3 Palliative age determination method

 We succeed in estimating individual ages from Strömgren photometry (Figueras et al. 1991; Masana 1994; Asiain et al. 1997) for a third of the sample (see Paper II). To rule out any bias coming from an analysis of this sub-sample alone, we propose an empirical palliative age estimation method based on the Hipparcos absolute magnitude and colour for the rest of the sample. On a first step, we use existing ages to draw a plot of ages versus (M_v, (B-V)). A primary age parameter is assigned as the mean age associated to a given range of (M_v, (B-V)) (Sect. 3.1). Then Strömgren age data are used a second time to assign a probability distribution of palliative ages as a function of the primary age parameter (Sect. 3.2).

3.1 Primary age parameter

 Strömgren ages are available for 1077 A-F type stars with spectral types later than A3 (B-V $\geq$ 0.08) because the metallicity cannot be determined for A0-A3 spectral types. With this sub-sample a relation between the absolute magnitude M_v, the colour indice B-V and the age has been calibrated on a grid to extrapolate a less accurate age for the rest of our sample (1900 stars). Since the relation cannot be calibrated with Strömgren ages for spectral types between A0 and A3, we add 30 known very young stars ($\sim$10⁷ yr) belonging to the Centaurus-Crux association (see Sect. 4) which have B-V $\leq$ 0.08. Consequences on the inferred age distribution are investigated in Sect. 3.2. Available Strömgren ages are averaged on a $200 \times 200$ grid, ranging from [-0.5, 2.5] in absolute magnitude and [-0.1, 0.6] in B-V. The result is smoothed and extrapolated to empty cells, where feasible, using a $10\times 10$ moving window. This process produces a primary age parameter (Fig. 2). In Fig. 3, Strömgren age indicators are plotted against this primary age parameter. Obviously, there is not a one to one relation between the primary age parameter and the Strömgren age. Very young Strömgren ages are the most affected by the degeneracy of this relation; while the bulk of primary age parameters older than ${\rm log(age)}=8.7$ are in relatively good agreement with Strömgren ages. Only 14 among the 1900 stars have (M_v,(B-V)) out of the calibrated grid and do not have primary age parameter. On the basis of Fig. 3, we can assign a probability distribution for the Strömgren age to each primary age parameter.

  

Figure 2: Primary age parameter versus (Mv,(B-V)) relationship calibrated with the Strömgren sub-sample. Grey levels represent the log(age) value

  

Figure 3: Primary age parameter versus Strömgren age

3.2 Palliative ages

 

  

Figure 4: Strömgren age distribution obtained with 1077 stars (full line) and palliative age distribution for the same sub-sample (dotted line)

  

Figure 5: Strömgren age distribution obtained with 1077 stars (full line) and palliative age distribution for the remainder, i.e. stars without Strömgren photometry (dotted line)

Given a primary age parameter i, Fig. 3 can be read as giving a discrete probability distribution P(j/i) of Strömgren ages j under i. P(j/i) is given by:

$$P(j/i)=\frac{n(i/j)}{q(i)}\\$$ (8)

where n(i,j) is the number of stars in cell (i,j) and

$$q(i)=\sum_{j}n(i,j).\\$$ (9)

These probabilities are discrete on a grid with an adaptive bin size to keep at least 3 stars per bin. This process produces a palliative age distribution which is free from possible biases affecting the sub-sample of stars with Strömgren photometry for stars with B-V $\geq$ 0.08.

The small deviations from the distribution of original Strömgren ages (Fig. 4) are due to finite bin steps used in the discretisation process. The palliative age distribution of the sub-sample without Strömgren photometry (Fig. 5) shows a great difference for very young ages with respect to the Strömgren age distribution. The great peak at $\sim$10⁷ yr partly results from the poor calibration of the relation between age and (M_v, (B-V)) for B-V $\leq$ 0.08. If the calibration of the relation is realized without the additional Centaurus-Crux stars, we would obtain a frequency, at $\sim$10⁷ yr, of $\sim$ 170 stars instead of $\sim$ 440 stars. It means that there are at least 170 stars of few 10⁷ year old and that the difference of $\sim$270 stars is composed of A0-A3 type stars which may be older. How older are they? Using theoretical isochrones from Bertelli et al. (1994), a relation between the total lifetime of a star at the turnoff point and the color indice (B-V) can be infered. We can show that the total lifetimes of A0-A3 type stars are between 4-6 10⁸ yr, if the metallicity of the stars are between Z=0.008 and Z=0.02. Taking the half lifetime of the star as the most probable age, we obtained, for these 270 stars, an age distribution in the range 2-3 10⁸ yr (${\rm log(age)}=8.3-8.5$). This distribution of these half lifetimes remains very well separated from the older stars in the sample. To summarize, the great peak at 10⁷ yr contains at least 170 stars of $\sim$10⁷ yr and 270 stars for which ages may spread up to 3 10⁸ yr. These stars belong to the younger part of the sample.

The palliative ages are statistical ages. Hence, they sometimes produce artifacts or young ghost peaks in some stream age distributions. Such dummy young peaks will always appear as the weak counterpart of a heavy peak around ${\rm log(age)}=8.7$. Nevertheless palliative ages permit to shed light on the age content of the phase space structures when Strömgren data are sparse.