A statistical analysis of the Two Dimensional XMM-Newton Group Survey: The impact of feedback on group properties

(abridged) We present a statistical analysis of 28 nearby galaxy groups from the Two-Dimensional XMM-Newton Group Survey (2dXGS). We focus on entropy and the role of feedback, dividing the sample into cool core (CC) and non cool core (NCC) systems, the first time the latter have been studied in detail in the group regime. The coolest groups have steeper entropy profiles than the warmest systems, and NCC groups have higher central entropy and exhibit more scatter than their CC counterparts. We compare the entropy distribution of the gas in each system to the expected theoretical distribution ignoring non-gravitational processes. In all cases, the observed maximum entropy far exceeds that expected theoretically, and simple models for modifications of the theoretical entropy distribution perform poorly. Applying initial pre-heating, followed by radiative cooling, generally fails to match the low entropy behaviour, and only performs well when the difference between the maximum entropy of the observed and theoretical distributions is small. Successful feedback models need to work differentially to increase the entropy range in the gas, and we suggest two basic possibilities. We analyse the effects of feedback on the entropy distribution, finding systems with a high measure of `feedback impact' to reach higher entropy than their low feedback counterparts and also to show significantly lower central metallicities. If low entropy, metal-rich gas has been boosted to large entropy in the high feedback systems, it must now reside outside 0.5r_500, to remain undetected. We find similar levels of enrichment in both high and low feedback systems, and argue that the lack of extra metals in the highest feedback systems points to an AGN origin for the bulk of the feedback, probably acting within precursor structures.

Knowledge of the dark matter-dominated potential, coupled with the entropy distribution of the gas within a system in hydrostatic equilibrium, completely defines the thermodynamic properties of the intracluster medium (ICM) (Voit et al. 2003). The advent of high quality X-ray data from Chandra and XMM-Newton have allowed detailed studies of the entropy properties of clusters (e.g. Pratt, Arnaud & Pointecouteau 2006;Morandi & Ettori 2007) and simulations have been extensively used to predict properties such as entropy profiles and the chemical enrichment of the ICM (see Borgani et al. 2008a,b for recent reviews). However, the important group regime has been less well explored, although significant progress is now being made in modelling feedback effects on the group scale (Davé, Oppenheimer & Sivanandam 2008). There is a parallel shortage of detailed observational evidence which bears on the processes shaping group evolution, and the aim of the present study is in part to address this.
Observations of galaxy clusters show a striking bimodality in their observed ICM properties, which leads to their classification into cool core (CC) and non cool core (NCC) systems. These classes are characterised by the observation (or not) of central positive temperature gradients. The open question in this field is how this apparent bimodality is produced. The short central cooling times observed in both types of system (Sanderson, Ponman & O'Sullivan 2006), causes difficulties for attempts to explain the two classes within the same framework of cluster evolution. However, recent attempts have been made to do just this (McCarthy et al. 2008), by invoking different levels of preheating before cluster collapse.
Recent results show strong similarities between CC clusters and CC groups. For example, the observed abundance gradients in CC clusters (De Grandi et al. 2004;Baldi et al. 2007) are also seen in the Chandra sample of 15 groups of Rasmussen & Ponman (2007), 14 of which were observed to have CCs. However, one crucial area for understanding the role of feedback that has not yet been probed is the nature of NCC systems in the group regime.
We are now entering an era where large studies of galaxy groups are possible with high quality X-ray data (Finoguenov et al. 2006(Finoguenov et al. , 2007Rasmussen & Ponman 2007;Sun et al. 2008). Although the sample analysed here was not selected in a statistical manner, we have an opportunity to gain considerable insight into the diversity of group properties, as this is the largest sample of groups to date with high quality XMM-Newton data, which has been analysed in a homogeneous way. One key advantage of our approach is that in the spectral analysis of this data, spherical symmetry was not an a priori requirement (see Finoguenov et al. 2006Finoguenov et al. , 2007. Additionally, due to the size of the sample, we have been able, for the first time, to separate the sample into CC and NCC groups, and also to look at the behaviour of groups as a function of temperature. This provides a unique opportunity to examine which processes drive the observed differences in group properties, by considering the behaviour which diverges from the mean relations.
The layout of the paper is as follows. In Section 2 we describe the sample, analysis and basic properties of the groups, in Section 3 we divide the sample to look at the mean properties of sub-samples of the data, and Section 4 explores the radial properties of the sample. We explore the entropy distributions and discuss the effects of feedback on the groups in Section 5, and we summarise our main results in Section 6.

GROUP SAMPLE
The Two-Dimensional XMM-Newton Group Survey (2dXGS) is an archival study of nearby (z < 0.024) galaxy groups, which were selected from the group catalogue of Mulchaey et al. (2003) and were chosen to have publicly available XMM-Newton data (Finoguenov et al. 2006(Finoguenov et al. , 2007. The complete sample selection and data analysis is described by Finoguenov et al. (2006Finoguenov et al. ( , 2007, hereafter referred to as F+06 and F+07 respectively. The former concentrates on the 'low' redshift sample (z < 0.012), and the latter concentrates on the 'high' redshift sample (0.012 < z < 0.024). These works studied the radial properties of individual groups and their deviations from the mean profiles, applying a novel spectral approach which we briefly describe below. Our aim is to bring these two samples together, supplemented by groups from the sample of Mahdavi et al. (2005) which were analysed using the same procedures, in an effort to undertake a full statistical study of the properties of a large sample of galaxy groups derived from high quality X-ray observations.
The full 2dXGS sample contains 25 nearby galaxy groups observed with XMM-Newton, which have been analysed in a homogeneous way. A summary of the data analysis procedure is given here; we refer the reader to F+06 and F+07, and references therein, for a full description of the XMM-Newton data, reduction and analysis. There are two main stages in the analysis following the initial XMM-Newton data reduction. Firstly, temperature and surface brightness maps were used to look at the overall structure of the group. The second part of the analysis extracted spectra from regions of contiguous surface brightness and temperature, and fitted single temperature hot plasma (APEC) models to the spectra, yielding the spectral properties of the group with no a priori assumption of the system being spherically symmetric. The abundances used were those of Anders & Grevesse (1989), and absorption was fixed at the Galactic value.
The result of this approach is to yield a series of spectrally derived parameters in both two-dimensional regions, and also from a more traditional spectral analysis of a series of concentric annuli. We can look at the radial properties of each group by assuming the characteristic radius for the measurement to be the mean radius of the region. Here we will concentrate mainly on using the properties derived from this novel analysis of 2d regions, except where otherwise stated. Due to the nature of the analysis, the customary spherical deprojection of the spectra cannot be performed. Instead, F+06 and F+07 determined the three dimensional gas properties in each of the analysed regions, by estimating the projection length of each region. We use here the derived gas properties from this approach, and refer readers to F+06, F+07 and references therein for more details on this procedure.
We define the mean temperature of each group to be that recovered from fitting a spectrum extracted from the radial range 0.1-0.3 r500 (F+06, F+07), so we restrict the analysis presented here to the 21 2dXGS groups with measured temperatures in this range. We have supplemented the 21 groups in the 2dXGS sample with seven groups with the highest data quality from Mahdavi et al. (2005), leading to a final sample of 28 galaxy groups. The Mahdavi et al. (2005) groups were analysed using the same two-dimensional procedure, but they have not also been analysed with the traditional annular approach which was also applied in the case of the 2dXGS groups. The groups in the Mahdavi sample were initially selected from the ROSAT All-Sky Survey/Center for Astrophysics Loose Systems (RASSCALS) of Mahdavi et al. (2000), and cover redshifts between 0.016 and 0.037. The mean temperatures for these groups have been re-extracted, to cover the 0.1-0.3 r500 region applied to the 2dXGS groups.

Group properties
The basic properties of the group sample appear in Tables 1 and 2, the first of which refers to the 2dXGS sample of F+06 and F+07, and the second of which shows the properties of the supplementary groups from Mahdavi et al. (2005). Column (1) shows the group name as expressed by F+06, F+07 and Mahdavi et al. (2005). We now describe the origin of the values shown in Tables 1 and 2. We assume a Hubble constant of H0 = 70 kms −1 Mpc −1 throughout.

2dXGS groups
Due to the overlap of 15 of the 2dXGS groups with the Group Evolution Multiwavelength Study (GEMS) sample of Osmond & Ponman (2004), the latter is our primary source for the group properties (shown in Table 1) supplementary to those provided by F+06 and F+07. Where distance measurements were available from Osmond & Ponman (2004), these were used. For the remaining groups, we re-scaled the distance measurements of Mulchaey et al. (2003) to our assumed Hubble constant (3C 499, NGC 507, NGC 2300 and NGC 4168). In the case of Hickson 51 and the Pavo group, it was necessary to estimate the distance from the redshift information provided in F+07.
X-ray luminosities are also drawn primarily from Osmond & Ponman (2004), which employed ROSAT PSPC data, and has the advantage of extending to larger radii than XMM-Newton data for many of these groups. We used the supplied β-model surface brightness fits of Osmond & Ponman (2004) to re-scale the X-ray luminosities from their original extraction radius to our r500 values. Individual extraction radii can be found in Osmond & Ponman (2004); they are typically ∼ 55 percent of our r500. In some cases, Osmond & Ponman could not fit a model to the surface brightness profile, and in these cases, a standard model of β = 0.5 and rcore = 6 kpc was assumed. This affects the following groups in our sample: Hickson 15, Hickson 92 and NGC 5171. For groups not in the GEMS sample, X-ray luminosities are from Mulchaey et al. (2003), but have been corrected in the following way for differences in the assumed Hubble constant, and to extrapolate to r500 in line with the X-ray luminosities of the GEMS groups. Where β-model surface brightness fits were available from Mulchaey et al. (2003) (NGC 507), we re-scaled the X-ray luminosities both for the assumption here of a lower Hubble constant, and to scale from the original extraction radius of 200 h −1 100 kpc to our values of r500. In two cases (3C 449, NGC 2300), the fitted core radii reported by Mulchaey et al. (2003) are lower than the resolution limit of the ROSAT PSPC instrument. In this case we set the core radius at this limit when scaling the X-ray luminosity to our value of r500. For NGC 4168, where only an upper limit on Lx was determined by Mulchaey et al. (2003) and hence no surface brightness modelling was undertaken, we report the upper limit corrected for the different Hubble constant only. In the case of Hickson 51, which is not in the GEMS sample, we used the X-ray luminosity for the group NRGb128 from the RASSCALS sample of Mahdavi et al. (2000), which is given as an equivalent designation for Hickson 51 in the NASA/IPAC Extragalactic Database (NED) 1 . We extrapolated this X-ray luminosity 1 http://nedwww.ipac.caltech.edu/ from the original extraction radius of 0.5 h −1 100 Mpc to our r500 assuming the standard β-model of Osmond & Ponman (2004), and we also scaled the X-ray luminosities to a Hubble constant of H0 = 70 kms −1 Mpc −1 . This is the same procedure we have used to determine X-ray luminosities for the majority of the Mahdavi et al. (2005) groups (see Section 2.1.2), so there is a consistency in our approach.
It was necessary to estimate the X-ray luminosity of Pavo from the work of Machacek et al. (2005), who fitted a β-model surface brightness profile to the extended emission outside the central galaxy (NGC 6876) with rcore = 196 ′′ and β = 0.3, yielding an Xray luminosity (in the energy band 0.5-2 keV) of log Lx = 41.8 within 463 ′′ , assuming a Hubble constant of 75 kms −1 Mpc −1 . We have extrapolated this model to our r500 and have corrected for the differences in the assumed Hubble constant, to quote the X-ray luminosity for Pavo shown in Table 1. The remaining caveat with this value is the slightly different energy band compared to the remainder of the sample, which were all derived from ROSAT data.
The velocity dispersion of each group comes from F+06 and F+07, except in the case of NGC 4636, for which F+06 could not determine the group membership satisfactorily (we refer the reader to this work for more information), and in this case, we quote the velocity dispersion of Osmond & Ponman (2004). The mean temperatures of the groups and the values of r500 were all derived by F+06 and F+07 using the method described in Section 2.
We also show selected optical properties of the central galaxy and the group. Values of D25, the diameter of the isophote where the B-band surface brightness is 25 mag arcsec −2 , are determined for the central group galaxy from the RC3 catalogue of de Vaucouleurs et al. (1991). We have applied the procedure of Osmond & Ponman (2004) in determining the B-band luminosity of the brightest group galaxy (BGG) and of the group, by extracting galaxies from NED within a projected radius of r500, centred on the group co-ordinates. For groups in the Osmond & Ponman (2004) sample, we used the group co-ordinates provided in this work, and for the remaining groups we use the NED co-ordinates corresponding to the group name. The group luminosities are 90 percent complete and we applied an absolute magnitude cut of MB = -16.32 to the group galaxies. The BGG is chosen to be the brightest galaxy within 0.25 r500 of the group centre. We refer the reader to Osmond & Ponman (2004) for more information on the applied method. In the case of the Pavo group, we have assumed the BGG to be NGC 6876 (Machacek et al. 2005); this is confirmed by our procedure for extracting B-band luminosities. Table 1 also gives the maximum radius in units of r500 to which spectral information is available, for each of the groups, to indicate the completeness of the spectral coverage. The final columns in Table 1 denote the subsamples to which each group belongs, in terms of their mean temperatures and core properties. These classifications are defined in Section 3.

Mahdavi et al. (2005) groups
Three of the Mahdavi et al. (2005) groups appear in the sample of Osmond & Ponman (2004). These are Hickson 97, SRGb119 (central galaxy NGC 741; Mahdavi et al. 2005) and NGC 5129. For these groups, distances are from Osmond & Ponman (2004), but in the remainder of cases the distances were estimated from the redshifts presented by Mahdavi et al. (2005). Similarly, the X-ray luminosities for these 3 groups are re-scaled from the original extraction regions presented in Osmond & Ponman (2004) to match our r500 values, as described in Section 2.1.1 for the groups in the Table 1. The basic properties of the 2dXGS groups in the sample. The groups have been classified by their mean temperature and their core properties (see Section 3). Distances and values of L X are from Osmond & Ponman (2004) unless otherwise stated. For data from Osmond & Ponman (2004), the X-ray luminosities have been extrapolated to our r 500 values (see text). L B,BGG and L B,grp are calculated as described in the text. Mean temperatures and values of r 500 are from F+06 and F+07. Velocity dispersions are also from F+06, F+07, except in the case of NGC 4636, where the value used is that from Osmond & Ponman (2004), and values of D 25 (the diameter of the isophote where the surface brightness is 25 mag/arcsec 2 in the B-band) are from RC3 (de Vaucouleurs et al. 1991), and refer to the brightest group galaxy (BGG). rmax denotes the maximum radius to which spectral information is available, in units of r 500 .
b Distance estimated from redshift given in Finoguenov et al. (2007). c Value from Mahdavi et al. (2000) rescaled using H 0 = 70 kms −1 Mpc −1 and extrapolated to our r 500 values using the standard β-model of Osmond & Ponman (2004) (see text for details). d Not available in RC3.
e Lx extrapolated to r 500 and corrected for H 0 = 70 kms −1 Mpc −1 from the measured X-ray luminosity and surface brightness profile of Machacek et al. (2005) (Note: energy band is 0.5-2 keV for this value). f D 25 from RC3 assuming central galaxy is NGC 6876 (Machacek et al. 2005). 2dXGS sample. For the remaining four groups, we use the X-ray luminosities from Mahdavi et al. (2000), which were extracted from the ROSAT All-Sky Survey, within a radius of 0.5 h −1 100 Mpc. Assuming the standard β-model of Osmond & Ponman (2004), we scale these X-ray luminosities both for H0 = 70 kms −1 Mpc −1 and to our radius of r500, in the manner described in Section 2.1.1. This ensures a degree of consistency between the 2dXGS and Mahdavi et al. (2005) samples.
Velocity dispersions for all seven systems come from Mahdavi et al. (2005), and mean temperatures and r500 values were re-derived from Mahdavi et al. (2005) in the radial range 0.1-0.3 r500. The B-band optical luminosities of the BGG (LB,BGG) and the group (LB,group) were determined using the same procedure as in Section 2.1.1. For the three GEMS groups, we used the co-ordinates provided by Osmond & Ponman (2004) for the NED search; we use the group co-ordinates from Mahdavi et al. (2005) for the remainder of the sample. Mahdavi et al. (2005) indicate that Abell 194 has no dominant galaxy, but our method finds NGC 541 to be the brightest galaxy within 0.25 r500, and we adopt this as the BGG. The central galaxy of RGH 80 (NGC 5098; Mahdavi et al. 2005) is listed in NED as a galaxy pair, but we find PGC 046515 to be the brightest galaxy within 0.25 r500 of the group centre, and we adopt this galaxy as the BGG in this system. Values of D25 were again extracted from the RC3 catalogue of de Vaucouleurs et al. (1991) for these systems. Table 2 also gives the maximum radius in units of r500 to which spectral information is available, for each of the groups, to indicate the completeness of the spectral coverage.

DIVIDING THE SAMPLE
The 2dXGS sample of groups has been presented previously (F+06, F+07) in two subsamples, distinguished by group distance, but it is also instructive to partition the sample on the basis of physical properties. We have divided the groups into two main subsamples, firstly on the basis of the mean temperature of the group, which acts as a proxy for the group mass, and secondly on the basis of whether or not the group has a CC. The classification of galaxy clusters into CC and NCC systems is well-known (e.g. Peres et al. 1998). These systems have been shown to exhibit different observational properties (e.g. Sanderson et al. 2006), and hence it is instructive to extend this classification to the group regime. Rasmussen & Ponman (2007) studied the temperature and abundance profiles of 15 groups Table 2. The basic properties of the groups in the sample originally from Mahdavi et al. (2005). Mean temperatures and values of r 500 have been re-derived from Mahdavi et al. (2005) in the radial range 0.1-0.3 r 500 . Distances have been estimated from the redshifts given in Mahdavi et al. (2005), and X-ray luminosities have been extrapolated to our r 500 values from Osmond & Ponman (2004) and Mahdavi et al. (2000). Values of L B,BGG and L B,grp are calculated as described in the text. D 25 values are from RC3 (de Vaucouleurs et al. 1991). rmax denotes the maximum radius to which spectral information is available, in units of r 500 .
b Values from Mahdavi et al. (2000) extracted within 0.5 Mpc, rescaled here for H 0 = 70 kms −1 Mpc −1 and extrapolated to our r 500 using the standard β-model of Osmond & Ponman (2004) (see text for details). c NGC 541 assumed as BGG (see text for details) as no dominant galaxy listed by Mahdavi et al. (2005). with Chandra data, 14 of which were noted to have a CC. The properties of groups that do not have CCs, and the connection between the overall state of the ICM and the core properties of groups, has not been investigated to date. A sample of the size presented here, although not statistically selected, offers an opportunity to probe the wider properties of groups compared to the presence (or not) of a CC.
To enable us to compare the radial profiles of groups directly, we scale the radii by r500, the radius within which the mean density of the group is 500 times the critical density. This was calculated in an iterative fashion using (F+06, F+07), The partition of the groups into subsamples based on their mean temperature and core properties is detailed below.

Temperature
We define the mean temperature of each group as that determined within the radial range 0.1-0.3 r500 (F+06, F+07). This removes the effect of any CC on the temperature determination, providing a robust measure of the mean temperature of the system. The median temperature across all groups was found to be 1.035 keV, and this was used to divide the groups into 2 subsamples, the Cool groups withT < 1.035 keV, and the Warm groups withT 1.035 keV. The temperature classification for each group appears in Tables 1 and 2.

Core properties
To distinguish between CC and NCC systems, one traditional approach is to consider the cooling time of the system (e.g. Peres et al. 1998). However, Sanderson et al. (2006) showed that in the cluster regime, systems with no central cooling can also exhibit the short central cooling times (< 5 Gyr) typically associated with CC systems. They defined the presence of a CC by considering the ratio of temperatures between an inner radius and an intermediate radial range, and we adopt the same approach here.
We define a group to have a CC if the mean temperature in the region 0.1-0.3 r500 is greater than the mean temperature in the range 0.00-0.05 r500, indicating a positive temperature gradient in the inner regions. If this was not the case, the group is classified as a NCC system. This classification is based wherever possible on results from the analysis of both the annular and the 2d regions reported in the earlier 2dXGS papers, to improve the radial coverage and allow a more thorough appraisal of the temperature profile. For the seven groups from the Mahdavi et al. (2005) sample, only the 2d information is available. The temperature profiles of the CC systems are shown in Figure 1, and those for the NCC systems are shown in Figure 2. In both cases, a weighted local regression fit to the data is also shown, to indicate the overall behaviour of the temperature profile whilst suppressing scatter. This fit was performed using the 'LOESS' function in version 2.5.1 of the R statistical environment package 2 (R Development Core Team 2008), hereafter referred to as R. The LOESS algorithm performs a weighted least squares fit in the local neighbourhood of each data point, where the size of the neighbourhood is defined to include a specified proportion of the data sample, and the distance to each neighbour is used to weight the least squares fit. For more information on the LOESS algorithm, we refer the reader to Cleveland, Grosse & Shyu (1992). For two systems, classification on the basis of the above criterion was problematical. Hickson 92 would be designated a CC system, since its profile shows a small drop in central temperature, albeit of rather low significance. However, this group (commonly known as Stephan's Quintet) is a system of galaxies undergoing multiple collisions, which are currently disturbing and heating its intergalactic medium (Trinchieri et al. 2003(Trinchieri et al. , 2005. It is far from being a typical CC system, in which radiatively cooling, dense gas is centred on a dominant early-type galaxy. We therefore reclassify it as an NCC system. In the case of Abell 194, the quality and resolution of the data did not permit us to extract a central spectrum within 0.05 r500. However, the temperature profile rises consistently inward to the innermost point, at ∼0.06 r500. We are therefore confident in classifying this group as a NCC, since all our CC systems show CC behaviour by this radius, whereas Abell 194 does not. The final sample therefore consists of 10 NCC groups,  Figure 1. The temperature profiles of the groups classified as being CCs. In the case of the 2dXGS groups, the data from both the annuli and two-dimensional regions has been used. The solid lines show weighted local regression fits to the data, see the text for details. The plots are arranged in order of ascending mean temperature, from top left to bottom right. Red tick marks show the radius of maximum temperature used in the calculation of the temperature drop, and the dashed red line shows a straight line fit (fitted in log-log space) to the temperature profiles in the core region. Continued overleaf. and 18 CC groups. Ten of the latter are included in the Chandra study of Rasmussen & Ponman (2007).
One interesting property of CC groups is the magnitude of the drop in temperature within the core. To estimate this, we found the radius of peak temperature, from the maximum in the local regression fits to the data. To be able to evaluate the temperature drop to the same small radius (0.01 r500) in all groups, we fitted a straight line in log-log space to the temperature within the radius of maximum temperature (red dashed lines in Figure 1), forcing the condition of passing through this maximum point. This fit was then used to evaluate the temperature at 0.01 r500, allowing groups with different radial sampling to be directly compared.
For our CC subsample, the ratio of the temperature at 0.01 r500 to the mean temperature of the system is found to be Tc/T = 0.69±0.16. This agrees with the result (0.58±0.14) of Rasmussen & Ponman (2007) within errors. However, the 'central temperatures' of (Rasmussen & Ponman 2007) were measured in the innermost radial bin of their analysis, which in most cases lay within 0.01r500. This may account for their slightly lower value of Tc/T .
We quantified the observed temperature drop as the difference between the maximum temperature and that interpolated at 0.01 r500. This quantity is temperature dependent, as hotter systems have the capacity to show larger temperature drops. To remove this dependence, we divided the temperature drop by the peak temperature of the system. Panel (a) of Figure 3 shows this fractional temperature drop versus mean temperature. We can test for possible correlation here using Kendall's rank order correlation coefficient τ , which is found to be 0.15 with a p-value (probability of chance occurrence) of 0.38, indicating no significant correlation within our sample. However, this is partly due to the low value of the central temperature decline in our hottest system, NGC 4073, and excluding this system yields τ = 0.27 and a p-value of 0.14, indicating a positive correlation at the >1σ level. The values of the fractional dip in Figure 3(a) are clearly for the most part lower than the value 0.6 which is typical in CC clusters (see Fig.7 of Sanderson et al. 2006), in agreement with the findings of Rasmussen & Ponman (2007). The CC group with the lowest fractional temperature drop (T drop / T peak ∼ 0.1) is SS2b153, the central galaxy of which is NGC 3411 (Mahdavi et al. 2005). This group has been interpreted as evidence of a CC system that has been re-heated by recent AGN activity (O'Sullivan et al. 2007).
The Chandra study of Rasmussen & Ponman (2007) found that the radius at which the temperature peaks is correlated with group temperature (and hence group size). We test for a correlation between the physical sizes of the CCs and the value of r500 (see Figure 3, panel (b)) in our sample. The value of Kendall's rank order correlation coefficient τ is 0.29, with a p-value of 0.10, so we have some evidence for the existence of a correlation: larger systems tend to have larger CCs, which is perhaps not a surprising result, and agrees qualitatively with the analysis of Rasmussen & Ponman (2007). Fitting a linear model using an orthogonal regression (Isobe et al. 1990) to allow for intrinsic scatter, we find a relationship between the size of the CC and the value of r500 of the following form, r(Tmax) = (0.14 ± 0.07)r500 − (1.0 ± 15.4) kpc.
( 2) where r(Tmax) is the radius of maximum temperature in kpc, and r500 is also measured in kpc. This relation yields a flatter slope than the analysis of Rasmussen & Ponman (2007), who found a slope of 0.20±0.02 r500, although the large error bars mean that the slopes of the two relations agree within errors. There are some differences in the methodology used to estimate r500 in the two studies, with the Rasmussen & Ponman (2007) values for the groups in common tending to be larger, which could contribute to the slightly different results obtained.
Our result has essentially zero intercept, so that r(Tmax) ≈ 0.14 r500. This is very similar to the ratio of the temperature peak radius to r500 found for clusters by Sanderson et al. (2006). Hence it seems that although the depth of the CC is less in groups than clusters, their size (scaled to r500) is similar. In Figure  3 panel (c), we scale the temperature peak radius to r500 and find no significant temperature trend in the scaled size of the CCs.
Comparing the global properties of the CC and NCC groups in our sample, the mean temperature of the NCC systems is found to be 0.89±0.08 keV, compared to 1.11±0.07 keV for the CC groups. Therefore, although the two populations overlap, the centroid of the distribution is significantly lower for the NCC subsample. Similarly, the mean X-ray luminosities of the subsamples are log LX = 42.70±0.12 erg s −1 for the CC groups, and log LX = 42.20±0.21 erg s −1 for the NCC systems. Of course, we are not dealing with a statistical sample of groups, so it is important not to over-interpret these results. However, one might expect selection effects to work in the opposite direction -cool, faint groups will be easier to detect if they have CCs -so our results provide tentative evidence that cooler groups are more likely to lack CCs. Moreover, it is interesting to note that three of the four coolest systems, with temperatures less than 0.7 keV, do not have CCs, which hints at the possibility of a lower temperature threshold for CC systems. Further study of low temperature groups (which are faint, and therefore hard to detect) is required to confirm whether there is indeed such a threshold.
To summarise, for our sample the fractional central drop in temperature is smaller than that seen in clusters. We see no clear trend in CC strength with mean temperature within our group sample, but the coolest systems (T < 0.7 keV) tend not to show CCs at all. We find a relation rather similar to that of Rasmussen & Ponman (2007) between the physical size of the CC region (defined as the radius of maximum temperature) and r500. Smaller groups (lower r500) exhibit smaller CCs, and the core size across our temperature range is typically 14% of r500, in good agreement with what is seen in clusters. . The temperature profiles of the groups classified as being NCCs. In the case of the 2dXGS groups, the data from both the annuli and two-dimensional regions have been used. The plots are arranged in order of ascending mean temperature, from top left to bottom right. The solid lines show weighted local regression fits to the data, see the text for details. We note here that no smooth fit could be found for the temperature profile of NGC 4168, hence a fit is not shown in this case.

RADIAL PROFILES OF GAS PROPERTIES
Here we look at the radial variation of deprojected gas properties: entropy, pressure and gas density. These quantities have been derived from a two dimensional analysis, rather than an annular approach, so we define the characteristic radius of each measurement as the mean of the two bounding radii of the spectral extraction region.

Entropy profiles
The entropy of a relaxed system, coupled with knowledge of the potential well, completely defines the properties of the intracluster medium (e.g. Voit et al. 2003). Gas entropy can provide useful insights into the non-gravitational processes shaping the ICM, since it remains unchanged when gas is simply moved around within a system. We define entropy as (e.g. Ponman, Sanderson & Finoguenov 2003), where T is the mean temperature of the system, and ne is the gas density, and S has units of keV cm 2 . This is related to the true thermodynamic definition of entropy via a logarithm and additive constant, and has the benefit of acting as a proxy for entropy which follows directly from two X-ray derived properties.
In the self-similar case, entropy simply scales with the virial temperature of the system. However, a modified entropy scaling of S /T 2/3 was found to perform well across a wide range of virial masses, from groups to clusters . We aim here to test the use of this modified entropy scaling across the group regime. Considering first the self-similar scaling (S ∝T ), Figure 4 shows power-law fits (in log-log space) to these scaled entropy profiles for both the Cool and Warm sub-samples, with their associated 68% confidence regions 3 . We note that few groups contribute beyond ∼0.5 r500, and hence results shown here beyond this radius should be treated with caution. If the self-similar scaling works then it should remove any systematic temperature dependence in the profiles, so that the fits to the Cool and Warm samples should coincide. In practice, one can see that outside the core the scaled profile for Cool groups lies significantly above that for Warm ones, indicating that a scaling less strong than S ∝T is required.

Optimum entropy scaling
We can investigate the optimum entropy scaling by looking at the entropy at a particular radius as a function of temperature. To each individual (unscaled) entropy profile, we fitted a power law model, to determine the entropy at 0.1 r500. In Figure 5 we plot the entropy at this characteristic radius as a function of the mean temperature of the system. We fit a straight line in log-log space to yield the optimum entropy scaling, which will simply be the slope of this relation. Due to the measurement errors on the mean temperature, and the errors on the entropy measurements, which are propagated from the errors on the power-law fits to each individual group, we fit the relation using the BCES generalisation of the orthogonal regression method presented by Akritas & Bershady (1996). This allows for both measurement errors (which may be correlated) and also for unknown intrinsic scatter in the relation. The narrow temperature range covered by the groups, coupled with the large dispersion in group properties, results in a large error in the slope of the S-T relation. We therefore combine our sample with the cluster data of Sanderson et al. (2009) and fit the relation using the combined group and cluster sample, to increase the temperature baseline.
As described by Akritas & Bershady (1996) and Isobe et al. (1990), the analytic form presented for the estimates of variance on the slope and intercept are only suitable for large samples, and more appropriately a bootstrap error analysis should be applied to describe the error behaviour of small samples. We have calculated both the analytical variance from Akritas & Bershady (1996), and the result from a bootstrap analysis. We extracted 100 bootstrap samples from our original data, and calculated the slope and intercept in each case, determining the mean and standard deviation from these results. Applying the orthogonal regression method yields a slope of 0.79±0.06 using the analytical variance; the slope resulting from our bootstrap analysis was found to be 0.78±0.06.
The recovered slopes are a little steeper than the 2/3 scaling of Ponman et al. (2003), although they are consistent with it at the 2 σ level. Our result is in good agreement with recent work of Sun et al. (2008), who find a slope of 0.78±0.12 at the somewhat larger radius of 0.15 r500, when applying a BCES y on x regression to their group sample. Since the S ∝ T 2/3 modified entropy scaling of Ponman et al. (2003) provides an acceptable representation of the entropy scaling from both the present study, and the Chandra study of Sun et al. (2008), we adopt this scaling for the remainder of this paper. However, we should bear in mind that the actual scaling now seems likely to be rather steeper thanT 2/3 .

Cool and Warm samples
Adopting the modified entropy scaling as discussed above, to remove the mean effects of group temperature on the entropy, we work with the scaled entropy S /T 2/3 , which has units of keV 1/3 cm 2 . Power-law models are fitted to radial profiles of the scaled entropy for the Cool and Warm subsamples separately. The resulting power-law fits and their error envelopes are shown in Figure 6. Where the entropy is only driven by gravitational processes, its radial variation is expected to be proportional to r 1.1 (Tozzi & Norman 2001). Figure 6 shows the observed profiles to be much flatter than this, as also found by Sun et al. (2008). This is consistent with observations of the inner regions of NCC clusters, although CC clusters are found to lie closer to the r 1.1 relation (Sanderson et al. 2009). The slopes of the entropy profiles fitted to the Cool and Warm groups are 0.77±0.03 and 0.65±0.02 respectively, indicating a steeper slope in the case of the cooler systems. Figure 6 also shows the 68% confidence bounds on the fits, arising from errors in the slope and intercept. It can be seen that outside the core, the scaled entropy for the Cool systems lies somewhat above that for the Warm groups, confirming the discussion from the pre- vious section, that the entropy scaling is actually rather steeper than S ∝T 2/3 .

Core properties
We now consider the entropy profiles of the groups, divided into the CC and NCC samples described in Section 3. For a sample of CC clusters, Donahue et al. (2006) find that the addition of a constant entropy pedestal of ∼10 keV cm 2 to a power-law entropy profile provides a better fit to their data than a simple power-law model. We use the notation S ′ to refer to scaled entropy, i.e. S ′ is equivalent to S /T 2/3 . The pedestal modification is of the following form, where S ′ 0 is the entropy pedestal, S ′ 500 is the normalisation of the profile at r500, and α is the index of the fit. The entropy pedestal modification of Donahue et al. (2006) is apparent within ∼10 kpc; the corresponding radial range in our data is ∼ 0.02-0.04 r500. However, within this radius there does not appear to be a flattening of the entropy profiles. This is exhibited in Figure 7, which shows (unweighted) local regression fits to the entropy profiles of the CC and NCC groups. We employed the 'LOESS' algorithm in R to perform the fit, and fitted the data in log-log space. The profiles of the CC groups continually decrease inwards towards a scaled entropy of ∼ 8 keV 1/3 cm 2 at radii within 0.01 r500. However, these fits are unweighted, so may be adversely affected by outlying points as their measurement errors are not accounted for. Instead, we directly fit the pedestal model of Equation 4 to determine if there is a statistical reason to accept a pedestal addition to the profile. We find suggesting the pedestal modification is not statistically required, as S ′ 0 is consistent with zero at the level of 1 σ. Numerical simulations that employ only radiative cooling and the effects of gravity have shown the value of the entropy pedestal in CC clusters to tend to zero as the system ages (Ettori & Brighenti 2008). In the coolest system considered (4 keV) by Ettori & Brighenti (2008), this occurs in approximately 2 Gyr, and the entropy profile of the system is also seen to steepen during this time. Unfortunately, we do not have enough radial coverage for all groups to test the success of a pedestal model on individual group profiles.
We can also see a suggestion of flattening in the entropy profiles of CC groups at larger radii, between 0.5 r500 and r500 (see Figure 7). There is also an apparent up-turn in the NCC profile at large radius. The top panel of Figure 7 shows the number of groups contributing at each radius. The effect seen in the CC profile at large radius is driven solely by the group RGH 80 as the radial coverage for the majority of CC groups extends only as far as 0.5 r500 (five groups each add one data point at radii larger than 0.5 r500). Similarly, the apparent up-turn in the NCC profile is driven by only 6 data points contributing outside 0.35 r500, and hence this feature should be regarded with caution. Therefore, comparing the profiles in Figure 7, the greatest difference between the CC and NCC profiles occurs within 0.15 r500, where the NCC systems show higher entropy levels than the CC systems. At approximately 0.15 r500 the profiles appear to converge, ignoring the artifacts introduced by the fits at large radius, as described above.
As we are interested in comparing the slopes of the CC and NCC samples directly, we proceed in Figure 8 to fit power-law models in log-log space to the CC and NCC samples separately. These take the form, which allows the derivation of the slope of the entropy profiles, allowing a comparison to that expected from gravitational effects alone. As already indicated in the locally weighted regression fits used in Figure 7, we see a significant difference in the observed slopes of the entropy profiles of the two samples. Figure 8 shows the 95% confidence bounds on the power-law fits, and at this level, the power-law fit to the NCC groups is flatter than that of the CC groups, within a radius of ∼0.15 r500. The fitted slope to the CC systems is 0.71±0.02, compared to 0.57±0.04 for the NCC groups, both flatter than the expectation for gravitational effects alone (Tozzi & Norman 2001). Outside ∼0.15 r500, the two powerlaw fits begin to converge, suggesting that the processes affecting the core have little effect at larger radii. As we will further show in Section 4.1.4, the NCC groups also show considerably more scatter in their entropies at small radii, evidenced by the broad confidence region in Figure 7, compared to their CC counterparts.

Individual groups
To look at the properties of individual groups, and how they relate to the mean trends, we fitted the scaled entropy profile for each group with a power-law model. In addition to the index of the fit, we determined the value of the entropy at three characteristic radii: 0.01 r500, 0.1 r500, and 0.5 r500. The values of each of these parameters is shown in Table 3. We also fitted a power-law to the com- Table 3. The results of the power-law fits to the scaled entropy profiles of individual groups. Shown are the fitted index, and the scaled entropy at 0.01 r 500 , 0.1 r 500 and 0.5 r 500 . The result for fitting a power-law to all the groups simultaneously is also shown. bined data from all groups, which yielded a slope of 0.70±0.02. The slope and entropy measurements for each group indicate how each group varies from the mean, or expected theoretical, trends. NCC clusters are seen to show more scatter in their entropy profiles at small radii (McCarthy et al. 2008). This trend is also seen in our group sample, as for CC groups the rms scatter in scaled entropy at 0.01 r500 is 6 keV 1/3 cm 2 , whereas for NCC groups it is three times larger: 18 keV 1/3 cm 2 . In contrast, at 0.1 r500, the profiles are much more comparable. The mean scaled entropy at 0.1 r500 for the CC groups is 70 keV 1/3 cm 2 , with an rms scatter of 26 keV 1/3 cm 2 , while for NCC systems the corresponding figures are 78 keV 1/3 cm 2 and 23 keV 1/3 cm 2 . This lends more support to the observation in Section 4.1.3 that the CC and NCC entropy profiles converge outside 0.15 r500. The fitted parameters for Hickson 15 show it to have a relatively steep entropy profile, with very low central entropy. These are unexpected properties for NCC groups. Conversely, the fit to NGC 5171 shows a very flat slope and high central entropy. However, both these groups are current mergers, and hence likely to be far from equilibrium. NGC 5171 has been studied in detail by Osmond & Ponman (2004), and HCG 15 shows signs of having undergone a recent merger (E. O'Sullivan, private communication). Moreover in both these groups, the two-dimensional data cover only a small radial range, and the entropy fit has to be extrapolated well beyond the data to provide the values at the inner and outer radii tabulated in Table 3. Anomalous results for these two systems are therefore not unexpected.

Pressure profiles
Applying the modified entropy scaling of Ponman et al. (2003), the gas density scales as ρ ∝T 1/2 , resulting in a scaling for pressure of P ∝T 3/2 . To compare across the group sample, we therefore scaled the pressure profiles byT −3/2 .
In an analogous manner to the analysis of the entropy profiles, we again divided the groups into the Cool and Warm samples, to see if there were any systematic differences in the pressure profiles. F+06 found that fitting a power-law to the pressure profiles did not yield a good fit, and they instead followed the approach of Sanderson, Finoguenov & Mohr (2005) in fitting a locally weighted regression curve to the data. This has the advantage of yielding a fit to the data that is independent of any model assumptions, and which also suppresses scatter. We follow the same approach here, and fit a non-parametric curve to the data (in log-log space) using the R function 'LOESS'. Figure 9 shows the fitted pressure profiles to the Cool and Warm group samples. The cooler groups are seen to exhibit steeper pressure profiles in the inner regions, most prominently within approximately 0.05 r500, rising to a larger value of scaled pressure at the innermost radius. Outside ∼ 0.15-0.2 r500, the fits to the two sub-samples begin to converge. It is possible that we are seeing the effect of the brightest group galaxy on the pressure profiles here, as the difference is most apparent within a small radius. The characteristic size of the BGG can be estimated using D25, the diameter of the isophote where surface brightness in the B-band is 25 mag/arcsec 2 . We converted values of D25 from the RC3 galaxy catalogue (de Vaucouleurs et al. 1991) into physical radii using the distance information given in Tables 1 and 2. We find the mean size of the BGGs to be 0.05±0.02 r500. Separating this into the Cool and Warm samples yields the same result, i.e. there is no system- atic difference in the sizes of the central galaxies as a fraction of r500. We postulate that the difference in pressure therefore arises from a difference in the dominance of the stellar mass component in the BGGs of Cool and Warm groups.
We can examine this idea by comparing the masses of the group cores, and comparing with the luminosities of their central galaxies. Assuming hydrostatic equilibrium, the total group mass within radius r is given by, where P is the pressure and ρ is the density of the gas. We evaluated Equation 7 for the Cool and Warm samples separately, using the 'canonical' fits to the pressure ( Figure 9) and gas density profiles (see Figure 11), and removing the applied temperature scaling. Within 0.05 r500 (roughly the optical extent of a typical BGG), the masses of the Cool and Warm systems are 2.1×10 11 M⊙ and 3.7×10 11 M⊙ respectively. Taking into account the difference in r500 between the Cool and Warm groups, these recovered masses are typical of brightest group galaxies at this radius (e.g. Humphrey et al. 2006). We can test the stellar dominance by calculating the massto-light ratio of the Cool and Warm systems. The mean B-band BGG luminosities (with standard errors) for the Cool and Warm samples are 4.9±0.9×10 10 LB,⊙ and 5.7±0.9×10 10 LB,⊙ respectively. Using the above masses then gives B-band mass-to-light ratios, within 0.05 r500, of 4.3 and 6.5 for the Cool and Warm groups respectively. This indicates increased dominance from the stellar mass in the cooler groups, which would lead to a more concentrated mass profile, and hence an excess pressure relative to the warmer groups.
Dividing the sample into the CC and NCC classes, the same fitting procedure can be applied, and the fitted pressure profiles of the sample split on the basis of their core properties are shown in  Figure 10. The CC groups exhibit somewhat higher pressure than the NCC groups, although the two profiles overlap at ∼ 0.15 r500.
The typical radius of maximum temperature was found to be ∼ 0.14 r500 (see Section 3.2), so the excess pressure is seen only within the region of the CC. This is to be expected, since hydrostatic equilibrium (equation 7) gives a steeper pressure gradient (for the same M (r)) when gas density is higher, as it is within CCs.

Gas density profiles
We also attempt to draw conclusions on the systematic differences between groups by examining the gas density profiles of the sample. The modified entropy and pressure scaling applied to the group sample so far implies a scaling for gas density of ρ ∝T 1/2 , which we apply to the sample. We follow an analogous procedure to examining the pressure profiles of the systems in examining the gas density profiles of the Cool and Warm systems. We performed locally weighted regression fits on the gas density profiles of the Cool and Warm groups separately, in log-log space, using the R 'LOESS' algorithm. The results of these fits are shown in Figure 11. The gas density profiles of the Warm and Cool groups again appear to diverge at approximately 0.05 r500, with the Cool groups showing a higher scaled central density compared to the Warm groups. As explained in Section 4.2, this is showing the effect of the central galaxy. The pressure of the Cool systems is more strongly peaked, and consequently, the gas density is affected in the same manner.
The same comparison can be made dividing the sample on the core properties of the groups, and this is shown in Figure 12. In line with the trends seen in the pressure profiles, the scaled density profiles of CC groups are higher than those of NCC groups, within a radius of ∼ 0.15 r500. This is showing the effects of radiative cooling in the CC systems, leading to a higher density and pressure in the region of the core (see Section 4.2).

ENTROPY DISTRIBUTIONS
Considerable insight into the heating and cooling processes affecting the intracluster medium can be gained from looking at how the entropy is distributed as a function of gas mass of the system (e.g. Voit et al. 2003). Having fitted power-laws to the scaled entropy profiles of all the groups individually (see Section 4.1.4), we used these alongside β-model fits to the gas density profiles to determine the gas mass in a series of radial shells. The shells were stepped in radius from 0.0 to 0.5 r500, with width 0.0005 r500, and we set the characteristic radius for a shell to be its mean radius. The fits to the entropy and gas density profiles were used to interpolate the entropy and density at the these characteristic radii. The gas mass in each shell, Mgas was determined as follows, where r is the characteristic radius of the shell, ρ is the density evaluated at radius r, and δr is the width of each shell, i.e. 0.0005 r500.
With entropy and gas mass then described by the same set of radial bins, we set up a series of scaled entropy bins of width 15 keV 1/3 cm 2 , and determined the total gas mass in each entropy bin. To be able to compare groups directly, we scaled the gas mass byT 2 to remove the dependence of gas mass on the mass of the system as a whole. This is a result of the applied entropy scaling which gives a gas density scaling ∝T 1/2 (see Section 4.3). In a particular shell of mass Mgas, Mgas = 4π r 2 ρ δr ∝ r 3 ρ.
From the virial theorem, the radius r is proportional toT 1/2 , and with the density ρ scaling asT 1/2 we find, Dividing the sample into CC and NCC systems, we can examine the entropy distribution in both types of system. Figure 13 shows a histogram of the gas mass as a function of entropy. This NRGb184 HCG97 SRGb119 Figure 13. Histogram of gas mass as a function of entropy for the CC groups, with each group colour-coded as shown in the legend. This shows the total gas mass in a particular entropy bin, indicating where the majority of the gas mass lies. We have used the modified entropy scaling ofT 2/3 to directly compare the systems, and have scaled the gas mass byT 2 (see text for details).  Pavo  HCG68  NGC5171  HCG15  HCG51  IC1459 NGC4168 A194 Figure 14. Histogram of gas mass as a function of entropy for the NCC groups, with each group colour-coded as shown in the legend. This shows the total gas mass in a particular entropy bin, indicating where the majority of the gas mass lies. We have used the modified entropy scaling ofT 2/3 to directly compare the systems, and have scaled the gas mass byT 2 (see text for details).
indicates where the majority of the gas lies in terms of its entropy value, for each of the CC groups. The shape of these histograms requires careful interpretation. Since entropy rises outwards, the peak in these histograms is produced as a consequence of the limiting upper radius of the gas mass calculation (0.5 r500). At this radius, some groups have reached a higher entropy level than others, which produces the high entropy tail in the plot. It is evident is that there are five CC systems (HCG 42, NGC 2300, NGC 2563 and SRGb119) where higher entropy values are reached at 0.5 r500 compared to the remainder of the CC sample. NGC 2300 reaches very high entropies (> 600 keV 1/3 cm 2 ), although the entropy profile is extrapolated from just 4 data points, all within 0.2 r500, so the result shown here should be interpreted with caution. The CC systems generally show a sharp increase in scaled gas mass with scaled entropy on the leading edge of the peak, and show a dominant peak at approximately 105-120 keV 1/3 cm 2 , indicating that similar entropy levels are reached by the groups at 0.5r500. The entropy distributions for the NCC systems are shown in Figure 14. In comparison to the CC systems, the NCC groups show a wider diversity in entropy distributions, as can also be seen in Figures 15 and 16. This is in agreement with the observed tight trends for the entropy profiles of CC groups, seen in Figure 8, whilst the NCC systems show a wider spread in their entropy properties.
The entropy histograms also show a lack of lower entropy gas in the NCC systems -in the CC systems the mean scaled gas mass with scaled entropy 90 keV 1/3 cm 2 is ∼ 9.8×10 10 M⊙ keV −2 , compared to ∼ 5.0×10 10 M⊙ keV −2 in the NCC systems, thus confirming the results seen in Section 4.1.3. Also, it is apparent that the entropy at which scaled gas mass peaks is slightly different, with the CC systems peaking in the bin 105-120 keV cm 2 , whereas the NCC histogram peaks at 120-135 keV cm 2 .
Since this analysis employs scaled entropy and scaled gas mass, the mean effect of system mass should have been scaled out. We find the mean total scaled gas mass, evaluated within 0.5 r500, for the CC and NCC groups to be (4.7±0.5)×10 11 M⊙ keV −2 and (4.1±0.5)×10 11 M⊙ keV −2 respectively. Therefore, there is no significant difference in the overall scaled gas content of these systems within 0.5 r500, indicating that the different core properties do not result from global differences in gas content.

Comparison to a theoretical entropy distribution
In an effort to ascertain the magnitude of the effects of feedback processes acting on the observed entropy distributions, we can compare these distributions to theoretical distributions where the effects of non-gravitational processes have been ignored. The analytical models of Voit et al. (2003) show that the radial distribution of entropy is approximately proportional to the enclosed gas mass if the gas accreted is cold and the accretion is smooth. Looking specifically at groups, Voit et al. (2003) postulate smoother accretion in groups relative to clusters, to explain the observed differences in the entropy profiles of systems spanning this range in mass. We can begin to evaluate the effects of feedback on the group gas by first computing the expected entropy histogram for a system where only gravitational processes are considered. The observed difference between this 'theoretical' distribution and the observed distribution can then be used as a probe of the impact of feedback processes.
We firstly require a baseline entropy profile S(r) for a system unaffected by non-gravitational processes. For this, we adopt the baseline entropy profile of Voit, Kay & Bryan (2005), derived from smoothed particle hydrodynamics (SPH) cosmological simulations of galaxy clusters, without any cooling or feedback processes. This has the form S(r) = 1.32 S200 r r200 where S200 is defined as S200 = 362 keV cm 2 T200 TX (12) where we have dropped the cosmological factors from Voit et al. (2005), which are negligible given the low redshift of our sample. For consistency within this work, we use S to denote entropy, equivalent to the K used by Voit et al. (2002) and Voit et al. (2005).
For the temperature TX we use our mean X-ray derived temperatures, and calculate T200 in the manner described by Voit et al. (2005). This requires knowledge of both r200 and the mass enclosed at this radius, M200, which is simply the volume at r200 multiplied by 200 ρcrit. The former can be calculated by determining a conversion factor from the known r500. We assume an NFW profile (e.g. Navarro, Frenk & White 1997), evaluated at an overdensity of 500. Sun et al. (2008) found a concentration at this overdensity of 4.2 for their sample of galaxy groups observed with Chandra and we proceed to use this concentration in the following analysis. We find for an NFW dark matter halo with this concentration, that r500 = 0.66 r200. This modifies Equation 11 to the following, Therefore, we use Equation 13 to describe the theoretical entropy profiles of our systems.
To calculate the gas mass, we assume, in line with Voit et al. (2002), that the gas density profile follows the dark matter density profile, scaling the latter by a factor Ω b / Ω dm , assuming Ω b = 0.022 h −2 (Voit et al. 2005) and Ω dm = 0.2. We then apply the same analysis procedure as for the observed entropy distributions, calculating first gas density and entropy as a function of radius, before determining the gas mass in each bin of entropy. We use scaled entropy and gas mass as for the observed histograms throughout our comparisons. Figures 15 and 16 show the observed and theoretical entropy distributions for the CC and NCC groups respectively. The rightmost bin of each distribution should be treated with caution, since due to our imposed cut-off at 0.5 r500, these bins are incomplete.
In all cases, the observed maximum entropy far exceeds the theoretically derived maximum entropy, suggesting substantial modification must have occurred to the theoretical distribution to yield the observed distribution. We now explore some simple heating and cooling prescriptions, to see to what extent they are able to reproduce the modified entropy distributions observed. Voit et al. (2002) consider three simple modifications to a baseline entropy distribution, which we summarise below:

Entropy modifications
(i) A truncation in the entropy distribution where gas is removed if its entropy falls below a threshold value corresponding to a given cooling time. This approximates the effects of cooling, where gas cools out to form stars, or is heated by feedback from proximate cooled gas (e.g. via supernova feedback), raising its entropy such that it convects to larger radii.
(ii) A shift in the entropy distribution, such that the entropy of all gas is increased by a certain baseline entropy, mimicing the effects of pre-heating.  Figure 15. The observed entropy distributions of the CC groups (shown in grey) and the calculated theoretical entropy distribution (shown in black). In all cases the observed distribution extends to a higher maximum entropy, compared to the theoretical distribution. The dashed line shows the entropy distribution resulting from a modification in the form of an entropy shift, followed by radiative cooling (see Section 5.2). Note the extended x axis in the case of NGC 2300, to show the high entropies reached by this group (this is due to a significant extrapolation, see Section 5). The groups are ordered in increasing mean temperature, from top left to bottom right, and all curves extend to 0.5 r 500 .
(iii) Lowering of the whole entropy distribution due to radiative cooling. The entropy distribution of the gas to the 3/2 power (i.e. S 3/2 ) is reduced by the 3/2 power of the critical entropy (Sc) across the whole entropy distribution. Gas which drops below zero entropy as a result of this modification is removed, as in model (i). The form of the entropy reduction is based on the approximation that the cooling function for gas in the group regime (T < 2 keV) has a temperature dependence Λ(T ) ∝ T −1/2 . The critical entropy Sc in models (i) and (iii) is some fraction of S200, the entropy at r200. In fact, following the discussion presented by Voit et al. (2002), we specify, calculating Sc for each group, where T is its mean temperature.
It can immediately be seen by considering the differences between the observed (grey) and theoretical (solid black) distributions in Figures 15 and 16 that simple truncation (model (i)) and shift (model (ii)) models are inadequate. Truncating the theoretical distribution at a critical entropy would lead to a loss of low entropy gas, but would not modify the shape of the distribution above the critical entropy, making no improvement in the agreement between the observed and theoretical distributions at high entropy, and making things worse at low entropy (where low S gas is actually observed in nearly all groups). If instead we add a constant (a 'shift') to the theoretical entropy distribution, we simply move this distribution in comparison to the observed distribution. This  Figure 16. The observed entropy distributions of the NCC groups (shown in grey) and the calculated theoretical entropy distribution (shown in black). Like the CC systems, in all cases the observed distribution extends to a higher maximum entropy compared to the theoretical distribution. The dashed line shows the entropy distribution resulting from a modification in the form of an entropy shift, followed by radiative cooling (see Section 5.2). The groups are ordered in increasing mean temperature, from top left to bottom right, and all curves extend to 0.5 r 500 .
will improve the agreement at high entropy, but would leave the distribution devoid of any low entropy gas. The radiative cooling modification implemented by Voit et al. (2002) reduces the theoretical entropy distribution S 3/2 by S 3/2 c and removes all gas with an entropy less than zero. The implications of such a model are clear; unlike the truncation and shift models, low entropy gas will remain. However, it is not possible to increase the maximum entropy reached by the theoretical model in this example, as whatever the value of Sc, the resulting entropy is a reduction of the original. Therefore, a simple radiative cooling model does not aid in matching the entropy profiles at the highest entropies reached in Figures 15 and 16.
It seems clear that a more sophisticated approach is required, and we can envisage a model which combines the simple modifications described above. To fix the high entropy behaviour, we first need a shift in entropy, which we allow to subsequently cool via the radiative modification to populate the low entropy end of the distribution. In physical terms, this model uses an entropy shift to mimic early pre-heating of the group gas, which subsequently cools through radiative cooling. In fact, we can calculate the exact level of the required entropy shift, S shif t , for the final maximum theoretical entropy to match the maximum entropy achieved by the observed distribution. Assuming the maximum observed entropy So,max is achieved by first adding a shift S shif t to the maximum theoretical entropy St,max, before subsequently cooling as described above by an amount S 3/2 c , we have, Rearranging Equation 15, we have Therefore, applying the shift S shif t before cooling by S 3/2 c will match the right hand edge of the entropy distribution. The dashed lines in Figures 15 and 16 show this entropy modification to the theoretical distribution. In the case of NGC 2300, we have extended the x axis to show the high entropy behaviour at 0.5 r500 (see Table 4.1.4), a result of fitting a power-law to a profile of only four data points, all located within 0.2 r500. Therefore, due to the extensive extrapolation the entropy distribution of NGC 2300 should be treated with caution.
Our applied "shift + cool" model is certainly more successful than models (i)-(iii), but it is able to produce a reasonable representation of the observed distribution in only a small number of cases. In the CC sample, NGC 4636 and RGH 80 are best represented by the applied model, however, Hickson 97 and SS2b153 also perform reasonably well. In the NCC sample, Pavo, Abell 194, NGC 5171 and 3C 449 all show a reasonable agreement between the observation and the applied model. However, the vast majority of groups show marked differences between the "shift + cool" model and the observed entropy distribution. In every case this disagreement is due to a lack of low entropy gas in the model. Conversely a higher gas mass than observed is usually present in the highest entropy bins. One caveat with our model is that the total gas mass within 0.5 r500 remains fixed. In reality, this value would change, due to movement of the gas as a consequence of the applied heating and cooling mechanisms. For example, if low entropy gas is removed from the distribution, higher entropy gas will flow in. Boosting the entropy of the gas would in reality cause it to expand, such that a proportion of the gas mass at high entropy would actually fall outside 0.5 r500, and the applied entropy shift will have been underestimated. Therefore, once the expansion of the gas is taken into account, the applied entropy shift needs to be somewhat larger than shown in Figures 15 and 16. This will exacerbate the difficulties of the model, which arise (see discussion below) from the large entropy boost required in many systems.
The required entropy shifts range between 115 keV cm 2 (RGH 80) and 567 keV cm 2 (NGC 2300). The models which recreate the observed entropy distribution most successfully are those where the value of the applied entropy shift is very similar to the critical entropy Sc (Equation 14) which governs the radiative cooling (Equation 15), since in these cases, the shift provides a good match at the high entropy end of the distribution, whilst the cooling is able to repopulate the low entropy end. For example, in NGC 4636, the applied entropy shift is 148 keV cm 2 compared to a critical entropy of 164 keV cm 2 and in RGH 80, the entropy shift is 115 keV cm 2 compared to a critical entropy of 109 keV cm 2 . The groups for which this works are those with the smallest difference in maximum entropy between the observed and theoretical distributions. In the majority of cases however, the required shift to match the high entropy end is much larger than the critical entropy, in which case the gas entropy is boosted to the point where cooling has little effect.

The effects of metallicity
The Voit et al. (2002) prescription for the critical entropy assumes that cooling has proceeded for 15 Gyr, and that the emissivity of the gas corresponds to a metallicity of 1/3 solar. However, in the group regime, emissivity is a strong function of metallicity (due to strong line cooling) and in CC groups there is a strong central abundance gradient (Rasmussen & Ponman 2007, Johnson et al., in prep.) leading to central metallicity close to solar values. We can assess the potential effects of such an abundance gradient on our 'shift + cool' model by instead assuming solar metallicity. This is assumed across the radial range, so will in fact be an over-estimate of the effects of a metallicity gradient. For a hot plasma at 1 keV, the emissivity scales up by a factor of ∼ 2.0 when the metallicity increases from 1/3 solar to solar. Equation 17 of Voit et al. (2002) describes the rate of change of entropy as follows, where our S is equivalent to the Voit et al. (2002) K. The Voit et al. (2002) model assumes the right hand side of this relation to remain constant, such that it can be simply integrated, Integrating over a Hubble time (tH) yields, Hence the effect of a change in emissivity, caused by an increase in metallicity, is to increase the value of Sc. The factor increase is equal to the factor increase in emissivity, to the power 2/3 (i.e. 2.0 (2/3) ≈ 1.6). In Figure 17 we show the size of the required entropy shift S shif t versus the critical entropy Sc in our 'shift + cool' model. Since almost all groups contain observable low entropy gas, the model can only be successful if Sc is similar to or larger than  Figure 17. The required entropy shift S shif t versus the critical entropy Sc for the CC (open circles) and NCC (filled squares) groups in the sample. The solid line is the line of equality, and the dashed line shows the same but re-normalised by the factor ∼1.6.
S shift . For the unmodified cooling threshold (solid line) it can be seen that only a few groups satisfy this requirement. However, with the factor 1.6 scaling in Sc, the model could be successful for at least half the groups, although approximately one third of the systems still require entropy shifts much greater than the critical entropy.
Our model is still very simple, of course. The uniform solar abundance will tend to overestimate the effects of cooling, as will the generous timescale of 15 Gyr allowed for the cooling to proceed. On the other hand, we have not allowed for the hierarchical merger history of the systems. This requires investigation with more sophisticated cosmological models. Nonetheless, we tentatively conclude that the magnitude of the entropy shift required in a significant fraction of groups is so high that our 'shift + cool' model will be unable to reproduce their properties.

Effects of feedback
We turn now to the link between modification of the entropy distribution and the feedback processes which are most likely responsible for this modification. If we make the assumption that the most prominent source of feedback in galaxy groups comes from the member galaxies, either through energy input from supernovae or feedback from active galactic nuclei (AGN), we can begin to quantify the likely impact of feedback. We construct a rough measure of 'feedback impact' using the ratio of the optical luminosity of the group to the thermal energy of the gas: fgrp = LB,group/(MgasTx), where Tx is the mean X-ray temperature of the system, and Mgas is the total gas mass at 0.5 r500 (derived as described in Section 5). The motivation here is that integrated feedback from supernovae should scale approximately with the stellar mass, and given the relationship (Magorrian et al. 1998) between the mass of supermassive black holes and the stellar mass of the spheroids they inhabit, integrated AGN feedback might also be expected to scale roughly with stellar mass. The effect of a given amount of injected energy on the intergalactic gas will depend on the factor by which it increases the existing total thermal energy, which motivates the denominator, (MgasTx), in our expression for fgrp. This measure of feedback impact therefore has units of LB,⊙ M⊙ −1 keV −1 . For clarity we will not show the units in the remainder of this section.
Our measure of feedback impact is admittedly imperfect. In the first place, the K-band luminosity would provide a better measure of stellar mass, since the B-band is strongly affected by young stellar populations. A full stellar population analysis would be even better, of course. However, we have consistent measurements of B-band luminosities for all the groups in our sample, and are encouraged to believe that biases due to young stars will not have a major impact, by the fact that X-ray bright groups have a low spiral fraction (Helsdon & Ponman 2003). A second respect in which our formula for fgrp is imperfect is that the denominator actually reflects the thermal energy of the gas after the feedback has had its effects, rather than before. We explore the effects of modifying the denominator at the end of this section, and find that it makes no qualitative difference to our conclusions.
Assuming that fgrp is monotonically related to feedback impact, we now use it to divide our sample into 3 subsets of (almost) equal size, covering ranges in feedback impact of 0.0 < fgrp 0.22, 0.22 < fgrp 0.6 and fgrp > 0.6. These bins contain 10 (8 CC + 2 NCC), 9 (6 CC + 3 NCC) and 9 (4CC + 5 NCC) groups respectively. Figure 18 shows the stacked entropy distributions for the CC (top row) and NCC (bottom row) groups, shown in order of increasing feedback impact from left to right. In both samples, we see a trend for an increase in the amount of gas at higher entropies as the feedback impact increases. There is also a suggestion of the peak of the distribution moving to higher entropies as the level of feedback impact increases. Qualitatively, this appears to indicate that the gas in groups with the largest capacity for feedback from their galaxy members (in the form of energy deposited into the intracluster medium from supernovae explosions or AGN) has been pushed to higher entropies. This is true regardless of whether the system has a CC or not.
Given the similarity of the entropy distributions in the highest feedback bins, we choose to combine these bins. Therefore, the sample now consists of a 'lower' feedback bin (fgrp 0.22) and a 'higher' feedback bin (fgrp > 0.22). Table 4 shows the break-down of our groups into the CC/NCC and high and low feedback classes. In brackets we show the expected number of groups in each of the four categories, given the relative ratios of CC to NCC groups, and high to low feedback systems (18 to 10, in each case) if the two classifications (CC/NCC and high/low feedback) were completely independent. This suggests that there is some tendency for the NCC systems to have a higher feedback impact than CC systems. The median feedback impact for the whole sample is 0.31, whereas for CC systems it is 0.27 and for NCC systems 0.62. We quantify the significance of the relationship between feedback and CC status by calculating how likely it is that there should be at least 8 groups in both the high feedback, NCC class, and the low feedback, CC class, given that only 6.4 systems were expected in each, using the binomial probability distribution. This probability is 6 percent, so we conclude that there is some evidence for a relationship between the two properties, but it is clear from Table 4 that the relationship is rather weak. In addition to affecting the energy of the gas in the intracluster medium, feedback processes can also inject and redistribute metals in the intracluster medium. Supernova feedback will introduce more metals into the intracluster medium, whilst AGN will not, so by examining the metal content of the gas in high and low feedback systems, we may hope to differentiate between the two sources of feedback. Figure 19 shows the metallicity profile of the lowest feedback (fgrp < 0.22) systems and the higher feedback systems (fgrp 0.22). We have stacked the abundance profiles of the two feedback sub-samples into equal radial bins. The lowest feedback systems extend to a smaller radius than the higher feedback systems. We therefore show the contribution at small radii from the lowest feedback systems as a single grey point. Over the majority of the profile, the low and high feedback systems are remarkably similar. Only in the innermost radial bin do the profiles differ significantly, with the higher feedback systems showing lower central metallicities than the lowest feedback systems. We estimated an approximate integrated metal fraction for each group by determining the product of metallicity and gas mass, summed over a series of radial shells, and dividing by the total gas mass. We restricted this calculation to within 0.3 r500, to ensure that the majority of groups have the appropriate radial spectral coverage to avoid significant extrapolation of the metallicity profile. For the low feedback systems, we find the mean metal fraction (with standard error) to be 0.23±0.04, whereas the mean metal fraction (with standard error) of the high feedback systems is 0.25±0.03. Hence, across this radial range, the integrated metal fraction of the two subsets is essentially identical. This is the case despite the abundance difference in the innermost radial bin since the metal fraction is dominated by shells at larger radii, where the majority of the gas lies. These results run counter to naive expectations. We would expect higher feedback systems to have experienced more supernova feedback, and therefore to show higher abundances and integrated metal mass fractions.
The fact that we see no increase in the integrated metal mass fraction in the highest feedback systems, coupled with the lower central metallicity of these groups, suggests that AGN may provide the dominant source of feedback, as supernova feedback would inject metals into the gas, along with the deposited energy, leading to extra enrichment. We have already seen in Figure 18, that there is a lack of low entropy gas in the high feedback systems, compared to the low feedback groups. We can examine whether high metallicity, low entropy gas has been boosted in entropy (generating high entropy gas with high metal abundance) by considering the distribution of metals as a function of entropy, rather than radius. In Figure 20, we show the stacked abundance as a function of entropy for the low (fgrp 0.22) and high (fgrp > 0.22) feedback systems. The samples have been divided into the same bins in entropy for comparison. As the low feedback systems extend to lower entropy, we show the remaining data points outside the range of comparison as an open grey circle in Figure 20. In the highest entropy gas, the gas in the low and high feedback systems is similarly enriched. The lowest entropy gas in the highest feedback systems is of lower metallicity, compared to gas at the same entropy in the lowest feedback systems. Hence, if in the high feedback systems, the high metallicity, low entropy gas has been boosted in entropy, it would need to be pushed beyond 0.5 r500 for us not to detect it in our combined data.
One concern in studying the relationship between our feedback impact parameter and gas entropy is that both are related to gas density. Systems with low gas density will have high entropy, and will also (via the entry of Mgas into our expression for fgrp) tend to have high feedback impact. As mentioned earlier, the denominator of the expression for fgrp should really be the total thermal energy in the gas before the action of feedback, rather than afterwards. Assuming that all systems would contain a cosmic ratio of gas to dark matter, in the absence of cooling and feedback, we can construct an alternative measure of feedback impact, as LB,group/(MtotTx). For self-similar systems at a given epoch, the total mass and characteristic temperature are related by Mtot ∝ T 3/2 . We therefore explore a modified definition of feedback impact of LB,group/T 5/2 x . Using this revised parameter to partition our group sample into high and low feedback classes, we find results very similar to those obtained with our earlier definition of feedback impact: gas with entropy greater than ∼30 keV (1/3) cm 2 is similarly enriched in the low and high feedback systems, and in the innermost entropy bin, the highest feedback systems reach lower central metallicity. We also find a trend for the higher feedback systems to reach higher entropies than the lowest feedback groups, as seen in Figure 18 for our unmodified measure of feedback impact. We conclude that the gas density dependence of our original feedback measure does not affect the results.

The σ − T relation
We can further investigate the impact of feedback by considering the quantity βspec, which is the ratio of the specific energy of the galaxies to the specific energy of the gas, where µ is the mean mass per particle, mp is the proton mass, k is Boltzmann's constant, and T is the temperature of the gas. If feedback raises the energy of the gas, then the consequence is to reduce βspec, as the subsequent specific energy of the gas would dominate over the specific energy of the galaxies. We can use the σ − T relation to investigate where groups lie with respect to βspec = 1. From a theoretical perspective we expect to see similar specific energy in gas and galaxies, unless feedback has had a major effect, and in the case of rich clusters, observations have shown that relaxed systems generally lie along a trend in the σ − T plane of the form σ ∝ T 0.5 , whilst groups appear to follow a steeper trend (see the discussion in Osmond & Ponman (2004) Figure 19. The stacked abundance profiles of the groups with the lowest level of feedback impact (fgrp < 0.22, see Section 5.3), shown as black filled squares and the groups with higher levels of feedback (fgrp 0.22), shown as red filled triangles. Each sample has been divided into equal bins of radius. Vertical error bars show the standard error in each abundance bin, and horizontal error bars show the width of each radial bin. The radial coverage of the low feedback systems extends to a smaller radius than the high feedback systems, and we show these remaining low feedback systems as a single grey open circle point. 1996) to fit a straight line to the σ−T relation for our group sample, yielding the fit: log10 σ = 0.9 (±0.3) log10 T + 2.57 (±0.02).
This method accounts for both the measurement errors and the unknown intrinsic scatter in the relation. Figure 21 shows the σ − T relation for the group sample, where we show the line corresponding to βspec = 1 as a solid line. Six of the ten NCC systems are seen to lie above the line of βspec = 1, whereas the majority of CC systems lie beneath this line. If the lack of central cooling in NCC systems resulted from strong feedback, as in the model of McCarthy et al. (2008), who invoke pre-heating prior to cluster collapse, then the specific energy of the gas should be higher than that of the galaxies in NCC groups, i.e. βspec < 1, whereas in practice, we see the exact opposite. This may indicate the importance of cluster merging in the formation of NCC systems, as in merging systems, the velocity dispersion can be boosted, which would in turn increase βspec. This is supported by the NCC group with the highest βspec, Hickson 92 (Stephan's Quintet), which is clearly a system currently undergoing multiple mergers (e.g. Trinchieri et al. 2003).

Discussion
We can connect the evidence of Section 5 to build a coherent picture relating galaxy group properties, entropy and the effects of feedback. Section 5.2 shows that modifications must have been made to the gas entropy distribution in these groups. Comparing with a theoretical distribution, which does not allow for any non-gravitational  Figure 19, but now as a function of scaled entropy. Each sample has been divided into five equal bins in scaled entropy. Vertical error bars show the standard error in each abundance bin, and horizontal error bars show the width of each entropy bin. The lowest feedback systems extend to a lower entropy, and we show these remaining low feedback systems as a single grey open circle point. processes, we find a higher maximum entropy in the observed entropy distributions. Simple entropy modifications are not sufficient to convert the theoretical distribution to the observed distribution, and we rule out simple entropy shift, truncation and radiative cooling models. Our model which mimics pre-heating followed by a bout of radiative cooling (our so-called 'shift + cool' model) only performs well in cases where the required entropy shift is modest. In many cases, the required large entropy shift suppresses cooling, making it impossible to populate the low entropy end of the observed distribution.
To avoid this problem, we require a model which applies a larger entropy shift to higher entropy gas. Two physical mechanisms which might achieve this are entropy amplification Voit et al. 2003) or episodic heating. In the former, when pre-heated gas hits an accretion shock there is an entropy boost, and higher entropy gas hitting the accretion shock gets a bigger boost than lower entropy gas. In the case of episodic heating, small bursts of heating could raise the group entropy, and subsequent cooling operates more efficiently in the lower entropy gas. Hence, over a number of cycles, the highest entropy gas could be boosted more than the lower entropy gas, as required to model our observed entropy distributions. The former mechanism could require up to two orders of magnitude less energy than the latter, depending on when the pre-heating takes place (McCarthy et al. 2008).
Defining feedback impact as LB,group/(MgasTx), we find gas of higher entropy in systems with higher feedback impact. However, there is no significant increase in metallicity in the highest feedback impact systems, and the metal mass fraction in the gas is similar to the lowest feedback systems. As our measure of feedback impact scales with the optical luminosity of the group, we would expect an increase in the number of supernovae, and hence an increase in the metallicity of the highest feedback systems. The lack of such a trend suggests that metals must have been removed from the gas within 0.5 r500 either by ejecting it to larger radii, or by locking up a larger fraction in stars or stellar remnants, and strongly favours AGN, rather than supernovae explosions, as the dominant source of feedback.
We see higher metallicity gas in the centres of the lowest feedback groups, but across the majority of the radial range, the metallicity profiles of the low and high feedback systems are comparable. The conclusion of Jetha et al. (2007) that central AGN within groups influence the gas properties of groups on a local, rather than a global, scale is supported by this observation, as the effect of feedback is noticeable only at the innermost radius. This contrasts with the results of Croston, Hardcastle & Birkinshaw (2005), who found evidence for radio sources affecting the global properties of the group gas, as represented by their location in the luminositytemperature plane.
In the highest entropy gas we study, whose entropy exceeds that expected due to gravitational collapse alone, similar levels of enrichment are reached in high and low feedback syatems. This again points to an AGN origin for the bulk of the feedback, though this feedback may well have taken place within precursor structures, rather than being driven by a central AGN within the assembled group.
Eight of the ten NCC systems have feedback impact >0.22, putting them in our highest feedback bin, and five of these have a value of βspec > 1. This is puzzling, since high feedback impact should indicate raise the specific energy of the gas, resulting in βspec < 1. The observed high βspec values could come about through merging, which may therefore be the most important mech-anism in the formation of NCC systems. It seems that there is a complex interplay between the gas and group properties in these systems.

CONCLUSIONS
We have constructed a sample of 28 galaxy groups from the Two-Dimensional XMM-Newton Group Survey (Finoguenov et al. 2006(Finoguenov et al. , 2007 and the RASSCALS groups of Mahdavi et al. (2005). We have statistically analysed the results of a novel two-dimensional spectral analysis technique applied by Mahdavi et al. (2005), Finoguenov et al. (2006 and Finoguenov et al. (2007) to high quality XMM-Newton data. This is the largest group sample that has been analysed in a consistent manner with XMM-Newton data, and the size of the sample allowed a division on the basis of the presence of a cool core. We find 18 groups to exhibit cool cores (CCs) and 10 to be classified as non cool cores (NCCs). This latter sample is the first of its kind in the group regime, and the analysis of these groups provides useful insights into the variation in the observed properties of these systems. We summarise our main results below: (i) We have measured the ratio of the central temperature drop (defined between the peak temperature and that at 0.01 r500) to the peak temperature in the CC systems. This fractional central decline in T is smaller than in clusters. Within our whole group sample, we find no significant trend with mean temperature, however, the hottest system in our sample (NGC 4073,T = 1.87 keV) has an unusually small temperature decline, and if this group is excluded we find a positive correlation between CC strength andT which is significant at the > 1σ level.
(ii) We find radially smaller CCs in the coolest systems, confirming the results of Rasmussen & Ponman (2007), and find no CC systems with temperatures less than ∼0.7 keV, indicating a possible lack of CC systems at lower temperatures.
(iii) We have investigated the relationship between temperature and entropy evaluated at 0.1 r500, incorporating the cluster sample of Sanderson et al. (2009). Applying a BCES orthogonal regression (Akritas & Bershady 1996) yields a slope of 0.79±0.06, in agreement with the Chandra sample of Sun et al. (2008). There is a large amount of scatter in the group data, and the fit is mostly constrained by the addition of the cluster points, leading us to adopt the modified entropy scaling (S ∝ T 2/3 of Ponman et al. 2003) for purposes of scaling group properties in this paper.
(iv) The entropy profiles of cool groups (defined as those with mean temperature below the median value for our sample of 1.035 keV) are significantly steeper than those of hotter groups. The slope of the entropy profiles of the Cool subsample is found to be 0.77±0.03, compared to 0.65±0.02 for the Warm group subsample.
(v) Comparing the entropy profiles of CC and NCC groups, we find there to be more scatter in the entropy profiles for the latter, which is consistent with the larger scatter shown in their temperature profiles. The central entropy, within ∼ 0.1 r500, is also flatter for the NCC systems, whereas in CCs, the entropy profiles decrease steadily into the centre of the system. There is no evidence for a central entropy pedestal, as required for clusters (Donahue et al. 2006), in the CC groups.
(vi) Examining the pressure and density profiles, we find a marked difference in the pressure profiles of Cool and Warm groups, with the scaled pressure profile of the former rising to higher central values than in warmer systems. We attribute this difference to the increased dominance of the stellar mass of the bright-est group galaxy in the centres of the cooler systems. Comparing CC and NCC systems, we see slightly steeper density and pressure profiles within the cores of CC systems, which is attributable to their central temperature decline. Outside the typical radius of the CC (∼0.1 r500), we find the two subsamples to have similar gas profiles.
(vii) We have investigated the entropy distribution of the groups by plotting the observed gas mass in bins of scaled entropy. The total gas mass within 0.5 r500 is comparable for the CC (4.7±0.5×10 11 M⊙ keV −2 ) and NCC groups (4.1±0.5×10 11 M⊙ keV −2 ). Comparing the observed entropy distributions to the theoretical expectation if non-gravitational processes are ignored, and find that the observed distributions generally reach much higher entropies within 0.5 r500. In trying to reconcile the differences between the observed and theoretical histograms, we find that simple modifications (shift, truncation or radiative cooling) are not sufficient to bring the theoretical distribution in line with that observed. A 'shift + cool' model which aims to match the high entropy behaviour of the two distributions performs well when the entropy shift is small, but in many cases the large entropy boost suppresses cooling, resulting in an entropy profile which lacks the low entropy gas required by the observations. This suggests a process whereby the entropy shift is larger for higher entropy gas. Potential mechanisms for achieving this are either through entropy amplification or episodic heating. We note that the former is likely to be more energy efficient.
(viii) We define the 'feedback impact' of a group, and find that systems with the highest feedback impact reach higher scaled entropies within 0.5 r500, regardless of whether or not they have CCs. We find higher metallicity in the central regions of the lowest feedback systems, but over the majority of the radial range, the metallicity profiles of the low and high feedback systems is comparable. If low entropy metal-rich gas in the highest feedback systems has been boosted in entropy, it has been pushed outside 0.5 r500. There is no evidence for an increase in the metal content with the level of feedback impact, which leads us to favour AGN, probably acting before group assembly, as the dominant source of feedback, rather than supernova explosions. We test for bias in our definition of feedback impact by scaling by the total mass of the system (approximated as ∝ T 3/2 ) rather than the gas mass. Changing our definition of feedback impact in this way does not affect our results.
(ix) Fitting the σ -T relation to the group sample yields a steep slope of 0.9±0.3. Six of the NCC groups lie above the βspec=1 line, which seems inconsistent with models which invoke feedback to eliminate CCs in these systems. We suggest that group-group mergers are more likely responsible for the elimination of CCs.

ACKNOWLEDGMENTS
We would like to thank for the anonymous referee for their very helpful comments. We thank Jesper Rasmussen for his very helpful suggestions and comments on the original manuscript, and Ian McCarthy for valuable discussions and for drawing our attention to the potentially important effects of metallicity gradients on the cooling model. We would like to thank Alastair Sanderson for providing the entropy data from his cluster sample, and for a number of useful discussions and suggestions. RJ acknowledges support from STFC/PPARC and the University of Birmingham. AF acknowledges support from BMBF/DLR under grants 50OR0207 and 50OR0204 to MPE. AF thanks the University of Birmingham for hospitality during his frequent visits. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.