A NEW METHOD TO CONSTRAIN SUPERNOVA FRACTIONS USING X-RAY OBSERVATIONS OF CLUSTERS OF GALAXIES

Most of the metals from oxygen through nickel residing in the ICM were synthesized by SN Ia or SN cc explosions. We provide a new XSPEC model called snapec to determine the total number of SN explosions and relative contributions of SN types to the metal content of the ICM using elemental yields provided by SN nucleosynthesis calculations in the literature. The expected elemental yields of SNe Ia and SNe cc from massive (10–50 M_☉) progenitor stars of various metallicity (0–1 A_☉) have been intensively studied by several authors (Woosley & Weaver 1995; Iwamoto et al. 1999; Nomoto et al. 2006). For instance, SN models predict that SNe Ia produce significant amounts of iron (Fe), nickel (Ni), and silicon (Si), while, for SNe cc, large quantities of oxygen (O), neon (Ne), and magnesium (Mg) are produced but very little Fe and Ni escapes the compact remnant.

snapec calculates the mass of the ith element (M^SNe_i) in terms of the number of SN Ia (N^Ia) and SN cc (N^cc) explosions that enrich the ICM and the yields per SNe Ia (y^Ia_i) and SNe cc (y^cc_i),

$$\begin{eqnarray} M_{i}^{{\rm SNe}} &= N^{{\rm Ia}}\, y^{{\rm Ia}}_{i} +N^{{\rm cc}}\, \big\langle y^{{\rm cc}}_{i}\big\rangle \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &= N^{{\rm SNe}}(1+R)^{-1}\big [R y_{i}^{{\rm Ia}}+\big\langle y_{i}^{{\rm cc}}\big\rangle \big], \end{eqnarray} \tag{ 2 }$$

where R is the ratio of SNe Ia to SNe cc (R = N^Ia/N^cc), and N^SNe is the total number of SN explosions N^SNe = N^Ia + N^cc. The SN yields y^Ia_i and y^cc_i are obtained from published SN models and stored in the snapec database in solar units. The database includes yields of 30 elements from SN Ia explosions of slow deflagration (W7, C-DEF), delayed detonation (WDD, CDDT, ODDT), and core degenerate scenarios (CDD) (Tsujimoto et al. 1995, hereafter T95; Iwamoto et al. 1999, hereafter I99; Maeda et al. 2010, hereafter M10), and from SN cc yields from an extensive range of progenitor masses (10–50 M_☉) and metallicities (0–1 times solar) (Tsujimoto et al. 1995; Iwamoto et al. 1999; Woosley & Weaver 1995, hereafter WW95; Nomoto et al. 2006, hereafter N06). snapec adopts solar abundances for metals with atomic number ⩽7 (e.g., helium, lithium, beryllium, boron, carbon, and nitrogen).

The snapec model allows a selection of solar abundance standards—the database incorporates the sets of Anders & Grevesse (1989), Lodders (2003), or Asplund et al. (2009). Since the majority of hydrogen and helium in the ICM was produced during the big bang, the standard hot big bang nucleosynthesis (BBN) model predicts the primordial helium abundance by unit mass Y_P to be 0.2565 ± 0.0010 (Izotov & Thuan 2010) which corresponds to 8.3% helium atoms in number assuming 0.3 solar abundances for metals (e.g., Markevitch 2007). The Asplund et al. (2009) solar abundances we use here predict 8.5% of helium which is consistent with BBN predictions.

The SN cc yields of the ith element, averaged over the IMF (ϕ), is

$$\begin{equation} \big\langle y^{{\rm cc}}_{i}\big\rangle =\frac{\int _{m_{1}}^{m_{2}} y^{{\rm cc}}_{i}(m) \phi (m)\, dm}{\int _{m_{1}}^{m_{2}}\phi (m)\, dm}, \end{equation} \tag{ 3 }$$

where m₁ and m₂ are the lower and upper limits for the masses of SN cc progenitors. Here we assume that stars with masses M ⩽ 10 M_☉ and M ⩾ 50 M_☉ do not explode as SNe cc. In this work we use the standard Salpeter IMF slope above 10 M_☉, ϕ = m^−2.35 (Salpeter 1955).

It is straightforward to transform Equation (2) to an expression for the abundance, by number relative to hydrogen, of the ith element. As long as the total mass fraction of metals is small and the He abundance is solar,

$$\begin{eqnarray} \frac{Z_{i}^{{\rm SNe}}}{Z^{\odot }_{i}} &= \frac{M_{i}^{{\rm SNe}}}{M_{i}^{\odot }} \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} &=\frac{N^{{\rm SNe}}}{M_{i}^{\odot }(1+R)}\,\big[R\, y_{i}^{{\rm Ia}}+\big\langle y_{i}^{{\rm cc}}\big\rangle \big]. \end{eqnarray} \tag{ 5 }$$

M^☉_i is the mass of the ith element for an ICM with solar abundances, i.e.,

$$\begin{equation} M_{i}^{\odot }\approx \frac{M_{{\rm ICM}}Z^{\odot }_{i}A_{i}}{m_{N}}, \end{equation} \tag{ 6 }$$

where Z^☉_i is the abundance of the ith element in solar units, A_i is the atomic mass of the ith element, and m_N = 2.27 × 10⁻²⁴ g/H-atom is the average nucleon mass per hydrogen atom (Asplund et al. 2009).

This XSPEC model, snapec, has five free parameters: gas temperature (kT_e), the total number of SN explosions (N^SNe) and the SN ratio (R) that enriches the ICM in the spectral region being analyzed, the cluster's redshift, and the XSPEC normalization factor identical to those defined for other thermal plasma models (see below). In addition, there are two fixed parameters, numerical indices that specify the SN Ia and SN cc models from which the yields are drawn. The predicted spectrum is constructed using the single-temperature thermal model (apec) (Smith et al. 2001) and AtomDB 2.0.1 (A. Foster et al. 2012, in preparation) to determine the continuum and line emission for the metal abundances calculated using Equation (5). This method has the advantage of simultaneously providing a single set of uncertainties on the SN ratio and the total number of SN explosions that enrich the ICM since the formation of the cluster. The results presented in this work are obtained using the Asplund et al. (2009) solar abundances.

snapec redefines the quantities in Equation (5) to express the abundances in terms the total number of SNe per 10¹² M_☉ of ICM plasma—i.e., rescaled to yield values appropriate for cluster cores. The total number of SNe is derived by multiplying by the ICM mass (in units of 10¹² M_☉) within the spectral extraction region. This may be estimated from the XSPEC normalization, which is proportional to the emission measure,

$$\begin{equation} N=\frac{10^{-14}}{4\pi \, (1+z)^{2}\, D_{A}^{2}\,}\int n_{e} n_{{\rm H}} dV, \end{equation} \tag{ 7 }$$

where n_e is the electron number density, D_A is the angular diameter distance, and z is the cluster's redshift.

XMM-Newton's high spectral resolution RGS instrument (wavelength bandpass ranging from 5 to 38 Å) can resolve strong X-ray lines and measure elemental abundances accurately in the compact central regions of nearby bright clusters of galaxies. We examined the RGS observations of the cool core cluster A3112 (see Table 1) using version 11.0 of the Science Analysis Subsystem (SAS) software. For details about RGS data reduction and analysis, see Bulbul et al. (2012, hereafter Paper I). We fit the combined RGS first- and second-order spectra using an absorbed snapec model in the 7–22 Å and 7–16 Å wavelength range, respectively. The absorption column density was fixed to the Leiden/Argentine/Bonn Galactic value (Kalberla et al. 2005). The N^SNe, R, redshift, kT_e, and normalization parameters were allowed to freely vary. The best-fit model parameters obtained from the snapec fits to RGS spectra for various SN models are shown in Table 2. Since, as in Paper I, fitting is conducted using the C-statistic and the C-statistic does not provide a direct estimate of the goodness of fit, we asses the goodness of fit using the corresponding χ² value. These final χ² values, obtained from the best fit determined by C-statistics, are reported in Table 2.

Table 2. The Best-fit Parameters of the snapec Model Obtained Using T95, I99, and M10 SN Yields and XMM-Newton RGS Spectra

SNe Ia	SNe cc	kTe	NSNe	R	$R^{{\rm Ia}}_{\%}$	χ2 of Source	χ2 of Bkgd
Model	Model	(keV)	(× 109)		(%)	Spectra	Spectra
						(2608 dof)	(1490 dof)
T95 (W7)	T95	3.69 ± 0.23	1.06 ± 0.33	0.52 ± 0.09	34.3 ± 6.4	2817.8	2026.8
I99 (W7)	I99	3.70 ± 0.23	1.06 ± 0.34	0.52 ± 0.09	34.2 ± 6.3	2818.7	2026.4
I99 (W70)	I99	3.74 ± 0.23	1.07 ± 0.33	0.50 ± 0.09	33.5 ± 6.2	2822.5	2025.5
I99 (WDD1)	I99	3.81 ± 0.23	1.15 ± 0.35	0.57 ± 0.10	36.2 ± 6.8	2835.8	2022.9
I99 (WDD2)	I99	3.80 ± 0.22	1.09 ± 0.34	0.48 ± 0.08	32.3 ± 5.9	2831.6	2023.6
I99 (WDD3)	I99	3.80 ± 0.21	1.06 ± 0.33	0.44 ± 0.08	30.3 ± 5.4	2829.5	2024.1
I99 (CDD1)	I99	3.81 ± 0.22	1.17 ± 0.35	0.59 ± 0.12	37.1 ± 7.1	2836.6	2022.8
I99 (CDD2)	I99	3.81 ± 0.22	1.08 ± 0.34	0.45 ± 0.08	31.2 ± 5.7	2831.5	2023.7
M10 (W7)	I99	3.70 ± 0.23	1.06 ± 0.33	0.51 ± 0.09	33.9 ± 6.4	2819.2	2826.2
M10 (CDEF)	I99	3.66 ± 0.30	1.28 ± 0.43	1.23 ± 0.27	56.3 ± 13.7	2809.5	2026.9
M10 (CDDT)	I99	3.33 ± 0.23	1.17 ± 0.41	1.34 ± 0.29	57.2 ± 14.5	2806.4	2026.8
M10 (ODDT)	I99	3.71 ± 0.23	1.12 ± 0.34	0.61 ± 0.11	37.8 ± 7.2	2823.6	2025.1

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We calculated the projected gas mass of A3112 confined in the central 38'' region corresponding to the effective RGS aperture (Paper I) based on the best-fit normalization parameter in the single-temperature thermal model (apec) fit to the RGS spectra using Equation (7). The apec normalization of 1.28 × 10⁻² gives a projected gas mass of (4.80 ± 0.70) × 10¹¹ M_☉ within the 38'' (∼52 kpc) region. We also calculated a spherical gas mass of (4.06 ± 0.43) × 10¹¹ M_☉ within the 38'' (∼52 kpc) which is consistent with the projected mass obtained from the apec normalization. This spherical gas mass was calculated using the deprojection analysis and modeling of the XMM-Newton EPIC observations of A3112 (see Bulbul et al. 2010, 2012). Thus, a conversion factor 0.41 is applied to the SNe per 10¹² M_☉ derived from spectral fits to obtain the total number of SNe, N^SNe parameters, which are also reported in Table 2.

We also convert the R parameter obtained from snapec fits into the fractional contribution of SNe Ia to the total number of SN explosions as follows:

$$\begin{eqnarray} R^{{\rm Ia}}_{\%}&=\frac{\mbox{SNe Ia}}{\mbox{(SNe\ Ia + SNe\ cc)}} \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} &=\frac{R}{1+R}. \end{eqnarray} \tag{ 9 }$$

We find that 30.3% ± 5.4% to 37.1% ± 7.1% of the total SN explosions are SNe Ia for models that use W7, CDD, and WDD yields. Application of M10 CDDT and CDEF SN Ia models implies a significantly higher fraction of SN Ia explosions (∼57% ± 14.5%). The reason for this difference is that M10 CDDT, CDEF SN Ia models have lower Fe abundance compared to the W7, W70, CDD, WDD, and ODDT models. Since Fe is mainly produced by SN Ia explosions, a lower abundance of Fe in SN Ia models in M10 CDDT, CDEF yields results in a higher number of SN Ia explosions and thus higher SN Ia fraction. The fraction of SN Ia explosions inferred using the I99 CDD and WDD models reported in Table 2 are consistent with those in de Plaa et al. (2007) based on elements accessible to the EPIC detectors: Si, S, Ar, Ca, Fe, and Ni. However, de Plaa et al. (2007) find a lower SN Ia fraction for W7 SN Ia models.

The SN models we use in this work produce equally good fits to RGS data (see Table 2). Therefore, global fits to the currently available high-resolution X-ray observations of clusters of galaxies do not allow one to distinguish between different types of SN models in an individual cluster. The RGS first- and second-order spectral fits obtained using the I99 SN Ia (W7) yields and Salpeter-IMF-averaged I99 SN cc yields are shown in Figure 1. We also show a close-up view of the fit in the RGS Fe-L bandpass obtained using the T95 SN Ia (W7) and Salpeter-IMF-averaged T95 SN cc yields in Figure 2. Figure 2 shows that the T95 SN yields accurately reproduce the RGS spectrum in the Fe-L bandpass.

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We examined the EPIC observations of the cool core cluster A3112 (see Table 1) using version 11.0 of the SAS software. The EPIC data processing and background modeling were carried out with the XMM-Newton Extended Source Analysis Software (XMM-ESAS) and methods (Kuntz & Snowden 2008; Snowden et al. 2008) described in Paper I. Since the EPIC spectra can constrain Si, S, Fe, and O abundances only in the innermost 30'' region (see Paper I), we examined only the central 30'' spectra in this work. We fit the cluster emission using the absorbed snapec model allowing the absorption column density, N^SNe, R, redshift, kT_e, and normalization parameters freely vary. The best-fit parameters of the instrumental and cosmic X-ray background model components were fixed to the best-fit values obtained from the overall fits and reported in Paper I. A conversion factor of 0.41 was again applied to the SNe per 10¹² M_☉ derived from EPIC spectral fits to obtain the N^SNe parameter. The best-fit model parameters obtained from the snapec fits to EPIC spectra for various SN nucleosynthesis models are shown in Table 3. These final χ² values, reported in Table 3, were obtained from the best fit determined by C-statistics.

Table 3. The Best-fit Parameters of the snapec Model Obtained Using T95, I99, and M10 SN Yields and XMM-Newton EPIC Spectra

SNe Ia	SNe cc	kTe	NSNe	R	$R^{{\rm Ia}}_{\%}$	χ2
Model	Model	(keV)	(× 109)		(%)	(1683 dof)
T95 (W7)	T95	3.49 ± 0.02	1.06 ± 0.02	0.40 ± 0.01	28.6 ± 1.0	2737.7
I99 (W7)	I99	3.50 ± 0.02	1.06 ± 0.03	0.40 ± 0.01	28.6 ± 1.0	2737.8
I99 (W70)	I99	3.50 ± 0.02	1.06 ± 0.02	0.39 ± 0.01	28.1 ± 1.0	2723.9
I99 (WDD1)	I99	3.47 ± 0.02	0.82 ± 0.02	0.76 ± 0.03	43.2 ± 2.9	2893.6
I99 (WDD2)	I99	3.49 ± 0.02	0.89 ± 0.02	0.51 ± 0.01	33.8 ± 1.0	2783.9
I99 (WDD3)	I99	3.47 ± 0.02	0.91 ± 0.11	0.42 ± 0.02	29.6 ± 1.9	2749.8
I99 (CDD1)	I99	3.41 ± 0.01	0.82 ± 0.11	0.81 ± 0.03	44.8 ± 2.9	2927.6
I99 (CDD2)	I99	3.50 ± 0.02	0.87 ± 0.11	0.49 ± 0.02	32.9 ± 1.9	2767.1
M10 (W7)	I99	3.51 ± 0.02	1.06 ± 0.01	0.40 ± 0.01	28.6 ± 1.0	2744.1
M10 (CDEF)	I99	3.54 ± 0.02	1.36 ± 0.11	0.77 ± 0.03	43.5 ± 2.9	2775.8
M10 (CDDT)	I99	3.39 ± 0.02	1.11 ± 0.10	1.28 ± 0.07	56.1 ± 6.5	3627.5
M10 (ODDT)	I99	3.49 ± 0.02	0.83 ± 0.13	0.82 ± 0.04	45.1 ± 3.8	2842.3

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The temperature, SN ratio (R), and the total number of SN measurements obtained from EPIC spectra are consistent with RGS results (Table 2) at the 1σ level. The 1σ difference is due to different temperature measurements reported by EPIC and RGS observations. Most of the SN nucleosynthesis models we use in this work produce equally good fits to EPIC data as in the case of RGS analysis. High signal-to-noise CCD resolution EPIC spectra yield stringent constraints on the model parameters (e.g., SN ratio (R) and the total number of SNe) but cannot distinguish between different types of SN nucleosynthesis models. Maeda et al. (2010) CDDT SN Ia models yield a worse fit to the EPIC spectra compared to Iwamoto et al. (1999), Tsujimoto et al. (1995), or Maeda et al. (2010) W7, CDEF, ODDT SN Ia models.

We find that fits to the EPIC spectra find a slightly wider range of SN type Ia fraction, so that 28.1% ± 1.0% to 44.8% ± 2.9% of the total SN explosions are SNe Ia for models that use W7, CDD, and WDD nucleosynthesis yields. As in the case of the RGS analysis, the M10 CDEF, CDDT SN Ia models imply a significantly higher fraction of SN Ia explosions (43.5% ± 12.9% to 56.1% ± 6.5%) as a result of lower Fe abundance compared to the W7, W70, CDD, WDD models. We also found that all W7 models from several simulations (e.g., T95, I99, and M10) produce similar goodness of fit, the total number of SNe, and SN Ia fraction. The snapec fits to EPIC MOS1, MOS2, and PN spectra performed using the Iwamoto et al. (1999) W70 SN Ia and Salpeter-IMF-weighted Iwamoto et al. (1999) SN cc metal yields are shown in Figure 3.

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The Japan/U.S. Astro-H Observatory is scheduled to launch in 2014 and will carry an X-ray calorimeter, the SXS, with a high energy resolution of ∼4.5 eV (Takahashi et al. 2010). The detector will be able to determine line widths and metal abundances with high precision, revolutionizing our understanding of physical processes (e.g., turbulence, the contributions of SN explosions to metal enrichment) in the ICM. We simulated 200 ks SXS observations of A3112 using the X-ray events simulator software (simx 1.2.1), based on the temperature and abundance measurements obtained from the snapec fits convolved with I99 SN Ia (W7) and I99 Salpeter-IMF-weighted SN cc yields. simx uses predefined detector response matrix, effective area, and predicted background for the SXS instrument to produce an event file by convolution with the telescope's point spread function.⁴ The SXS spectrum is extracted using the FTOOL xselect based on the event file produced with the simx software.

We then fit the simulated SXS spectrum with the snapec model convolved with various SN Ia and SN cc yields for variable temperature, N^SNe, R, redshift, and normalization. Figures 4 and 5 show the soft X-ray band and a close-up view of the Fe-K bandpass for fits to the simulated SXS spectrum of A3112 obtained assuming the same SN yields used to create the spectrum. The relative contributions of SN Ia and SN cc products to the spectrum are shown as green and blue lines in Figures 4 and 5, respectively. The figures illustrate how the significant amounts of Fe and Mg produced by SN Ia explosions, and large quantities of O synthesized in SN cc explosions, are manifest in the spectrum. The best-fit parameters and goodness of fits obtained using the I99, T95, and M10 SN yields are shown in Table 4. The features from metals primarily produced by SN Ia explosions such as S, Ar, Ca, and Ni that were inaccessible to the EPIC and RGS now become important diagnostics for Astro-H data analysis. As a result of the accurate abundance measurements of these metals, uncertainties in the N^SNe and R parameters are reduced, and they may be determined more robustly and with higher precision (see Table 4).

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As expected, good fits are found using SN models that have similar elemental yields, e.g., using T95 (W7) or I99 (W70) SN Ia models, to those of the I99 (W7) SN Ia and SN cc yields adopted to produce the simulated Astro-H spectrum (see Table 4). On the other hand, the significantly higher χ² values obtained from fits with SN models that have markedly different yield patterns, e.g., I99 CDD and I99 WDD models, demonstrate that future Astro-H observations of galaxy clusters will allow one to distinguish between different SN models and provide fundamental constraints on SN nucleosynthesis.

Within its virial radius, a sufficiently massive cluster is approximately a closed box. For a global efficiency of converting gas to stars ε_sf, the total mass in stars formed is

$$\begin{equation} M_{*\rm form}=\varepsilon _{\rm sf}\, M_{\rm bary}, \end{equation} \tag{ 10 }$$

where M_bary is the total baryon mass. At the present time the total mass in stars, whether contained in individual cluster galaxies (including the BCG) or associated with intracluster light (ICL), is

$$\begin{equation} M_{*}=M_{\rm bary}\, \varepsilon _{\rm sf}\, (1-R_{*}), \end{equation} \tag{ 11 }$$

and the mass in gas is

$$\begin{eqnarray} M_{\rm gas}& = M_{\rm bary}-M_{*} \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} & =M_{\rm bary}\left[1-\varepsilon _{\rm sf}(1-R_{*})\right], \end{eqnarray} \tag{ 13 }$$

where R_* is the mass return fraction. We can safely neglect the distinction between the mass in gas and the mass in the ICM for a cluster as rich as A3112 where the gas mass in galaxies is relatively small, and henceforth equate M_gas with M_ICM. The star formation efficiency, in terms of the observable M_ICM/M_*,

$$\begin{equation} \varepsilon _{\rm sf}=(1-R_{*})^{-1}(1+M_{\rm ICM}/M_{*})^{-1}. \end{equation} \tag{ 14 }$$

The total number of SN cc and Type Ia explosions can be expressed as

$$\begin{equation} N^{{\rm cc}}=\eta ^{{\rm cc}}\, M_{*\rm form} \end{equation} \tag{ 15 }$$

and

$$\begin{equation} N^{{\rm Ia}}=\eta ^{{\rm Ia}}M_{*\rm form}, \end{equation} \tag{ 16 }$$

where η^cc and η^Ia are, respectively, the specific numbers of SN cc and SN Ia explosions per star formed. Connecting η^cc and η^Ia to the snapec model parameters defined in Section 2,

$$\begin{equation} R=\frac{\eta ^{{\rm Ia}}}{\eta ^{{\rm cc}}}, \end{equation} \tag{ 17 }$$

and

$$\begin{equation} N^{{\rm SNe}}=\eta (1-R_{*})^{-1}M_{*}, \end{equation} \tag{ 18 }$$

where η = η^II + η^Ia. We note that these refer to all of the SNe while in Section 2; N^SNe and R parameters refer only to SNe that enrich the ICM.

Adopting a "diet Salpeter" IMF that produces relatively fewer low-mass stars (Bell & de Jong 2001), Maoz & Mannucci (2011) estimated η^Ia to be ∼0.002. Adopting the mass return fraction and specific SN cc rate for the diet Salpeter IMF: R_* ∼ 0.35 (Fardal et al. 2007; O'Rourke et al. 2011), η^Ia ∼ 0.002, and η^II ∼ 0.008 (Maoz & Mannucci 2011; Botticella et al. 2012), one predicts R ∼ 0.25, which corresponds to $R^{{\rm Ia}}_{\%}=0.2$, and η ∼ 0.01. For clusters in the A3112 mass range (e.g., M₅₀₀ ∼ 3.0 × 10¹⁴ M_☉, Paper I), recent analysis finds, typically, M_*/M_ICM ∼ 0.1 (Balogh et al. 2011; Lin et al. 2012). Therefore, from Equation (18), the estimated total number of SN explosions per 10¹² M_☉ of ICM is ∼1.54 × 10⁹. Multiplying by the correction factor of 0.41 appropriate to the core of A3112, N^SNe becomes ∼6.3 × 10⁸. Comparing with our results in Table 2, we may conclude that the inner 52 kpc of A3112 has been enriched by more SNe, with a higher percentage of SNe Ia, than is true globally for a typical cluster of its mass.

The galactic mass in clusters is dominated by early-type galaxies that form their stars quickly. This results in the well-established enhancement in [α/Fe], the abundance ratio of α-elements to Fe (expressed as the logarithm with respect to solar)—i.e., SNe cc are preferentially locked up in stars (Loewenstein 2004). In their investigation of the giant elliptical galaxy NGC 4472, Loewenstein & Davis (2010) found that a ratio of SNe Ia to total SNe of N^Ia*/N^SNe* ∼ 0.11 (N^cc*/N^SNe* ∼ 0.89) and a number of SNe per mass in (present-day) stars of N^SNe/M_* ∼ 0.0083 resulted in [α/Fe]_* ∼ 0.25 and Z_Fe* ∼ 1 (as observed in this particular galaxy, but typical of the class) assuming yields from Kobayashi et al. (2006). This enables us to estimate the lock-up corrections, η^cc* and η^Ia*, to the total specific numbers of SNe per star formed. This provides, in turn, the corresponding values available to enrich the ICM, η^Ia_ICM and η^cc_ICM:

$$\begin{eqnarray} \eta ^{{\rm Ia*}}=&\,\left(\frac{N^{{\rm Ia*}}}{M_*}\right)(1-R_{*}) \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} =&\,6.0\times 10^{-4}Z_{{\rm Fe*}}, \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} \eta ^{{\rm cc*}}=&\,\left(\frac{N^{{\rm cc*}}}{M_{*}}\right)(1-R_{*}) \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} =& 4.8\times 10^{-3}Z_{{\rm Fe*}}, \end{eqnarray} \tag{ 22 }$$

$$\begin{equation} \eta ^{{\rm Ia}}_{{\rm ICM}}=\eta ^{{\rm Ia}}-\eta ^{{\rm Ia*}}=1.4\times 10^{-3} \,{M}_{\odot }^{-1}, \end{equation} \tag{ 23 }$$

and

$$\begin{equation} \eta ^{{\rm cc}}_{{\rm ICM}}=\eta ^{{\rm cc}}-\eta ^{{\rm cc*}}=3.2\times 10^{-3} \,{M}_{\odot }^{-1}. \end{equation} \tag{ 24 }$$

That is, the metal production from ∼60% of SNe cc, and ∼30% of SNe Ia, must be locked up in stars to enrich them to solar Fe abundances and [α/Fe]_* ∼ 0.25. This results in revised predictions of R ∼ 0.30—consistent with our RGS results for A3112, but an even lower expected total number of SNe—N^SNe ∼ 2.9 × 10⁸ SNe.

Sand et al. (2012) recently estimated a specific SN Ia rate in cluster galaxies of r_SNIa ∼ 0.04 SNuM⁵ within R₂₀₀ (∼1 Mpc) (see also Maoz et al. 2010) that may be directly injected into the ICM by the r < 52 kpc stellar population. The implied level of SNIa available for enrichment accumulated over time τ is

$$\begin{equation} N^{{\rm Ia}}=10^{-4}M_{*}({<}{\mathrm 52} \,{\rm kpc})\left(\frac{r_{{\rm SNIa}}}{0.1\; {\rm SNuM}}\right)\left(\frac{\tau }{10^9\,{\rm yr}}\right), \end{equation} \tag{ 25 }$$

where the stellar mass M_* refers to the r < 52 kpc region, and r_SNIa should now be interpreted as an average over τ. We use the Hernquist approximation (Hernquist 1990) to deVaucouleurs profiles for the A3112 BCG and ICL components and estimate M_* = 3.6 × 10¹¹ M_☉ within 52 kpc—i.e., M_* = 0.88M_ICM. A large star-to-gas ratio, M_* ∼ M_ICM, and a long timescale τ > 10⁹ yr (with the accompanying increase in r_SNIa) would be required for this to be a significant source of enrichment.

For most of the yield sets we consider, the level of ICM enrichment for the central (4.06 ± 0.70) × 10¹¹ M_☉ requires ∼1.1 × 10⁹ SNe, with SNe Ia accounting for ∼ one-third of the total. Although our SN parameters are derived for the inner 52 kpc, the high gas mass and temperature imply that this gas is primarily intracluster in origin. If we account for the fraction of SN products locked up in cluster galaxy stars, which is higher for SNe cc based on the measured stellar [α/Fe] enhancement, we can explain the proportions of SNe Ia and SNe cc that we infer enriched the ICM. However, the total number of SNe we infer is greater than expected by a factor of ∼4 for a typical star-to-gas ratio, M_*/M_ICM ∼ 0.1. These conclusions are altered by direct injection of SN Ia ejecta in the BCG only if accumulated over a large fraction of a Hubble time and are exacerbated for the M10 CDEF and CDDT models. Evidently, the stellar population responsible for enriching this material is characterized by enhanced efficiency in producing SNe, and/or the mass in stars responsible for the enrichment, M_* > 0.3M_ICM.

Larger estimates of M_*/M_ICM ∼ 0.2 at the A3112 mass range may be found in Gonzalez et al. (2007; see also Langaná et al. 2011). Indeed A3112 was investigated by Gonzalez et al. (2007), from which we may derive M_*/M_ICM ∼ 0.27. The implication is that there is substantial ICL component unaccounted for in other studies (see also Bregman et al. 2010) and that the star formation efficiency is very high (Equation (14)).

Even if of intracluster origin, the enrichment of this inner cluster gas is likely to reflect the somewhat special conditions in the cluster core where high initial overdensity may result in enhanced efficiency of galaxy and star formation. Thus, the effective value of M_*/M_ICM may very well be larger than is representative of the ICM as a whole. At the same time, for rich galaxy clusters in general, the global ICM shows enrichment beyond what is expected based on the stars we see in galaxies today (Portinari et al. 2004; Loewenstein 2006; Maoz et al. 2010; Bregman et al. 2010).

Another possible mechanism contributing to the greater than expected number of SNe found in our fits is strong early enrichment of the lowest-entropy X-ray-emitting gas stripped from early galaxies. The heavier metals found in gas stripped at large radii will settle into the cluster core due to the central gravitational potential, while the stars that produced them remain at larger radii. Therefore, a significant fraction of metals residing in the cluster core that we observe today may not have originated from the local stellar population (Million et al. 2011). In this scenario, the apparent total expected number of SNe will be enhanced relative to that inferred from M_*/M_ICM. A more extensive study of a larger sample is needed to fully address this issue.