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Astron. Astrophys. 326, 249-256 (1997)

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5. Energy requirements

One of the main objectives of this kind of atmospheric modelling is to constrain the amount of chromospheric heating required to sustain the atmosphere of the stars, i.e., to balance the radiative losses due to the different spectral lines and continua. Here we compute the radiative cooling rate [FORMULA] (ergs cm-3 s-1), i.e. the net amount of energy radiated at a given depth by the atmosphere, which is given by


In this paper we computed the contributions due to [FORMULA], H, He I, Mg I and II, Ca I and II, Fe I, Si I, Na I, Al I and CO. The overall results and the most important individual contributions are presented in Fig. 6. A positive value implies a net loss of energy (cooling), and a negative value represents a net energy absorption (heating).

[FIGURE] Fig. 6a and b. Total radiative cooling rate for our models of Gl 588 (upper panel) and of Gl 628 (lower panel) and the most important contributions to it. Thick line: total rate; dotted line: CO; short-dashed line: Mg II lines; dot-dashed line: H total; long-dashed line: [FORMULA] free-free; thin line: [FORMULA] bound-free.

As can be seen, the cooling rate for both stars is very similar, well within the error in the calculations due to differences in the depth grid. The larger temperature of our model for Gl 628 for m between 10-2 and 10-3 does not imply a significantly larger radiative loss, since the larger contribution, that of CO, does not vary substantially. In fact, it can be seen in the figure that the cooling rate in the photosphere and the low chromosphere is determined mainly by the balance between [FORMULA] and CO cooling.

In the temperature minimum region, the [FORMULA] cooling rate is negative, driving the total rate to negative values. In this case there is a missing cooling agent, a fact that was already noted for the Sun, and for AD Leo in Paper I. However, it should be pointed out that almost all cooling is due to CO, which is the only molecule included in the calculations. It seems reasonable to expect that, should other molecules be computed in detail, the results would show a similar behaviour as the CO calculations, bringing this region closer to radiative balance.

This results can be compared with the ones shown in Fig. 9 of Paper I, which shows that the dMe star AD Leo has a much larger cooling rate in the high chromosphere, which implies that much more heating is needed to sustain the higher temperature (see Fig. 1). In the mid and high-chromosphere the energy balance is determined by the hydrogen cooling rate and the Mg II rate.

In Fig. 7 we show a detail of the cooling rates in the high chromosphere, that can be compared with Fig. 10 of Paper I. We note that line cooling is positive for all three stars, and it is the most important radiative loss for the cooler stars. In the transition region, for temperatures larger than 15000 K, [FORMULA] is the most important cooling agent, whereas at lower temperatures the Balmer lines become dominant. In this region, cooling in the [FORMULA] line is three times larger than in [FORMULA], and ten times larger than for [FORMULA] to H8 together. As the temperature drops further, Mg II h and k lines become important. Ca II H and K cooling, on the other hand, is an order of magnitude smaller than Mg II cooling.

[FIGURE] Fig. 7. Detail of the chromospheric cooling rates for our models: Gl 588 (upper panel), Gl 628 (lower panel). Thick line: total rate; long-dashed line: Mg II; dot-dashed line: Balmer lines; dotted line: Ly- [FORMULA].
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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998