3. Distributions of velocity dispersions
This simple analytical result has been checked by means of a Monte Carlo study. We have randomly generated a sample of LMC stars. The progenitor mass was drawn in the range according to a Salpeter law. The age of formation was drawn in the range -12 Gyr according to the stellar formation history favoured by Geha et al. (1998). The vertical velocity dispersion was then evolved in time from formation up to now according to Wielen's (1977) relation:
This purely diffusive relation is known to be inadequate to describe velocity dispersions in our Galaxy (Edvardsson et al. 1993). We will however use it in our model, as heating processes in the LMC may be different than those in the galaxy. The LMC is indeed subject to tidal heating by the Milky Way (Weinberg 1999) and has most probably suffered encounters with the SMC . Although this simple relation lacks a theoretical motivation, it will be shown to account for several features of the velocity distributions in the LMC , without being at variance with any observation. The initial velocity dispersion was taken to be 10 km s-1, and the diffusion coefficient in velocity space along the vertical direction to be 300 km2 s-2 Gy- 1 so that our oldest stars have a vertical velocity dispersion reaching up to km s-1. For each star, the actual vertical velocity was then randomly drawn, assuming a Gaussian distribution with width .
In order to compare our Monte Carlo results with the Zaritsky et al. (1999) measurements of the radial velocities of LMC clump stars, we selected two groups of stars according to their position in the HR diagram. Following Zaritsky et al., we use their colour index
so that the RC population is defined by with a magnitude whereas the VRC stars have the same colour index C and brighter magnitudes . In order to infer the colours and magnitudes of the stars that we generated, we used the isochrones computed by Bertelli et al. (1994) for a typical LMC metallicity and helium abundance of and .
A random sample of 190 stars that passed the VRC selection criteria is presented in Fig. 1 where the vertical velocities are displayed. This histogram may be compared to Fig. 10 of Zaritsky et al. (1999) where no VRC star is found with a velocity in excess of 60 km s-1. With the full statistics, our Monte Carlo generated a population of 2,900 VRC objects whose vertical velocity distribution has a RMS of 18 km s-1. The agreement between the Zaritsky et al. observations and our Monte Carlo results is noteworthy. The average age of our VRC sample is 0.87 Gyr.
We also selected a random sample of 75 RC stars whose velocity distribution is featured in Fig. 2. Even with a diffusion coefficient as large as km2 s- 2 Gy-1 so as to comply with a large LMC self-lensing optical depth, our full statistics of 18,000 RC objects has a velocity dispersion of 23 km s-1. This is slightly below the value of km s-1 quoted by Zaritsky et al. Observations are nevertheless fairly scarce with only 75 RC stars. When Zaristsky et al. fitted a Gaussian to the RC radial velocity distribution featured in the Fig. 11 of their paper, they obtained a 95% C.L. dispersion of km s-1 with a large uncertainty. Our Monte Carlo velocity dispersion of 23 km s-1 is definitely compatible with that result. We infer an average age for the RC population of 1.8 Gyr to be compared to our analytical result of 1.95 Gyr. This agrees well with Beaulieu and Sackett's conclusion that isochrones younger than 2.5 Gyr are necessary to fit the red clump. Notice finally that our age estimates for these various clump populations are in no way related to LMC kinematics. They merely result from the postulated Salpeter IMF, the Geha et al. preferred stellar formation history and the Bertelli et al. isochrones.
With this model, 70% in mass of the LMC disk consists of objects whose vertical velocity dispersion is in excess of 25 km s-1, although the average vertical velocity dispersion of RC stars, for instance, is only 23 km s-1.
What about the other measurements? The velocity dispersion of PNs has been found equal to 19.1 km s-1 (Meatheringham et al. 1988). These authors estimate that the bulk of the PNs have an age near 3.5 Gyr. They also note that younger objects are present down to an age of order Gyr. Meatheringham et al. come finally to the conclusion that the indicative age of the PN population is 2.1 Gyr. This value agrees well once again with our analytical estimate. Our Monte Carlo gives a slightly larger value of 2.4 Gyr for the age of the PNs, with a velocity dispersion of 24.7 km s-1. Because the observed sample contains 94 objects, the measured value of 19.1 km s-1 suffers presumably from significant uncertainties.
Quite interesting also are the measurements by Hughes et al. (1991) of the velocity dispersions of LPVs as a function of their age. Their sample of 63 "old" LPVs has a velocity dispersion of km s-1. For the bulk of the LMC populations, we obtain an average velocity dispersion of km s-1. The problem at stake is actually the age of those old LPVs. These stars indeed display an age-period relation. However, Hughes et al. derived this relation from kinematics considerations, using precisely Eq. 9, and postulating the same diffusion coefficient as in the Milky Way. They thus inferred an average age of 9.5 Gyr. Finding instead the position of these stars in a colour-magnitude diagram and using LMC isochrones would have led to a clean determination of the age-period relation. A direct determination of the age of LPVs is nevertheless spoilt by a few biases. Some LPVs are carbon stars and the ejected material around them may considerably dim their luminosities. These stars may also pulsate on an harmonic of the fundamental mode. Both effects lead to an under-determination of their luminosity and hence to an overestimate of their age (Menessier 1999). As a matter of fact, Groenewegen and de Jong (1994) conclude that LMC stars whose progenitor mass is less than 1.15 never reach the instability strip on the AGB. This yields an upper limit on the age of LPVs of Gyr, in clear contradiction with the average age of 9.5 Gyr inferred by Hughes et al. for old LPVs.
Finally, Schommer et al. (1992) have obtained a velocity dispersion of km s-1 for 9 old LMC clusters. Their large error of 10 km s-1 is due to the small size of the sample. It is not clear whether or not these clusters have formed in the disk. If they nevertheless had, they would have undergone a fairly restricted orbital heating with respect to the LMC stars. Those systems and the giant molecular clouds have actually comparable masses and the energy exchange between them does not result in a significant increase of the velocity dispersion of the clusters unlike what happens to the stars.
© European Southern Observatory (ESO) 1999
Online publication: October 14, 1999