Folks,
I submitted the following paper to ApJ and put a copy on Astro-Ph: Article-id: 0804.2241. Its main purpose was to develop the tools for rapid forward modeling of loops observed by XRT and AIA.
Best regards,
Piet
Scaling Laws and Temperature Profiles for Solar and Stellar Coronal Loops with Non-uniform Heating
P.C.H. Martens
Abstract:
The bulk of solar coronal radiative loss consists of soft X-ray emission from quasi-static loops at the cores of Active Regions. In order to develop diagnostics for determining the heating mechanism of these loops from observations by coronal imaging instruments, I have developed analytical solutions for the temperature structure and scaling laws of loop strands for a wide range of heating functions, including footpoint heating, uniform heating, and heating concentrated at the loop apex. Key results are that the temperature profile depends only weakly on the heating distribution -- not sufficiently to be of significant diagnostic value -- and that the scaling laws survive for this wide range of heating distributions, but with the constant of proportionality in the RTV scaling law ($P_{0}L, \thicksim, T_{max}^3$) depending on the specific heating function. Furthermore, quasi-static analytical solutions do not exist for an excessive concentration of heating near the loop footpoints, a result in agreement with recent numerical simulations. It is demonstrated that a generalization of the solutions to the case of a strand with a variable diameter leads to only relatively small correction factors in the scaling laws and temperature profiles for constant diameter loop strands. A quintet of leading theoretical coronal heating mechanisms is shown to be captured by the formalism of this paper, and the differences in thermal structure between them may be verified through observations. Preliminary results from full numerical simulations demonstrate that, despite the simplifying assumptions, the analytical solutions from this paper are stable and accurate.
-------------------------------------------------------------------- Piet Martens Tel: 617-496-7769 Center for Astrophysics Fax: 617-496-7577 60 Garden Street, MS 58 Cell: 617-999-0353 Cambridge, MA 02138 pmartens@cfa.harvard.edu --------------------------------------------------------------------
Dear Piet,
Congratulations to the new analytical paper on loop profiles. I just came back from a 3-week trip to France and saw your new paper which I am going to read. Incidentally, I also worked on some analytical approximations for loop profiles as well as time profiles of heated and cooling loops. Perhaps we can compare some of the analytical solutions in both papers. I think the two papers are quite complementary, but I had the same motivation, to derive some analytical expressions that provide simple tools for rapid forward modeling of loops observed by XRT and AIA. You probably have also seen Jim Klimchuk's EBTEL paper which also goes in this direction (just accepted by ApJ). I give you the URL of my preprint below and we could think if a comparison of some cases could be useful to corroborate each others approach.
Cheers, Markus
Aschwanden,M.J. and Tsiklauri, D. 2008, The Astrophysical Journal, ... (subm) URL1="http://www.lmsal.com/~aschwand/eprints/2008_hydroapprox.pdf" Analytical approximations in explicit form for the hydrodynamic evolution of heated and cooling coronal loops
On Apr 16, 2008, at 10:25 AM, Petrus Martens wrote:
____________________________________________ Dr. Markus J. Aschwanden Solar & Astrophysics Laboratory Lockheed Martin Advanced Techology Center Org. ADBS, Bldg. 252 3251 Hanover St., Palo Alto, CA 94304, USA Phone: 650-424-4001, FAX: 650-424-3994 URL: http://www.lmsal.com/~aschwand/ e-mail: aschwanden@lmsal.com _______________________________________ ____________________________________
Hey Markus,
Thanks for the kind words. Right now I am focusing on the numerical verification mentioned in the paper, plus, most important, on supporting Trae in the final writing for his thesis -- that mostly concerns flaring loops.
I hope we can get together sometime at a meeting and compare notes. Perhaps some future Santorini-like meeting to move into the XRT/AIA era.
Cheers,
Piet
Markus J. Aschwanden wrote:
Hi Piet,
I am just reading your new paper on analytical solutions of loop profiles, which overlaps and complements my new paper on analytical approximations of loop dynamics. Here a few overlapping aspects:
(1) In Eq.(28) you show that the RTV scaling law survives in your analytical solution, which fully agrees with my finding (Eq.81-82 in my paper), even for different non-uniform heating functions. For the constant of proportionality, I find also slightly different values for different heating functions (Eq.82), basically scaling with the Mach number of the upflows.
(2) Also the second scaling law (your Eq.31) is reproduced in my paper in an equivalent way (Eq. 20 in my paper), with slightly different values for the constant for non-uniform heating.
(3) The expression for the loop top temperature (your Eq.35) is also reproduced in Eq.23 in my paper.
(4) Your analytical solutions for the temperature profiles (Fig.3 in your paper) are nearly identical to the approximations I derive in my paper (Fig.2), for uniform as well as non-uniform heating.
So, all the overlapping parts in your and my paper seems to agree quite well, although my approach is based on some approximations (such as neglecting the radiative loss during the heating phase). On the other side, my approach is complementary in the sense that it gives also an approximation for the density profile n(s), as well as for the time evolution of temperature T(s,t) and density n(s,t) during the heating and cooling phase. So, both papers provide some analytical tools in a complementary way, which might be useful for quick forward-modeling of loops observed with Hinode and/or STEREO.
Congrats, again, to your very useful analytical study. Cheers,
Markus
____________________________________________ Dr. Markus J. Aschwanden Solar & Astrophysics Laboratory Lockheed Martin Advanced Techology Center Org. ADBS, Bldg. 252 3251 Hanover St., Palo Alto, CA 94304, USA Phone: 650-424-4001, FAX: 650-424-3994 URL: http://www.lmsal.com/~aschwand/ e-mail: aschwanden@lmsal.com _______________________________________
loops@solar.physics.montana.edu