The Mean Moment of Inertia for Irregularly Shaped Phobos and Its Application to the Constraint for the Two-Layer Interior Structure for the Martian Moon
Konrad Willner
Bruce W. Banerdt
Sami Asmar
Round 1
Reviewer 1 Report
Review Manuscript „The mean moment of inertia for irregularly shaped Phobos and its application to the constraint for the two-layer interior structure for the Martian moon”
The manuscript describes yet another method to derive information about the interior of the Martian moon Phobos based on known quantities like bulk density, volume and principle moment of inertia or gravity potential coefficients.
The modelled two-layer interior is no new model – as is described within the manuscript. What is new is the way to approach the modelling. The findings agree with other publications and confirm our current knowledge of the interior modelling.
Overall, the manuscript is concise and describes the approach generally well. However, when it comes to the actual modelling, some information is missing. That is, which parameters are varied within the thousands of cycles that are described. Are these really just the densities and the radius? One would expect that for an optimization of three parameters, less effort is necessary.
The conclusions drawn are well argued based on the findings. Though, I am no expert on that, the argument that the earlier accretion of the core leads to a stronger compression in comparison to the outer layer found as result of the study, appears weak from my point of view. This is because the rotation rate is rather high in the case of Phobos and centrifugal forces play a role here too – pointing in the opposite direction when than the forces needed to compress material.
The manuscript needs some review – also from a language point of view. Some parts are not easily understood and, in some points, I see doublings of statements right after one another.
Detailed comments are noted below.
Lines 18/19: A core radius of 8.2 km would mean that the core almost reaches the surface. Though the mean radii are larger, local smaller radii are observed in shape models. Are there indications in image observations where this might be?
Line 33/34: “… suggesting its feature of a relatively old satellite.” It needs to be taken care with such statements. Schmedemann et al., 2014 relied on crater counting and determined ages of 3.5Ga to 4.3Ga which I would consider old. However, Ramsley & Head, 2017 argue that the surface could be a result of secondary impact cratering resulting in a much younger age than derived from crater counting. In any case please add a reference here to make sure you point out where this assumptions is based upon.
Line 76: “The interior structure of Phobos was ever ….” Suggested rephrasing:
“In the past, the interior structure of Phobos was modeled ….”
The sentence mentioned above is not quite correct. Many publications mention that due to no other information a homogeneous mass distribution is assumed. Other even try to model a more diverse interior structure. Willner (2010) did approach a multilayer modelling based on the density and forced libration observations. You mention yet other publications further along the manuscript. Hence, rephrasing of this sentence should be considered.
Line 94: It should read here “… data from the European Mars Express Mission ….”
Line 99: The link of reference [23] points to the digital terrain model in form of a gridded, map projected 2D grid. But not to a 3D representation as shown in Figure 1. Figure 1 also appears to be a different shape model than the one of the given references.
Lines 108-109: “In addition, we found the two-layer model is appropriate to formulate the modeled moment of inertia and is shown in Figure 2.” – OK but how did you find this? Somehow the explanation is missing. It is just a statement.
Lines 110 – 111: “In the two-layer model, Phobos was assumed to be composed by two layers: “ – it says twice that the two layer model is applied for Phobos here. Please rephrase.
Line 175/176 & Eq. (12): It is not quite clear what x really is. Is that a vector? Or are the single parameters estimated separately? This is a point that could be elaborated on in more detail. It is not very clear if all parameters are optimized together or if the search for an optimum parameter set is a subsequent approach.
Line 176: “… parameters needs … “ rephrasing necessary
Line 180 and Line 184: “The estimated parameters were sampled with an equal probability distribution.” Is basically repeated in line 184 with “… ) were sampled using an equal probability distribution.”
Figures 3 & 4: Why do the frequencies differ across the graphs? Is the sum of all occurrences the same ?
Line 212: “The paragraphs describe…” - > which paragraphs. Rephrasing necessary
Line 223: “to insight into how ….” -> To evaluate how changes… Rephrasing necessary
Some sections and parts need revision of the English language. I noted some points above in the comments and suggestions for authors box. This might not be all occurances of improvable language
Author Response
Response to Reviewer 1 Comments
The manuscript describes yet another method to derive information about the interior of the Martian moon Phobos based on known quantities like bulk density, volume and principle moment of inertia or gravity potential coefficients.
The modelled two-layer interior is no new model – as is described within the manuscript. What is new is the way to approach the modelling. The findings agree with other publications and confirm our current knowledge of the interior modelling.
Point 1: Overall, the manuscript is concise and describes the approach generally well. However, when it comes to the actual modelling, some information is missing. That is, which parameters are varied within the thousands of cycles that are described. Are these really just the densities and the radius? One would expect that for an optimization of three parameters, less effort is necessary.
Response 1: As we stated in the introduction section, the interior structure of Phobos remains an open issue due to insufficient measurements. In our study, we used the recently updated gravity coefficients of degree-2 and mean density to explore the possible interior structure of Phobos. In contrast to previous studies that suggested a lighter core (Guo et al., 2021) or even a negative density gradient (Dmitrovskii et al., 2022), our study found only a heavier core than the outer layer of Phobos. Furthermore, our methodology accounted for the irregular shape of Phobos, which is particularly relevant for small celestial bodies.
In Table 1, we provided the ranges of varying parameters such as core radius, core density, and the outer layer density. While the reviewers may expect more optimized parameters for the real interior structure of Phobos, the limited available data is not sufficient to constrain the quite complex interior structure of this moon. Our trade-off model, which includes two layers, is designed to detect the potential core of Phobos. We acknowledge that the mean moment of inertia is sensitive to the density distribution of a target body, but our approach provides a good reference for similar studies on the interior structure of Phobos.
Point 2: The conclusions drawn are well argued based on the findings. Though, I am no expert on that, the argument that the earlier accretion of the core leads to a stronger compression in comparison to the outer layer found as result of the study, appears weak from my point of view. This is because the rotation rate is rather high in the case of Phobos and centrifugal forces play a role here too – pointing in the opposite direction when than the forces needed to compress material.
Response 2: We appreciate the reviewer's comment regarding the likely effect of centrifugal forces on Phobos' interior structure. We deleted that the assumed scenario of re-accretion that the earlier formed core would have large density resulted from high pressure assumption. We corrected the section of conclustion and abstract accordingly. As to the study of centrifugal force’s effect on the interior structure, this matter was beyond the scope of our present study. We will consider incorporating centrifugal forces into our model and testing their effects in future research.
Point 3: The manuscript needs some review – also from a language point of view. Some parts are not easily understood and, in some points, I see doublings of statements right after one another.
Response 3: Thanks for providing valuable feedback on our manuscript. We have carefully reviewed and revised the content, removing any redundant or repetitive information. We hope that this updated version meets the recommendations made by the reviewer.
Detailed comments are noted below.
Point 4: Lines 18/19: A core radius of 8.2 km would mean that the core almost reaches the surface. Though the mean radii are larger, local smaller radii are observed in shape models. Are there indications in image observations where this might be?
Response 4: Using the mean value of core radius close to 8.2 km, we have replaced Figure 2. Please take a look at Figure 2 on page 4 for more detail.
Point 5: Line 33/34: “… suggesting its feature of a relatively old satellite.” It needs to be taken care with such statements. Schmedemann et al., 2014 relied on crater counting and determined ages of 3.5Ga to 4.3Ga which I would consider old. However, Ramsley & Head, 2017 argue that the surface could be a result of secondary impact cratering resulting in a much younger age than derived from crater counting. In any case please add a reference here to make sure you point out where this assumptions is based upon.
Response 5: The study of Schmedemann has been cited in our manuscript as follows
‘Phobos’ surface is heavily cratered, with many craters spread over its surface, likely suggesting its feature of a relatively old satellite [2].’.
Please take a look on lines 33-34 on page 1 for more detail.
Point 6: Line 76: “The interior structure of Phobos was ever ….” Suggested rephrasing: “In the past, the interior structure of Phobos was modeled ….”. The sentence mentioned above is not quite correct. Many publications mention that due to no other information a homogeneous mass distribution is assumed. Other even try to model a more diverse interior structure. Willner (2010) did approach a multilayer modelling based on the density and forced libration observations. You mention yet other publications further along the manuscript. Hence, rephrasing of this sentence should be considered.
Response 6: We have replaced the expression as follows
‘Due to the lack of information on Phobos' interior structure, it has been previously assumed to be homogeneous. However, other models have proposed a multilayer structure based on density and forced libration observations [10] or a rubble pile composition that includes voids and/or water ice [1,19-21].’
Please take a look on lines 80-81 on page 2 for more detail.
Point 7: Line 94: It should read here “… data from the European Mars Express Mission ….”
Response 7: We have corrected this sentence accordingly. Please take a look on Line 98 on page 3 for more detail.
Point 8: Line 99: The link of reference [23] points to the digital terrain model in form of a gridded, map projected 2D grid. But not to a 3D representation as shown in Figure 1. Figure 1 also appears to be a different shape model than the one of the given references.
Response 8: Citation [23] provided DTM data in latitude and longitude format. To produce the three-dimensional shape data presented in Figures 1-2, we converted this data into rectangular coordinates (x, y, and z).
Point 9: Lines 108-109: “In addition, we found the two-layer model is appropriate to formulate the modeled moment of inertia and is shown in Figure 2.” – OK but how did you find this? Somehow the explanation is missing. It is just a statement.
Response 9: We have added the explanation in Line 112-120 on page 3 as follows
‘We opted for a two-layer model when estimating Phobos' internal structure for several reasons. Firstly, the limited constraint information available for Phobos makes multi-layer models highly uncertain. Secondly, given the small volume of Phobos, it is unlikely that large-scale differentiation processes occurred in its interior. Finally, we accounted for Phobos' irregular shape when calculating its mean moment of inertia. Models with more than three layers would increase complexity and hinder the estimation of model parameters. Therefore, based on these considerations, we only used a two-layer model to estimate Phobos' mean moment of inertia. The two-layer models are shown in Figure 2.’
Please take a look at these Lines 112-120 on Page 3 fore more detail.
Point 10: Lines 110 – 111: “In the two-layer model, Phobos was assumed to be composed by two layers: “ – it says twice that the two layer model is applied for Phobos here. Please rephrase.
Response 10: We have rephrased the expression as follows
‘Our two-layer model assumed that Phobos is composed of an inner solid core with a density of rc, and an outer layer that is likely a mixture of rock and ice with a density of rm.’
Please take a look on Lines 120-122 on Page 3 for more detail.
Point 12: Line 175/176 & Eq. (12): It is not quite clear what x really is. Is that a vector? Or are the single parameters estimated separately? This is a point that could be elaborated on in more detail. It is not very clear if all parameters are optimized together or if the search for an optimum parameter set is a subsequent approach.
Response 12: The parameter x represents the parameters to be estimated, such as the core radius, core density, and outer layer density. We use x to represent these parameters for convenience in describing the sensitivity function S(f, x). The sensitivity function is estimated by Equation (12) for different estimated parameters (e.g., rc, rc and rm).
Point 13: Line 176: “… parameters needs … “ rephrasing necessary
Response 13: We have rephrased the sentences as follows
‘The best values for these parameters are those that result in misfit values close to zero, as a close-to-zero misfit indicates a good fit between the model and observed data.’
Please take a look on Lines 186-188 on Page 6 for more detail.
Point 14: Line 180 and Line 184: “The estimated parameters were sampled with an equal probability distribution.” Is basically repeated in line 184 with “… ) were sampled using an equal probability distribution.”
Response 14: We have removed the sentence at Line 180 in the original text to avoid repetition.
Point 15: Figures 3 & 4: Why do the frequencies differ across the graphs? Is the sum of all occurrences the same ?
Response 15: The reason for the difference in frequency counts between Figures 3 and 4 is due to the plotting process. In order to generate a clearer display, we removed the data points with very small frequencies close to zero, which caused some slight differences in the frequency counts for parts of the two models that are greater than zero.
Point 16: Line 212: “The paragraphs describe…” - > which paragraphs. Rephrasing necessary
Response 16: We have replaced the phrase “The paragraphs” as “Figure 3 and Figure 4”. Please take a look on Line 224 on Page 7 for more detail.
Point 17: Line 223: “to insight into how ….” -> To evaluate how changes… Rephrasing necessary
Response 17: We have rephrased the expression as follows
‘If the estimated parameters are not sensitive to the misfit function, their optimized values will be meaningless. Therefore, we used Eq. (12) to estimate the sensitivities for these estimated parameters based on the optimized values listed in Table 2’.
Please take a look on Lines 235-237 on Page 8 for more detail.
Author Response File: 
Author Response.docx
Reviewer 2 Report
The authors investigate some interior properties of Phobos. In particular, by applying a 2-layer model, they provide estimates of the moon mean density and moment of inertia, in terms of frequency distributions. This methodology can be extended to other irregularly-shaped bodies of the solar system (provided the necessary observations are available). This study suggests the presence of a density gradient inside the moon, from the more dense deep interior toward the less dense surface. Based on these results, the authors could provide some additional discussion to highlight their significance. For example, how does the ice fraction in the moon "mantle" compare with estimates from the surface of Mars? Would the higher density of the "core" be related more to the lower ice fraction or more to the lower porosity? Could the author think of a mechanism to produce an ice-poor core and a ice-rich mantle? Would it be possible to foresee a scenario in which the "core" may actually represent a primitive "planetesimal" on which solids were added during the "re-accretion" process?
The article is clear and generally well-written. It could, however, use an English language revision to correct nouns (e.g., "histograms" instead of "paragraphs") and verbs in some places.
Author Response
Response to Reviewer 2 Comments
The authors investigate some interior properties of Phobos. In particular, by applying a 2-layer model, they provide estimates of the moon mean density and moment of inertia, in terms of frequency distributions. This methodology can be extended to other irregularly-shaped bodies of the solar system (provided the necessary observations are available). This study suggests the presence of a density gradient inside the moon, from the more dense deep interior toward the less dense surface. Based on these results, the authors could provide some additional discussion to highlight their significance. For example,
Point 1: how does the ice fraction in the moon "mantle" compare with estimates from the surface of Mars? Would the higher density of the "core" be related more to the lower ice fraction or more to the lower porosity? Could the author think of a mechanism to produce an ice-poor core and a ice-rich mantle? Would it be possible to foresee a scenario in which the "core" may actually represent a primitive "planetesimal" on which solids were added during the "re-accretion" process?
Response 1: As stated in our manuscript, previous studies have suggested that Phobos may possess either a denser or lighter core, as well as positive or negative density gradients (Guo et al., 2021; Dimitrovskii et al., 2022). However, Guo et al. (2021) demonstrated that a lighter core was more likely, and Dimitrovskii et al. (2022) showed that both negative and positive density gradients were possible, as well as a rubble pile structure. Our study, on the other hand, indicates that Phobos is more likely to have a denser core than outer layer. The difference in results between our study and previous work may be due to differences in the methodologies used to detect the interior structure of Phobos. We believe that with more measurement data available in the future, the interior structure of Phobos can be better constrained.
Our two models gave a mean core radius close to 8.2 km, a mean core density around 2500 kg×m-3, and a mean outer layer density close to 1400 kg×m-3. We further tested the sensitivities of the three estimated parameters, which indicated that they are all sensitive to the misfit function and thus the optimized values are robust. Therefore, our study provides a good reference for constraining the interior structure of Phobos, and can be extend to the constraint of interior structure of other small bodies with irregular shapes. Assuming that the density difference between the core and outer layer is associated with water-ice content, we found that the outer layer has an approximate water-ice content of 11%, with a rock density of 2200 kg×m-3, while the core has a lower water-ice content of around 2.4%, with a rock density of 3000 kg×m-3. We provided this example solely to illustrate the relationship between density difference and water-ice content. Whether the density difference is associated with porosity is beyond the scope of our discussion, but we intend to study this issue in our future work.
Our finding of a denser core than outer layer in Phobos provides important information for studying its interior structure, particularly for understanding its formation. Whether the core revealed by our study represents a primitive "planetesimal" is beyond the scope of discussion in our work, but we plan to explore this subject in future research. We also corrected the sections of abstract and conclusion accordingly. Please take a look at the sections of abstract and conclusions in the updated manuscript for more detail.
Author Response File: Author Response.docx
Reviewer 3 Report
This work is interesting and could be a nice addition to the literature regarding the interior structure of irregular bodies and perhaps even provide constraints on the origin of Phobos. But there are some glaring omissions in the description of the method, and the discussion and conclusions seem unfocused and vague.
The paragraph covering lines 183-191purport to describe the crux of the analysis in which the most likely coupled parameters of density and radius are estimated, but it has no detail whatsoever. it is impossible to evaluate the validity of the parameter estimation without understanding the method.
Similarly, lines 223-233 describe a sensitivity analysis on the varables. But It is difficult for me to evaluate this sensitivity analysis, as “sensitivity” is never even defined, and conclusions are stated without support, e.g., "However, a 1% increase in core density for 230 Model-II would decrease the sensitivity of S2(f, ?c) by 2.4%. Overall, the estimated parameters are sensitive to changes in the misfit function. These inversed parameters are thus 232 reliable in this study." Why are they reliable? Are these percentages significant? Why or why not?
There are many places where similar conclusions are made without support:
lines 193-201: How do we know that 1500 particles are sufficient to “estimate the best fit”? How do we know that 1000 iterations are sufficient to “ensure stability”? How do we know that the PSO calculation has been performed enough times to “reduce the impact of random noise”?
lines 203-206: Why don’t the libration amplitudes of Burmeister et al. work in this analysis? What is the justification for ignoring this data? Do you consider their results to be in error?
Finally, the conclusions are very weak. What are the implications for the origin of Phobos (capture vs. accretion in place)? How do the derived densities fit with compaction or fracturing? Why are large quantities of water ice invoked? Where did it come from?
A few miscellaneous minor issues:
line 92: Thomas (1989) reference is missing.
Figure 3 and 4 captions: The descriptions for a) b) and c) are all scrambled. For each figure they should be a) core radius, b) outer density, c) core density
There are a number of places where the English is awkward or the text does not make sense grammatically. Here are a few examples:
lines 76-78: I am not sure what this sentence means: "The interior structure of Phobos was ever modeled as homogeneous, heavily fractured from Stickney impact, locally compressed, and a rubble pile consisted with voids and/or water ice."
lines 80-82: "The positive density gradient indicated that the core-density was greater than the mantle density, while the negative gradient suggested that the core-density was lower than the mantle density." This is very awkward and unclear.
lines 98-99: "The DTM derived by Willner et al. [1], which is available online [23] and is shown in Figure 1." I am not sure what the authors are trying to say.
line 160: "constant variables" should be "constants".
line 212: I think “paragraphs” should be “Figures 3 and 4” in order for this sentence to make sense.
line 223: I think “insight into” should be “investigate” in order for this sentence to make sense.
lines 243-245: This sentence does not make any sense to me: "Meanwhile, Guo et al. [19] identified exceptions to the prevailing pattern of a lighter core, finding that in some cases, the Phobos core is denser (?c=2200 kg m-3) than the outer layer (?m=1810 kg m-3).
lines 245-246: I think the authors are trying to say that based on these findings, an interior density increasing with depth is a likely configuration of Phobos.
Author Response
Response to Reviewer 3 Comments
Point 1: this work is interesting and could be a nice addition to the literature regarding the interior structure of irregular bodies and perhaps even provide constraints on the origin of Phobos. But there are some glaring omissions in the description of the method, and the discussion and conclusions seem unfocused and vague.
Response 1: We are delighted to hear that the reviewer is interested in our work. After reviewing the comments as well as those of other reviewers, we have made revisions to the conclusion section in the udpated manuscript. Please refer to the updated conclusion for a more detailed understanding of our findings. Thank you for your valuable feedback and input.
Point 2: The paragraph covering lines 183-191 purport to describe the crux of the analysis in which the most likely coupled parameters of density and radius are estimated, but it has no detail whatsoever. it is impossible to evaluate the validity of the parameter estimation without understanding the method.
Response 2: Our models both involve two governing equations but require estimation of three parameters: the core radius, the core density, and the outer layer density. This results in an underdetermined equation that can only be solved using nonlinear algorithms to produce a distribution of possible values for the estimated parameters. In practice, we identify the optimal values for these parameters by selecting those with the maximum frequency distribution, as presented in Figures 3 and 4. We calculated the observed mean denstiy from the mass and volume listed in Table 1. The observed values of the mean moment of inertia factor were determined using the gravity coefficients of degree-two through Equation (3).
For Model 1, we obtained the mean density and mean moment of inertia factor using Equations (5) and (9), respectively. For Model 2, we obtained these values using Equations (7) and (10), respectively. We then calculated the joint residual of the mean density and moment of inertia factor using Equation (11), and retained parameter values for which f<0.1. The frequency distribution of all possible values was subsequently obtained, resulting in the distribution plots shown in Figures 3 and 4.
We hope that this explanation clarifies our computational process and provides a greater understanding of our methodology.
Point 3: Similarly, lines 223-233 describe a sensitivity analysis on the varables. But It is difficult for me to evaluate this sensitivity analysis, as “sensitivity” is never even defined, and conclusions are stated without support, e.g., "However, a 1% increase in core density for 230 Model-II would decrease the sensitivity of S2(f, ?c) by 2.4%. Overall, the estimated parameters are sensitive to changes in the misfit function. These inversed parameters are thus 232 reliable in this study." Why are they reliable? Are these percentages significant? Why or why not?
Response 3: Equation (12) defines sensitivity as a function of the fitting residual f and the estimated parameters x, such as the core radius, core density, and outer layer density. A 1% change in any estimated parameter x will result correspondingly in a change in Equation (12). If the sensitivity value S calculated using Equation (12) is less than 1%, it indicates that the estimated parameter is not sensitive to the fitting residual, meaning that the inverted parameter has no meaning. Conversely, a sensitivity value greater than 1% indicates that the inverted parameter is meaningful and can be accepted.
To illustrate this concept, we can calculate the sensitivity of the estimated parameters in Table 2 for Model 2. For example, with a core radius rc of 8.2 km, a core density r of 2573 kg×m-3, and an outer layer density rm of 1340 kg×m-3, we can substitute these values into Equation (12) to obtain a sensitivity value of -2.4. This indicates that a 1% increase in core density would correspond to a 2.4% decrease in the fitting residual. This confirms that the core density is sensitive to the fitting residual, and that the optimal values of the estimated parameters for Model 2 in Table 2 are acceptable.
Thank you for your feedback. We are glad that our explanation of the definition of sensitivity and its calculation was helpful.
There are many places where similar conclusions are made without support:
Point 4: lines 193-201: How do we know that 1500 particles are sufficient to “estimate the best fit”? How do we know that 1000 iterations are sufficient to “ensure stability”? How do we know that the PSO calculation has been performed enough times to “reduce the impact of random noise”?
Response 4: Our study involved multiple experiments and comparisons, which showed that the frequency distribution of more than 1500 particles remained consistent, as illustrated in Figures 3 and 4. Based on these findings, we concluded that 1500 particles were sufficient to obtain stable results, and we included this proposal in our paper. We further tested the case of 1000 iterations and confirmed that the resulting frequency distribution was the same as that in Figures 3 and 4, reaffirming the stability of our results.
To address the concerns raised by the reviewers, we performed additional experiments with more than 1500 particles and more than 1000 iterations and presented the results in our updated manuscript. Based on these results, we confirmed that PSO with a population size of 1500 particles and 1000 iterations was capable of producing stable and reliable outcomes. The revised content in the updated manuscript is as follows
‘Our study involved multiple experiments and comparisons, which showed that the frequency distribution of more than 1500 particles remained consistent. We further tested the case of 1000 iterations and confirmed that the resulting frequency distribution was the same as the case of 1000 iteration. We thus considered the PSO method with a population size of 1500 particles and 1000 iterations in our study.’
Please take a look on Lines 205-210 on Page 7 for more detail.
Point 5: lines 203-206: Why don’t the libration amplitudes of Burmeister et al. work in this analysis? What is the justification for ignoring this data? Do you consider their results to be in error?
Response 5: In our manuscript, we have addressed the issue of selecting libration amplitudes. We found that the amplitudes provided by Burmeister et al. did not result in an approximate normal distribution similar to Figures 3 and 4. Instead, these amplitudes generated a uniform distribution, which could not be used to find optimized values for the estimated parameters. Consequently, we abandoned the use of Burmeister's libration amplitude for estimating the core radius, core density, and outer-layer density.
Determining accurate libration amplitudes and estimating the propagated errors associated with this parameter is outside the scope of our study at present, since there are currently no enough data to do this work. However, we acknowledge that this remains a topic for future exploration as additional data become available from other studies. We hope that this issue can be closely examined and explored further in the future.
Point 6: Finally, the conclusions are very weak. What are the implications for the origin of Phobos (capture vs. accretion in place)? How do the derived densities fit with compaction or fracturing? Why are large quantities of water ice invoked? Where did it come from?
Response 6: As stated in our manuscript, previous studies have suggested that Phobos may possess either a denser or lighter core, as well as positive or negative density gradients (Guo et al., 2021; Dimitrovskii et al., 2022). However, Guo et al. (2021) demonstrated that a lighter core was more likely, and Dimitrovskii et al. (2022) showed that both negative and positive density gradients were possible, as well as a rubble pile structure. Our study, on the other hand, indicates that Phobos is more likely to have a denser core than outer layer. The difference in results between our study and previous work may be due to differences in the methodologies used to detect the interior structure of Phobos. We believe that with more measurement data available in the future, the interior structure of Phobos can be better constrained.
Our two models gave a mean core radius close to 8.2 km, a mean core density around 2500 kg×m-3, and a mean outer layer density close to 1400 kg×m-3. We further tested the sensitivities of the three estimated parameters, which indicated that they are all sensitive to the misfit function and thus the optimized values are robust. Therefore, our study provides a good reference for constraining the interior structure of Phobos, and can be extend to the constraint of interior structure of other small bodies with irregular shapes. Assuming that the density difference between the core and outer layer is associated with water-ice content, we found that the outer layer has an approximate water-ice content of 11%, with a rock density of 2200 kg×m-3, while the core has a lower water-ice content of around 2.4%, with a rock density of 3000 kg×m-3. We provided this example solely to illustrate the relationship between density difference and water-ice content, and whether the density difference is associated with compaction or fracturing is beyond the scope of discusion in our study. We intend to delve into this topic in our future research.
Point 7: line 92: Thomas (1989) reference is missing.
Response 7: We have added the study of Thomas in reference numbered as 22. Please take a look on the section of reference on Page 10 for more detail.
Point 8: Figure 3 and 4 captions: The descriptions for a) b) and c) are all scrambled. For each figure they should be a) core radius, b) outer density, c) core density
Response 8: We have corrected the captions for Figures 3 and 4. Please take a look at Figure 3 and Figure 4 on Page 7 for more detail.
Comments on the Quality of English Language
There are a number of places where the English is awkward or the text does not make sense grammatically. Here are a few examples:
Point 9: lines 76-78: I am not sure what this sentence means: "The interior structure of Phobos was ever modeled as homogeneous, heavily fractured from Stickney impact, locally compressed, and a rubble pile consisted with voids and/or water ice."
Response 9: We have corrected this sentence as follows
‘Due to the lack of information on Phobos' interior structure, it has been previously assumed to be homogeneous. However, other models have proposed a multilayer structure based on density and forced libration observations [10] or a rubble pile composition that includes voids and/or water ice [1,19-21].’
Please take a look on Lines 78-81 on Page 2 for more detail.
Point 10: lines 80-82: "The positive density gradient indicated that the core-density was greater than the mantle density, while the negative gradient suggested that the core-density was lower than the mantle density." This is very awkward and unclear.
Response 10: We have corrected this sentence as follows
‘The positive density gradient in our study indicates that the density of the core was greater than that of the outer layer. Conversely, the negative gradient suggests that the density of the core was lower than that of the outer layer.’
Please take a look on Lines 83-86 on Page 2 for more detail.
Point 11: lines 98-99: "The DTM derived by Willner et al. [1], which is available online [23] and is shown in Figure 1." I am not sure what the authors are trying to say.
Response 11: For the purpose of illustrating the shape of Phobos, we have cited the digital terrain model (DTM) provided by Willner et al. Figure 1 displays the DTM to provide the readers with a visual representation of Phobos' shape.
Point 12: line 160: "constant variables" should be "constants".
Response 12: We have corrected the phrase “constant variables” as “constant”. Please take a look on Line 170 on Page 6 for more detail.
Point 13: line 212: I think “paragraphs” should be “Figures 3 and 4” in order for this sentence to make sense.
Response 13: We have corrected accordingly. Please take a look on Line 224 on Page 7 for more detial.
Point 14: line 223: I think “insight into” should be “investigate” in order for this sentence to make sense.
Response 14: Considering all the reviewers’ suggestions, we have corrected the sentence where the phgraph located as follows
‘If the estimated parameters are not sensitive to the misfit function, their optimized values will be meaningless. Therefore, we used Eq. (12) to estimate the sensitivities for these estimated parameters based on the optimized values listed in Table 2.’
Please take look on Lines 236-237 on page 8 for more detail.
Point 15: lines 243-245: This sentence does not make any sense to me: "Meanwhile, Guo et al. [19] identified exceptions to the prevailing pattern of a lighter core, finding that in some cases, the Phobos core is denser (?c=2200 kg m-3) than the outer layer (?m=1810 kg m-3).
Response 15: We have corrected this sentence as follows
‘In addition, even though the study of Guo et al. [21] demonstrated that Phobos’ core density was lower than that of its outer layer, they also discovered instances where the core density (rc=2200 kg×m-3) was greater than that the outer layer (rm=1810 kg×m-3).’
Please take a look on Lines 255-258 on Page 8 for more detail.
Point 16: lines 245-246: I think the authors are trying to say that based on these findings, an interior density increasing with depth is a likely configuration of Phobos.
Response 16: Yes, we want to covery the possibility of Phobos’ core density being higher than that of the outer layer. We have accordingly replaced the word ‘possible’ with the word ‘likely’. Please take a look on Line 258-259 on Page 8 for more detail.
Author Response File: Author Response.docx
Reviewer 4 Report
Review report “The mean moment of inertia for irregularly shaped Phobos and 2 its application to the constraint for the two-layer interior struc-3 ture for the Martian moon” by Zhen Zhong, Qilin Wen, Jianguo Yan, and Lijun Pang, submitted to Remote Sensing
This manuscript explains the role of the inner structure of asteroid Phobos in the physical properties, particularly focusing in the mean moment of inertia. A basic model is proposed, and the physical approach used is correct. To assess the thermal inertia of an asteroid-like satellite is not trivial at all and already points to the relevance of the work presented.
Asteroids have irregular shapes, but the study of their physical properties, particularly the rotational ones constitutes a big challenge. The main reason is our current lack of understanding about its formation and evolution, also in Phobos case. In fact, to solve the caveat, Phobos will be the target of the return sample mission called Martian Moons eXploration by the Japanese agency (JAXA). That mission will analyze its physico-chemical properties from orbit, and also having ground truth from the samples collected to be studied in terrestrial laboratories, giving clues about the origin of Mars’ satellite.
I missed in the introduction a few more paragraphs dealing with the relevance of collisional gardening in the origin of the regolith covering asteroids. In fact, the outer layer of Phobos could be a direct consequence of the processing by impactors. The regolith layer of an asteroid of tens of km in diameter like Phobos might be several kilometers thick. There are interesting reviews that should be cited here. For example, Beitz et al. (2016) modelled impact gardening, and matched it with current chondrite evidence. These authors concluded that the size of the biggest crater in an impact shattered asteroid must be a good estimation of the depth of the regolith layer. If so, the presence of Stickney may indicate that there is not a primordial “core”, but the entire Phobos is a rubble pile. Please discuss it.
I wonder if perhaps you could consider a more complex 3 layers model formed by: an inner primordial remnant inside (if it exists see below), a large pile of large rubble on top of it, and a collisionally gardened outer surface. If so, do you think that such structure will affect significantly the results?
Concerning the formation of Phobos, the authors should echo the proposed formation model of Madeira et al. (2023). Some models discussed in that manuscript propose that Phobos formed after a giant impact that produced a debris disk around Mars. If so, perhaps Phobos could be a rubble pile, not necessarily fitting your basic model of core and rubble/ice. In such a case, would you expect differences in the moment of inertia?
Other comments:
Line 36: It could be good introducing how complex could be the origin of Phobos. The exact dynamic model is still unknown, but a work deserves to be cited in the introduction. Hesselbrock & Minton (2017) showed that a recycling process may happen during the reassembling of Phobos, by which the progenitors perhaps were destroyed into a Roche-interior ring and reaccreted several times. I think that it should be presented in the introduction to exemplify the plausible complexity behind the formation of Phobos.
Line 39: “To accurately determine the interior structure of Phobos, a direct exploration of Phobos is needed to constrain its moment of inertia.” It is not fully true, you should also cite the role of radar has in improving our understanding of asteroid and comet structure.
Line 184: You used a “equal probability distribution.”. Please explain. Did you try other distributions? If so, the outcome is significantly different?
Line 261: Perhaps you could state that your results (average density and water fraction) could be consistent with the expectations for hydrated carbonaceous chondrite groups. A couple of papers to cite here could be: Macke et al. (2011) & Trigo-Rodríguez et al.. (2019).
Finally, I found that the conclusions’ section should be better organized. Please enumerate your main findings and describe the main implications of them. Please also try to emphasize what is really novel compared with previous work.
In consequence, I think that this manuscript is a nice piece of work. In any case, some sections should be slightly revised before being considered for publication. My recommendation is a minor review, leaving to the editor the possibility to send me the revised version for a final checking.
References
Beitz E. et al. (2016) The collisional evolution of undifferentiated asteroids and the formation of chondritic meteoroids, Ap J. 824, 29 pp.
Hesselbrock A.J. and D.A. Minton (2017) An ongoing satellite–ring cycle of Mars and the origins of Phobos and Deimos. Nature Geoscience 10, 266–269.
Macke R.J. et al. (2011) Density, porosity, and magnetic susceptibility of carbonaceous chondrites. Meteoritics & Planetary Science 46, 1842–1862.
Madeira G. et al. (2023) Exploring the Recycling Model of Phobos Formation: Rubble-pile Satellites. The Astron. J. 165, id.161, 22 pp.
Trigo-Rodríguez, J.M. et al. (2019) Accretion of water in carbonaceous chondrites: current evidence and implications for the delivery of water to early Earth, Space Science Reviews 215:18, 27 pp.
Comments for author File: 
Comments.pdf
Some sentences are too long and could be shortened a bit
Author Response
Response to Reviewer 4 Comments
This manuscript explains the role of the inner structure of asteroid Phobos in the physical properties, particularly focusing in the mean moment of inertia. A basic model is proposed, and the physical approach used is correct. To assess the thermal inertia of an asteroid-like satellite is not trivial at all and already points to the relevance of the work presented.
Asteroids have irregular shapes, but the study of their physical properties, particularly the rotational ones constitutes a big challenge. The main reason is our current lack of understanding about its formation and evolution, also in Phobos case. In fact, to solve the caveat, Phobos will be the target of the return sample mission called Martian Moons eXploration by the Japanese agency (JAXA). That mission will analyze its physico-chemical properties from orbit, and also having ground truth from the samples collected to be studied in terrestrial laboratories, giving clues about the origin of Mars’ satellite.
I missed in the introduction a few more paragraphs dealing with the relevance of collisional gardening in the origin of the regolith covering asteroids. In fact, the outer layer of Phobos could be a direct consequence of the processing by impactors. The regolith layer of an asteroid of tens of km in diameter like Phobos might be several kilometers thick.
Point 1: There are interesting reviews that should be cited here. For example, Beitz et al. (2016) modelled impact gardening, and matched it with current chondrite evidence. These authors concluded that the size of the biggest crater in an impact shattered asteroid must be a good estimation of the depth of the regolith layer. If so, the presence of Stickney may indicate that there is not a primordial “core”, but the entire Phobos is a rubble pile. Please discuss it.
Response 1: We have included the study of Beitz et al. (2016) in reference number 20 and cited it on Line 268 on page 8. The rubble pile model is one potential structure that we considered; however, we found it difficult to model when considering the mean moment of inertia alone. Therefore, we developed a differentiated model that incorporates both the moment of inertia and the mean density of Phobos. From the frequency distributions, we found a possible core density larger than that of the outer layer of Phobos. From the frequency distributions as shown in Figures 3 and 4 in the manuscript, we found a possible core density larger than that of the outer layer of Phobos. This finding aligns with recent studies by Dmitrovskii et al. (2022) and Guo et al. (2021), which also suggest the presence of a large core. If our results deviate from the actual structure of Phobos, it may be due to the unconstrained gravity coefficients of degree-2. We do not deny the previous studies but only discovered a denser core than the outer layer by considering the current gravity coefficients of degree-2 in the differentiated model. To avoid misunderstanding, we have modified our expressions in the abstract as follows
“These values have a high sensitivity on the misfit function, implying a higher density likely inside Phobos than the outer layer. Given that the large core-density was associated with ice content, it suggested that the fractional ice content in the outer layer is approximately 11% with a rock density of 2200 kg·m-3, while the content in the core is lower at 2.4% with a rock density of 3000 kg·m-3.”
Please take a look on the part of abstract for more detial.
Point 2: I wonder if perhaps you could consider a more complex 3 layers model formed by: an inner primordial remnant inside (if it exists see below), a large pile of large rubble on top of it, and a collisionally gardened outer surface. If so, do you think that such structure will affect significantly the results?
Response 2: We have actually tested such a case of a 3-layer model, but as the previous study by Yan et al. (2015) indicated, the thickness of the crust is not sensitive to the result. Therefore, we did not consider such a complex model with three layers. We have added the reason to select the two-layer model on Lines 112-118 on Page 3, which is as follows
‘We opted for a two-layer model when estimating Phobos' internal structure for several reasons. Firstly, the limited constraint information available for Phobos makes multi-layer models highly uncertain. Secondly, given the small volume of Phobos, it is unlikely that large-scale differentiation processes occurred in its interior. Finally, we accounted for Phobos' irregular shape when calculating its mean moment of inertia. Models with more than three layers would increase complexity and hinder the estimation of model parameters.’
Please take a look on these lines on Page 3 for more detail.
Point 3: Concerning the formation of Phobos, the authors should echo the proposed formation model of Madeira et al. (2023). Some models discussed in that manuscript propose that Phobos formed after a giant impact that produced a debris disk around Mars. If so, perhaps Phobos could be a rubble pile, not necessarily fitting your basic model of core and rubble/ice. In such a case, would you expect differences in the moment of inertia?
Response 3: Our research does not extend to the study of Phobos' formation, and we wish to reiterate that we do not dispute the findings of other studies, particularly those related to the rubble pile model. Nevertheless, based on the current Phobos second-degree gravity coefficient, the layered model suggests a more substantial core than a lighter one. Recently, alternative models have been proposed which suggest the presence of a heavier core or negative density gradient; examples include research by Dmitrovskii et al. (2022) and Guo et al. (2021). Furthermore, the differentiated model's findings still serve as a frame of reference for our understanding of Phobos' interior structure. We plan to conduct further studies on Phobos' potential formation in the future, once essential data become available.
Other comments:
Point 4: Line 36: It could be good introducing how complex could be the origin of Phobos. The exact dynamic model is still unknown, but a work deserves to be cited in the introduction. Hesselbrock & Minton (2017) showed that a recycling process may happen during the reassembling of Phobos, by which the progenitors perhaps were destroyed into a Roche-interior ring and reaccreted several times. I think that it should be presented in the introduction to exemplify the plausible complexity behind the formation of Phobos.
Response 4: We have cited this study of Hesselbrock & minton (2017) on Lines 37-39, and highlighted the complex origion of Phobos. This cited study is also added in reference number 6. The introduction is as follows:
“It was likely originated from a recycling process in which the progenitors from impacting process perhaps were destroyed into a Roche-interior ring and reaccreted several times [6].”
Please take a look on these Lines 37-39 on page 1.
Point 5: Line 39: “To accurately determine the interior structure of Phobos, a direct exploration of Phobos is needed to constrain its moment of inertia.” It is not fully true, you should also cite the role of radar has in improving our understanding of asteroid and comet structure.
Response 5: We have modified this sentencs as follows:
‘While radar data can providde some import information on the interior structure of Phobos, a direct exploration of the moon is crucial to accrurately determine its moment of inertia’.
Please take a look on Lines 40-41 on Page 1 for more detail.
Point 6: Line 184: You used a “equal probability distribution.”. Please explain. Did you try other distributions? If so, the outcome is significantly different?
Response 6: Yes, the outcome will be difference for other distribution. Studying on the different distribution is out the range of discussion in our work. We considered an equal probability distribution for the estimated parameters according to previous study of Yan et al. (2015). The estimated parameters include the core radius, the core density and the outer layer density.
Point 7: Line 261: Perhaps you could state that your results (average density and water fraction) could be consistent with the expectations for hydrated carbonaceous chondrite groups. A couple of papers to cite here could be: Macke et al. (2011) & Trigo-Rodríguez et al.. (2019).
Response 7: We have accordingly cited these two studies in our updated manusrcipt as follows
‘The mean value of rock density and the estimated content of water ice are close to previous studies [38-39]’, which can be found on Line 274 on page 8. These two cited studies are supplemented in reference numbered as 38 and 39, respectively.
Point 8: Finally, I found that the conclusions’ section should be better organized. Please enumerate your main findings and describe the main implications of them. Please also try to emphasize what is really novel compared with previous work.
Response 8: In our study, the novel aspect lies in our consideration of the irregular shape of celestial bodies, which is more appropriate for detecting the internal structure of small bodies. We highlight this point in the conclusion section, stating that, ' Compared to previous studies using similar methods, our methodology takes into account the irregular shape of celestial bodies. Consequently, our models are more suitable for estimating the internal structure of small celestial bodies by leveraging the mean moment of inertia and mean density.' For further details, please refer to Lines 305-310 on Page 9.
Point 9: In consequence, I think that this manuscript is a nice piece of work. In any case, some sections should be slightly revised before being considered for publication. My recommendation is a minor review, leaving to the editor the possibility to send me the revised version for a final checking.
Response 9: Thanks for your valuable feedback. We have made revisions to our manuscript based on the recommendations, with the exception of a few suggestions that were beyond the scope of discussion in our work. We have done our best to ensure that these modifications meet the requirements from reviewer, and we hope that the reviewer find them satisfactory.
Author Response File: Author Response.docx
Reviewer 5 Report
This is an excellent paper that addresses with sound scientific methodology and derivations a relevant research topic on the exploration of the possible interior structure of Phobos. I was very impressed with the presentation and results and recommended accepting it for publication.
The authors were exhaustive in examining previous methods and selecting an appropriate method, examining the moment of inertia with updated gravity results, and modifying a method previously successfully used by other authors for spherical bodies to be applied to irregularly shaped bodies like Phobos.
A two-layer model is appropriate for a small body with evidence for density variation. Choosing a spherical shape model for the core, in this case, is an acceptable simplification. Proposing a conclusion based on the derived data that a potential core formed with a larger mass that was "subsequently compressed to have lower porosity" is a sound possibility that should be made available in the literature for future researchers to examine further.
I found the data you referenced in PDS without a problem.
Congratulation.
Author Response
Response to Reviewer 5 Comments
This is an excellent paper that addresses with sound scientific methodology and derivations a relevant research topic on the exploration of the possible interior structure of Phobos. I was very impressed with the presentation and results and recommended accepting it for publication.
The authors were exhaustive in examining previous methods and selecting an appropriate method, examining the moment of inertia with updated gravity results, and modifying a method previously successfully used by other authors for spherical bodies to be applied to irregularly shaped bodies like Phobos.
A two-layer model is appropriate for a small body with evidence for density variation. Choosing a spherical shape model for the core, in this case, is an acceptable simplification. Proposing a conclusion based on the derived data that a potential core formed with a larger mass that was "subsequently compressed to have lower porosity" is a sound possibility that should be made available in the literature for future researchers to examine further.
Response: We greatly appreciate the fair evaluation of our paper provided by the reviewer. We will further constrain the internal structure of Phobos with the lates data in future research.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
The manuscript has much improved - both from a perspective of content and language.
Though a few small language improvements could be done. Please have a careful read of the proof and try to improve the language further.
