Square Roots of Kepler Ellipses, Electrons and Rydberg Atoms in a Magnetic Field


  • Yu. P. Stepanovsky National Science Center “Kharkiv Institute of Physics and Technology”




planets, Kepler problem, Rydberg atoms, magnetic field


Young Kepler’s daring ideas on the structure of the Solar system are applied to the analysis of planetary distances in the exoplanetary system HD 10180. Using Zhukovsky’s transformation, the essence of the spinor regularization of Kepler’s problem is explained as extracting the square root of an ellipse and using a Kepler eccentric anomaly instead of the usual time. The achievements of Kharkiv radio astronomers in the search for radio recombination lines of Rydberg carbon atoms at the UTR-2 radio telescope are considered. A generalized spinor regularization of the Kepler problem is used to analyze the energy spectra of Rydberg hydrogen atoms in a magnetic field.


G. Chr. Lichtenberg. Vermischte Schriften, Band 2 (Dieterich, 1801).

P. Nowak. Hodowanie Troglodyt'ow (Kronos, 2014).

V.I. Arnold. Do You Need Mathematics at School? (MCNMO, 2004) (in Russian).

N.M. Karamzin History of the Russian State, Vol. 1 (Terra, 1998) (in Russian).

Galileo Galilei. Opere, Volume II (Nicolo Bettoni e Comp, 1832).

Galileo Galilei. Il Saggiatore (Roma, 1623).

Yu.A. Bely. Tycho Brahe (Nauka, 1982) (in Russian).

Yu.A. Bely. Johannes Kepler (Nauka, 1971) (in Russian).

V.E. Thoren, J.R. Christianson. The Lord of Uraniborg: A Biography of Tycho Brahe (Cambridge Univ. Press, 1990) [ISBN: 978-0-521-35158-4]. https://doi.org/10.1017/CBO9780511665417

M. Caspar. Kepler (Dover, 1993) [ISBN: 0-486-67605-6].

V. Steklov. Galilei (GIZ, 1923) (in Russian).

A. Koestler. The Sleepwalkers: A History of Man's Changing Vision of the Universe (Hutchinson, 1959) [ISBN: 0-14-019246-8].

J. Keplerus. Mysterivm Cosmographicvm (Gruppenbachius, 1596).

J. Keplerus Astronomia Nova AITIOΛOΓITOΣ (V¨ogelin, 1609). https://doi.org/10.5479/sil.126675.39088002685477

J. Keplerus. Ioannis Keppleri Harmonices Mvndi (Lincii, 1619) https://doi.org/10.5479/sil.135810.39088002800316

F. Arago. Biographies of Famous Astronomers, Physicists, and Geometers. Vol. 1 (RCD, 2000) (in Russian).

A. Einstein. Mein Weltbild (Ullstein Taschenbuch, 2005) [ISBN: 978-3-548-36728-6].

W. Pauli, C. Jung. Atom and Archetype: The Pauli/Jung Letters 1932-1958 (Princeton Univ. Press, 2001) [ISBN: 9780415120784].

A. Pais. The Genius of Science: A Portrait Gallery (Oxford Univ. Press, 2000) [ISBN: 9780198506140]. https://doi.org/10.1063/1.1325238

W. Pauli. The influence of archetypal ideas on the scientific theories of Kepler. In: W. Pauli. Writings On Physics And Philosophy (Springer, 1994) [ISBN: 978-3-662-02994-7]. https://doi.org/10.1007/978-3-662-02994-7_23

M.M. Nieto. The Titius-Bode Law of Planetary Distances: Its History and Theory (Pergamon Press, 1972) [ISBN: 978-1483126944].

M.A. Blagg. On a suggested substitute for Bode's Law. Mon. Not. R. Astron. Soc. 73, 414 (1913). https://doi.org/10.1093/mnras/73.6.414

V.I. Arnold. Huygens and Barrow, Newton and Hooke (Birkh¨auser, 2012) [ISBN: 978-3-0348-9129-5].

I. Newton. Philosophiae Naturalis Principia Mathematica (Londini, 1687). https://doi.org/10.5479/sil.52126.39088015628399


P. Ackroyd. Newton (Nan A. Talese, 2008) [ISBN: 9780385507998].

N.A. Beliaev, K.I. Churiumov. Halley's Comet and Its Observation (Nauka, 1985) (in Russian).

T. Levi-Civita. Trajettorie singolari ed urti nel problema ristretto dei tre carpi. Ann. Math. 9, 1 (1903). https://doi.org/10.1007/BF02419867

P. Kustaanheimo. Spinor regularization of Kepler motion. Ann. Univ. Turkuensis A. 73, 3 (1964).

N. Joukowsky. ¨ Uber die Konturen der Tragfl¨achen der Drachenflieger. Z. Flugtechnik 1, 281 (1910), 3, 81 (1912).

E. Stiefel, G. Scheifele. Linear and Regular Celestial Mechanics (Springer, 1971) [ISBN: 978-3-642-65029-1]. https://doi.org/10.1007/978-3-642-65027-7

A.S. Bakai, Yu.P. Stepanovsky. Adiabatic Invariants (Naukova Dumka, 1981) (in Russian).

I.S. Shklovsky. Mind, Life, Universe (Janus-K, 1996) (in Russian).

G.E. Gorelik. The Soviet Life of Lev Landau through the Eyes of Eyewitnesses (Vagrius, 2009) [ISBN: 978-5-98264-035-2] (in Russian).

L.D. Landau, E.M. Lifshits. Theoretical physics. Quantum mechanics. Part 1. Nonrelativistic theory (GITTL, 1948) (in Russian).

E. Fermi. ¨ Uber die magnetischen momente der atomkerne. Zeits. f¨ur Physik 60, 320 (1930). https://doi.org/10.1007/BF01339933

E. Fermi. Magnetic moments of atomic nuclei. Nature 125, 16 (1930). https://doi.org/10.1038/125016a0

C.G. Darwin. The wave equations of the electron. Proc. Roy. Soc. Lond. A118, 654 (1928). https://doi.org/10.1098/rspa.1928.0076

E. Fermi, E. Segr'e. Zur theorie der hyperfeinstrukturen. Zeits. f¨ur Physik 82, 729 (1933). https://doi.org/10.1007/BF01334120

I. Pomeranchuk, Y. Smorodinsky. On the energy levels of systems with Z > 137. J. Phys. USSR 9, 97 (1945).

L. Fermi. Atomi in famiglia (Mondadori, 1954).

J.E. Nafe, E.B. Nelson, I.I. Rabi. The hyperfine structure of atomic hydrogen and deuterium. Phys. Rev. 71, 914 (1947). https://doi.org/10.1103/PhysRev.71.914

H.M. Foley, P. Kusch. On the intrinsic moment of the electron. Phys. Rev. 73, 412 (1948). https://doi.org/10.1103/PhysRev.73.412

J. Schwinger. On quantum electrodynamics and the magnetic moment of the electron. Phys. Rev. 73, 416 (1948). https://doi.org/10.1103/PhysRev.73.416

D. Hanneke, S. Fogwell, G. Gabrielse. New measurement of the electron magnetic moment and the fine structure constant. Phys. Rev. Lett. 100, 120801 (2008). https://doi.org/10.1103/PhysRevLett.100.120801

T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio. Revised value of the eighth-order electron g − 2. Phys. Rev. Lett. 99, 110406 (2007). https://doi.org/10.1103/PhysRevLett.99.110406


S.G. Karshenboim. Precision physics of simple Atoms: QED tests, nuclear structure and fundamental constants. Phys. Rept. 422, 1 (2005). https://doi.org/10.1016/j.physrep.2005.08.008

E.U. Condon, G.H. Shortley The Theory of Atomic Spectra (Cambridge Univ. Press, 1935).

I.B. Khriplovich, A.I. Milstein, S.S. Petrosyan. Nuclear structure corrections to deuterium hyperfine splitting. Phys. Lett. B 366, 13 (1996). https://doi.org/10.1016/0370-2693(95)01354-7

I.S. Shklovsky. Cosmic Radio Waves (Harvard Univ. Press, 1960). https://doi.org/10.1063/1.3056747

W.W. Holloway, Jr. and R. Novick. Determination of the hyperfine structure of atomic nitrogen by optical rientation. Phys. Rev. Lett. 1, 367 (1958). https://doi.org/10.1103/PhysRevLett.1.367

J.D. Kraus. Radio Astronomy (Cygnus-Quasar Books, 2005).

A.A. Konovalenko, L.G. Sodin. Neutral 14N in the interstellar medium. Nature 283, 360 (1980). https://doi.org/10.1038/283360a0

D.H. Blake, R.M. Crutcher, W.D.Watson. Identification of the anomalous 26.131-MHz nitrogen line observed towards Cas A. Nature 287, 707 (1980). https://doi.org/10.1038/287707a0

A.A. Konovalenko, L.G. Sodin. The 26.13 MHz absorption line in the direction of Cassiopeia A. Nature 294, 135 (1981). https://doi.org/10.1038/294135a0

V.M. Kontorovich, S.Ya. Braude. Radio Waves Tell about the Universe (Fizmatlit, 2011) (in Russian).

B.P. Ryabov. Jovian decametric emission. Multiscale dynamic spectra. Radio Phys. Radio Astron. 6 (1), 103 (2001).

Academician S. Ya. Braude in the memoirs of contemporaries (Radio Astronomical Institute of the NAS of Ukraine, 2005) (in Russian).

A.A. Konovalenko. I.S. Shklovsky and low-frequency radio astronomy. Radio Phys. Radio Astron. 22 (1), 7 (2017). https://doi.org/10.15407/rpra22.01.007

A. Konovalenko, L. Sodin, V. Zakharenko, et al. The modern radio astronomy network in Ukraine: UTR-2, URAN and GURT. Exper. Astron. 42 11 (2016).

M.A. Gordon, R.L. Sorochenko. Radio Recombination Lines (Springer, 2009). https://doi.org/10.1007/978-0-387-09691-9

P.S. Ehrenfest - A.F. Ioffe: Scientific Correspondence (1907-1933) (Nauka, 1990) (in Russian).

L.D. Landau, E.M. Lifshitz. Mechanics (Nauka, 1988) [ISBN: 5-02-013850-9] (in Russian).

L.D. Landau, E.M. Lifshits. Field Theory (Nauka, 1988) [ISBN: 5-02-014420-7] (in Russian).

V. Fock. Bemerkung zur quantelung des harmonischen oszillators im magnetfeld. Zeits. f¨ur Physik 47, 446 (1928). https://doi.org/10.1007/BF01390750

I.I. Rabi. Das freie Elektron im homogenen magnetfeld nach der diracschen theorie. Zeits. f¨ur Physik 49, 507 (1928). https://doi.org/10.1007/BF01333634

W. Wilson. The quantum theory and electromagnetic phenomena. Proc. Roy. Soc. London A 102, 478 (1923). https://doi.org/10.1098/rspa.1923.0007

C.G. Darwin. The diamagnetism of the Free electron. Math. Proc. Cambr. Phil. Soc. 27, 86 (1931). https://doi.org/10.1017/S0305004100009373

L. Landau. Diamagnetismus der metalle. Zeits. f¨ur Physik 64, 629 (1930). https://doi.org/10.1007/BF01397213

W. Pauli. ¨ Uber gasentartung und paramagnetismus, Zeits. f¨ur Physik 41, 81 (1927). https://doi.org/10.1007/BF01391920

H.-J. van Leeuwen. Probl'emes de la th'eorie 'electronique du magn'etisme. J. Phys. Radium 2 (12), 361 (1921). https://doi.org/10.1051/jphysrad:01921002012036100

N. Bohr. Collected Works, Vol. 1, Early Works (1905-1911) (Elsevier, 1972).

Baptiste Savoie. A rigorous proof of the Bohr-van Leeuwen theorem in the semiclassical limit. Rev. Math. Phys. 27, 1550019 (2015). https://doi.org/10.1142/S0129055X15500191

L. Kouwenhoven, C. Marcus. Quantum dots. Physics World 11(6), 35 (1998). https://doi.org/10.1088/2058-7058/11/6/26

D. Bimberg, M. Grundmann, N.N. Ledentsov. Quantum Dot Heterostructures (Wiley, 1999) [ISBN: 978-0-471-97388-1].

E. Drigho-Filho, S. Kuru, J. Negro, L.M. Nieto. Superintegrability of the Fock-Darwin system. Ann. Phys. 383, 101 (2017). https://doi.org/10.1016/j.aop.2017.05.003

V.M. Galitsky, B.M. Karnakov, V.I. Kogan. Problems in Quantum Mechanics (Nauka, 1992) (in Russian).

K. von Klitzing, G. Dorda, M. Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494 (1980). https://doi.org/10.1103/PhysRevLett.45.494

Y. Aharonov, A. Casher. Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field. Phys. Rev. A 19, 2461 (1979). https://doi.org/10.1103/PhysRevA.19.2461

Yu.P. Stepanovsky. Fractional quantum Hall effect. Electromagn. Phenom. 1, 427 (1998).

E.M. Purcell. Electricity and Magnetism (Cambridge Univ. Press, 2011) [ISBN: 978-1-107-01360-5].

R.J. Fonck, F.L. Roesler, D.H. Tracy. Comparison of atomic quasi-Landau spectrum with semiclassical strongfield-mixing models. Phys. Rev. A 21, 861 (1980). https://doi.org/10.1103/PhysRevA.21.861

S. Feneuille. Semiempirical scaling laws for adiabatic energy levels of highly excited hydrogen atom in high magnetic fields. Phys. Rev. A 26, 673 (1982). https://doi.org/10.1103/PhysRevA.26.673

F. Cooper, A. Khare, U. Sukhatme. Supersymmetry in Quantum Mechanics (World Scientific, 2001) [ISBN: 978-981-238-650-2]. https://doi.org/10.1142/4687

Yu.P. Stepanovsky. Hydrogen atom in a magnetic field, superWKB quantization and Majorana's equation. In: Problems of Theoretical Physics (Naukova Dumka, 1991) (in Russian).

A. Krot'ko, Yu. Stepanovsky. Highly excited two-dimensional hydrogen atom in magnetic field. Visnyk Lviv Univ. Ser. Fiz. 39, 54 (2006).

Yu. P. Stepanovsky. A hydrogen atom in an external field as an anharmonic oscillator. Ukr. J. Phys. 32, 1316 (1987).

I.E. Ovcharenko, Yu. P. Stepanovsky, On some properties of 2-D Weyl equation for charged massless spin 1/2 particle. Probl. Atom. Sci. Techn. 3 (1), 56 (2007).

Le Syst'eme international d'unit'es (SI)/The International System of Units (SI) (BIPM, 2019).

Yu. P. Stepanovsky. "Nobel" physical phenomena and some concepts of modern physics. In: Fundamental Problems of Precision Theory (Nauka, 2001) [ISBN: 5-02-024947-5].

S.J. Orfanidis. Electromagnetic Waves and Antennas (Rutgers Univ., 2016).

C.J.A.P. Martins. The status of varying constants: a review of the physics, searches and implications. Rep. Prog. Phys. 80, 126902 (2017). https://doi.org/10.1088/1361-6633/aa860e



How to Cite

Stepanovsky, Y. P. (2020). Square Roots of Kepler Ellipses, Electrons and Rydberg Atoms in a Magnetic Field. Ukrainian Journal of Physics, 65(10), 835. https://doi.org/10.15407/ujpe65.10.835



General physics