Abstract
The asteroid proper elements computed from an analytical secular perturbation theory are thought to be of inferior accuracy with respect to the synthetic proper elements computed by means of a purely numerical procedure. To verify this, we carried out two different tests: one based on a plain comparison of the analytical and synthetic proper elements to check whether they differ by more than the corresponding errors of synthetic elements, and the other consisting in direct determination of instabilities of analytical elements themselves. In both tests, we used much larger samples than in the previous attempts. The results clearly confirm the superior quality of synthetic values. We also present a new synthetic method to determine the locations of secular resonances, based on a polynomial fit of the values of asteroid secular rates g, s, and subsequent computation of resonance frequencies on a grid in the phase space of proper elements. Plots showing the contour lines corresponding to selected small values of resonant divisors serve to visualize the resonance locations. We analyzed in detail the accuracy of the fit and of the secular resonance locations, comparing them with the previously available positions determined by means of the analytical theory; we found that the two methods give qualitatively the same results, but that the results of the new synthetic method appear to be more reliable. In view of the above results, we conclude that the regular maintenance of the catalog of analytical proper elements can be discontinued in the future.
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Notes
The median standard deviations of synthetic proper elements amount, respectively, to \(\sim 10^{-6}\), \(\sim 2\times 10^{-4}\), \(\sim 10^{-4}\).
An obvious advantage of the polynomial fit method is the possibility to easily add higher-degree monomials, without making the method much more complex.
References
Carpino, M., Milani, A., Nobili, A.M.: Long-term numerical integrations and synthetic theories for the motion of the outer planets. Astron. Astrophys. 181, 182–194 (1987)
Carruba, V.: The (not so) peculiar case of the Padua family. Mon. Not. R. Astron. Soc. 395, 358–377 (2009). https://doi.org/10.1111/j.1365-2966.2009.14523.x
Carruba, V.: The stable archipelago in the region of the Pallas and Hansa dynamical families. Mon. Not. R. Astron. Soc. 408, 580–600 (2010). https://doi.org/10.1111/j.1365-2966.2010.17146.x
Carruba, V., Michtchenko, T.A.: A frequency approach to identifying asteroid families. Astron. Astrophys. 475, 1145–1158 (2007). https://doi.org/10.1051/0004-6361:20077689
Carruba, V., Huaman, M.E., Domingos, R.C., Santos, C.R.D., Souami, D.: Dynamical evolution of V-type asteroids in the central main belt. Mon. Not. R. Astron. Soc. 439, 3168–3179 (2014). https://doi.org/10.1093/mnras/stu192. arXiv:1401.6332
Carruba, V., Vokrouhlický, D., Nesvorný, D.: Detection of the Yarkovsky effect for C-type asteroids in the Veritas family. Mon. Not. R. Astron. Soc. 469, 4400–4413 (2017). https://doi.org/10.1093/mnras/stx1186
Ferraz-Mello, S.: The convergence domain of the Laplacian expansion of the disturbing function. Celest. Mech. Dyn. Astron. 58, 37–52 (1994). https://doi.org/10.1007/BF00692116
Hori, G.: Theory of general perturbation with unspecified canonical variable. Publ. Astron. Soc. Jpn. 18, 287 (1966)
Knežević, Z.: Asteroid family identification: history and state of the art. In: Chesley SR, Morbidelli A, Jedicke R, Farnocchia D (eds) Asteroids: New Observations, New Models, IAU Symposium, vol 318, pp 16–27, https://doi.org/10.1017/S1743921315008728 (2016)
Knežević, Z.: Computation of asteroid proper elements: recent advances. Serb. Astron. J. 195, 1–8 (2017). https://doi.org/10.2298/SAJ170407005K
Knežević, Z., Milani, A.: Synthetic proper elements for outer main belt asteroids. Celest. Mech. Dyn. Astron. 78, 17–46 (2000)
Knežević, Z., Milani, A.: Proper element catalogs and asteroid families. Astron. Astrophys. 403, 1165–1173 (2003). https://doi.org/10.1051/0004-6361:20030475
Knežević, Z., Milani, A., Farinella, P., Froeschle, C., Froeschle, C.: Secular resonances from 2 to 50 AU. Icarus 93, 316–330 (1991). https://doi.org/10.1016/0019-1035(91)90215-F
Knežević, Z., Froeschlè, C., Lemaitre, A., Milani, A., Morbidelli, A.: Comparison between two theories of asteroid proper elements. Astron. Astrophys. 293, 605–612 (1995)
Knežević, Z., Tsiganis, K., Varvoglis, H.: The dynamical portrait of the Veritas family region. In: Warmbein B (ed) Asteroids, Comets, and Meteors: ACM 2002, ESA Special Publication, vol 500, pp 335–338 (2002)
Lemaitre, A., Morbidelli, A.: Proper elements for highly inclined asteroidal orbits. Celest. Mech. Dyn. Astron. 60, 29–56 (1994). https://doi.org/10.1007/BF00693091
Michel, P., Froeschlé, C.: The location of linear secular resonances for semimajor axes smaller than 2 AU. Icarus 128, 230–240 (1997). https://doi.org/10.1006/icar.1997.5727
Michtchenko, T.A., Lazzaro, D., Carvano, J.M., Ferraz-Mello, S.: Dynamic picture of the inner asteroid belt: implications for the density, size and taxonomic distributions of real objects. Mon. Not. R. Astron. Soc. 401, 2499–2516 (2010). https://doi.org/10.1111/j.1365-2966.2009.15825.x
Milani, A.: The Trojan asteroid belt: proper elements, stability, chaos and families. Celest. Mech. Dyn. Astron. 57, 59–94 (1993). https://doi.org/10.1007/BF00692462
Milani, A.: The dynamics of the Trojan asteroids. In: Milani A, di Martino M, Cellino A (eds) Asteroids, Comets, Meteors 1993, IAU Symposium, vol 160, p. 159 (1994)
Milani, A., Gronchi, G.F.: Theory of Orbit Determination. Cambridge University Press, Cambridge (2010)
Milani, A., Knežević, Z.: Secular perturbation theory and computation of asteroid proper elements. Celest. Mech. Dyn. Astron. 49, 347–411 (1990). https://doi.org/10.1007/BF00049444
Milani, A., Knežević, Z.: Asteroid proper elements and secular resonances. Icarus 98, 211–232 (1992). https://doi.org/10.1016/0019-1035(92)90091-K
Milani, A., Knežević, Z.: Asteroid proper elements and the dynamical structure of the asteroid main belt. Icarus 107, 219–254 (1994). https://doi.org/10.1006/icar.1994.1020
Milani, A., Cellino, A., Knežević, Z., Novaković, B., Spoto, F., Paolicchi, P.: Asteroid families classification: exploiting very large datasets. Icarus 239, 46–73 (2014). https://doi.org/10.1016/j.icarus.2014.05.039. arXiv:1312.7702
Milani, A., Knežević, Z., Spoto, F., Cellino, A., Novaković, B., Tsirvoulis, G.: On the ages of resonant, eroded and fossil asteroid families. Icarus 288, 240–264 (2017). https://doi.org/10.1016/j.icarus.2016.12.030
Milani, A., Knežević, Z., Spoto, F., Paolicchi, P.: Asteroid cratering families: recognition and collisional interpretation. Astron. Astrophys. 622, A47 (2019). https://doi.org/10.1051/0004-6361/201834056
Morbidelli, A.: Asteroid secular resonant proper elements. Icarus 105, 48–66 (1993). https://doi.org/10.1006/icar.1993.1110
Morbidelli, A., Henrard, J.: Secular resonances in the asteroid belt—theoretical perturbation approach and the problem of their location. Celest. Mech. Dyn. Astron. 51, 131–167 (1991a). https://doi.org/10.1007/BF00048606
Morbidelli, A., Henrard, J.: The main secular resonances \(\nu _6, \nu _5\) and \(\nu _{16}\) in the asteroid belt. Celest. Mech. Dyn. Astron. 51, 169–197 (1991b). https://doi.org/10.1007/BF00048607
Nesvorný, D., Morbidelli, A.: Three-body mean motion resonances and the chaotic structure of the asteroid belt. Astron. J. 116, 3029–3037 (1998). https://doi.org/10.1086/300632
Nesvorny, D., Bottke, W.F., Dones, L., Levison, H.F.: The recent breakup of an asteroid in the main belt region. In: AAS/Division for Planetary Sciences Meeting Abstracts #34, AAS/Division for Planetary Sciences Meeting Abstracts, vol 34, p. 02.08 (2002)
Nesvorný, D., Bottke, W.F., Levison, H.F., Dones, L.: Recent origin of the solar system dust bands. Astrophys. J. 591, 486–497 (2003). https://doi.org/10.1086/374807
Nobili, A.M., Milani, A., Carpino, M.: Fundamental frequencies and small divisors in the orbits of the outer planets. Astron. Astrophys. 210, 313–336 (1989)
Novaković, B., Tsiganis, K., Knežević, Z.: Chaotic transport and chronology of complex asteroid families. Mon. Not. R. Astron. Soc. 402, 1263–1272 (2010). https://doi.org/10.1111/j.1365-2966.2009.15970.x. arXiv:0910.5857
Schubart, J.: Additional results on orbits of Hilda-type asteroids. Astron. Astrophys. 241, 297–302 (1991)
Tsiganis, K., Knežević, Z., Varvoglis, H.: Reconstructing the orbital history of the Veritas family. Icarus 186, 484–497 (2007). https://doi.org/10.1016/j.icarus.2006.09.017
Tsirvoulis, G., Novaković, B.: Secular resonances with Ceres and Vesta. Icarus 280, 300–307 (2016). https://doi.org/10.1016/j.icarus.2016.06.024
Vokrouhlický, D., Brož, M., Morbidelli, A., Bottke, W.F., Nesvorný, D., Lazzaro, D., Rivkin, A.S.: Yarkovsky footprints in the Eos family. Icarus 182, 92–117 (2006). https://doi.org/10.1016/j.icarus.2005.12.011
Williams, J.G.: Secular Perturbations in the Solar System. Ph.D. thesis, University of California, Los Angeles (1969)
Yoshikawa, M.: A simple analytical model for the secular resonance nu6 in the asteroidal belt. Celest. Mech. 40, 233–272 (1987). https://doi.org/10.1007/BF01235843
Yuasa, M.: Theory of secular perturbations of asteroids including terms of higher orders and higher degrees. Publ. Astron. Soc. Jpn. 25, 399 (1973)
Acknowledgements
This paper is dedicated to the memory of Andrea Milani, who passed away in the course of its preparation. It is very appropriate that the paper appears in the CM&DA Golden Topical Collection issue, as it was our intention with this paper to round off and terminate the research on asteroid analytical proper elements which we commenced with the first paper devoted to this topic published in CM&DA nearly 30 years ago. Gratitude goes to the referees V. Carubba and A. Morbidelli for many useful suggestions that helped improving the original manuscript. ZK acknowledges support through the Projects F-187 of the Serbian Academy of Sciences and Arts, and OI176011 of the Ministry of Education, Science and Technological Development of Serbia.
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This article is part of the topical collection on 50 years of Celestial Mechanics and Dynamical Astronomy Guest Editors: Editorial Committee.
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Knežević, Z., Milani, A. Are the analytical proper elements of asteroids still needed?. Celest Mech Dyn Astr 131, 27 (2019). https://doi.org/10.1007/s10569-019-9906-4
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DOI: https://doi.org/10.1007/s10569-019-9906-4