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.     

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.     

Spatial and temporal variations of solar coronal loops

Skylab EUV observations of an active region near the solar limb were analyzed. Both cool (T < 10⁶ K) and hot (T > 10⁶ K) loops were observed in this region. For the hot loops the observed intensity variations were small, typically a few percent over a period of 30 min. The cool loops exhibited stronger variations, sometimes appearing and disappearing in 5 to 10 min. Most of the cool material observed in the loops appeared to be caused by the downward flow of coronal rain and by the upward ejection of chromospheric material in surges. The frequent EUV brightenings observed near the loop footpoints appear to have been produced by both in situ transient energy releases (e.g. subflares) and the infall/impact of coronal rain. The physical conditions in the loops (temperatures, densities, radiative and conducting cooling rates, cooling times) were determined. The mean energy required to balance the radiative and conductive cooling of the hot loops is approximately 3 × 10^–3 erg cm^–3 s^–1. One coronal heating mechanism that can account for the observed behavior of the EUV emission from McMath region 12634 is heating by the dissipation of fast mode MHD waves.     

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.     

The Nature of Blinkers and the Solar Transition Region

Solar plasma that exists at around 10⁵ K, which has traditionally been referred to as the solar transition region, is probably in a dynamic and fibril state with a small filling factor. Its origin is as yet unknown, but we suggest that it may be produced primarily by one of five different physical mechanisms, namely: the heating of cool spicular material; the containment of plasma in low-lying loops in the network; the thermal linking of cool and hot plasma at the feet of coronal loops; the heating and evaporating of chromospheric plasma in response to a coronal heating event; and the cooling and draining of hot coronal plasma when coronal heating is switched off. We suggest that, in each case, a blinker could be produced by the granular compression of a network junction, causing subtelescopic fibril flux tubes to spend more of their time at transition-region temperatures and so to increase the filling factor temporarily.     

The structure of coronal arcades and the formation of solar prominences

The temperature and density are obtained for coronal plasma in thermal and hydrostatic equilibrium and located in a force-free magnetic arcade. The isotherms are found to be inclined to the magnetic field lines and so care should be taken in inferring the magnetic structure from observed emission.When the coronal pressure becomes too great, the equilibrium ceases to exist and the material cools to form a quiescent prominence. The same process can be initiated at low heating rates when the width or shear of the arcade exceeds a critical value.We suggest that the prominence should be modelled as a dynamic structure with plasma always draining downwards. Material is continually sucked up along field lines of the ambient arcade and into the region lacking a hot equilibrium, where it cools to form new prominence material.     

The temperature-density structure of coronal loops in hydrostatic equilibrium

The temperature and density structure are computed for a comprehensive set of coronal loops that are in hydrostatic and thermal equilibrium. The effect of gravity is to produce significant deviations from the usual uniform-pressure scaling law (T(pL) ^1/3) when the loops are taller than a scale height. For thermally isolated loops it lowers the pressure throughout the loop, which in turn lowers the density significantly and also the temperature slightly; this modifies the above scaling law considerably. For more general loops, where the base conductive flux does not vanish, gravity lowers the summit pressure and so makes the radiation decrease by more than the heating. This in turn raises the temperature above its uniform pressure value for loops of moderate length but lowers it for longer loops. A divergence in loop cross-section increases the summit temperature by typically a factor of 2, and decreases the density, while an increase in loop height (for constant loop length) changes the temperature very little but can halve the density.One feature of the results is a lack of equilibrium when the loop pressure becomes too large. This may explain the presence of cool cores in loops which originally had temperatures below 2 × 10⁶ K. Loops hotter than 2 × 10⁶ K are not expected to develop cool cores because the pressure necessary to produce non-equilibrium is larger than observed.     

The Prominence-Corona Transition Region in transverse magnetic fields

An emission measure analysis is performed for the Prominence-Corona Transition Region (PCTR) under the assumption that the cool matter of quiescent filaments is contained in long, thin magnetic flux loops imbedded in hot coronal cavity gas. Consequently, there is a transition region around each thread.Comparison of the model and observations implies that the temperature gradient is perpendicular to the magnetic lines of force in the lower part of the PCTR (T < 10⁵ K). It is shown that in this layer the heating given by the divergence of the transverse conduction fails to account for the observed UV and EUV emission by several orders of magnitude. It is, therefore, suggested that the heating of these layers could be due to dissipation of Alfvén waves.In the high-temperature layers (T 10⁵ K), where the plasma 1, the temperature gradient is governed by radiative cooling balancing conductive heating from the surrounding hot coronal gas. Also in these outer layers the presence of magnetic fields reduces notably the thermal conduction relative to the ideal field-free case. Numerical modelling gives good agreement with observed DEM; the inferred value of the flux carried by Alfvén waves, as well as that of the damping length, greatly support the suggested form of heating. The model assumes that about 1/3 of the volume is occupied by threads and the rest by hot coronal cavity matter.The brightness of the EUV emission will depend on the angle between the thread structure and the line of sight, which may lead to a difference in brightness from observations at the limb and on the disk.     

The XUV structure of solar active regions

XUV spectroheliograms of 2 active regions are studied. The images are due to lines emitted at temperatures between 8 x 10⁴ K and 2 x 10⁶ K and thus are indicative of transition region and coronal structures. The hot coronal lines are formed solely in loop structures which connect regions of opposite photospheric magnetic polarity but are not observed over sunspots. Transition region lines are emitted in plages overlying regions of intense photospheric magnetic field and in loops or loop-segments connecting such regions. The hot coronal loops are supported hydrostatically while only some of the transition zone loops are. The coronal and transition zone loops are distinctly separated and are not coaxial. A comparison of direct measurements of electron densities using density sensitive line ratios with indirect measurements using emission measures and path lengths shows the existence of fine structures of less than a second of arc in transition region loops. From a similar analysis, hot coronal loops do not have any fine structure below about 2 seconds of arc.     

Hard X-ray imaging of a solar gradual hard X-ray burst on April 1, 1981

An intense solar X-ray burst occurred on April 1, 1981. X-ray images of this gradual hard X-ray burst were observed with the hard X-ray telescope aboard the Hinotori satellite for the initial ten minutes of rise and maximum phases of the burst. The hard X-ray images (13–29 keV) look like a large loop without considerable time variation of an elongated main source during the whole observation period. The main X-ray source seems to lie along a ridge of a long coronal arcade 2 × 10⁴ km above a neutral line, while a tangue-like sub-source may be another large coronal loop although the whole structure of the X-ray source looks like a large semi-circular loop. Both nonthermal and hot thermal (3–4 × 10⁷ K) electrons are contributing to the source image. The ratio of these components changed in a wide range from 2.3 to 0.4 during the observation, while the image was rather steady. It suggests that both heating and accelerations of electrons are occurring simultaneously in a common source. Energetic electrons of 15–30 keV would be collisionally trapped in the coronal magnetic loops with density of the order of 10¹¹ cm^–3.     

The relationship between UV and X-ray active region structures

Yohkoh and the Coronal Diagnostic Spectrometer (CDS) on the Solar and Heliospheric Observatory (SOHO) jointly observed two brightenings in active region NOAA 7981 on 6 August 1996. Combining the UV data from CDS with information from the high time resolution coronal images obtained with the Soft X-ray Telescope (SXT) on Yohkoh, provides us with important information on the relationship between the transition region and corona. Our observations show that cool plasma (T_e = 2.2 x 10^-5 K) can lie at the same altitude as the hot coronal plasma (T_e = 1–4 x 10⁶ K). The lower temperature structure is not formed from the cooling of the hotter coronal loop. We are also able to observe a low temperature cut-off of T_e = 1–4 x 10⁶ K for a loop which repeatedly brightened over the period of approximately one day.     

The growth of filaments by the condensation of coronal arches

A model of filament formation based on the condensation of coronal arches is described. The condensation results from initiating the radiative instability within an arch by superimposing a transient energy supply upon the steady state heating mechanism. The transient energy supply increases the density within the arch so that when it is removed the radiative losses are sufficient to lead to cooling below the minimum in the power loss curve.Times from the initial formation of the condensation to its temperature stabilization as a cool filament have been calculated for various initial conditions. They lie in the range 10⁴ to 10⁵ s with the majority of the time spent above a temperature of 1 × 10⁶ K.Under the assumption that the condensation of a single arch forms an element of the filament, a complete filament requires the condensation of an arcade of loops. Using experimentally derived parameters, filament densities of 10¹¹ to 10¹² cm^–3 can be obtained.     

Morphology and spatial distribution of XUV and X-ray emissions in an active region observed from Skylab

We studied the morphology and spatial distribution of loops in an active region, using coordinated observations obtained with both the S082A XUV spectroheliograph and the S056 grazingincidence X-ray telescope on Skylab. The active region loops in the temperature range 5 × 10⁵ –3 × 10⁶ K fall basically into two distinctive groups: the hot loops with temperatures 2–3 × 10⁶ K as observed in coronal lines and X-rays, and the relatively cool loops with temperature 5 × 10⁵ –1 × 10⁶ K as observed in transition-zone lines (Ne vii, Mg ix). The brightest hot coronal loops in the active region are mostly low-lying, compact, closely-packed, and show greater stability than the transition-zone loops, which are fewer in number, large, and slender. The observed aspect ratio of the hot coronal loops is in the range of 0.1 and 0.2, which are almost two orders of magnitude larger than those for the Ne vii loops. Brief discussion of the MHD stability of the loops in terms of the aspect ratio is presented.     

Validity of the isobaric assumption to the solar corona

Many coronal heating mechanisms have been suggested to balance the losses from this tenuous medium by radiation, conduction, and plasma mass flows. A previous paper (Walsh, Bell, and Hood, 1995) considered a time-dependent heating supply where the plasma evolved isobarically along the loop length. The validity of this assumption is investigated by including the inertial terms in the fluid equations making it necessary to track the sound waves propagating in a coronal loop structure due to changes in the heating rate with time. It is found that the temperature changes along the loop are mainly governed by the variations in the heating so that the thermal evolution can be approximated to a high degree by the simple isobaric case. A typical isobaric evolution of the plasma properties is reproduced when the acoustic time scale is short enough. However, the cooling of a hot temperature equilibrium to a cool one creates supersonic flows which are not allowed for in this model.     

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.     

Thermal nonequilibrium: A trigger for solar flares?

In this paper, we suggest that a solar flare may be triggered by a lack of thermal equilibrium rather than by a magnetic instability. The possibility of such a thermal nonequilibrium (or catastrophe) is demonstrated by solving approximately the energy equation for a loop under a balance between thermal conduction, optically thin radiation and a heating source. It is found that, if one starts with a cool equilibrium at a few times 10⁴ K and gradually increases the heating or decreases the loop pressure (or decreases the loop length), then, ultimately, critical metastable conditions are reached beyond which no cool equilibrium exists. The plasma heats up explosively to a new quasi-equilibrium at typically 10⁷ K. During such a thermal flaring, any magnetic disruption or particle acceleration are secondary in nature. For a simple-loop (or compact) flare, the cool core of an active-region loop heats up and the magnetic tube of plasma maintains its position. For a two-ribbon flare, the material of an active-region (or plage) filament heats up and expands along the filament; it slowly rises until, at a critical height, the magnetic configuration becomes magnetohydrodynamically unstable and erupts violently outwards. In this case thermal nonequilibrium acts as a trigger for the magnetic eruption and subsequent magnetic energy release as the field closes back down.     

Dissipation of Longitudinal Oscillations in Stratified Nonisothermal Hot Coronal Loops

We investigate the damping of longitudinal (i.e., slow or acoustic) waves in nonisothermal, hot (T≥ 5.0 MK), gravitationally stratified coronal loops. Motivated by SOHO/SUMER and Yohkoh/SXT observations, and by taking into account a range of dissipative mechanisms such as thermal conduction, compressive viscosity, radiative cooling, and heating, the nonlinear governing equations of one-dimensional hydrodynamics are solved numerically for standing-wave oscillations along a magnetic field line. A semicircular shape is chosen to represent the geometry of the coronal loop. It was found that the decay time of standing waves decreases with the increase of the initial temperature, and the periods of oscillations are affected by the different initial footpoint temperatures and loop lengths studied by the numerical experiments. In general, the period of oscillation of standing waves increases and the damping time decreases when the parameter that characterises the temperature at the apex of the loop increases for a fixed footpoint temperature and loop length. A relatively simple second-order scaling polynomial between the damping time and the parameter determining the apex temperature is found. This scaling relation is proposed to be tested observationally. Because of the lack of a larger, statistically relevant number of observational studies of the damping of longitudinal (slow) standing oscillations, it can only be concluded that the numerically predicted decay times are well within the range of values inferred from Doppler shifts observed by SUMER in hot coronal loops.     

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.     

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.