4. Alternative constraints
From the discussion in the previous section it must be concluded that it is unlikely that dMe stars loose mass at rates of . If the mass loss rate is in reality lower, then the radio, JCMT and IRAS data points require alternative explanations. For the radio data these are readily available. The common interpretation of radio emission from dMe stars is gyro-synchrotron emission from non-thermal particles (or coherent emission). This interpretation is supported by the results of Benz and Alef (1991) who discuss intercontinental VLBI observations of YZ CMi at 1.7 GHz. They found that the source was not resolved at this frequency and found a radio diameter of or . The derived brightness temperature was with a lower limit of . This brightness temperature already rules out the possibility that the radio emission is caused by a stellar wind. If YZ CMi would be loosing mass at a rate of , as suggested by MDRM, then the effective radius at 1.7 GHz would amount to
with the gaunt factor at 1.7 GHz. A source of this size would certainly have been resolved with intercontinental VLBI. Alternatively, we can argue that at 1.7 GHz has to be smaller than the found by Benz and Alef. This gives an upper limit for the mass loss rate which is compatible with the VLBI results
This upper limit automatically implies that the wind is optically thin at radio frequencies. The frequently observed nonthermal emission from dMe stars implies that this emission is not absorped by a wind. Requiring that the optical depth of the wind at e.g. 6 cm is less than unity gives
with the gaunt factor at 5 GHz. Comparing Eqs. (18) and (19) shows that they have the same form but that the upper limit set by the VLBI observations is slightly more restrictive.
Additional constraints on the mass loss rate can be obtained by considering the observed fluxes at other frequencies. E.g. from the data presented by Doyle (1989) it follows that the flux from YZ CMi in the IUE-SWP band-pass () amounts to (in erg s-1). Güdel et al. (1993) give for the flux in the ROSAT-PSPC band-pass () a flux of . We have calculated the emissivities in these band-passes for the temperatures listed in Table 1 using the Utrecht spectral code SPEX (Kaastra and Mewe, 1993, Mewe and Kaastra, 1994). From the fact that any emission from a stellar wind must not exceed the observed fluxes, a maximum value for the permitted emission measure can be derived. By using the expression for EM, as given in Table 2, this can be translated into an upper limit for . Another constraint follows from the fact that the wind must not contribute substantially to the interstellar absorption between the star and the observer. There have been no reports that spectral fits of EUV and X-ray data from dMe stars require abnormal column densities. Interstellar absorption is caused by photo-ionization. In the expression for the neutral hydrogen column density given in Table 2 we have only considered the presence of hydrogen. In that way the expression for yields only an lower limit to the absorption by a wind. By assuming that the absorption by the wind must not exceed the canonical value an additional constraint for follows.
In Fig. 5 we show the upper limits for as follow from 1) VLBI (Eq. (18)), 2) the neutral hydrogen absorption , 3) the observed flux in the IUE - SWP band-pass and 4) the observed flux in the ROSAT - PSPC band-pass. It is important to emphasize that each curve, related to a specific instrument, provides the best constraint in a specific temperature range, e.g. for a hot stellar wind () the neutral hydrogen column density would be a rather poor constraint; in this case it is better to use the radio and/or X-ray data. At IUE and VLBI provide good constraints. The figure shows that, as expected, at low temperatures () the neutral hydrogen absorption provides the most stringent constraint. The upper limit for the VLBI source size gives the most important constraint while IUE and ROSAT also provide reasonably useful constraints. Note that the latter constraints were derived assuming that the emission by the wind equals at most the observed flux. If one were to use as a constraint that the allowable wind emission is only a fraction f of the observed flux (in order not to mask the coronal emission), then the IUE and ROSAT curves in Fig. 5 must be multiplied by a factor .
All curves in Fig 5 are calculated for winds without an acceleration region (). The constraints set by the IUE and ROSAT fluxes are proportional so that the presence of an acceleration region near the star can considerably reduce the upper limits for the mass loss shown in the figure. The same applies for the constraint set by the neutral hydrogen column density which (roughly) scales with . Furthermore, all curves are for a terminal wind velocity of 100 km/s. Higher velocities of the wind lead to a proportionally higher vlaue for the allowable mass loss rate.
At low temperatures a large number of atoms in the wind is not fully ionized. Therefore the permitted value for the mass loss rate at temperatures is likely to be at least a factor two lower than indicated in the figure because we did not consider the presence of He, C, N and O in the expression for . In-between the IUE and ROSAT curves, data from the Extreme Ultraviolet Explorer EUVE can probably result in additional constraints given the wavelength range covered by the EUVE. For YZ CMi no EUVE data are however available to us. Also more refined constraints can be obtained by considering the observed fluxes from individual lines in e.g. IUE spectra. For the moment we conclude that if the wind temperature is in the range , the mass loss rates of dMe stars must be . At higher temperatures a safe upper limit is . At these rates the winds will not result in an observable signal in the radio - IR range. Because the frequency at which the spectrum becomes optically thin is proportional to the mass loss rate, will be at least a factor 100 lower than the values given in Table 3. At radio frequencies the flux by the wind is reduced by at least a factor (Eq. (12)).
The only way to infer the presence of such tenuous winds is by considering the effect the wind has on the strength of strong spectral lines at EUV wavelengths. Schrijver et al. (1994) pointed out that strong lines can be subject to resonant scattering. Although scattering does not result in photon destruction, except in the case of branching, these authors demonstrated that, if an asymmetry is introduced between the emitting volume and the scattering volume, the photon flux towards the star can be increased. The photons can then be destroyed at the stellar surface. In the case of a chromosphere/corona embedded in a stellar wind the required asymmetry follows naturally. The photons emitted in the chromosphere or corona are then scattered in the tenuous wind and a fraction is subsequently destroyed upon impact at the stellar surface. Weak lines are not affected by scattering and therefore this effect can lead to a detectable difference between the ratio of line intensities of weak and strong lines and the expected ratio. The optical depth (at the line centre) for scattering is given by
The constants can be found in Schrijver et al. for a number of strong lines. In general their values are in the range . The electron column density of the scattering medium is given by . In the case of a stellar wind we have
where the last step applies to YZ CMi. These expressions show that for mass loss rates of the order of the optical depth for scattering can become unity leading to a detectable effect. On the other hand it can be argued that if the mass loss rate would be of the order of , the optical depth for resonant scattering would be . This would have a dramatic effect on the strong lines in EUV and X-ray spectra which would be strongly attenuated. There is no observational evidence that this occurs. An illustration of the effect of photon scattering, for inferring the presence of a tenuous wind from Procyon, can be found in Schrijver et al. (1996).
Finally we note that Fig. 5 is compiled under the assumption that the wind is isothermal and that the ionization balance reflects the kinetic temperature in the wind. If the temperature of the wind would drop with radial distance, while the ionization balance would be determined by a much higher freezing-in temperature, a different situation arises. In that case we would be dealing with a cool wind which is still e.g. fully ionized (`over-ionized'). This results in a modification of the VLBI constraint. The reason why the VLBI curve turns upward near is that at low temperatures the plasma contains many neutrals and µ becomes large in Eq. (18). If we would be dealing with a cool wind, which is over-ionized because of freezing-in at the base, then µ and in Eq. (18) must be evaluated at the much higher ionization temperature. Taking and (see Table 1) changes the VLBI constraint to
with T now the kinetic temperature of the wind. This shows that for cool over-ionized winds the VLBI constraint almost coincides with the curve at . Of course, when over-ionization due to a frozen-in ionization balance occurs the curve is not relevant anymore because there are no neutrals. But at the same time the VLBI constraint becomes as restrictive as the original constraint at low temperatures.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998