5 Line velocities

Another property of the emission line spectra that is useful for comparison with theoretical models of Mira atmospheres is the velocity of the emission lines. This line velocity should represent the velocity of the emission region associated with the shock wave passing through the Mira atmosphere.

Figure 20: Absolute line fluxes plotted against phase in SScl. Note the nearly equal fluxes for H$_\zeta$, H$_\gamma$ and H$_\eta$ at phase $\phi = 0.02$. At phase $\phi =0.75$ the fluxes of newly appearing emission lines are shown. The Balmer line fluxes H $_\gamma ^{F}$, H $_\delta ^{F}$, H $_\zeta ^{F}$ and H$_\eta ^{F}$from Fox et al. (1984) are plotted in grey. At $\phi =-0.07$ H $_\gamma ^{F}$ and H $_\zeta ^{F}$ are coincident while H $_\gamma ^{F}$ and H$_\eta ^{F}$ are equal at $\phi =0.0$ and also nearly coincide with H$_\eta$

In Fig. 24, we plot the velocities (relative the stellar center-of-mass) of representative emission lines against the phase of the pulsation cycle. The velocities are defined at half-height of the line profiles and are obtained from all stars in the sample. When overlying absorption is dominant (for example, at early phases of the H$\delta$ line), no velocity measurement was made.

The plot clearly shows that when the shock emerges from deep in the photosphere, the post-shock emission region has a measured outward velocity of $\sim$10-12 km s^-1. This result is in good agreement with the velocity measured in the infrared (Hinkle 1978; Hinkle et al. 1982, 1984) for deep pulsating layers when converted to center-of-mass velocities (Wood 1987). As phase advances, the emission line velocity decreases until it becomes essentially zero around minimum light. Note that the material whose velocity is measured by the emission lines is associated with the near-shock zone: this is quite different from the deeper, infall zone whose velocity is measured near minimum light by the infrared spectra.

Figure 21: Left: Absolute line fluxes plotted against phase in RRSco. Note that at phase $\phi = 0.29$ the fluxes of SiI 4102Å, FeI 4202Å and FeI 4307Å are nearly equal. At phase $\phi =0.58$ the fluxes of newly appearing emission lines are shown. Balmer line fluxes measured by Fox et al. (1984) are plotted in grey and marked with H $_\gamma ^{F}$, H $_\delta ^{F}$, H $_\zeta ^{F}$ and H$_\eta ^{F}$ (fluxes for H $_\gamma ^{F}$ and H $_\zeta ^{F}$ are coincident at $\phi =-0.04$). Right: Absolute line fluxes plotted against phase in RAql. Note that there are coinciding points at phase $\phi = 0.01$ (MgI 3832Å and FeI 3852Å) as well as $\phi =0.32$ (H$_\zeta$, FeI 4307Å and MgIÅ). The Balmer line fluxes from Fox et al. (1984) are plotted in grey (fluxes for H $_\gamma ^{F}$, H $_\delta ^{F}$, H $_\zeta ^{F}$ are nearly equal at $\phi =-0.01$)

Figure 22: Left: Absolute line fluxes plotted against phase in RCar. Note that there are coincident points for the phases $\phi =-0.09$ (MgI 3838Å and FeI 3852Å), $\phi = 0.11$ (MgI 3829Å and FeI 4202Å) as well as at phase $\phi =0.39$ (FeI 4202Å and MgI 4571Å; FeII 4583Å and [FeII]7F 4287Å; FeI 4461Å and [FeII]7F 4359Å). Right: Balmer line fluxes compared to those of Fox et al. (1984), which are plotted in grey and marked with H $_\gamma ^{F}$, H $_\delta ^{F}$, H $_\zeta ^{F}$ and H$_\eta ^{F}$(coincident fluxes: $\phi =-0.37$ H $_\gamma ^{F}$ and H $_\delta ^{F}$, $\phi =0.16$ H $_\gamma ^{F}$ and H$_\eta ^{F}$ and $\phi =0.21$H $_\gamma ^{F}$ and H $_\delta ^{F}$)

At first sight, it is surprising that in the average Mira the apparent velocity of the emission lines approaches zero half a cycle after the shock emerges from the deep photosphere. Since the lines are still in emission, the shock must still be propagating outward so that the post-shock material, from which the line emission presumably originates, should show a positive outward velocity. (We note in passing that since pulsation in the outer layers can be quite irregular (e.g. Bessell et al. 1996), in an individual cycle the shock could be stalled or even reversed by infalling material from a previous cycle, but, on average, shocks must progress outward.) By the time of minimum light, the shock is far above the photosphere (defined for the present discussion to be at optical depth one). For example, in a typical fundamental mode Mira model (e.g. see the models of Bessell et al. 1996 or Hofmann et al. 1998), the photosphere is at $\sim$240 $R_{\odot}$ while the shock is at $\sim$420 $R_{\odot}$. In a simple geometric model for emission from such a system, neglecting absorption above the photosphere, the emission lines would have square profiles going from + $v_{\rm shock}$ to $-0.82\times v_{\rm shock}$, where $v_{\rm shock}$ would be $\sim$5 km s^-1: only the emission with the most negative velocity is hidden behind the star. We would therefore expect the line emission to be centered close to velocity zero, as observed. Detailed models for the transfer of line photons originating from the shock are needed to make quantitative estimates of shock velocities high in the Mira atmosphere.

Figure 23: Simplified time sequence for the observed emission lines and their total fluxes (in arbitary units)

Figure 24: The velocities of representative emission lines plotted against the phase of the pulsation cycle