Astron. Astrophys. 333, 619-628 (1998)

## 5. Discussion

### 5.1. How many X-ray quiet proper motion candidates might be PMS stars?

All proper motion selected stars observed have magnitudes in the range . To get a rough idea about the bolometric luminosities of these stars, we note that assuming a distance of 145 pc and a mean extinction of , this magnitude range transforms into a range of bolometric luminosities of for a K5 star. Of course, earlier or later spectral types would give lower or higher luminosities, respectively. In the following, we restrict our analysis to this magnitude range. The total number of X-ray quiet proper motion candidates in this magnitude range is 442.

We observed 115 of these 442 X-ray quiet proper motion candidates and could not find any PMS stars among them. This information can be used to derive an upper limit on the possible number of PMS stars in the X-ray quiet proper motion candidate sample. For a statistical description of the problem we can use the hypergeometric distribution: We have a sample of stars, from which we have randomly selected and observed stars. If we assume that there are M PMS stars among the N stars, then the probability P of finding k PMS stars among the n observed stars is given by

With we find for and for . This means that at the 90% (99%) confidence level the number of PMS stars in the sample of X-ray quiet proper motion candidates is smaller than 8 (15).

### 5.2. Completeness of the X-ray selected sample

If we want to estimate the completeness of our X-ray selected sample, we have to take two effects into account. First, Fig. 4 shows that the distribution of proper motions of the new PMS stars is considerably wider than our (very strict) selection criterion for the X-ray quiet proper motion members. This means that our selection criterion was somewhat too strict, since stars with proper motions similar to those of the new PMS stars also have to be regarded as possible members. Thus the number of possible proper motion members is probably larger than 442.

On the other hand, it is important to realize that not all stars with proper motions indicating membership actually are related to Upper Sco. It is clear that some fraction of our proper motion sample, like every proper motion sample, will be made up of field stars, having similar proper motions as the member stars just by chance. This fraction will be larger when the mean proper motion of the association is very near to the mean of the background population of field stars, i.e., when the convergent point is very near to the solar antapex.

Unfortunately, this is the case for the Upper Sco association. In the direction of Upper Sco the reflex motion of the Sun corresponds to proper motions of mas/yr for pc, mas/yr for pc, and mas/yr for pc. This means that field stars at distances between 100 pc and 200 pc tend to have proper motions very similar to the Upper Sco members and thus our sample of proper motion candidates contains many field stars. This effect can be seen in Fig. 4. However, with the available data it is not possible to disentangle the field stars from the proper motion members.

Nevertheless, we can try to roughly estimate the number of possible members in a very conservative way in order to get a reliable upper limit for the fraction of PMS stars among them. We note that all our proper motion candidates and all new PMS stars have and mas/yr. There are 1518 stars fulfilling this requirement and we regard these stars as possible members. In order to take the field star pollution at least partially into account, we subtract the number of 512 stars with and mas/yr. In this way we can roughly correct for the pollution with field stars showing random proper motions. The resulting number of 1006 possible members is a secure upper limit to the number of actual proper motion members, since it still is contaminated by field stars due to the solar reflex motion.

We now can perform the same kind of calculation as above and find that with 90% confidence this sample of 1006 possible members contains no more than 18 PMS stars. This is a secure upper limit on the number of X-ray quiet PMS stars in the magnitude range considered. The number of known PMS stars in this magnitude range is 56. This means that the X-ray selected sample contains at least 75% of all PMS stars.

It should be noted that the completeness of 75% is a very conservative lower limit; our sample is probably complete to a considerably higher degree. On the other hand, we already know that our RASS selected sample cannot be 100% complete, since we have found 2 PMS stars among the stars selected from pointed ROSAT observations. Our best guess for the completeness of the RASS selected sample is 80% - 90%. It would be very hard to obtain a better estimate of the completeness since this would require optical spectroscopy of many hundred stars.

© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998