Analytical Approximations to Hydrostatic Solutions and Scaling Laws of Coronal Loops | PHYS 4007, Study Guides, Projects, Research of Physics

Download Analytical Approximations to Hydrostatic Solutions and Scaling Laws of Coronal Loops | PHYS 4007 and more Study Guides, Projects, Research Physics in PDF only on Docsity! ANALYTICAL APPROXIMATIONS TO HYDROSTATIC SOLUTIONS AND SCALING LAWS OF CORONAL LOOPS Markus J. Aschwanden and Carolus J. Schrijver LockheedMartin Advanced Technology Center, Solar and Astrophysics Laboratory, Department L9-41, Building 252, 3251 Hanover Street, Palo Alto, CA 94304; aschwanden@lmsal.com Received 2002 February 14; accepted 2002May 22 ABSTRACT We derive accurate analytical approximations to hydrostatic solutions of coronal loop atmospheres, applicable to uniform and nonuniform heating in a large parameter space. The hydrostatic solutions of the temperature TðsÞ, density neðsÞ, and pressure profile pðsÞ as a function of the loop coordinate s are explicitly expressed in terms of three independent parameters: the loop half-length L, the heating scale length sH , and either the loop-top temperature Tmax or the base heating rate EH0. The analytical functions match the numer- ical solutions with a relative accuracy ofd10
2–10
3. The absolute accuracy of the scaling laws for loop base pressure p0(L, sH , Tmax) and base heating rate EH0(L, sH , Tmax), previously derived for uniform heating by Rosner et al., and for nonuniform heating by Serio et al., is improved to a level of a few percent.We generalize also our analytical approximations for tilted loop planes (equivalent to reduced surface gravity) and for loops with varying cross sections. There are many applications for such analytical approximations: (1) the improved scaling laws speed up the convergence of numeric hydrostatic codes as they start from better initial values, (2) the multitemperature structure of coronal loops can be modeled with multithread concepts, (3) line-of-sight integrated fluxes in the inhomogeneous corona can be modeled with proper correction of the hydrostatic weighting bias, (4) the coronal heating function can be determined by forward-fitting of soft X-ray and EUV fluxes, or (5) global differential emission measure distributions dEM=dT of solar and stellar coronae can be simulated for a variety of heating functions. Subject headings: hydrodynamics — stars: coronae — Sun: corona 1. INTRODUCTION Accurate density and temperature models are funda- mental tools to explore the physical processes of plasma heating and cooling in solar and stellar coronae. The tre- mendous increase of imaging data in soft X-rays and extreme ultraviolet (EUV) produced by the Yohkoh, SoHO, TRACE, ROSAT, ASCA, Chandra, and Newton spacecraft have stimulated modeling efforts in an unpre- cedented way. The modeling of coronal loops with hydrostatic solutions, which ensure the basic physical conservation laws of mass, momentum, and energy, how- ever, is computation-expensive with numeric codes, par- ticularly for large sets to model entire stellar coronae. Therefore, appropriate analytical expressions for their density and temperature profiles are highly desirable. At this time, no analytical solutions are known for the hydrostatic equations, except for an approximate temper- ature function in the special case of uniform heating, constant cross section, and zero gravity (Rosner, Tucker, & Vaiana 1978; Kuin & Martens 1982). In this study we use a numerical code to compute some 1000 hydrostatic solutions in a large parameter space, for uniform as well as for nonuniform heating functions, and develop accu- rate analytical approximations by fitting them to the numerical solutions. By the same token, we quantify also the well-known scaling laws of loop base pressure and heating rate as derived earlier by Rosner et al. (1978) and Serio et al. (1981), but with higher precision, and add loop expansion as a parameter. The new analytical for- mulation consists of explicit expressions as a function of five independent parameters and can conveniently be applied to forward-fitting of coronal data and statistical studies of solar and stellar atmospheres (e.g., Schrijver & Aschwanden 2002). The content of the paper includes a definition of the hydrodynamic equations as they are used here (x 2), a brief description of a numerical code that is used to calculate the exact hydrostatic solutions (x 3), the derivation of analytical approximations and more accurate scaling laws (x 4), gener- alizations for very short heating scale lengths, inclined loops, and loops expanding with height (x 5), and a discus- sion of applications (x 6). 2. HYDRODYNAMIC EQUATIONS We define the quantities of the time-independent hydro- dynamic equations used in this study, which have been used with slightly different notations, approximations, assump- tions, and degree of completeness in previous work (e.g., Parker 1958; Rosner et al. 1978; Priest 1982; Mariska et al. 1982; Craig & McClymont 1986; Klimchuk, Antiochos, & Mariska 1987; Klimchuk & Mariska 1988; Withbroe 1988; Bray et al. 1991). The one-dimensional, time-independent (d=dt ¼ 0) hydrodynamic equations involve the equations of mass con- servation, 1 A d ds ðnvAÞ ¼ 0 ; ð1Þ the momentum equation, mnv dv ds ¼
dp ds þ dpgrav dr dr ds
 ; ð2Þ The Astrophysical Journal Supplement Series, 142:269–283, 2002 October # 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A. 269 and the energy equation (expressed in conservative form), 1 A d ds ðnvA½
enth þ
kin þ
grav þ AFCÞ ¼ EH þ ER ; ð3Þ where s is the distance along the loop measured from the solar surface, r is the radial distance to Sun center, AðsÞ is the loop cross section, m is the average particle mass, nðsÞ is the particle density, vðsÞ is the velocity of a single fluid, pðsÞ is the gas pressure, pgravðrÞ is the gravitational pressure,
enthðsÞ is the enthalpy,
kinðsÞ is the kinetic energy,
gravðrÞ is the gravitational potential, FCðsÞ is the conductive flux, EHðsÞ is the volumetric heating rate, and ERðsÞ is the volu- metric radiative loss rate. The mass density (also called  ¼ mn) of a fully ionized gas is defined by mnðsÞ ¼ meneðsÞ þminiðsÞ  lmpneðsÞ ; ð4Þ with mi ¼ lmp the average ion mass (i.e., l  ð10 1þ 1 4Þ=11 ¼ 1:3 for a coronal composition of H : He ¼ 10 : 1), mp ¼ 1:67 10
24 g the proton mass, and the density nðsÞ is assumed to be equal for electrons and ions (n ¼ ne ¼ ni) in a fully ionized gas. The total pressure pðsÞ of a fully ionized gas is defined by the ideal gas law and relates to the (electron) density nðsÞ by pðsÞ ¼ ½neðsÞ þ niðsÞkBTðsÞ  2nðsÞkBTðsÞ ; ð5Þ where kB ¼ 1:38 10
16 erg K
1 is the Boltzmann constant and TðsÞ is the electron temperature. The enthalpy energy
enthðsÞ comprises the heat energy acquired (or lost) at constant volume, plus the work done against the pressure force when the volume changes, and is defined by
enthðsÞ ¼ 5 2 kBTðsÞ ; ð6Þ the kinetic energy
kinðsÞ is
kinðsÞ ¼ 1 2mv 2ðsÞ ; ð7Þ the gravitational potential
gravðrÞ is
gravðrÞ ¼
GMm r ¼
mg R2 r
 ; ð8Þ with the solar gravitation g ¼ GM=R2 ¼ 2:74 10 4 cm s
2 and solar radius R ¼ 6:96 1010 cm. The differential gravitational pressure, used in the momentum equation (2), is dpgrav dr ðrÞ ¼
GMmn r2 ¼
mng R2 r2
 : ð9Þ The next term of the energy balance equation describes the divergence of the conductive flux, which in a one-dimen- sional flux tube model is FCðsÞ ¼
T 5=2ðsÞ dTðsÞ ds   ¼
2 7  d ds  T7=2ðsÞ  ; ð10Þ with  ¼ 9:2 10
7 erg s
1 cm
1 K
7/2 the Spitzer conduc- tivity. The most unknown term is the volumetric heating rate EHðsÞ along the loop, which crucially depends on assump- tions on the physical heating mechanism. Many previous loop models assumed uniform heating, EHðsÞ ¼ const (e.g., Rosner et al. 1978), for sake of simplicity. Here we parame- terize the heating function with two parameters: with the base heating rate EH0 and an exponential scale length sH , as it was introduced by Serio et al. (1981), EHðsÞ ¼ E0 exp
s sH
 ¼ EH0 exp
s
s0 sH
 : ð11Þ While Serio’s base heating rate E0 refers to the photosphere (at s ¼ 0), we introduce a base heating rate EH0 that refers to the same reference height s ¼ s0 as we will refer the base temperature T0, the base pressure p0, and the base density n0. This Ansatz allows us to model nonuniform heating localized above the loop footpoints from arbitrary small heating scale lengths (sH5L) up to the limit of uniform heating (sH4L). Alternative parameterizations of heating functions that are suitable for loop-top heating have been used elsewhere (e.g., Priest et al. 2000; MacKay et al. 2000). The radiative losses ERðsÞ are proportional to the square of the electron density, n2eðsÞ, multiplied with a temperature- dependent function
(T) (Tucker &Koren 1971), ERðsÞ ¼
n 2 eðsÞ
½TðsÞ ; ð12Þ which was approximated by Rosner et al. (1978) by piece- wise power laws [see Appendix A in Rosner et al. 1978 for the definition of
ðTÞ]. For a discussion of other calcula- tions of the radiative loss function and consequences on the hydrostatic solutions see x 4.5. The one-dimensional parameterization of loops with a distance coordinate s involves an angle ðsÞ between the magnetic field line (defining a loop) and the radial direction r. The simplest geometry employs semicircular loops, for which the height hðsÞ in the loop plane relates to the loop distance s by hðsÞ ¼ rðsÞ
R ¼ 2L  sin s 2L
 ; ð13Þ with L the loop half-length. The derivative ðdh=dsÞ defines then the cosine of the angle  used in the momentum bal- ance equation (2), dr ds
 ¼ dh ds
 ¼ cos s 2L
 ¼ cos : ð14Þ For the variation of the loop cross section AðsÞ along the loop coordinate s we follow the line-dipole model of Vesecky, Antiochos, & Underwood (1979). In their model the inner and outer field line of a loop intersect in the lowest subphotospheric point, where the line dipole is buried, while the cross section varies as a sin2 function, expanding by a factor of C from the photosphere to the loop apex. Because we are using a semicircular geometry for the loops, only loops with an expansion factor of  ¼ 2 can be accommo- dated in the geometry of Vesecky et al. (1979). To allow for an arbitrary large range of expansion factors C in semicircu- lar loops we generalize the model of Vesecky et al. (1979) by relaxing the condition of a zero cross section at the sub- photospheric anti-apex point (s ¼
L). We define a general- ized cross section function AðsÞ ¼ A0 sin 2  2 sþ ssub Lþ ssub
 ; ð15Þ where the zero cross section point is located at position 270 ASCHWANDEN & SCHRIJVER Vol. 142 assumption of constant pressure in the RTV and MKB model, while the remaining discrepancy is attributed to dif- ferent approximations in the radiative loss function, as dis- cussed below. We are motivated to employ the same formalism for a larger parameter space, since our Ansatz of the temperature parameterization (eq. [19]) reproduces the numerical solution with extremely high accuracy (d10
3) and has a simpler analytical form than the formulations given in equations (17) and (18). We fitted the analytical expression TðsÞ (eq. [17]) with the three free variables a, b, s0 to all of our over 1000 numerical solutions TnumðsÞ in the entire parameter space of L, sH , Tmax and found that the power indices a and b essentially depend only on a single parameter, the ratio L=sH , but have no dependence on the maximum temperature Tmax or the parameters L and sH separately. In other words, the solu- tion of the temperature function is invariant in TðsÞ=Tmax and s=L. The proportionality of TðsÞ=Tmax is also evident in the analytical solutions of Rosner et al. (eq. [17]) and Martens et al. (eq. [18]). We found that the dependence of the temperature power indices aðL=sHÞ and bðL=sHÞ can best be fitted with the empirical functions: aðL;SHÞ ¼ a0 þ a1 L sH
a2 ; ð20Þ bðL;SHÞ ¼ b0 þ b1 L sH
b2 : ð21Þ The best fits are shown in Figure 3, for the subset of hydro- static solutions with a maximum temperature of Tmax ¼ 3 MK, sorted by the parameter L=sH . The best-fit coefficients are given in Table 1. The similarity of the coefficients con- firms that there is no significant dependence on the loop-top temperature Tmax, and thus a and b are independent of Tmax. We run our analytical approximation of the temperature function (eqs. [19]–[21]) through all 1000 numerical solu- tions with the same coefficients and find a relative accuracy of ½TðsÞ=TnumðsÞ
1d10
2 10
3. Thus, the parameteriza- tion of TðsÞ given with equations (19)–(21) provides us a simple analytical formulation of the temperature function that is sufficiently accurate in the entire parameter space and can be used as a powerful tool to solve the hydrostatic equations. 4.2. The Pressure Function pðsÞ After we have obtained a suitable approximation of the temperature function TðsÞ, we have a much easier way to determine the pressure function pðsÞ, because we can directly integrate the momentum equation (eq. [2]), where the density nðsÞ can be substituted by the ideal gas law nðsÞ ¼ pðsÞ=2kBTðsÞ (eq. [5]), so that the momentum equa- tion contains only the unknown pressure function pðsÞ, dpðsÞ dh ¼
pðsÞ 0 106 K TðsÞ   1þ hðsÞ R  
2 ð22Þ 0 20 40 60 80 100 Loop length coordinate s[Mm] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 T em pe ra tu re T [M K ] AS (numerical) AS (analytical) RTV (analytical) MKB (analytical) L = 100 Mm Tmax = 3 MK uniform heating Fig. 2.—Hydrostatic solution of a uniformly heated loop with a loop- top temperature of Tmax ¼ 3 MK and a half-length of L ¼ 100 Mm are shown (top panel ) from our numerical code (crosses), the analytical solution of Rosner, Tucker, & Vaiana (RTV: dashed line), the analytical solution of Martens, Kankelborg, & Berger (MKB; dashed-dotted line), and the best fit with our analytical function (AS), TðsÞ ¼ Tmaxf1
½ðL
sÞ=ðL
s0Þ 2:0074g0:2858 (solid line). The base pres- sure in the RTV model had to be adjusted by a factor of pRTV0 ¼ p0  0:2985 to match our numeric solution. 0.01 0.10 1.00 10.00 L/s_H 1.5 2.0 2.5 3.0 3.5 4.0 a( L/ s H ) a0= 2.00029 a1= 0.343010 a2= 1.41843 a =a0+a1*(L/s_h)a2 dT<0.005 N = 262 0.01 0.10 1.00 10.00 L/s_H 0.24 0.26 0.28 0.30 0.32 0.34 b( L/ s H ) b0= 0.322623 b1= -0.00771525 b2= 0.903699 b =b0+b1*(L/s_h)b2 dT<0.005 N = 262 0.01 0.10 1.00 10.00 L/s_H 0.65 0.70 0.75 0.80 0.85 0.90 q λ (L /s H ) c0= 0.699276 c1= 0.0241457 c2= 1.24209 q =c0+c1*(L/s_h)c2 dT<0.005 N = 262 Fig. 3.—Fits of the power indices aðL=sHÞ (eq. [20]; top panel ) and bðL=sHÞ (eq. [21]; middle panel ) in the temperature function TðsÞ (eq. [19]), and for the scale height factor qðL=sHÞ (eq. [27]; bottom panel ) in the pres- sure function pðL=sHÞ (eqs. [25] and [26]), shown for 262 numeric solutions withTmax ¼ 3MK. No. 2, 2002 HYDROSTATIC APPROXIMATIONS OF CORONAL LOOPS 273 with the reference scale height for a 1 MK temperature plasma defined by 0 ¼ 2kBð106 KÞ lmpg ¼ 4:6 109 cm : ð23Þ In the derivation of Serio’s scaling law, the pressure func- tion was approximated by an exponential function with a constant pressure scale height p, which neglects the temper- ature variation TðsÞ along the loop and the radial variation of gravity. Because neglect of these effects leads to devia- tions in the determination of the base pressure p0 and the related scaling law, we retain these dependencies in the form of a height-dependent pressure scale height pðsÞ. Integrat- ing the differential equation (eq. [22]) by applying the mean- value theorem to the temperature dependence, ln pðsÞ p0
 ¼
Z h h0 1 0 106 K Tðs½h0Þ 1þ h0ðsÞ R  
2 dh0 
1 0 1 q 106 K Tðs½hÞ Z h h0 1þ h0ðsÞ R  
2 dh0 ; ð24Þ we can integrate the pressure function pðsÞ ¼ p0 exp
hðsÞ
h0 pðsÞ   ; ð25Þ which has the following height-dependent scale height pðsÞ: pðsÞ ¼ 0 TðsÞ 106 K   1þ hðsÞ R   qðL; sHÞ : ð26Þ The factor qðL; sHÞ corrects for the mean-value theorem of the temperature dependence TðsÞ, for which we know that it depends primarily on the ratio (sH=L), according to equa- tions (19)–(21) (see also Fig. 3, bottom). Therefore, the only thing left to do for an accurate analytic solution is the numerical determination of the coefficient q as a function of (sH=L). Again, we make an Ansatz with three variables c0; c1; c2 for this function, similar to equations (19)–(21), qðL; sHÞ ¼ c0 þ c1 L sH
c2 ð27Þ and determine these free variables by fitting the analytical pressure function pðsÞ (eq. [25]) parameterized with height- dependent scale heights pðsÞ (eq. [26]) to the numerical sol- utions of pðsÞ to find a best fit of the variables c0; c1; c2. The best fit to a subset of numerical solutions with Tmax ¼ 3MK yields the values c0 ¼ 0:699, c1 ¼ 0:024, c2 ¼ 1:240 (see Fig. 3, bottom). In previous work we approximated these correc- tion factor with a constant value, q  0:75 (eq. [13] in Aschwanden et al. 2000b), which essentially represents c0. We find similar coefficients for the data sets of Tmax ¼ 1, 5, and 10 MK, which are listed in Table 1. Based on these results we have now an accurate analytical approximation of the hydrostatic solutions, specified by the temperature profile TðsÞ (eqs. [19]–[21]), the pressure profile pðsÞ (eqs. [25]–[27]), from which follows also the density profile according to the ideal gas law, and nðsÞ ¼ pðsÞ kBTðsÞ : ð28Þ The only missing scaling parameters are the base pressure p0 and the base heating rate EH0, which have to be determined via scaling laws as a function of the independent parameters (L, sH , Tmax). 4.3. The Scaling Laws So far we obtained analytical solutions as a function of five unknown parameters, L, sH , Tmax, EH0, and p0. While only three parameters are independent (e.g., L, sH , Tmax), the dependence of the other two parameters TABLE 1 Best-Fit Coefficients in Analytical Approximations of Hydrostatic Solutions TemperatureTmax Coefficient 1MK 3MK 5MK 10MK EquationNumber a0 ................... 2.098 2.000 2.055 2.026 38 a1 ................... 0.258 0.343 0.328 0.298 a2 ................... 1.565 1.418 1.649 1.570 b0 ................... 0.320 0.323 0.329 0.309 39 b1 ...................
0.009
0.008
0.009
0.009 b2 ................... 0.877 0.902 0.852 0.928 c0 ................... 0.693 0.699 0.700 0.707 45 c1 ................... 0.026 0.024 0.014 0.029 c2 ................... 1.199 1.240 2.427 0.915 d0 ................... 1452 1416 1428 1506 53 d1 ...................
0.074
0.087
0.064
0.036 d2 ...................
0.030
0.044 0.000 0.001 d3 ...................
0.001
0.003
0.023
0.021 d4 ................... 0.015 0.043 0.010 0.011 e0 ................... 0.686 10
6 0.831 10
6 0.808 10
6 0.707 10
6 54 e1 ................... 0.558 0.848 0.847 0.685 e2 ...................
0.423
0.707
0.634
0.403 e3 ................... 0.548 0.057
0.058 0.063 e4 ................... 0.156 0.365 0.361 0.145 274 ASCHWANDEN & SCHRIJVER Vol. 142 (e.g., p0 and EH0) on the three independent parameters is specified by two scaling laws, i.e., p0(L, sH , Tmax) and EH0(L, sH , Tmax). These two scaling laws have been derived in Rosner et al. (1978) and Serio et al. (1981) by integrating the energy equation in two different ways: (1) as spatial integral R f ðsÞds and (2) as temperature inte- gral R f ðTÞdT after substituting the conductive flux vari- able FCðTÞ. Rosner et al. (1978) derived the scaling laws under the fol- lowing assumptions and approximations: 1. Constant pressure, pðsÞ ¼ p0. 2. Uniform heating, sH ¼ 1. 3. Radiative loss function is approximated with single power law,
ðTÞ 
0T
1=2, with
0 ¼ 10
18:81 erg cm3 s
1. 4. Auxiliary function fHðTÞ5 fRðTÞ, i.e., R T5=2EHðTÞdT5 R T5=2ERðTÞdT . 5. Neglect height dependence of solar gravitation gðhÞ ¼ gð1þ h=RÞ
2 (eq. [9]). 6. Footpoint in photosphere (s0 ¼ 0). The scaling laws can generally be expressed as a function of the independent variables [L0, sH , Tmax] by p0ðL0; sH ;TmaxÞ ¼ 1 L0 Tmax S1
3 ; ð29Þ EH0ðL0; sH ;TmaxÞ ¼ L
2 0 T 7=2 maxS2 : ð30Þ where we denoted the footpoint-apex distance by L0 ¼ L
s0. Under the assumptions and approximations listed above, Rosner et al. (1978) derived the following constants for the expressions S1 and S2: SRTV1 ¼ 1400 ; ð31Þ SRTV2 ¼ 0:95 10
6 : ð32Þ We show the comparison of the RTV scaling laws with our numerical solutions as a function of L=sH in Figure 4 (top panels). The scaling law for the base pressure agrees with the numerical solutions within ðpRTV0 =p0Þ  0:9 0:1 for near- uniform heating (L=sHd1). The scaling law for the base heating rate agrees with the numerical solutions within ðERTVH0 =EH0Þ  0:8 0:3 for near-uniform heating (L=sHd1Þ but yields too low heating rates down to frac- tions of 0.2 for short heating scale lengths (at L=sHd2). Serio et al. (1981) generalized the RTV scaling laws for variable pressure (owing to gravity) and nonuniform heat- ing but retained the other approximations from the deriva- tion of Rosner et al. (1978). Thus, Serio’s derivation is subject to the same set of assumptions and approximations except the first two: 1. The pressure is assumed to be an exponential function of the loop length, pðsÞ  p0 expð
s=spÞ, with sp ¼ 0TMK. This approximation neglects the temperature variation TðsÞ along the loop, and thus the variation of the pressure scale height pðsÞ (see eq. [26]). 2. The height dependence in the pressure function is approximated by the semicircular loop coordinate s, pðhÞ  pðsÞ. Note that the numerical calculations of coefficients , , 0, 0 (eqs. [3.7]–[3.8] in Serio et al. 1981) are optimized based on numerical solutions in some (unspecified) parame- ter space, which probably covers a different parameter regime than our numerical solutions. Moreover, Serio et al. calculate hydrostatic solutions for loops with an expansion factor of 5, while we calculate cases for constant as well as expanding cross sections separately. Serio’s scaling laws have the same basic dependence on L0 and Tmax as the RTV laws (eqs. [29]–[30]) but differ in the scaling law expressions S1 and S2 (eqs. [31]–[32]), SSerio1 ¼ 1400 exp
0:08 L0 sH
0:04 L0 sp
 ; ð33Þ SSerio2 ¼ 0:95 10
6 exp 0:78 L0 sH
0:36 L0 sp
 ; ð34Þ where we denoted sp ¼ 0TMK and TMK ¼ Tmax=106 MK. While Serio’s generalization accounts for nonuniform heat- ing and pressure variation, the approximations made in the derivation lead to differences from the proper numerical sol- utions, as shown in Figure 4 (middle row) relative to our exact numerical solutions. The agreement with our numeri- cal solutions of p0 and EH0 are within 1:0dp Serio 0 =p0d1:4 and 0:9dESerioH0 =EH0d1:3, with some extreme deviations down to ESerioH0 =EH0e0:2. In order to achieve a higher level of accuracy between the numerical hydrostatic solutions and the scaling law approx- imations, we add two additional correction terms to Serio’s expressions (eqs. [33] and [34]), leading to five coefficients for each of the two scaling laws, called di and ei, i ¼ 0; . . . ; 4, respectively: SAS1 ¼ d0 exp d1 L0 sH þ d2 L0 sp
 þ d3 L0 sH þ d4 L0 sp   ; ð35Þ SAS2 ¼ e0 exp e1 L0 sH þ e2 L0 sp
 þ e3 L0 sH þ e4 L0 sp   : ð36Þ We determine these 10 coefficients di and ei by minimizing the differences of the scaling law expressions p0ðdiÞ (eqs. [29] and [35]) [and EH0ðeiÞ (eqs. [30] and [36])] to the numerical solutions pnum0 [and E num H0 ] from our 1000 numerical runs, which cover the parameter space of [Tmax ¼ 1 10 MK, L ¼ 4 400 Mm, sH ¼ 4 400 Mm]. The best-fit values of the coefficients di and ei, i ¼ 0; . . . ; 4 are tabulated in Table 1, for different temperatures Tmax ¼ 1, 3, 5, and 10MK. If one uses just one set of coefficients (say from Tmax ¼ 3:0 MK in Table 1) for a larger temperature range, the accuracy of the scaling laws is about d5% in the temperature of T ¼ 2 5 MK and degrades to 10%–20% in the temperature range of T ¼ 1 10 MK. For a higher accuracy in the order of a few percent, a spline interpolation of the coefficients (given in Table 1) as a function of Tmax is recommended. These empirical scaling laws (eqs. [35] and [36]) provide a best fit to the numerical solutions within an accuracy of a few per- cent (see Fig. 4, bottom panels). The functional dependence of these scaling laws is shown in Figure 5 for EH0(L, sH , Tmax), and in Figure 6 for p0(L, sH , Tmax), respectively. 4.4. Choice of Independent Parameters ½L; sH ;EH0 The analytical formulation of the scaling law derived by Serio et al. (1981) requires the independent parameter set [L; sH ;Tmax], because the pressure scale height sp ¼ 0ðTmax=106 K) depends on Tmax, so that the second scaling law (eqs. [30] and [34]) can only be expressed explic- itly for EH0ðL; sH ;TmaxÞ, but not explicitly in the form of TmaxðL; sH ;EH0Þ. The same is also true for our modified No. 2, 2002 HYDROSTATIC APPROXIMATIONS OF CORONAL LOOPS 275 dances yields a somewhat higher temperature function and a massively enhanced density and pressure function (Fig. 8, black curves), while the solution based on coronal abundan- ces yields a lower density and pressure (Fig. 8, gray curves), due to the enhanced radiative losses of the iron element at temperatures around T  1 MK. Thus, this uncertainty by about a factor of 2 in the assumptions on elemental abun- dances far outweighs the inaccuracy of our analytical approximation to the numerical hydrostatic solutions within the few percent level. 5. GENERALIZED HYDROSTATIC SOLUTIONS In this section we generalize our analytical approxima- tions of the hydrostatic solutions to allow applications in a wider range of observational circumstances. We generalize the solutions for extremely small heating scale lengths, for inclined loops, for loops with variable cross sections, and for slow velocity flows. These generalizations have been tested in part of the previously used parameter space, which covers a temperature range of T ¼ 1 10MK. 5.1. Small Heating Scale Heights In the previous sections we covered a large parameter space with spatial scales in the range of 4–400Mm for L and sH , but we excluded extremely small heating scale lengths, say with sHdL=3. Already Serio et al. (1981) subdivided hydrostatic solutions into two classes with the same crite- rion: class I are loops with the temperature maximum at the loop top (which is the case for sHeL=3), and class II are loops with the temperature maximum at some intermediate position between the loop top and footpoints (which is the case for sHdL=3Þ. We show numerical solutions of hydro- static temperature profiles from sH ¼ L down to sH ¼ L=25 in Figure 9 (solid lines). The temperature maximum clearly moves downward the loop with decreasing heating scale length ratio sH=L, and a larger segment of the loop becomes near isothermal. Because our previous temperature approx- imation with a generalized ellipse function xa þ x1=b ¼ 1 has its maximum by definition at the loop top, the same approx- imation cannot represent loops with an intermediate tem- perature maximum. However, a correction can be added that makes the analytical approximation valid down to extremely short heating scale lengths of sHeL=25. Because the temperature solution was found to be nearly invariant with respect to the normalized temperature TðsÞ=Tmax and spatial coordinate z ¼ ðs
s0Þ=ðL
s0Þ, the correction term scales only with the ratio sH=L. We found a good approxi- mation within the d1% level (Fig. 9, dashed lines) with the Temperature 0 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0 1.2 T (s ) [M K ] Tmax= 1 MK Lloop= 40 Mm sheat=317 Mm Density 0 10 20 30 40 0 5.0•108 1.0•109 1.5•109 2.0•109 n( s) [c m -3 ] Pressure 0 10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 p( s) [d yn e cm -2 ] Chromospheric abundances (Meyer) Coronal abundances (Feldman) Conductive flux 0 10 20 30 40 -1.2•105 -1.0•105 -8.0•104 -6.0•104 -4.0•104 -2.0•104 0 2.0•104 F C (s ) Fig. 8.—Comparison of hydrostatic solutions computed for two differ- ent radiative loss functions: for the RTV six–power-law approximation (Fig. 7, thick line) and chromospheric abundances according to Meyer (1985), and for the same radiative loss function and coronal abundances according to Feldman (1992). The difference in the solution is shown in gray. 0.0 0.2 0.4 0.6 0.8 1.0 Loop length normalized (s-s0)/(L-s0) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 T em pe ra tu re n or m al iz ed T /T m ax sH/L= 1.000 sH/L= 0.794 sH/L= 0.631 sH/L= 0.501 sH/L= 0.398 sH/L= 0.316 sH/L= 0.251 sH/L= 0.200 sH/L= 0.158 sH/L= 0.126 sH/L= 0.100 sH/L= 0.079 sH/L= 0.063 sH/L= 0.040 Heating scale height sH = 4 Mm Loop lengths L = 4-100 Mm Fig. 9.—Hydrostatic solutions for extremely short heating scale lengths, sH=L ¼ 0:04; . . . ; 1:0. The numeric solutions are shown in solid lines, and the analytical approximation (eqs. [40] and [41]) in dashed lines. 278 ASCHWANDEN & SCHRIJVER Vol. 142 following corrected temperature function: z ¼ L
s L
s0
 ; ð38Þ T 0ðsÞ ¼ Tmax½1
z ab 1þ 0:510 log L sH
 ð1
zÞz5   : ð39Þ Because our analytical approximations of the hydrostatic solutions and scaling laws are all expressed in terms of the temperature solution (Table 2), the standard temperature approximation TðsÞ can simply be replaced by the corrected function T 0ðsÞ in this formalism, and the analytical approxi- mations for the pressure function pðsÞ and density nðsÞ as well as the resulting scaling laws will automatically be cor- rected as a function of the improved temperature function T 0ðsÞ. This correction, however, needs only to be applied for extremely short heating scale lengths, in the range of sH < L=3. 5.2. Inclined Loops Most of the observed coronal loops have some inclination of the average loop plane with respect to the vertical on the solar surface. For instance, a bundle of 30 stereoscopically reconstructed active region loops were found to have an almost uniform distribution of inclination angles in the range of ¼
49 . . .þ69 (Aschwanden et al. 1999). While the gravitational scale height is strictly measured in vertical direction, the effective scale height in the loop plane varies with the cosine of the vertical scale height, so we can define an effective gravity component along the loop, geff ¼ g cos : ð40Þ TABLE 2 Summary of Analytical Formulae to Calculate Hydrostatic Solutions and Scaling Laws Description Formula Constants: Height of loop base......................... s0 ¼ 1:3 108 cm Temperature at loop base ............... T0 ¼ 2:0 104 K Solar radius .................................... R ¼ 6:96 1010 cm Solar gravity................................... g ¼ 2:74 104 cm s
2 Spitzer conductivity........................  ¼ 9:2 10
7 erg s
1 cm
1K
7/2 Independent variables: Loop half-length ............................ L (cm) Heating scale length ....................... sH (cm) Loop-top temperature.................... Tmax (K) Base heating rate ............................ EH0 (ergs cm
3 s
1) Loop plane inclination angle .......... h (deg) Loop expansion factor ................... C 1 Choice 1: [L, sH ,Tmax, h, C]............. Choice 2: [L, sH ,EH0, h, C].............. CTmax  55:2 E0:977H0 L 2 0 exp 
0:687ðL0=sHÞ  2=7 Dependent parameters: Half-loop length above base ........... L0 ¼ L
s0 Loop height.................................... h1 ¼ 2L=ð Þ Subphotospheric zero point............ ssub ¼ L ð=2Þ= arcsinð1=1=2Þ
1½ 
1 Equivalent heating scale length....... sH ¼ sH 1þ ð
1ÞðsH=LÞ½ 
1=2 if (sH L, C 1) sH ¼ sH ð
1Þ½ 
1=2 if (sH > L, C> 1) Temperature index 1....................... a ¼ a0 þ a1 L0=sH  a2 Temperature index 2....................... b ¼ b0 þ b1 L0=sH  b2 Scale height factor .......................... q ¼ c0 þ c1 L0=sH  c2 Effective gravity.............................. geff ¼ g cos Effective scale height....................... 0 ¼ 2kB106½K=lmpgeff   ¼ 4:6 109 1= cos ð Þ cm Serio scale height ............................ sp ¼ 0 Tmax=106 Kð Þ Scaling law factor 1 ........................ S1 ¼ d0 exp  d1ðL0=sHÞ þ d2ðL0=spÞ  þ d3ðL0=sHÞ þ d4ðL0=spÞ Scaling law factor 2 ........................ S2 ¼ e0 exp  e1ðL0=sHÞ þ e2ðL0=spÞ  þ e3ðL0=sHÞ þ e4ðL0=spÞ Base heating rate (for Choice 1) ...... EH0 ¼ L
20 T 7=2 maxS2 Base pressure.................................. p0 ¼ L
10 ðTmax=S1Þ 3 Analytical approximations: Normalized length coordinate ........ zðsÞ ¼ ðL
sÞ=ðL
s0Þ Height (in loop plane)..................... h0ðsÞ ¼ h1 sin s=h1ð Þ Loop cross section area .................. AðsÞ ¼  sin2  ð=2Þðsþ ssubÞ=ðLþ ssubÞ  Temperature (if sH=L > 0:3) .......... TðsÞ ¼ Tmax 1
z a½ b Temperature (if sH=L 0:3) .......... TðsÞ ¼ Tmax½1
z ab 1þ 0:510 log L=sHð Þð1
zÞz5½  Conductive flux .............................. FCðsÞ ¼
TðsÞ 5=2½dTðsÞ=ds Pressure scale height....................... pðsÞ ¼ 0½TðsÞ=106 K½1þ h0ðsÞ=Rq Pressure.......................................... pðsÞ ¼ p0 expf
½h0ðsÞ
h0ðs0Þ=pðsÞg Density........................................... nðsÞ ¼ ½pðsÞ=2kBTðsÞ No. 2, 2002 HYDROSTATIC APPROXIMATIONS OF CORONAL LOOPS 279 The conceptually simplest method to include the inclination of the loop plane in hydrostatic solutions is to use the effec- tive gravity component for calculations of hydrostatic solu- tions in the inclined loop plane. All we have to do is to replace the gravity g by the effective gravity geff in the equation for the reference scale height 0 in equation (26) and to rename the height variable hðsÞ in equation (13) by h0ðsÞ to indicate that h0 is measured in the loop plane, which is then related to the vertical height h by h ¼ h0 cos . The correction of the effective scale height 0 is then carried to the pressure scale height pðsÞ (eq. [26]) and the resulting pressure pðsÞ (eq. [25]) and density function neðsÞ (eq. [28]). This generalization for inclined loop planes is implemented in the summary of analytical formulae in Table 2. 5.3. Loops with Expanding Cross Sections Vesecky et al. (1979) computed hydrostatic solutions for loops with expansion factors and found that the tempera- ture profile does not change much, but the densities increase somewhat for higher expansion factors C. So far we dis- cussed only hydrostatic solutions of loops with constant cross sections. However, we developed a numeric code (x 3) that can compute hydrostatic solutions for loops with arbi- trary cross sections, e.g., characterized by an expansion fac- tor C (eqs. [15] and [16]). We compute now a few cases with such expansion factors of  ¼ 1, 2, 5, 10. The resulting hydrostatic solutions for the temperature TeðsÞ and density neðsÞ are shown in Figure 10. From the temperature solu- tions we see that higher loop expansion factors C (Fig. 10) have about the same effect on temperature profiles as shorter heating scale lengths sH (Fig. 9). We can therefore define an equivalent heating scale length sH by comparing the heating power PðsÞ integrated over the loop cross sec- tion AðsÞ in a loop with a constant cross section, PðsÞ ¼ EHðsÞA0, with that of an expanding loop, PðsÞ ¼ EHðsÞA ðsÞ, EH0 exp
h sH
 A0 ¼ E  H0 exp
h sH
 AðsÞ : ð41Þ At the footpoints we can use a (linear) first-order Taylor expansion of the heating function EHðsÞ (eq. [11]) and the area cross section functionAðsÞ (eq. [15]), EHðsÞ  EH0
1
h sH þ . . .  ; ð42Þ EHðsÞ  E  H0
1
h sH þ . . .  ; ð43Þ AðsÞ  A0 1þ ð
1Þ h L þ . . .   : ð44Þ Inserting these first-order expansions into the equivalence equation (eq. [49]) we find the following relation: s ;foot H ¼ sH .h 1þ ð
1Þ sH L i ; for sH L ; L=½ð
1Þ ; for sH4L : ( ð45Þ This reduced heating scale length compensates for the diverging cross section at the loop footpoints, while the cross section is nearly constant at the loop top (given the sin2-function in eq. [15] for the cross section variation). Thus, the temperature profile at the loop top is similar to that of constant cross sections, and no compensation is needed in the heating scale length, s ;top H  sH : ð46Þ Comparing the numerical hydrostatic solutions of loops with variable cross sections with those of constant cross sec- tions, we found that the overall behavior of the equivalence heating scale length sH is reproduced rather well by the geo- metric mean from the loop-top and footpoints, i.e., sH  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ;foot H s ;top H q : ð47Þ Thus, a useful approximation of the equivalent heating scale height sH to compensate for the loop expansion factor C is sH ¼ sH ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ ð
1ÞðsH=LÞ p ; for sH L;   1 ; L ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð
1Þ p ; for sH > L;  > 1 : 8 > > < > : ð48Þ 0.0 0.2 0.4 0.6 0.8 1.0 Loop length normalized (s-s0)/(L-s0) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 T em pe ra tu re n or m al iz ed T /T m ax Γ = 1 Γ = 2 Γ = 5 Γ = 10 Loop expansion factor L = 40 Mm sH = 20 Mm 0.0 0.2 0.4 0.6 0.8 1.0 Loop length normalized (s-s0)/(L-s0) 0 5.0•108 1.0•109 1.5•109 2.0•109 D en si ty n e [c m -3 ] Fig. 10.—Hydrostatic solutions of the temperature TðsÞ (top) and den- sity neðsÞ (bottom) for loops with geometric expansion factors  ¼ 1, 2, 5, 10. The exact numeric solutions are plotted with solid lines, the analytical approximations with dashed lines. Note that the analytical approximations are accurate to within a few percent for  ¼ 1 10. 280 ASCHWANDEN & SCHRIJVER Vol. 142