3 The variety of cross-correlation profiles and their frequency of occurrence

For each of the 315 spectra, we compute the CCFs with the K0III and the M4V default templates. A large variety of CCFs are observed among LPVs, ranging from the classical single-peak profile to much more complex profiles (asymmetrical peak, double peak, "noisy'' profiles, etc.). We developed an automatic classification procedure which classifies the CCFs according to their shape and contrast into a finite number of archetypes, thus avoiding any subjectivity. Table 3 provides for each observation the CCF code for the K0- and M4-templates, together with the night number, the Julian date, the phase and, when appropriate, the heliocentric radial velocities as derived by a single or double gaussian fit of the CCF. The different kinds of CCFs observed are as follows:

• Single peak: the most common profile observed is the classical single peak (Fig. 1a). In this case, the minimum of the CCF is supposedly the radial velocity of the star, at least for the non-variable stars. For variable stars, the observed velocity might be perturbed by atmospheric motions and may thus not correspond to the center-of-mass velocity (it is worth noting that these intrinsic radial-velocity variations render the search for binaries among LPVs extremely difficult; see Sect. 4.4).

  Single-peak profiles occur at all phases. 54% of the LPV sample stars showed at least once this kind of profile with the K0-template (28% of the total number of observations). These values amount respectively to 89% and 74% with the M4-template. This cool template thus yields single-peak, well-contrasted CCFs much more often than the K0-template. These profiles are coded "1p'' in Table 3;

• Double peak and asymmetrical peak: the cross-correlation technique permits to very clearly reveal the doubling of the lines despite the severe crowding of the LPV spectra (Fig. 1b). Sometimes, the profile only exhibits an asymmetry, as if the double peak was about to appear (or to disappear) (Fig. 1c).

  A double or asymmetrical peak was observed for 50% of the LPV stars (28% of the total number of observations) with the K0-template. Most of them were not known in the literature to exhibit the line-doubling phenomenon. The doubling is essentially observed around maximum light (between phases -0.1 and 0.3), as can be seen in Fig. 2. With the M4-template, only 23% of sample stars (10% of the total number of observations) showed these kinds of profiles (see Sect. 2.2 of Paper II for a discussion of the physical origin of this difference between the K0- and the M4-templates), and the phases range from -0.2 to 0.6. These profiles are coded "2p'' (double peak) or "ap'' (asymmetrical peak) in Table 3;

• Doubtful single/double peak and noisy profile: the profiles cannot always be unambiguously classified as single or double peak, because of the poor contrast and/or the complex shape of the CCFs. Such profiles can only be classified as "doubtful single profiles'' or "doubtful double profiles'' (Figs. 1d and 1e, coded 1p? and 2p?, respectively, in Table 3). In some cases, the contrast of the CCF is so low that the cross-correlation profile does not even yield any clear dip; no radial velocity can be derived from such noisy CCFs (Fig. 1f), identified as "noisy profiles'' (coded "np'' in Table 3). The distinction between "doubtful single/double-peak profile'' and "noisy profile'' is somewhat arbitrary (compare Figs. 1e and 1f) and depends upon the particular (and somewhat arbitrary) values adopted for the parameters involved in our automatic classification procedure. No attempt to derive radial velocities was made with these kinds of profiles.
  
   

   

  Table 2: Log of observations.

  Date of	Julian Date	Night
  Observations	(2450000+)	Number
  1998 Aug., 05-06	1031.5	N1
  1998 Sep., 03-04	1060.5	N2
  1998 Sep., 28-29	1085.5	N3
  1998 Oct., 06-07	1093.5	N4
  1998 Dec., 23-24	1171.5	N5
  1999 Jan., 26-27	1205.5	N6
  1999 Feb., 23-24	1233.5	N7
  1999 Apr., 22-23	1291.5	N9
  1999 May, 19-20	1318.5	N10
  1999 Jul., 05-06	1365.5	N12
  1999 Jul., 26-27	1386.5	N13
  1999 Sep., 02-03	1424.5	N14
  1999 Sep., 28-29	1450.5	N15
  1999 Nov., 16-17	1499.5	N16
  1999 Dec., 16-17	1529.5	N17
  2000 Jan., 10-11	1554.5	N18
  2000 Feb., 22-23	1597.5	N19
  2000 Feb., 23-24	1598.5	N20
  2000 Apr., 17-18	1652.5	N22
  2000 May, 18-19	1683.5	N23
  2000 Jun., 20-21	1716.5	N24
  2000 Jul., 11-12	1737.5	N25
  2000 Aug., 09-10	1766.5	N26
  2000 Aug., 10-11	1767.5	N27

  
  

  
  

  Figure 1: Examples of the different kinds of CCFs observed: a) single peak: RCet, night N7, phase 0.96, mask M4V; b) double peak: RBoo, night N7, phase 0.98, mask K0III; c) asymmetrical peak: XMon, night N22, phase 0.94, mask K0III; d) doubtful single peak: RCet, night N18, phase 0.89, mask K0III; e) doubtful double peak: RCet, night N4, phase 0.30, mask K0III; f) noisy profile: RCnc, night N3, phase 0.96, mask K0III.

  
  
  

  Figure 2: Phase distributions (i) for the observations which exhibit a double or an asymmetrical peak with the K0-template (solid line) and (ii) for the total set of observations (dashed line).

  
  

  It must be noted that the general structure of the CCFs presented in Figs. 1d and 1e is in fact strikingly similar, the only difference being the respective contrasts of the leftmost and second-to-leftmost peaks. Moreover, a given star (like R Cet; see Table 3 and Fig. 1) may evolve from 1p? CCFs to 2p? CCFs and even to 2p at different phases of its light cycle, thus suggesting that the 1p? and 2p? CCFs of the kind displayed in Figs. 1d and 1e may be intrinsically similar and should in fact be classified together (that conclusion may in fact even be extended to some "np'' profiles, like the one displayed in Fig. 1f whose general structure resembles that of Figs. 1d and 1e). However, in this first analysis of the statistics of the line-doubling phenomenon, it was decided to adopt conservative classification criteria, at the risk of excluding physically-sound data. In particular, the line-doubling statistics presented in Figs. 2, 6-9 relies only on CCFs classified as "2p'' or "ap'' and excludes CCFs classified as "2p?'' and "np''. The possible physical implications of this conservative choice are discussed further in Sect. 4.3.1.1.

  Doubtful or noisy profiles were obtained at least once for 61% of the LPV sample stars (44% of the total number of observations) with the K0-template. Although these profiles were found at all phases, they are preferentially observed around minimum light (when the star is fainter and cooler). With the M4-template, the above percentages turn to only 17% of the stars (most of which are the S-type and C-type stars of the sample) and 16% of the total number of observations.

Figure 2 presents the distribution in phase of the double-peak profiles as compared to the total number of observations. The striking features exhibited by Fig. 2 are (i) the very sharp rise in the fraction of double-peak profiles at phase -0.1, (ii) the total absence of double-peak profiles between phases 0.4 and 0.7, and (iii) the fraction of double-peak profiles remains almost constant ($\sim$40%) between phases -0.1 and 0.3.

Another conclusion that can be drawn at this point concerns the interest of the cross-correlation technique for dynamical studies: as already pointed out in 1, the cross-correlation technique provides a powerful tool to extract double lines despite the severe crowding of the spectra of giant stars (see however the discussion of Sect. 4.3.2.1 relative to late-type LPVs). The study of the spectral variations associated with the pulsation of LPV stars has no more to be restricted to the few clean near-IR spectral lines. The much richer visible spectrum is now accessible as well to these studies, opening great potentialities as illustrated for instance by the tomographic technique described in Paper II.