The “Drake Equation” of Exomoons—A Cascade of Formation, Stability and Detection

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper provides a valuable review of the formation, stability, and detectability of exomoons, which will undoubtedly serve as a useful reference in the field. However, as a review paper, I would encourage the authors to incorporate additional research, particularly on the topics of formation and stability.

Major problems

1. Regarding formation scenarios (Section 3), could you elaborate on the limitations and challenges associated with different scenarios for regular satellites?

2. How does the migration of planets impact the formation of satellites?

3. In terms of stability, it would be beneficial to provide some quantified survival rates. This should not be limited to tidal models (refer to https://iopscience.iop.org/article/10.1088/1538-3873/abfe04#paspabfe04t1, Fig 3), but could also consider other models such as photo-evaporation (refer to https://iopscience.iop.org/article/10.3847/0004-637X/833/1/7, column “Orbit planet” of Table 2). It would be best to display these rates on the same chart. 

4. For Equation 1, are there any quantified theoretical or simulated results for f_formed (the fraction of moons forming in planet systems in general), especially for regular satellites in solar-like star systems?

5. Currently, the paper only contains eight figures and tables, half of which do not summarize results or reiterate key findings. I suggest adding more critical figures and tables, preferably those that summarize and distill important information. Including key results from previous studies would also be beneficial.

 

Minor problems

1. Please check Equation 3.

2. For Figure 1, could you clarify the meaning of the solid and dashed lines, as well as the y-axis, and check if the blue dashed line represents the Hill radius or γ R_Hill?

3. It seems unnecessary to include the images in Table 1. Could you clarify the meanings of "regular" and "irregular"? And why are there two question marks?

4. Please check the references as there are some strange symbols appearing, such as "!" (line 328) and "?" (line 620).

5. On line 347, it seems "Figure 2" should actually be "Figure 1."

 

I look forward to seeing the revised manuscript. Thanks!

 

Comments on the Quality of English Language

Good quality of English.

Author Response

This paper provides a valuable review of the formation, stability, and detectability of exomoons, which will undoubtedly serve as a useful reference in the field. However, as a review paper, I would encourage the authors to incorporate additional research, particularly on the topics of formation and stability.

Major problems

1. Regarding formation scenarios (Section 3), could you elaborate on the limitations and challenges associated with different scenarios for regular satellites?
Answer: We summarized the advantages and the limitations, restrictions on the three main regular moon formation scenarios in subsect. 3.1.1., 3.1.2., as well as 3.1.3.

2. How does the migration of planets impact the formation of satellites?
Answer: We can investigate this question from two points of view: 1) during planet migration the moon system of the given planet is endangered by the perturbations of the close encounter with other planets, resulting in the disintegration of the partially formed/newly completed moon system; 2) migrating giant planets can perturb the nearby planetesimals/planetary embryos orbit, or scatter into other parts of the young planetary system; if the scattered objects cross the orbit of planets, it can lead to collisions with the planets (see a Moon-forming giant impact-like event).

Namouni (2010) found that if the planetary migration time is longer than the period of a given moon, then the migration has no direct influence on the moon's orbit. If the planet migrates inward even closer to the host star, the size of the planet's Hill sphere decreases, thus, the formed or the forming moons become unstable (especially the long-period moons).  

3. In terms of stability, it would be beneficial to provide some quantified survival rates. This should not be limited to tidal models (refer to https://iopscience.iop.org/article/10.1088/1538-3873/abfe04#paspabfe04t1, Fig 3), but could also consider other models such as photo-evaporation (refer to https://iopscience.iop.org/article/10.3847/0004-637X/833/1/7, column “Orbit planet” of Table 2). It would be best to display these rates on the same chart. 
These results have been incorporated into the paper, in the last paragraph of Sect. 4. We decided not to plot the results on our figures, because of the very different physics in the background, and the very different nature of the tested systems (mono versus multiple satellites) would be very confusing in the same figure. We however discussed that the limit of tidal stability is more outwards (~0.4 AU) than for the limit of evaporation stability (0.1 AU), therefore for mono satellite systems, accounting for the tidal formalism limit itself will be sufficient.

 

4. For Equation 1, are there any quantified theoretical or simulated results for f_formed (the fraction of moons forming in planet systems in general), especially for regular satellites in solar-like star systems?
It is currently impossible to give a somewhat orienting estimate for f_formed, especially in the moon size range that can be promising for a detection. We emphasized this statement in the text, and added the appropriate references.

5. Currently, the paper only contains eight figures and tables, half of which do not summarize results or reiterate key findings. I suggest adding more critical figures and tables, preferably those that summarize and distill important information. Including key results from previous studies would also be beneficial.
This paper has gone through a major extension, following the suggestions of all the three referees. We hope that the new additions will contribute to this issue.

 

Minor problems

1. Please check Equation 3.
Answer: Done.

2. For Figure 1, could you clarify the meaning of the solid and dashed lines, as well as the y-axis, and check if the blue dashed line represents the Hill radius or γ R_Hill?

The caption of the figure has been extended. Indeed, the blue dashed line represents gamma*Hill radius, with a selection of gamma=⅓. These have been corrected in the figure.

3. It seems unnecessary to include the images in Table 1. Could you clarify the meanings of "regular" and "irregular"? And why are there two question marks?
The definition of regular and irregular moons were given in the text, under subsection 3.1. Due to the suggestion of several Referees, we have relocated this part as the final section of Sect 3, and added extensions to emphasize more that the actual definitions are seen here.

Concerning the figure at the Table, it has of course no scientific interpretation. However, eye catchers are commonly accepted in e.g. conference posters and slides, and we decided to fill in the otherwise empty space here accordingly. It also gives the table an emphasis (it is the most summarizing and for content, the most exact table in the paper anyways), and also offers a graphical element if the authors or the Readers wish to use the table in a slide or poster later.

4. Please check the references as there are some strange symbols appearing, such as "!" (line 328) and "?" (line 620).
Answer: the reference list has been updated. We no longer use the outdated \bibitem{} format.

5. On line 347, it seems "Figure 2" should actually be "Figure 1."
Answer: Done.
 

I look forward to seeing the revised manuscript. Thanks!

 

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

 

This paper provides a “Drake-equation” for exomoon detection, in the form of a series of cascades of formation, stability and detection. The paper introduces a background of the current scientific understanding of exomoon formation, exomoon stability, and exomoon detection hopes. The authors do a good job of summarizing the history of exomoon searches and outline a pathway forward towards potentially discovering confirmed exomoons. The paper presents the formula for the expected number of confirmed exomoon observation by multiplying the number of exoplanets detected by a given observing technique by (1) the fraction of moonsforming in planet systems in general, (2) the fraction of moons surviving by the time of observation, and (3) the probability of an observation of the moon based on a given observation strategy. By drawing the parallel to the Drake-equation, which was used to outline a conference on the search for extraterrestrial life, by representing the factors that play a role in the search for extraterrestrial life, the paper does not quantitatively estimate the probability of an exomoon detection.

 

The paper overall is written clearly, although there are some sections (particularly section 6.3) that could be shortened, reorganized, and made clearer. Overall, I enjoyed reading the paper, and believe that the paper would be a nice contribution to the field of exomoon astronomy. Therefore, I believe that after some revisions, the paper would warrant publication.

 

I have several questions about some of the decisions made and ways in which the authors see the paper being best utilized. I think by making this intentionality more clear the paper could be strengthened.

 

Major Questions / Suggestions:

1. Is there an inherent benefit in drawing such a strong comparison between the Drake-equation and this formalism for exomoon detection? I ask this because the Drake-equation is somewhat infamous for being misused and I wonder if drawing such a dominant parallel between the two (ie. in the title) could diminish the perceived utility of the paper? Further, the Drake-equation was first presented as the outline for a SETI conference and I wonder if the results in this paper may be more directly applicable if the cascading structure of exomoon detection was framed as yield calculations for particular formation pathways. Thinking of comparisons of exoplanet surveys that tend to focus on different stars, and especially given different formation pathways, do you think it would be helpful if this was framed as yield calculations for particular formation pathways rather than one all-encompassing equation, which would allow observers to provide information about (ie rule out or confirm) specific formation channels? Another approach could be to frame the three cascading probabilities as the three filters of exomoon detection. I think this could be beneficial, as the authors point out, that we know much more about the different pathways of formation, stability, and detection of exomoons than we currently do for the original Drake-equation. For example, this version of the equation could allow an observer to say “our survey for cold-Jupiters with X formation channel predicts Y exomoon detections and we found Z exomoons, therefore something is awry with our models.” Lastly, as you mention, in this formula, the fractions are not independent of one another, while the Drake-equation fractions are independent. By presenting the formula as a yield prediction, an observer could choose parameters in their search based on their targets and thus create a mutually independent version of your formula.

2. The paper would be strengthened by a larger discussion of false positives and how false positives might affect the use of this equation. In any survey, the number of detections is a sum of true positives and false positives (e.g. the false positive rate of validated transiting exoplanets is 1% or less). Some consideration of the impact into the yield here would be beneficial.

3. I really liked the discussion of different detection methods that have been used in exomoon searches and could be possibly used in the future for exomoon detection. I think the paper would be strengthened by a brief discussion of the benefits of combining multiple detection methods. Combining multiple detection methods would be helpful in 2 primarily ways:

1. Different detection methods are often complementary in several ways (transit detections provide radii, TTVs provide masses, RVs provide masses, astrometry provide masses, etc.) Additionally, by combining multiple detection methods will often have different detection biases that can be utilized to probe different regions of parameter space. 

2. By observing an exomoon candidate with different independent detections methods, one can decrease the likelihood of false positive signals.

A parallel could be made to the history of exoplanet discovery – where by combining radial velocity and transit methods of the early exoplanets (such as 51 Peg b) astronomers were able to both confirm the exoplanet detections and also learn that the early RV detections of hot-Jupiters were not a good representation of the ultimate distribution that begun to be revealed in exoplanet demographics.

4. There are several places where the writing in the text leans a little towards opinion and speculation. I would recommend removing this from the text and point out several examples in the minor comments section below.

 

Minor Comments:

 

1. Lines 3-4: The sentence starting with “Following the “cascade” structure….” is confusing and should be reworded.

2. Line 8 and Line 12: Back to back sentences start with “This way” … perhaps reword one of them?

3. Line 28: Perhaps for the discussions of tidal parameter estimation should also briefly mention the recent paper by Kisare and Fabrycky (2023) titled “Tidal Dissipation in Satellites Prevents Hill Sphere Escape”.

4. Line 39: “...moons that seen in our Solar…” → “...moons that exist in our Solar…” 

5. Line 40: I would remove or reword the end of the sentence starting at “...while we do…”

6. Lines 35-42: I think there could be a deeper discussion of the difficulties of detecting exomoons compared to exoplanets. One such discussion of the inherent difficulties is in Cassese and Kipping (2022).

7. Line 50: please add description of how you define “enough large” moon – if it means large enough to be detected, isn’t this a part of #3? 

8. Line 84: I would replace the “:” after performed with a “.” And start a new sentence “It also assumes that…”

9. Line 111: Perhaps it would be helpful to list the 24 independent variables and a short description of what you mean by “run above at least 8 variables.”

10. Line 143-145: I think #4 is not really an “unsolved problem” like #1-3, but rather describes #1-3, and so would be better places before the “:” in line 136.

11. Line 146: maybe define “regular”

12. Line 208: spelling of “spreding”

13. Line 243: maybe define “irregular”

14. Equation 5: does this assume a non-eccentric moon? If yes, maybe be explicit about this assumption.

15. Figure 1: duplicate “a” in the 4th line

16. Line 327: perhaps orbital separation is a better word than radius here, for clarity.

17. Line 337: not sure what you mean by “on the prize” here?

18. Line 347: I think you mean to reference Figure 1.

19. Line 351: “was” → “has”

20. Lines 351-365: maybe point to Figure 1 again.

21. Line 366: please elaborate what you mean by tidal evolution being “the most important key” for an observable moon

22. Section 5.1: missing reference to Figure 2 in the text.

23. Line 393: I would add references to:

1. Gordon and Agol, 2022: “Analytic Light Curve for Mutual Transits of Two Bodies Across a Limb-darkened Star”

2. Kipping 2011: “LUNA: an algorithm for generating dynamic planet–moon transits”

24. Line 397: I would also cite Kipping 2021 “The Exomoon Corridor: Half of all exomoons exhibit TTV frequencies within a narrow window due to aliasing”

25. Line 407: Some of the exomoon candidates discussed in this section aren’t “proven false detections” so I would recommend re-labeling this section.

26. Line 424: I don’t believe that there was an exomoon candidate in the Kepler-1513b system, but rather a TTV signal that was identified as being plausibly consistent with an exomoon. Perhaps a better explanation would say “A TTV signal consistent with an exomoon was identified in Kepler-1513b (Kipping and Yahalomi, 2022), however, the TTV signal was subsequently shown to be better represented by a second planet in the system (Yahalomi et al., 2023).” Or something similar to this.

27. Line 423: “exo-moon” → “exomoon”

28. Line 458: “abd” → “and”

29. Figure 3: Are the transit depths of planet occulted and planet eclipsed equal to each other? Are the transit depths of companion occulted and companion eclipsed the equal to each other? It is hard to tell due to the trend in the out-of-event part of the curve and so would likely be helpful to explicitly state and explain in the caption.

30. Line 482: “mire” → “more”

31. Figure 4: maybe this is just preference, but perhaps the histogram would be easier to interpret if detection, formation, and dynamics for each year weren’t stacked on top of each other but were rather next to each on the x-axis.

32. Line 489:  “exo-moon” → “exomoon”

33. Line 609:  “exo-moon” → “exomoon”

34. Line 611:  “exo-moon” → “exomoon”

35. Line 530: I would be explicit when you say “Due to dynamical constraints…”

36. Line 540: a citation would be helpful here.

37. Line 564: “The exomoon studies had seen…” → “Exomoon studies have seen….”

38. Section 6.3: I believe that this section can be reformatted and the wording improved. If you would like to keep the bulleted format, I would suggest using bullet points, a numbering, or something equivalent. 

1. The first open question in “Formation” is more of a detection question, not a formation question?

2. Line 587: “How well the…” → “How well do the…”

3. The observations section doesn’t seem to be discussing open questions like the other 2 sections.

4. Line 599: “(very probably) false positive detections that have been already advertised.” Describing these exomoon candidates as very probably false positive detections is not supported by the current literature. My understanding is that there is an active debate and it is possible that they are false positives, but one can’t say that they are very probably false positives based on current evidence. Further, I wouldn't say that the exomoon candidates were “advertised.” Something like published or discussed or discovered is more appropriate, as “advertised” implies a non-scientific process of desimination.

5. Line 607: Please fix the beginning of the sentence starting with “is a reasonable hope…”

6. I understand that this is outside the scope of this paper, but perhaps it could be worth mentioning that this formalism could be used in the future by observers to predict how many moons Roman and JWST will detect? Maybe in the future questions for observations section word work well.

39. Line 615: “During the phe past…” → “During the past…”

Thank you for sharing this interesting work, and I look forward to seeing a revised version. I think this will be an interesting contribution to the exomoon literature.

 

Comments on the Quality of English Language

Most of my suggestions regarding the quality of the English language in the paper are described above. To repeat, I believe that overall the paper is written well, and that some minor clarifications would strengthen the paper. I think that particularly Section 6.3 could be re-written and really drive home the significant areas where discoveries in formation and stability and advances in observational capabilities could improve our exomoon detection capabilities in the near future.

Author Response

We thank the referee for reviewing our paper.

Dear Authors,
 
This paper provides a “Drake-equation” for exomoon detection, in the form of a series of cascades of formation, stability and detection. The paper introduces a background of the current scientific understanding of exomoon formation, exomoon stability, and exomoon detection hopes. The authors do a good job of summarizing the history of exomoon searches and outline a pathway forward towards potentially discovering confirmed exomoons. The paper presents the formula for the expected number of confirmed exomoon observation by multiplying the number of exoplanets detected by a given observing technique by (1) the fraction of moonsforming in planet systems in general, (2) the fraction of moons surviving by the time of observation, and (3) the probability of an observation of the moon based on a given observation strategy. By drawing the parallel to the Drake-equation, which was used to outline a conference on the search for extraterrestrial life, by representing the factors that play a role in the search for extraterrestrial life, the paper does not quantitatively estimate the probability of an exomoon detection.
 
The paper overall is written clearly, although there are some sections (particularly section 6.3) that could be shortened, reorganized, and made clearer. Overall, I enjoyed reading the paper, and believe that the paper would be a nice contribution to the field of exomoon astronomy. Therefore, I believe that after some revisions, the paper would warrant publication.
 
I have several questions about some of the decisions made and ways in which the authors see the paper being best utilized. I think by making this intentionality more clear the paper could be strengthened.
 
Major Questions / Suggestions:

Is there an inherent benefit in drawing such a strong comparison between the Drake-equation and this formalism for exomoon detection? I ask this because the Drake-equation is somewhat infamous for being misused and I wonder if drawing such a dominant parallel between the two (ie. in the title) could diminish the perceived utility of the paper? Further, the Drake-equation was first presented as the outline for a SETI conference and I wonder if the results in this paper may be more directly applicable if the cascading structure of exomoon detection was framed as yield calculations for particular formation pathways. Thinking of comparisons of exoplanet surveys that tend to focus on different stars, and especially given different formation pathways, do you think it would be helpful if this was framed as yield calculations for particular formation pathways rather than one all-encompassing equation, which would allow observers to provide information about (ie rule out or confirm) specific formation channels? Another approach could be to frame the three cascading probabilities as the three filters of exomoon detection. I think this could be beneficial, as the authors point out, that we know much more about the different pathways of formation, stability, and detection of exomoons than we currently do for the original Drake-equation. For example, this version of the equation could allow an observer to say “our survey for cold-Jupiters with X formation channel predicts Y exomoon detections and we found Z exomoons, therefore something is awry with our models.” Lastly, as you mention, in this formula, the fractions are not independent of one another, while the Drake-equation fractions are independent. By presenting the formula as a yield prediction, an observer could choose parameters in their search based on their targets and thus create a mutually independent version of your formula.

The Drake-equation was originally written to express the structure of a problem, in the form of steps of a chain of events. The cascade equation we use in this paper exactly follows this philosophy. We indeed build up the directions of argumentation along this event chain and rating the individual steps within it. The “Drake-equation” in the title expresses this choreography, and we decided to keep this title. It can be a matter of taste at a level, but in fact Reviewer 3 seemed to liked the “Drake” narrative quite well, and even suggested to widen the narrative more heavily in this direction, e.g. in the discussion with going back to the factors in the cascade equation to round up the narrative of the paper.

The comments about the “original” Drake-equation are exact, and we share the opinion of Referee #2 in all these regards. 

 

 

The paper would be strengthened by a larger discussion of false positives and how false positives might affect the use of this equation. In any survey, the number of detections is a sum of true positives and false positives (e.g. the false positive rate of validated transiting exoplanets is 1% or less). Some consideration of the impact into the yield here would be beneficial.

We have added a new subsection 5.2. about the false detections and detection statistics in general.

 

 

I really liked the discussion of different detection methods that have been used in exomoon searches and could be possibly used in the future for exomoon detection. I think the paper would be strengthened by a brief discussion of the benefits of combining multiple detection methods. Combining multiple detection methods would be helpful in 2 primarily ways:

 

 

Different detection methods are often complementary in several ways (transit detections provide radii, TTVs provide masses, RVs provide masses, astrometry provide masses, etc.) Additionally, by combining multiple detection methods will often have different detection biases that can be utilized to probe different regions of parameter space. 

 

 

By observing an exomoon candidate with different independent detections methods, one can decrease the likelihood of false positive signals.

 

A parallel could be made to the history of exoplanet discovery – where by combining radial velocity and transit methods of the early exoplanets (such as 51 Peg b) astronomers were able to both confirm the exoplanet detections and also learn that the early RV detections of hot-Jupiters were not a good representation of the ultimate distribution that begun to be revealed in exoplanet demographics.

There are several places where the writing in the text leans a little towards opinion and speculation. I would recommend removing this from the text and point out several examples in the minor comments section below.

We added a subsection about multiple detections to the manuscript. Also, we went through the following list meticulously to make sure that all these points have been removed or clarified properly.

 

 
Minor Comments:
 

Lines 3-4: The sentence starting with “Following the “cascade” structure….” is confusing and should be reworded.

This part have been reworded.

 

 

Line 8 and Line 12: Back to back sentences start with “This way” … perhaps reword one of them?
Answer: We reworded the second sentence.

 

 

Line 28: Perhaps for the discussions of tidal parameter estimation should also briefly mention the recent paper by Kisare and Fabrycky (2023) titled “Tidal Dissipation in Satellites Prevents Hill Sphere Escape”.
Answer: The reference is included now.

 

Line 39: “...moons that seen in our Solar…” → “...moons that exist in our Solar…” 
Answer: Done.

 

Line 40: I would remove or reword the end of the sentence starting at “...while we do…”
This half sentence has been removed

 

Lines 35-42: I think there could be a deeper discussion of the difficulties of detecting exomoons compared to exoplanets. One such discussion of the inherent difficulties is in Cassese and Kipping (2022).

This paper and two additional references have been added.

 

 

Line 50: please add description of how you define “enough large” moon – if it means large enough to be detected, isn’t this a part of #3? 

A clarifying footnote has been added about it.

 

 

 

Line 84: I would replace the “:” after performed with a “.” And start a new sentence “It also assumes that…”
Answer: Done.

 

Line 111: Perhaps it would be helpful to list the 24 independent variables and a short description of what you mean by “run above at least 8 variables.”

These are now listed as the first paragraph of the subsection “The parameter domain”

 

Line 143-145: I think #4 is not really an “unsolved problem” like #1-3, but rather describes #1-3, and so would be better places before the “:” in line 136.

The layout has been modified. #4 is unnumbered now, but it is a bulleted item, showing that it is an important aspect of all the other questions #1-3.

 

 

Line 146: maybe define “regular”
The definition of regular and irregular moons were given in the previous version of the text under subsection 3.1. Due to the suggestion of several Referees, we have relocated this part as the final section of Sect 3, and added extensions to emphasize more that the actual definitions are seen here.

 

 

Line 208: spelling of “spreding”
Answer: Done.

 

Line 243: maybe define “irregular”
Answer: Done, see comment #10

 

Equation 5: does this assume a non-eccentric moon? If yes, maybe be explicit about this assumption.

A clarifying footnote has been added about the eccentricity.

 

 

Figure 1: duplicate “a” in the 4th line
Answer: Done.

 

Line 327: perhaps orbital separation is a better word than radius here, for clarity.
Answer: Done.

 

Line 337: not sure what you mean by “on the prize” here?

This sentence has been rephrased.

 

Line 347: I think you mean to reference Figure 1.
Answer: Done.

 

Line 351: “was” → “has”
Answer: Done.

 

Lines 351-365: maybe point to Figure 1 again.
Answer: The reference is added, but for Fig 2 (Fig 1 in the previous version).

 

Line 366: please elaborate what you mean by tidal evolution being “the most important key” for an observable moon.

This sentence has been rephrased.

 

 

Section 5.1: missing reference to Figure 2 in the text.
Answer: Reference to Fig 2 was on line 396, and therefore it was not missing.

 

Line 393: I would add references to:

The suggested references have been added.

 

 

Gordon and Agol, 2022: “Analytic Light Curve for Mutual Transits of Two Bodies Across a Limb-darkened Star”
Answer: Done.

 

Kipping 2011: “LUNA: an algorithm for generating dynamic planet–moon transits”
Answer: Done.

 

Line 397: I would also cite Kipping 2021 “The Exomoon Corridor: Half of all exomoons exhibit TTV frequencies within a narrow window due to aliasing”
Answer: Done.

 

Line 407: Some of the exomoon candidates discussed in this section aren’t “proven false detections” so I would recommend re-labeling this section.

Has been reworded to possible false detections.

 

 

Line 424: I don’t believe that there was an exomoon candidate in the Kepler-1513b system, but rather a TTV signal that was identified as being plausibly consistent with an exomoon. Perhaps a better explanation would say “A TTV signal consistent with an exomoon was identified in Kepler-1513b (Kipping and Yahalomi, 2022), however, the TTV signal was subsequently shown to be better represented by a second planet in the system (Yahalomi et al., 2023).” Or something similar to this.

This part has been rephrased.

 

 

Line 423: “exo-moon” → “exomoon”

Answer: Done.

 

Line 458: “abd” → “and”

Answer: Done.

 

 

Figure 3: Are the transit depths of planet occulted and planet eclipsed equal to each other? Are the transit depths of companion occulted and companion eclipsed the equal to each other? It is hard to tell due to the trend in the out-of-event part of the curve and so would likely be helpful to explicitly state and explain in the caption.

The figure shows a symbolic representation that we made clear in the caption and the text.

 

 

Line 482: “mire” → “more”

Answer: Done.

 

Figure 4: maybe this is just preference, but perhaps the histogram would be easier to interpret if detection, formation, and dynamics for each year weren’t stacked on top of each other but were rather next to each on the x-axis.

We have considered this possibility and decided to stay with the current version of the plot, because its envelope shows the cumulative intensity of the related ``exomoon science’’ together.

 

 

Line 489:  “exo-moon” → “exomoon”

Answer: Done.

 

Line 609:  “exo-moon” → “exomoon”

Answer: Done.

 

Line 611:  “exo-moon” → “exomoon”

Answer: Done.

 

Line 530: I would be explicit when you say “Due to dynamical constraints…”

This sentence has been rephrased.

 

 

Line 540: a citation would be helpful here.
Answer: Added citation.

 

Line 564: “The exomoon studies had seen…” → “Exomoon studies have seen….”
Answer: Done.

 

Section 6.3: I believe that this section can be reformatted and the wording improved. If you would like to keep the bulleted format, I would suggest using bullet points, a numbering, or something equivalent. 

 

 

The first open question in “Formation” is more of a detection question, not a formation question?

This part has been rephrased.

 

 

Line 587: “How well the…” → “How well do the…”

Answer: Done.

 

The observations section doesn’t seem to be discussing open questions like the other 2 sections.

 

 

Line 599: “(very probably) false positive detections that have been already advertised.” Describing these exomoon candidates as very probably false positive detections is not supported by the current literature. My understanding is that there is an active debate and it is possible that they are false positives, but one can’t say that they are very probably false positives based on current evidence. Further, I wouldn't say that the exomoon candidates were “advertised.” Something like published or discussed or discovered is more appropriate, as “advertised” implies a non-scientific process of desimination.
Answer: There is an ever-increasing evidence against the currently known exomoon candidates (see e.g. Heller et al 2023, the most recent comprehensive study stating that exomoons around Kepler-1625b and Kepler-1708b are unlikely). While we changed the word “advertised”, we argue that the current exomoon candidates are all false positives, as is demonstrated in many other works. 

 

 

Line 607: Please fix the beginning of the sentence starting with “is a reasonable hope…”
Answer: Done.

 

I understand that this is outside the scope of this paper, but perhaps it could be worth mentioning that this formalism could be used in the future by observers to predict how many moons Roman and JWST will detect? Maybe in the future questions for observations section word work well.
Indeed, we considered this paper as a theoretical summary, without explicit analysis of specific missions. We added, however, an extension about feasibility studies and their proper evaluation, which can help the practical estimates for a generic instrument in the future.

 

Line 615: “During the phe past…” → “During the past…”
Answer: Done.
 

Thank you for sharing this interesting work, and I look forward to seeing a revised version. I think this will be an interesting contribution to the exomoon literature.

Reviewer 3 Report

Comments and Suggestions for Authors

Review Drake equation for moons 

 

Dear Editors,

 

I have thoroughly perused the manuscript titled "The 'Drake equation' of exomoons – a cascade of formation, stability, and detection". Broadly speaking, I find it to be a rather ingenious method of addressing and reviewing the issue of exomoon detection. The manuscript intensively examines various key multidisciplinary processes that must be fulfilled to achieve the landmark discovery of exomoons, or more precisely, a population of moons that likely remain hidden within the extensive databases of exoplanet transits, radial velocities, and astrometry.

 

Given its specialised review of the current state of the art, I deem the manuscript well-suited to the interests of your journal. However, due to its inherent complexity, certain sections require further elaboration. For instance, the section on the evolution and stability of these systems could be enhanced with more recent literature, as there have been significant developments in this area. The same applies to the detectability section, which would benefit from the inclusion of additional aspects related to the detection of these systems.

 

Furthermore, I would advise the authors to meticulously revise the manuscript for textual accuracy, as it contains numerous typographical errors, missing spaces, and commas. I have also observed several incomplete or erroneous references that warrant rectification.

Abstract:

 

I recommend a careful revision of the abstract to enhance its clarity and correctness. Specifically, please consider amending grammatical missteps such as "applicated" to "applied", and adding the necessary preposition in "instead the original scope" to "instead of the original scope". Additionally, it may benefit the narrative to streamline the final sentence for greater readability. I would like to commend the intriguing premise of structuring the detectability problem of exomoons in the manner of the Drake Equation, which underscores the interdisciplinary significance of your research in astrophysics.

Introduction:

 

In this section, which introduces and substantiates the manuscript, I have no specific comments. The section effectively clarifies that the manuscript's objective is not to provide an estimate but to present an organised approach to the problem of exomoon detection, justifying the use of an expression analogous to the Drake equation. However, there are some missing references, for example, in item one, line 25.

 

Section 2:

 

This section introduces the structure of the problem through an equation estimating the number of detectable exomoons and describes each term of the estimate. It outlines the involved parameter space and estimates the number of relevant parameters that would allow for an estimation of this quantity. The authors contend that the terms presented in Equation 1 are independent. However, it occurs to me that there might be processes interlinking these terms in various ways. For instance, the 'demise' of a satellite colliding with another due to chaotic dynamics or tidal migration could increase the detectability of one, forming circumlunar rings, which would significantly affect planetary transits by increasing the satellite's apparent surface area (see Sucerquia et al., 2022). Nevertheless, as a preliminary approach, the terms are quite independent and succinctly describe the issue at hand.

 

Regarding Equation 2, it is unclear to me what Np represents.

 

Sec 2.1: There are some missing references, for example, the one on line 118.

 

Sec 3: Concerning formation models, an early indicator of moon formation might be the detection of synestias, which could even result from a collision between a planet and its star.

 

Sec 3.2.2: The text stating, "Because of the need to dissipate energy during the process, capturing a single major body to a stable orbit around a planet is difficult (but can happen through chaotic orbits, [44]) and is considered to happen extremely rarely," overlooks an alternative stabilization mechanism to chaos, namely, interaction and subsequent orbital circularization through tidal interactions. See, for example, Porter and Grundy 2011.

 

Section 4:

 

It is imperative to note that current studies, albeit significant, have limitations. Primarily, they do not take into account changes in the planet's rheological properties. This includes constant dissipative properties of the planetary interior, like the K2/Q term, and the shrinking planetary radius (Fortney et al., 2007; Ogilvie, 2013; Guenel, Mathis & Remus, 2014). These factors are vital in the tidal evolution of a moon, especially in the orbital evolution of more massive satellites, where an increase in the semi-major axis is always observed, heightening the risk of ejection (Alvarado, 2017; Sucerquia, 2019, 2020). Additionally, models often treat satellites as point masses, overlooking tidal deformations in the moons and the dissipation of orbital energy. This simplification could miss critical aspects, especially if lunar interiors are fluid.

 

Section 5.1:

 

For a more comprehensive understanding of the search for exomoons in transit signals, the inclusion of Transit Duration Variations (TDVs), Transit Timing Variations (TTVs; Kipping et al., 2012, 2013, 2014), and Transit Radius Variations (TRVs; Rodenbeck, Heller & Gizon, 2020) should be considered. These methods allow for the estimation of key exomoon characteristics, such as their lunar semi-major axis, mass, and size. Discussing these effects would significantly enrich this section.

 

Section 5.2.1:

 

I propose a minor but significant terminological modification: changing "The extra radial velocity of a planet orbiting its parent star" to "The extra radial velocity semi-amplitude of a planet orbiting its parent star" for greater accuracy.

 

Section 5.2.4:

 

Instead of "light curve", I suggest using "phase curve" to avoid confusion, as the configurations in figure 3 are not related to transits. A brief explanation of how phase curves differ from light curves could be beneficial.

 

Section 5.2:

 

Beyond the mentioned techniques, it is crucial to consider alternative methods for exomoon detection. This includes observation of volcanic events, detection of planetary rings, and observation of gaps caused by moons, as seen in Saturn (Zuluaga et al., 2022; Sucerquia, 2022), or direct detection of binary planets (Lazzoni et al., 2023). These techniques would complement and enrich the current landscape. Also relevant is a discussion on how classifying satellites as planets, from a geological perspective, could influence the Drake equation for exomoons.

 

Section 7:

Finally, to bring a coherent closure to the manuscript, I suggest revisiting the proposed Drake equation for exomoons. A discussion that reevaluates this equation, considering the data and concepts presented in the manuscript, would provide an integrative and reflective conclusion on the topic.

Comments on the Quality of English Language

Regarding the use of English in the manuscript, I observed several typographical errors throughout the text, some of which are readily apparent while others are more subtle. These errors, though minor, can impact the readability and professional presentation of the work. Initially, I started to list these errors individually, but I soon realized that a more efficient approach would be to recommend a thorough review of the entire manuscript. This comprehensive revision would ensure that such typographical inaccuracies are identified and corrected, enhancing the overall quality of the text. Such meticulous attention to detail will undoubtedly contribute to the clarity and effectiveness of the manuscript's communication.

Author Response

We thank the referee for their review of our paper.

Dear Editors,

 

I have thoroughly perused the manuscript titled "The 'Drake equation' of exomoons – a cascade of formation, stability, and detection". Broadly speaking, I find it to be a rather ingenious method of addressing and reviewing the issue of exomoon detection. The manuscript intensively examines various key multidisciplinary processes that must be fulfilled to achieve the landmark discovery of exomoons, or more precisely, a population of moons that likely remain hidden within the extensive databases of exoplanet transits, radial velocities, and astrometry.

 

Given its specialised review of the current state of the art, I deem the manuscript well-suited to the interests of your journal. However, due to its inherent complexity, certain sections require further elaboration. For instance, the section on the evolution and stability of these systems could be enhanced with more recent literature, as there have been significant developments in this area. The same applies to the detectability section, which would benefit from the inclusion of additional aspects related to the detection of these systems.
Answer: Formation models and references are updated with recent results.
 

Furthermore, I would advise the authors to meticulously revise the manuscript for textual accuracy, as it contains numerous typographical errors, missing spaces, and commas. I have also observed several incomplete or erroneous references that warrant rectification.

We have used Writefull widget in overleaf to find and correct or rephrase ~150 textual instances.

 

 

Abstract:

 

I recommend a careful revision of the abstract to enhance its clarity and correctness. Specifically, please consider amending grammatical missteps such as "applicated" to "applied", and adding the necessary preposition in "instead the original scope" to "instead of the original scope". Additionally, it may benefit the narrative to streamline the final sentence for greater readability. I would like to commend the intriguing premise of structuring the detectability problem of exomoons in the manner of the Drake Equation, which underscores the interdisciplinary significance of your research in astrophysics.

Answer: Done.

Introduction:

 

In this section, which introduces and substantiates the manuscript, I have no specific comments. The section effectively clarifies that the manuscript's objective is not to provide an estimate but to present an organised approach to the problem of exomoon detection, justifying the use of an expression analogous to the Drake equation. However, there are some missing references, for example, in item one, line 25.

 We went through the manuscript to include the missing references and also added quite a few new ones.

Section 2:

 

This section introduces the structure of the problem through an equation estimating the number of detectable exomoons and describes each term of the estimate. It outlines the involved parameter space and estimates the number of relevant parameters that would allow for an estimation of this quantity. The authors contend that the terms presented in Equation 1 are independent. However, it occurs to me that there might be processes interlinking these terms in various ways. For instance, the 'demise' of a satellite colliding with another due to chaotic dynamics or tidal migration could increase the detectability of one, forming circumlunar rings, which would significantly affect planetary transits by increasing the satellite's apparent surface area (see Sucerquia et al., 2022). Nevertheless, as a preliminary approach, the terms are quite independent and succinctly describe the issue at hand.

Indeed, the rings that can be formed as an outcome of satellite-satellite collisions are well observable indicators for former moons in the system. This discussion has been extended under 3.2.1, with adding several new references as well. We also emphasized more in the Introduction that this paper mostly concentrates on systems with a single dominant moon.
 

Regarding Equation 2, it is unclear to me what Np represents.

It is intended to note that the sum goes for all known exoplanets. This annotation has now been revised, and replaced with a lower + upper indexing of the summation sigma.

 

Sec 2.1: There are some missing references, for example, the one on line 118.
Added Sasaki et al. (2012)
 

Sec 3: Concerning formation models, an early indicator of moon formation might be the detection of synestias, which could even result from a collision between a planet and its star.
We completed subsect. 3.2.1. with a paragraph on the synestias based on the referee's suggestion.

 

Sec 3.2.2: The text stating, "Because of the need to dissipate energy during the process, capturing a single major body to a stable orbit around a planet is difficult (but can happen through chaotic orbits, [44]) and is considered to happen extremely rarely," overlooks an alternative stabilization mechanism to chaos, namely, interaction and subsequent orbital circularization through tidal interactions. See, for example, Porter and Grundy 2011.
Answer: Based on the suggested literature, subsect. 3.2.2. is extended now with an additional paragraph.
 

Section 4:

 

It is imperative to note that current studies, albeit significant, have limitations. Primarily, they do not take into account changes in the planet's rheological properties. This includes constant dissipative properties of the planetary interior, like the K2/Q term, and the shrinking planetary radius (Fortney et al., 2007; Ogilvie, 2013; Guenel, Mathis & Remus, 2014). These factors are vital in the tidal evolution of a moon, especially in the orbital evolution of more massive satellites, where an increase in the semi-major axis is always observed, heightening the risk of ejection (Alvarado, 2017; Sucerquia, 2019, 2020). Additionally, models often treat satellites as point masses, overlooking tidal deformations in the moons and the dissipation of orbital energy. This simplification could miss critical aspects, especially if lunar interiors are fluid.

These references have been added to the manuscript in a new subsection, with the appropriate discussions.

 

Section 5.1:

 

For a more comprehensive understanding of the search for exomoons in transit signals, the inclusion of Transit Duration Variations (TDVs), Transit Timing Variations (TTVs; Kipping et al., 2012, 2013, 2014), and Transit Radius Variations (TRVs; Rodenbeck, Heller & Gizon, 2020) should be considered. These methods allow for the estimation of key exomoon characteristics, such as their lunar semi-major axis, mass, and size. Discussing these effects would significantly enrich this section.

Answer: We added Kipping et al (2009) that explains the TDVs and Rodenbeck et al (2020) to explain the Transit Depth Variations.
 

Section 5.2.1:

 

I propose a minor but significant terminological modification: changing "The extra radial velocity of a planet orbiting its parent star" to "The extra radial velocity semi-amplitude of a planet orbiting its parent star" for greater accuracy.

 Answer: Done.

Section 5.2.4:

 

Instead of "light curve", I suggest using "phase curve" to avoid confusion, as the configurations in figure 3 are not related to transits. A brief explanation of how phase curves differ from light curves could be beneficial.

 Answer: Done.

Section 5.2:

 

Beyond the mentioned techniques, it is crucial to consider alternative methods for exomoon detection. This includes observation of volcanic events, detection of planetary rings, and observation of gaps caused by moons, as seen in Saturn (Zuluaga et al., 2022; Sucerquia, 2022), or direct detection of binary planets (Lazzoni et al., 2023). These techniques would complement and enrich the current landscape. Also relevant is a discussion on how classifying satellites as planets, from a geological perspective, could influence the Drake equation for exomoons.

These techniques have been included with the references to them.

 These possibilities have been added to the list of alternative techniques ().

Section 7:

Finally, to bring a coherent closure to the manuscript, I suggest revisiting the proposed Drake equation for exomoons. A discussion that reevaluates this equation, considering the data and concepts presented in the manuscript, would provide an integrative and reflective conclusion on the topic.

The Discussion have been extended according to this suggestion.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have made significant improvements in addressing my concerns. With a few minor corrections , it will be ready for publication. The issues mainly pertain to formatting and language, which need to be adjusted to the correct format or reorganized. Line number corresponds to the manuscript of the supplementary file.

1. Line 124: "involving tidal interactions [e.g. 27] and the equations of [27]."
2. Line 344: "(The note can be ... in detail)."
3. Line 364: "where the γ factor..." (remove the first-line indentation).
4. Lines 494-498: It is unnecessary to divide this into multiple lines, and punctuation at the end needs to be added.
5. Equation (13).
6. Equations 14-16: Missing or incorrect punctuation.
7. Line 629: "5.4.1. Combination of detection methods."(Check the section number)
8. Lines 643-647: "(note here ... Solar System)."
9. Line 800: "Full elaboration of 5.1.: Sz. K." (It should be 5.1)
10. Section 4.1: Condense it into no more than two paragraphs, without separating it into a separate section.

Please carefully review the manuscript once again.

Comments on the Quality of English Language

Good quality of English.

Author Response

Dear Editors,

We addressed the comments of Referee 1. The replies are encoled below, and all changes are marked via latexdiff.

_________________________________________________

The authors have made significant improvements in addressing my concerns. With a few minor corrections , it will be ready for publication. The issues mainly pertain to formatting and language, which need to be adjusted to the correct format or reorganized. Line number corresponds to the manuscript of the supplementary file.
Line 124: "involving tidal interactions [e.g. 27] and the equations of [27]."
Line 344: "(The note can be ... in detail)."
Line 364: "where the γ factor..." (remove the first-line indentation).
Lines 494-498: It is unnecessary to divide this into multiple lines, and punctuation at the end needs to be added.
Equation (13).
Equations 14-16: Missing or incorrect punctuation.
Line 629: "5.4.1. Combination of detection methods."(Check the section number)
n 4.1: Condense it into no more than two paragraphs, without separating it into a separate section.
Lines 643-647: "(note here ... Solar System)."
Line 800: "Full elaboration of 5.1.: Sz. K." (It should be 5.1)

>> All these points have been corrected.

SectioPlease carefully review the manuscript once again.

>> We have run the WriteFull AI based corrector again to locate potential phrasing issues and to help clarify the phrasings.
         

Reviewer 3 Report

Comments and Suggestions for Authors

Dear Editor,

 

Upon reviewing the latest version of the manuscript titled "The 'Drake Equation' of Exomoons – A Cascade of Formation, Stability, and Detection," I am pleased to note a significant improvement. The manuscript adeptly clarifies the complexities involved in the detection of moons, providing a comprehensive and extensive review of the literature that aptly frames the current challenges in this field.

 

I believe the manuscript is now in a robust state and well-prepared for publication in the journal. However, I would suggest to address some minor coments listed below and the addition of a paragraph or a few sentences in the discussion section addressing how the approach taken in this study influences the original framework of the Drake Equation. The original equation does not directly consider moons as potential habitats for life. Yet, the insights offered in this manuscript could impact the estimation of the number of potentially habitable planets ($n_e$). It would be valuable to explore whether the inclusion of moons could potentially increase $n_e$, thereby affecting the estimated likelihood of encountering extraterrestrial civilizations. Such a discussion would not only enrich the manuscript but also provide a broader perspective on the implications of exomoon research in the search for extraterrestrial life.

 

Best regards,

 

Minor comments:

 

• Please provide the reference indicated where it says "(see Simon et al. (REF) for further details)", line 510.

• I noticed that there are some author annotations in the text; please remove them and check the capitalization (sect. 6).

• In relation to the "large exomoons" identified during transit observations, it is noteworthy to consider that these findings have been alternatively interpreted as instances of 'chronomoons'. These are essentially moons of conventional size which are accompanied by rings, contributing to more pronounced transit depths. Highlighting this phenomenon could significantly enrich the ongoing discourse on the detectability of such celestial bodies. (see https://academic.oup.com/mnras/article/512/1/1032/6461099)

Comments on the Quality of English Language

A careful review of the manuscript is advised to correct minor errors, of which only a few were noted.

Author Response

Dear Editors,

We addressed all question from Referee 3. Our replies are enclosed below. All changes are marked via latexdiff.

__________________________________
Dear Editor,
 
Upon reviewing the latest version of the manuscript titled "The 'Drake Equation' of Exomoons – A Cascade of Formation, Stability, and Detection," I am pleased to note a significant improvement. The manuscript adeptly clarifies the complexities involved in the detection of moons, providing a comprehensive and extensive review of the literature that aptly frames the current challenges in this field.
 
I believe the manuscript is now in a robust state and well-prepared for publication in the journal. However, I would suggest to address some minor coments listed below and the addition of a paragraph or a few sentences in the discussion section addressing how the approach taken in this study influences the original framework of the Drake Equation. The original equation does not directly consider moons as potential habitats for life. Yet, the insights offered in this manuscript could impact the estimation of the number of potentially habitable planets ($n_e$). It would be valuable to explore whether the inclusion of moons could potentially increase $n_e$, thereby affecting the estimated likelihood of encountering extraterrestrial civilizations. Such a discussion would not only enrich the manuscript but also provide a broader perspective on the implications of exomoon research in the search for extraterrestrial life.
 
>> We have added a new subsection 6.4 about this issue.

 

 
Minor comments:
 

Please provide the reference indicated where it says "(see Simon et al. (REF) for further details)", line 510.

>>The reference has been added

 I noticed that there are some author annotations in the text; please remove them and check the capitalization (sect. 6).

>>This point has been polished.

In relation to the "large exomoons" identified during transit observations, it is noteworthy to consider that these findings have been alternatively interpreted as instances of 'chronomoons'. These are essentially moons of conventional size which are accompanied by rings, contributing to more pronounced transit depths. Highlighting this phenomenon could significantly enrich the ongoing discourse on the detectability of such celestial bodies. (see https://academic.oup.com/mnras/article/512/1/1032/6461099)

>>This reference has been added.