Analytical model of static coronal loops
J. Dudík^{1}  E. Dzifcáková^{1,2}  M. Karlický^{2}  A. Kulinová^{1,2}
1  Department of Astronomy, Physics of the Earth and Meteorology, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská Dolina F2, 842 48 Bratislava, Slovak Republic
2  Astronomical Institute of the Academy of Sciences of the Czech Republic, Fricova 298, 251 65 Ondrejov, Czech Republic
Received 9 September 2008 / Accepted 23 March 2009
Abstract
By solving the energyequilibrium equation in the stationary case, we derive analytical formulae in the form of scaling laws for nonuniformly heated and gravitationally stratified coronal loops. The heating is assumed to be localized in the chromosphere and to exponentially decrease with increasing distance along the loop strand. This exponential behavior of the heating and pressure profiles implies that we need to use the meanvalue theorem, and in turn fit the meanvalue parameters of the scaling laws to the results of the numerical simulations. The radiativeloss function is approximated by a powerlaw function of the temperature, and its effect on the resulting scaling laws for coronal loops is studied. We find that this effect is more important than the effect of varying loop geometry. We also find that the difference in lengths of the different loop strands in a loop with expanding crosssection does not produce differences in the EUV emission of these strands significant enough to explain the observed narrowness of the coronal loops.
Key words: hydrodynamics  Sun: corona  Sun: UV radiation  stars: coronae
1 Introduction
Since the first observations of the solar corona in the extreme ultraviolet (EUV) and soft Xrays, it has been undeniably clear that the corona is a highly structured environment. The basic structural blocks observed mainly in active regions are coronal loops  thin, archlike, or elongated cylindrical structures delineating magnetic field lines. Coronal loops are anchored in the solar chromosphere at one or both ends. The presence of EUV and Xray emission from these coronal loops means that the loops are hot  their temperatures reach several million K. The fact that the physical mechanism(s) heating the corona has still not been successfully identified constitutes the coronal heating problem (e.g., Klimchuk 2006).
One approach to constraining the coronal heating problem is the direct physical modeling of the temperature and density structure of the coronal loops. Analytical models are an invaluable tool for accomplishing this task, even if they employ a number of simplifying assumptions. The most frequent assumption is that of static energy equilibrium. Further assumptions include the uniform pressure throughout the loop (e.g., Kuin & Martens 1982; Chiuderi et al. 1981; Rosner et al. 1978; Craig et al. 1978; Martens 2009) and uniform energy input (Vesecky et al. 1979), while others relax one or both of these assumptions (e.g., Aschwanden & Schrijver 2002; Wragg & Priest 1979; Serio et al. 1981). The stability of the static solutions have been extensively studied (e.g., Craig et al. 1982; Hood & Priest 1979,1980; Chiuderi et al. 1981; Antiochos 1979). Aschwanden & Tsiklauri (2009) demonstrated the feasibility of analytical models even for nonstatic situations corresponding to impulsively heated loops with subsequent cooling.
The usual solution to the energy balance equation in the static case is written in the form of scaling laws due to the overspecification of the boundary conditions (Martens 2009). The three most seminal papers on scaling laws are those of Rosner et al. (1978), Serio et al. (1981), and Aschwanden & Schrijver (2002). The first two of these papers assume a simplified radiativeloss function in the form of the temperature powerlaw with fixed parameters, while the latest uses the radiativeloss function of Tucker & Koren (1971). The effect of the radiativeloss function parameters on resulting scaling laws was studied by Kuin & Martens (1982) and Martens (2009), although only for the loops with uniform pressure. None of these papers treat the effects of varying loop geometry. However, the effect of the varying loop crosssectional area on the temperature and density profiles of the coronal loops has been extensively studied (Aschwanden & Schrijver 2002; Martens 2009; Vesecky et al. 1979) and found to be small, even though the changes in temperature and density profiles act cumulatively to increase the differential emission measure near the loop apex (Vesecky et al. 1979).
We derive new analytical scaling laws for nonuniformly heated and gravitationally stratified coronal loops. These scaling laws are explicitly dependent on the parameters of the radiativeloss function, making them the most general scaling laws for coronal loops existing to date.
This paper is organized as follows. The assumptions and derivation of new scaling laws for coronal loops is given in Sect. 2. Discussion of different effects on the equilibrium solutions or resulting EUV emission is given in Sect. 3, which includes studies of the effect of the loop geometry (Sect. 3.2), the effect of the radiativeloss function parameters (Sect. 3.3), and the effect of the expanding loop crosssection on the EUV emission in a given TRACE filter (Sect. 3.5). Our conclusions are summarized in Sect. 4.
2 Hydrostatic scaling laws
2.1 Hydrostatic equations and assumptions
To construct an analytical model of a coronal loop, several assumptions about the nature of energy equilibrium, chemical composition, and gravitational stratification of the solar corona must be made. These assumptions are discussed below.
For stellar coronae, the gas pressure is significantly higher than the radiation pressure. The total kinetic pressure p is then related to the electron density
and (electron) temperature T by the equation of state
where JK^{1} is the Boltzmann constant and q is the particle fraction number per one electron. The value of q is given by the chemical composition and ionization equilibrium in the star's corona. In the simplest case of fully ionized hydrogen plasma, q = 2. This value has been used by previous authors (Aschwanden & Schrijver 2002; Rosner et al. 1978; Serio et al. 1981).
The hydrostatic balance equation
together with the equation of state in Eq. (1) gives the pressure stratification of the solar atmosphere. Here and represents the solar radius and surface gravity, respectively, kg is the mass of a hydrogen atom, and is the hydrogen mass fraction per one electron. The last quantity is related to the mean molecular weight as . For solar corona, . The pressure p at height h above the solar photosphere is given by the solution of Eq. (2)
Here h_{0} denotes the height of the upper chromosphere  transitionregion boundary and is the pressure scale height
If the nearisothermal approximation ( ) holds, the previous equation reduces to
where the term is usually very close to unity, since .
For a nonisothermal atmosphere, the following analytical approximation is applicable (Aschwanden & Schrijver 2002)
where the correction factor is in general a function of the temperature and also other loop parameters, but is not a function of the height h (Aschwanden & Schrijver 2002, Eqs. (26) and (27)).
Since the magnetic field usually dominates the force balance in the solar corona, it can be safely assumed that the crossfield transportation effects are negligible and the frozenin and forcefree approximations hold well in this environment. Hydrostatic balance along a given magnetic field line (loop strand; Martens 2009) can then be expressed in terms of the pressure and temperature dependence p = p(s), T = T(s), where s is the position along the field line, measured upwards from the photospheric footpoint s = 0 to the highest position on the field line (apex) s_{1}. The correspondence between p(s) and p(h) is given by the field line geometry, i.e., the mapping .
The energy balance along a magnetic field line is determined by the balance of energy sources and sinks. In the stationary case it is given by the equilibrium between radiative losses ,
heating sources ,
and the divergence of thermal conductive flux
,
which can act both as a source or as a sink, depending on the temperature structure. The equation of energy balance in the stationary case is
For the purposes of this study, we assume a simple onepiece powerlaw radiativeloss function (e.g., Kuin & Martens 1982; Martens et al. 2000). For a Maxwellian plasma, it has the form
where and are two parameters characterizing the dependence of the radiativeloss function on temperature T. This radiativeloss function is similar to the one used by Rosner et al. (1978) and Serio et al. (1981) in deriving their respective scaling laws. The values of Wm^{3} K^{1/2} and (Kuin & Martens 1982) are commonly assumed in deriving these scaling laws (e.g., Priest 1982). Here we relax these assumptions by treating both and as parameters of the radiativeloss function.
The form of the volumetric heating function
is still unknown. It depends critically on the assumed physical heating mechanism(s). Here we make use of the parameterized form of the heating function introduced by Serio et al. (1981) and implemented by Aschwanden & Schrijver (2002) as
This type of the heating function has two parameters: , the volumetric heating rate at the position s_{0} (corresponding to the height h_{0}) of the chromospheric footpoint of the magnetic field line, and , the exponential heating scalelength measured along the field line. Obviously, the maximum of the heating function is localized at the footpoint s_{0} and corresponds to the physical heating mechanisms that release heat energy mainly in the transition region and lower corona.
Thermal conductive flux
at position s along a given magnetic field line is given by (Spitzer 1962)
where Wm^{1} K^{7/2} is the Spitzer thermal conduction coefficient. Since thermal conduction across the magnetic field can be neglected, this equation allows one to solve Eq. (7) as a onedimensional problem.
Further assumptions about the thermal conduction profile along a magnetic field line are
which mean that the magnetic field line is thermally isolated and its temperature profile T(s) is monotonic and has extrema only at both the footpoint and apex. These assumptions are necessary for integrating the temperature profile in both the temperature and spatial domains. In terms of a coronal loop, the assumptions given by Eqs. (11), (12) and (13) translates into a temperature maximum at the loop apex and a temperature minimum at the loop footpoint. The monotonical increase in temperature from footpoint to apex is ensured by the assumption in Eq. (13). The assumption in Eq. (11) means that the thermal conduction from the coronal loop into the chromosphere is negligible, i.e., the loop does not contribute to the heating of the underlying chromosphere. Equation (12) is the necessary, although not sufficient condition for loop symmetry. For the purposes of this paper, a geometrically symmetrical coronal loop of halflength L=s_{1} heated equivalently at both footpoints will be assumed. The loop is treated as a slender fluxtube along the chosen magnetic field line, which defines the loop geometry, i.e., the loop is represented by a onedimensional loop strand (Martens 2009). The effect of the varying loop crosssection is neglected. This question is discussed later in Sect. 3.5.
2.2 Derivation of the scaling laws
Equation (7) can be integrated to obtain analytical formulae describing coronal loops. The assumptions (11), (12), and (13) allow us to integrate from the footpoint s_{0} to the loop apex s_{1} = L in both the temperature and geometrical domains. The derivation here proceeds in a way analogous to Priest (1982).
First we rewrite the pressure stratification of the solar atmosphere given by Eq. (5) or Eq. (6) in terms of the loop coordinate s and simplify it to the following form, which is equivalent to Eq. (9):
where denotes the pressure scale length measured along the coronal loop. The discussion about the relation of to the pressure scale height is given in Sect. 3.1.
In the onedimensional form of Eq. (7), the divergence operator
.
changes to
.
We multiply this equation by the factor
,
then substitute into this equation the Eqs. (8), (9) and (10), and then substitute the electron density
by pressure according to Eqs. (1) and (14) to obtain
Direct integration of this equation over the temperature range from the apex (coronal) temperature T_{1} to the lower transitionregion temperature T_{0} is impossible, since the pressure scalelength is directly dependent on the temperature and height profiles, i.e., (cf. Eq. (35)) and is not in general a natural number. However, the meanvalue theorem (Appendix A) can be used to obtain the following expression
where and are constants to be determined. The lefthand side of Eq. (16) equals zero because of the assumptions (11) and (12).
In the case of a typical coronal loop,
.
If
,
then all terms containing T_{0} can be neglected. In the case of
,
the (coronal) loop apex temperature T_{1} would be strongly coupled to the temperature of the chromosphere. In the following,
will always be assumed. Neglecting T_{0} immediately results in the following expression for the base pressure
To obtain a second relation describing the coronal loop, Eq. (15) is integrated again from T_{1} to a temperature T', , i.e., as the function of the integral boundary. Substituting for from Eq. (17), using the meanvalue theorem and rearranging the equation leads to
where and are a pair of coefficients that are in general functions of T'. In the following, the expression
will be used. Dividing the Eq. (18) by , extracting the squareroot, and integrating it again from T_{0} to T_{1} in the temperature domain and from s_{0} to L in the corresponding spatial domain will, after substituting x = T'/T_{1}, lead to
where the expressions , , and have been used. For , and . The represents here the mean value of when integrated over s in the entire coronal portion of the loop. Similarly, stands for the mean value of . This equation is analogous to Eq. (6) of Kuin & Martens (1982), which relates the radiativelossweighted values of temperature T_{i} and density n_{i} in a uniformly heated loop with homogeneous pressure. In our case, Eq. (20) relates the apex temperature T_{1} to the base pressure p_{0} when assuming a pressurestratified loop and nonuniform heating.
The value of T_{0} with respect to T_{1} can again be neglected. The integral function
which exists only if , is bounded and easily calculated numerically, despite the divergence of the subintegral function for . We note that P is in general a function of x, since and are functions of T'. However, since little assumption about the temperature profile T(s) has been made, there is no information about the function P(x). Thus, P is assumed to be constant for to some values and :
Figure 1: Top: the function, plotted for and . Bottom: the function plotted for the same range in . 

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The condition
results in formal bounding for the heating scaleheight:
The shape of the function is depicted in Fig. 1. The special case of , corresponding to the scaling laws of Rosner et al. (1978), was already derived by Priest (1982, p. 2378).
If the set of variables (L_{0},
,
), where
L_{0} = Ls_{0} is considered to be independent and T_{1} and p_{0} to be dependent variables, the scaling laws (17) and (20) can be easily rewritten to obtain the following expressions for T_{1} and p_{0}:
Alternatively, the scaling laws can be written for the choice of as independent and as dependent variables, i.e., in the form compatible with Eqs. (29) and (30) of Aschwanden & Schrijver (2002). In this form, the scaling laws become
while the scaling law expressions S_{1} and S_{2} in our version (DDKK) take the form
=  (28)  
=  (29) 
From Eq. (26), it is clear that the dependency of base pressure p_{0} on the apex temperature T_{1} is modified by the presence of the radiativeloss function parameter . For , the temperature exponent is equal to 3. The dependency of p_{0} on S_{1} of the inverse thirdpower is retained, although the S_{1} itself depends on .
Equation (22) can be used to obtain
where . Defining , , , and gives
Table 1: Bestfit model coefficients in scaling laws expressions (32) and (31).
This form of the scaling laws for coronal loops is only a formal one, and for the completion one needs to determine the values of P, P', , , , and . Unfortunately, without directly assuming the function T(s), their values cannot be determined analytically. Aschwanden & Schrijver (2002) found good analytical approximations of the function T(s) that depend on the geometrical properties of the loop, i.e., values of L and , and also the apex temperature T_{1} (cf. their Eqs. (19)(21) and Table 1). This means that the values of P, P', , , , and can depend on the loop parameters L, , and T_{1}, i.e., they can change from loop to loop.
At this point an adequate approximation is required. The simplest assumption that can be made is that these parameters are constants for the entire ranges of L and , while they (obviously) must change with the apex temperature T_{1}. In other words, we assume that the temperature profile does not change much with L and . For this assumption, one can expect that the value of P will be close to unity, since P must approach unity for low values and shorter L, where the scaling laws of Rosner et al. (1978) hold. To find the values of P, P', , , , and a numerical method for solving Eq. (7) with the appropriate boundary conditions must be invoked. The values of P, P', , , , and can then be determined by direct fitting of the scaling laws (26) and (27) to the results of numerical simulations, for the assumption that is the pressure scalelength corresponding to the apex temperature T_{1}.
Figure 2: Scaling law for the base pressure p_{0} (Eqs. (26) and (31)) and base heating rate (Eqs. (27) and (32)) as a functions of L, , and T_{1}, for , and Wm^{3} K^{1/2}. Lines corresponding to a given are terminated at points where . 

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To do this, we used the numeric code developed by Aschwanden & Schrijver (2002), which is part of the hydro package of the SolarSoft environment (Freeland & Handy 1998), running as a layer on top of the Interactive Data Language (IDL). The code enables one to choose from several implemented radiativeloss functions and then calculate solutions to Eq. (7). The chosen values of T_{0} and T_{1} together with Eqs. (11) and (12) represent the boundary conditions of a given loop. A fixed choice of T_{1} removes the difficulty of being a function of apex temperature. The value of s_{0} is determined by the height of the chromosphere together with the assumed loop geometry. We used the values of h_{0} = 1.3 Mm and K for the upper chromosphere, and L and acted as independent variables. The code returns the values of and p_{0}. Assuming a semicircular loop geometry and executing the code multiple times for different choices of L and , we constructed a numerical grid of and p_{0} as functions of L and . The values of P, P', , , , and were then determined by the LevenbergMarquardt method (Marquardt 1963), which minimizes the difference between the grid of numerical models and the scaling laws given by Eqs. (26) and (27) together with Eqs. (31) and (32). We indicate the parameter values for the choice of , Wm^{3} K^{1/2} (Kuin & Martens 1982), and 4 Mm Mm, which is the same parameter space used by Aschwanden & Schrijver (2002). Since the computed numerical values of p_{0} and span two and four orders of magnitude, respectively, the minimization was performed in logarithmic space. The results are summarized in Table 1.
For , the points in parameter space were excluded from the fitting procedure, because the numerical code often failed to converge here. The fitting accuracy of scaling laws is 40% for and 15% for p_{0}. The accuracy is higher for lower values and rapidly decreased towards the boundary. This is because for higher values, the function T(s) changes rapidly (cf. Fig. 9 of Aschwanden & Schrijver 2002). The accuracy of the numerical solutions is less than 2%.
The resulting dependency of p_{0} and on L, , and T_{1} is plotted in Fig. 2. In Fig. 3, the scaling laws are compared with numerical solutions and previously derived scaling laws of Rosner et al. (1978), Serio et al. (1981) and Aschwanden & Schrijver (2002). These plots are compiled in the same way as Figs. 46 of Aschwanden & Schrijver (2002) to facilitate direct visual comparison.
Figure 3: Comparison of ``DDKK'' scaling laws (Eqs. (26), (27) together with Eqs. (31) and (32)) with the numerical solutions ( top row) and previously derived scaling laws of Rosner et al. (1978, second row), Serio et al. (1981, third row), and Aschwanden & Schrijver (2002, bottom row). Average ratios and standard deviations are given in each figure for each temperature separately. 

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Figure 4: Comparison of the numerical results for the base pressure for elliptical and circular loop geometries. Left column: elliptical, portraitoriented loops vs. semicircular loops. Right column: elliptical, landscapeoriented loops vs. semicircular loops. Computed numerical values are denoted by the ``+'' sign. Different lines in plots stands for different values of heating scale length . From left to right: Mm, 5.5 Mm, 7.7 Mm, 10.7 Mm, 14.9 Mm, 20.7 Mm, 28.7 Mm, 40 Mm, 55.5 Mm, 77.2 Mm, 107.3 Mm, 149.1 Mm, 207.2 Mm, 287.9 Mm and400 Mm. Maximum plotted values are approximately 1.9. 

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Figure 5: Comparison of the numerical results for the base pressure p_{0} and base heating rate for the two radiativeloss functions and with Wm^{3}, , Wm^{3} K^{1/2} and . The black lines in both pictures correspond to the estimated ratio values. Computed numerical values are denoted by the ``+'' sign. Gray lines connect points with constant values of the heating scale length . From left to right: Mm, 5.5 Mm, 7.7 Mm, 10.7 Mm, 14.9 Mm, 20.7 Mm, 28.7 Mm, 40 Mm, 55.5 Mm, 77.2 Mm, 107.3 Mm, 149.1 Mm, 207.2 Mm, 287.9 Mm and400 Mm. Maximum plotted values are approximately 1.0. Higher values are not plotted due to abundance of numerical errors for the case of . 

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Figure 6: Same as in Fig. 5, but for the two radiativeloss functions and with Wm^{3} K^{2/3}, . Maximum plotted values are about 1.9. 

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Slightly higher accuracy can be achieved by relaxing the assumption that ,
,
,
and
are constants. If these parameters are allowed to vary slightly with L and ,
the scaling law expressions in Eqs. (31) and (32) can be expanded into Taylor series. Neglecting terms higher than linear leads to
where the expressions , and have been used. Once again, we assume that P, P', , , , and are constants to be determined by fitting the results of numerical simulations. The fitting accuracy is 2535% for and 79% for p_{0}. We note that the scaling law expressions (33) and (34) differ from the empirical scaling law expressions of Aschwanden & Schrijver (2002, Eqs. (35) and (36) therein). The correction terms in our scaling law expressions (33) and (34) are strongly dependent on L_{0} and . The correction terms in the empirical scaling law expressions of Aschwanden & Schrijver (2002), which contain a weaker dependency on L_{0} and , allow for higher fitting accuracy. This accuracy is reported to be 5% by these authors. However, inspection of their Fig. 4 bottom left panel suggests that the accuracy is closer to 10% for . This accuracy could also be achieved in our case, if we would fit their scaling law expressions (35) and (36) with the results of numerical simulations. We note that the difference between the numerical simulations of Aschwanden & Schrijver (2002) and the numerical simulations computed here is in the type of radiativeloss function addopted. While they used the radiativeloss function computed by Tucker & Koren (1971), we use the onepiece powerlaw approximation of this function given by Kuin & Martens (1982) (cf. Fig. 7 in Aschwanden & Schrijver 2002). We conclude that the correction terms in scaling law expressions containing terms linear in , and , as suggested by Aschwanden & Schrijver (2002), are more appropriate to reproducing the numerical simulations, although they are merely empirical, since they have little physical justification. However, our derivation justifies the d_{0} and e_{0} parameters of Aschwanden & Schrijver (2002) in terms of the parameter P' and the integral function in Eq. (21).
3 Discussion
3.1 Hydrostatic scaleheight
We note that the pressure scalelength
is not equal to the pressure scaleheight
.
The pressure scalelength
is defined as the distance along the loop between points of pressure
p(s_{0}) = p_{0} and
.
In contrast, pressure scaleheight
is the vertical distance between the points s_{0}, corresponding to height h_{0}, and
,
corresponding to height
.
Since for a given loop geometry, the relation h = h(s) and the inverse relation s = s(h) is known, using the Eqs. (6) and (14), we can write approximately
With the knowledge of the loop apex height h_{1}, we can estimate to be
This equation was used in constructing the grid of numerical models in Sect. 2.2 for the assumption of semicircular loop geometry.
3.2 Elliptical loops  a case study
In this section we perform a case study of the importance of different loop geometries on the resulting distributions of base pressure p_{0} and volumetric heating rate as functions of (L, , T_{1}). The effects of varying loop geometry on hydrostatic loop solutions has not been studied previously the literature. Any possible effects on the resulting emission models were neglected (e.g., Schrijver et al. 2004).
We computed a grid of numerical models identical to those in Sect. 2.2, but for loops of semielliptical shape. The semimajor to semiminor axis ratio of these loops was chosen to equal 2, which corresponds to an eccentricity of . In the first case, the semielliptical loops are oriented in portrait form, i.e., the photospheric footpoint baseline of the semielliptical loop is about a factor of 1.541 shorter than the photospheric footpoint baseline of the semicircular loop with the same halflength L, while the apex is located 1.298times higher. In the second studied case, the semielliptical loops are oriented landscape, i.e., the photospheric footpoint baseline of a such semielliptical loop is 1.298times longer than the baseline of the semicircular loop with the same halflength, while the apex is located times lower.
For the purpose of constructing the grid of numerical models, we modified the and routines of the hydro package by changing the employed h = h(s) function, which is for semielliptical loops given by the elliptic integral of the second kind, together with the component of gravitational force acting along the loop. We computed the grid of numerical models for four apex temperatures T_{1} = 1, 3, 5, and 10 MK. Unfortunately, we found that the numerical grid for the latest apex temperature was impossible to construct, because we encountered serious problems related to the spatial distribution of points s along loops. The problem appears for the entire range of recommended spatial distribution parameters (Aschwanden, private communication) and is left unresolved. However, computations for the first three apex temperatures were not affected by this problem. The results for the base heating rate for these apex temperatures are displayed in Fig. 4. Taking into account that the error in the numerical simulations is less than 2%, we conclude that the effect of elliptical loop geometry is 13% for the base heating rate for loops with . We note that because of Eq. (17), the relative changes in the base pressure p_{0} are expected to scale as the square root of the changes in the base heating rate . Thus, the semielliptical loop geometry causes less than a 6% difference in the base pressure p_{0}.
The total changes
in the radiative output of a loop due to the different loop geometry then scale as
We are unable to study the effects of elliptical loop geometry for higher values of the ratio, since the code fails to converge for more often in the case of semielliptical loops than for semicircular loops. However, the overall trend is evident from Fig. 4. With the increasing ratio, the differences related to geometry start to rise, reaching maxima for Mm for T_{1} = 1 MK. The maxima move progressively to higher values for increasing apex temperature. The effect is also more pronounced for landscapeoriented loops than for portraitoriented loops.
3.3 Effect of radiativeloss function
We now study the effects of different radiativeloss functions on the density and temperature structure of the coronal loops. On the one hand, Eq. (26) together with Eq. (31) show the explicit dependence of the base pressure on the parameters
and
of the radiativeloss function (8). If we assume two different radiativeloss functions,
and
,
we obtain the following expression for the ratio of the resulting base pressures
(37) 
where P_{1} and P_{2} can in general be different values for different radiativeloss functions. Similarly, the ratio of the resulting base heating rates is given by the expression (Eqs. (27) and (32)).
On the other hand, Eqs. (27) and (32) suggest that the volumetric base heating rate does not depend on , and depends only weakly on through the function. This result is confusing, since changes in total radiative output could in general require adjustments to the total energy input so that the assumed energy balance given by Eq. (7) is kept throughout the loop, even if the heating scalelength remains fixed, i.e., the volumetric heating rate could in general be dependent on and .
To study this in more detail, we constructed a grid of numerical models using the radiativeloss functions
and
with
Wm^{3},
and
Wm^{3} K^{2/3},
,
which are the last two parts of the Rosner et al. (1978, Appendix A therein) analytical fit to the radiativeloss function of Raymond & Smith (1977). The first approximation is valid within the temperature range 10^{5.75} K
< T < 10^{6.3} K, and the second is valid for 10^{6.3} K
< T < 10^{7} K. We thus construct the grid of numerical models using the radiativeloss function
for the apex temperature T_{1} = 1 MK and the second radiativeloss function
for the apex temperatures T_{1} = 3 and 5 MK. Considering P = 1, the estimated values of p_{0} with respect to the values of
are
for the cases of T_{1} = 1, 3, and 5 MK, respectively. The estimated ratios of base heating rates for these three temperatures are 1.330, 0.919, and 0.919.
The results of numerical simulations using the two radiativeloss functions and are depicted in Figs. 5 and 6. It can be seen that for the case of , the pressures are close to their estimated values given by Eq. (39), while for , which deviates more significantly from the case , the results for are slightly further from the expected value. In all cases, the differences from the expected values given by Eq. (39) increases with increasing ratio.
The values of the ratios and are wideranging, making the expected ratio values only a rough estimate. There are large deviations from the expected value for both strongly nonuniform heated loops ( ) and large loops ( Mm). The ratio of base heating rates for the large loops approaches unity for all three studied apex temperatures.
The discrepancies between results of numerical models and the expected values can be explained only in terms of the dependence of the T(s) function on the assumed radiativeloss function. For the fixed values of T_{1}, L and , any change in the T(s) function will result in adjustments to the parameters P, P', , , , and then implicitly in p_{0} and . It can then be expected that the discrepancies will be in general larger for the heating rate ratios than for the base pressure ratios, because , (Table 1). However, the exact dependence of the T(s) function on assumed parameters of the radiativeloss function is not known except for the cases of uniform pressure and heating rate (Kuin & Martens 1982) and uniform pressure and heating explicitly dependent on temperature (Martens 2009). The reconstruction of the temperature profile from Eq. (18) is difficult because of the unknown dependence of and on T'. We are thus left with direct construction of grids of numerical models in evaluating the effect of the radiativeloss function on the resulting distributions of p_{0} and .
3.4 Scaling law for apex temperature
In this section we return to Eq. (24), which gives the apex temperature as a function of , and . The choice of , and as independent parameters is more natural from the perspective of modeling coronal emission in EUV and soft Xray (e.g., Schrijver et al. 2004), since the coronal emission in the EUV or soft Xray filter is given by the product of and the filter response (Mok et al. 2005, and references therein), the latter being a function of both the temperature and electron density.
The values of , and appearing in Eqs. (24) and (25) together with the values of and were reconstructed from the values of , and and are listed in Table 2.
Table 2: Coefficients in scaling laws (24) and (25) reconstructed from values in Table 1.
Using the numerically obtained values of , it is possible to reconstruct the apex temperature T_{1} as function of , and . The results are shown in Fig. 7, which suggests that the resulting apex temperatures T_{1} are within 20% of their expected values.
We note that the expression is negative for all studied apex temperatures. The inequality sign in the condition (23) must then be reversed, so that it is always satisfied.
Figure 7: Scaling law for the apex temperature T_{1} as function of L, and (Eq. (24)) reconstructed from the values of Table 2. Points for which the numerical simulations did not converge have been removed form the plot. 

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3.5 Loops with expanding crosssection
We now focus our attention on the effect of the expanding crosssection of the loop. We examine in particular the concept of a loop as an expanding magnetic fluxtube. We utilized the geometry of expanding loops with a buried dipole of Aschwanden & Schrijver (2002), which generalizes the expanding loop model of Vesecky et al. (1979) for an arbitrary large range of loop expansion factors . The loop expansion factor is defined to be , where A_{1} and A_{0} are the apex and photospherical crosssections of the loop, respectively. The geometrical model of the expanding loop of Aschwanden & Schrijver (2002) employs the semicircular geometry in an expanding toroidal shell around the central, semicircular magnetic field line or loop strand. The central vertical cut through the loop containing the central magnetic field line is depicted in Fig. 3.14 right panel of Aschwanden (2004).
Previous authors (Aschwanden & Schrijver 2002; Martens 2009; Vesecky et al. 1979) claimed that the effect of expanding loop crosssection on the temperature profile T(s) is small. However, after considering the expanding geometry of Aschwanden & Schrijver (2002), it is clear that the inward (lowermost) and outward (topmost) field lines or loop strands of the expanding fluxtube have different lengths, and because of Eqs. (24) and (25) they must have different apex temperatures and base pressures; i.e., , . This means that the emission from these two field lines observed in EUV or Xray filters must differ. To study this, we computed the ratio of the inward loop strand to the outward loop strand apex emissions in two TRACE filters, 171 and 195 Å.
The length
and
of the coronal portion of the inward and outward loop strands are given by
where , are the radii of the circles corresponding to these loop strands, , , and and are the heights of the centers of these circles above the photosphere. They are given by
where , and the value of is given by Eq. (16) of Aschwanden & Schrijver (2002). We computed the emission ratios for the values of , Mm^{2} and 2.07 Mm^{2} corresponding to 1 1 (TRACE resolution, Handy et al. 1999) and 1.98 1.98 (SOHO/MDI resolution, Scherrer et al. 1995), respectively.
We assume that L_{0} is the length of the coronal portion of the central strand and the heating parameters and do not change for the entire loop. The computation of the emission ratio proceeds as follows. First, we choose the apex temperature of the central loop strand from the interval MK, 1.5 MK. Equation (27) then gives the base heating rate for the chosen central strand apex temperature. The lengths and are then computed using Eqs. (40). These values were inserted into Eqs. (24) and (25), which provide the apex temperatures , and the base pressures and as functions of , and . The apex densities and were computed using Eqs. (14), (36), (4), and (1). The computed values of , , , and were then used together with the TRACE filter response (Mok et al. 2005) for 171 and 195 Å filters to obtain the emissions in the corresponding filter.
Figure 8: Computed ratios of emission at apexes of the inward (i) and outward (o) loop strands as function of and A_{0}. Left column: emission ratios for the 171 Å TRACE filter, Right column: emission ratios for the 195 Å TRACE filter. Top row: Case of uniform heating, Bottom row: case of nonuniform heating. 

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The emission ratio is expected to increase with increasing , A_{0}, and decreasing L_{0}, because of the increasing difference between and . We also note that if the apex temperature of the central strand is close to the temperature for which the filter response reaches its maximum (1 MK for the 171 Å filter, and 1.5 MK for the 195 Å filter), then the computed emission ratio should be close to unity; this is because the apex temperatures of the inward and outward strands and will correspond to roughly equal values of the filter response. The situation will be different for temperatures where the filter response is decreasing ( MK for 171 Å filter) or increasing ( MK for 195 Å filter), where the difference between the and would increase the observed emission ratio.
We present the results for the case of uniform heating (L = 40 Mm, Mm; Fig. 8 top row) and nonuniform heating (L = 40 Mm, Mm; Fig. 8 bottom row). For both these cases, the ratio of the computed emission is too low to explain the observed narrowness of coronal loops (López Fuentes et al. 2006; Klimchuk 2000; Watko & Klimchuk 2000). The only case where the emission ratio could suffice to explain the observed narrowness of the coronal loops is the case of short ( Mm), uniformly heated loops, since the ratio reaches a factor of 5 or more for these loops. We thus conclude that in general the difference in lengths of the inward and outward loop strands cannot explain the observed spatial profile of coronal loops. For stationary, nearpotential coronal loops obeying the scaling laws, the parameters of the heating function within a magnetic fluxtube with photospheric crosssections corresponding to the TRACE resolution (1 ) must vary rapidly to produce a coronal loop with the observed narrowness.
4 Conclusions
We have derived an analytical set of scaling laws between loop apex temperature T_{1}, base pressure p_{0}, base heating rate , loop halflength L, and heating scale length for geometrically symmetrical, nonuniformly heated, and gravitationally stratified onedimensional loop strands that are thermally isolated with monotonically increasing temperature profile. For analytical treatability, the derivation of the scaling laws employed the meanvalue theorem, which directly implied the need of fitting the analytical results to the numerical simulations under the assumption of constant meanvalue parameters. This strict assumption in turn leads to moderate accuracy of the scaling laws with respect to the numerical simulations. Higher accuracy can be achieved only by employing Taylor expansion or additional empirical terms. The latter provides higher precision, and ensures that the scaling laws of Aschwanden & Schrijver (2002) are the most precise scaling laws existing to date, even though their empirical terms have no analytical background. However, the presence of d_{0} and e_{0} parameters is justified in terms of our parameter P' and the function.
We performed a case study of the semielliptical loop geometries. The difference in loop geometry results only in relatively small changes in p_{0} and , of the order of less than approximately 6% for p_{0} and about twice as much for . The results are more sensitive to the parameters of the radiativeloss function. The base pressure p_{0} depends explicitly on both and , while the dependence of the base heating rate on these parameters is apart from the presence of the function by means of the temperature profile. However, the effect of radiativeloss function is nonnegligible and thus a correct powerlaw approximation to the radiativeloss function must be taken before the scaling laws are applied, e.g., in forward modeling of coronal emission. It is interesting that the base heating rate for long, uniformly heated loops is almost independent of the details of the radiativeloss function.
The effect of an expanding loop crosssection on the resulting EUV emission in TRACE 171 Å and 195 Å filters was studied. We conclude that the different lengths of the inward and outward loop strand are insufficient to explain the spatial narrowness of coronal loops except for the case of small, uniformly heated loops, i.e., if the parameters of the heating function did not vary in space, both inward and outward loop strands would be visible in a given TRACE EUV filter.
Appendix A: Meanvalue theorem
The meanvalue theorem for integration (e.g., Bartsch 1977) states that for the integrable function f(x) and continuous function g(x),
,
there exists a number
,
for which the following relation holds:
We demonstrate the use of this theorem in evaluating the integral J defined by (cf. Eq. (15))
Here we must take into account that because of the assumptions presented in Eqs. (11), (12), and (13), the temperature profile T(s) monotonically increases, which means that there exists a unique and monotonic inverse function s = s(T). Similarly, due to the h = h(s) dependence, there exists a unique h = h(T) function.
Using the meanvalue theorem it is clear that there exists some
for which the integral (A.2) equals to
Since the function h(T) is monotonic, there must exist some and , such that , and defined by
Thus the integral J equals to
where and the parameter is defined as
Equation (36) can be used to convert Eq. (A.6) into the final result
We note that the correction factor depends on apex temperature T_{1}, loop halflength L, and also on heating scale length (Eq. (27) and Table 1 of Aschwanden & Schrijver 2002). The temperature profile T(s) can also be dependent on L and , which means that the , and parameters are functions of , and thus .
Acknowledgements
The authors are thankful to the referee, P. Martens, for his comments that helped to clarify several issues in the manuscript. We are indebted to M. Aschwanden for incorporating the hydro package into the SolarSoft library, thus providing us with a powerful and versatile tool to study the equilibrium solutions for coronal loops. We are also thankful for helpful and lengthy discussions regarding the software. We would also like to extend our thanks to J. Klacka for helpful discussion regarding the function. This work was supported by Scientific Grant Agency, VEGA, Slovakia, Grant No. 1/0069/08, Grant IAA300030701 of the Grant Agency of the Academy of Sciences of the Czech Republic and ESAPECS project No. 98030. J.D. also acknowledges the financial support of the Comenius University Grant No. 414/2008. The Solar and Heliospheric Observatory (SOHO) is a project of international cooperation between ESA and NASA. The Transition Region and Coronal Explorer (TRACE) is a mission of the StanfordLockheed Institute for Space Research, and part of the NASA Small Explorer program.
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All Tables
Table 1: Bestfit model coefficients in scaling laws expressions (32) and (31).
Table 2: Coefficients in scaling laws (24) and (25) reconstructed from values in Table 1.
All Figures
Figure 1: Top: the function, plotted for and . Bottom: the function plotted for the same range in . 

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In the text 
Figure 2: Scaling law for the base pressure p_{0} (Eqs. (26) and (31)) and base heating rate (Eqs. (27) and (32)) as a functions of L, , and T_{1}, for , and Wm^{3} K^{1/2}. Lines corresponding to a given are terminated at points where . 

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In the text 
Figure 3: Comparison of ``DDKK'' scaling laws (Eqs. (26), (27) together with Eqs. (31) and (32)) with the numerical solutions ( top row) and previously derived scaling laws of Rosner et al. (1978, second row), Serio et al. (1981, third row), and Aschwanden & Schrijver (2002, bottom row). Average ratios and standard deviations are given in each figure for each temperature separately. 

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In the text 
Figure 4: Comparison of the numerical results for the base pressure for elliptical and circular loop geometries. Left column: elliptical, portraitoriented loops vs. semicircular loops. Right column: elliptical, landscapeoriented loops vs. semicircular loops. Computed numerical values are denoted by the ``+'' sign. Different lines in plots stands for different values of heating scale length . From left to right: Mm, 5.5 Mm, 7.7 Mm, 10.7 Mm, 14.9 Mm, 20.7 Mm, 28.7 Mm, 40 Mm, 55.5 Mm, 77.2 Mm, 107.3 Mm, 149.1 Mm, 207.2 Mm, 287.9 Mm and400 Mm. Maximum plotted values are approximately 1.9. 

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In the text 
Figure 5: Comparison of the numerical results for the base pressure p_{0} and base heating rate for the two radiativeloss functions and with Wm^{3}, , Wm^{3} K^{1/2} and . The black lines in both pictures correspond to the estimated ratio values. Computed numerical values are denoted by the ``+'' sign. Gray lines connect points with constant values of the heating scale length . From left to right: Mm, 5.5 Mm, 7.7 Mm, 10.7 Mm, 14.9 Mm, 20.7 Mm, 28.7 Mm, 40 Mm, 55.5 Mm, 77.2 Mm, 107.3 Mm, 149.1 Mm, 207.2 Mm, 287.9 Mm and400 Mm. Maximum plotted values are approximately 1.0. Higher values are not plotted due to abundance of numerical errors for the case of . 

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In the text 
Figure 6: Same as in Fig. 5, but for the two radiativeloss functions and with Wm^{3} K^{2/3}, . Maximum plotted values are about 1.9. 

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In the text 
Figure 7: Scaling law for the apex temperature T_{1} as function of L, and (Eq. (24)) reconstructed from the values of Table 2. Points for which the numerical simulations did not converge have been removed form the plot. 

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In the text 
Figure 8: Computed ratios of emission at apexes of the inward (i) and outward (o) loop strands as function of and A_{0}. Left column: emission ratios for the 171 Å TRACE filter, Right column: emission ratios for the 195 Å TRACE filter. Top row: Case of uniform heating, Bottom row: case of nonuniform heating. 

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In the text 
Copyright ESO 2009