Analytical Approximations to Hydrostatic Solutions and Scaling Laws of Coronal Loops

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 T(s), density ne(s), and pressure profile p(s) 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 numerical solutions with a relative accuracy of ≲10-2-10-3. 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.