Abstract
The rotation period is one of the fundamental physical characteristics of asteroids. It can be determined from photometric measurements by standard methods of time-series period analysis or by creating a physical model of an asteroid with the rotation period being one of the fitted parameters. We used the latter approach to determine the sidereal rotation period for more than 5000 asteroids, out of which about 1600 are those for which their period was not known. We processed photometric measurements of about 100,000 asteroids from the ATLAS survey with the light curve inversion technique in the Asteroids@home project to search for the best-fit rotation period. This was repeated 25 times with randomly resampled—bootstrapped—data. For thousands of asteroids, their best-fit period was the same for most of the bootstrapped data sets; thus, their rotation period was determined with a high degree of reliability.
1 Introduction
Asteroid photometry is a simple yet powerful tool to reveal some basic physical properties of observed objects. Time-resolved photometry—a light curve—provides a direct measurement of the rotation period. The vast majority of asteroid rotation periods (currently around 30,000 in the Asteroid Lightcurve Database, LCDB, of Warner et al., 2009)1 have been determined from light curve analysis.
When an asteroid is observed over a longer time interval (years), its light curves change as the aspect and the solar phase angle change. If the coverage of geometries is sufficient (which usually requires several apparitions for a main-belt asteroid), the evolving shape of the light curves uniquely defines the direction of the rotation axis and the convex shape of the asteroid, together with the sidereal rotation period (Kaasalainen and Ďurech, 2020). The process of reconstruction of asteroid shape and spin is called light curve inversion, and it can be done almost routinely if there is a sufficient amount of observations (Kaasalainen et al., 2001, 2002).
Apart from classical light curves that are true “curves” showing how the brightness evolves with time, there are also photometric observations that are sparse with respect to the rotation period. Thus, instead of a curve, we have individual sparse-in-time brightness measurements. This data type are typically produced by sky surveys and they can be used the same way as light curves for the shape and spin reconstruction of asteroids (Kaasalainen, 2004).
With the light curve inversion, the shape and spin of an asteroid are found by fitting a model (described by the rotation period P, the direction of the spin axis in ecliptic coordinates (λ, β), and parameters of a convex shape) to data. The best model is found by scanning the period/pole parameter space with the standard χ2 measure used to define the best agreement between the model and the data. Hundreds of models have been derived from dense photometry (Wang et al., 2015; Warner et al., 2017; Husárik, 2018; Marciniak et al., 2018; Franco and Pilcher, 2020, for example) and thousands from sparse photometry (Ďurech et al., 2009; Hanuš et al., 2013; Ďurech et al., 2016; Ďurech and Hanuš, 2018; Ďurech et al., 2020, for example). Sparse photometry is available from large sky-surveys (Pan-STARRS, ATLAS, Catalina, Gaia, ZTF, etc.) essentially for all asteroids. Because the photometric accuracy and the number of data points are usually not sufficient to derive a reliable model, the success rate of inversion of sparse data is low. However, as we show in this paper even when sparse data are not abundant enough to derive a reliable full spin/shape model, the rotation period can be derived uniquely.
This work aims to derive sidereal rotation periods of asteroids that have photometric data from the ATLAS survey. In our previous work (Ďurech et al., 2020), we used the same data to derive full shape/spin models.
2 ATLAS Photometry
In this work, we used the same data set as Ďurech et al. (2020). The data come from the Asteroid Terrestrial-impact Last Alert System (ATLAS) telescopes located in Hawaii (Tonry et al., 2018b,a; Smith et al., 2020; Heinze et al., 2018) and consist of photometric measurements collected from June 2015 to October 2018 in orange (o, 560–820 nm) and cyan (c, 420–650 nm) filters. The original data set consisted of photometry of about 180,000 asteroids. However, we selected only asteroids with at least 100 observations, which reduced the total number of objects to about 100,000.
We processed this data set at Asteroids@home project (Ďurech et al., 2015)—the best-fit sidereal rotation period was searched for at an interval of 2–1,000 h, and ten initial spin axis directions were tried for each trial period. The spin and shape parameters then converged to a local minimum in χ2. The global minimum in χ2 then defined the best-fit sidereal rotation period PA. This part of the work was, in fact, ready because we used the periodograms computed already by Ďurech et al. (2020). In our previous work (Ďurech et al., 2020), we selected the global minimum in χ2, tested its significance with respect to other global minima, checked the reliability of the shape model using the same processing pipeline as Ďurech et al. (2018), and reported the shape models with their rotation poles and periods. In our new approach, we concentrated only on rotation periods. We used a bootstrap (BS) method to resample the observations randomly and determine the period and its uncertainty via a Monte Carlo approach. We repeated the period scan for each BS realization, generated a new periodogram, and tested the robustness of the original best-fit period PA.
3 Bootstrap
For each asteroid, we created 25 bootstrapped samples of the original data set in both filters independently (we randomly selected the same number of measurements) and repeated the period search, i.e., we computed other 25 periodograms at Asteroids@home. From each periodogram, we selected the best-fit period defined as having the minimum χ2 value. This way, we obtained for each asteroid (95 789 in total) 25 best periods , i = 1, … , 25, from bootstrapped data. We decided to compute 25 BS samples as a compromise between the robustness of our analysis and the computational time spent on Asteroids@home.
The motivation for this approach was our expectation that if the best period is always the same for all BS samples, it is likely to be the actual rotation period. On the other hand, if the original period PA is not found in resampled BS data, it is likely just a random value not related to the real rotation period. Figure 1 shows the distribution of NBS, which is the number of cases when the best BS period was the same as the original period PA. By “the same” we mean that their relative difference was not larger than 1%. In other words, for each asteroid, we define
FIGURE 1
If every BS sample gives the same best-fit period, then NBS = 25. If no agrees with the original PA, then NBS = 0. As we can see in Figure 1, for about 20,000 asteroids, the best period in each BS sample is just a random value that is, not related to the original period—the number of agreements NBS is zero. The majority (about 70,000) asteroids have no more than five cases in which PBS agrees with PA, their NBS ≤ 5. As expected, the number of asteroids with some value of NBS decreases with increasing NBS until the maximum NBS = 25. The number of asteroids for which all 25 BS samples give the same period as the original data is ∼. The number of observations plays a crucial role in the robustness of period determination—the mean number of data points for asteroids in each histogram bar is shown in Figure 1; it increases with increasing NBS. The minimum number of observations was 100. Asteroids with uncertain periods (NBS ≲ 5) have, on average, less than 200 observations; those with almost certain period determination (NBS ≳ 20) have more than 300.
In Figure 2, we show a comparison between periods PA determined from the original ATLAS data and periods PDB taken from the LCDB of Warner et al. (2009) with the uncertainty tag U = 3 (release from June 2021, 3830 asteroids with U = 3). The uncertainty tag evaluates the reliability of the period and U = 3, the highest value, means that the period is unambiguous and uniquely determined. Asteroids with high NBS (red points, 20 ≤ ≤ NBS ≤ 25) lie primarily on the diagonal, which means that the periods PLCDB and PA are the same. For medium-reliability period determinations (blue points, 10 ≤ NBS ≤ 15), many PA values do not agree with the LCDB period, and often there is an alias of 24 or 48 h; or PA is half of PDB. For the low-reliability group (grey points, 1 ≤ NBS ≤ 6), there is no apparent relation between the two periods—PA periods are just arbitrary values not reliable at all.
FIGURE 2
3.1 Reliability of Period Determination
Out of the total sample of ∼ asteroids, 1784 have NBS = 25, i.e., all BS realizations lead to the same best-fit period, so these are those asteroids with the most reliable period determination. Out of them, 904 also have a period compiled in the LCDB with U = 3, which enables us to compare our ATLAS periods with independent values. We assume that PDB and PA agree if their relative difference is less than 5%. For 885 cases, the ATLAS period agrees with PDB. However, 19 asteroids have different periods; they are listed in Tables 1, 2. We checked the LCDB records and original publications to decide which period was the correct one. In 14 cases, we concluded that the ATLAS period was correct; in four cases, the LCDB period was correct; and in one case, the asteroid is tumbling, so there is no single rotation period. So in 885 cases out of 904, PA and PDB are the same (within a 5% interval), and we assume that these periods are correct—they are real rotation periods. The number of incorrect LCDB periods in our sample is 14 out of 904, so the probability that for a randomly chosen asteroid PDB is not correct (for U = 3) is 1.6%. In other words, we estimated that the probability of a period in the LCDB with U = 3 being correct is .
TABLE 1
| NBS | NA | NU3 | Nsame | psame [%] | [%] | |||
|---|---|---|---|---|---|---|---|---|
| 25 | 1784 | 904 | 885 | 97.9 | 4 | 14 | 1.6 | 99.5 |
| 24 | 795 | 231 | 224 | 97.0 | 2 | 4 | 1.8 | 98.6 |
| 23 | 667 | 165 | 159 | 96.4 | 3 | 2 | 1.3 | 97.9 |
| 22 | 639 | 114 | 109 | 95.6 | 4 | 1 | 0.9 | 97.2 |
| 21 | 642 | 114 | 105 | 92.1 | 93.6 | |||
| 20 | 666 | 99 | 91 | 91.9 | 93.4 | |||
| 19 | 752 | 106 | 90 | 84.9 | 86.3 | |||
| 18 | 696 | 72 | 56 | 77.8 | 79.0 | |||
| 17 | 757 | 74 | 60 | 81.1 | 82.4 |
For nine groups of asteroids with different NBS, the table lists the number NA of asteroids with a given NBS, the number NU3 of those that have U = 3 period record in the LCDB, the number Nsame of asteroids with the PA and PDB being the same (±5%), the probability psame = Nsame/NU3, the number of wrong ATLAS periods (Table 2), the number of wrong LCDB periods (Table 2), and the probability that PA is correctly determined.
TABLE 2
| Asteroid | PA (h) | σBS (h) | PDB (h) | A | DB | Comment |
|---|---|---|---|---|---|---|
| NBS = 25 | ||||||
| 496 Gryphia | 24.515 2 | 0.000 5 | 1072 | ✓ | ✓ | The LCDB rotation period is outside the search interval. It is likely to be the correct period because the light curves of Pilcher et al. (2017) are flat with no apparent period of 24 h |
| 526 Jena | 11.876 4 | 0.000 4 | 9.474 | Our PA value is the same as the period determined by Ďurech et al. (2019) and is consistent also with light curves of Barucci et al. (1994), although they determined the period to 9.474 h | ||
| 571 Dulcinea | 189.12 | 0.03 | 126.3 | According to Stephens (2011), this asteroid is a tumbler so there in no unique rotation period | ||
| 818 Kapteynia | 17.461 | 0.003 | 16.35 | ✓ | Our period is likely to be correct because it is consistent with other sparse photometry (ASAS-SN, Hanuš et al., 2021). The LCDB period of 16.35 h (Stephens, 2002) might be just a result of incorrect folding of separate light curves | |
| 893 Leopoldina | 12.599 3 | 0.000 3 | 14.115 | ✓ | Ďurech et al. (2020) derived a full model with the same period from ATLAS data | |
| 1248 Jugurtha | 12.190 42 | 0.000 03 | 12.91 | ✓ | Our period is the same as an independent determination of Ďurech et al. (2016) | |
| 1332 Marconia | 32.123 | 0.001 | 19.16 | ✓ | Our period is the same as those derived by Devogèle et al. (2017), Ďurech et al. (2019) | |
| 1536 Pielinen | 33.119 | 0.009 | 66.22 | ✓ | The half period of 33 h was found using a convex model, the ellipsoidal model gives the correct period of 66 h. Also a full model derived from the same ATLAS data by Ďurech et al. (2020) has period of 66 h | |
| 1586 Thiele | 3.296 293 | 0.000 009 | 3.086 | ✓ | Our period is correct, the LCDB one is based on only two nights of Childers and Church (2007), likely incorrectly phased | |
| 1684 Iguassu | 9.143 4 | 0.000 4 | 6.415 6 | ✓ | Our period is confirmed by independent results of Waszczak et al. (2015) | |
| 1786 Raahe | 30.168 | 0.002 | 18.72 | ✓ | Our period is correct—confirmed by independent sparse data and also Behrend’d database | |
| 3066 McFadden | 32.752 5 | 0.000 5 | 13.798 | ✓ | Pál et al. (2020) independently confirmed our period | |
| 3409 Abramov | 8.503 61 | 0.000 04 | 7.791 | ✓ | Erasmus et al. (2020) derived the same period form the same ATLAS data, other sparse photometry also confirms this period | |
| 3422 Reid | 3.218 274 | 0.000 009 | 2.91 | ✓ | Pál et al. (2020) found the same period | |
| 3507 Vilas | 4.755 0 | 0.000 03 | 3.959 | ✓ | The same period was found by Erasmus et al. (2020). Ďurech et al. (2020) derived a full model | |
| 5132 Maynard | 3.609 0 | 0.000 1 | 3.902 | ✓ | Independent confirmation of PA by Pál et al. (2020) | |
| 6192 Javiergorosabel | 39.317 | 0.002 | 78.85 | ✓ | Our period is the same as Pál et al. (2020) but it corresponds to one-peak light curve, ellipsoidal model gives double period 78 h | |
| 9033 Kawane | 2.882 69 | 0.000 02 | 5.765 6 | ✓ | The same as above, false half period | |
| 37 635 1993 UJ1 | 662.7 | 0.3 | 600 | ✓ | The periods are similar although their difference is larger than 5% | |
| NBS = 24 | ||||||
| 520 Franziska | 8.251 38 | 0.000 3 | 16.507 | ✓ | ✓ | Wrong half period |
| 740 Cantabia | 32.141 0 | 0.006 | 64.453 | Pál et al. (2020) and also Hanuš et al. (2021) report our PA, Stephens et al. (2010) reports a value that is, twice larger | ||
| 1227 Geranium | 17.268 3 | 0.004 | 12.363 | ? | ? | Ďurech et al. (2020) derived the same period and a full model but the shape has an unrealistic triangular pole-on silhouette. Not clear which period is the correct one |
| 1960 Guisan | 7.846 66 | 0.004 | 8.46 | ✓ | Two low-quality light curves of Binzel (1987) probably incorrectly phased | |
| 2791 Paradise | 16.348 6 | 0.005 | 9.81 | ✓ | Hanuš et al. (2016) reports period of 9.81 h (on a limited search interval) but Behrend’s web reports 16.361 h | |
| 4797 Ako | 3.870 091 | 0.000 02 | 4.085 | ✓ | Ďurech et al. (2020) derived a full model with the period close to that of Bennefeld et al. (2009) | |
| 11 087 Yamasakimakoto | 6.279 556 | 0.000 04 | 4.536 9 | ✓ | Pál et al. (2020) confirm our period | |
| NBS = 23 | ||||||
| 1539 Borrelly | 23.831 2 | 0.01 | 15.922 | ✓ | ✓ | Behrend’s web and also Polakis (2020) confirm our period |
| 2425 Shenzhen | 9.838 246 | 0.000 02 | 14.715 | ✓ | Confirmed by other sparse data and Ďurech et al. (2016) | |
| 2895 Memnon | 3.760 08 | 0.000 1 | 7.516 | False half solution, Ďurech et al. (2020) derived a unique model from the same data | ||
| 7783 1994 JD | 15.908 7 | 0.01 | 31.83 | ✓ | False half period | |
| 10 704 1981 RQ1 | 3.754 01 | 0.000 05 | 7.507 | ✓ | False half period. Erasmus et al. (2020) have the correct double period from the same data | |
| 18 582 1997 XK9 | 107.712 1 | 0.01 | 114 | ✓ | ✓ | We consider these periods to be the same within their uncertainty intervals |
| NBS = 22 | ||||||
| 282 Clorinde | 49.362 3 | 0.02 | 6.42 | ✓ | ✓ | Confirmed by Ďurech et al. (2020) and Bonamico and van Belle (2021) |
| 518 Halawe | 7.159 08 | 0.000 3 | 14.31 | False half period | ||
| 1949 Messina | 3.390 64 | 0.000 2 | 3.649 1 | ✓ | False period | |
| 7937 1990 QA2 | 3.116 85 | 0.000 2 | 6.23 | ✓ | Probably false half period | |
| 10 037 1984 BQ | 7.854 744 | 0.000 07 | 6.748 2 | ✓ | Probably incorrect PA | |
List of asteroids with NBS ≥ 22 for which the period PA that we obtained from ATLAS data disagrees with the period PDB in the LCDB. The formal uncertainty of PA is σBS. By “✓,” we mark our decision if either ATLAS period (A) of LCDB period (DB) is correctly determined. Behrend’s web is a database of asteroid light curve observations and rotation periods available at https://obswww.unige.ch/behrend/page-cou.html.
We compared ATLAS and LCDB periods also for asteroids with NBS = 24, 23, 22 and inspected the discrepant cases (Table 2). The results are summarized in Table 1. The probability that for a given asteroid both periods are correct is a product of probabilities that PA is correct and that PDB is correct. From the analysis of the group of asteroids with NBS = 25, we know that , so we can compute . The table lists these probabilities for NBS ≥ 17. Probability is for NBS ≥ 22, for NBS = 21 or 20 it is around 93%, and it drops below 90% for NBS < 20.
The analysis above depends on the total number of BS samples, however, not critically. If, for some reason, we had only 22 BS samples in total, then for NBS = 22 (according to Table 1), NA = 3885, NU3 = 1414, Nsame = 1377, psame = 97.4%, ,, and . So our new estimate of the probability would be almost the same as before and would be somewhere between previous values of probabilities determined for NBS = 22, 23, 24, 25.
3.2 Periods From ATLAS Data
In total, the sample of asteroids with consists of 5126 period determinations (we excluded those listed in Table 2 as not correct and other 55 asteroids that were affected by large systematic errors in their input photometric data). We consider the probability high enough to publish these periods; they are provided as Supplementary Material to this paper. In Table 3, we list a small fraction of the results as an example. The uncertainty σBS of the rotation period is determined as a standard deviation of values that agree with PA. In many cases, this error is unrealistically small. Therefore, as a conservative upper limit of the period uncertainty, we define , where Δ is the length of the time interval covered by the data. This uncertainty of the sidereal rotation period corresponds to a shift of 1/20 in the rotation phase over the interval Δ (Kaasalainen, 2004). If σBS is significantly (several times) larger than σmax and PA is close to 24 h, it is a strong indication that the detected period is not the true rotation period of the asteroid but a false alias period related to 1-day sampling of the data. As can be seen in Figure 3 on a histogram of periods, for asteroids with asteroids with σBS/σmax > 10, there is an excess of those with the rotation period close to 24 h. These are mostly false-positive solutions that consistently yield the same rotation period of ∼ h for all BS samples, but it is a bias caused by the observations being carried out at one location. The 1-day pattern in the data is inevitable for all BS samples.
TABLE 3
| Asteroid | PA | σBS | σmax | NBS | |
|---|---|---|---|---|---|
| (h) | (h) | (h) | |||
| 13 | Egeria | 7.046 34 | 0.000 8 | 0.000 9 | 22 |
| 24 | Themis | 8.374 15 | 0.000 6 | 0.000 2 | 23 |
| 26 | Proserpina | 13.105 4 | 0.002 | 0.000 3 | 21 |
| 32 | Pomona | 9.447 690 | 0.000 08 | 0.000 2 | 24 |
| 33 | Polyhymnia | 18.609 12 | 0.000 8 | 0.000 7 | 25 |
| 34 | Circe | 12.174 549 | 0.000 1 | 0.000 3 | 24 |
| 37 | Fides | 7.332 7 | 0.001 | 0.000 1 | 21 |
| 42 | Isis | 13.582 72 | 0.000 1 | 0.000 4 | 23 |
| 48 | Doris | 11.889 92 | 0.000 6 | 0.000 3 | 22 |
| 49 | Pales | 20.708 14 | 0.000 3 | 0.000 9 | 25 |
| ⋮ | |||||
| 217101 | 2001 XM29 | 2.709 0 | 0.000 1 | 0.000 06 | 23 |
| 217298 | 2004 JY4 | 38.02 | 0.07 | 0.01 | 20 |
| 222655 | 2001 XW186 | 3.365 68 | 0.000 4 | 0.000 1 | 21 |
| 231 865 | 2000 SY318 | 3.016 51 | 0.000 2 | 0.000 08 | 23 |
| 250436 | 2003 WT137 | 8.050 50 | 0.000 9 | 0.000 7 | 21 |
| 267090 | 1999 VS198 | 8.952 3 | 0.003 | 0.000 7 | 23 |
| 270324 | 2001 XV96 | 88.095 5 | 0.08 | 0.08 | 23 |
| 350872 | 2002 PG43 | 19.710 | 0.03 | 0.007 | 20 |
| 373534 | 2001 TR169 | 5.063 67 | 0.000 6 | 0.000 5 | 20 |
| 411201 | 2010 LJ14 | 114.233 02 | 0.008 | 0.03 | 23 |
For asteroids with NBS ≥ 20, the table lists their rotation period PA, the standard error σBS estimated from bootstrap, the uncertainty σmax of the period estimated from the length of the observing interval, and the number NBS of cases when PA was the same as . The probability of PA being correct is directly related to NBS according to Table 1. The complete table with more than 5000 records is available as Supplementary Material.
FIGURE 3
For a part of asteroids with our ATLAS-based period determination, their rotation period was already known, sometimes also with a corresponding shape model. Namely, there are 3526 asteroids for which some period is reported in the LCDB; however, the number of reliable periods with U = 3 is only 1616. For 1600 asteroids, we derived their rotation period for the first time.
In Figure 4, we show a similar plot as Pál et al. (2020), namely the comparison of distribution of periods from the LCDB, TESS, and our ATLAS results. Although there is an apparent lack of long periods in our results when compared with TESS results of Pál et al. (2020), ATLAS sparse photometry can be used for an efficient determination of rotation periods of the order of hundreds of hours. Recent results of Erasmus et al. (2021) show that ground-based surveys are capable of detection rotation periods even longer than thousand hours. However, for the majority of asteroids in our sample, we were not able to determine their rotation period, so it is not possible to use the derived periods for statistical studies without properly accounting for bias.
FIGURE 4
4 Conclusion
We have derived sidereal rotation periods for more than 5000 asteroids; for more than 1600, it is the first period determination. The reliability of these periods is , so some periods can be incorrect, but the whole sample is a significant increase in the number of asteroids with a known rotation period. The method of bootstrapping the original data is simple to implement, although computationally demanding. The same approach can also be used to new ATLAS data, ideally combined with other sparse photometry, for example, from Gaia Data Release 3.
Statements
Data availability statement
The data analyzed in this study is subject to the following licenses/restrictions: ATLAS photometry. Requests to access these datasets should be directed to https://atlas.fallingstar.com.
Author contributions
JĎ—ATLAS data processing, bootstrap, interpretation; MV—processing of bootstrapped periodograms, interpretation; RV—technical administration of the Asteroids@home project; NE—ATLAS scientist.
Funding
This work has been supported by the grant 20-08218S of the Czech Science Foundation. This work has made use of data from the Asteroid Terrestrial-impact Last Alert System (ATLAS) project. ATLAS is primarily funded to search for near earth asteroids through NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575; byproducts of the NEO search include images and catalogs from the survey area. The ATLAS science products have been made possible through the contributions of the University of Hawaii Institute for Astronomy, the Queen’s University Belfast, the Space Telescope Science Institute, and the South African Astronomical Observatory (SAAO), and the Millennium Institute of Astrophysics (MAS), Chile.
Acknowledgments
We greatly appreciate the contribution of tens of thousands of volunteers who joined the Asteroids@home BOINC project and provided their computing resources. This research has made use of IMCCE’s Miriade VO tool.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fspas.2022.809771/full#supplementary-material
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Summary
Keywords
asteroids, photometry, surveys, light curves, bootstrap
Citation
Ďurech J, Vávra M, Vančo R and Erasmus N (2022) Rotation Periods of Asteroids Determined With Bootstrap Convex Inversion From ATLAS Photometry. Front. Astron. Space Sci. 9:809771. doi: 10.3389/fspas.2022.809771
Received
05 November 2021
Accepted
17 January 2022
Published
22 February 2022
Volume
9 - 2022
Edited by
Daniel Hestroffer, Université de Sciences Lettres de Paris, France
Reviewed by
Alan Harris, NASA Jet Propulsion Laboratory (JPL), United States
Jianguo Yan, Wuhan University, China
Updates
Copyright
© 2022 Ďurech, Vávra, Vančo and Erasmus.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Josef Ďurech, durech@sirrah.troja.mff.cuni.cz
This article was submitted to Astrostatistics, a section of the journal Frontiers in Astronomy and Space Sciences
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