The Statistical Analysis of Exoplanet and Host Stars Based on Multi-Satellite Data Observations

Round 1

Reviewer 1 Report

The paper is well written. The major remark is that the coefficients of the fits at figures should contain error estimates and some  minor changes to make the error estimates smaller and noticeable. E.g. The equivalent functions y(x)=a+b*x=y_mean+b*(x-x_mean) have the same shape, as well as the slope statistical error = accuracy, but the value a=y(0) is an extrapolated one, which is typically well outside the data, and has a large error estimate, which may then lead to incredibly large error estimates of the accuracy of the approximation. E.g. here is a graph of the data in a relatively small interval ~4. What is the sense of the approximation starting with zero? Not for this data, but for a "live" example. Have a linear approximation of a salary, say, from 1990 to 2024. The value "a" in a linear approximation will correspond to 0-th year (yes, we know, that DC starts from 1, 0 is missing). No sense. Not too much sense for the data in a range 4-5 to be extrapolated to 0. So it would be great to change approximations, to make confidence interals (e.g. a review 2020kdbd.book..191A or other cited sources). If testing hypotheses on a presence of a statistical dependence. one may be recommended to make ANOVA - the analysis of variances with a FAP = False alarm probability estimate.

So I would recommend the paper after minor changes

Author Response

We thank the referee for the comments on the paper. We have tried to alleviate his or her concerns in the revised version of the paper. Our point-by-point response to his or her concerns is given below. Relevant changes in the manuscript have been made with the bold font. In our response below the referee’s comments are reproduced in bold.

1. On line 137: Do the author mean that the formation rate is 2.02%, i.e., 0.0202? If so, the percentage sign should be added (in the abstract and other parts of the text as well where the rate is mentioned).

Response: Agree, this is a typo.

2. On line 141 and after: Can the lower formation rate for more distant planets be affected by observational biases? If not, out to what distances are observation datasets complete?

Response: We think the lower formation for more distant planets can be affected by observational. But in the manuscript, we did not consider this factor, and we also mentioned the sources of error in the formula for calculating the planetary production rate in our paper.

3. On line 144 and after: The high formation rate at close distances may be affected by orbital migration. Would the analysis be able to quantify this process? If so, how? If not, it should be clearly stated.

Response: Sorry, we not. Thank you for your suggestion. In the samples we used, the factor of orbit migration was not taken into account. We have consulted relevant materials and found that there are only corresponding models but not necessarily accurate theories on the orbit migration of both so far. So we didn't consider this factor. Nevertheless, we are very grateful for your suggestion.

4. Line 163: Is "Michael" a reference? Please, correct.

Response: Yes. We have made the revisions in the literature accordingly.

5. Line 222: Is f(Z) the formation rate of giant planets? If so, please clarify the statement.

Response: Agree, f(Z) is the formation of giant planets, we have made annotations in the manuscript. Z is stellar metallicity.

6. Line 236 and after. The argument provided in the text for the impact of log(g) is problematic. log(g) is determined by stellar mass and radius. For planets that orbit many stellar radii away from the star (i.e., when tidal interactions with the star are negligible), the radius of the star is irrelevant (as a direct effect). Therefore, the argument that the authors provide may be misleading. Their argument may actually apply to the stellar mass. Both core formation and accretion of gas on giant planets, in general, are not related to stellar radius and therefore the connection to log(g) is not justifiable in this context. I would suggest that the author make a connection with stellar mass rather than with log(g). In either case, the discussion needs to be corrected. A statement such as: "An excessively high gravitational acceleration in the surrounding environment 245 could inhibit the gas from converging towards this core..." is generally incorrect (say beyond a distance of ~0.1AU).
7. Paragraph starting at 259. Since the argument again concerns log(g) the same advice as above applies. The connection may be done to stellar mass, not to log(g). The problem is that stellar gravity at the orbital location of the planet is important, however, that force is not connected to the surface gravity on the star.
8. Paragraph starting at 265. I would suggest removing the paragraph. Note that (in Newtonian gravity) the gravity field of a star depends only on its mass and distance, not log(g).
9. Sections 4.2.1 and 4.2.2. The same suggestion applies regarding the use of log(g) in the analysis. The connection should be made to stellar mass or stellar type.

• Paragraph starting at 265. I would suggest removing the paragraph. Note that (in Newtonian gravity) the gravity field of a star depends only on its mass and distance, not log(g).

Response: Thank you very much for your suggestions. We have referenced your suggestions and verified them.

For the giant planet, firstly, we plotted a three-dimensional scatter plot of stellar mass, giant planet period, and formation rate, with colors representing the magnitude of stellar mass (as shown in fig1). We employed the same data processing method as described in the manuscript for handling, fitting three variables: stellar mass, planet period, and formation rate. Ultimately, the fitted equation obtained is:

            , (1)

 is giant planet period,  is stellar mass.

Due to the relatively low distinguishability of stellar mass in terms of color, we plotted contour lines for stellar mass to facilitate the observation of the relationship between stellar mass and formation rate. Specifically, contour lines were drawn for =0.5, 1.5, 2.5, 3.5, and 4.5. It can be observed that lower stellar mass correlates with higher formation rates. To validate this conclusion, we selected the interval with the highest sample quantity, namely 300 to 700 days, and then used the fitted equation (1) to fit the scatter plot of stellar mass and formation rate. In the fig1, 500 represents the median value of the selected period range. However, we observed that the fitting results for stellar mass and formation rate were relatively poor compared to the fitting plot of  and formation rate presented in the manuscript. The distribution of points for stellar mass and formation rate was relatively scattered, whereas the plot for and formation rate exhibited a more concentrated distribution, indicating a stronger correlation.

Fig1. The relationship between the orbital periods of giant planets, stellar mass and their formation rates and the relationship between the stellar mass and their formation rates.

For minor planet, we also plotted three-dimensional scatter plots of minor planet orbital period, production rate, and host star mass. The color in the figure represents the magnitude of the stellar mass. As shown in Figure 2, there is a clear distinction in colors among the minor planet. We also drew contours of stellar mass,=0.5, 1, and 1.5. From the figure, it can be observed that it still supports the argument that lower stellar mass corresponds to higher production rates. Subsequently, we performed fitting on these three variables, with the fitting formula as follows:

, (2)

  is planet period,  is stellar mass.

To validate the accuracy of the relationship between stellar mass and formation rate, we selected the interval with the largest sample size, namely the data points within the orbital period range of 3 to 7 days. We plotted a scatter plot of stellar mass against formation rate and performed fitting using the formula (2) derived earlier. In the fig2, 5 represents the median of the selected interval. As the figure, it can be observed that the fitting for asteroids is much better than for gas giants, but still not as concentrated as the points for . Therefore, we tentatively conclude that the production rate is not solely dependent on stellar mass. However, the above argument is insufficient to support our thesis in the paper.

Fig2. The relationship between the orbital periods of minor planets, stellar mass and their formation rates and the relationship between the stellar mass and their formation rates.

  To further strengthen the verification, we separately plotted histograms of the difference between the actual and theoretical values of stellar mass and log g for gas giants and asteroids. The actual values are the stellar mass and log g values from the data, while the theoretical values are derived from our fitting formula.

For giant planet:

For minor planet:

Finally, we plot the hist plot is:

Fig3. The histogram of the difference between the actual and theoretical values for giant planet.

Fig4. The histogram of the difference between the actual and theoretical values for minor planet.

As Fig3 and Fig4, it is evident that the error in stellar mass is larger compared to the error in . Therefore, in this sample, we believe that , rather than stellar mass, remains the variable most closely related to the formation rate. So, the results demonstrate that while stellar mass does influence the production rate, we cannot ignore the impact of stellar radius on the production rate.

• Line 252 and after. From a formation perspective, the argument of long distances, long formation times may be correct. However, in the solar system, the largest planets are also the most distant ones. Some comment is needed in that paragraph highlighting this issue.

Response: We have taken your advice into consideration and consulted relevant literature, incorporating explanations into the manuscript. We greatly appreciate it.

• Figures 6-9: What does a planet formation rate of 1 mean? One planet per star? Please provide explanation in the captions.

Response: No, 1 is a ratio rather than a numerical value, and we have made the necessary corrections in the manuscript.

• Line 304 and after. Please, provide reference suggesting that Sub-Neptunes and warm-Neptune are more likely to form in place rather than to migrate. Typically, these planets belong to the mass range where tidal interactions with circumstellar material are strongest.

Response: We have incorporated your suggestions and identified relevant literature to strengthen the arguments in our paper.

• Section 4.2.3. Note that if the planet mass is proportional to planet radius, it implies that the average density of the planet is proportional to the radius squared. The authors may want to comment on this result.

Response: Thank you for your suggestion, we have explained it in the article.

15. Figure 10. What do the plus signs indicate in the plot? I would suggest adding the solar system planets to this plot as well, as a

Response: The plus sign indicates taking the bin, we have explained it in the article. Then we took your advice into consideration, we included asteroids within the solar system on the asteroid mass-radius fitting chart for reference. We found that the planets within the solar system basically followed the trend of positive correlation between mass and radius, which is undoubtedly a good result. However, due to some errors in the observation samples we used and the difference between the planetary system and the solar system, our fitting line is not very close to the data points. Nevertheless, it can still generally explain the trend of positive correlation.

• The equivalent functions y(x)=a+b*x=ymean+b*(x-xmean) have the same shape, as well as the slope statistical error = accuracy, but the value a=y(0) is an extrapolated one, which is typically well outside the data, and has a large error estimate, which may then lead to incredibly large error estimates of the accuracy of the approximation. E.g. here is a graph of the data in a relatively small interval ~4. What is the sense of the approximation starting with zero? Not for this data, but for a "live" example. Have a linear approximation of a salary, say, from 1990 to 2024. The value "a" in a linear approximation will correspond to 0-th year (yes, we know, that DC starts from 1, 0 is missing). No sense. Not too much sense for the data in a range 4-5 to be extrapolated to 0. So it would be great to change approximations, to make confidence interals (e.g. a review 2020kdbd.book..191A or other cited sources). If testing hypotheses on a presence of a statistical dependence. one may be recommended to make ANOVA - the analysis of variances with a FAP = False alarm probability estimate.

Response: Thank you for your suggestion. In the process of data processing, we did not consider a little errors, but in the Python fitting method we used for data processing, there is a calculation of relative errors. If we consider the error issue now, it will undoubtedly be a huge workload, and we cannot complete it within the limited revision period. Nevertheless, we still want to thank you again for pointing out the error issue and suggesting a correction. We will conduct relevant research on this in the future. Thank you again.

Author Response File: Author Response.docx

Reviewer 2 Report

Dear Editor,

I read the manuscript by Tang et al. on the statistical analysis of exoplanets and host stars. The manuscript provides useful trends in observations that can surely be useful in a number of applications in exoplanet science. The paper would benefit from proofreading since the presentation of some parts of the text could be improved. Moreover, some explanations of statistical trends either require more discussion or need to be corrected. Detailed comments are provided below.

On line 137: Do the author mean that the formation rate is 2.02%, i.e., 0.0202? If so, the percentage sign should be added (in the abstract and other parts of the text as well where the rate is mentioned).

On line 141 and after: Can the lower formation rate for more distant planets be affected by observational biases? If not, out to what distances are observation datasets complete?

On line 144 and after: The high formation rate at close distances may be affected by orbital migration. Would the analysis be able to quantify this process? If so, how? If not, it should be clearly stated.

Line 163: Is "Michael" a reference? Please, correct.

Line 222: Is f(Z) the formation rate of giant planets? If so, please clarify the statement.

Line 236 and after. The argument provided in the text for the impact of log(g) is problematic. log(g) is determined by stellar mass and radius. For planets that orbit many stellar radii away from the star (i.e., when tidal interactions with the star are negligible), the radius of the star is irrelevant (as a direct effect). Therefore, the argument that the authors provide may be misleading. Their argument may actually apply to the stellar mass. Both core formation and accretion of gas on giant planets, in general, are not related to stellar radius and therefore the connection to log(g) is not justifiable in this context. I would suggest that the author make a connection with stellar mass rather than with log(g). In either case, the discussion needs to be corrected. A statement such as: "An excessively high gravitational acceleration in the surrounding environment 245 could inhibit the gas from converging towards this core..." is generally incorrect (say beyond a distance of ~0.1AU).

Line 252 and after. From a formation perspective, the argument of long distances, long formation times may be correct. However, in the solar system, the largest planets are also the most distant ones. Some comment is needed in that paragraph highlighting this issue.

Paragraph starting at 259. Since the argument again concerns log(g) the same advice as above applies. The connection may be done to stellar mass, not to log(g). The problem is that stellar gravity at the orbital location of the planet is important, however, that force is not connected to the surface gravity on the star.

Paragraph starting at 265. I would suggest removing the paragraph. Note that (in Newtonian gravity) the gravity field of a star depends only on its mass and distance, not log(g).

Line 273 until the end of section. Again, I would suggest replacing log(g) with stellar mass and repeat the analysis based on mass. Also, as mentioned above, planet migration may affect the final orbit of a planet. If the authors are discarding this process, they should clearly state this at the beginning of the analysis.

Figures 6-9: What does a planet formation rate of 1 mean? One planet per star? Please provide explanation in the captions.

Line 304 and after. Please, provide reference suggesting that Sub-Neptunes and warm-Neptune are more likely to form in place rather than to migrate. Typically, these planets belong to the mass range where tidal interactions with circumstellar material are strongest.

Sections 4.2.1 and 4.2.2. The same suggestion applies regarding the use of log(g) in the analysis. The connection should be made to stellar mass or stellar type.

Section 4.2.3. Note that if the planet mass is proportional to planet radius, it implies that the average density of the planet is proportional to the radius squared. The authors may want to comment on this result.

Figure 10. What do the plus signs indicate in the plot? I would suggest adding the solar system planets to this plot as well, as a reference.

Conclusions. The concluding section should be revised, reflecting the changes suggested above.

See above.

Author Response

Response to Referee’s comments on manuscript

We thank the referee for the comments on the paper. We have tried to alleviate his or her concerns in the revised version of the paper. Our point-by-point response to his or her concerns is given below. Relevant changes in the manuscript have been made with the bold font. In our response below the referee’s comments are reproduced in bold.

1. On line 137: Do the author mean that the formation rate is 2.02%, i.e., 0.0202? If so, the percentage sign should be added (in the abstract and other parts of the text as well where the rate is mentioned).

Response: Agree, this is a typo.

2. On line 141 and after: Can the lower formation rate for more distant planets be affected by observational biases? If not, out to what distances are observation datasets complete?

Response: We think the lower formation for more distant planets can be affected by observational. But in the manuscript, we did not consider this factor, and we also mentioned the sources of error in the formula for calculating the planetary production rate in our paper.

3. On line 144 and after: The high formation rate at close distances may be affected by orbital migration. Would the analysis be able to quantify this process? If so, how? If not, it should be clearly stated.

Response: Sorry, we not. Thank you for your suggestion. In the samples we used, the factor of orbit migration was not taken into account. We have consulted relevant materials and found that there are only corresponding models but not necessarily accurate theories on the orbit migration of both so far. So we didn't consider this factor. Nevertheless, we are very grateful for your suggestion.

4. Line 163: Is "Michael" a reference? Please, correct.

Response: Yes. We have made the revisions in the literature accordingly.

5. Line 222: Is f(Z) the formation rate of giant planets? If so, please clarify the statement.

Response: Agree, f(Z) is the formation of giant planets, we have made annotations in the manuscript. Z is stellar metallicity.

6. Line 236 and after. The argument provided in the text for the impact of log(g) is problematic. log(g) is determined by stellar mass and radius. For planets that orbit many stellar radii away from the star (i.e., when tidal interactions with the star are negligible), the radius of the star is irrelevant (as a direct effect). Therefore, the argument that the authors provide may be misleading. Their argument may actually apply to the stellar mass. Both core formation and accretion of gas on giant planets, in general, are not related to stellar radius and therefore the connection to log(g) is not justifiable in this context. I would suggest that the author make a connection with stellar mass rather than with log(g). In either case, the discussion needs to be corrected. A statement such as: "An excessively high gravitational acceleration in the surrounding environment 245 could inhibit the gas from converging towards this core..." is generally incorrect (say beyond a distance of ~0.1AU).
7. Paragraph starting at 259. Since the argument again concerns log(g) the same advice as above applies. The connection may be done to stellar mass, not to log(g). The problem is that stellar gravity at the orbital location of the planet is important, however, that force is not connected to the surface gravity on the star.
8. Paragraph starting at 265. I would suggest removing the paragraph. Note that (in Newtonian gravity) the gravity field of a star depends only on its mass and distance, not log(g).
9. Sections 4.2.1 and 4.2.2. The same suggestion applies regarding the use of log(g) in the analysis. The connection should be made to stellar mass or stellar type.

• Paragraph starting at 265. I would suggest removing the paragraph. Note that (in Newtonian gravity) the gravity field of a star depends only on its mass and distance, not log(g).

Response: Thank you very much for your suggestions. We have referenced your suggestions and verified them.

For the giant planet, firstly, we plotted a three-dimensional scatter plot of stellar mass, giant planet period, and formation rate, with colors representing the magnitude of stellar mass (as shown in fig1). We employed the same data processing method as described in the manuscript for handling, fitting three variables: stellar mass, planet period, and formation rate. Ultimately, the fitted equation obtained is:

            , (1)

 is giant planet period,  is stellar mass.

Due to the relatively low distinguishability of stellar mass in terms of color, we plotted contour lines for stellar mass to facilitate the observation of the relationship between stellar mass and formation rate. Specifically, contour lines were drawn for =0.5, 1.5, 2.5, 3.5, and 4.5. It can be observed that lower stellar mass correlates with higher formation rates. To validate this conclusion, we selected the interval with the highest sample quantity, namely 300 to 700 days, and then used the fitted equation (1) to fit the scatter plot of stellar mass and formation rate. In the fig1, 500 represents the median value of the selected period range. However, we observed that the fitting results for stellar mass and formation rate were relatively poor compared to the fitting plot of  and formation rate presented in the manuscript. The distribution of points for stellar mass and formation rate was relatively scattered, whereas the plot for and formation rate exhibited a more concentrated distribution, indicating a stronger correlation.

Fig1. The relationship between the orbital periods of giant planets, stellar mass and their formation rates and the relationship between the stellar mass and their formation rates.

For minor planet, we also plotted three-dimensional scatter plots of minor planet orbital period, production rate, and host star mass. The color in the figure represents the magnitude of the stellar mass. As shown in Figure 2, there is a clear distinction in colors among the minor planet. We also drew contours of stellar mass,=0.5, 1, and 1.5. From the figure, it can be observed that it still supports the argument that lower stellar mass corresponds to higher production rates. Subsequently, we performed fitting on these three variables, with the fitting formula as follows:

, (2)

  is planet period,  is stellar mass.

To validate the accuracy of the relationship between stellar mass and formation rate, we selected the interval with the largest sample size, namely the data points within the orbital period range of 3 to 7 days. We plotted a scatter plot of stellar mass against formation rate and performed fitting using the formula (2) derived earlier. In the fig2, 5 represents the median of the selected interval. As the figure, it can be observed that the fitting for asteroids is much better than for gas giants, but still not as concentrated as the points for . Therefore, we tentatively conclude that the production rate is not solely dependent on stellar mass. However, the above argument is insufficient to support our thesis in the paper.

Fig2. The relationship between the orbital periods of minor planets, stellar mass and their formation rates and the relationship between the stellar mass and their formation rates.

  To further strengthen the verification, we separately plotted histograms of the difference between the actual and theoretical values of stellar mass and log g for gas giants and asteroids. The actual values are the stellar mass and log g values from the data, while the theoretical values are derived from our fitting formula.

For giant planet:

For minor planet:

Finally, we plot the hist plot is:

Fig3. The histogram of the difference between the actual and theoretical values for giant planet.

Fig4. The histogram of the difference between the actual and theoretical values for minor planet.

As Fig3 and Fig4, it is evident that the error in stellar mass is larger compared to the error in . Therefore, in this sample, we believe that , rather than stellar mass, remains the variable most closely related to the formation rate. So, the results demonstrate that while stellar mass does influence the production rate, we cannot ignore the impact of stellar radius on the production rate.

• Line 252 and after. From a formation perspective, the argument of long distances, long formation times may be correct. However, in the solar system, the largest planets are also the most distant ones. Some comment is needed in that paragraph highlighting this issue.

Response: We have taken your advice into consideration and consulted relevant literature, incorporating explanations into the manuscript. We greatly appreciate it.

• Figures 6-9: What does a planet formation rate of 1 mean? One planet per star? Please provide explanation in the captions.

Response: No, 1 is a ratio rather than a numerical value, and we have made the necessary corrections in the manuscript.

• Line 304 and after. Please, provide reference suggesting that Sub-Neptunes and warm-Neptune are more likely to form in place rather than to migrate. Typically, these planets belong to the mass range where tidal interactions with circumstellar material are strongest.

Response: We have incorporated your suggestions and identified relevant literature to strengthen the arguments in our paper.

• Section 4.2.3. Note that if the planet mass is proportional to planet radius, it implies that the average density of the planet is proportional to the radius squared. The authors may want to comment on this result.

Response: Thank you for your suggestion, we have explained it in the article.

15. Figure 10. What do the plus signs indicate in the plot? I would suggest adding the solar system planets to this plot as well, as a

Response: The plus sign indicates taking the bin, we have explained it in the article. Then we took your advice into consideration, we included asteroids within the solar system on the asteroid mass-radius fitting chart for reference. We found that the planets within the solar system basically followed the trend of positive correlation between mass and radius, which is undoubtedly a good result. However, due to some errors in the observation samples we used and the difference between the planetary system and the solar system, our fitting line is not very close to the data points. Nevertheless, it can still generally explain the trend of positive correlation.

• The equivalent functions y(x)=a+b*x=ymean+b*(x-xmean) have the same shape, as well as the slope statistical error = accuracy, but the value a=y(0) is an extrapolated one, which is typically well outside the data, and has a large error estimate, which may then lead to incredibly large error estimates of the accuracy of the approximation. E.g. here is a graph of the data in a relatively small interval ~4. What is the sense of the approximation starting with zero? Not for this data, but for a "live" example. Have a linear approximation of a salary, say, from 1990 to 2024. The value "a" in a linear approximation will correspond to 0-th year (yes, we know, that DC starts from 1, 0 is missing). No sense. Not too much sense for the data in a range 4-5 to be extrapolated to 0. So it would be great to change approximations, to make confidence interals (e.g. a review 2020kdbd.book..191A or other cited sources). If testing hypotheses on a presence of a statistical dependence. one may be recommended to make ANOVA - the analysis of variances with a FAP = False alarm probability estimate.

Response: Thank you for your suggestion. In the process of data processing, we did not consider a little errors, but in the Python fitting method we used for data processing, there is a calculation of relative errors. If we consider the error issue now, it will undoubtedly be a huge workload, and we cannot complete it within the limited revision period. Nevertheless, we still want to thank you again for pointing out the error issue and suggesting a correction. We will conduct relevant research on this in the future. Thank you again.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

May be accepted in a present form

Reviewer 2 Report

None

Proof reading may improve some parts of the text.