Thermal equilibria of coronal magnetic loops with non-constant cross-sectional area

Equations of thermal equilibrium along coronal loops are solved in the absence of gravity but where the cross-sectional area changes along the loop. The footpoint temperature is assumed to be 2 × 10⁴ K. Several fundamental types of solution are found, namely hot loops, cool loops, hot-cool loops (where the footpoints and summits are cool but the intermediate parts are hotter) and warm loops (cool along most of their lengths except the summits). On increasing the cross-sectional area the summit temperature generally increases slightly except for warm loops where no increase in temperature is recorded and hot-cool loops where a dramatic increase in summit temperature may occur. The cool and hot-cool loops may model elementary fibril structures within prominences.     

Thermal equilibria of coronal magnetic loops

Equations of thermal equilibrium along coronal loops with footpoint temperatures of 2 × 10⁴ K are solved. Three fundamentally different categories of solution are found, namely hot loops with summit temperatures above about 4 × 10⁵ K, cool loops which are cooler than 8 × 10⁴ K along their whole length and hot-cool loops which have summit temperatures around 2 × 10⁴ K but much hotter parts at intermediate points between the summit and the footpoints. Hot loops correspond to the hot corona of the Sun. The cool loops are of relevance for fibrils, for the cool cores observed by Foukal and also for active-region prominences where the magnetic field is directed mainly along the prominence. Quiescent prominences consist of many cool threads inclined to the prominence axis, and each thread may be modelled as a hot-cool loop. In addition, it is possible for warm loops at intermediate summit temperatures (8 × 10⁴K to 4 × 10⁵ K) to exist, but the observed differential emission measure suggests that most of the plasma in the solar atmosphere is in either the hot phase or the cool phase. Thermal catastrophe may occur when the length or pressure of a loop is so small that the hot solution ceases to exist and there are only cool loop solutions. Many loops can be superimposed to form a coronal arcade which contains loops of several different types.     

Steady models for the hard X-ray loops in the solar corona

In some gradual hard X-ray bursts with high intensity, hard X-ray source (15–40 keV) is steadily located in the corona along with softer X-ray source (5–10 keV).Two stationary models, high density and high temperature models, are proposed to solve the difficult problem of confinement of hot (or nonthermal) plasma in the direction of the magnetic field along the loops in the corona. In both models, an essential point is that the effective X-ray source is composed of fine dense filamentary loops imbeded in a larger rarefied coronal loop, and the electron number density in the filaments is so high as 10¹¹–10¹² cm^-3. If the density is so high heat conduction can be as reasonably small as of the order of 10²⁷ erg s ^-1 for the given emission measures of observed X-rays, since the required cross-sectional area is small and also classical conduction is valid. Collisional confinement of thermal tail, and nonthermal electrons if any, up to 50–60 keV in the filaments is also possible, so that the hard X-ray images can be loop like structure instead of double source (foot points).High density model is applicable to the coronal filamentary loops with temperature T _m < 5 × 10⁷ K at the loop summit. The heat flow from the summit downwards is lost almost completely by the radiation from the loop during the conduction to the foot points. A continuous energy release is assumed near the summit to maintain the stationary temperature T _m, and pressure balance is maintained along the loop. In this model, the number density at the summit is given by n _m - 10⁶ T _m ² /s_m, where s _m is the length of the loop from the summit to the foot point, and the distribution of temperature and density along the loop are given by T = T _m(s/s_m)^1/3 and n = n _m(s/s_m)^-1/3, respectively.High temperature model is applicable to the filamentary loops with higher temperature up to about 10^8.5 K and comparatively lower number density as 10¹¹ cm^-3 for the requirement of magnetic confinement of the hot plasma in radial direction. The radiation from the loop is negligibly small in this model so that the heat flux is nearly conserved down to the foot points. In this case, temperature gradient is smaller than that of the high density model, depending on the tapering of the magnetic bottle.In both models, the differential emission measure is maximum at the highest temperature T _m and the brightness distribution along the loop shows a maximum around the summit of the loop if some magnetic tapering is taken into account.     

The effects of Alfvén waves on heating plasma in post-flare loops

In investigating the effects of collision Alfvén waves on the heating of a cool-type solar loop, like the post-flare loop, models are proposed, and the distributions of ion or electron density, temperature, pressure, and wave energy density are simulated. We assumed the magnetic field strength in the loop is about 100 G and found that Alfvén waves can propagate through the whole loop, that is to say, the decay length of collision Alfvén waves which we consider can reach to the height or length of the loop. Thus, the Alfvén wave heating is a considerable heating mechanism in cool loops. And we also found that the variations of density, pressure, and wave energy density are more significant than those of the temperature. In the whole loop, the temperature is of the order of 10⁴ K. In comparison with other parameters, the temperature can be considered as homogeneous; hence, the heat conductive flux in the simulations is omitted.     

Emitting region of sodium lines in solar prominences

We have calculated the emission spectra of hydrogen and sodium atoms in the cool part of prominence models which satisfy simultaneously the constraints of radiative transfer, statistical equilibrium and charge-particle conservations.In the considered range of our model parameters, emission strengths of H and Nai D lines increase with the temperature and the total number density. Low-pressure models raise the ionization rate highly but yield very weak Nai D line intensities, since these model prominences contain small amounts of free electrons and sodium atoms which have a deep relation with the formation of sodium lines. We find that sodium D lines should be emitted in the high pressure region of prominences, and that their intensities are difficult to attain in the cool core of any model prominence with a temperature as low as 4000 K. In order to explain consistently the spectral emissions of H and Nai D lines observed in quiescent prominences, a total number density higher than 4 x 10¹¹ cm^-3 and a temperature over 5000 K are required at least in the cool part of prominences.Contributions from the Kwasan and Hida Observatories, University of Kyoto, No. 282.     

The temperature structure and pressure balance of magnetic loops in active regions

EUV observations show many active region loops in lines formed at temperatures between 10⁴K and 2×l0⁶K. The brightest loops are associated with flux tubes leading to the umbrae of sunspots. It is shown that the high visibility of certain loops in transition region lines is due principallly to a sharp radial decrease of temperature to chromospheric values toward the loop axis. The plasma density of these cool loops is not significantly greater than in the hot gas immediately surrounding it. Consequently, the internal gas pressure of the cool material is clearly lower. The hot material immediately surrounding the cool loops is generally denser than the external corona by a factor 3–4. When the active region is examined in coronal lines, this hot high pressure plasma shows up as loops that are generally parallel to the cool loops but significantly displaced laterally. In general the loop phenomenon in an active region is the result of temperature variations by two orders of magnitude and density variations of around a factor five between adjacent flux tubes in the corona.     

Non-equilibrium ionization in coronal loops

The ionization conditions in coronal loops are investigated in the temperature range 2 × 10⁵–2 × 10⁶K, assuming velocity, density and temperature distributions computed for a siphon model of a pure hydrogen plasma. Use is made of the set of the carbon ions as an example of the general behaviour of the ions characteristic of that temperature range. It is found that the deviation from equilibrium ionization is large for subsonic-supersonic flow if the density is less than 5 × 10⁹cm^–-3, with the exception of the lower part of the first leg of very cool loops (T 2 × 10 K). With this exception cooler loops, given their larger density drop along the axis, show deviations from ionization equilibrium more easily than hotter ones, in spite of their lower flow velocity. We conclude that the possibility of a non-equilibrium state must be taken into account when deducing from measurements of line intensities the temperature of loops in which a flow may occur.Now at Institute for Plasma Research, Stanford University, as an E.S.A. Fellow.     

MHD equilibrium and stability properties of a bipolar current loop

A study is made of equilibrium and stability properties of a semi-toroidal current loop imbedded in a high temperature plasma. The loop carries a toroidal current density J _t and poloidal current density J _p. By explicity including the global curvature of the loop, the net Lorentz and pressure forces acting along the major radius are calculated. Requirement of equilibrium force-balance gives rise to conditions that must be satisfied by the physical parameters and geometry. On the basis of these conditions, we deduce a class of equilibrium semi-toroidal current loops satisfying c ^#X2212;1 J × B ? ▽p = 0. It is found that the averge pressure inside the loop is less than the ambient coronal pressure in equilibrium. Furthermore, this class of equilibria is shown to be stable to a number of destructive MHD modes. The theoretical results are discussed in the context of solar bipolar current loops.     

Relation between cool and hot post-flare loops of 26 June 1992 derived from optical and X-ray (SXT-Yohkoh) observations

We have analyzed the physical conditions of the plasma in post-flare loops with special emphasis on dynamics and energy transport using SXT-data (hot plasma) and optical ground-based data from Pic du Midi, Wrocaw, and Ondejov (cool plasma). By combining the H observations with the SXT images we can understand the relationship between cool and hot plasmas, the process of cooling post-flare loops and the mechanism which maintains the long duration of these loops. Using recent results of NLTE modeling of prominence-like plasmas, we derive the emission measure of cool H loops and this gives us a realistic estimate of the electron density (2.2 × 10¹⁰ cm^–3). Then, by comparing this emission measure with that of hot loops derived from SXT data, we are able to estimate the ratio between electron densities in hot and cool loops taking into account the effect of geometrical filling factors. This leads to the electron density in hot loops 7 × 10⁹ cm^–3. We also derive the temperature of hot X-ray loops ( 5.5 × 10⁶ K), which, together with the electron density, provides the initial values for solving the time-dependent energy balance equation. We obtain the cooling times which are compared to a typical growth-time of the whole loop system ( 2000 s). In the legs of cool H loops, we observe an excess of the emission measure which we attribute to the effect of Doppler brightening (due to large downflow velocities).     

Structure and dynamics of cool flare loops

MSDP observations of the 16 May, 1981 two-ribbon flare are used to study the physical structure and the dynamical behaviour of cool flare loops. The loops have been detected in the H line just after the flare maximum and they appeared in absorption against the disk. Using the first-order differential cloud model (DCM1) technique, we derive empirically some basic plasma parameters at 15 points along one loop leg. The flow velocities and the true heights have been reconstructed with respect to a geometrical projection. Subsequently, detailed non-LTE models of cool loops have been constructed in order to fit H source function values previously derived from DCM1 analysis. It is demonstrated that this source function is rather sensitive to the radial component of the flow velocity (the so-called Doppler brightening) and to enhanced irradiation of the loops from the underlying flare ribbons. In this way, we have been able to estimate quantitatively all plasma parameters which determine the physical structure of cool loops (i.e., the temperature, pressure, density), as well as the momentum-balance condition within the loops. For these dark loops we have arrived at relatively low gas pressures of the order of 0.1–0.5 dyne cm^-2 with corresponding electron densities around 10¹¹ cm^-3. Pressure-gradient forces have been found to be of small importance in the momentum-balance equation, and thus they cannot explain departures from a free-fall motion found in our MSDP data analysis. We propose three possible solutions to this problem.     

Thermal conduction and modeling of static stellar coronal loops

We have modeled stellar coronal loops in static conditions for a wide range of loop length, plasma pressure at the base of the loop and stellar surface gravity, so as to describe physical conditions that can occur in coronae of stars ranging from low mass dwarfs to giants as well as on a significant fraction of the Main-Sequence stars.Three alternative formulations of heat conduction have been used in the energy balance equation, depending on the ratio ₀/L _Tbetween electron mean free path and temperature scale height: Spitzer's formulation for ₀/L _Tless than 2 × 10^–3, the Luciani, Mora, and Virmont non-local formulation for ₀/L _Tbetween 2 × 10^–3 and 6.67 × 10^–3 and the limited free-streaming formulation for ₀/L _Tlarger than 6.67 × 10^–3.We report the characteristics of all loop models studied, and present examples to illustrate how the temperature and density stratification can be drastically altered by the different conductivity regimes. Significant differences are evident in the differential emission measure distribution vs temperature, an important observable quantity. We also show how physical conditions of coronal plasma, and in particular thermal conduction, change with stellar surface gravity.We have found that, for fixed loop length and stellar gravity, a minimum of loop-top plasma temperature occurs, corresponding to the highest value of base plasma pressure for which the limited free-streaming conduction occurs. This value of temperature satisfies the appropriate scalingT 10^–9 L g, in cgs units.     

Siphon flows in coronal loops: I. Adiabatic flow

It is now known that the corona is filled with a multitude of loop-like structures. The likelihood of these loops being in static equilibrium is small and so this paper explores the possibility of steady isothermal or adiabatic flows, driven by a pressure difference between the loop feet. For a symmetric loop the flow becomes supersonic at the summit and is then retarded by a shock-wave at some point on the downflowing leg. The effect of adiabatic flow is to lower both pressure and temperature by at least a factor of two and so provide a possible explanation for the cool cores that are sometimes observed in coronal loops. Asymmetric loops, whose cross-sectional area increases or decreases in the flow direction, are found to possess a wide range of both subsonic and shocked flows. Converging loops have subsonic flows if the pressure difference between the footpoints is small, but shocked flows if the pressure difference is large enough. Diverging loops exhibit only shocked flows towards a low pressure footpoint, but can have either subsonic or shocked flow towards a high pressure footpoint. Flows in diverging loops can therefore be either accelerated or decelerated.     

Thermal equilibria of coronal magnetic arcades

A coronal magnetic arcade can be thought of as consisting of an assembly of coronal loops. By solving equations of thermal equilibrium along each loop and assuming a base temperature of 2 × 10⁴ K, the thermal structure of the arcade can be found. By assuming a form for the plasma pressure in the arcade, the possible thermal structures can be shown to depend on three parameters. Arcades can contain hot loops with summits hotter than 400 000 K, cool loops at temperatures less than 80 000 K along their lengths, hot-cool loops with cool summits and cool footpoints but hotter intermediate portions, and warm loops, cooler than 80 000 K along most of their lengths but with summits as hot as 400 000 K. For certain arcades, there exist regions where more than one kind of loop is possible. If the parameters describing the arcade are varied, it is possible for non-equilibrium to occur when a type of solution ceases to exist. For example, hot or warm loops can cease to exist so that only cool solutions are possible when the arcade size or pressure is decreased, while warm or cool loops may give way to hot-cool loops when the heating is reduced or the pressure is increased.     

Physical parameters in long-decay coronal enhancements

X-ray images have been studied quantitatively to determine electron temperature and density as functions of time in two long-decay X-ray enhancements (LDE's). This is the first study of the X-ray emission from LDE's to include all corrections for scattering and vignetting. Derived electron density is about twice that found by Vorpahl et al. (1977) and by Smith et al. (1977) in the same events. Our results are combined with those for two other LDE's to find their general characteristics. The LDE's all had the form of arcades of very bright loops which were 1–3 × 10⁶ K hotter at the apices than along the legs. This temperature structure was maintained for at least 8 hr in each case. From this it is inferred that continual heating was taking place at the loop apices. Each LDE was preceded by a filament eruption and a white-light transient. Each was associated with a loop prominence system (LPS) composed of cool (T _e < 10⁵ K) loops nested 2–8 × 10³ km below the hot LDE loops. And, although the energy release rates in the four events varied greatly even 4 hr after onset, they all had similar growth rates (loop height vs time 1 km s^–1). Event lifetimes were very long, from 24 to 72 hr. After a survey of published models, it is concluded that only a magnetic reconnection model (e.g., Kopp and Pneuman, 1976) is consistent with these observations of the LDE-LPS phenomenon.     

On the distribution of material as a function of temperature in the post-flare loop system of 12 August 1970

Observations of the post-flare loop system formed after the east limb proton flare of 12 August 1970 include (a) sets of filtergrams from which photographic subtractions have been constructed and (b) spectra from which a distribution of electron density as a function of temperature for three coronal regions are derived. The filtergrams show no indications of radial velocities in excess of 80 km/s. The spectra indicate an increase in density at the tops of the loops with most of the material at a relatively cool temperature: N 6.0 × 10¹⁰, T = 3 × 10⁵K. The distribution functions obtained for areas just above and just below loops indicate a lower electron density and the presence of material at high temperatures, N 2.0 × 10¹⁰ and T 2.6 × 10⁶K (above the loops) and T _e > > 4.4 × 10⁶K for material below the loops.     

Thermal equilibria of isobaric coronal magnetic arcades

A coronal magnetic arcade can be thought of as consisting of an assembly of coronal loops. By solving equations of isobaric thermal equilibrium along each loop and assuming a base temperature of 2 × 10⁴ K, the thermal structure of the arcade can be found. The possible thermal equilibria can be shown to depend on two parameters L ^* p ^* and h ^*/p ^* representing the ratios of cooling (radiation) to condu and heating to cooling, respectively. Arcades can contain four types of loops: hot loops with summits hotter than 400000 K; cool loops at temperatures less than 80000 K along their lengths; hot-cool loops with cool summits and cool footpoints but hotter intermediate portions; and warm loops, cooler than 80000 K along most of their lengths but with summits as hot as 400000 K. Two possibilities for coronal heating are considered, namely a heating that is independent of magnetic field and a heating that is proportional to the square of the local magnetic field. When the arcade is sheared the thermal structure of the arcade may change, leading in some cases to non-equilibrium or in other cases to the formation of a cool core.     

Mass flow in loop type coronal transients

The white light coronagraph on Skylab observed many loop type coronal transients. These loops travel through the coronagraph's field of view (2–6R ) over a period of a few hours, after which the legs of the loops usually remain visible for a few days. In this paper we investigate the temporal changes in density and mass per unit length measured along the legs of such loops during the several days after the initial eruption. Examination of 8 transients shows that the mass and density in the legs decrease during the few hours after the top of the loop has travelled beyond the coronagraph's field of view. The mass and density then increase slowly, during the next one half to one day, then decrease again over approximately the same period. These changes are generally shown to be too rapid to be explained by solar rotation, indicating that the transient legs have a lifetime of only a few days.The results of a detailed study of the transient of 10 August 1973 are compared with the results from theoretical calculations. For the top of the loop a one-dimensional flow problem is solved, assuming a balance between gravity, inertia, and pressure gradients. The legs are modeled by a flow in a tube of constant cross section. Models for the flow in the legs were calculated under the assumption that the mass distribution is close to hydrostatic equilibrium. Using these models we can estimate that approximately 5 × 10¹⁴ g of material flow outward through the legs of this transient. We also find that the best fit to the observed average density gradient is obtained with a temperature of 1.7 × 10⁶ K.On leave from Max-Planck Institut für Physik und Astrophysik, Munich, Germany.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

POST-FLARE LOOPS OF 26 JUNE 1992 – IV. Formation and Expansion of Hot and Cool Loops

Observations of the post-flare loops after the X3.9 flare which occurred on 25 June, 1992 at 20:11 UT by the Yohkoh/SXT in X-rays, as well as in H obtained at 5 different observatories, have provided a unique, longest ever, set of data for a study of the relationship between the hot and cool post-flare loops as they evolve. At any given time, the altitude difference between the hot X-ray loops of 6–7× 10⁶ K and the cool H loops of 1.5× 10⁴ K is related to the expansion rate of the loop systems and their cooling time. Therefore, measurements of the expansion rate and relative height of hot and cool loops can provide direct observational values for their cooling times. We measured the altitude of hot and cool loops for 15 and 19 hours, respectively, and found that the cooling time increased as the density of the loops decreased. We found a reasonably good agreement between the observed cooling times and those obtained from model calculations, although the observed values were always somewhat longer than the theoretical ones. Taking into account evolutionary effects, we also found similar shapes and configurations of hot and cool loops during the entire observing period and confirmed that, at any time, hot loops are at higher altitude than cool loops, suggesting that cool loops indeed evolve from hot loops. These results were used to check the validity of the reconnection model.     

The post flare loops observed at the total eclipse of February 16, 1980

A post flare loop system was observed on the west limb at the total solar eclipse of February 16, 1980 in Kenya. Analyzing the monochromatic images and the flash spectra, we obtained the following results: (1) the lower part of the post flare loop system is characterized mainly by distinct cool loops of H and Fe x 6374. Fe x 6374 emitting plasma (T _e = 1.0 × 10⁶ K) is highly concentrated in the loops. The 6374 loops are broader in diameter and located very close to but a little higher than the corresponding H loops. The electron densities of the dense part in H and Fe x 6374 loops are 10¹¹ cm^-3 and 6 × 10⁹cm^-3, respectively; (2) the Ca xv emitting region (3.5 × 10⁶ K) is confined to the upper part of the post flare loops. The electron density of this hot region is estimated as 8 × 10⁹ cm^-3 from the Ca xv line intensity ratio, I(5694)I(5445). These observational results led us to construct an empirical model of the post flare loop system which is consistent with the reconnection model of Kopp and Pneuman (1976).Contributions from the Kwasan and Hida Observatories, University of Kyoto, No. 267.     

The formation of flare loops by magnetic reconnection and chromospheric ablation

Slow-mode shocks produced by reconnection in the corona can provide the thermal energy necessary to sustain flare loops for many hours. These slow shocks have a complex structure because strong thermal conduction along field lines dissociates the shocks into conduction fronts and isothermal subshocks. Heat conducted along field lines mapping from the subshocks to the chromosphere ablates chromospheric plasma and thereby creates the hot flare loops and associated flare ribbons. Here we combine a non-coplanar compressible reconnection theory with simple scaling arguments for ablation and radiative cooling, and predict average properties of hot and cool flare loops as a function of the coronal vector magnetic field. For a coronal field strength of 100 G the temperature of the hot flare loops decreases from 1.2 × 10⁷ K to 4.0 × 10⁶ K as the component of the coronal magnetic field perpendicular to the plane of the loops increases from 0% to 86% of the total field. When the perpendicular component exceeds 86% of the total field or when the altitude of the reconnection site exceeds 10⁶km, flare loops no longer occur. Shock enhanced radiative cooling triggers the formation of cool H flare loops with predicted densities of 10¹³ cm^–3, and a small gap of 10³ km is predicted to exist between the footpoints of the cool flare loops and the inner edges of the flare ribbons.