Review Reports - Estimation of Host Galaxy Extinction for SNe Ia

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The overall presentation needs critical rewording.

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

We thank you for extremely useful and constructive comments, they greatly helped us to improve the paper.
We have made all required corrections and responded all questions raised.
All changes are highlighted in red in the text.
The details of the revisions are given below, point-by-point.

Please, note that corrections required by other reviewers are also made and highlighted in the text.

We are ready to answer your questions if you still have them.
Thank you very much again for your cooperation,
With best regards,
Oleg Malkov, on behalf of co-authors

----------------------

The report on the article "Estimation of host galaxy extinction for SNe Ia"
by O.Malkov, A. Sytov, G. Zhao, and Z. Zhong

The article describes a technique of determination of distances to extragalactic SNe Ia.
This issue is of fundamental significance for measurement of the cosmological parameters: its
mass-energy composition which governes the evolution of the cosmic scale factor. Besides,
scrutinizing of the spatial distribution of SNe Ia on large scales and their connection with
the host galaxies is also important for dynamical and chemical evolution of galaxies.
In this regard the overall premise sounds interesting and might be of common interests
for astrophysics and cosmology communities. I have though a few comments needed before
acceptance the manuscript in the Galaxies journal.

Main points:

1. Abstract: the sentence "In this paper, we generate, based on MaNGA data, a set of
galaxies..., distribute the parameters of the galaxies... and also specify the distribution
of interstellar matter in them" sounds hard to understand. A more explicit explanation
of what is understood by "we generate", "distribute the parameters" and "specify the
distribution" would be helpful for getting a clear impression about the approach used in
the manuscript. Moreover, as can be seen later on page 5 (first line) by "specifying the
distribution" the authors mean, in particular, the identical structural parameters a0 =
1.6 mag kpc-1 and \beta = 0.116 kpc for all galaxies in the sample. Without additional
explanations such an assumption appears to be an over-simplification. Within this
approach, the authors consider in fact a single late-type galaxy with fixed parameters
of the interstellar medium. In this sence, the abstract conflicts with the content of the
paper.

==> You are absolutely right. We have described the procedure inaccurately.
In fact, we do NOT distribute of interstellar matter (though we distribute parameters of the galaxies, R and h),
we just use a simple model of ISM distribution, the same for all galaxies.
This is now reflected in the Abstract.

2. Introduction: in 4th paragraph on page 2 the authors state their goal "the possibility
of statistically estimating the magnitude of interstellar extinction in the host galaxy".
From general view, one of the most interesting issues of the anticipated results is the
accuracy of such estimations as compared either with the intergalactic extinction, or
with the accuracy needed for measurements of the cosmological parameters. From this
point it would worth to indicate these numbers: AV ~ 0.1 (Menard etal 2010) and
m-M ~ 0.15 (Riess etal 1998), correspodingly.

==> These references are now inserted in section Conclusion, 2nd paragraph.

3. Sec 2.1: 4th paragraph in the Section: what precisely does mean "absorbing material
extends"? How exactly this boundary is defined with regard to Av? Speaking in the
simples case: what is the boundary value of the exponent in Eq (1) exp(-d sin |b|/\beta)?
It would be also helpful for understanding to draw a schematical representation of the
"boundary".

==> To determine the distance d from SN to the "boundary" of the host galaxy (Eqs.(3,4))
we use five parameters: galactic coordinates l,b of the observer in the host galaxy coordinate system,
radius R and height h of the host galaxy, and distance r of SN from the center of the host galaxy.
They can be illustrated in a diagram (see parameters.jpg attached),
but we are not sure that it is advisable to insert this into the text...

4. Sec 2.1: my impression is that the Section needs a more organized structure. As can
be understood, the first two paragraphs describe the Parenago formula in Eq (1) and
its asymptotics Eq (2) for the Milky Way galaxy. Next (after Eq 2) paragraph deals
already with a configuration of the observer and a SN Ia in the host galaxy, as seen
from the context and from Eq (3). How the parameters of the host galaxy (vertical
and radial scales of the disk, orientation with respect to the observer) are eliminated
from Eq (3)? How b and l in Eqs (3) and ((4) relate to the latitude and longitude in
the observer's galaxy and in the SN Ia host galaxy? It is not quite obvious from the
presentation. Additional comments and a schematic illustration are needed.

==> Section 2.1 is now divided into two sections. Parameters of the host galaxy (R, h),
coordinates of the observer's galaxy in the host galaxy coordinate system (l,b )
are given in the illustration parameters.jpg (see above).

5. Sec 2.1: the next two paragraphs contain a brief description of SNe Ia distributions in
their hosts in form of a spherical exponential distribution \propto exp(-r/R). The variable
R is met further in text and Figures below. What should readers understand by the
R's in different cases? Is the SN Ia radial scale R for different host galaxies invariant
or it changes from one to another galaxy? These have to be answered and clearly
explained in order not to come into conflict with the abstract.

==> In our model the variable R describes the host galaxy size, varies from galaxy to galaxy,
and has nothing to do with supernova's location in the host galaxy.

6. Sec 3, the first two paragraphs: while calculating the extinctions AV the authors use
the fixed values a0 = 1.6 mag kpc -1 and \beta = 0.116 kpc-1 for all sampled galaxies.
The reasons of making this particular choice should be pronounced.

==> A reference to Sharov's 1963 paper has been inserted.

7. The essence of the results is illustrated in Fig 3. The left panel shows that the modeled
AV peaks at AV ~ 0.3 with a spread of \DeltaAV ~ 0.4. This is more or less expected by
order of magnitude for the MW galaxy seen from outside. If speaking about possible
implication of the suggested method in measurements of the cosmological parameters
with high redshift SNe Ia, it would be interesting to compare the obtained \Delta AV with
the variations of m-M < \le 0.15 in the first experiments by Riess etal 1998. How the
authors can comment their results from this point of view? Whether the presented
technique is sufficiently efficient and robust for estimation of distances to SN Ia in
such statistical modeling?

==> Yes, thank you once more. We now compare our results with results of other authors,
see section Results (last paragraph), section Conclusion (2nd paragraph).

Minor points:

1. page 3, 3d paragraph: what is precisely meant by the "method of inverse functions"?
A reference or a brief explanation would be helpful.

==> The correct term for this method is "Inverse transform sampling", we have corrected this in paper.

2. page 3, 6th line from bottom: the "Vallue Added Catalogues" needs a definition or a
mentioning explicitly in the beginning of the paragraph.

==> In IFS surveys like the MaNGA survey, the tables containing various parameter properties of the target
obtained after processing the raw data are usually called "value added catalogs." Therefore, we would like
to retain this common term, but for a better description, we would like to revise the sentence
"Here we use these Value Added Catalogues to select edge-on late-type galaxies (LTGs) and..." to
"Here we use the MaNGA PyMorph photometric Value Added Catalogue (MPP-VAC-DR17)
and the MaNGA Deep Learning Morphological Value Added Catalogue (MDLM-VAC-DR17) from \cite{2022MNRAS.509.4024D}
to select edge-on late-type galaxies (LTGs) and...".

3. Eq. (6) suggests that the R and h are statistically independent. I think, it would
make sence to mention this circumstance.

==> It is now mentioned in the text.

4. Fig 1: what is the source of the errors for R and h in the right panel.
I think that the manuscript requires careful major revision in order to make the obtained
results more convincing and their presentation readable.

==> The errors for R and h I have provided come from the above Value Added Catalogues and error propagation.
Correspondingly, we have revised the last sentence of the second paragraph in Sec 2.2 to
"and histograms of their uncertainties, which are derived from the above Value Added Catalogue and error propagation,
are shown in the same figure, right panel."

----------------------

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

As the reviewer, I want to make an overall comment about the paper -- this does not impact the quality of the paper as written, but perhaps provide a future direction. w

While an estimate of the extinction that a SN Ia's light suffers as it travels to us is interesting and the authors have done a solid job in developing an estimate, it is not clear how their estimate improves our measures of the distances to these objects. The approach introduces an additional uncertainty seemingly without improving distance values. Certainly a user could now 'quantify' a distance range, but that was already possible. The approach would be enhanced considerably if there was a link to some quantity that, even statistically, would help pin down whether any particular SN Ia had an extinction of 0.1 or 0.5.

Comments on the Quality of English Language

abstract, line 4: 'than' should be 'that'

page 2, 2nd paragraph, line 3: Though the problem ... circumstances.
this is not a complete sentence: recommend removing 'Though'

last sentence: remove 'also' as it is redundant with 'recently'

page 2, just above equation 1: 'visial' should be 'visual'

Author Response

Dear reviewer,

Please, note that corrections required by other reviewers are also made and highlighted in the text.

We are ready to answer your questions if you still have them.
Thank you very much again for your cooperation,
With best regards,
Oleg Malkov, on behalf of co-authors

----------------------

Comments and Suggestions for Authors

As the reviewer, I want to make an overall comment about the paper -- this does not impact the quality of the paper as written, but perhaps provide a future direction. w

==> Yes, unfortunately you are right.
There is not as much practical benefit from our calculations as we would like.
The only thing we say in our paper is that when estimating the distance to a SN Ia, it is necessary to increase the extinction value and, consequently, the resulting distance will become smaller.
Indeed, it would be good to introduce a "some quantity that, even statistically, would help pin down whether any particular SN Ia had an extinction of 0.1 or 0.5".
Unfortunately, there are very few candidates for the role of such a "some quantity": the brightness of the SN Ia and its coordinates.
However, in some cases, another parameter is known from observations: the apparent distance of the SN Ia from the center of the host galaxy (if this galaxy is visible).
In our next paper, we plan to do exactly what you recommend: get an estimate of the extinction as a function of this distance.
The brightness of the SN Ia also looks promising as such a parameter.
Thanks again for the idea.

Comments on the Quality of English Language

abstract, line 4: 'than' should be 'that'

==> Corrected.

page 2, 2nd paragraph, line 3: Though the problem ... circumstances.
this is not a complete sentence: recommend removing 'Though'

==> Thank you for your recommendation. Corrected.

last sentence: remove 'also' as it is redundant with 'recently'

==> Corrected.

page 2, just above equation 1: 'visial' should be 'visual'

==> Corrected.

----------------------

Reviewer 3 Report

Comments and Suggestions for Authors

The paper presents a model for statistical estimation of
interstellar extinction suffered by type Ia Supernovae.
The problem of accounting for extinction is among the most
significant for using SNe Ia as distance indicators, and the refereed
paper provides new information to this field of research.

The model presented in the paper involves an accurate calculation of
the light pass length in a galaxy and modeling of size distribution of
spiral galaxies based on the MaNGA survey. The results are the model
distributions of SNe Ia by the extinction value, by the radial distance
and distance to the galaxy border.

The model is simplified, it does not include the distribution of
SNe by the height above the galactic plane and difference of
extinction coefficient with radius of the galaxy and morphological type.
It may be noted that Hackobyan et al. (MNRAS 471, 1390, 2017)
found out that SNe Ia in spiral galaxies are located in thick disk,
which in our Galaxy has height scale about 0.7-0.9 kpc. Besides, SNe Ia
may also be found in the bulge population of spirals.

It may be interesting to compare the results of the paper with
another models of extinction distribution (e.g. Riello and Patat 2005,
MNRAS 362, 671) and observed distributions (e.g. Holwerda et al. 2015,MNRAS 446, 3768).

Abstract: "the value of extinction in the SN Ia host galaxy remains
unknown and therefore neglected" - there are several methods for
determination of extinction, which are used in many works.

Introduction, line 8 from above: it is not clear what SNe are called
"regular". According to Li et al. 2011 (MNRAS 412, 1441, 1473)
the rate of core-collapse SNe (CCSNe) in galaxies similar to our
Milky Way Galaxy ia about
1/50 years, while SNe Ia are about 4 times less frequent, with rate
1/200 years.

Comments on the Quality of English Language

Some inadequate terms: "consistent explosions", "regular supernovae", "informing the distance ladder", "muddying the issue".

Author Response

Dear reviewer,

Please, note that corrections required by other reviewers are also made and highlighted in the text.

We are ready to answer your questions if you still have them.
Thank you very much again for your cooperation,
With best regards,
Oleg Malkov, on behalf of co-authors

----------------------

Comments and Suggestions for Authors

==> We thank you for the very valuable information.
It is now included in the text (section Conclusion).

==> Thank you for bringing these interesting works to our attention.
Comparing our results with them shows a qualitative agreement.
An insertion has been made in the text (end of section Results).

Abstract: "the value of extinction in the SN Ia host galaxy remains
unknown and therefore neglected" - there are several methods for
determination of extinction, which are used in many works.

==> "remains unknown and therefore neglected" is replaced by "is determined with much more uncertainty"
(than extinction in the Milky Way Galaxy).

Introduction, line 8 from above: it is not clear what SNe are called "regular".

==> Here "regular" means "regular Type II", see explanation below.

According to Li et al. 2011 (MNRAS 412, 1441, 1473)
the rate of core-collapse SNe (CCSNe) in galaxies similar to our
Milky Way Galaxy ia about
1/50 years, while SNe Ia are about 4 times less frequent, with rate
1/200 years.

==> The link to the Li et al. 2011 has been inserted, and the text has been adjusted accordingly.

Comments on the Quality of English Language

Some inadequate terms: "consistent explosions", "regular supernovae", "informing the distance ladder", "muddying the issue".

==> All these terms have been corrected, see below:

"consistent explosions"

==> The word "consistent" has been removed

"regular supernovae"

==> Replaced by the "regular Type II supernovae".
This term can be found in literature. See, e.g., Yaron et al.'s paper
"Confined dense circumstellar material surrounding a regular type II supernova"
(2017NatPh..13..510Y).

"informing the distance ladder"

==> Rephrased: "construction of a distance ladder".

"muddying the issue"

==> Rephrased: "The situation is complicated by the fact"

----------------------

Reviewer 4 Report

Comments and Suggestions for Authors

This paper adresses an important challenge for cosmology: modelling the expected distribution of absorption for Type Ia supernovae in distant galaxies. This has a potential impact for all standardization scheme which rely on modelling the distribution of extinction for producing cosmological contours. The results are not limited to Type Ia supernovae, but can be employed for any kind of bright object located in the zone of high stellar densities in spiral galaxies. Even if the analysis is presented in this paper is rather simplistic, it paves the road for more detailed models in the future.

Main comment:

The results are quite interesting, especially the distribution shown figure 3 on the left. Please improve the figure with error bars.

Does the distribution follow an approximate functional form (as is seems to be the case at least for Av>0.2) ? For improved impact for the paper, the authors should consider proposing a fit to the histogram. They should also provide comparison with an exponential law used commonly by bayesian models (see Mandel 2022MNRAS.510.3939M eq. 25)

Also, the authors should compare their findings with other similar work, for instance Duarte 2023A&A...680A..56D

Minor comment:

Abstract:

Source of luminosity variations like « metallicity, age, … » are cited in the abstract, but not addressed in the main text. These sources are not central to the paper, but quoting them puts the work on extinction into perspective. They should therefore be included in the introduction, with proper citation.

1. Introduction

l.4-8 : the explosion mechanism is more complex than just reaching the Chandrasekhar mass. See for instance Livio 2018PhR...736....1L . Please improve the text accordingly.

l.10 : « happen roughly » -> were detected. This is likely that the detection in the MW is quite inefficient, one expects a higher rate when comparing with other galaxies. For a recent discussion about the rates, see for instance 2024A&A...689A.203P (not necessary to add this as a citation as it is not central to this paper)

l.20 « More modern ». Older method were not presented. -> Modern (drop More)

l.37 Separating intrinsic variabilities from dust seminal paper is up to my knowledge Chotard 2011A&A...529L...4C. Your reference 15 even if interesting seems to lack proper recognition by the community (only 2 citations as seen on ADS)

2. Description of the model

There is an abundance of parameters. If page count allows, I suggest to provide a representative schema to help the reader figuring out what the various parameters are, for improved readability.

l.49 visial -> visual

l.80 please mention the redshift range of galaxies from Manga

Eq. 5 : there are 3 different x. Label them x_1,2,3 to avoid confusion.

Figure 1:

The red dots are likely hiding the blue dots. As a result, on cannot see if the two distributions agree or if the « red » distribution is much broader. Please change this (for instance, using iso-density contours or any other suitable method)

3. Results

See main comment.

l.118: please provide a reference with respect to the chosen values of a0 and beta.

l.139 : among the simplifications, the Parenago law has limitations, Please cite for instance Arenou 1992A&A...258..104A as example of a more complete model.

Author Response

Dear reviewer,

Please, note that corrections required by other reviewers are also made and highlighted in the text.

We are ready to answer your questions if you still have them.
Thank you very much again for your cooperation,
With best regards,
Oleg Malkov, on behalf of co-authors

----------------------

Comments and Suggestions for Authors

Main comment:

The results are quite interesting, especially the distribution shown figure 3 on the left. Please improve the figure with error bars.

==> Histogram uncertainty can be estimated as Poisson noise level $\sigma_i/N = \pm\sqrt(n_i)/N$ (normalized).
It is now added to the text.

==> Yes, thank you, it is a perfect idea. We present approximation functions for left and right wings
of the distribution, shown in Fig.3 (left panel).

Also, the authors should compare their findings with other similar work, for instance Duarte 2023A&A...680A..56D

==> It is rather difficult to compare our results with those of Duarte et al. 2023,
since they did not directly calculate the Av values.
However, we compared our results with other works (see last paragraph in section Results).

Minor comment:

Abstract:

Source of luminosity variations like " metallicity, age, ... " are cited in the abstract, but not addressed in the main text. These sources are not central to the paper, but quoting them puts the work on extinction into perspective. They should therefore be included in the introduction, with proper citation.

==> It is now included in the introduction.

1. Introduction

l.4-8 : the explosion mechanism is more complex than just reaching the Chandrasekhar mass. See for instance Livio 2018PhR...736....1L . Please improve the text accordingly.

==> The text is corrected and the reference has been inserted.

l.10 : " happen roughly " -> were detected. This is likely that the detection in the MW is quite inefficient, one expects a higher rate when comparing with other galaxies. For a recent discussion about the rates, see for instance 2024A&A...689A.203P (not necessary to add this as a citation as it is not central to this paper)

==> The text is corrected. Thanks for the useful link to Palicio et al., we still inserted it into the text.

l.20 " More modern ". Older method were not presented. -> Modern (drop More)

==> Corrected.

==> A reference to Chotard et al.'s 2011 paper has been inserted.

2. Description of the model

There is an abundance of parameters. If page count allows, I suggest to provide a representative schema to help the reader figuring out what the various parameters are, for improved readability.

==> Actually, to determine the distance d from SN to the "boundary" of the host galaxy (Eqs.(3,4))
we use only five parameters: galactic coordinates l,b of the observer in the host galaxy coordinate system,
radius R and height h of the host galaxy, and distance r of SN from the center of the host galaxy.
They can be illustrated in a diagram (see parameters.pdf attached),
but we are not sure that it is advisable to insert this into the text...

l.49 visial -> visual

==> Corrected.

l.80 please mention the redshift range of galaxies from Manga

==> The redshift range of our sample galaxies from MaNGA survey is around 0.013<~ z <~0.083.
And we have added this sentence in front of the last sentence of the second paragraph in section 2.2.

Eq. 5 : there are 3 different x. Label them x_1,2,3 to avoid confusion.

==> Done.

Figure 1:

The red dots are likely hiding the blue dots. As a result, on cannot see if the two distributions agree or if the " red " distribution is much broader. Please change this (for instance, using iso-density contours or any other suitable method)

==> We drew a picture with iso-density contours, but in our opinion,
it looks even worse than the original plot.
We are now proposing a version that uses empty circles instead of filled ones.

3. Results

See main comment.

l.118: please provide a reference with respect to the chosen values of a0 and beta.

==> A reference to Sharov's 1963 paper has been inserted.

l.139 : among the simplifications, the Parenago law has limitations, Please cite for instance Arenou 1992A&A...258..104A as example of a more complete model.

==> References to Arenou et al.'s 1992 and to the works of other authors are inserted into the text.

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

As one of the first reader of your manuscript I do not feel impressed of the results. My point is that the presentation and the results articulated in the "conclusions" sound inconsistent.

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

We thank you again for extremely useful and constructive comments, they greatly helped us to improve the paper.
We have made all required corrections and responded all questions raised.
All changes are now highlighted in magenta in the text.
The details of the revisions are given below, point-by-point.

We are ready to answer your questions if you still have them.
Thank you very much again for your cooperation,
With best regards,
Oleg Malkov, on behalf of co-authors

----------------------

The report on the article "Estimation of host galaxy extinction for SNe Ia" by O.
Malkov, A. Sytov, G. Zhao, and Z. Zhong

In the revised version several corrections and additional are made, the improve the overall
presentation. A few important issues are though remained to be thoroughly articulated, in
order to make the consideration and the obtained results more obvious and convincing for
the reader.
In red are unanswered comments from my previous review.

1. Abstract: the sentence "In this paper, we generate, based on MaNGA data, a set of
galaxies..., distribute the parameters of the galaxies... and also specify the distribution
of interstellar matter in them" still sounds hard to understand. Perhaps, a more
explicite note that the paper performs s statistical modeling, would be helpfull in this
regard.

==> Abstract is corrected.

2. Sec 2.1: 4th paragraph in the Section: what precisely does mean "absorbing material
extends"? How exactly this boundary is defined with regard to Av? Speaking in the
simples case: what is the boundary value of the exponent in Eq (1) exp(-d sin |b|/\beta)?
It would be also helpful for understanding to draw a schematical representation of the
"boundary". I still think that schematical illustration is necessary for the reader to
figure out what is this conical surface? how the conical model affects the amount of
absorbimg material with respect to a disk-like ISM?

==> Schematical illustration is inserted in the text. Now it is Fig.1.

Another issue, the sencitivity of the results with respect to the exact extent of the
"boundary" does also matter: for instance, how the results of the modeling and the
conclusions can vary if h is replaced by h \pm \Delta h? Whether their final conclusions are
stable with regard to such variations of the boundary? Simple estimates would worth,
particularly because extended dust halos are rather commong in spiral galaxies, as
demonstrated by edge-on galaxies (see, e.g, NGC891 case).

==> We are taking the first step in order to get preliminary, pilot results.
Our model, therefore, has to be simplified. The discussion about extended dust halos
(which is very useful, thank you again) is inserted in the Conclusion section.

3. Sec 2.1: Still (see above) schematic illustration woud be helpful to make Eq (3)
readable.

==> See above.

4. Sec 3, the first two paragraphs: while calculating the extinctions AV the authors use
the fixed values a0 = 1.6 mag kpc -1 and \beta = 0.116 kpc-1 for ALL sampled galaxies.
The reasons of making this particular choice should be pronounced. I am afraid, simple
referring to the paper by Sharov (1967) for the Milky Way is not sufficient to use these
a0 and \beta as unconditionally representative numbers. This simplification should be at
least clearly pronounced in the Abstract, otherwise it sounds misleading.

==> The corresponding fragment of the Abstract has been paraphrased.

5. The essence of the results is illustrated in Fig 3. The authors stress that their
model predicts Ahost > 0.15 at the level 90%, and to some extent is consistent with
m - M \le 0.15 obtained in the first experiments by Riess etal 1998. However, the
senses of these inequalities Ahost > 0.15 and m - M \le 0.15 differ and this difference
deserves at least mentioning.

==> Yes, we are well aware that senses of these inequalities differ,
and that Ahost and m-M are different parameters.
But you are right, mentioning parameter m-M here may confuse the reader.
Consequently, we preferred to remove the link to the Reiss's paper.

An additional issue in the Conclusion is connected with the distribution of SNe Ia
in the plane of their spiral host galaxies. This statement seems to conflict with the
spherical SNe Ia distribution f (r) \propto exp (-r/R) stated in Sec. 2.2. Moreover, SNe Ia
are known to be distributed mostly in thick disks or in spherical populations. This
apparent contradiction between the model decsribed above and the conclusions makes
the presentation very confusing and inconsistent.
I am afraid the manuscript requires additional revision to improve overall presentation
and pronounce the obtained results more consistent

==> No, in our model we assume that the supernova is always located in the galactic disk, and
the distribution you quoted [f (r) \propto exp (-r/R)] describes the distance of the supernova
from the galactic center (see new Fig.1). The fact that in reality SNe Ia in spiral galaxies
can be located in thick disk, is noted in the Conclusion section.

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Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

I hope to see your new approach with a more thorough approach. The problem you are attacking deserves it. In the present form it would worth i) to stress explicitly that the scale height for SNIa in their hosts is assumed to be=0 (page 4, 1st paragraph), and ii) perhaps, the second paragraph in Conclusions would also sound appropriately in 1st paragraph of page 4 - this is though up to you.