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356 CHAPTER 9. CORONAL HEATING

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Heating Energy Requirement

We start to analyze the coronal heating problem by inquiring first about the energy

requirements. A coronal heating sourceEH has to balance at least the two major loss

terms of radiative lossER

and thermal conductionEC

, as we specified in the energyequation (3.6.2) for a hydrostatic corona,

EH(x) ER(x) EC(x) = 0, (9.1.1)where each of the terms represents an energy rate per volume and time unit (erg cm3

s1 ), and depends on the spatial location x. Because the corona is very inhomoge-

neous, the heating requirement varies by several orders of magnitude depending on thelocation. Because of the highly organized structuring by the magnetic field (due to the

low plasma-parameter in the corona), neighboring structures are fully isolated and

can have large gradients in the heating rate requirement, while field-aligned conduc-tion will smooth out temperature differences so that an energy balance is warranted

along magnetic field lines. We can therefore specify the heating requirement for each

magnetically isolated structure separately (e.g., a loop or an open fluxtube in a coronal

hole), and consider only the field-aligned space coordinates in each energy equation,

as we did for the energy equation (3.6.2) of a single loop,

EH(s) ER(s) EC(s) = 0 . (9.1.2)Parameterizing the dependence of the heating rate on the space coordinates with an

exponential function (Eq. 3.7.2) (i.e., with a base heating rate EH0 and heating scalelengthsH), we derived scaling laws for coronal loops in hydrostatic energy balance,

which are known as RTV laws for the special case of uniform heating without gravity

(Eqs. 3.6.1415), and have been generalized by Serio et al. (1981) for nonuniform

heating and gravity (Eqs. 3.6.1617). It is instructional to express the RTV law as a

function of the loop density ne and loop half lengthL, which we obtain by inserting

the pressure from the ideal gas law,p0 = 2nekBTmax, into Eqs. (3.6.1415),

Tmax103 (neL)1/2 (9.1.3)EH0 2 1017n7/4L1/4 (9.1.4)

This form of the RTV law tells us that the heating rate depends most strongly on the

density,EH0 n7/4, and very weakly on the loop length L. Actually, we can re-trieve essentially the same scaling law using a much simpler argument, considering

only radiative loss, which is essentially proportional to the squared density (Eq. 2.9.1),

EH0 ER = n2 (T) 1022n2 (erg cm3 s1 ) (9.1.5)ee

where the radiative loss function can be approximated by aconstant (T ) 1022 [erg cm3 s1 ] in the temperature

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