The sun’s "surface" temperature of 6000 K is easily deduced from its blackbody spectrum. As technology has evolved, researchers have slowly gained the ability to measure the sun’s temperature as a function of distance from its center. Intuitively, one would expect the temperature to decrease radially, since the sun generates heat at its core. However, measurements show that the temperature decreases to a minimum of 4500 K in the chromosphere, and then proceeds to increase rapidly, until it reaches more than 2x10⁶ K in the corona (Figure 1). This paradox, the solar heating problem, remains unresolved decades after its discovery. Of the four solar atmospheric regions, photosphere, chromosphere, transition region (TR), and corona, the TR is the site of much of the temperature increase. In traditional models, this large temperature increase occurs in a relatively short distance (~1000 km), creating a huge temperature gradient, and therefore massive heat transfer.

Figure 1. Temperature profile for the solar atmosphere.

Despite the range of temperatures, all the plasma in the TR is hot enough for hydrogen to ionize, or lose its electron. Since an atom without electrons cannot emit energy, the TR does not give off the H a emission, which is typical of the chromosphere and can be measured from Earth. Instead, the main emissions from the TR are from heavier elements, such as He, C, N, O, Ne, Mg, S, and Si, and are in the extreme-ultraviolet (EUV) part of the electromagnetic spectrum. Since the earth’s atmosphere is effective at blocking out EUV, measurements of the TR are possible only from space. Since these measurements are difficult and expensive, modeling of the TR is much more theoretical than similar work for other regions. Despite the difficulties, various missions, such as the Solar and Heliospheric Observatory (SOHO), Yohkoh, Skylab, and the Multi-spectral Solar Telescope Array (MSSTA), have acquired enough data on the sun’s emission to show that the TR gives off a characteristic spectrum.

Researchers once believed that the TR was the interface between plasma at chromospheric temperatures and plasma at coronal temperatures. One such model held that the interface of spicules, jet-like structures comprised of plasma at chromospheric temperatures, and ambient structure-less plasma at coronal temperatures generated the TR emission (Athay 1984). This model was disproved by observations, because no structure-less plasma exists at coronal temperatures. As the resolution in observations improved, loop-like structures became distinguishable, and another model was developed. This model held that TR emission was generated by the interface of the chromosphere with coronal-temperature loop structures (Vesecky et al. 1979). Such loop structures consist of flowing plasma that is confined along the arc-like magnetic field lines in the solar atmosphere because of the constituent ions’ and electrons’ electric charge. They vary in size and the largest are located in the corona. Unfortunately, this model was shown to be impossible, because such loops do not have enough material at TR temperatures to account for the magnitude of the TR emission (Rabin and Moore 1984).

Abandoning the interfacial models, the cool loop model, based on loop structures with peak temperatures ranging continuously through that of the TR, was proposed (Antiochos and Noci 1986; Dowdy et al. 1986). Though some observations suggest that this model may be valid, calculations based on theory show that loops with certain peak temperatures are not stable (Cally and Robb 1991). Hence, the plausibility of this model is still under debate, and, as of now, no self-consistent model of the entire TR exists.

A more successful approach to studying the TR is to examine the heat transfer through the region. Two physically plausible heat distribution mechanisms, ambipolar diffusion and turbulent thermal conduction, have been incorporated into models. At lower transition region (LTR) temperatures, when the net flow of plasma is low, magnetic fields cause ions to diffuse downwards and the neutral atoms to diffuse upwards. This is ambipolar diffusion, a powerful phenomenon that allows neutral hydrogen to be found at temperatures as high as 7×10⁵ K, much higher than the predicted value of 2×10⁵K, which consequently causes more energy to be radiated away over a larger range of temperatures (Fontenla et al. 1991). Also at LTR temperatures, plasma flow can become turbulent, resulting in turbulent thermal conduction (Cally 1990). In a series of papers. Fontenla et al. (1990, 1991, 1993, 2002) derived models (which they designated A, C, F, and P), based on the ambipolar diffusion mechanism. These models describe the different regions of the lower transition region (LTR, 2×10⁴ K < T < 10⁵ K) extremely well. The regions within the LTR are distinguished by their different magnetic field properties, which cause different plasma behavior. The regions exist in all four thermally categorized regions of the solar atmosphere.

In this paper, we present models which correctly describe the entire TR by deriving loop models located in the upper transition region (UTR, 10⁵ K < T < 10⁶ K). These are consistent with the Fontenla et al. (1990, 1991, 1993, 2002) models of the LTR-UTR boundary. A complete model for the TR is created by linking our model with the Fontenla et al. (1990, 1991, 1993, 2002) models for the LTR. We compare our models to observations, including the intensity of emission given at typical UTR lines, temperature-sensitive diagnostic line ratios, and the differential emission measure (DEM), which is a way to measure the amount of emitting material as a function of temperature. Figure 2 shows the observed DEM used here, along with measurements done by Landi and Chiuderi Drago (2003).

Figure 2. Observed DEM curve for the solar atmosphere, 104 K < T < 107 K, taken from Landi and Chiuderi Drago (2003). Gridlines were added at log T = 5 and log T = 6 to show the section of the DEM we are interested in reproducing.

This study is a continuation of the work of Oluseyi et al. (1999a, b), who showed that quasi-static, "lukewarm" loop structures with temperature maxima between 5x10⁵ K and 9x10⁵ K, satisfying observationally motivated initial conditions can account for UTR emissions and may contribute substantially to LTR emission via thermal conduction (Oluseyi et al. 1999a, b). Their study was motivated by the observation of unexpected diffuse emission spanning the entire solar disk in images of the solar atmosphere taken in the 171-175 Å bandpass by MSSTA. Emission in that bandpass is dominated by Fe ix/x lines, which are most efficiently produced by active plasma between 1x10⁶ K and 1.1x10⁶ K. However, subsequent observations showed that, because two cooler oxygen lines existed within that particular bandpass, the diffuse emission is actually from structures in the UTR.

Figure 3. TRACE H Lya image of the LTR.	Figure 4. TRACE Fe ix/x 171-175 Å image of the UTR.	Figure 5. Superposition of figures 3 and 4, with white areas showing correlation.

Here, we present new observations that justify correlations between the structures of the LTR and those of the UTR that are apparent in two images taken by the Transition Region and Coronal Explorer (TRACE). Figure 3 shows the LTR, Figure 4 the UTR, and Figure 5 the result of setting the contrasts of both photographs to maximum and then superimposing. Bright areas in Figure 5 show overlap of structure. We believe this is evidence that at least some structures in the LTR and UTR are interconnected; hence, we will create models, located in the UTR, which extend the very successful Fontenla et al. (1990, 1991, 1993, 2002) models of the LTR.

Figure 6. An MSSTA image of the solar corona in the 171-175 Å bandpass showing loop structures (left); a sketch of the loop model used in this analysis (right).

(1)

where s is the distance along the loop, F_c is the conductive heat flux per unit cross-sectional area, e is the energy input per unit volume, and is the radiative energy loss per unit volume (Figure 6). The loops are assumed to have constant cross-section, an assumption supported by numerous observations by Oluseyi et al. (1999b). In a sufficiently hot plasma, where the temperature of the electrons is >10⁵ K, we can assume full ionization and apply the classic Spitzer conductivity,

(2)

where k ~10^-6. Of all three Fontenla et al. (1990, 1991, 1993, 2002) models (A, C, and F), the lowest value of is model A’s 8.996x10⁴ K. Therefore, we can assume that the Spitzer conductivity equation is appropriate for all loops. For the radiative loss function L (T) (ergs cm³ s^-1), we use the approximation comprised of power laws of the form

, (3)

each valid on a sub-interval of the relevant temperature range, where and M are constants, joined continuously. Since the loops we consider are typically small (L < 10⁹ cm), we can assume that all loops are shorter than the gravitational scale height, and hence at constant pressure along the full length of the loops. This allows us to write the equation of state as

constant. (4)

Upon substitution of (3) and (4) into (1), we get

, (5)

(6)

For our models it is assumed that the temperature maximum, where the conductive heat flux disappears, is located at the loop apex; i.e. L = s(T_MAX), and F_c(T_MAX) = 0. To get an expression linking the temperature with distance along the loop, (2) can be integrated after a separation of variables to

In making sure that the loops are not geometrically degenerate (loop diameter very near loop half length) we impose a one-to-10 ratio between the loop diameter and the loop half-length. The resulting loop diameters are about 10⁷–10⁸ cm. However, to account for the lack of direct observations of lukewarm loops, the thickness of the structures must be below the one arc second resolution limit of the various instruments that have taken exposures of the TR. Hence, while we impose the above-mentioned one-to-10 ratio, we set the diameters that exceed one arc second (~7 x 10⁷ cm) to one arc second. The diameters directly affect the projected area of the loops and therefore the calculated absolute intensities. An underestimation of a loop diameter would cause an underestimation of the projected area of the loop, and therefore an overestimation of all absolute intensities from the loop. By similar reasoning, an overestimation of the loop diameter would cause an underestimation of all absolute intensities from the loop. This means our derived absolute intensities are only loosely constrained, but the ratios of them are unaffected. Ideally we would like to place a restriction on the loop half-lengths for a similar reason (L < 5´ 10⁸ cm). However, because we only have one free parameter, e, we can only restrict either the maximum temperature or the loop half-length. We have decided it is more important to keep the maximum temperatures above 2.5´ 10⁵ K due to stability issues, rather than keep the loop half-lengths below 1.5´ 10⁹ cm, which is three times the preferred value. Additionally, we check each loop for a non-negative radiative flux, F_R = e L – F_c0, because we believe it is possible for a loop to satisfy all restrictions mentioned above and yet have a negative flux, which is physically impossible.

We carried out the necessary computations with the programmable math software package Maple. To derive a loop model of a certain type for a given e , we first bisected the radicand from (9) to find the temperature at which the conductive heat flux disappears (dT/ds = 0). Equation (10), with , was then evaluated for s(T), T₀ < T < T_MAX, using Maple’s integration command int, and the loop half-length L, temperature and density profiles (figures 7 and 8), radiative flux, and the conductive to radiative flux ratio were determined for the loop.

In a preliminary study, we modeled quasi-static loop structures using the temperature and conductive flux as given by the corresponding Fontenla et al. (1990, 1991, 1993, 2002) models (A, C, and F) with the LTR-UTR boundary as boundary conditions. The pressures from the Fontenla et al. (1990, 1991, 1993, 2002) models were used as an additional constraint; hence, for each loop, e becomes the only free parameter. The loops were named according to the model type and peak temperature, which ranges between 3´ 10⁵ K and 9´ 10⁵ K. For example, the loop A3 uses the values from the Fontenla et al. (1990, 1991, 1993, 2002) model type A and has a peak temperature of 3´ 10⁵ K. Table 1 lists the parameters and some derived values for each of these loops.

Emission at 13 major O iii, O v, O vi, and Ne v lines within the temperature range were calculated using the emissivity values as given in Landini and Fossi (1990) with 25 data points spaced evenly along the temperature variation for each loop half-length. For each loop, we computed the amount of disk coverage necessary at each O v and O vi line to match the observed intensities from Vernazza and Reeves (1978). To further test the accuracy of our loops, we also compared four temperature-sensitive diagnostic line ratios: O v (629.7 Å /172.17 Å), O vi (1031.95 Å /173.03 Å), (1031.95 Å /184 Å), and (150.1 Å /184 Å).

For the four loop models that match data exceptionally well, we computed the respective DEM given by with 250 data points spaced evenly throughout the temperature variation in each loop half-length. Lastly, we plotted the temperature maxima versus the pressure multiplied by the loop half-length for our loops, to see if they match the scaling law proposed by Rosner et al. (1978).

Figure 7 . Temperature profiles for representative models

Figure 8 . Density profiles for representative models

Figure 9 . Plot of emissivity for oxygen lines used in our analysis, as a function of log T. Shown in pink is the O vi 629.7 Å line; it is much brighter than the others.

The results from the four loop models and four temperature sensitive diagnostic line ratios are reported in Figure 10. The calculated DEMs from the four loop models that match data exceptionally well, A6, A7, C8, and C9 are shown in Figure 11. The temperature maxima versus pressure multiplied by loop half-length for our loops are reported in Figure 12, and show remarkable consistency with the scaling law proposed by Rosner et al. (1978).

Figure 10 . A plot of the residuals for all computed line ratios. The residual is the computed ratio divided by the observed ratio; a residual of one would denote a perfect match.

Figure 11. DEM graphs of representative models.

Figure 12. A plot of the scaling law for representative models. The scaling law relates the pressure, temperature maximum, and loop half-length. If the scaling law is correct, then any two of the three parameters would uniquely define the third.

Since the Fontenla et al. (1990, 1991, 1993, 2002) models, and hence their derived values, are based on separate regions of the solar atmosphere, and the Malinovski and Heroux (1973) measurements are for full disk, we do not expect our numbers to match precisely with theirs. Rather, we take into account the inherent temperature range of each region (bright network more hot than average quiet sun, etc.) to predict if we overestimate or underestimate the values given by Malinovski and Heroux (1973). For example, an F loop based on values from a very bright network would consist of plasma at relatively high temperatures. This means that, for F loops, lines such as O vi 1037.65 Å, which are produced most efficiently at lower temperatures, will have lower emission than that measured by Malinovski and Heroux (1973). Given this interpretation, our results show reasonable agreement with their measurements.

We had originally hoped that the DEM curves would be shaped such that it would be possible to, by having a certain combination of these loops, reproduce the observed DEM curve in the temperature range of the UTR. Unfortunately, it is obvious that, because none of our models have a DEM with an upturn for T £ ~3´ 10⁵K, there is no way we can reproduce the desired upturn with only a combination of these loops (Figure 2).

Oluseyi et al. (1999a, b) showed that there are three different domains of the e -- parameter space that can yield loop models which match observations well. Of the three resulting classes of loops, the radiative type loops (radiative flux at least two times bigger than conductive flux) yielded DEM’s which contain the upturn we want. Since our study fixed two of the three parameters, we are not able to explore parameter space as fully. None of our loops, based on the radiative flux to conductive flux ratio, are radiative. Moreover, even though the Fontenla et al. (1990, 1991, 1993, 2002) models correctly describe the LTR, their derived parameter values at the LTR-UTR boundary are not the only potentially valid ones. Hence, our failure to reproduce the DEM curve does not disprove the existence of loop structures in the UTR. Indeed, they have been observationally confirmed to exist. It may be possible for these same structures, with different boundary conditions, to be responsible for emission in the UTR, or some other type of structure to be responsible for most of the observed emission.

In deriving our loop model, we only used constraints as suggested by measurements. This means our study can be improved with more realistic modeling conditions, one of which is to relax the constraint of a constant volumetric energy input. The relevant equations (1)-(9) will only be slightly complicated upon the replacement of e , the constant volumetric energy input, with, an exponentially decreasing volumetric energy input with scale height s_H. After such a replacement, however, (10) will become recursive in s(T), and advanced numerical techniques will have to be employed to derive models from it.

Conclusion

This study is a small step in attempting to explain the observed solar UTR with the "lukewarm" loop (LWL) structures as proposed first by Oluseyi et al. (1999a). Our LWL models were able to match the diagnostic line ratios very well, and gave disk coverages that more closely matched observed values than those given by the original models generated by Oluseyi et al. (1999a). Even though our models failed to reproduce the DEM curve in the temperature range of the UTR, as we have noted earlier, this by no means disproves the existence of LWL structures. Most importantly, we have discovered that there does exist a unique class of loops, namely those whose main mode of heat loss is radiation (as compared with conduction), which will always give a DEM with the upturn our models lacked. It is possible that the strict boundary conditions we impose in order to match our models to the successful LTR models of Fontenla et al. (1990, 1991, 1993, 2002) prevent us from reproducing the DEM. Oluseyi et al. (1999b) showed that there are at least three classes of LWL structures. All of these have been observed to exist; thus, the observed DEM likely arises from contributions from all three classes, while our models all belong to the same the class of structures (where radiation and conduction both contribute significantly to heat loss). Additional components such as the UTR component of coronal structures, dynamically evolving structures, and open-field structures (plasma structures that are only rooted at one end, as opposed to LWLs which are rooted on both ends) are also likely contributors.

The excellent match the LWL structures have with observed diagnostic line ratios indicates that they likely dominate the UTR and that the distribution of plasma with temperature mirrors that of the LWLs. We hope that this research will serve to establish LWLs as a potential main contributor to the observed UTR emission, and thereby encourage more research on them and on the UTR — until eventually even this most tenuous and elusive part of the solar atmosphere can be understood.

The author would like to acknowledge Lawrence Berkeley National Laboratory, especially the Center for Science and Engineering Education (CSEE), its High School Student Research Participation Program, Dr. Rolland Otto, and Mr. Paul Robinson for making this work possible. The author would also like to thank Dr. Robert Cahn, Mr. Robert Hasson, Mitch Garcia, Mr. David Locke, and Nate Yuen for helpful discussions about this project. Lastly, the author would like to express her great appreciation to Dr. Hakeem Oluseyi for volunteering many hours over the course of this research project, and for his willingness to explain things for the 10th time. This work was funded by the Department of Energy.

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