10 Ages and distances

   

Table 3: Fundamental parameters of NGC 6134 and NGC 3680: interstellar reddening, metallicity, age (Gyr), and distance (kpc). The estimated standard errors are given in parenthesis

Cluster	E(b-y)	[Fe/H]	Age	d
NGC 6134	0.263	0.28	0.69	1.41
	(0.004)	(0.02)	(0.10)	(0.06)
NGC 3680	0.048	0.09	1.48	1.10
	(0.011)	(0.02)	(0.15)	(0.05)

We have used theoretical isochrones calculated using the recent programme by Pols et al. (1998). We have fitted isochrones which rely on standard physics and models which also include overshooting. The programme by Pols et al. (1998) provides V and B-V colours which rely on synthetic spectra by Kurucz et al. (1992). The conversion from the observed Strömgren indices to effective temperatures is made by applying the calibration of Alonso et al. (1996). Note that this calibration includes a term proportional to c₀, hence only stars for which the c₀ index have been inferred are plotted (as described in Sect. 6).

Figure 10: The figures show Pols theoretical isochrones that best fit NGC 6134 and NGC 3680. Dashed lines are standard models while solid lines are models which include overshooting. In the left figure is also plotted a part of the standard model isochrone when shifted -0.75 in V, i.e. the approximate location of the binary star sequence. In the plot to the left the box symbols mark the $\delta$ Scuti stars. In both plots are given the parameters of the model at the turnoff for the overshoot isochrones ( $T_{\rm eff}, M_V$, and $M/M_\odot$) and log age (A), metallicity ([Fe/H]), and distance modulus (V-MV)

We calculate a range of isochrones with the metallicity found in Sect. 8. The isochrones are shifted vertically to fit the lower end of the main sequence, and then the age is changed to fit the turnoff stars. The result for both standard models (dashed line) and overshooting models (solid line) is shown in Fig. 10.

It seems that the isochrones from models including overshoot describe the extended hook in NGC 6134 better than the standard model isochrones. No firm conclusion can be given here for NGC 3680 from the limited sample of stars. However, previous studies of this cluster by Nordström et al. (1997) (Strömgren by photometry) and Kozhurina-Platais et al. (1997) (BV photometry) both agree that isochrones that include some degree of overshoot are needed to explain the observed CMD.

The derived fundamental parameters of the open clusters using the Strömgren indices (metallicity and interstellar reddening) and the age and distance found when using isochrones that include overshoot are quoted in Table 3.

Nordström et al. (1997) have performed isochrone fitting using several models, and also found that isochrones which include some amount of overshooting are needed. They find an age of $1.45\pm0.3$ Gyr when fitting an overshoot isochrone to 12 stars around the turnoff, which is in excellent agreement with our result, i.e. age $1.48\pm0.3$ Gyr. Nordström et al. (1997) determine the distance by fitting the Hyades cluster to the lower main sequence of NGC 3680 and find $V-M_V = 10.5\pm0.3$, which agrees with our somewhat smaller value $V-M_V = 10.2\pm0.2$, which was found by shifting the isochrone to fit the lower main sequence. Kozhurina-Platais et al. (1997) found age $1.6\pm0.15$ Gyr and $V-M_V = 10.2\pm0.3$. Note that they used a somewhat higher value for the interstellar reddening and also used solar metallicity isochrones. It is important to note that the present results and those by Nordström et al. (1997) and Kozhurina-Platais et al. (1997), are obtained by using different isochrone codes. It is reassuring that our results for the age and distance of NGC 3680 agree within the errors.

   

Table 4: Astrophysical parameters for the six $\delta$ Scuti stars in NGC 6134

ID$_{\rm B}$	ID$_{\rm F}$	V	b-y	m1	$\beta$	E(b-y)	$M/M_\odot$	$T_{\rm eff}$	$L/L_\odot$
508	397	13.548	0.422	0.159	2.784	0.256	1.99	7795	18.0
574	348	12.452	0.461	0.104	2.738	0.271	2.41	7476	49.4
616	159	13.179	0.415	0.136	2.778	0.252	2.13	7869	25.3
679	87	13.547	0.424	0.144	2.758	0.236	1.99	7760	18.0
853	161	11.973	0.477	0.120	2.742	0.264	2.47	7370	76.8
906	9	12.266	0.486	0.181	2.698	0.227	2.46	7226	58.7

Kjeldsen & Frandsen (1991) fitted standard model isochrones to their observed CMD of NGC 6134 and found distance modulus V-M_V = 11.25 and age 0.89 Gyr which we will now compare with our results, i.e. $V-M_V = 10.75\pm0.20$ and age $0.69\pm0.10$. Twarog et al. (1997) have determined V-M_V=11.1when using the BV photometry by Kjeldsen & Frandsen (1991) but with different interstellar reddening (Sect. 7) and metallicity (Sect. 8).

The reason for the somewhat higher distance modulus found by Kjeldsen & Frandsen (1991) is only partly explained by the somewhat lower value of E(b-y) found by us, yielding a difference in V of $\Delta V = 4.3 \times 0.059 = 0.25$. Kjeldsen & Frandsen (1991) used Solar metallicity isochrones, i.e. isochrones with somewhat higher M_V and higher $T_{\rm eff}$ (fixed age). On the other hand this was standard isochrone models, i.e. isochrones with higher M_V and lower $T_{\rm eff}$ (fixed age). Consequently, these differences alone do not explain the large difference in distance modulus or age. We note that Kjeldsen & Frandsen (1991) only had a very limited sample of $\sim60$ stars. They also had no individual determination of the interstellar reddening, as we now have, in order to exclude stars that are clearly not members of the cluster. The difference in the quality and quantity of the two data sets explain the differences in the determined age and distance.