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Phase Space Analysis: The Equilibrium of the Solar Magnetic Cycle

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We present the results of a statistical study of the solar cycle based on the analysis of the superficial toroidal magnetic field component phase space. The magnetic field component used to create the embedded phase space was constructed from monthly sunspot number observations since 1750. The phase space was split into 32 sections (or time instants) and the average values of the orbits on this phase space were calculated (giving the most probable cycle). In this phase space it is shown that the magnetic field on the Sun’s surface evolves through a set of orbits that go around a mean orbit (i.e., the most probable magnetic cycle that we interpret as the equilibrium solution). It follows that the most probable cycle is well represented by a van der Pol oscillator limit curve (equilibrium solution), as can be derived from mean-field dynamo theory. This analysis also retrieves the empirical Gnevyshev – Ohl’s rule between the first and second parts of the solar magnetic cycle. The sunspot number evolution corresponding to the most probable cycle (in phase space) is presented.

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  1. Berhanu, M., Monchaux, R., Fauve, S., Mordant, N., Pétrélis, F., Chiffaudel, A., Daviaud, F., Dubrulle, B., Marié, L., Ravelet, F., Bourgoin, M., Odier, Ph., Pinton, J.-F., Volk, R.: 2007, Eur. Phys. Lett. 77, 59001.

  2. Bracewell, R.N.: 1953, Nature 133, 512.

  3. Chaplin, W.J., Elsworth, Y., Miller, B.A., Verner, G.A., New, R.: 2007, Astrophys. J. 659, 1749.

  4. Dubrulle, B., Blaineau, P., Mafra Lopes, O., Daviaud, F., Laval, J.-P., Dolganov, R.: 2007, New J. Phys. 9, 308.

  5. Houdek, G., Chaplin, W.J., Appourchaux, T., Christensen-Dalsgaard, J., Däppen, W., Elsworth, Y., Gough, D.O., Isaak, G.R., New, R., Rabello-Soares, M.C.: 2001, Mon. Not. Roy. Astron. Soc. 327, 483.

  6. Libbrecht, K.G., Woodard, M.F.: 1990, Nature 345, 779.

  7. Marsh, N., Svensmark, H.: 2003, Space Sci. Rev. 107, 317.

  8. Mininni, P.D., Gomez, D.O., Mindlin, G.B.: 2001, Solar Phys. 201, 203.

  9. Parker, E.N.: 1955, Astrophys. J. 121, 491.

  10. Polygiannakis, J.M., Moussas, X.S.: 1996, Solar Phys. 163, 193.

  11. Pontieri, A., Lepreti, F., Sorriso-Valvo, L., Vecchio, A., Carbone, V.: 2003, Solar Phys. 213, 195.

  12. Priest, E.R.: 1984, Solar Magnetohydrodynamics, D. Reidel, Dordrecht.

  13. Reid, G.C.: 2000, Adv. Space Res. 94(1/2), 1.

  14. Rogers, L.M., Richards, M.T., Richards, D.St.P.: 2006, astro-ph/0606426.

  15. Schatten, K.H.: 2003, Adv. Space Res. 32(4), 451.

  16. Solanki, S.K.: 2003, Astron. Astrophys. Rev. 11, 153.

  17. Solanki, S.K., Krivova, N.A.: 2005, Solar Phys. 224, 197.

  18. Usoskin, I.G., Mursula, K.: 2003, Solar Phys. 218, 319.

  19. Walther, G.: 1999, Astrophys. J. 513, 990.

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Correspondence to I. Lopes.

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Passos, D., Lopes, I. Phase Space Analysis: The Equilibrium of the Solar Magnetic Cycle. Sol Phys 250, 403–410 (2008).

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  • Solar cycle: models
  • Sunspots: magnetic fields
  • Sunspots: statistics