Skip to main content
Log in

Transport Characteristics of the Near-Surface Layer of the Nucleus of Comet 67P/Churyumov–Gerasimenko

  • Published:
Solar System Research Aims and scope Submit manuscript

Abstract

The paper considers the free molecular flow of gas through the dusty porous surface layer of a comet nucleus. The study is based on computer models of generation of a porous medium and Knudsen gas diffusion. We consider various types of homogeneous and heterogeneous layers constructed from nonintersecting spheres, including layers that contain microcracks or inner cavities. Using the test-particle method, we quantitatively estimate the free path distribution function, layer permeability, and effective kinetic characteristics of sublimation products passing through a nonisothermal porous layer. In addition, in this approach, we consider the volumetric absorption of visible solar radiation in the near-surface absorbing layer. Simple approximation expressions are obtained for all the transport characteristics under study, which makes it possible to estimate the characteristics with sufficient accuracy for practical applications in the physics of comets. The results will be used to construct new consistent models of energy transfer in the near-surface layer of a comet nucleus and, first of all, to analyze the results of the observations of comet 67P/Churyumov‒Gerasimenko.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.

Similar content being viewed by others

REFERENCES

  1. Bentley, M.S., Torkar, K., Jeszenszky, H., Romstedt, J., Schmied, R., and Mannel, T., Cometary dust at the nanometre scale – the MIDAS view after perihelion, European Planetary Science Congress, Nantes, France, Sept. 27–Oct. 2, 2015, id. EPSC2015-441.

  2. Bhatia, S.K., Capillary network models for transport in packed beds: consideration of pore aspect ratio, Chem. Eng. Commun., 1996, vol. 154, pp. 183–202.

    Article  Google Scholar 

  3. Bird, G.A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford: Oxford Univ. Press, 1994, 2nd ed.

    Google Scholar 

  4. Blum, J. and Schrapler, R., Structure and mechanical properties of high-porosity macroscopic agglomerates formed by random ballistic deposition, Phys. Rev. Lett., 2004, vol. 93, pp. 115503.1–115503.4.

  5. Blum, J., Gundlach, B., Krause, M., Fulle, M., Johansen, A., Agarwal, J., von Borstel, I., et al., Evidence for the formation of comet 67P/Churyumov–Gerasimenko through gravitational collapse of a bound clump of pebbles, Mon. Not. R. Astron. Soc., 2017, vol. 469, p. S755.

    Article  Google Scholar 

  6. Boudreau, B.P., The diffusive tortuosity of fine-grained unlithified sediments, Geochim. Cosmochim. Acta, 1996, vol. 60, pp. 3139–3142.

    Article  ADS  Google Scholar 

  7. Carman, P.C., Flow of Gases through Porous Media, London: Butterworths, 1956.

    MATH  Google Scholar 

  8. Choukroun, M., Keihm, S., Schloerb, F.P., Gulkis, S., Lellouch, E., Leyrat, C., von Allmen, P., et al., Dark side of Comet 67P/Churyumov-Gerasimenko in Aug.–Oct. 2014. MIRO/Rosetta continuum observations of polar night in the southern regions, Astron. Astrophys., 2015, vol. 583, id. A28.

  9. Clausing, P., Über die Strömung sehr verdünnter Gase durch Röhren von beliebiger Länge, Ann. Phys. Ser., 1932, vol. 12, pp. 961–989.

    Article  ADS  Google Scholar 

  10. Davidsson, B.J.R. and Skorov, Y.V., On the light-absorbing surface layer of cometary nuclei. I. Radiative transfer, Icarus, 2002, vol. 156, pp. 223–248.

    Article  ADS  Google Scholar 

  11. Davidsson, B.J.R. and Skorov, Y.V., On the light-absorbing surface layer of cometary nuclei. II. Thermal modeling, Icarus, 2002, vol. 159, pp. 239–258.

    Article  ADS  Google Scholar 

  12. Davidsson, B.J.R. and Skorov, Y.V., A practical tool for simulating the presence of gas comae in thermophysical modeling of cometary nuclei, Icarus, 2004, vol. 168, pp. 163–185.

    Article  ADS  Google Scholar 

  13. Della Corte, V., Rotundi, A., Fulle, M., Gruen, E., Weissman, P., Sordini, R., Ferrari, M., Ivanovski, S., Lucarelli, F., Accolla, M., Zakharov, V., Mazzotta, EpifaniE., Lopez-Moreno, J.J., Rodriguez, J., Colangeli, L., Palumbo, P., Bussoletti, E., Crifo, J.F., Esposito, F., Green, S.F., Lamy, P.L., McDonnell, J.A.M., Mennella, V., Molina, A., Morales, R., Moreno, F., Ortiz, J.L., Palomba, E., Perrin, J.M., Rietmeijer, F.J.M., Rodrigo, R., Zarnecki, J.C., Cosi, M., Giovane, F., Gustafson, B., Herranz, M.L., Jeronimo, J.M., Leese, M.R., Lopez-Jimenez, A.C., and Altobelli, N., GIADA: Shining a light on the monitoring of the comet dust production from the nucleus of 67P/Churyumov-Gerasimenko, Astron. Astrophys., 2015, vol. 583, id. A13. https://doi.org/10.1051/0004-6361/201526208

  14. Dullien, F.A., Porous Media, Fluid Transport and Pore Structure, NY: Academic Press, 1991.

    Google Scholar 

  15. El-Maarry, M.R., Groussin, O., Keller, H.U., Thomas, N., Vincent, J.-B., Mottola, S., Pajola, M., et al. Surface morphology of comets and associated evolutionary processes: A review of Rosetta’s observations of 67P/Churyumov-Gerasimenko, Space Sci. Rev., 2019, vol. 215, p. 36.

    Article  ADS  Google Scholar 

  16. Enzian, A., Cabot, H., and Klinger, J., A 2 1/2 D thermodynamic model of cometary nuclei. I. application to the activity of comet 29P/Schwassmann-Wachmann 1, Astron. Astrophys., 1997, vol. 319, pp. 995–1006.

    ADS  Google Scholar 

  17. Fanale, F.P. and Salvail, J.R., An idealized short-period comet model—Surface insolation, H2O flux, dust flux, and mantle evolution, Icarus, 1984, vol. 60, pp. 476–511.

    Article  ADS  Google Scholar 

  18. Fornasier, S., Hasselmann, P.H., Barucci, M.A., Feller, C., Besse, S., Leyrat, C., Lara, L., et al., Spectrophotometric properties of the nucleus of comet 67P/Churyumov-Gerasimenko from the OSIRIS instrument onboard the ROSETTA spacecraft, Astron. Astrophys., 2015, vol. 583, id. A30.

  19. Fornasier, S., Hoang, V.H., Hasselmann, P.H., Feller, C., Barucci, M.A., Deshapriya, J.D.P., Sierks, H., et al., Linking surface morphology, composition, and activity on the nucleus of 67P/Churyumov-Gerasimenko, Astron. Astrophys., 2019, vol. 630, id. A7.

  20. Fulle, M., Della Corte, V., Rotundi, A., Weissman, P., Juhasz, A., Szego, K., Sordini, R., Ferrari, M., Ivanovski, S., Lucarelli, F., Accolla, M., Merouane, S., Zakharov, V., Mazzotta Epifani, E., López-Moreno, J.J., Rodríguez, J., Colangeli, L., Palumbo, P., Grün, E., Hilchenbach, M., Bussoletti, E., Esposito, F., Green, S.F., Lamy, P.L., McDonnell, J.A.M., Mennella, V., Molina, A., Morales, R., Moreno, F., Ortiz, J.L., Palomba, E., Rodrigo, R., Zarnecki, J.C., Cosi, M., Giovane, F., Gustafson, B., Herranz, M.L., Jeronimo, J.M., Leese, M.R., López-Jiménez, A.C., and Altobelli, N., Density and charge of pristine fluffy particles from comet 67P/Churyumov-Gerasimenko, Astrophys. J. Lett., 2015, vol. 802, no. 1, id. L12. https://doi.org/10.1088/2041-8205/802/1/L12

  21. Fulle, M., Marzari, F., Della, CorteV., Fornasier, S., Sierks, H., Rotundi, A., Barbieri, C., Lamy, P.L., Rodrigo, R., Koschny, D., Rickman, H., Keller, H.U., Lopez-Moreno, J.J., Accolla, M., Agarwal, J., A’Hearn, M.F., Altobelli, N., Barucci, M.A., Bertaux, J.-L., Bertini, I., Bodewits, D., Bussoletti, E., Colangeli, L., Cosi, M., Cremonese, G., Crifo, J.-F., Da Deppo, V., Davidsson, B., Debei, S., De Cecco, M., Esposito, F., Ferrari, M., Giovane, F., Gustafson, B., Green, S.F., Groussin, O., Grun, E., Gutierrez, P., Guttler, C., Herranz, M.L., Hviid, S.F., Ip, W., Ivanovski, S.L., Jerónimo, J.M., Jorda, L., Knollenberg, J., Kramm, R., Kührt, E., Küppers, M., Lara, L., Lazzarin, M., Leese, M.R., López-Jiménez, A.C., Lucarelli, F., Mazzotta Epifani, E., McDonnell, J.A.M., Mennella, V., Molina, A., Morales, R., Moreno, F., Mottola, S., Naletto, G., Oklay, N., Ortiz, J.L., Palomba, E., Palumbo, P., Perrin, J.-M., Rietmeijer, F.J.M., Rodríguez, J., Sordini, R., Thomas, N., Tubiana, C., Vincent, J.-B., Weissman, P., Wenzel, K.-P., Zakharov, V., and Zarnecki, J.C., Evolution of the dust size distribution of comet 67P/Churyumov-Gerasimenko from 2.2 au to perihelion, Astrophys. J., 2016, vol. 821, no. 1, id. 19. https://doi.org/10.3847/0004-637X/821/1/19

  22. Hilchenbach, M., Kissel, J., Langevin, Y., Briois, C., von Hoerner, H., Koch, A., Schulz, R., Silén, J., Altwegg, K., Colangeli, L., Cottin, H., Engrand, C., Fischer, H., Glasmachers, A., Grun, E., Haerendel, G., Henkel, H., Höfner, H., Hornung, K., Jessberger, E.K., Lehto, H., Lehto, K., Raulin, F., Le Roy, L., Rynö, J., Steiger, W., Stephan, T., Thirkell, L., Thomas, R., Torkar, K., Varmuza, K., Wanczek, K.-P., Altobelli, N., Baklouti, D., Bardyn, A., Fray, N., Krüger, H., Ligier, N., Lin, Z., Martin, P., Merouane, S., Orthous-Daunay, F.R., Paquette, J., Revillet, C., Siljestrom, S., Stenzel, O., and Zaprudin, B., Comet 67P/Churyumov-Gerasimenko: Close-up on dust particle fragments, Astrophys. J. Lett., 2016, vol. 816, no. 2, id. L32. https://doi.org/10.3847/2041-8205/816/2/L32

  23. Groussin, O., Attree, N., Brouet, Y., Ciarletti, V., Davidsson, B., Filacchione, G., Fischer, H.-H., et al., The thermal, mechanical, structural, and dielectric properties of cometary nuclei after Rosetta, Space Sci. Rev., 2019, vol. 215, p. 29.

    Article  ADS  Google Scholar 

  24. Grün, E., Gebhard, J., Bar-Nun, A., Benkhoff, J., Dueren, H., Eich, G., Hische, R., Huebner, W.F., Keller, H.U., and Klees, G., Development of a dust mantle on the surface of an insolated ice-dust mixture – Results from the KOSI-9 experiment, J. Geophys. Res., 1993, vol. 98, pp. 15091–15104.

    Article  ADS  Google Scholar 

  25. Gulkis, S., Allen, M., von Allmen, P., Beaudin, G., Biver, N., Bockelée-Morvan, D., Choukroun, M., Crovisier, J., Davidsson, B.J.R., Encrenaz, P., Encrenaz, T., Frerking, M., Hartogh, P., Hofstadter, M., Ip, W.-H., Janssen, M., Jarchow, C., Keihm, S., Lee, S., Lellouch, E.L., Leyrat, C., Rezac, L., Schloerb, F.P., and Spilker, T., Subsurface properties and early activity of comet 67P/Churyumov-Gerasimenko, Science, 2015, vol. 347, no. 6220, id. aaa0709. https://doi.org/10.1126/science.aaa0709

  26. Gundlach, B., Skorov, Y.V., and Blum, J., Outgassing of icy bodies in the solar system – I. The sublimation of hexagonal water ice through dust layers, Icarus, 2011, vol. 213, p. 710.

    Article  ADS  Google Scholar 

  27. Heat and Gas Diffusion in Comet Nuclei, Huebner, W.F., Benkhoff, J., Capria, M.-T., Coradini, A., De Sanctis, C., Orosei, R., and Prialnik D., Eds., Noordwijk, Netherlands: ESA Publications Division, 2006.

    Google Scholar 

  28. Kaufmann, E. and Hagermann, A., Experimental investigation of insolation-driven dust ejection from Mars’ CO2 ice caps, Icarus, 2017, vol. 282, pp. 118–126.

    Article  ADS  Google Scholar 

  29. Kaufmann, E., Kömle, N.I., and Kargl, G., Laboratory simulation experiments on the solid-state greenhouse effect in planetary ices, Icarus, 2006, vol. 185, pp. 274–286.

    Article  ADS  Google Scholar 

  30. Keller, H.U., Arpigny, C., Barbieri, C., Bonnet, R.M., Cazes, S., Coradini, M., Cosmovici, C.B., Delamere, W.A., Huebner, W.F., Hughes, D.W., Jamar, C., Malaise, D., Reitsema, H.J., Schmidt, H.U., Schmidt, W.K.H., Seige, P., Whipple, F.L., and Wilhelm, K., First Halley multicolour camera imaging results from Giotto, Nature, 1986, vol. 321, pp. 320–326.

    Article  ADS  Google Scholar 

  31. Kloss, C., Goniva, C., Hager, A., Amberger, S., and Pirker, S., Models, algorithms and validation for opensource DEM and CFD-DEM, Prog. Comput. Fluid Dyn., Int. J., 2012, vol. 12, nos. 2–3, pp. 140–152.

    Google Scholar 

  32. Knudsen, M., Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren, Ann. Phys., 1909, vol. 333, pp. 75–130. https://doi.org/10.1002/andp.19093330106

    Article  MATH  Google Scholar 

  33. Kömle, N.I. and Steiner, G., Temperature evolution of porous ice samples covered by a dust mantle, Icarus, 1992, vol. 96, pp. 204–212.

    Article  ADS  Google Scholar 

  34. Kömle, N.I., Steiner, G., Dankert, C., Dettleff, G., Hellmann, H., Kochan, H., Baguhl, M., Kohl, H., Kölzer, G., Thiel, K., and Öhler, A., Ice sublimation below artificial crusts: Results from comet simulation experiments, Planet. Space Sci., 1991, vol. 39, pp. 515–524.

    Article  ADS  Google Scholar 

  35. Kossacki, K.J., Spohn, T., Hagermann, A., Kaufmann, E., and Kührt, E., Comet 67P/Churyumov-Gerasimenko: Hardening of the sub-surface layer, Icarus, 2015, vol. 260, pp. 464–474.

    Article  ADS  Google Scholar 

  36. Langevin, Y., Hilchenbach, M., Ligier, N., Merouane, S., Hornung, K., Engrand, C., Schulz, R., Kissel, J., Rynö, J., and Eng, P., Typology of dust particles collected by the COSIMA mass spectrometer in the inner coma of 67P/Churyumov-Gerasimenko, Icarus, 2016, vol. 271, pp. 76–97.

    Article  ADS  Google Scholar 

  37. Macher, W., Kömle, N.I., Skorov, Y.V., Rezac, L., Kargl, G., and Tiefenbacher, P., 3D thermal modeling of two selected regions on comet 67P and comparison with Rosetta/MIRO measurements, Astron. Astrophys., 2019, vol. 630, id. A12.

  38. Markin, V., Capillary equilibrium in porous media. Communication 3. Characteristics of porous media, Izv. Akad. Nauk SSSR, 1965, vol. 11, pp. 1967–1975.

    Google Scholar 

  39. Markkanen, J. and Agarwal, J., Scattering, absorption, and thermal emission by large cometary dust particles: Synoptic numerical solution, Astron. Astrophys., 2019, vol. 631, id. A164.

  40. Marschall, R., Rezac, L., Kappel, D., Su, C.C., Gerig, S.-B., Rubin, M., Pinzón-Rodríguez, O., and 25 colleagues, A comparison of multiple Rosetta data sets and 3D model calculations of 67P/Churyumov-Gerasimenko coma around equinox (May 2015), Icarus, 2019, vol. 328, p. 104–126.

    Article  ADS  Google Scholar 

  41. Mekler, Yu., Prialnik, D., and Podolak, M., Evaporation from a porous cometary nucleus, Astrophys. J., 1990, vol. 356, pp. 682–686.

    Article  ADS  Google Scholar 

  42. Mottola, S., Attree, N., Jorda, L., Keller, H.U., Kokotanekova, R., Marshall, D., and Skorov, Y., Nongravitational effects of cometary activity, Space Sci. Rev., 2020, vol. 216, id. 2.

  43. Pätzold, M., Andert, T., Hahn, M., Asmar, S.W., Barriot, J.-P., Bird, M.K., Häusler, B., Peter, K., Tellmann, S., Grun, E., Weissman, P.R., Sierks, H., Jorda, L., Gaskell, R., Preusker, F., and Scholten, F., A homogeneous nucleus for comet 67P/Churyumov-Gerasimenko from its gravity field, Nature, 2016, vol. 530, no. 7588, pp. 63–65.

    Article  ADS  Google Scholar 

  44. Pätzold, M., Andert, T.P., Hahn, M., Barriot, J.-P., Asmar, S.W., Häusler, B., Bird, M.K., et al., The nucleus of comet 67P/Churyumov-Gerasimenko – Part I: The global view—nucleus mass, mass-loss, porosity, and implications, Mon. Not. R. Astron. Soc., 2019, vol. 483, pp. 2337–2346.

    Article  ADS  Google Scholar 

  45. Rotundi, A., Sierks, H., Della Corte, V., Fulle, M., Gutierrez, P.J., Lara, L., Barbieri, C., Lamy, P.L., Rodrigo, R., Koschny, D., Rickman, H., Keller, H.U., López-Moreno, J.J., Accolla, M., Agarwal, J., A’Hearn, M.F., Altobelli, N., Angrilli, F., Barucci, M.A., Bertaux, J.L., Bertini, I., Bodewits, D., Bussoletti, E., Colangeli, L., Cosi, M., Cremonese, G., Crifo, J.-F., Da Deppo, V., Davidsson, B., Debei, S., De Cecco, M., Esposito, F., Ferrari, M., Fornasier, S., Giovane, F., Gustafson, B., Green, S.F., Groussin, O., Grün, E., Güttler, C., Herranz, M.L., Hviid Stubbe, F., Ip, W., Ivanovski, S., Jerónimo, J.M., Jorda, L., Knollenberg, J., Kramm, R., Kührt, E., Küppers, M., Lazzarin, M., Leese, M.R., López-Jiménez, A.C., Lucarelli, F., Lowry, S.C., Marzari, F., Epifani, E.M., McDonnell, J.A.M., Mennella, V., Michalik, H., Molina, A., Morales, R., Moreno, F., Mottola, S., Naletto, G., Oklay, N., Ortiz, J.L., Palomba, E., Palumbo, P., Perrin, J.-M., Rodriguez, J., Sabau, L., Snodgrass, C., Sordini, R., Thomas, N., Tubiana, C., Vincent, J.-B., Weissman, P., Wenzel, K.-P., Zakharov, V., and Zarnecki, J.C., Dust measurements in the coma of comet 67P/Churyumov-Gerasimenko inbound to the Sun, Science, 2015, vol. 347, no. 6220, id. aaa3905. https://doi.org/10.1126/science.aaa3905

  46. Sagdeev, R.Z., Blamont, J., Galeev, A.A., Moroz, V.I., Shapiro, V.D., Shevchenko, V.I., and Szegő, K., Vega spacecraft encounters with comet Halley, Nature, 1986, vol. 321, pp. 259–262.

    Article  ADS  Google Scholar 

  47. Schloerb, F.P., Keihm, S., von Allmen, P., Choukroun, M., Lellouch, E., Leyrat, C., Beaudin, G., et al., MIRO observations of subsurface temperatures of the nucleus of 67P/Churyumov-Gerasimenko, Astron. Astrophys., 2015, vol. 583, id. A29.

  48. Schulz, R., Hilchenbach, M., Langevin, Y., Kissel, J., Silen, J., Briois, C., Engrand, C., Hornung, K., Baklouti, D., Bardyn, A., Cottin, H., Fischer, H., Fray, N., Godard, M., Lehto, H., Le Roy, L., Merouane S., Orthous-Daunay, F.-R., Paquette, J., Rynö, J., Siljeström, S., Stenzel, O., Thirkell, L., Varmuza, K., and Zaprudin, B., Comet 67P/Churyumov-Gerasimenko sheds dust coat accumulated over the past four years, Nature, 2015, vol. 518, pp. 216–218.

    Article  ADS  Google Scholar 

  49. Skorov, Y.V. and Rickman, H., A kinetic model of gas flow in a porous cometary mantle, Planet. Space Sci., 1995, vol. 43, pp. 1587–1594.

    Article  ADS  Google Scholar 

  50. Skorov, Y.V., Kömle, N.I., Markiewicz, W.J., and Keller, H.U., Mass and energy balance in the near-surface layers of a cometary nucleus, Icarus, 1999, vol. 140, pp. 173–188.

    Article  ADS  Google Scholar 

  51. Skorov, Y.V., Markiewicz, W.J., Basilevsky, A.T., and Keller, H.U., Stability of water ice under a porous nonvolatile layer: Implications to the south polar layered deposits of Mars, Planet. Space Sci., 2001, vol. 49, pp. 59–63.

    Article  ADS  Google Scholar 

  52. Skorov, Y.V., van Lieshout, R., Blum, J., and Keller, H.U., Activity of comets: Gas transport in the near-surface porous layers of a cometary nucleus, Icarus, 2011, vol. 212, pp. 867–876.

    Article  ADS  Google Scholar 

  53. Skorov, Y.V., Rezac, L., Hartogh, P., and Keller, H.U., Is near-surface ice the driver of dust activity on 67P/Churyumov-Gerasimenko, Astron. Astrophys., 2017, vol. 600, id. A142.

  54. Spohn, T., Knollenberg, J., Ball, A.J., Banaszkiewicz, M., Benkhoff, J., Grott, M., Grygorczuk, J., Hüttig, C., Hagermann, A., Kargl, G., Kaufmann, E., Kömle, N., Kührt, E., Kossacki, K.J., Marczewski, W., Pelivan, I., Schrödter, R., and Seiferlin, K., Thermal and mechanical properties of the near-surface layers of comet 67P/Churyumov–Gerasimenko, Science, 2015, vol. 349, no. 6247, id. aaa0464. https://doi.org/10.1126/science.aab0464

  55. Steiner, G., Two considerations concerning the free molecular flow of gases in porous ices, Astron. Astrophys., 1990, vol. 240, pp. 533–536.

    ADS  Google Scholar 

  56. Vincent, J.-B., Farnham, T., Kührt, E., Skorov, Y., Marschall, R., Oklay, N., El-Maarry, M.R., et al., Local manifestations of cometary activity, Space Sci. Rev., 2019, vol. 215, art. no. 30.

    Article  ADS  Google Scholar 

Download references

Funding

V. Reshetnyk was supported by a grant from the Ministry of Education and Science of Ukraine 20BF051-02 [0120U102178] “Wave processes and effects in active resonant layered plasma media and metamaterials.” L. Rezac was supported by project DFG-392267849.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yu. Skorov.

Additional information

Translated by M. Chubarova

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Reshetnyk, V., Skorov, Y., Vasyuta, M. et al. Transport Characteristics of the Near-Surface Layer of the Nucleus of Comet 67P/Churyumov–Gerasimenko. Sol Syst Res 55, 106–123 (2021). https://doi.org/10.1134/S0038094621020040

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0038094621020040

Keywords:

Navigation