We thank the reviewer for critical comments. The present work represents a major advance
in studies of collisional families as it more than doubles the number of cases known previously.
All new families reported here are statistically significant (see below). It was not our original intention
to publish a comprehensive list of new families here - the present paper was originally thought
to be a description of the proper element catalog - but given reviewer's comments we made an
additional effort to produce a more complete list. The families identified here will be characterized in
detail elsewhere (this effort will need many months to finish). Below we explain how we revised
the present manuscript. 

Comment:
The paper "Catalog of Proper Orbits for 1.25 Million Main Belt Asteroids
and a Preliminary Search for New Collisional Families" by Nesvorny et al.
presents a new set of asteroids' proper orbital elements and identification
of new asteroid families. Though useful, the new set of proper elements is
largely based on the previous methodology, bringing a little scientific novelty.
Nevertheless, the part related to asteroid families is interesting and could
provide enough scientific novelty to the paper. Therefore, generally speaking,
the results, especially the identification of new families, are relevant to
the scientific community, and the topic covered in the paper is worth investigating.

Answer:
It is true that the methodology is based on the Fourier approach to the computation of proper elements
(Sidlichovsky & Nesvorny 1997, Knezevic & Milani 1999). This makes sense because the
Fourier approach is the the most powerful method developed so far. We speed up and
improve the computation by using the symplectic integrator/corrector and a long Fourier window,
employ the Frequency Modified Fourier Transform, which is arguably the best method
to pull out planetary frequencies, and massively parallelize the computation on 12,000
cores of the NASA Pleiades Supercomputer. The manuscript reports many novel findings, including:
1) 136 (!) new collisional families are identified, several families are studied in detail
2) we identify and characterize the youngest asteroid family found to date (age 16-17 kyr!)
3) we make available to the scientific community the largest catalog of proper elements 
4) the precision of proper elements is (slightly) improved over the previous catalogs 
5) we explain how the use of symplectic integrator and massive parallelization
will allow us to compute proper elements for 7 million main belt asteroids predicted to be
found by the Rubin observatory
Moreover, it makes perfect sense to publish this work in ApJS because the proper element catalog
is an extensive dataset, which is ideal for the Supplement series.  

Comment:
Having said that, I found that the paper presents mostly unfinished work,
lacking many details on the methodology and making many claims without presenting
the evidence. The only exception is the analysis of small groups presented at
the end of the paper, which is well-written and mostly provides the necessary
information.

Answer:
We made an additional effort to improve the manuscript. See below for our answer to specific
comments of the reviewer. 

Comment:
The novelty and significance of the newly identified families are overstated without
proper verification against existing data. A rigorous cross-check with established
databases is necessary to confirm these are indeed new discoveries.

Answer:
We thank the reviewer for pointing five cases of families that were incorrectly reported as
new. We made an additional effort to more carefully check the new families against previous
publications (see below). We also establish the statistical significance of the new families
(see below). Yes, this paper reports a discovery of 136 statistically significant families. 
This is both novel and significant (no overstatement here).

Comment:
Therefore, in its present form, the paper does not meet publication standards,
and, in my opinion, a significant revision is needed before it can be published
in ApJS.

Answer:
The manuscript has been significantly revised and improved. See below for our detailed answer to the
specific comments of the reviewer.

Comment:
Section 1: Introduction
"... but the ever increasing computer power now allows the proper elements to be computed
to a greater precision numerically (Knežević & Milani 2000)."
Here I suggest also citing Knezevic & Milani, CeMDA 2019, along with a given reference.

Answer:
Citation of Knezevic & Milani (2019) was added. We also added Sidlichovsky & Nesvorny (1996)
who pioneered the use of Fourier analysis for the computation of asteroid proper elements/frequencies.

Comment:
The claim about scalability to larger datasets needs clarification or removal, as it suggests an
improvement in computational methodology that the paper does not substantiate. If I understand it
correctly, the total computing time needed to obtain proper elements was above 750,000 CPU hours,
right? If so, it looks like the requirement typically encountered in the classical computation of
synthetic proper elements. This is fine on the one side. However, on the other side, a statement
made at the end of the section, "The methodology described below is readily scalable to ~10^7 bodies
and can be used to compute proper orbits for the large volume of main-belt asteroids expected to be
discovered by the Rubin Observatory in the next decade (Schwamb et al. 2023)." is highly misleading
and should be removed. It implies that there is an improvement making the computation of proper
elements easier, which is, however, not justified anyhow.

Answer:
We describe a practical method that can be used to compute proper elements for the large amount of data
expected from the Rubin observatory. This has two parts. First, the symplectic integrator/corrector
speeds up the N-body integration by a factor of ~5 (the exact factor depends on the timestep, 1-5 days)
Second, the massive parallelization on 10,000+ cores will allow us to compute the proper elements in
under a month. This is now clarified at the end of the last section:

"The methodology described here offers a practical method to compute asteroid proper elements
for the large amount of data expected from the Rubin observatory (Schwamb et al. 2023). There are two
improvements. First, the symplectic integrator/corrector speeds up the computation of proper elements
by a factor of $\sim$5. Second, the massive parallelization on 10,000+ cores of the NASA Pleiades
Supercomputer (as demonstrated here) will allow us to compute the proper elements for $10^7$ main-belt
asteroids in under a month of wall-clock time."

Comment:
Section 2: Methods
"We do not distinguish between the numbered and unnumbered (single- or multi-opposition) bodies, but
the information about the quality of the osculating orbits (the number of oppositions) is propagated
to the final catalog."

Computing proper elements also for single-opposition asteroids is, in some cases, fine. However, putting them
into the catalog could have a negative effect. This reminds me of another paper presenting rotational periods for
asteroids derived from sparse photometry. In this work, the authors present many values as a "minimum period."
The results ended up at the JPL, and many people are using them as given, completely neglecting the actual
value of this information, which in most cases has nothing to do with the real rotational period.
Therefore, I strongly suggest excluding the single-opposition asteroids from the catalog. They
represent only a tiny portion of the catalog while having very questionable scientific value.

Answer:
The catalog lists the number of oppositions in column 11. The user can decide whether to discard asteroids
with 1, 2 or any number of oppositions. We do not see any advantage to removing single opposition asteroids
from the catalog as this would limit and not enhance the information contained there. We added a warning
about single opposition orbits in section 3:

"The single-opposition orbits must be used with care. While in most cases the single-opposition orbits
are good enough to obtain reliable proper elements, there are also instances where this is not the case.
We leave the decision to the user who can easily apply any cut based on the values listed in column 11."

Comment:
The authors included all the major planets in the dynamical model used to compute proper elements. While
this should, in principle, increase the accuracy with respect to the model where Mercury is not considered,
I feel it can actually have the opposite effect. This is because the 7-day time step, though appropriate for
other bodies, it is too short for Mercury. With only ~12 points per orbit, precise propagation is doubtful,
meaning that the orbit of Mercury, and consequently, its influence on the other bodies, is not properly
reconstructed, resulting likely in an additional source of uncertainties in the proper elements. Still, this
should not be a major issue, and I'm certainly not asking to redo integrations without Mercury. However, this
should be acknowledged in the paper.

Answer:
The time step was incorrectly stated in the manuscript. In the main integrations, we used a time step equal
to 0.003 yr, or about 1.1 days, now corrected. This is conservatively short. According to our tests, the
proper elements can be accurately computed with the symplectic method for a timestep up to about 5 days.
These longer timesteps will be considered for LSST. 

If we had done the simulations without Mercury we would expose ourselves to the easy criticism that our
proper elements are not good enough because the integration did not include Mercury. This is not the way
forward. We checked that the orbit of Mercury is stable and does not show any unusual behavior. For example,
the semimajor axis shows oscillations around a fixed value (no diffusion). We also recalculated the
proper elements of the first 1000 asteroids from an integration where the we halved the time step and found
that the proper elements and their errors were practically identical to those obtained from the original
integration. This demonstrates that the time step is not an issue. We added a clarification to section 2:

"We tested different timesteps. The one that was selected for the
main integrations is conservatively short. We checked that the integrated orbit of Mercury is
stable and does not show any unusual behavior. The semimajor axis of Mercury shows oscillations
around a fixed value (i.e., no diffusion) and Mercury's eccentricity/inclination evolution
closely follows expectations. The proper frequencies of Mercury are correctly recovered from the
integration. We also recalculated the proper elements of the first 1000 asteroids from an integration
where we halved the time step and found that the proper elements and their errors were practically
identical to those obtained from the original integration."

Comment:
The methodology described on page 4 needs to be extended with some specific examples, and also, in some places,
additional details should be provided. For instance, "For orbits near mean-motion and secular resonances,
however, which may be affected by orbital chaos, we observed splitting of the proper term into a number of
Fourier terms with similar frequencies. In these cases, the amplitude of the largest proper term usually
corresponds to the minimum of (e or sin(i)) oscillations, which is inconvenient because the normal proper
elements are desired to be close to the mean value of osculating elements."
An example of this behavior would be very useful here.

Answer:
Sure, we added two figures in the Appendix that illustrate the frequency splitting in a specific example
(asteroid 5 Astraea), and discuss the specific values of frequencies and amplitudes that we obtain in this
case. See Sidlichovsky & Nesvorny (1996) for the first instance where this problem was reported.

Comment:
Similarly, "Short Fourier intervals could be used to reduce problems with the proper term splitting, but the
optimal interval length is unknown a priori. Also, as long intervals should be used to define more stable proper
elements in most cases, one would be tempted to use the Fourier interval that flexibly adjusts from case to case.
Unfortunately, according to our tests, it is not obvious how to define robust criteria for the variable interval
length." Again, the test results should somehow be presented in the paper.

Answer:
We have done tons of these tests to optimize the strategy (weeks of work) and showing a few examples would
not do proper justice to this. We are uncertain whether the reviewer disagrees with our statement and/or
why he/she thinks these intermediate results should be presented in the paper. Is that really that interesting
for the reader? We added a short description in the appendix to illustrate the proper term splitting in
the example of (5) Astraea.   

Comment:
"The uncertainties of ap , ep and sin ip were obtained as the RMS of the proper elements computed from different
intervals within the 10-Myr integration time span." Additional information on how uncertainties are calculated,
such as the choice of intervals for these calculations, would enhance the transparency and reproducibility of
the results.

Answer:
Sure, we used 5 equally spaced intervals that cover the whole integration timespan. The following text was added:

"We tested different choices and, for the final catalog, used five equally spaced intervals that cover the whole
integration timespan. Using more intervals slows down the calculation but does not significantly improve the
estimate of uncertainties."

Comment:
In relation to the previous comments, I suggest including in this section a schematic diagram showing the steps
performed to compute the proper elements. That would make the procedure much more straightforward to follow.

Answer:
Flow diagrams are useful for complicated projects with many teams/team members who need to efficiently communicate.
This is not the case here. We therefore prefer to pass on this suggestion. 

Comment, 
Section 3: Catalog
"It is clear from these figures that the asteroid belt shows complex orbital structure (asteroid families,
resonances, etc.), which is not obvious from the distribution of osculating elements (Figs. 1C-D). That is,
in fact, the chief motivation behind calculating the proper elements: in the proper element space, various
orbital structures come into focus."
Prior studies that have observed intricate orbital structures should be acknowledged to link
the current findings within the existing literature. (e.g. Milani & Knezevic, Icarus 1994; ).

Answer:
Sure, reference to Milani & Knezevic 1994 was added here. Other relevant references are already cited in the
introduction.   

Comment:
"In about 64% of cases the precision is better than 10 m/s, which should be satisfactory for the identification
of even very compact families."
Though this statement could be correct, an explanation of why this precision should be satisfactory would be
welcome here.

Answer:
OK, we added that the smallest distance cutoff used in HCM is ~10 m/s, such that if the precision of proper
elements is better than 10 m/s, this is good enough. We added:

"The HCM cutoff used in the identification of the most compact asteroid families is $d_{\rm cutoff}\simeq 10$ m/s.
Thus, even the most compact asteroid families can be identified if the precision of proper elements is better
than 10 m/s."

Comment:
Section 4: Asteroid families
In my view, the paper does not bring enough scientific novelty to warrant publication without the results
on families. Therefore, although the preliminary results are welcome, the results presented in this section
must meet certain standards.

Answer:
This section was revised and significantly improved. There are 136 new (statistically significant) families
reported here. This is both novel and significant. See below for our answer to reviewer's specific comments. 

Comment:
Unfortunately, the section is written well below the journal's standards. The methodology used to
identify the families is not explained at all. Instead, the authors refer to other papers or simply
claim that something is "obvious" without providing any evidence. Moreover, the list of "new" families
includes groups that have been identified in prior studies.

Answer:
These issues have been corrected (see below). As for the methodology, we referred the reader to Nesvorn\'y
et al. (2015) where the methodology was described in detail. Given the reviewer's comment we added new
section 2.2 where we explain things in detail.

Comment:
"For example, in the left panels of Fig. 4, there are two streaks for ap > 3.176 au, one with slightly
higher and one with slightly lower proper eccentricities, that appear to diverge from each other with
increasing semimajor axis. The upper streak with ep ≃ 0.065-0.07 has lower proper
inclinations (sin ip ≃ 0.154-0.158)."
I fully agree that the structure of the Veritas family, as obtained with the new set of proper elements,
is well-defined and interesting. However, although the presentation of a larger number of objects highlights
some features more clearly, it does not reveal any new patterns not already visible in existing data from
the AFP. Especially, two streaks mentioned by the authors are also visible in the AFP data. This should
be clearly acknowledged.

Answer:
Note that the original text did not claim that this is a new feature. To the contrary. The two left panels
mentioned above are the new catalog *and the AFP catalog* - both show the same feature. We now explicitly say
that these streaks are visible in the AFP catalog. Text added:

"(these streaks were already visible in the AFP data)"

Comment:
"All together, we found 7 new families in the inner belt, 34 new families in the middle belt, 15 new families
in the pristine zone, and 29 new families in the outer belt:"
I'm not sure what cross-checking was performed by the authors to verify if an identified family is new.
However, a quick comparison with results from Novakovic et al. (2011) and Carruba et al. (2019) shows that
there are groups labeled here as new, but are actually already proposed by other authors. E.g. (260) Huberta,
(1390) Abastumani, (1312) Vasar, and (194) Prokne. Also (20674) is a well-known family containing active
asteroid 331P/Gibbs (P/2012 F5)... given that I suspect many families listed here as new had already been
identified. I urge the authors to do a job properly and put verified and reliable information in the paper.

Answer:
Thanks for this input. Indeed the reviewer is correct that the five families mentioned above were known
previously. Please note that this part was significantly revised. Tables 3-7 report a complete list of
new families from this work. We first established that the detected family was not reported in the synthesis
of asteroid families published in the Asteroids IV book (Nesvorny et al. 2015). We than searched recent
publications to establish whether individual families were or were not reported previously. Citations previous works were
added in Tables 3-7. There are 153 new families with respect to Nesvorny et al. (2015) and 136 new
families with respect to all previous work.

Comment
"In all these cases, the new families clearly stand out from the background such that their significance is undisputed."
I do not know what this sentence is based on, given that no evidence is presented except for the plots,
which are far from clear evidence. I do not have a specific reason not to believe in the statement,
but science should be based on evidence, not on beliefs.

Answer:
OK. We performed two statistical tests to demonstrate that the new families identified here are statistically
significant. The first test involves counting all know asteroids in a box in proper a,e,i around the identified family.
This gives us the number of asteroids N1 (both family and background). We then distribute N bodies in the box and run
HCM to identify the maximum number of asteroids N2 that can be grouped together at the velocity cutoff V_cut, where
V_cut was the cutoff velocity originally used to pull out the family from the catalog. We repeat this process 1000
times and find, in all tested cases, that N2 is always much smaller than N3, where N3 is the number of real
family members identified at cutoff V_cut. This demonstrates that new families are significant at (at least) the
99.9% level. The second test involves created a box above and below the real family in the proper inclination;
these two boxes represent the local family background. We count the number of bodies in each box, this gives n1
(upper box), n2 (middle box with the family) and n3 (lower box), where n2 is always larger than n1 and n3 (because
the middle box contains the family). We than establish how likely it is, just from random statistical fluctuation,
with the background density (n1+n2+n3)/3, to obtain n>=n2 in the middle box. These probabilities are exceedingly low.  

New section 2.2 describes these methods, revised section 4 reports the results. We added new Table 8 and
additional text:

"In the great majority of cases, the new families clearly stand out from the background such that their
significance is undisputed. We first illustrate the methods described in Sect. 2.2 for
the (3787) Aivazovskij  (pristine zone) and (4291) Kodaihasu (outer belt) families. These new
families are compact but not exceedingly so; they represent typical
cases of compact families identified here. (3787) Aivazovskij has 12 members identified with
$d_{\rm cut}=15$ m/s, (4291) Kodaihasu has 16 members identified with $d_{\rm cut}=20$ m/s. 
Applying the HCM-based method described in Sect. 2.2, we establish that both families are
significant at least at the 99.9\% level. In the case of (3787) Aivazovskij, there are 12 bodies
in the box around the family ($n_2=12$), and no bodies in boxes directly below or above ($n=12$).
The probability that this happens by chance is only $\simeq 2.6\times10^{-6}$. In the case of (4291) Kodaihasu,
there are 16 bodies in the box around the family ($n_2=16$), and no bodies in boxes directly
below or above ($n=16$). The probability that this happens by chance is only $\simeq 3\times10^{-8}$
($10^8$ trials used here). This demonstrates that the two families are statistically significant. 

We applied the box method (Sect. 2.2, Rozehnal et al. 2016) to all families identified here. In 137 cases,
representing $\simeq 90$\% of the total of 153, the number of trials ($10^7$) was insufficient to distinguish
the statistical significance from 1. This shows that these families are significant at least at
5 sigma level. Table \ref{\tab6} shows the statistical significance for the remaining 16 families.
Most of these families are compact and contain a small number of members; these small families are more
likely to be produced by statistical fluctuations. Still, in most cases, the probabilities reported
in Table \ref{\tab6} are comfortably small. In addition, many of the compact families show clustered
orbital longitudes and past convergence (see below). This includes the family around (9332) 1990SB1.
The statistical significance of these families would greatly increase if these properties were taken
into account.

The two most problematic families are (240) Vanadis (17 members, 0.003 probability), (22766) 1999AE7
(5 members, 0.01 probability) and (77882) 2001 SV124 (316 members, 0.05 probability). We prefer
to report these families here but acknowledge that these cases need a more detailed analysis and/or
confirmation from additional data. It is useful to include them here to see whether these borderline
case will or will not be confirmed. The (77882) 2001 SV124 is one of only two larger families (with
more than 20 members) -- the other one being (106) Dione -- that is listed in Table \ref{tab6}."


Comment:
Figs. 10 and 12: Please indicate in the captions how many bodies are included in each computation.

Answer:
Done. The plots included all known members in the two families, 17 in Figure 10 and 9 in Figure 12.
There are 16 and 8 lines in these figures, respectively, because the angles are plotted relative to the
largest member. 

We thank the reviewer for critical comments.