simply as a result of projection effects, given the results on the temperature profile shown in Fig. 5(a).]

Table 3.

The best-fitting parameter values and 1π (and 90 per cent; in parentheses) confidence limits determined from the multiphase analysis of the central 100 kpc region in the extended 0.3–7.0 keV band. The mass deposition rates from the cooling flow (M) are in units of M_⊙ yr⁻¹. Column densities (N_H) and intrinsic column densities (ΔN_H) are in units of 10²⁰ atom cm⁻². The limits on the edge energy (E_edge) determined with absorption model (v) are in keV. Other details as in Table 1.

For our preferred cooling-flow emission models including intrinsic absorption, which provide the most consistent physical description of the spectral and imaging data for the central 100 kpc region (Section 6), we measure an integrated mass deposition rate . Fig. 5(b) shows the projected temperature profile for Abell 2390, corrected for the effects of the cooling flow using spectral model C2(iii). (For the results determined with spectral model A in the band have been used.) We see that correcting for the effects of the cooling flow does not have a major impact on the temperature profile measured in the central regions of the cluster. Assuming that the isothermal cooling flow model provides a reasonable description of the data, the drop in the central temperature shown in Figs 5(a) and (b) should then reflect the mass distribution in the cluster core.

The data for Abell 2390 do not provide firm constraints on the nature of the intrinsic absorption in the cluster. Using the partial-covering absorption model (iv), we find that high covering fractions are preferred. For the edge-like absorption model (v), the lower limit on the edge energy is essentially unconstrained: for emission models C1(v) and C2(v), however, the upper limit on the edge energy is inconsistent with the O i K edge of oxygen , suggesting that the absorption is unlikely to be due to oxygen-rich dust grains (e.g. Arnaud & Mushotzky 1998; Allen et al. 2000b) or a warm, ionized absorber (Buote 2000).

Assuming that the intrinsic absorption is due to cold gas lying in a uniform screen in front of the cooling flow, using spectral model C2(iii), we measure . The mass of absorbing gas implied by this model is then , where r_abs is the radial extent of the absorber in kpc and ΔN_H is the equivalent hydrogen column density in units of 10²¹ atom cm⁻². For (Section 6.2) we obtain [although Allen & Fabian (1997) and Wise & Sarazin (2001) argue that, for a geometry in which the absorbing material is distributed throughout the X-ray-emitting region, the true mass is likely to be a few times higher]. This mass is in reasonable agreement with the mass expected to have been accumulated by the cooling flow within the same radius over its lifetime: (the factor 2 in the denominator arises from the assumption that the integrated mass deposition rate, M˙, grows approximately linearly with time).

4.5 Spectral deprojection analysis

The results discussed in the previous subsections are based on the analysis of projected spectra. In order to determine the effects of projection on the spectral results, we have also carried out a simple deprojection analysis of the Chandra spectral data.

For this analysis we have used the same annular regions and have assumed that the emission from each spherical shell (the shells are defined by the same inner and outer radii as the annular regions) is isothermal and absorbed by the Galactic column density (spectral model A). The fit to the outermost annulus is used to determine the temperature and emission measure in the outermost spherical shell. The contribution from that shell to each inner annulus is then determined by purely geometric factors (e.g. Kriss, Cioffi & Canizares 1983). The fit to the second annulus inwards is used to determine the parameters for the second spherical shell, and so forth, working inwards. Thus, the spectral model for the ith annulus working inwards is the weighted sum of i absorbed, isothermal models. This parallels the usual image deprojection procedure (e.g. Fabian et al. 1981).

We have used the xspec code and Chandra data in the band. The data for all eight annular spectra were fitted simultaneously in order to determine the parameter values and confidence limits correctly. The spectral model used therefore has free parameters (where n is the total number of annuli), corresponding to the temperature and emission measure in each spherical shell and the overall emission-weighted metallicity (the metallicity is linked to the same value at all radii). Note that we have not attempted to correct for residual emission from gas at radii beyond the outermost annulus since the steeply rising surface brightness profile of the cluster causes this emission to have a negligible effect on the results.

The temperature profile determined with the spectral deprojection method is shown in Fig. 7.

Figure 7.

The predicted deprojected temperature profile (grey curve) determined from 100 Monte Carlo simulations using the best-fitting NFW mass model (with , and see Section 5). The predicted profile has been binned to the same spatial resolution as the spectral deprojection results (open points; Section 4.3), which are also shown overlaid. The agreement between the deprojected spectral results and best-fitting NFW mass model predictions (reduced for DOF) indicates that the NFW mass model provides a good description of the spatially resolved Chandra spectra.

4.6 Comparison with previous work

The mean ambient temperature for Abell 2390 of , determined from the combined analysis of the data in the range, is in good agreement with the previous results of from Allen (1998) based on ASCA observations, and from Böhringer et al. (1998) based on a joint analysis of ASCA and ROSAT Position Sensitive Proportional Counter (PSPC) data. [Both Allen (1998) and Böhringer et al. (1998) accounted for the effects of the central cooling flow in their modelling of the integrated cluster spectra. Böhringer et al. (1998) also examined a simpler, single-temperature emission model with which they measured a lower mean emission-weighted temperature of The ambient temperature for Abell 2390 measured with Chandra is also in good agreement with the predicted value of ∼12.0 keV using the cooling-flow-corrected relation of Allen & Fabian (1998); we assume a bolometric luminosity of ∼ as measured by ASCA since the Chandra observations do not cover the whole of the cluster.

The best-fitting mass deposition rate from the cooling flow of , determined from the Chandra spectrum for the central 100 kpc region, is lower than previous measurements based on the analysis of integrated spectra for the whole cluster from ASCAAllen 2000), from joint Böhringer et al. 1998) and from BeppoSAXEttori, Allen & Fabian 2001) observations (although the results are marginally consistent at the ∼95 per cent confidence level). In part, this difference is a result of the fact that the ambient gas temperature in the centre of the cluster, corrected for the effects of the cooling flow, is lower than the mean value measured at larger radii (Fig. 5b). If this drop in the central ambient temperature is not accounted for (as was the case in the previous ASCA and BeppoSAX studies, which could not spatially resolve the cooling flow from the hotter, outer cluster gas), then the cooler, ambient gas in the cluster core will also tend to be modelled as part of the cooling flow, and the total mass deposition rate will be overestimated. This effect is illustrated by the fact that a fit with spectral model C2(iii) to a single Chandra spectrum covering the entire central 500 kpc (radius) of the cluster gives , and , in good agreement with the previous ASCA, ROSAT and BeppoSAX results.¹ However, a fit to the central 200 kpc radius region (the maximum possible size of any cooling flow; Section 6) gives , in good agreement with the value listed in Table 3. These results, and the consistent findings from the spectral, image deprojection and X-ray colour profile analyses of the innermost 100 kpc presented in Section 6, highlight the need for detailed spatially resolved spectroscopy when attempting to study the properties of cooling flows in distant clusters.

Finally, we note that the excess column density acting on the cooling flow component measured with spectral model C1(iii) of is consistent with the previous measurement of from ASCA, using the same model (Allen 2000).

5 Measurement of the cluster mass profile

5.1 The mass model

The observed X-ray surface brightness profile (Fig. 3a) and deprojected temperature profile (Fig. 7) may together be used to determine the X-ray gas mass and total mass profiles in the cluster. For this analysis, we have used an updated version of the image deprojection code developed in Cambridge (see e.g. White, Jones & Forman 1997 for details). A variety of simple parametrizations for the cluster mass distribution were examined, to establish which could provide an adequate description of the Chandra data. For those mass models providing reasonable fits, the best-fitting parameter values were determined using a simple iterative technique.² Spherical symmetry and hydrostatic equilibrium are assumed throughout.

We find that a good fit for six degrees of freedom (DOF)] to the Chandra data can be obtained using an NFW mass model

 

(1)

where ρ(r) is the mass density, is the critical density for closure and

 

(2)

with a scale radius and a concentration parameter (68 per cent confidence limits). The normalization of the mass profile may also be expressed in terms of an equivalent velocity dispersion, (with r_s in units of Mpc). The equivalent velocity dispersion associated with the best-fitting X-ray mass model of (68 per cent confidence limit) is in good agreement with the robust, optically determined value of (Borgani et al. 1999; from a re-analysis of the data of Carlberg et al. 1996).

We note that the best-fitting values of c, σ and r_s are correlated. Fixing in the NFW model, we determine a concentration parameter and an equivalent velocity dispersion (68 per cent confidence limit keeping r_s fixed.) For , we obtain and . Fixing , we obtain and . Note, however, that the mass distributions within the central 1 Mpc radius are similar in all three cases.

The best-fitting NFW mass model, with and , has a virial radius and an integrated mass within this radius of . The relationship between M₂₀₀, r_s and c for Abell 2390 is consistent with that expected for such a massive cluster formed at a redshift (e.g. Eke, Navarro & Frenk 1998).

The deprojected X-ray gas temperature profile implied by the best-fitting NFW mass model (given the observed surface brightness profile) is shown overlaid on the deprojected spectral results in Fig. 7. The model results have been rebinned to the same resolution as the spectral data.

Finally, we note that the NFW mass model provides a significantly better fit to the Chandra data for Abell 2390 than a singular isothermal sphere for seven DOF]. However, a variety of other, two-parameter models including a softened isothermal sphere for six DOF with , a King approximation to an isothermal sphere for six DOF with and a full isothermal sphere [equation 4–125 of Binney & Tremaine (1987): for six DOF with also provide good descriptions of the Abell 2390 mass profile.

5.2 Comparison of X-ray and lensing mass measurements

Abell 2390 is one of the best-studied lensing clusters (e.g. Pelló et al. 1991; Kassiola, Kovner & Blandford 1992; Narasimha & Chitre 1993; Pierre et al. 1996; Squires et al. 1996; Bezecourt & Soucail 1997; Frye & Broadhurst 1998; Pelló et al. 1999). The cluster exhibits an unusual, strongly lensed ‘straight arc’ approximately 38 arcsec (174 kpc) away from the nucleus of the central galaxy (Pelló et al. 1991) in addition to many other arcs and arclets (e.g. Bezecourt & Soucail 1997; Pelló et al. 1999). Pierre et al. (1996) present a two-component mass model for the central regions of the cluster and measure a projected mass within the radius defined by the brightest arc of . Squires et al. (1996) present a weak lensing analysis of the cluster and determine an azimuthally averaged mass profile covering the central ∼4.3 arcmin (1.2 Mpc).

Fig. 8 shows the projected mass profile determined from the Chandra X-ray data, with the Squires et al. (1996) weak lensing results and Pierre et al. (1996) strong lensing results overlaid. For the X-ray analysis, we assume that the NFW mass models extend out to r₂₀₀ in each case. (The limits on the X-ray results are the maximum and minimum masses at each radius for the range of NFW models with the 68 per cent confidence contour in the plane.) The agreement between the X-ray and lensing mass results in Fig. 8 is reasonable at all radii studied. The mean scatter of the lensing results about the best-fitting X-ray mass profile within the central 1 Mpc region is < 20 per cent. The agreement between the independent lensing and X-ray mass measurements, together with the consistent results on the equivalent X-ray and observed optical galaxy velocity dispersions, confirms the validity of the hydrostatic assumption used in the X-ray analysis and suggests that the mass profile in Abell 2390 has been robustly determined.

Figure 8.

A comparison of the projected total mass determined from the Chandra X-ray data (Section 5,) with the strong lensing result of Pierre et al. (1996; filled circle) and the weak lensing results of Squires et al. (1996; open triangles). The best-fitting NFW X-ray mass model is shown as the dotted curve (parameters , . The limits on the X-ray results (the maximum and minimum masses at each radius for the range of NFW models with the 68 per cent confidence contour in the r_s–σ plane) are shown as solid curves.

We note that, at small radii, the strong lensing mass of Pierre et al. (1996) slightly exceeds (at ∼1σ significance) the best-fitting value determined from the Chandra data within . This difference, albeit marginal, may be related to the residual substructure on these scales seen in the X-ray (Section 3.1) and optical images and strong lensing mass map (Pierre et al. 1996), and could indicate a slight enhancement of the central lensing mass due to an alignment of the dominant mass clumps in the cluster and/or the presence of additional, non-thermal pressure support of the X-ray gas in the cluster core.

5.3 The X-ray gas mass fraction

The X-ray gas to total mass ratio as a function of radius, ƒ_gas(r), determined from the Chandra data is shown in Fig. 9. We find that the best-fitting ƒ_gas value rises with increasing radius with at (conservative limits determined by combining the 1σ errors on the integrated gas mass at each radius with the uncertainties in the total mass distribution shown in Fig. 8 in quadrature). This value is consistent with the previous measurement of per cent (1σ limits) at from Ettori & Fabian (1999) using ROSAT PSPC and ASCA data and assuming strict isothermality in the X-ray gas.

Figure 9.

The ratio of the X-ray gas mass to total gravitating mass as a function of radius. The three curves show the best-fitting value (dotted curve) and conservative confidence limits (solid curves; determined by combining the 1σ errors on the integrated gas mass at each radius with the uncertainties in the total mass distribution shown in Fig. 8). At we measure .

Following the usual arguments, which assume that the properties of clusters provide a fair sample of those of the Universe as a whole (e.g. White, Efstathiou & Frenk 1993; White & Fabian 1995; Evrard 1997; Ettori & Fabian 1999; Bahcall et al. 1999), we may use our result on the X-ray gas mass fraction in Abell 2390 to estimate the total matter density in the Universe, Ω_m. Assuming that the luminous baryonic mass in galaxies in Abell 2390 is approximately one-fifth of the X-ray gas mass (e.g. White et al. 1993; Fukugita, Hogan & Peebles 1998) and neglecting other possible sources of baryonic dark matter in the cluster, we obtain , where Ω_b is mean baryon density in the Universe and h₅₀ is the Hubble constant in units of 50 km s⁻¹ Mpc⁻¹. For (O'Meara et al. 2000), . (Accounting for any additional, dark baryonic matter in the cluster would lower the measured value of Ω_m. Likewise, if the value of ƒ_gas increases towards larger radii in Abell 2390, the true value for Ω_m will be lower than our quoted result.)

6 The properties of the cooling flow

6.1 Radial properties of the cluster gas

The results on the electron density and cooling time as a function of radius, determined from the image deprojection analysis using the best-fitting NFW mass model, are shown in Fig. 10. Within the central 500 kpc radius, the electron density profile can be parametrized for 52 DOF) by a β profile with a core radius , and a central density (1σ errors). The core radius for the electron density distribution is slightly smaller than the value measured directly from the projected surface brightness profile under the assumption of strict isothermality in the X-ray gas (Section 3.2). Indeed, the evidence for any flat central core in the electron density distribution is marginal and a simple broken power-law model, with and slopes interior and exterior to the break radius of and , provides as good a fit to the electron density profile in the central 500 kpc region for 51 DOF).

Figure 10.

The results on (a) the electron density and (b) the cooling time, determined from the X-ray image deprojection analysis using the best-fitting NFW mass model. Error bars are the 1σ errors determined from 1000 Monte Carlo simulations. A Galactic column density of and a metallicity of 0.4 solar are assumed.

For an assumed Galactic column density of we measure a central cooling time [i.e. the mean cooling time within the central 1.97 arcsec (or ∼9 kpc) bin] of , and a cooling radius, at which the cooling time first exceeds a Hubble time, of . (Errors on the central cooling time are the 10 and 90 percentile values from 100 Monte Carlo simulations. The upper and lower confidence limits on the cooling radii are the points where the 10 and 90 percentile values exceed and become less than the Hubble time, respectively.)

6.2 X-ray colour profile analysis

We have constructed an X-ray ‘colour’ profile for the cluster in order to determine the size of the central region within which the cooler spectral components are concentrated. Two separate images were created in the energy bands and and in the rest frame of the source) with a 1.97 arcsec (four raw detector pixels) pixel scale. These soft and hard X-ray images were background-subtracted and flat-fielded (taking full account of the spectral energy distributions of the detected photons). All significant point sources were masked out and excluded from the analysis. Azimuthally averaged surface brightness profiles for the cluster were then constructed in each energy band, centred on the overall peak of the X-ray emission (Section 3). The X-ray ‘colour’ profile formed from the ratio of the surface brightness profiles in the soft and hard bands is shown in Fig. 11.

Figure 11.

The X-ray colour profile, formed from the ratio of the counts in the and bands, as a function of radius. At large radii the ratio is approximately constant and consistent with a cluster temperature of . Within a ‘break’ radius of , however, the colour ratio rises sharply. The grey curve shows the predicted colour profile for a cooling flow with an effective age of ∼3 Gyr from the image deprojection analysis (Section 6). We use the best-fitting NFW mass model and assume that the cooling gas is intrinsically absorbed by a column density of , as determined with spectral model C1(iii).

From examination of Fig. 11 we see that at large radii the observed X-ray colour ratio is approximately constant, with a mean value of (1σ error determined from a fit to the data between radii of 150 and 300 kpc). By comparison with simulated spectra we find that this result is consistent with an isothermal plasma with a temperature . (We assume a metallicity and Galactic absorption.) Within a ‘break’ radius of (1σ errors determined from a χ² fit with a broken power-law model), however, the colour ratio rises sharply, indicating the presence of significantly cooler gas.

6.3 Analysis of the mass deposition profile

The outermost radius at which cooling occurs may also be expected to be associated with a ‘break’ in the X-ray surface brightness profile and, more evidently, the mass deposition profile determined from the deprojection code. The mass deposition profile from the cooling flow, which is a parametrization of the X-ray luminosity distribution in the cluster core (see e.g. White et al. 1997), is shown in Fig. 12. Fitting this profile with a simple broken power-law model, we determine a break radius of (1σ errors), in reasonable agreement with the break radius determined from the fit to the X-ray colour profile (Section 6.2) and the electron density distribution (Section 6.1). The best-fitting broken power-law model is shown overlaid on the mass deposition profile in Fig. 12.

Figure 12.

The mass deposition rate (MDR) in units of M_⊙ yr⁻¹, determined from the image deprojection analysis using the best-fitting NFW mass model. The dotted line is the best-fitting broken power-law model (Section 6.3). Error bars are the 10 and 90 percentile values from 1000 Monte Carlo simulations. A Galactic column density of and a metallicity of 0.4 solar are assumed.

The slopes of the mass deposition profile internal and external to the break radius are and , respectively. Accounting only for absorption due to cold gas with the nominal Galactic column density, we determine an integrated mass deposition rate within the break radius of . If we also account for the presence of intrinsic absorption, with the properties determined using spectral model C2(iii), the mass deposition rate within the break radius rises to , in good agreement with the spectral result for the central 100 kpc of .

6.4 The age of the cooling flow

Allen et al. (2001a) discuss a number of methods that may be used to estimate the effective ages of cooling flows from X-ray data. (Such ages are estimates of the times that new, undisturbed cooling flows would require to evolve to their present states and are likely to be related to the intervals between major merger events in cluster cores.) Essentially, these methods identify the age of a cooling flow with the cooling time of the X-ray gas at the break radius in either the X-ray colour or deprojected mass deposition profile.

Using Fig. 10(b), we see that the cooling time of the X-ray gas at the break radius in the X-ray colour profile lies in the range . [The cooling time at the break radius is measured from a least-squares fit to the data in Fig. 10(b) over the range using a power-law model.] If we instead identify the age of the cooling flow with the cooling time at the break radius in the mass deposition profile in Fig. 12, we infer an age of . (In both cases we assume that no intrinsic absorption acts beyond the outer edge of the cooling flow, which is reasonable if the absorbing matter is accumulated by the flow.)

In summary, we see that the X-ray colour profile, image deprojection analysis and spectral data provide consistent results on the properties of the cooling flow in Abell 2390, indicating a mass deposition rate in the range and an effective age of .

6.5 On the rising ambient temperature profile within the cluster core

In principle, the results on the X-ray gas temperature and surface brightness within the cluster core may be used to distinguish between an NFW model for the dark matter distribution and alternative models with a steeper central cusp (e.g. Moore et al. 1999). The results for Abell 2390 presented here are certainly consistent with an NFW profile (see Fig. 7). However, given the complexity of the gas within the central 100 kpc and the relatively short exposure time of the present observation (which limits the number of independent spectra that can be extracted from the region of the cluster core), we do not attempt to explore such models here. This issue will be better addressed with future, deeper observations of nearer, brighter systems.

The fact that the ambient temperature profile, corrected for the effects of the cooling flow, rises with increasing radius throughout the central ∼200 kpc is interesting. In detail, the density and temperature structure in a cluster core will depend on the thermal history of the gas, as well as on the underlying dark matter distribution, and, subject to pressure equilibrium and convective stability, may be flat, increase or decrease with radius. As discussed in Section 4.6, the presence of relatively cool, ambient gas in the central regions of Abell 2390 may have led to overestimates of the mass deposition rate in previous studies with ASCA and BeppoSAX in which this region was not spatially resolved.

There are several reasons why relatively cool, dense gas might exist beyond the outer edge of the present-day cooling flow in Abell 2390. The first is that a pre-existing cooling flow may have been disturbed several Gyr ago. Cooler, denser gas could then have been spread out over several 100 kpc, with the cooling flow having so far only re-established (or maintained) itself within the inner ∼. A second possibility is that the densest gas from the cores of infalling subclusters may have been stripped and deposited over time in the core of the main cluster without strong shocking (Fabian & Daines 1991). The situation seen in the Chandra data for Abell 2142 (Markevitch et al. 2000) is reminiscent of this, with a sharp drop in temperature from to (and a corresponding rise in density) observed at , moving inwards from north-west of the cluster centre. Such a sharp drop in temperature at these radii cannot be due to any current cooling flow in the cluster.

7 Conclusions

The main conclusions from this work may be summarized as follows:

• (i)

  We have measured the distribution of mass in Abell 2390 using Chandra X-ray observations. The mass profile can be well modelled by an NFW profile with a scale radius and a concentration parameter (68 per cent confidence limits). The normalization of the mass profile may also be expressed in terms of an equivalent velocity dispersion, , in good agreement with the optically determined value of (Borgani et al. 1999).

• (ii)

  The best-fitting Chandra mass model is in good agreement with independent measurements from strong and weak lensing studies. The mean scatter between the X-ray and lensing values within the central radius is < 20 per cent.

• (iii)

  The X-ray gas to total mass ratio rises with increasing radius within the central 0.9 Mpc, with at . Following the usual arguments, this result on the X-ray gas mass fraction may be converted into a constraint on the mean mass density in the Universe, .

• (iv)

  The X-ray gas temperature rises with increasing radius within the central , and then remains approximately isothermal with out to .

• (v)

  The azimuthally averaged surface brightness profile exhibits a small core radius and a clear break at , where the slope changes abruptly from to .

• (vi)

  The best-fitting mass deposition rate from the cooling flow, determined in a consistent manner from the spectral and imaging Chandra data, lies in the range . This value is lower than previous estimates based on integrated ASCA and BeppoSAX spectra for the entire cluster, which could not resolve the drop in the central, ambient gas temperature. We estimate an effective age for the cooling flow of .

Acknowledgments

We thank G. Squires for communicating his weak lensing results, R. Mushotzky for helpful comments, and P. Thomas for helpful comments and discussions. We also thank R. Johnstone and R. Schmidt for coding and discussions regarding the analysis of Chandra data. We acknowledge the support of the Royal Society.

¹

Note that although the fit with spectral model C2(iii) to the Chandra spectrum for the central 500 kpc (radius) region overestimates the mass deposition rate from the cooling flow, the measured temperature is in good agreement with the true value at large radii in the cluster (Fig. 5b). Thus, previous ASCA and BeppoSAX studies which attempted to account for the effects of cooling flows on integrated cluster spectra (e.g. Allen 1998; Böhringer et al. 1998) may have overestimated the mass deposition rates from the cooling flows but are likely to have provided reasonable estimates of the mean cluster temperatures.

²

Given the observed surface brightness profile and a particular parametrized mass model, the deprojection code is used to predict the temperature profile of the X-ray gas. (We use the median model temperature profile determined from 100 Monte Carlo simulations.) The predicted temperature profile is rebinned, using an appropriate flux weighting, and compared with the results from the spectral deprojection analysis (Fig. 7). The χ² difference between the observed and predicted temperature profiles is then calculated. The parameters for the mass model are stepped through a regular grid of values in r_s and σ (or alternatively r_s and c) to determine the best-fitting values (which give the minimum χ²) and confidence limits. For Abell 2390 the final grid used ranges from 0.1 to 2.5 Mpc in r_s and from 500 to 2500 km s⁻¹ in σ.