X-RAY AND OPTICAL FLUX RATIO ANOMALIES IN QUADRUPLY LENSED QUASARS. II. MAPPING THE DARK MATTER CONTENT IN ELLIPTICAL GALAXIES

As mentioned above, the aim of the SER algorithms is to improve the spatial resolution of the X-ray image. They do this by using only split-pixel events so there is necessarily a loss of signal, and this needs to be weighed against the improved resolution. To quantify the signal loss, we measure the number of 0.3–8 keV counts in the standard image, SER image, and SER-CO image in a 63 square region around the lensed quasars. The distributions of the fraction of counts in the SER and SER-CO images compared to the standard image are shown in Figure 2. The SER algorithm is able to utilize about 75% of the events on average, while the SER-CO image can utilize only about 25%.

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The advantage of the SER algorithms is demonstrated in Figure 3, which shows a standard image of PG 1115+080 made at the default resolution along with the standard, SER, and SER-CO images made at 00492 pixel⁻¹. The blended close pair in the standard image is separated into two distinct peaks in the SER image.

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This improvement in resolution can be quantified by the width parameters of the Gaussian and β models. Using the values from all observations of all systems, we find that the best-fit Gaussian FWHM is about 006 smaller on average in the SER images compared to the standard images. It is only about another 002 smaller on average in the SER-CO images. The best-fit β-model r₀ is about 012 smaller on average in the SER images compared to the standard images, but also only about another 002 smaller on average in the SER-CO images. The results are nearly identical for the βc model: 012 smaller in the SER images and only another 002 smaller in the SER-CO images. This small gain in resolution of ∼002 of the SER-CO images comes at a large price in signal loss.

We compared the best-fit amplitudes of a specific model in the standard, SER, and SER-CO images and found reasonably good agreement among all four models. There tended to be more outliers in the SER-CO image fits, likely due to decreased signal, but most amplitudes agreed within 1σ uncertainties. This good agreement is reassuring, but the real aim of exploring the SER and SER-CO images is the possibility of better constraining the amplitudes, i.e., reducing the uncertainty in the best-fit model amplitudes.

In all cases, the model fits to the SER-CO have larger amplitude errors on average, again most likely due to the large reduction in signal inherent in using that algorithm. In the Gaussian, β-, and βc-model fits, the amplitude errors in the SER fits and the standard fits are comparable (13% for SER versus 12% for standard in the Gaussian fits and 15% versus 13% in both the β- and βc-model fits), whereas the amplitude errors in the SER fits are somewhat smaller than those in the standard fits with the PSF model (12% versus 17%).

The most important feature of a model is how well it represents the data. To explore this for the four models, we show histograms of the reduced "cstat" statistic in Figure 4. The mean values are almost identical for all models, but the Gaussian ones tend to have tails to higher values of the reduced statistic than the others.

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Another way to visualize the goodness of the image fit is by comparing the radial profiles of the data with each of the models. To illustrate this, we choose the observation that has the highest number of counts, which is ObsID 9181 of RX J1131−1231. The radial profile of each quasar image from the center to 05 is shown in Figure 5. Overlaid are the radial profiles from the best-fit models of each type. The Gaussian models are universally too squat, and the PSF models are often a poor fit in the center. The β- and βc-model profiles do the best job of matching the data at all radii. Note that these are not fits to the radial profiles; rather, they are radial profiles of the best-fit models overlaid on radial profiles of the data.

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Considering the fit statistics and radial profiles, the β and βc models appear to be the best choices to represent the data. Given the tight correlation seen in Figure 1, it is not surprising that both models give nearly identical results. We have a slight preference for the βc models because of the reduction in free parameters.

Although the SER-CO fits have smaller reduced statistics on average, they have larger uncertainties and more outliers and are therefore not the ideal choice. Between the standard and SER images, the uncertainties are comparable, as are the reduced statistics. We favor the SER images for the task at hand because we believe that the increase in effective spatial resolution will provide higher fidelity results for the systems where the separation between quasar images is far less than 1''.

The results from the βc-model fits to the SER images are given in Table 1. Each amplitude and uncertainty is reported as a fraction of the total, defined as the sum of the four amplitudes. In many cases the amplitude errors are asymmetric.

The four images of each quasar are either saddle points or minima of the light travel time surface. We denote the higher magnification minimum as the "HM" image and the lower magnification minimum as the "LM" image. Likewise for the saddle point images, the higher magnification saddle point is "HS", and the lower magnification saddle point is "LS." We have previously modeled all of these lens systems (Pooley et al. 2007; Blackburne et al. 2011) to determine the local convergence κ and shear γ for each of these images, which also gives the "macrolensing" magnification of each image. These parameters given in Table 2 are provided by the models presented in Blackburne et al. (2011).

These large-scale lens models can give only the total κ at the site of each image without regard to the form of the matter present. We generate a series of 12 custom microlensing maps for each image by assuming that some fraction of κ is in a clumpy component (stars) and the rest is in a smooth component (dark matter). We use a logarithmic sequence of stellar fractions (S_j): 1.47%, 2.15%, 3.16%, 4.64%, 6.81%, 10%, 14.68%, 21.5%, 31.62%, 46.4%, 68.13%, and 100%.

In total, 672 microlensing maps were produced using the "microlens" ray-tracing code (Wambsganss 1990, 1999; Wambsganss et al. 1990). These magnification maps are constructed in the source plane, and their centers are referenced to the location of one of the quasar images. They show the effects of microlensing magnification (due to the sum of all the microimages) for a source location anywhere within the map. The mean macrolensing magnification, due to the smooth lensing potential, has been subtracted off. Each map is 2000 × 2000 pixels, with an outer scale of 20 r_Ein and a pixel size of 0.01 r_Ein, where r_Ein is the Einstein radius of a microlensing star of average mass. The stars are drawn from a mass function similar to the well-known one of Kroupa (2001). The mass function runs from 0.08 M_☉ to 1.5 M_☉ with a break at 0.5 M_☉ and logarithmic slopes of −1.8 and −2.7 below and above the break, respectively. The average mass of a microlensing star is 0.247 M_☉, and the stellar mass above and below which 50% of the mass lies is 0.335 M_☉.

Figure 6 shows portions of each of the four microlensing maps (HM, HS, LM, and LS) produced for PG 1115+080 for stellar fractions of both 10% and 100% to illustrate the differences among the microlensing maps. For each map, a histogram of the logarithm of the magnification values is made and normalized, and this is used as the probability distribution for microlensing effects P(μ_{i, j}|S_j) where μ_{i, j} = log₁₀(micromag_{i, j}), i ∈ {HM, HS, LM, LS}, and S_j is one of the stellar fractions listed above. For convenience, we also define m_{i, j} = −μ_{i, j}. These normalized histograms are shown in the bottom panels of Figure 6.

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Our goal is to determine the probability of each stellar fraction S_j for a lensing galaxy. Our measurements of the X-ray fluxes of the four images divided by their respective macrolensing magnifications give four estimates of the intrinsic flux F_{X, intr} of the quasar. We use conditional probability to express P(S_j) as

$$\begin{equation} P(S_j) = \sum _\mathcal {X}P(S_j | \mathcal {X}_\mathrm{HM},\mathcal {X}_\mathrm{HS},\mathcal {X}_\mathrm{LM},\mathcal {X}_\mathrm{LS}) P(\mathcal {X}), \end{equation} \tag{ 2 }$$

where $\mathcal {X}= \log _{10}(F_\mathrm{X,intr} / F_\mathrm{norm})$ and $\mathcal {X}_i$ indicates the estimate of $\mathcal {X}$ from image i. We choose F_norm = 10⁻¹⁴ erg cm⁻² s⁻¹, which has no effect on the analysis.

We use Bayes's theorem to express

$$\begin{eqnarray} &&\textit {P}(\textit {S}_j | \mathcal {X}_\mathrm{HM},\mathcal {X}_\mathrm{HS},\mathcal {X}_\mathrm{LM},\mathcal {X}_\mathrm{LS}) \nonumber\\ &&\quad= \frac{P(\mathcal {X}_\mathrm{HM},\mathcal {X}_\mathrm{HS},\mathcal {X}_\mathrm{LM},\mathcal {X}_\mathrm{LS}|S_j)P_\mathrm{pr}(S_j)}{\sum _j P(\mathcal {X}_\mathrm{HM},\mathcal {X}_\mathrm{HS},\mathcal {X}_\mathrm{LM},\mathcal {X}_\mathrm{LS}|S_j)P_\mathrm{pr}(S_j)}\qquad\quad \end{eqnarray} \tag{ 3 }$$

where P_pr(S_j) is the a priori probability of S_j and the denominator is a normalization term. We take P_pr(S_j) to be uniform and combine it with the denominator as the constant A in what follows. We compute it by ensuring that ∑_jP(S_j) = 1.

Because the four $\mathcal {X}_i$ are physically distinct, their probabilities are independent from each other, and we can express

$$\begin{equation} P(\mathcal {X}_\mathrm{HM},\mathcal {X}_\mathrm{HS},\mathcal {X}_\mathrm{LM},\mathcal {X}_\mathrm{LS}|S_j) = \prod _i P(\mathcal {X}_i|S_j). \end{equation} \tag{ 4 }$$

Substituting Equations (3) and (4) into Equation (2), we arrive at

$$\begin{equation} P(S_j) = A \sum _\mathcal {X}\prod _i P(\mathcal {X}_i|S_j) P(\mathcal {X}) \end{equation} \tag{ 5 }$$

and what remains is to calculate $P(\mathcal {X}_i|S_j)$ and $P(\mathcal {X})$.

The probability of the intrinsic flux of a quasar $P(\mathcal {X})$ can be determined from the number counts obtained from deep studies of the X-ray background. We use the results from Giacconi et al. (2001) that $N(>\mbox{$F_{\rm X}$}) \sim \mbox{$F_{\rm X}$}^{-0.85}$, but we note this has little impact on the analysis. A uniform distribution would produce nearly identical results.

We estimate the intrinsic flux F_{X, intr} from the measured flux (f_{X, i}) of an image and the lensing effects, both macrolensing and microlensing. The measured flux is

$$\begin{equation} f_{\mathrm{X},i} = F_\mathrm{X,intr} \times \mathcal {M}_i \times 10^{\mu _{i,j}}, \end{equation} \tag{ 6 }$$

where $\mathcal {M}_i$ is the macro-magnification of image i. We define

$$\begin{equation} x_i = \log _{10}([f_{\mathrm{X},i}/F_\mathrm{norm}]{/}\mathcal {M}_i) \end{equation} \tag{ 7 }$$

which allows us to write

$$\begin{equation} \mathcal {X}_i = x_i + m_{i,j}. \end{equation} \tag{ 8 }$$

Because the probability of the sum of two random variables is the convolution of their individual probabilities, we can express

$$\begin{eqnarray} P(\mathcal {X}_i|S_j) & =& P(x_i + m_{i,j} | S_j)= P(x_i) * P(m_{i,j}|S_j), \end{eqnarray} \tag{ 9 }$$

where P(x_i) comes from the uncertainties on the flux measurements of the images and the P(m_{i, j}|S_j) are the reverse of P(μ_{i, j}|S_j), the normalized histograms of the microlensing maps, as discussed above.

We do not measure the f_{X, i} directly, though; rather, we obtain them by multiplying the total flux of all four images (via spectral fitting) and the individual fractions of the total (via two-dimensional image fitting). We define

$$\begin{equation} T = \log _{10}(F_\mathrm{X,tot}/F_\mathrm{norm}) \end{equation} \tag{ 10 }$$

and

$$\begin{equation} r_i = \log _{10}(\mathrm{frac}_i / \mathcal {M}_i) \end{equation} \tag{ 11 }$$

so that

$$\begin{equation} x_i = r_i + T. \end{equation} \tag{ 12 }$$

Again, using the property of the sum of two random variables, we express

$$\begin{eqnarray} P(x_i) & = P(r_i + T) = P(r_i) * P(T), \end{eqnarray} \tag{ 13 }$$

where the probability distributions r_i are asymmetric Gaussians with standard deviations equal to the 1σ uncertainties in the image fractions (Table 1) and T is a symmetric Gaussian with a standard deviation equal to the uncertainty in f_{X, tot} (Table 1). Introducing notation G_j and using Equations (9) and (13), we have

$$\begin{eqnarray} G_j & =\prod _i P(\mathcal {X}_i|S_j)\nonumber \\ & = \prod _i P(r_i) * P(T) * P(m_{i,j}|S_j) \end{eqnarray} \tag{ 14 }$$

which can be seen in the bottom panels of Figure 6 using values from ObsID 363.

All of the above has been worked out for a single observation of a system, but several systems have been observed multiple times with Chandra. We combine these multiple observations using conditional probability:

$$\begin{equation} P(S_j) = \sum _k P(S_j|\mathrm{obs}_k) P(\mathrm{obs}_k), \end{equation} \tag{ 15 }$$

where we take P(obs_k) as a weighting factor (normalized to unity) that combines two measures of the effectiveness of the observation to provide unique and useful information.

The first ingredient in P(obs_k) concerns the uniqueness of the information from the observation. Over time, the proper motions of the lensing galaxy and background quasar, as well as the internal motions of the microlensing stars, can be thought of as an effective motion of the source through the field of the microlensing map (Wyithe et al. 2000). The more time between observations, the higher the chance that the source is in a different enough region of the map to be considered an independent sampling of it. We therefore include a term in P(obs_k) proportional to how isolated in time the observation is, defined as the sum of the intervals between the observation and all other observations.

The second ingredient in P(obs_k) is based on the quality of the information that the observation provides. Observations which yield tight constraints on the individual fractions and the total flux consequently give much better defined probability functions for the stellar fraction (we point out specific examples below). We use the measured uncertainties (Table 1) on the fractions (symmetrized) and the total flux to calculate this. Our full expression is

$$\begin{equation} P(\mathrm{obs}_k) = B\left(\sum _{l\ne k} \left|t_k - t_l\right| \right) \ \prod _i \frac{\mathrm{frac}_{i,k}}{\sigma _{\mathrm{frac}_{i,k}}}\ \frac{F_{\mathrm{X,tot},k}}{\sigma _{F_{\mathrm{X,tot},k}}}, \end{equation} \tag{ 16 }$$

where t_k is the epoch of observation k and B is a normalization constant such that ∑_kP(obs_k) = 1.

Using Equations (9), (13), and (15), we can express Equation (5) in terms of observables, the microlensing magnification map histograms, and the intrinsic flux probability from the deep field quasar number counts:

$$\begin{equation} P(S_j) = \sum _k A_k \left(\sum _\mathcal {X}G_{j,k} P(\mathcal {X})\right) P(\mathrm{obs}_k). \end{equation} \tag{ 17 }$$

These probabilities are plotted for each of the 14 lensing galaxies in Figure 7 as functions of the stellar mass fraction.

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The effect of poorly constrained image fluxes is easily seen in the nearly flat probability distribution of SDSS 1138+0314, which has only one Chandra observation, in which the average uncertainty of the image fractions is ∼70%. Compare this to HE 0230−2130, which also has only one Chandra observation, but in which the average image fraction uncertainty is ∼20%.

We would like to consider each lensing galaxy as a typical member of an ensemble, each with roughly the same configuration such that we are probing the matter content at roughly the same radial distance R from the center of the lensing galaxy. To calculate these distances in physical units, we use the angular measurements of the images and galaxies available on the CASTLES Web site⁸ along with the redshifts to the lensing galaxies, z_l.

We take the arithmetic mean of the four impact parameters where the images form in the lensing galaxy as a characteristic radial distance, R_c. Most of the systems have R_c within a factor of a few of each other except for Q 2237+0305, in which the images form at a mean R_c of 0.7 kpc, about an order of magnitude less than the mean R_c of 6.6 kpc of the other systems (see Figure 8). We exclude Q 2237+0305 from the rest of the analysis. We note that, had we used the geometric means instead, the numbers would be very similar. The images in Q 2237+0305 form at a geometric mean radial distance of 0.7 kpc, and the geometric mean of the radial distances for the rest of the ensemble is 6.1 kpc.

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Unfortunately, z_l is not known for HE 1113−0641, H 1413+117, and WFI J2026−4536 so R cannot be calculated for the images of these systems. There have been indications of lenses of H 1413+117 at redshifts of 0.8, 1.4, and 1.7 (Magain et al. 1988; Kneib et al. 1998; Faure et al. 2004). Morgan et al. (2004) estimate a redshift of 0.4 for the lensing galaxy of WFI J2026−4536, and Blackburne et al. (2008) estimate z_l = 0.7 for HE 1113−0641. These values are all comparable to the redshifts of the other lensing galaxies, unlike Q 2237+0305 with z_l = 0.04, and it is reasonable to assume that their impact parameters are also comparable. We therefore include them in the joint analysis. The radial distances of the images in these systems are shown with outlined symbols in the top of Figure 8 assuming z_l of 0.7, 0.8, and 0.4 for HE 1113−0641, H 1413+117, and WFI J2026−4536, respectively; they are not included in the histogram in the bottom panel of Figure 8.

For the 10 systems other than Q 2237+0305 with known z_l, their mean impact parameters R_c are within a factor of 2.5 of each other. If we consider all individual R in these 10 systems, the spread is nominally a factor of 14. Excluding the two extrema (the LS image in B 1422+231 at R = 1.2 kpc and the LM image in RX J0911+0551 at R = 17 kpc), the spread in R among the 10 systems with known z_l is only a factor of 3.2. As we discuss in Section 6.2, this is a small enough range in R_c and R that the ratio of stellar matter to dark matter is expected to vary by only 1.6 over this interval, and we feel comfortable combining the individual results to obtain an ensemble result for a mean R_c of 〈R_c〉 = 6.6 kpc.

We form the joint probability function of the ensemble—excluding Q 2237+0305—by multiplying together the individual probability functions of the 13 lensing galaxies (shown in Figure 7), and normalizing. The results are displayed in Figure 9, which shows the joint probability distribution of the ensemble for the percentage of matter in stars at a mean impact parameter of 6.6 kpc. The highest peak of this discrete distribution occurs at 6.8% stellar matter (93.2% dark matter), and the interpolated peak occurs at 6.3% ± 0.3% stellar matter (93.7% dark matter).

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When exploring the results of the individual observations shown in Figure 7, we noticed that RX J1131−1231 had one of the highest probabilities for a 100% stellar fraction, after Q 2237+0305. This is surprising given that the first Chandra observation of RX J1131−1231 displayed a strong signature of significant dark matter presence: a highly suppressed saddle point image (Blackburne et al. 2006). We examined the RX J1131−1231 probabilities on an observation by observation basis and found that, indeed, the first observation (ObsID 4814) strongly favored a stellar fraction of 22%. The other observations favored higher stellar fractions, either with a roughly flat distribution above 22% stars or a strong peak at 100% stars.

We noticed a correlation that those observations with a flat distribution above 22% were the ones that had lower LS fractions, and the handful of observations (six) that peaked at 100% stars were the ones with an LS fraction >0.05 in Table 1. We separately analyzed these two groups, and the results are shown, along with ObsID 4814, in Figure 10.

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RX J1131−1231 has the most X-ray observations and displays interesting behavior. The HS image evolved from being strongly demagnified by microlensing to being strongly magnified by microlensing, and the LS image shows microlensing variations of over a factor of two. It may be that our snapshot analysis is not appropriate for such complex behavior. Our treatment of each observation separately and combination of their weighted results discards information on temporal evolution. An analysis that assesses the probability of a certain stellar fraction, S_j, to produce the entirety of the observations, similar to the Bayesian III method in Pooley et al. (2009), may be more appropriate but is beyond the scope of this work.

Although we have some minor concerns about the robustness of the RX J1131−1231 stellar fraction probabilities, their effect on our joint analysis shown in Figure 9 is minor. If we perform the analysis without RX J1131−1231, we see that the most probable stellar fraction is slightly lower (the interpolated peak is at 4.6% ± 0.2%), and the probability of 100% stars is near zero (Figure 11).

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Our measurements of the stellar mass fraction pertain to the impact parameter, R, that the quad images make with respect to the lensing galaxy and span a range of ∼3–11 kpc. For any given impact parameter, R, a large range of radial distances in three dimensions (i.e., for all r > R) is probed within the lensing galaxy. Since the percentage of mass in stars is expected to decrease with increasing r, we would like to ascertain whether our ensemble average likelihood distribution for the dark matter fraction (see Figure 9) is well defined, or whether we should expect to see a decreasing progression of star fraction with increasing mean impact parameter, R_c.

Following Koopmans et al. (2009) and Schwab et al. (2010), we express the three-dimensional light density in an elliptical galaxy as

$$\begin{equation} I(r) = I_{\rm S0} \left(\frac{r}{r_0}\right)^{-\delta }, \end{equation} \tag{ 18 }$$

where I_S0 and r₀ are constants for a given galaxy, and δ is a more nearly universal constant which Schwab et al. (2010) determined to be

$$\begin{equation} \delta = 2.4 \pm 0.11 \end{equation} \tag{ 19 }$$

based on 54 lenses from the SLACS survey (e.g., Treu et al. 2006; Koopmans 2006; Gavazzi et al. 2007). The value of 0.11 is supposed to represent the rms variation in δ among different galaxies, rather than an uncertainty in the mean value of δ. We assume that this power law holds over the radial interval r ≃ 1–10 kpc. We also assume that the stellar mass function and evolutionary states of the stars are distance independent, so that I(r) also represents the stellar mass density.

Similarly, Koopmans et al. (2009) and Schwab et al. (2010) took the total mass density to be of the form

$$\begin{equation} \rho (r) = \rho _0\left(\frac{r}{r_0}\right)^{-\alpha } \end{equation} \tag{ 20 }$$

with constants that are analogous to those in Equation (4); α is found to be

$$\begin{equation} \alpha = 1.96 \pm 0.08 \end{equation} \tag{ 21 }$$

again based on the SLACS survey (e.g., Treu et al. 2006; Koopmans 2006; Gavazzi et al. 2007; Koopmans et al. 2009; Schwab et al. 2010). Similarly, the value of 0.08 is supposed to represent an rms variation from galaxy to galaxy, rather than an uncertainty in the mean. In this expression, ρ represents both the dark matter and stellar contributions to the mass density.

The observations determine the most probable stellar fraction S, which is the fraction in stars of the total column density C_tot along the line of sight at impact parameter R. We can integrate expressions (18) and (20) to obtain

$$\begin{equation} \frac{C_\mathrm{stars}}{C_\mathrm{tot}} = \frac{I_{\rm S0}}{\rho _0} \frac{\Gamma ((\delta -1)/2)}{\Gamma ((\alpha -1)/2)}\frac{\Gamma (\alpha /2)}{\Gamma (\delta /2)} \left(\frac{r_0}{R}\right)^{\delta - \alpha }, \end{equation} \tag{ 22 }$$

where the only radial dependence is in the final factor, R^{α − δ}. For the nominal values for δ and α listed above, this reduces to

$$\begin{equation} S(R) = \frac{C_\mathrm{stars}}{C_\mathrm{tot}} \simeq 0.77\, \frac{I_{\rm S0}}{\rho _0} \left(\frac{r_0}{R}\right)^{0.44}. \end{equation} \tag{ 23 }$$

Therefore, for a range of mean impact parameters from R_c = 3.9 to 9.5 kpc, we expect the stellar fraction S to vary by only a factor of ∼1.5 due to the dependence on the impact parameter. Given that this is roughly the resolution of our logarithmic grid of a dozen values of S and that our sample is modest in size, we would not expect our results to be sensitive to the range in R_c.

Nevertheless, we divided the 10 lens with known z_l into two groups: those with R_c < 7 kpc and those with R_c ⩾ 7 kpc. The second group initially contained RX J1131−1231, but we removed it from the following analysis given the issues discussed in Section 6.1. We ran a joint analysis separately on these two groups, and the results are shown in Figure 12. As expected, the group with smaller 〈R_c〉 has a most probable stellar fraction S that is larger than the group with larger 〈R_c〉.

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To see how well this result agrees quantitatively with Equation (23), we use a fitting function to determine the precise location of the peak of the probability distribution of each group. The first group, with 〈R_c〉 = 4.9 kpc, peaks at S₁ = 5.9% ± 0.5%. The second group, with 〈R_c〉 = 8.4 kpc, peaks at S₂ = 4.5% ± 0.2%. Based on the impact parameter ratio of 1.7, Equation (23) predicts a stellar fraction ratio of 1.3. Somewhat remarkably, given the modest size of our samples, S₁/S₂ = 1.3 ± 0.13.

We also note the most likely stellar fraction of 100% for Q 2237+0305 (see Figure 7). The lensing galaxy in this system is much closer than the others at z_l = 0.04 and consequently has a much smaller R_c of 0.7 kpc. It is the only system with a most likely stellar fraction of 100%, and this is in qualitative agreement with expectations. Quantitatively, it is larger than suggested by Equation (23), but this may be due to either of Equations (18) or (20) not being valid at such a small impact parameter.