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Astron. Astrophys. 319, 578-588 (1997)

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3. Applications to observations

Fig. 1 shows that the flux distributions of the three objects are very similar. Therefore we start with a discussion of the flux distributions in general terms before turning to the specific objects. From the observed flux at radio frequencies (Eq. (12)) it follows that


where [FORMULA] is the mass loss rate in units of [FORMULA], [FORMULA] is the terminal wind velocity in units of 100 km/s, [FORMULA] is the observed flux at 6 cm ([FORMULA]) in units of [FORMULA]. [FORMULA] is the gaunt factor at [FORMULA] as given in Table 1. Because for all objects [FORMULA] is of the order unity, Eq. (15) shows that the radio fluxes imply mass loss rates of the order of [FORMULA] as was inferred by MDRM who took [FORMULA]. Also it follows that the wind temperature has to be higher than [FORMULA] because otherwise µ is so large (see Table 1) that unrealistic mass loss rates ([FORMULA]) result.

An important assumption made by MDRM is that the observed power-law distribution [FORMULA] in the radio - IR frequency range is indicative of a stellar wind. This assumption is based on the fact that similar power-laws are observed for early type stars loosing mass. There is however an important difference between the above inferred mass loss rate for dMe stars and that of early type stars. The latter have mass loss rates in the range [FORMULA] which is a factor [FORMULA] larger than the data for dMe stars suggest. This has important consequences for the flux distribution because the lower mass loss rates of dMe stars will result in smaller optical depths. This leads to a modification of the spectrum and in the following we show that, if dMe stars loose mass at a rate of [FORMULA], the resulting flux distribution is not a power-law in the radio - IR range. Apart from the fact that the optical depths in the winds differ, there is a second difference between early and late type stars: for early type stars [FORMULA] while for dMe stars the reverse holds.

Because [FORMULA] occurs at frequency [FORMULA] we can use for [FORMULA] and [FORMULA] the Rayleigh-Jeans approximation while in Eq. (9) [FORMULA]. From Eqs. (8) and (15) it follows that


In order to discuss the flux distribution it is useful to consider the optical depth along the line of sight passing through the center of the star [FORMULA]. Cassineli et al. (1977) have argued that the emission originates from [FORMULA]. It is convenient to introduce the optical depth [FORMULA] with H defined in Table 2. At some frequency [FORMULA], [FORMULA] will become unity. Because [FORMULA], the transition between optically thick and optically thin will occur rapidly near this frequency. Unit optical depth corresponds to [FORMULA] or


For [FORMULA] (constant velocity wind) [FORMULA]. At frequencies below [FORMULA] the wind is optically thick and has a [FORMULA] spectral distribution while at higher frequencies the optically thin approximation Eq. (13) applies. Eq. (13) shows that, in the optically thin part of the spectrum, the contribution by the wind is almost independent of frequency, at frequencies [FORMULA], apart from a small variation caused by the frequency dependence of the gaunt factor. We define [FORMULA]. At [FORMULA] the wind emission drops rapidly as [FORMULA].

In Fig. 3 we show the calculated emission from the star and the wind (for [FORMULA]) as follows from numerical integration of Eq. (11) for [FORMULA] and [FORMULA] (thick solid lines). Also indicated are the black-body distributions from the star and the wind, the separate contributions by the star and the wind to the observed flux and the optically thin approximation (Eq.  13). The data points for YZ CMi are also plotted. The figures clearly show that up to frequency [FORMULA] the flux distribution is that of a mass loosing star, as assumed by MDRM, but at frequency [FORMULA] the wind contribution becomes relatively flat. At higher frequencies the contribution by the star starts to dominate the spectrum. In the range [FORMULA] the variation of the wind contribution is only caused by the gaunt factor. In Fig. 3a frequency [FORMULA] corresponds to [FORMULA] above which frequency the wind contribution drops exponentially. Note that at frequencies [FORMULA] the optically thin approximation (Eq.  13) is very accurate. For low temperatures of the wind, like e.g. [FORMULA], the black-body emission by the star is slightly reduced due to wind absorption but at those frequencies the wind emission dominates anyway. Figs. 3a and 3b show that frequency [FORMULA] goes up as the temperature of the wind increases. Because [FORMULA], apart from a weak dependence on the Gaunt factor, reducing the wind temperature to a value lower than [FORMULA], like e.g. [FORMULA], does not improve the fit to the data points. The reason is that at frequency [FORMULA] the black body emission of the wind varies as [FORMULA] and is therefore only weakly dependent on the wind temperature. At the same time, for higher values of [FORMULA], the effective radius of the source [FORMULA] becomes smaller. Together these effects result in that the flux distributions of winds with [FORMULA] resemble very much the flux distribution shown in Fig. 3a. We conclude that the observed flux distribution cannot be reconciled with that of a stellar wind because the IRAS - and, depending on the wind temperature, also the JCMT -data points are not fitted at all.

[FIGURE] Fig. 3a and b. Flux distributions for constant velocity winds with temperatures of a [FORMULA] and b [FORMULA]. The dots are the observed fluxes for YZ CMi (Fig. 1a). The thick solid line is the resulting distribution from the wind and the star (Eq. (11)). The thin solid line is the contribution by the wind which dominates at low frequencies. The dashed-triple dot line is the contribution by the star (first term in Eq. (11)). The dashed line is the black-body curve for the star and the dotted line that for the wind. For these curves the area is taken equal to the stellar disk. The dashed-dotted line is the optically thin approximation (Eq. (13)) which is very accurate at frequencies above the frequency [FORMULA].

A possible way out would be to increase the optical depth so that [FORMULA] is found at IR frequencies. This can be accomplished by allowing the wind to accelerate over some distance (so increasing H in Eq. (17)). Fig. 3 shows that [FORMULA] has to be increased by at least a factor 100 in order to have the turn-over frequency near the IRAS points and that cool winds are preferable. Eq. (17) and Table 2 show that [FORMULA]. Therefore either [FORMULA] or [FORMULA] has to be large, or both. This has however a strong effect on the emission measure which scales as [FORMULA]. The strong increase of the emission measure which occurs when the wind is allowed to accelerate has a dramatic effect on the flux distribution at high frequencies. To illustrate this we show in Fig. 4 a the flux distributions for six combinations of the parameters [FORMULA] and [FORMULA]. For each combination we give in Table 3 the values of the frequencies at which the emission becomes optically thin ([FORMULA]), at which the effective radius equals the acceleration region ([FORMULA] at [FORMULA]) and at which the effective radius equals the radius of the star ([FORMULA] at [FORMULA]). The combination [FORMULA], [FORMULA] (case 1) corresponds to the example given in Fig. 3a. In cases 2 - 4 we have kept the acceleration radius at two stellar radii and varied [FORMULA] while for cases 5 and 6 the acceleration radius is at five stellar radii.

[FIGURE] Fig. 4a and b. a Flux distributions for six combinations of [FORMULA] and [FORMULA] and a wind temperature of [FORMULA]. The data points are those for YZ CMi. Case 1 corresponds to Fig. 3a. b Contributions to the total wind emission for case 6 in panel (a). The solid curve is the total emission from the wind. The curves labeled 2, 3 and 4 correspond to the second, third and fourth term on the left hand side of Eq. (11). Curve [FORMULA] is the flux from a black-body with a area equal to the stellar disk. Curve [FORMULA] is the flux from a black-body with a area equal to the acceleration region minus the stellar disk area. At frequency [FORMULA] the effective radius equals the size of acceleration region. At frequency [FORMULA] the emission becomes optically thin and at frequency [FORMULA] the effective radius equals the stellar radius.


Table 3. The values for [FORMULA], [FORMULA] and [FORMULA] for the models shown in Fig. 4.

The figure clearly shows that as [FORMULA] becomes higher the emission measure increases. This results in a (strong) increase of the flux at high frequencies. For all combinations of [FORMULA] and [FORMULA] either the IR data points are not fitted or there is too much flux at frequencies above [FORMULA]. The flux distributions can be explained by considering the contributions by the different terms in Eq. (11). Because the IRAS data points are best fitted when [FORMULA] and [FORMULA] we show in Fig. 4 b the contributions by the different terms in Eq. (11) as an illustration. We start with the last term in Eq. (11) (curve 4 in Fig. 4 b) which dominates at low frequencies and results in a [FORMULA] distribution. The part in the curly brackets can be approximated as [FORMULA] for [FORMULA] and as [FORMULA] for [FORMULA]. This implies that near [FORMULA] the contribution by this term will change. [FORMULA] corresponds to [FORMULA]. The frequency at which the effective radius equals the acceleration radius ([FORMULA], see Table 2) corresponds to [FORMULA]. Combining these expressions we see that [FORMULA] corresponds to a frequency [FORMULA]. Above this frequency the contribution by the last term in Eq. (11) becomes flatter and becomes proportional to [FORMULA]. This shows that for larger values of [FORMULA] this term becomes relatively smaller at high frequencies. The flattening of the contribution above frequency [FORMULA] is clearly visible in curve 4 in Fig. 4 b.

The second and third term of Eq. (11) are shown as curves 2 and 3 in Fig. 4 b. At low frequencies, at which the emission is optically thick, these terms equal [FORMULA] and [FORMULA] respectively. Their sum equals the flux from a black-body at temperature [FORMULA] and with a surface area set by the size of the acceleration region. Above frequency [FORMULA] the outer parts of the acceleration region become increasingly optically thin. This has the effect that towards higher frequencies one observes the flux from a black-body [FORMULA] with a decreasing surface area (curve 3). At frequency [FORMULA] the emission becomes optically thin. This occurs before frequency [FORMULA] (at which [FORMULA]) is reached. In the frequency range [FORMULA] the second and third term in Eq. (11) dominate. Because at these frequencies the flux is proportional to [FORMULA], with [FORMULA], the flux can become relatively strong (cases 4 and 6 in Fig. 4 a). Table 3 shows that in the consecutive cases 2, 3, 5, 6 and 4 the difference between [FORMULA] and [FORMULA] becomes larger. Fig. 4 a shows that larger differences between these frequencies are accompanied by stronger emission. The reason is that we have [FORMULA] contrary to the situation in winds near early type stars.

The arguments used by MDRM to explain the power-law distributions shown in Fig. 1 were based on the theory for stellar winds from early type stars. Above we demonstrated that this theory cannot be applied to the (possible) winds of late type stars. The two fundamental differences are: 1) the derived mass loss rates for dMe stars from the radio data are factors [FORMULA] smaller than for early type stars. This implies that the frequency at which the emission becomes optically thin ([FORMULA]), and the frequency at which [FORMULA], are found at lower frequencies. This results in a deviation from the power-law spectrum. MDRM assumed the presence of a power-law distribution between the radio and IR data points. 2) the winds in dMe stars are characterized by [FORMULA]. This has the consequence that at IR frequencies the emission can become very strong. Even more important is the fact that [FORMULA] results in strong emission from the wind in the Wien part of the stellar black-body distribution. Although the flux distributions in the radio - IR range can be fitted by assuming the presence of a wind acceleration region, these models predict too much flux at frequencies [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998