Solar Physics

, Volume 291, Issue 12, pp 3659–3683 | Cite as

Improved Determination of the Location of the Temperature Maximum in the Corona

  • J. F. LemaireEmail author
  • K. Stegen


The most used method to calculate the coronal electron temperature [\(T_{\mathrm{e}} (r)\)] from a coronal density distribution [\(n_{\mathrm{e}} (r)\)] is the scale-height method (SHM). We introduce a novel method that is a generalization of a method introduced by Alfvén (Ark. Mat. Astron. Fys. 27, 1, 1941) to calculate \(T_{\mathrm{e}}(r)\) for a corona in hydrostatic equilibrium: the “HST” method. All of the methods discussed here require given electron-density distributions [\(n_{\mathrm{e}} (r)\)] which can be derived from white-light (WL) eclipse observations. The new “DYN” method determines the unique solution of \(T_{\mathrm{e}}(r)\) for which \(T_{\mathrm{e}}(r \rightarrow \infty) \rightarrow 0\) when the solar corona expands radially as realized in hydrodynamical solar-wind models. The applications of the SHM method and DYN method give comparable distributions for \(T_{\mathrm{e}}(r)\). Both have a maximum [\(T_{\max}\)] whose value ranges between 1 – 3 MK. However, the peak of temperature is located at a different altitude in both cases. Close to the Sun where the expansion velocity is subsonic (\(r < 1.3\,\mathrm{R}_{\odot}\)) the DYN method gives the same results as the HST method. The effects of the other free parameters on the DYN temperature distribution are presented in the last part of this study. Our DYN method is a new tool to evaluate the range of altitudes where the heating rate is maximum in the solar corona when the electron-density distribution is obtained from WL coronal observations.


Solar corona Coronal electron temperature Solar wind 



The authors wish to acknowledge Clément Botquin for his efficient assistance in the early work of programming. The open-source computer codes to compute and plot DYN temperature distributions are in FORTRAN and IDL. Copies are available through We wish to thank the Royal Belgian Institute for Space Aeronomy (BIRA-IASB), the Royal Observatory of Belgium (ROB), and BELSPO (Federal Public Planning Service Science Policy) for their support. We acknowledge discussions with Viviane Pierrard (BIRA-IASB), Marius Echim (BIRA-IASB), Hervé Lamy (BIRA-IASB), and Yuriy Voitenko (BIRA-IASB). We are grateful to Jack Scudder (University of Iowa) for his constructive remarks and suggestions concerning the first version of this work, which had been posted 16 December 2011 at arXiv . He encouraged us to submit this work to a refereed international Journal. We appreciate his interest in our work.

Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.


  1. Alfvén, H.: 1941, On the solar corona. Ark. Mat. Astron. Fys. 27, 1. zbMATHGoogle Scholar
  2. Allen, C.W.: 1947, Interpretation of electron densities from coronal brightness. Mon. Not. Roy. Astron. Soc. 107, 426. ADSCrossRefGoogle Scholar
  3. Aschwanden, M.J.: 2009, Physics of the Solar Corona, Springer, New York. ISBN 3-540-30765-6. Google Scholar
  4. Baumbach, S.: 1937, Strahlung, Ergiebigkeit und Electronendichte der Sonnenkorona. Astron. Nachr. 263, 121. ADSCrossRefGoogle Scholar
  5. Brandt, J.C.: 1970, Introduction to the Solar Wind, Freeman, San Francisco. Google Scholar
  6. Brandt, J.C., Michie, R.W., Cassinelli, J.P.: 1965, Interplanetary gas, X: coronal temperatures, energy deposition, and the solar wind. Icarus 4, 19. ADSCrossRefGoogle Scholar
  7. Chapman, S.: 1957, Notes on the solar corona and the Terrestrial Ionosphere. Smithson. Contrib. Astrophys. 2, 1. ADSCrossRefGoogle Scholar
  8. Cranmer, S.R., Kohl, J.L., Noci, G., Antonnucci, E., Tondello, G., Huber, M.C.E., et al.: 1999, An empirical model of a polar Coronal Hole at solar minimum. Astrophys. J. 511, 481. ADSCrossRefGoogle Scholar
  9. de Jager, C.: 1979, Structure and dynamics of the solar atmosphere. Handb. Phys. 52, 80. ADSGoogle Scholar
  10. de Patoul, J., Foulon, C., Riley, P.: 2015, 3D electron density distributions in the solar corona minima: assessment for more realistic solar wind modeling. Astrophys. J. 814, 68. ADSCrossRefGoogle Scholar
  11. Deforest, C.E., Lamy, P.L., Llebaria, A.: 2001, Solar plumes lifetime and Coronal Hole expansion: determination from long-term observations. Astrophys. J. 560, 490. ADSCrossRefGoogle Scholar
  12. Doyle, J.G., Teriaca, L., Banerjee, D.: 1999, Coronal Hole diagnostics out to 8 RS. Astron. Astrophys. 349, 956. ADSGoogle Scholar
  13. Ebert, R.W., McComas, D.J., Elliott, H.A., Forsyth, R.J., Gosling, J.T.: 2009, Bulk properties of the slow and fast solar wind and interplanetary coronal mass ejections measured by Ulysses: three polar orbits of observations. J. Geophys. Res. 114, A01109.  DOI. ADSCrossRefGoogle Scholar
  14. Echim, M.M., Lemaire, J., Lie-Svendsen, O.: 2011, A review of solar wind modeling: kinetic and fluid aspects. Surv. Geophys. 12, 53. arXiv.  DOI. Google Scholar
  15. Edlén, B.: 1942, An attempt to identify the emission lines in the spectrum of the solar corona. Ark. Mat. Astron. Fys. 28B(1), 1. Google Scholar
  16. Fisher, R., Guhathakurta, M.: 1995, Physical properties of polar coronal rays and holes as observed with the Spartan 201-01 coronograph. Astrophys. J. Lett. 477, L139. ADSGoogle Scholar
  17. Gibson, S.E., Fludra, A., Bagenal, F., Biesecker, D., Del Zanna Bromage, B.: 1999, Solar minimum streamer densities and temperatures vising Whole Sun Month coordinated data sets. J. Geophys. Res. 104, 9691. ADSCrossRefGoogle Scholar
  18. Habbal, S.R., Esser, R., Arndt, M.B.: 1993, How reliable are Coronal Hole temperatures deduced from observatons? Astrophys. J. 413, 435. ADSCrossRefGoogle Scholar
  19. Kayshap, P., Banerjee, D., Srivastava, A.K.: 2015, Diagnostics of a Coronal Hole and the adjacent quiet Sun by the hinode/EUV Imaging Spectrometer (EIS). Solar Phys. 290, 2889.  DOI. ADSCrossRefGoogle Scholar
  20. Kohl, J.L., Noci, G., Cranmer, S.R., Raymond, J.C.: 2006, Ultraviolet spectroscopy of the extended solar corona. Astron. Astrophys. Rev. 13(1:2), 31.  DOI. ADSCrossRefGoogle Scholar
  21. Kopp, R.A., Holzer, T.E.: 1976, Dynamics of Coronal Hole regions, I steady polytropic flows with multiple critical points. Solar Phys. 49, 43.  DOI. ADSCrossRefGoogle Scholar
  22. Koutchmy, S.: 1977, Study of the June 30, 1973 trans-polar Coronal Hole. Solar Phys. 51, 399. ADSCrossRefGoogle Scholar
  23. Krieger, A.S., Timothy, A.F., Roelof, E.C.: 1973, A Coronal Hole and its identification as the source of a high velocity solar wind Stream. Solar Phys. 29, 505.  DOI. ADSCrossRefGoogle Scholar
  24. Lemaire, J.: 2010, Convective instability of the solar corona: why the solar wind blows. In: Makismovic, M., Issautier, K., Meyer-Vernet, N., Moncuquet, M., Pantellini, P. (eds.) CP1216, Twelfth Inter. Solar Wind Conference, AIP, St Malo, 20.  DOI. Google Scholar
  25. Lemaire, J., Scherer, M.: 1969, Le champ électrique de polarisation dans l’exosphère ionique polaire. C. R. Math. 269B, 666. Google Scholar
  26. Lemaire, J., Scherer, M.: 1970, Model of the polar ion exosphere. Planet. Space Sci. 18, 103. ADSCrossRefGoogle Scholar
  27. Lemaire, J., Scherer, M.: 1973, Kinetic model of the solar and polar winds. Rev. Geophys. Space Phys. 11, 427. ADSCrossRefGoogle Scholar
  28. Levine, R.H., Altschuler, M.D., Harvey, J.W.: 1977, Solar sources of the interplanetary magnetic field and solar wind. J. Geophys. Res. 82, 106. ADSCrossRefGoogle Scholar
  29. Meyer-Vernet, N.: 2007, Basics of the Solar Wind, Cambridge University Press, Cambridge. ISBN-13: 978-0-521-81420-1. CrossRefGoogle Scholar
  30. Minnaert, M.: 1930, On the continuous spectrum of the corona and its polarisation. Zeits. Astrophyz. 1, 209. ADSzbMATHGoogle Scholar
  31. Munro, R.H., Jackson, B.V.: 1977, Physical properties of a polar Coronal Hole from 2 to \(5\, \mathrm{R}_{\mathrm{S}}\). Astrophys. J. 213, 874. ADSCrossRefGoogle Scholar
  32. Parenti, S., Bromage, B.J.I., Poletto, G., Noci, G., Raymond, J.C., Bromage, G.E.: 2000, Characteristics of solar coronal streamers: element abundance, temperature and density from coordinated CDS and UVCS SOHO observations. Astron. Astrophys. 363, 800. ADSGoogle Scholar
  33. Parker, E.N.: 1958, Dynamics of the interplanetary gas and magnetic fields. Astrophys. J. 128, 664. ADSCrossRefGoogle Scholar
  34. Parker, E.N.: 2010, Kinetic and hydrodynamic representations of coronal expansion and the solar wind. In: Makismovic, M., Issautier, K., Meyer-Vernet, N., Moncuquet, M., Pantellini, P. (eds.) CP1276, Twelfth Intern. Solar Wind Conference, AIP, St Malo, 3.  DOI. Google Scholar
  35. Pierrard, V., Borremans, K., Stegen, K., Lemaire, J.F.: 2014, Coronal temperature profiles obtained from kinetic models, and from brightness measurements during solar eclipses. Solar Phys. 298, 183. arXiv.  DOI. ADSCrossRefGoogle Scholar
  36. Pottasch, S.R.: 1960, Use of the equation of hydrostatic equilibrium in determining the temperature distribution in the outer solar atmosphere. Astrophys. J. 131, 68. ADSCrossRefGoogle Scholar
  37. Saito, K.: 1970, A non-spherical axisymmetric model of the solar corona of minimum type. Ann. Tokyo Astron. Obs. 12, 53. ADSGoogle Scholar
  38. Schuster, M.: 1880, On polarization of the solar corona. Mon. Not. Roy. Astron. Soc. 60, 35. Google Scholar
  39. Scudder, J.D.: 1992, Why all stars possess circumstellar temperature inversions. Astrophys. J. 398, 319. ADSCrossRefGoogle Scholar
  40. Sitter, E.C., Guhathakurta, M.: 1999, Semi-empirical two-dimensional magneto hydrodynamic model of the solar corona and interplanetary medium. Astrophys. J. 523, 812. ADSCrossRefGoogle Scholar
  41. van de Hulst, H.C.: 1950, The electron density of the solar corona. Bull. Astron. Inst. Neth. 11, 135. ADSGoogle Scholar
  42. van de Hulst, H.C.: 1953, The chromosphere and the corona. In: Kuiper, G. (ed.) The Sun, University of Chicago Press, Chicago, 207. Google Scholar
  43. Wang, Y.-M.: 1995, Empirical relationship between the magnetic field and mass and energy flux in the source region of the solar wind. J. Geophys. Res. 105, 25133. ADSCrossRefGoogle Scholar
  44. Wang, Y.M., Jr. Sheeley, N.R.: 1990, Solar wind speed and coronal flux-tube expansion. Astrophys. J. 355, 726. ADSCrossRefGoogle Scholar
  45. Wang, Y.M., Jr. Sheeley, N.R.: 2006, Sources of the solar wind at Ulysses during 1990 – 2006. Astrophys. J. 653, 708. ADSCrossRefGoogle Scholar
  46. Wilhelm, K., Marsch, E., Dwivedi, B.N., Hassler, D.M., Lemaire, P., Gabriel, A.H., Huber, M.C.E.: 1998, The solar corona above polar Coronal Holes as seen by SUMER on SOHO. Astrophys. J. 500, 1023. ADSCrossRefGoogle Scholar
  47. Wilhelm, K., Abbo, L., Auchère, F., Barbey, N., Feng, L., Gabriel, A.H., Giordano, S., Imada, S., Liebbaria, A., Matthaeus, W.H., Poletto, G., Raouafi, N.-E., Suess, S.T., Teriaca, L., Wang, Y.-M.: 2011, Morphology, dynamics and plasma parameters of plumes and inter-plume regions in solar Coronal Holes. Astron. Astrophys. Rev. 19, 35.  DOI ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Royal Belgian Institute for Space AeronomyBrusselsBelgium
  2. 2.Royal Observatory of BelgiumBrusselsBelgium

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