## 3. Applications to observationsFig. 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 is the mass loss rate in units of
, is the terminal wind
velocity in units of 100 km/s, is the observed
flux at 6 cm () in units of
. is the gaunt factor
at as given in Table 1. Because for all
objects is of the order unity, Eq. (15)
shows that the radio fluxes imply mass loss rates of the order of
as was inferred by MDRM who took
. Also it follows that the wind temperature has
to be higher than because otherwise
An important assumption made by MDRM is that the observed power-law distribution 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 which is a factor 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 , 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 while for dMe stars the reverse holds. Because occurs at frequency we can use for and the Rayleigh-Jeans approximation while in Eq. (9) . 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 . Cassineli et al. (1977) have
argued that the emission originates from . It
is convenient to introduce the optical depth
with For (constant velocity wind) . At frequencies below the wind is optically thick and has a 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 , apart from a small variation caused by the frequency dependence of the gaunt factor. We define . At the wind emission drops rapidly as . In Fig. 3 we show the calculated emission from the star and
the wind (for ) as follows from numerical
integration of Eq. (11) for and
(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
the flux distribution is that of a mass
loosing star, as assumed by MDRM, but at frequency
the wind contribution becomes relatively flat.
At higher frequencies the contribution by the star starts to dominate
the spectrum. In the range the variation of
the wind contribution is only caused by the gaunt factor. In
Fig. 3a frequency corresponds to
above which frequency the wind contribution
drops exponentially. Note that at frequencies
the optically thin approximation (Eq. 13) is very accurate. For
low temperatures of the wind, like e.g. , 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
goes up as the temperature of the wind
increases. Because , apart from a weak
dependence on the Gaunt factor, reducing the wind temperature to a
value lower than , like e.g.
, does not improve the fit to the data points.
The reason is that at frequency the black body
emission of the wind varies as and is
therefore only weakly dependent on the wind temperature. At the same
time, for higher values of , the effective
radius of the source becomes smaller. Together
these effects result in that the flux distributions of winds with
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
A possible way out would be to increase the optical depth so that
is found at IR frequencies. This can be
accomplished by allowing the wind to accelerate over some distance (so
increasing
The figure clearly shows that as becomes
higher the emission measure increases. This results in a (strong)
increase of the flux at high frequencies. For all combinations of
and either the IR data
points are not fitted or there is too much flux at frequencies above
. The flux distributions can be explained by
considering the contributions by the different terms in Eq. (11).
Because the 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 and respectively. Their sum equals the flux from a black-body at temperature and with a surface area set by the size of the acceleration region. Above frequency 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 with a decreasing surface area (curve 3). At frequency the emission becomes optically thin. This occurs before frequency (at which ) is reached. In the frequency range the second and third term in Eq. (11) dominate. Because at these frequencies the flux is proportional to , with , 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 and becomes larger. Fig. 4 a shows that larger differences between these frequencies are accompanied by stronger emission. The reason is that we have 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 smaller than for early type stars. This implies that the frequency at which the emission becomes optically thin (), and the frequency at which , 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 . This has the consequence that at IR frequencies the emission can become very strong. Even more important is the fact that 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 . © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |