Coronal Heating Problem

Transcription

1 PHY 690C Project Report Coronal Heating Problem by Mani Chandra, Arnab Dhabal and Raziman T V (Y6233) (Y7081) (Y7355) Mentor: Dr. M.K. Verma 1

2 Contents 1 Introduction 3 2 The Coronal Heating Problem Coronal losses Regions in the corona Review of heating processes Heating by Magnetohydrodynamic Turbulence Is the spectrum Kolmogorov or Kraichnan? Kolmogorov flux Kraichnan flux Comparison Conclusion 9 5 Acknowledgements 9 References 10 2

3 1 Introduction The sun can be divided into six zones[1]. In the order of the distance from the sun, these are the core, the radiative zone, the convective zone, the photosphere, the chromosphere and the corona. The core has a temperature of 13 million K and generates energy by nuclear fusion. Energy is transferred from the core to the photosphere via the intermediate zones, radiative zone and convective zone which are at lower temperatures than the core. The photosphere is the visible region of the sun. This zone from which we receive the visible spectrum directly is at a temperature between 4500K and 6000K. Fig. 1 : Illutration of the structure of the sun Source: Wikimedia Commons Above this is the chromosphere which is about 2000 km thick and can have temperatures upto 20,000K. Corona, the plasma atmosphere of the sun, extends to millions of kilometers. The temperature in the corona ranges from one to three million K. The explanation of the abnormally high temperatures in the corona is the focus of the coronal heating problem. 3

4 2 The Coronal Heating Problem The coronal heating problem tries to study how the corona can be more than two orders of magnitude hotter than the photosphere. Ever since it was first established in 1939 that the corona is hotter than the visible regions of the sun, this has been among the perplexing problems in astrophysics[2]. Seven decades since, a satisfactory resolution to the problem is yet to be achieved. Explanation of the coronal heating problem requires the understanding of two aspects: The source of the energy required for heating and the mechanism of the heating. The source is not so much of a point of contention as the heating mechanism[2]. The turbulent photospheric and subphotospheric motions at coronal loop footpoints can provide the energy required for heating up the corona. The primary difficulty is in identifying the mechanism that can transport this energy into the corona and convert it to heat in such a small length scale. Fig. 2 : Temperature variation of the upper layers of the sun Credit: NASA. Source: Wikimedia commons Over the years, many candidates have been put forward as a possible mechanism for heating. The mechanisms can be broadly classified into AC (wave heating) and DC (impulsive heating) processes depending on the types of electric currents that are pumped at the footpoints of coronal loops in the process[3]. DC currents are those with variation timescales much longer than the Alfvén transit time along the coronal loop. On the other hand the AC currents vary at much shorter timescales. Both types of mechanisms may be operating simultaneously with differing importance in different regions of the corona. 4

5 2.1 Coronal losses The radiative losses of the corona come from three sources[4]: Emission in resonance lines of ionized metals Radiative recombinations due to the most abundant coronal ions Bremstrahlung radiation at high temperatures The total radiation loss per unit volume is given by L = n e n H P (T ) where n e and n H are the number densities of electrons and hydrogen ions in the plasma respectively. Using the value n e = n H = cm 3 and P (T ) = for the temperature range that corona is in, we get a value of W/m 3. This evaluates to about 300W per unit area of the solar surface. 2.2 Regions in the corona Corona can be divided into three regions: active regions, quiet-sun regions and coronal holes. Active regions are ensembles of loop structures connecting points of opposite magnetic polarity in the photosphere. All phenomena directly linked to magnetic fields happen in the active regions. Coronal holes are the regions of the corona which are dark in X-rays and do not emit much of radiation. Fast solar wind leaves the corona through the coronal holes. Aschwanden ([5]) gives the heating requirements for these regions as: Active regions : W/m 2 Quiet-sun regions : W/m 2 Coronal holes : W/m 2 Overall, the active regions make up 82.4%, the quite-sun regions 17.2% and the coronal holes only 0.4% of the total heating requirement. As the Alfvén timescales are short in the active regions, DC processes will be dominant in them. On the other hand, coronal holes have longer Alfvén timescales making AC processes more important. The simple calculation above seems to fall near the boundary of the heating requirement for the active region and the quiet-sun region. 5

6 2.3 Review of heating processes Here we give short descriptions of the proposed heating processes, mainly based on the papers by Narain and Ulmschneider ([6] and [7]) Acoustic wave heating This model attributes heating to the dissipation of acoustic shock waves. Surface convection zones generate a spectrum of acoustic waves. Rapid density decrease in the outer solar atmosphere results in increase of wave amplitude, leading to shocks. Shock dissipation in turn heats the corona. Acoustic energy can also be dissipated in the absence of waves by radiation damping or ionisation pumping. Fast and slow magnetoacoustic body waves Alfvén-mode, slow-mode and fast-mode waves are formed in the bulk of a compressible fluid in a magnetic field. Slow-mode and fast-mode waves are called magnetoacoustic waves as these are compressible modes. Fast mode waves can dissipate energy via Landau damping whereas slow mode waves can dissipate via shocks. Alfvén body waves Incompressible Alfvén-mode waves travel parallel and antiparallel to the magnetic field direction in the bulk of the fluid. They can dissipate energy by a variety of mechanisms such as mode coupling, phase mixing, resonance heating, viscous heating, turbulent heating and Landau damping. Surface Alfvén waves Surface Alfvén waves can exist at fluid boundaries. Resonant absorption is the mechanism proposed for coronal heating by the surface waves Currents Currents and magnetic fields also carry energy. Current sheets are more important in heating than volume currents. Dissipation mechanisms include Joule heating and magnetic reconnection. Nanoflares are another possible dissipation mechanism. 6

7 3 Heating by Magnetohydrodynamic Turbulence Magnetohydrodynamic turbulence is the more accepted mechanism for coronal heating. Footpoint motions in the photosphere feed energy into the large scale modes. The energy is transferred to the small scale modes through MHD turbulence. 3.1 Is the spectrum Kolmogorov or Kraichnan? To find out which of the models of MHD turbulence is operational in the case of coronal heating, we compute the energy fluxes as predicted by the two models and compare them with the power requirement of the corona Kolmogorov flux The Kolmogorov energy spectrum is given by E(k) Cɛ 2 3 k 5 3 Here ɛ is the energy dissipation rate per unit mass. ke(k) Cɛ 2 3 k 2 3 Now E(k)dk is the total energy in the system, which can be related to the velocity. U 2 Cɛ 2 3 k 5 3 Thus we get the energy dissipation rate per unit mass as a function of velocity and the length of the coronal loops that feed energy into the corona. ɛ U 3 L The power dissipated per unit area is found to be where L 0 is the thickness of the corona P ρu 3 L o L Putting in numbers : ρ = kg/m 3, U = 50km/s, L 0 = m, L = km We get P 400W/m 2 7

8 3.1.2 Kraichnan flux Following the same procedure as above from the Kraichnan spectrum we have E(k) A(ɛB 0 ) 1 2 k 3 2 ke(k) A(ɛB 0 ) 1 2 k 1 2 U 2 ɛ B 0 2 k 1 2 ɛ U 4 P B 0 L ρu 4 L o LB 0 The Kraichnan flux is found to be U B 0 times the Kolmogorov flux. In the active regions, we have a large magnetic field around 1000G. This gives P 0.2W/m Comparison The heating requirement for the active region of the corona is W/m 2. The Kolmogorov flux found above falls in this range. The Kraichnan flux, on the other hand, is lower than the requirement for even coronal holes. Even if we assume a lower magnetic field of 100G, that increases the power only by a factor of 10, taking it upto 2W/m 2. Here it is to be noted that the mean magnetic field of the sun is of the order of 1G. If we put this number in, we would get a power dissipation rate comparable to that of the Kolmogorov flux. However, the processes that provide the source of energy for coronal heating operate in the active regions and the magnetic field there is of the order of G. So a value of 100G at least is in order to be used in calculation. Also to be noted is that most of the heating requirement of the corona comes from the active region. During solar minima when the active region area reduces considerably, the coronal heating requirement falls by two orders of magnitude[5]. Hence the calculations need to focus on the active regions. Under this assumption, it is seen that the Kolmogorov spectrum is sufficient to power the active regions of the corona while the Kraichnan spectrum is insufficient. However, this analysis is quite oversimplified. A more accurate analysis is required to ascertain the MHD spectrum that is really operational in the case of coronal heating. 8

9 4 Conclusion The understanding of exact mechanism by which coronal heating occurs remains incomplete. MHD turbulence is one of the more widely accepted of the mechanisms proposed. From the simplified analysis given, it is seen that Kolmogorov MHD flux might be sufficient to satisfy the coronal heating requirements whereas the Kraichnan flux falls short. However, a more accurate analysis is required to ascertain the finding. 5 Acknowledgements We would like to thank our mentor Prof. M.K. Verma for guiding us through this project. We are also grateful to our classmates for their feedback during presentation sessions. 9

10 References [1] Wikipedia. Sun Wikipedia, the free encyclopedia, [Online; accessed 14-April- 2011]. [2] P.; Matthaeus Oughton, S.; Dmitruk. Coronal Heating and Reduced MHD. Turbulence and Magnetic Fields in Astrophysics. Edited by E. Falgarone, and T. Passot., Lecture Notes in Physics, vol. 614, p [3] Daniel O.; Martens Milano, Leonardo J.; Gomez. Solar Coronal Heating: AC versus DC. Astrophysical Journal v.490, p , 20 November [4] Wikipedia. Coronal radiative losses Wikipedia, the free encyclopedia, [Online; accessed 14-April-2011]. [5] Markus J. Aschwanden. An Evaluation of Coronal Heating Models for Active Regions Based on Yohkoh, SOHO, and TRACE Observations. The Astrophysical Journal, Volume 560, Issue 2, pp [6] P. Narain, U.; Ulmschneider. Chromospheric and coronal heating mechanisms. Space Science Reviews, vol. 54, Dec. 1990, p [7] P. Narain, U.; Ulmschneider. Chromospheric and Coronal Heating Mechanisms II. Space Science Reviews, Volume 75, Issue 3-4, pp