% Dear Dr Mulchaey: We thank the referee for a constructive and helpful report on our paper (ApJ MS#75213). We apologize for not responding sooner due to professional constraints as well as the recent birth of a child. Below please find our response the report and a description of the changes we have made. These include changes suggested by the referee (listed below under part 1), as well as some extra improvements (listed below under part 2). We trust that the revised paper can be accepted for prompt publication. Shardha Jogee & co-authors ************************************************************************** Part 1: Changes made to the paper in response to the referee's report The points of the referee have been labeled P1 P2, etc for easy referencing. === Major issues: P1. > The aim of the paper is to identify candidates for a recent merger of mass > ratio M1/M2>1/10, so involving 2 (or more) interacting galaxies/components. > Reading the paper, and looking at the examples proposed in Fig. 2, I > understand that in the majority of the cases the single galaxies/components > are not resolved in the multiband > photometry used to measure mass, or in l=2800 COMBO-17 band used to measure > UV-SFR and in the 24micron photometry. Hence just one mass and one estimate > of the SFR for the whole system are available. Is this correct? On the > contrary, for how many interacting systems both colliding components are > photometrically resolved and have distinct mass measurements (like case 1 in > Fig. 2)? How these "resolved" cases > are treated in Fig. 1, 12 and 17? Are the different components shown > separately? I think that for homogeneity with "unresolved cases" in Fig. 1 > they should be represented by a single point, with the mass being the sum of > the two masses, the color being the some luminosity-weighted average of the > two colors. How the "resolved" cases are treated when counting the > interacting systems, deriving the galaxy interaction statistics? For > example, I think that the 2 > galaxies in panel 1 of figure 2 should count just as one interacting system. > Moreover, how these cases are treated when studying the contribution of > mergers on the star formation? In my opinion, the sum of the star formation > of the two galaxies must be taken, to be consistent to cases in which the > two interacting systems are not resolved, and one total star formation is > available for the whole system. Do the authors agree on that? What about > potential mergers in > which the two components have two distinct mass measurements, both below the > limit of 2.5x10^10 Msun, but for which M1+M2>2.5x10^10 Msun? They should be > included in the sample, I think. Summarizing, it is not clear to me how > photometrically "resolved" and "unresolved" cases are > treated throughout the paper, and if in that respect the sample of merging > candidates is homogeneous. The referee raises here important issues related to the treatment of interacting galaxies and mergers. We address this by first explaining in detail below the classification system, and then answering each point made by the referee. The sections 3.1 and 3.2 have also been edited to clarify the classification system along the lines below. The visual classification system we adopted for identifying interacting galaxies was aimed at setting a procedural framework that allows merger fractions and rates from observations and the theoretical models (outlined in section 4.5) to be defined in similar ways and to be readily compared. Specifically: 1) The theoretical models (HOD, SAM, SPH) track galaxies with a stellar mass M*>Mcut, which have experienced a merger of relevant mass ratio M1/M2 within the last visibility timescale t_vis, between times (t_obs-t_vis) and t_obs, where - t_obs is the time corresponding to the observed redshift z. - t_vis is the timescale over which morphological distortions are visible after a merger and we adopt a nominal value of ~0.5 Gyr (see section 4.3) - M_cut is the cutoff mass for the merger, which is 2.5e10 Mo for the high mass sample S1 and 1.0e9 Mo for the intermediate mass sample S2 (section 2) - The mass ratio is 1/41/10 for major+minor mergers The merger fraction and rate are calculated in the HOD, SAM, and SPH models, as outlined in section 4.5 2) Analogous to the theoretical models, the goal of the visual classification system is to identify systems with M*>= Mcut, which show evidence of having experienced a merger of mass ratio M1/M2>1/10 within the last visibility timescale t_vis, between time (t_obs-t_vis) and time t_obs. These mergers are denoted as type 'Int' in the paper. In practice, they consist of three types of mergers, which we denote as Int-1, Int-2a, and Int-2b and define below. These three types of systems encompass young to advanced mergers, and are identified/handled in different ways, as described below. I) Mergers of type Int-1 Mergers of type Int-1 primarily represent advanced mergers of mass M*>=M_cut, which appear as single ACS systems and are likely associated with a merger of mass ratio >1/10 that occurred betwen times (t_obs-t_vis) and t_obs. Systems of type Int-1 are identified empirically based on the following criteria: i) They show morphological distortions, which are similar to those seen in simulations of mergers of mass ratio >1/10, and remain visible over the visibility timescale t_vis. The distortions include arcs, shells, ripples, tidal tails, large tidal debris, highly asymmetric light distributions, double nuclei inside a common body, etc. The presence of such distortions is considered indicative of a past merger that occurred between times (t_obs-t_vis) and t_obs. (We make an extra test to verify that the distortions are caused by a past merger, rather than a present tidal interaction, by verifying that such systems do not have a distorted companion of similar spectrophotometric redshift within 40 kpc). ii) They appear as a single distorted system , rather than 2 individually recognizable galaxies in ACS images of PSF~0.1" (corresponding to 380 pc at z~0.24 and 750 pc at z~0.80). This suggests the system is an advanced merger where the 2 progenitor galaxies have had time to coalesce into a single ACS system by time t_obs. Such advanced mergers have a single redshift (Wolf et al 2004), stellar mass M* (Borch et al. 2006), and UV-based star formation rate SFR_UV (Bell et al. 2005, 2007) determined from the COMBO-17 ground-based data of resolution 1.5". This lack of resolved COMBO-17 data for the individual progenitor galaxies that led to the remnant is not a concern because we are only concerned in the analysis with the stellar mass and SFR of the merger. Furthermore, the condition M*>=M_cut is expected to apply to the merger, and not to the progenitors, in both model and observations. The majority of the mergers we identify are of type Int-1. Examples in Fig. 2 are cases 2, 3, 4, 5, 6, 7, 9, 10 11, and 12. For advanced mergers of type Int-1 with a single (M*,z), the evidence for a merger of mass ratio >1/10 does not come from a measured stellar mass ratio M1/M2, but instead is inferred from the presence of the afore-mentioned morphological distortions, which are seen in simulations of mass ratio M1/M2>1/10. Without individual masses M1 and M2, the separation of these mergers into major and minor is not possible in every case, since the morphological disturbances induced depend not only on the mass ratio of the progenitors, but also on the orbital geometry (prograde or retrograde), the gas mass fraction, and structural parameters (e.g., Mihos \& Hernquist 1996; Struck 1997; Naab \& Burkert 2001; Mihos et al. 1995, di Matteo et al. 2007). Instead, we can only separate such mergers into three groups: clear major mergers clear minor mergers, and ambiguous cases of 'major or minor' merger as follows: - The class of clear major mergers includes systems that host fairly unique tell-tale morphological distortions characteristic of a major merger, such as 2 tails of equal lengths, 2 nuclei of similar luminosity, (e.g. case 6 in Fig 2) and a train-wreck morphology (e.g.,case 12 in Fig. 2). - The class of clear minor merger includes the merger systems where the outer disk (identified based on the presence of disk features, such as spiral arms and bars) has clearly survived the recent past merger. Examples include case 2 of a warped disk in Fig. 2. The reasoning behind classifying such remnants as a minor merger is that the outer disk of a galaxy survives a minor merger, but not a major merger. This statement is true except in the extreme case of a pure gas disk with very low star formation efficiency, as such a disk survives a major merger -- see paper by Robertson et al. 2004. However, the massive systems at z<1 that we are considering in this study are extremely unlikely to be pure gas disks. One further criterion is applied. When identifying minor mergers through the presence of a distorted surviving outer disk , we take care to check that the candidate is not in the early phases of a merger, which would destroy the disk in the near future, on a timescale ~tvis. We do this by avoiding those galaxies, which have both a distorted outer disk AND a close companion of similar redshift (within the photometric redshift accuracy) and mass ratio >1/4. - The class of ambiguous 'major or minor' merger is assigned to systems hosting distortions, which could be due to both a major and a minor merger. Examples in Fig. 2 are cases 3, 4, 5, 7, 8, 10, and 11. II) Mergers of type Int-2 Mergers of type Int-2 primarily represent young mergers, which appear as very close pairs of overlapping galaxies (VCPOG) in ACS images, have M*>=M_cut, and are likely associated with a merger mass ratio >1/10 that occurred between times (t_obs-t_vis) and t_obs Systems of type Int-2 are identified empirically based on the following criterion: (i) ACS images show 2 individually recognizable galaxies whose bodies overlap to form a common continuous envelope of light, and whose centers have a small separation d<20 kpc. One or both of the galaxies often have morphological distortions. These properties suggest that the 2 progenitor galaxies have recently merged, at a time close to t_obs, but have not yet coalesced into a single ACS system of type Int-1 It is important to note that we are NOT concerned here with pairs of galaxies with a wide separation d>= tvis * v (where v is the relative speed between the 2 progenitor galaxies) because such systems represent potential *future* mergers that will occur in the next discrete time step of (t_obs+t_vis). Instead , here, we are only interested very close overlapping pairs of galaxies with separation d<< tvis * v, which represent young mergers that have occurred at a past time close to t_obs, and which will likely coalesce into a single ACS system well before the next discrete time step of (t_obs+t_vis). For t_vis of 0.5 Gyr and v~00 km/s, d<< (t_vis* v) translated to d << 50 kpc. We thus only consider VCPOG where the galaxies overlap and have a separation < 20 kpc, which corresponds to d<5.3" at z~0.24 and d<2.8" are z~0.80. One caveat in handling systems of type Int-2 is that some of the VCPOGs may be chance projections rather than real gravitationally bound mergers. However this uncertainty does not significantly affect our result as the vast majority (>80%) of the mergers in our study are of type Int-1 rather than Int-2. Furthermore, the likelihood of chance projection is mitigated due to the fact that we are considering pairs of very small separation. The COMBO-17 ground-based data of resolution ~1.5" will resolve a fraction of the VCPOG that make up systems of type Int-2. Thus, we divide the latter systems can into sub-classes Int-2a and Int-2b: II-a) Mergers of type Int-2a These VCPOG are not resolved by COMBO-17 data. Thus, only one stellar mass, redshift, and UV-based SFR are available for the pair. The lack of resolved COMBO-17 data for each galaxy in the pair is not a problem because we are only concerned with the mass and SFR_UV of the progenitor. The criterion M*>Mcut also applies to the mass of the merger rather than the progenitor. Since only one mass is available for the entire merged system, the evidence for a merger of mass ratio >1/10 in systems of type Int-2a does not come from a measured mass ratio, but from the presence of morphological distortions, which are seen in simulations of mass ratio M1/M2>1/10. The same approach described above for systems of type Int-1 is used here. II-b) Mergers of type Int-2b These VCPOG are resolved by COMBO-17 data such that stellar masses M1 and M2, as well as redshifts and UV-based SFRs, are available for both galaxies in the pair. In this case, the evidence for a merger of mass ratio >1/10 comes directly from the measured mass ratio M1/M2. The SFR and mass of the merger is considered as the sum of that of its two progenitor galaxies One caveat should be noted regarding the completeness of mergers of type Int-2b. Strictly speaking, the criterion M*>M_cut applies to the merger mass (M1+M2) rather than to M2 or M1 individually. For M_cut ~2.5e10, counting all major mergers of type Int-2b with 1/480%) of the mergers in our study are of type Int-1 rather than Int-2. Based on the above description of the mergers of type Int-1,Int2-a and Int2-b traced by the observations, here are answers to the specific questions of the referee: > Reading the paper, and looking at the examples proposed in Fig. 2, I > understand that in the majority of the cases the single galaxies/components > are not resolved in the multiband photometry used to measure mass, or > in l=2800 COMBO-17 band used to measure UV-SFR and in the 24micron photometry. > Hence just one mass and one estimate of the SFR for the whole system are available. > Is this correct? For systems of type Int-1 and Int2-a, only one stellar mass M*, redshift z, and SFR_UV are available in the combined ACS-COMBO-17 catalog. > How are these "resolved" cases treated in Fig. 1, 12 and 17? > How the "resolved" cases are treated when counting the > interacting systems, deriving the galaxy interaction statistics? Systems of type Int-2b are counted as one merger (e.g., when comparing to models) but as 2 interacting galaxies in the old Figures 1, 12, and 17, since each member of the pair counts as one interacting galaxy, with a separate (M*,z) in the source statistics of the combined ACS-COMBO-17 source extraction catalog. Note that in the new version of the paper Fig 12 is now Fig. 13, and the old Fig. 17 has been removed because it is not critical and we face a shortage of space after adding new extra figures 17 to 21, to meet the referee's point P16, under minor issues. > Moreover, how these cases are treated when studying the contribution of > mergers on the star formation? For systems of type Int-2b, the sum of the SF in the pair members is counted as the contribution of that merger to the SF. > What about potential mergers in > which the two components have two distinct mass measurements, both below the > limit of 2.5x10^10 Msun, but for which M1+M2>2.5x10^10 Msun? They should be > included in the sample See above description of systems of type Int-2b, where we address this point. P2. > More detail is needed in my opinion about the sed fitting procedure and > stellar mass determination, that is a key point for the whole paper. How do > the complexity, the distorted morphology and the multiplicity of these > galaxies impact the multi-band photometry and finally the mass measurement? > Are the star formation histories used for the fitting complex enough to > reproduce multi-phase systems like interacting galaxies? The SED fitting procedure and derivation of stellar masses are described extensively in Borch \etal (2006). We have added in the current paper, paragraph 2 in section 2, which addresses the issues raised by the referee, including the nature of the SED templates, the star formation histories, the IMF dependence, and the typical errors on the stellar masses. It is also worth noting that while the stellar masses are certainly associated with uncertainties introduced by bursting star formation histories, the Borch \etal (2006) templates do include bursts explicitly, thus compensating for the worst of the uncertainties. In this respect, the use of stellar masses is better than the use of B-band of V-band luminosities in defining the samples of interest. P3. > In section 4.3 authors claim that on average each massive galaxy has > undergone 0.7 mergers of mass ratio > 1/10 over the redshift interval > z~0.24-0.8. Of these, 0.2 are major mergers. I think that these estimates > can be easily used to quantify the contribution of major (and minor) mergers > to the evolution of the mass function for massive galaxies between z=0.8 and > z=0.24. This is an issue of tremendous interest today, and could be a good > complement to the study of the > impact of galaxy interactions on the average star formation history of the > universe presented in Sect. 4.6. We agree with the referee that constraints on the mass function for massive galaxies would be interesting. However this topic is beyond the scope of the current paper, which is already near the page limit of ApJ and addresses three major issues: the empirical merger history of galaxies over z=0.2 to 0.8, comparison of the data to a suite of different hierarchical models, and the impact of galaxy interactions on the average star formation history. This topic will be addressed in several future papers, focusing on different mass ranges and merger types by the first author and co-authors. > Finally, just a "style" annotation: many phrases are repeated, unchanged, at > different positions of the paper. It could be a personal feeling, but I > think that this does not facilitate the reading. We have revised the text to reduce the repetition. ======== Minor issues: P4. > Section 1. > > The introduction is not well balanced between the overview of the literature > results and the outline of the paper. I suggest to move the discussions in > points 4 and 5 before the outline, that possibly has to be shortened. We have addressed the quantitative points (P5, P6, P7, P17 below) raised by the referee on section 1 , and also expanded the overview of the literature. We also rearranged some paragraphs and shortened the outline. However, we did not change the remaining layout of the introduction as this is a somewhat subjective issue, and we feel that the current presentation of the outline, along with a description of how this study complements existing work, is effective in putting this study in context. P5 > Moreover, I suggest to expand the part presenting results about the > evolution of the merging rate. In my opinion, it is interesting to know > whether the merging fraction/rate evolves with redshift or is constant, but > even more important is to quantify the impact of the merging events on the > evolution of massive galaxies. In other words, not only the slope of the > evolution is important, but even the zero point. > Note that the cited papers do not agree on that: for example, Lin et > al. 2004 say that just 9% of the massive ETG experienced a major merger > since z=1.2; on the contrary, Bell et al. 2006 claim that ~50% of all > galaxies with present-day masses M>510^10 Msun have undergone a major > merger since z=0.8. Discuss briefly these differences. The difference between the results of Bell et al. (2006) and Lin et al. (2004) are addressed in the footnote 9 of Bell et al. (2006), which states the following: "Differences between our determination and that of Lin et al.(2004) have a factor of 1.3 contribution from inclusion of our data and that of the 2dFGRS and Le Fevre et al. (2000), a factor of 2 difference following Section 5.4, a factor of 1.2 difference in the redshift integration method, a factor of 1.25 difference in timescale (our 0.4 Gyr vs. their 0.5 Gyr), and finally a difference of 1.3 from the fraction of their pairs deemed to be real (from our sample we estimated 65\% of the pairs are physical; they argued 50\% following the low redshift analysis of Patton et al.): in all, a factor of five difference in inferred remnant density would be expected, and indeed our estimate is five times larger than theirs. " We have added in section 1, paragraph 3, a discussion of the above differences. P6. > For the sake of completeness, add perhaps de Ravel et al. 2008, recently > appeared on astro-ph. This reference has been added. P7 > A small inaccuracy: Cassata et al. 2005 actually claimed a mild increase of > the merger rate, even within large errors due to the small sample. This statement (for z<1) been corrected in section 1, paragraph 3. P8 > Section 2: Is the initial sample containing 4740 selected down to > R=24(Vega)? I can guess it is, but is not clearly said in Section 2. Yes, it is. This point has been clarified in section 2, with the statement " a sample of 4766 galaxies selected down to $R_{\rm Vega} \le$ 24, over $z\sim$~0.24--0.80". The samples S1 and S2 are derived by applying mass cuts of M*>=2.5e10 and M*>=1e9, respectively, to this sample of galaxies. P9 > section 3.2: > I find a bit misleading that, introducing Int-1&2 cases, authors describe > the effect of the merging event just on one of the 2 galaxies involved in > the interaction, since the aim of the paper is to identify interacting > systems and not single galaxies involved in a merger. I would rather say > something like "a SYSTEM (not a galaxy) is assigned > to class Int-1 if the interacting galaxies exhibit strong > morphological distortion...". And "Int-2 systems involve two galaxies fairly > symmetric, with overlapping envelopes of light, and masses satisfying the > criterion M1/M2>1/10..." Do the authors agree on that? Yes. The description of merging systems has been clarified in section 3, as outlined in point P1, under 'Major issues'. P10 > Section 3.3: > Attention: case 6 in Fig. 2 is reported both as an example of ambiguous > major/minor merger and as an example of major merger. This has been corrected to indicate major merger only. P11 > Again, when describing major mergers, it is not clear to me why authors > concentrate on one of the two galaxies interacting rather than on the > whole system. This issue has been clarified in point 1, under 'Major issues', as well as in section 3. We do consider the two interacting galaxies in a likely merging pair as one merger in the merger statistics. P12 > Fig.2. What is the size of the postage stamps? I would put the size in > arcsec in the caption (if it is the same for all the galaxies, as I guess) > and the size in kpc in each panel, together with the redshift. This can help > the reader to have an idea about the physical scale of the interaction. > Case 1. The two galaxies shown constitute just one merging system, so they > must weight as one system in the merging rate statistics. Is this correct? A size bar of 1", the redshfit and the corresponding kpc conversion have been added to Fig. 2 (and also to Fig.~3 for consistency). Yes, case 1 in Fig. 2 counts as one merging system in the merger statistics, and the description of merging systems has been clarified, as outlined in point 1, under 'Major issues'. P13. > Section 4. Section 4.1 & 4.2: At the moment, authors first discuss fig. 4, > then they pass to fig. 5, an finally they come back to fig.4. Moreover, the > reader can understand that redshift dependent systematics concern only the > visual classification, while in may opinion they affect also CAS > measurements. I think that this can be avoided moving the part describing > the redshift-dependent systematic effects at the beginning or at the > end of section 4, perhaps in a separated subsection. We agree with the referee that redshift-dependent systematic effects affect both the visual classification and CAS. However, we have proceeded to make this point clear in a somewhat different way and preserved many aspects of the current organization of the text in section 4.1 & 4.2 for the following reasons: a) The visually-based interaction fraction is shown in Fig. 4 and described in section 4.1. The tests for SB dimming and bandpass shifting described in section 4.1, Figure 5, and Table 3, primarily test the impact of these redshift-dependent systematic effects on the visual classes, and not on CAS. Testing the impact of SB dimming and bandpass shifting on the CAS-based results would necessitate a different suite of tests (see point c), which are described at the end of section 4.2. b) From the tests on the visual classes in section 4.1, Figure 5 and Table 3, the main message we want the reader to get at the end of section 4.1 is that the visually based classes are reliable within the error bar of ~26% already assigned to them, and they can therefore be used for testing the CAS-based results, which are then outlined in the next section 4.2. c) In section 4.2, the tests on CAS illustrated in Figures 7, 8, and 9 focus on a comparison of galaxies classified as interactions or mergers by CAS versus the visual classification. The 2 problems uncovered here relate to heavy contamination (i.e. 44% to 80% of the systems picked by the CAS criterion are actually visually classified as non-interacting) and modest recovery (i.e the CAS criterion picks 50% to 70% of the systems visually classified as interacting). The effect of bandpass shifting on CAS is illustrated in Fig. 9 and discussed at the end of section 4.2: the contamination level rises sharply at z>0.5, when the rest-frame wavelength falls below 4000 A in the near UV. The reader is referred to Taylor et al. (2007) for modified CAS criteria in the near-UV. P14 > Fig.5 Add at least the size in arcsec in the caption. Say which systems are > classified as interacting. A size bar of 1" and the corresponding kpc measure has been added. The galaxies are all at similar redshift of z~0.7 so that 1"~7 kpc. The interacting galaxies and mergers are noted in the caption. P15 > Section 4.4 Fig.10. I do not understand how the samples plotted in the > figure are chosen. Why do the authors just plot Lotz et al. and Conselice et > al. and not the other samples discussed in section 4.4? The studies plotted initially are all based primarily on using morphological distortions to trace interacting/merging galaxies. The other studies mentioned in section 4.4 are based on close pair fraction and focus on major mergers. We have included an expanded version of Fig. 10 to include the latter studies too. P16 > Section 4.6 & 4.7 Why do the authors here discuss just sample S2 and not > even the massive one? Not enough statistics? Larger uncertainties on SFR? > Not enough 24 um detections? What about using the 24-micron stacking > technique? The high mass (M*/Mo>=2.5e10) sample S1 has significantly smaller number statistics than the intermediate mass (M*/Mo>=1.0e9)sample S2, but the results are essentially similar for both samples on the issue of the average SFR and the integrated SFR density. The figures below have been added for the high mass sample: Fig 15 = Average SFR (UV, UV+IR, UV+IR-stacked) using visual galaxy types Fig 17 = SFR density (UV, UV+IR, UV+IR-stacked) using visual galaxy types Fig.19 = Average SFR (UV, UV+IR, UV+IR-stacked) using CAS-based galaxy types Fig 21 = SFR density (UV, UV+IR, UV+IR-stacked) using CAS-based galaxy types They complement Figures 14, 16, 18, and 20 for the intermediate mass sample: P17 > Authors state that the evolution of the cosmic SFR density over z~0.24-0.8 > is predominantly shaped by non-interacting galaxies. Note that a similar > result was already found by Lotz et al. 2008: cite it here and in the > introduction. We have added a reference to Lotz et al. (2008), but note that the latter study did not address in detail the issue of the cosmic SFR density over z~0.24-0.8. ************************************************************************** Part 2: Extra changes made to the paper 1. Figures 1,7, and 13 have been modified to show only galaxies with stellar masses >= 1e9 solar masses because this is the relevant mass range over which the sample S2 is defined (see section 2) and over which the analysis and results of the paper apply. Furthermore, over this mass range, the visual types were re-checked in detail. (In the earlier version of the paper, even galaxies with stellar masses below 1e9 solar masses were plotted, although they are not part of the analysis.) 2. Section 4.2 We previously showed (paragraph 5+6 and Fig. 8) the recovery fraction and contamination fraction produced when the CAS criterion is used to select interacting/merging systems in the intermediate mass (M*/Mo >= 1e9) sample. For the sake of completeness, we have now added the corresponding numbers for the high mass sample (M*/Mo>=2.5e10) in paragraphs 5+6 of section 4.2. This addition is in the spirit of the comment made by the referee (in point 13 of "Minor issues") to include results for M*/Mo>=2.5e10 in other section. 2. Section 4.5 Previously, we only showed a comparison (current Fig. 12) of the empirical merger rate R to and the merger rate predicted by different theoretical models of galaxy evolution in the context of a Lambda-CDM cosmology. We have now also added a comparison of the empirical vs theoretical merger fraction f in Fig. 11, as the merger fraction is more directly related to the observations. We made sure modelers treated galaxy mergers in a way that paralleled the observational procedure: for instance, multiple mergers separated by a time larger than the visibility timescale were counted as multiple mergers in both data and models. The conclusions on the agreement between data vs model for the merger fraction f, are similar to the earlier conclusions on data vs model for the merger rate R. **************************************************************************