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Astron. Astrophys. 361, 167-174 (2000)

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4. Discussion

4.1. The relation between light and velocity changes

From the figures it is obvious that the light and velocity changes in semiregular variables are correlated. Most of the light curve irregularities, especially variability in the length of a cycle, are physically linked to a "semiregular" velocity curve. A relation between the size of light and velocity change is indicated by comparison of the velocity amplitude and light amplitude listed in GCVS4 (Hinkle et al. 1997). By using the light curves there is the possibility for a more direct comparison of these two quantities.

Might it be possible to quantify the qualitative impression of the relation between light and velocity changes? Such an attempt is of course restricted by the limited accuracy both in the light curve and in the velocity data, and the small time span covered by the observations. Therefore we consider the following result as preliminary. The comparison of light and velocity amplitudes by Hinkle et al. (1997) gives a mean ratio between light and velocity change of approximately 4 km s-1/mag for the semiregular variables. However, our comparison of simultaneously obtained velocity and light measurements shows that the ratio ranges from 3 to 7 km/s/mag. (For V450 Aql and g Her we did not derive this quantity as velocity and light changes do not correlate as well as the other objects in our sample.) Most of the scatter will originate from the lower time resolution of the velocity data leaving considerable uncertainty in shifting and scaling the velocity curve onto the light curve. While the order of the mean value of both investigations is the same, the large scatter we found in the more detailed comparison indicates a more complex relation between velocity and light change as the latter will be influenced by other parameters, for example temperature variations. Data on bolometric light variations and simultaneous information on the temperature change would be necessary to allow a better quantification of such a relation. Furthermore, due to the obvious changes from cycle-to-cycle the definition of a mean amplitude and an amplitude range will be the more appropriate parameter.

Cummings et al. (1998) presented high-precision relative radial velocities measured in the optical wavelength region for 31 red giants. They also made broad-band photometric observations to differentiate between possible causes of velocity variations. The observed light and radial velocity variations were either in inverted phase, or shifted by a significant percentage (up to 30-40%) of the cycle-length. What they found is in good agreement with our results.

4.2. Indications for the cause of the variability

When identifying the mechanism responsible for the observed variations two features visible in the data must be taken into account. First, all SRVs show their maximum velocity at light minimum and their minimum velocity at light maximum. Second, most of the SRVs investigated here show the puzzling feature that the measured CO velocities are all more negative then the stellar velocity (Lebzelter 1999).

4.2.1. Velocity shift

The observed velocity shift between the high excitation CO lines and the systemic velocity has been mentioned already in Lebzelter (1999). For some stars the velocity reference point has been obtained from blue-optical spectra. While it is not clear whether this spectral range is useable to determine the systemic velocity or not, we note that the same effect is observed for stars, for which the center-of-mass velocity has been derived from microwave thermal CO observations.

Several mechanisms are thinkable for producing such a velocity shift. Some of them have already been discussed by Lebzelter (1999). Pulsation may well produce considerable velocity shifts and asymmetries between maximum outflow and infall velocity. It has been shown from model calculations of AGB stars that traveling waves introduced by the pulsation can persist longer than one light cycle in the stellar atmosphere (e.g. Höfner & Dorfi 1997). These effects are correlated with the strength of the pulsation. As the objects discussed here are mainly low amplitude objects it is not clear yet, if a velocity shift of correct size can be produced by pulsation alone. Models of late type pulsating stars calculated by Bowen (1988) indicate the existence of a warm "chromospheric" layer of outflowing gas (called "calorisphere" by Willson & Bowen 1986). If the CO lines we use in this investigation are formed in such a layer then its outward motion may well explain the observed velocity shift. However, the high excitation CO lines are quite weak, so it is more likely that they are formed deep inside the stellar atmosphere. Furthermore, such a layer is not found in the models by Höfner & Dorfi (1997), therefore its existence is still a matter of debate. A further possible mechanism responsible for the velocity shift are large convective cells on the stellar surface as suggested by Lebzelter (1999). However, detailed modeling of variability due to stellar convection is not yet available for AGB stars.

In the following we will discuss the correlation between light and velocity changes in the light of the possible reasons accounting for the velocity shift.

4.2.2. Pulsation

In this section we discuss the case when the pulsation is considered to be the dominant factor in producing stellar light and velocity changes. In the case of variations driven only by pulsation an assumption must be made to deal with the systematic difference between the observations and the center-of-mass velocity. Possible reasons have been mentioned above and in Lebzelter (1999). However, the further discussion does not depend on what reason is chosen.

Superficially both the light curves and radial velocity curves are in agreement with radial pulsation being the main reason for the light variations. The observed properties, e.g. radial velocity minima occurring very close to maximum light, are typical in radially pulsating stars (RR Lyrae-, high-amplitude [FORMULA] Scuti- and Cepheid variables). In radially pulsating stars the light- and radial velocity curves are in some cases perfectly mirror images (see e.g. Bersier et al. 1994a, 1994b). Our velocity curves are plotted in an inverted form, therefore, they show exactly the same behavior as other radial pulsators. This means that the star reaches its maximum light during the expansion.

However, such a `mirror behavior' suggests that the light variation is caused by radius variation only. From miras we know that the visual light change in late-type stars is dominated by the temperature change via the variations in the strengths of the TiO bands. The effect of the TiO bands is exactly the opposite way, leading to a minimum flux (in the visual) when the star is largest, i.e. has its minimum temperature. In that case a phase shift between the CO line forming layers and the layers producing the visual continuum is necessary to explain the observations reported in this paper. This is in agreement with results from present AGB star models proposing extended atmospheres where different spectral features can originate at completely different radii from the stellar center. Unfortunately, no simultaneous temperature determinations exist for the SRVs of our sample.

Accepting this pulsational approach, we estimated the order of magnitude of the relative radius variation caused by the radial motion. A well observed example is W Cygni. W Cygni's light curve is dominated by two periods ([FORMULA] days, [FORMULA] days, Kiss et al. 1999). These periods, their ratio and theoretical calculations by Ostlie & Cox (1986) imply the following stellar parameters: [FORMULA], [FORMULA], [FORMULA]. The adopted model gives [FORMULA] days and [FORMULA] days for the fundamental and first overtone mode, respectively. Recent models including the coupling with convection (Xiong et al. 1998) give [FORMULA] days and [FORMULA] days (first and third overtones) for [FORMULA], [FORMULA] and [FORMULA]. In either case the periods fit for a stellar radius of [FORMULA]. Considering the data subset around JD 2446000, the radial velocity of W Cygni changed approximately 2 km s-1 in 70 days which gives an order-of-magnitude estimate for the corresponding radial displacement about [FORMULA]. That is 5 percent relative radius change being a reasonable value for a radial pulsator. This is, of course, a very oversimplified estimation. We stress that several different layers with different velocities might contribute to the final CO line profiles and hence to the measured velocities. Therefore radius variations calculated from these velocity variations have to be taken with care (see e.g. Hinkle et al. 1982for a discussion). However, the most important consequence of our simple estimate is that radial pulsation is a realistic explanation for a set of semiregular variables.

4.2.3. Convective cells

As a second approach we assume that the outer atmospheric layers do not pulsate at all. The velocity variations might then originate from convective motion. According to Schwarzschild (1975) and other authors only a small number of convective cells cover the surface of an evolved red giant. Lebzelter (1999) suggested that they might be responsible for a systematic blueshift of the measured velocities. This is based on the assumption that we mainly see the outflowing matter which is hotter and therefore brighter than the matter flowing back onto the star. This outflow has to be by no means constant from the (distant) observers view. While the individual motion within each granule on a star like the sun will average out over the whole stellar disk, the situation is different for the red giants due to the small number of cells on the surface. Beside velocity variations within each individual cell the movement of cells on the stellar surface as well as their evolution will lead to significant variations both in light and velocity integrated over the stellar disk. Velocity variations caused by convective motion are also in agreement with the observed coincidence of maximum outflow velocity and light maximum. If the number of convective cells on the surface increases, the area of matter falling back onto the star, which forms the borders of these cells, increases. Therefore the fraction of the surface covered by infalling matter changes the brightness and the convective velocity shift in the same way. For a velocity variation within an individual convective cell the same effect is achieved as a higher velocity will carry hot matter into higher regions of the atmosphere leading to a brightening of the whole star. To quantify this, detailed simulations are needed. The time scale of the observed variations for the short period objects is in agreement with the convective time scale of about 40 days as noted by Antia et al. (1984).

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Online publication: September 5, 2000