4 The radial velocity curve and the H $\mathsf{\alpha}$ line variation

4.1 Radial velocity curve of MM Herculis

The radial velocity measurements (RV) of MM Her were obtained by cross-correlation of each echelle order of MM Her spectra with spectra of the bright radial velocity standard star $\alpha$ Ari, whose radial velocity is -14.3 kms^-1(Evans 1979). For this purpose we used FXCOR, which is one of the tasks of IRAF.

The wavelength ranges for the cross-correlation of MM Her spectra were selected in order to exclude the H$\alpha$ and Na I D₂ lines, which are contaminated by chromospheric emission. The spectral regions heavily affected by telluric lines (e.g. the $\lambda~6276-\lambda~ 6315$ band of O₂) were also excluded.

Figure 6: Sample of Cross Correlation Functions (CCFs) between MM Her and template spectra ($\alpha$ Ari) at different phases (dots). The Gaussian fits to the two peaks are displayed with a dashed line for the K0 IV component and with a dotted line for the G2 IV one.

Figure 6 shows examples of cross-correlation functions (CCFs) at various orbital phases. RVs of the two components have been obtained by a two-Gaussian fit to the CCFs.

The radial velocity measurements, listed in Table 3 together with their standard errors, are weighted means of the individual values deduced from each order. The usual weight  $W_{\rm i}=\frac{1}{\sigma_{\rm i}^2}$ has been given to each measure.

The standard errors of the weighted means have been computed on the basis of the errors  $\sigma_{\rm i}$ in the RV values for each order according the usual formula (see e.g. Topping 1972). The latter are computed by FXCOR according to the fitted peak height and the antisymmetric noise as described by Tonry & Davis (1979).

The observational points and the associated errors are displayed in Fig. 7 as a function of orbital phase (dots for the hotter star and open circles for the cooler one) as determined by means of the ephemeris based on the photometric times of primary eclipse (Eq. (1)). The sinusoidal solution (dashed and solid lines for the K0 and the G2 component respectively in Fig. 7), determined for a circular orbit, fits very well all the observations.

The orbital parameters of the system, derived from the radial velocity solution and from the solution of the light curve (Evren 1987b), are listed in Table 4, where c and h indices refer to the cooler and the hotter component of the system, respectively.

From blue-violet coudé spectrograms, Imbert (1971) found a systemic velocity of $-50.8\,\pm\,1.0~{\rm km\,s^{-1}}$. He deduced the velocities of the cooler component only from the Ca  II H & K emission cores. Popper (1988) published radial velocity measurements essentially based on Lick blue-violet spectrograms. He measured the RV of the hotter star, which contributes about the 80% of the total light at these wavelengths, by means of several absorption lines, while he used the Ca  II H & K emission lines for the cooler component, with the addition of few measurements performed onto red orticon spectra. From his own RV's and a re-analysis of Imbert's data, he adopted $K_{\rm h}=74.0~{\rm km\,s^{-1}}$, $K_{\rm c}=70.5~{\rm km\,s^{-1}}$ and $\gamma=-51.5\,\pm\,1.0~{\rm km\,s^{-1}}$. Hall & Ramsey (1992), using 13 echelle spectra, found the systemic velocity to be $-54.2\,\pm\,3.3~{\rm km\,s^{-1}}$. Our systemic velocity ( $\gamma=-52.6\,\pm\,0.2~{\rm km\,s^{-1}}$) is consistent, within the errors, with previous determination reported above. So, it appears that the maximum variation of $\gamma$ from 1967 (epoch of the first RV curve) up to now does not exceed $3.4~{\rm km\,s^{-1}}$.

The radial velocity amplitudes $K_{\rm h}=72.88~{\rm km\,s^{-1}}$, and $K_{\rm c}=68.78~{\rm km\,s^{-1}}$ we obtain are a little smaller than the Imbert (1971) and Popper (1988) ones, but in good agreement with Hall & Ramsey (1992) values. Our mass ratio q=0.944 leads to masses a little different from one another and as found by previous authors, but in better agreement with the spectral type (G2 V, K0 IV) and the evolutionary stage of the two components, i.e. the G2 component is still on the main sequence while the more massive K0 star has already become a subgiant.

Figure 7: The radial velocity curve of MM Her for 1999. Open circles: RV's of the cool component. Filled circles: RV's of the hotter one. Crosses: RV values obtained near the secondary eclipse, when the spectral lines of the two components are heavily blended. The error bars are always smaller than the symbol dimension and are barely visible inside the open circles. The best-fit solutions are drawn with dashed and continuous line for the cool and hot component, respectively.

 

 

Table 3: Radial velocity measurements of MM Her.

HJD	Phase	$V_{\rm c}$	$V_{\rm h}$
(2400000+)		(km s-1)	(km s-1)
51380.4224	0.2558	$15.76 \pm 0.85$	$-126.03 \pm 0.25$
51381.4758	0.3881	$-7.48 \pm 0.86$	$-101.55 \pm 1.26$
51382.3496	0.4979	--	$-52.53 \pm 0.41$
51382.3914	0.5032	--	$-53.09 \pm 0.26$
51382.4332	0.5084	--	$-54.00 \pm 0.59$
51384.3957	0.7550	$-120.59 \pm 0.25$	$20.72 \pm 1.30$
51385.4555	0.8881	$-97.21 \pm 0.89$	$-3.72 \pm 1.55$
51386.3972	0.0064	$-49.54 \pm 0.54$	-
51386.4466	0.0126	$-48.31 \pm 0.64$	-
51388.4165	0.2601	$17.02 \pm 1.11$	$-125.23 \pm 1.54$
51390.3442	0.5022	--	$-53.17 \pm 0.44$
51390.3956	0.5087	--	$-53.65 \pm 0.45$
51422.3309	0.5205	--	$-53.17 \pm 0.28$
51423.3422	0.6475	$-105.92 \pm 0.43$	$6.75 \pm 0.70$
51427.2964	0.1443	$2.88\pm 0.58$	$-111.13 \pm 0.62$
51427.3484	0.1508	$4.21\pm 0.81$	$-112.17 \pm 0.92$

 

 

Table 4: New orbital parameters of MM Her.

$\gamma$ (kms-1)	$-52.59 \pm 0.19$
$K_{\rm h}$ (kms-1)	$72.88 \pm 0.23$
$K_{\rm c}$ (kms-1)	$68.78 \pm 0.24$
$M_{\rm h}/M_{\rm c}$	$0.944 \pm 0.005$
$a_{\rm h}$sin i (km)	$7.978~10^{6} \pm 2.5\times 10^{4}$
$a_{\rm c}$sin i (km)	$7.529~10^{6} \pm 2.7\times 10^{4}$
$M_{\rm h}\sin^{3} i~(M_{\rm\odot})$	$1.139 \pm 0.009$
$M_{\rm c}\sin^{3} i~(M_{\rm\odot})$	$1.206 \pm 0.009$

4.2 H $\mathsfsl{\alpha}$ emission

The H$\alpha$ line is an important indicator of chromospheric activity. It shows itself as an emission feature above the continuum in very active stars (e.g. II Peg, V711 Tau, UX Ari, XX Tri, AR Psc); in less active stars only a filled-in absorption line is observed.

The situation is more complex in a double-lined system in which both spectra are simultaneously seen and shifted at different wavelength, according to the orbital phase. Therefore, in order to extract a valid information about the chromospheric contribution a comparison is needed with a "synthetic'' spectrum constructed with two stellar spectra that mimic the two components of the system in absence of activity.

For MM Her, the complete filling of the H$\alpha$ line coming from the cooler component was first emphasized by Hall & Ramsey (1992) by means of the spectral synthesis technique. They also found in the difference spectra a faint emission feature corresponding to the G component at phases of good wavelength separation of the two lines, and a small extra-absorption at $\phi=0.034$.

Montes et al. (1997) observed in MM Her a relevant emission excess from the H$\alpha$ line of the cooler component, but no emission coming from the hotter one. They have no spectra near the primary eclipse.

We observed MM Her during eleven nights, acquiring a total of 16 spectra. In addition to MM Her, some inactive stars of spectral type similar to that of each component of the system have been observed. We have chosen 51 Peg (G2.5 IV) to reproduce the hotter component, and $\delta$ Eri (K0 IV) for the cooler one. The spectra of 51 Peg and $\delta$ Eri have been broadened by convolution with the appropriate rotational profile ( $v\sin i= 10~{\rm km\,s^{-1}}$ and $18~{\rm km\,s^{-1}}$ for the hot and cool component, respectively, Strassmeier et al. 1993a) and then have been co-added, properly weighted and Doppler-shifted according to the RV solution derived in the previous subsection.

The contributions of the two stars to the combined spectrum at the H$\alpha$wavelength have been evaluated from the relative areas of the Gaussians fitted to the cross-correlation peaks of each component (see Fig. 6). Average contributions of 0.40 and 0.60, for the hotter and cooler star respectively, have been obtained. These weights are in agreement with those derived from photometric elements (radii and effective temperatures) of the WINK solutions by Evren (1987b), in which a temperature of 5770 K has been assigned to the hotter component, in agreement with its spectral type and color indices.

During the eclipses these weights have been properly corrected, taking into account the light contribution of each star at each phase.

Furthermore, since the light of the cooler active star changes with phase due to starspots, the relative weights of the two components to the observed spectrum change accordingly. In order to also take into account this effect, we evaluated the true weights, deriving them from the simultaneous out-of-eclipse R light curve, corrected for the light offset of the inactive G star. The final weights of the cool component ($w_{\rm C}$) for each observation phase are listed in Table 5. To define the net H$\alpha$ emission of the two components we have subtracted the synthetic spectrum from each MM Her spectrum. In the difference spectrum the absorption lines cancel out and the excess emission of the cool component in the H$\alpha$ core appears as a positive residual well above the noise (Fig. 8). In this figure, a sample of H$\alpha$ profiles at different phases around the two quadratures is shown. In the left-hand panels the observed spectra are displayed by a thick line, while the thin line reproduces the synthetic ones. In the right-hand panels the differences are shown. The phases of observations and the wavelength of the H$\alpha$centers of the hot and cool component are also marked. Figure 9 displays a sample of spectra near to the two eclipses, together with the configuration of the system at the observational phase, as viewed by an observer from Earth.

In the difference profile (Figs. 8, 9) we do not find any evidence of emission from the G2 V component, in agreement with Montes et al. (1997). The slight emission from the G2 V component claimed by Hall & Ramsey (1992) may result from the choice of the inactive standard star $\kappa$ Del (G5 IV), whose spectrum does not well represent the G component of MM Her. Moreover, it seems that they do not take into account the change of the relative weights of the two stars due to the intrinsic variability of the K0 star; at least there is no mention made in their paper.

Our spectrum at phase 0.013, the closest to the one at 0 $\hbox{$.\!\!^{\scriptscriptstyle\rm p}$ }$034, in which Hall & Ramsey (1992) found evidence of extra-absorption, does not show any sign of excess absorption. We do not find evidence of extra-absorption at any other phase.

 

 

Table 5: H$\alpha$ equivalent widths measurements of MM Her obtained in 1999.

HJD (2400000+)	Phase	$w_{\rm C}$	EW (Å)
51380.4224	0.2558	0.574	0.664$\,\pm\,$0.095
51381.4758	0.3881	0.582	0.665$\,\pm\,$0.125
51382.3496	0.4979	0.410	0.415$\,\pm\,$0.085
51382.3914	0.5032	0.421	0.486$\,\pm\,$0.125
51382.4332	0.5084	0.446	0.495$\,\pm\,$0.139
51384.3957	0.7550	0.600	0.749$\,\pm\,$0.124
51385.4555	0.8881	0.587	0.741$\,\pm\,$0.121
51386.3972	0.0064	0.953	1.328$\,\pm\,$0.191
51386.4466	0.0126	0.813	0.995$\,\pm\,$0.208
51388.4165	0.2601	0.574	0.634$\,\pm\,$0.085
51390.3442	0.5022	0.413	0.383$\,\pm\,$0.105
51390.3956	0.5087	0.448	0.477$\,\pm\,$0.097
51422.3310	0.5205	0.550	0.548$\,\pm\,$0.094
51423.3422	0.6475	0.599	0.751$\,\pm\,$0.076
51427.2964	0.1443	0.571	0.616$\,\pm\,$0.072
51427.3484	0.1508	0.571	0.653$\,\pm\,$0.086

The net equivalent width (EW) of the H$\alpha$emission has been evaluated on the difference spectra integrating along the residual emission profile. The errors on the measured EW were estimated determining the S/N in two windows on the right and left-hand side of H$\alpha$ in the difference spectrum and multiplying it by the width of the integration range.

Figure 8: Sample of H$\alpha$ spectra of MM Her acquired around the two quadratures. In the left-hand panels the observed spectra are displayed by a thick line, while the synthetic ones are reproduced by thin lines. In the right-hand panels the differences are shown. The phases of observations and the wavelength of the H$\alpha$centers of the hot and cool component are also marked.

Figure 9: Sample of H$\alpha$ spectra of MM Her acquired near the two eclipses. The system configuration, as viewed from the Earth, is sketched on the right-hand side.

Equivalent widths (EW) calculated from emission excess are listed, together with their corresponding JD, phase, and weights, in Table 5.

Since these EWs are relative to the local continuum that is a mixture of the G and K star spectra whose weights vary with phase, to put the EW in a uniform scale in units of the K-star continuum, we need to correct these values dividing by its actual contribution to the composite spectrum ($w_{\rm C}$). The corrected EWs are plotted versus orbital phase in Fig. 10. Different symbols have been used for in-eclipse values.

From Fig. 10 the EW measurements seem to vary with phase by about 0.15 Å, or about 14%, i.e. of the same size of the errors.

However, the average distribution of the corrected emission EW values (Fig. 10, upper panel) seems to indicate a lower emission at the first quadrature (phases $0.0 \div 0.5$) and a little higher emission at the second quadrature (phases $0.5\div 1.0$). It is not clear if this difference is evidence of a real rotational modulation or the result of other effects.

In the lower panel of Fig. 10 we display the V light curve of MM Her obtained at the Ege University Observatory in 1999, at about the same time of the spectrographic observations. We can notice that the phase variation of the H$\alpha$ emission is similar to that of the V light curve, i.e. the emission is higher at the phases at which the star is brighter. If this direct correlation is real, one has to suppose that the H$\alpha$ faculae are more concentrated on the brighter hemisphere of the K star, and also that the V light maximum is due to faculae contribution. Although this anti-correlation will be clarified by UV observations, interpreting the Mg II or C IV line flux may indicate not only the presence of long-lived optical spot groups on one hemisphere of the cool component but also a corresponding long-lived plage region. This, however, is in contrast with the blueing in the B-V at minimum light (Paper I) attributed to bright faculae overlaying the cool spots.

We have already analyzed the contemporaneous variation of H$\alpha$ emission and the photometric wave in RS CVn binaries, finding in many cases that the higher emission occurs at the wave minimum, suggesting a close association of chromospheric plages with photospheric spots (Catalano et al. 1996, 2000). A similar solar-like scenario has been shown by several authors for active binaries (e.g. Weiler 1978; Bopp & Talcott 1978; Ramsey & Nations 1984; Rodonò et al. 1987; Doyle et al. 1989; Strassmeier 1994) as well as for single stars (Strassmeier et al. 1993b; Frasca et al. 2000b). But also cases of stars with a well defined photometric wave, that generally show normal solar-type correlation and sometimes do not show any H$\alpha$ modulation, have been found (see Catalano et al. 2000).

Although a positive correlation between photometric wave and H$\alpha$ emission in the present data of MM Her may not be excluded, we would like to remark that a similar difference of H$\alpha$ emission at the two quadratures has been found for the G2 IV component of AR Lac (Frasca et al. 2000a). In the AR Lac case the effect was associated with the extra-absorption effect clearly detected in the H$\alpha$ profile before and during primary eclipse. In the case of MM Her, evidence of extra-absorption have been found by Hall & Ramsey (1992) after the primary eclipse. The lower emission of the K0 IV component at the first quadrature seems to be consistent with extension of absorbing matter above the trailing hemisphere of that star.

Another interesting similarity/difference between MM Her and AR Lac is given by the behaviour of the H$\alpha$ emission. Both systems have components of similar spectral types (G2 V-IV/K0 IV) but different orbital period. In AR Lac both components display chromospheric activity in Ca II, Mg II and H$\alpha$ (see e.g. Frasca et al. 2000a). This fact could be explained by the stronger dynamo action on the hotter component of AR Lac, because the orbital and rotational period, the system being synchronous, is shorter than that of MM Her (2 days for AR Lac versus 8 days for MM Her). But according to this picture, we should expect a stronger H$\alpha$ emission also from the K component of AR Lac, while the reverse is true. The excess emission EW from the K0 IV of AR Lac after renormalization to the star continuum is about 0.3 Å (the value reported in Frasca et al. 2000a is referred to the system combined continuum), while the emission of the cool component of MM Her is about 1.1 Å. The two stars are very similar, the mass and radius of the K0 star being $M=1.27~M_{\odot}$and $R=2.89~R_{\odot}$ for MM Her (Evren 1987b), and $M=1.33~M_{\odot}$and $R=2.72~R_{\odot}$ for AR Lac (Frasca et al. 2000a). So, the only significant difference is the orbital period, i.e. the system separation. For some reason the tidal effect tends to suppress the chromospheric H$\alpha$ emission, the rotation period working in the opposite sense to what we expect from dynamo theory.

Figure 10: H$\alpha$ EW a) and V light b) variation of MM Her in 1999. In the upper panel, data acquired near the primary and secondary eclipse are plotted with open squares and asterisk, respectively.