Stellar Chromospheric Activity and Age Relation from Open Clusters in the LAMOST Survey

The LAMOST telescope is a special reflecting Schmidt telescope (Cui et al. 2012; Zhao et al. 2012; Luo et al. 2015). Its primary mirror (Mb) is 6.67 m × 6.05 m and its correcting mirror (Ma) is 5.74 m × 4.40 m. It adopts an innovative active optics technique with 4000 optical fibers placed on the focal surface. It can obtain spectra of 4000 celestial objects simultaneously, which makes it the most efficient spectroscope in the world. In 2019 June, the LAMOST official website⁷ released five data releases (DR5_v3) to international astronomers. DR5_v3 has 9,026,365 spectra for 8,183,160 stars, 153,863 galaxies, 52,453 quasars, and 637,889 unknown objects. These spectra cover the wavelength range of 3690–9100 Å with a resolution of 1800 at the 5500 Å. DR5_v3 also provides stellar parameters such as effective temperature (T_eff), metallicity ([Fe/H]), surface gravity (log g), and radial velocity (RV) for millions of stars. The typical error for T_eff, [Fe/H], log g, and RV are 110 K, 0.19 dex, 0.11 dex, and 4.91 km s⁻¹, respectively (Gao et al. 2015). In this work, we measure CA indices $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ and $\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }$ for the Ca ii K and Hα lines. Our used spectra and stellar parameters including T_eff, [Fe/H], log g, and RV are all from DR5_v3.

Cantat-Gaudin et al. (2018) provided 401,448 member stars with membership probability of 1229 open clusters in Gaia DR2. We select those member stars with membership probability >0.6. In addition, Melotte 25 (Hyades) is added from Röser et al. (2019). The celestial coordinates of these member stars are used to cross-match with the LAMOST general catalog. Only dwarfs (log g > 4.0) with 4000 K < T_eff < 7000 K and signal-to-noise ratios (S/N) that satisfied some limits are selected. For a star with multiple spectra, only the spectrum with highest S/N is retained. For the Ca ii K line, 1240 spectra of 89 open clusters with S/N g (S/N in g band) >30 remain. For the Hα line, 1305 spectra of 93 open clusters with S/N r (S/N in r band) >50 remain. Table 1 lists these open clusters. The information about member stars can be found on online materials. The ages of these open clusters are from literatures as shown in Table 1. For most open clusters, their ages are from Kharchenko et al. (2013). However, the ages of eight open clusters are not found in literatures, which are not used to derive CA–age relations. Cluster ages are given as log t, where t is in units of year. We calculate average [Fe/H] for each open cluster as shown in Table 1.

Table 1.  Open Clusters

Name	J2000RA	J2000DEC	log t	References	${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{LA}}$	$[\mathrm{Fe}/{\rm{H}}]\_{\mathrm{std}}_{\mathrm{LA}}$
NGC_2264	100.217	9.877	6.75	1	0.234
Collinder_69	83.792	9.813	6.76	1	−0.014	0.1723
IC_348	56.132	32.159	6.78	1	−0.024	0.1755
NGC_1333	52.297	31.31	6.8	1	0.196
ASCC_16	81.198	1.655	7.0	1	−0.102	0.1223
Stock_8	81.956	34.452	7.05	1	−1.189	1.0445
ASCC_21	82.179	3.527	7.11	1	0.053	0.0515
Collinder_359	270.598	3.26	7.45	1	0.034	0.1754
ASCC_19	81.982	−1.987	7.5	1	−0.123	0.088
Stephenson_1	283.568	36.899	7.52	1	0.2	0.0565
NGC_1960	84.084	34.135	7.565	1	−0.246	0.0853
Alessi_20	2.593	58.742	7.575	1	0.057	0.1974
Dolidze_16	78.623	32.707	7.6	1	−0.012	0.036
IC_4665	266.554	5.615	7.63	1	0.09	0.0831
Melotte_20	51.617	48.975	7.7	1	0.041	0.1049
NGC_2232	96.888	−4.749	7.7	1	0.019	0.1087
ASCC_13	78.255	44.417	7.71	1	−0.086
ASCC_114	324.99	53.997	7.75	1	−0.127
ASCC_6	26.846	57.722	7.8	1	−0.039
FSR_0904	91.774	19.021	7.8	1	−0.178	0.0585
Alessi_19	274.741	12.311	7.9	1	0.073	0.0512
ASCC_105	295.548	27.366	7.91	1	0.059	0.0405
Trumpler_2	39.232	55.905	7.925	1	−0.027	0.0365
ASCC_113	317.933	38.638	7.93	1	−0.011
NGC_7063	321.122	36.507	7.955	1	−0.093	0.0064
NGC_7243	333.788	49.83	7.965	1	−0.068	5.0E-4
ASCC_29	103.571	−1.67	8.06	1	−0.234	0.008
NGC_7086	322.624	51.593	8.065	1	0.004
Alessi_37	341.961	46.342	8.125	2	0.057	0.0213
Melotte_22	56.601	24.114	8.15	1	0.006	0.1241
Alessi_Teutsch_11	304.127	52.051	8.179	2	0.029
NGC_2186	93.031	5.453	8.2	1	−0.233
NGC_2168	92.272	24.336	8.255	1	−0.062	0.0759
Gulliver_20	273.736	11.082	8.289	2	−0.097	0.013
FSR_0905	98.442	22.312	8.3	1	−0.154
NGC_1647	71.481	19.079	8.3	1	0.025	0.0718
ASCC_11	53.056	44.856	8.345	1	−0.081	0.037
NGC_1912	82.167	35.824	8.35	1	−0.162	0.0342
NGC_2301	102.943	0.465	8.35	1	−0.02	0.0675
NGC_744	29.652	55.473	8.375	1	−0.213
NGC_1039	40.531	42.722	8.383	1	−0.027	0.1309
Stock_10	84.808	37.85	8.42	1	−0.09	0.0736
ASCC_108	298.306	39.349	8.425	1	−0.046	0.0589
Stock_2	33.856	59.522	8.44	1	−0.056	0.0694
Czernik_23	87.525	28.898	8.48	1	0.157
ASCC_23	95.047	46.71	8.485	1	−0.032	0.074
FSR_0985	92.953	7.02	8.5	1	0.056
Stock_1	294.146	25.163	8.54	1	0.048
NGC_1528	63.878	51.218	8.55	1	−0.145	0.0415
NGC_2099	88.074	32.545	8.55	1	−0.017	0.0549
Ferrero_11	93.646	0.637	8.554	2	−0.103	0.044
NGC_7092	322.889	48.247	8.569	1	−0.29
NGC_1342	52.894	37.38	8.6	1	−0.155	0.0809
NGC_1907	82.033	35.33	8.6	1	−0.18
NGC_2184	91.69	−2.0	8.6	1	−0.074	0.082
NGC_1750	75.926	23.695	8.617	2	−0.017	0.0846
ASCC_12	72.4	41.744	8.63	1	−0.085
NGC_6866	300.983	44.158	8.64	1	0.019	0.067
NGC_1582	67.985	43.718	8.665	1	−0.072	0.0742
Roslund_6	307.185	39.798	8.67	1	0.024	0.0797
NGC_1662	72.198	10.882	8.695	1	−0.111	0.0865
Alessi_2	71.602	55.199	8.698	1	−0.01	0.0618
ASCC_41	116.674	0.137	8.7	1	−0.093	0.0774
Collinder_350	267.018	1.525	8.71	1	−0.034	0.0923
ASCC_10	51.87	34.981	8.717	1	−0.02	0.049
NGC_2548	123.412	−5.726	8.72	1	−0.026	0.0624
NGC_1758	76.175	23.813	8.741	2	−0.013	0.0523
NGC_1664	72.763	43.676	8.75	1	−0.082	0.0602
NGC_1708	75.871	52.851	8.755	1	−0.065	0.0236
NGC_6633	276.845	6.615	8.76	1	−0.098	0.0373
NGC_2281	102.091	41.06	8.785	1	−0.033	0.1
IC_4756	279.649	5.435	8.79	1	−0.087	0.0803
NGC_2194	93.44	12.813	8.8	1	−0.008
NGC_1545	65.202	50.221	8.81	1	−0.001
Dolidze_8	306.129	42.3	8.855	1	0.025
Melotte_25	66.725	15.87	8.87	3	−0.003	0.13
NGC_1817	78.139	16.696	8.9	1	−0.205	0.0936
NGC_2355	109.247	13.772	8.9	1	−0.248	0.0538
NGC_2632	130.054	19.621	8.92	1	0.187	0.1053
King_6	51.982	56.444	8.975	1	−0.055
NGC_6811	294.34	46.378	9.0	4	−0.08	0.0768
NGC_1245	48.691	47.235	9.025	1	−0.186
NGC_752	29.223	37.794	9.13	1	−0.067	0.0825
Koposov_63	92.499	24.567	9.22	1	−0.016
NGC_7789	−0.666	56.726	9.265	1	−0.145	0.1199
NGC_2112	88.452	0.403	9.315	1	−0.138	0.0311
NGC_2420	114.602	21.575	9.365	1	−0.278	0.0462
NGC_2682	132.846	11.814	9.535	1	−0.003	0.0566
Gulliver_22	84.848	26.368			−0.225
Gulliver_25	52.011	45.152			−0.009
Gulliver_6	83.278	−1.652			−0.049	0.1697
Gulliver_60	303.436	29.672			−0.12	0.002
Gulliver_8	80.56	33.792			−0.266	0.0745
RSG_1	75.508	37.475			−0.014	0.0834
RSG_5	303.482	45.574			0.098	0.093
RSG_7	344.19	59.363			0.014	0.012

Note. The first column is name of open clusters. The second and third columns are mean R.A. and decl. (J2000) of member stars and they are in units of (^◦). Mean R.A. and decl. of all clusters but Melotte 25 are from Cantat-Gaudin et al. (2018), while the coordinates of Melotte 25 are from Dias et al. (2014). The fourth column is ages of clusters, which are represented by log t, where t is in units of yr. The fifth column is references from which the ages are cited: (1) Kharchenko et al. (2013), (2) Bossini et al. (2019), (3) Gossage et al. (2018), (4) Sandquist et al. (2016). However, the ages of eight open clusters are not found in literatures. The sixth and seventh columns are the mean value and standard deviation of [Fe/H] of each open cluster. Some clusters have no standard deviation, which means they are represented by only one member star. Note that Stock 8 has [Fe/H]_LA = −1.189. The cluster has only two member stars, of which one has [Fe/H] = −2.23. This star should not be a member star of the cluster. We do not calculate its log R' values.

Download table as:  ASCIITypeset images: 1 2

$$\begin{eqnarray}&&\mathrm{EW}=\int \displaystyle \frac{f(\lambda )-f({\lambda }_{c})}{f({\lambda }_{c})}d\lambda \end{eqnarray} \tag{ 1 }$$

As an example, the EW of the Ca ii K line is measured by using Equation (1). Here, f(λ_c) denotes the pseudo-continuum flux. To measure the EW of the Ca ii K line, we integrate the line flux from 3930.2 to 3937.2 Å. The pseudo-continuum flux f(λ_c) is estimated by interpolating the flux between 3905.0–3920.0 Å and 3993.5–4008.5 Å. These values for Ca ii K and Hα can be found in Table 2.

Figure 1 shows the spectra of an active star (dotted line) and an inactive star (solid line). V and R represent wavelengths of violet and red pseudo-continua, respectively. The labels "CaK" and "Hα" illustrate the wavelength regimes of Ca ii K and Hα lines, respectively. EW are measured from RV corrected spectra. Figure 2 plots EW_CaK versus T_eff and ${\mathrm{EW}}_{{\rm{H}}\alpha }$ versus T_eff. Color is used to represent for the ages of open clusters.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We use a simple Monte Carlo simulation to obtain the error of EW. Detailed information can be found in Appendix A. For the Ca ii K line, the average error of EW_CaK is about 0.1 Å when T_eff > 5500 K. The average error of EW_CaK is about 0.2 Å at T_eff = 4500 K. For the Hα line, the average error of ${\mathrm{EW}}_{{\rm{H}}\alpha }$ is about 0.04 Å.

From Figure 2(a), it is clear that as T_eff decreases, EW_CaK first decreases and then increases. Figure 2(b) shows that as T_eff decreases EW_Hα increases. As for the two panels, at high T_eff range (T_eff > 6500 K), member stars of different age populations mix. As T_eff decreases, young member stars tend to have larger EW than old member stars, which conforms to our expectation. The scatter of EW increases as T_eff decreases. Most member stars in our sample have T_eff > 5500 K.

To measure ${\mathrm{EW}}_{\mathrm{CaK}}^{{\prime} }$ and $\mathrm{EW}{{\prime} }_{{\rm{H}}\alpha }$, basal lines are needed. The LAMOST official website provides the AFGK type (spectral type) star catalog. In this catalog, stars that satisfy 4000K < T_eff < 7000 K, log g > 4.0, and S/N g > 30 (S/N r > 50), the same limitations as member stars, are selected. Further, only stars with −0.8 < [Fe/H] < 0.5 are selected because at poor [Fe/H] range ([Fe/H] ≤ −0.8) there is a negative correlation between EW_CaK and [Fe/H]. We calculate their EW_CaK and EW_Hα by the same method described in Section 3.1. Those stars with EW > 10 Å or EW < −10 Å are excluded. Figure 3 shows EW_CaK versus T_eff and EW_Hα versus T_eff. This plot includes 1,563,898 stars for EW_CaK and 1,581,197 stars for EW_Hα. Few stars are located at about 4570 K. This may be caused by a defect in the LAMOST pipeline at this T_eff.

Zoom In Zoom Out Reset image size

Because [Fe/H] has more effects on EW_CaK than EW_Hα, we classify stars into three classes according to [Fe/H]: −0.8 < [Fe/H] < −0.2, −0.2 ≤ [Fe/H] < 0.1, and 0.1 ≤ [Fe/H] < 0.5 before determining the basal lines for EW_CaK. For each class, we rebin the stars on T_eff with a bin width of 50 K. In each bin, a 10% quantile is calculated in EW_CaK. Then, five order polynomial is fitted to these quantiles as shown in the left panel of Figure 3. The solid lines are fitting curves. From the left panel we can see that the three basal lines corresponding to three different [Fe/H] ranges have some differences. When T_eff > 6000 K, the basal line of the poor [Fe/H] range is above that of the rich [Fe/H] range. Generally speaking, poor [Fe/H] stars have larger EW_CaK than rich [Fe/H] stars because for poor [Fe/H] stars' metallic lines are relatively shallow. Note that our EW is negative for an absorption line. For Hα, we directly rebin the stars on T_eff with a bin width of 50 K without [Fe/H] classification and the basal line is obtained by the same method as above. The right panel of Figure 3 shows the basal line for EW_Hα. Then for member stars, ${\mathrm{EW}}_{\mathrm{CaK}}^{{\prime} }$ and $\mathrm{EW}{{\prime} }_{{\rm{H}}\alpha }$ are obtained by using Equation (2). Stars whose $\mathrm{EW}^{\prime} \gt 0$ are retained in our study.

$$\begin{eqnarray}&&\mathrm{EW}^{\prime} =\mathrm{EW}-{\mathrm{EW}}_{\mathrm{basal}}.\end{eqnarray} \tag{ 2 }$$

After obtaining EW', Equations (3) and (4) are used to calculate the excess fractional luminosity R'. In Equation (3), ATLAS9 model atmospheres⁸ are used to calculate χ of grid points. F(λ) is flux at the stellar surface. We choose λ = 3950.5 Å for Ca ii K and λ = 6560.0 Å for Hα. σ is the Stefan–Boltzmann constant. Turbulent velocity (vturb = 2.0 km s⁻¹) and mixing length parameter (1/H = 1.25) are adopted for the model spectra. We calculate χ in terms of T_eff, [Fe/H] and log g. T_eff ranges from 4000 K to 7000 K in steps of 250 K. The values of log g range from 4.0 to 5.0 in steps of 0.5. [Fe/H] ranges from −2.5 to 0.5 in steps of 0.5 plus a value [Fe/H] = 0.2. Three dimensional linear interpolation is used to obtain the corresponding χ for each star. Finally, excess fractional luminosity R' is obtained by using Equation (4).

$$\begin{eqnarray}&&\chi =\displaystyle \frac{F(\lambda )}{\sigma {T}_{\mathrm{eff}}^{4}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&R^{\prime} =\mathrm{EW}^{\prime} \times \chi .\end{eqnarray} \tag{ 4 }$$

With the above procedures, log R' value for each member star is obtained. We perform a cross-match between our sample and the Simbad database,⁹ and exclude those stars labeled as "Flare*," "pMS*," "RSCVn," "SB*," "EB*WUMa," "EB*," "EB*Algol", and "EB*betLyr" by Simbad. Flare stars and binaries can affect CA level (Curtis 2017; Fang et al. 2018). In Appendix B, we simply discuss the impact of binaries. The result is shown in Figure 4. There are 1091 member stars in 82 open clusters with $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ values and 1118 member stars in 83 open clusters with $\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }$ values. Some open clusters are represented by only one or two stars. The Melotte 22 and NGC 2632 have over 100 member stars. From Figures 4(a) and (b), young member stars have similar log R' values as old member stars when T_eff > 6500 K. As T_eff decreases, young member stars tend to have larger log R' than old member stars, which conforms to our expectation. This phenomena is alike to Figure 2.

Zoom In Zoom Out Reset image size

We use a simple Monte Carlo simulation to obtain the error of log R'. Detailed information can be found in Appendix A. For the Ca ii K line, $\sigma (\mathrm{log}\,{R}_{\mathrm{CaK}}^{{\prime} })$ has a large scatter when T_eff > 6000 K (see Figure 10). $\sigma (\mathrm{log}\,{R}_{\mathrm{CaK}}^{{\prime} })$ is about 0.05 dex and 0.15 dex at 5500 K and 4500 K. For the Hα line, the distribution of $\sigma (\mathrm{log}\,{R}_{{\rm{H}}\alpha }^{{\prime} })$ has a large scatter from 0.0 to 0.5 dex at all T_eff range (see Figure 10).

There are some member stars that should be noticed. In Figure 4(b), we can see that four member stars surrounded by a rectangle box have very high $\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }$ values, which belong to the same open cluster: NGC 2112 (log t = 9.315). We check their spectra and find that they have very strong balmer emission lines. The open cluster is in the direction of the famous H ii region known as Barnard's loop (Haroon et al. 2017). We suspect that the high $\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }$ values of this cluster are caused by interstellar medium (ISM). In panels (a) and (b) of Figure 4, some member stars have very low values so that they will pull down the mean value of log R' within an open cluster and increase the scatter obviously. So we exclude those stars whose $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }\lt -6.40$ for the Ca ii K line and those stars whose $\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }\lt -7.00$ for the Hα line in the following study.

Mamajek & Hillenbrand (2008) derived CA–age relation by using a traditional indicator $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$, which was derived from S-values in the Mount Wilson HK project (Vaughan et al. 1978; Noyes et al. 1984). We cross-match our results with Table 5 of Mamajek & Hillenbrand (2008). The comparison between our results and theirs is shown in Figure 5. The crossing match sample only includes three open clusters: Melotte 20, Melotte 22, and NGC 2682. From Figure 5, as $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ or $\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }$ decreases, $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ also decreases. However, for those stars whose CA indices are low, our indices show a little larger scatter than theirs, which might be caused by different data-processing methods. For stars whose EW are close to the basal lines, their EW' are close to zero and the $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ and $\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }$ values discern more when taking the logarithm. In Appendix C, we list a table (Table 4) to illustrate it.

Zoom In Zoom Out Reset image size

4.2.1. log R' versus log t in Different T_eff Ranges

The mean value of $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ ($\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }$) for each open cluster is calculated. Figure 6 plots this mean value versus the age of each cluster. The left-top corner gives the T_eff range of member stars chosen to calculate mean value. Clusters that have only one star are not displayed in this plot. From Figure 6(a) we find that when stellar age log t < 8.5 (0.3 Gyr), the mean value of $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ starts to decrease slowly as age increases. Then, after log t = 8.5, the mean value decreases rapidly until log t = 9.53 (3.4 Gyr). Soderblom et al. (1991) pointed out that the evolution of CA for a low-mass star may be going through three stages: a slow initial decline, a rapid decline at intermediate ages (∼1–2 Gyr), and a slow decline for old stars like the Sun. Although there are some differences on age ranges of each stage, our conclusion is consistent with that of Soderblom et al. (1991) for the two former stages. In our sample, the number of old open clusters (log t > 9.0) is small and the age is only extended to log t = 9.53, so it is hard to see whether there is a slow decline for old stars like the Sun. From Figure 6(b), the mean value of $\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }$ decreases from nearly log t = 6.76 (5.7 Myr) to log t = 9.53 (3.4 Gyr). Although it does not show the trend: a slow initial decline and then a rapid decline, we can see the same trend as Ca ii K if we divide the T_eff range as done below.

Zoom In Zoom Out Reset image size

From Figure 6, we can see some open clusters deviate from the location of our expectation or have a large error bar. In panel (a), three old open clusters (NGC 7789, NGC 2112, and NGC 2420) show a little larger mean values. Their average [Fe/H] are relatively poor compared to young open clusters. For example, NGC 2420 has average [Fe/H] equal to −0.278 ± 0.0462 (see Table 1). Poor [Fe/H] stars have relatively shallower metallic lines than rich [Fe/H] stars, leading to large ${\mathrm{EW}}_{\mathrm{CaK}}$ and $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$, which may be the reason of these larger mean values. Some open clusters have one or two member stars whose $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ values are very low, so that they pull down the mean values, like Collinder 69 and Collinder 359. We see that NGC 1817 has a very large error bar. This cluster has five member stars and all stars are with T_eff > 6500 K. Three of them have large $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ values, the other two have low $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ values. The difference between the two groups is about 1 dex. In panel (b), NGC 2112 has a very large mean value. The reason is discussed above.

The scatter of log R' within an open cluster is large. In addition to measurement error, there are many other physical factors contributed to the scatter. Within an open cluster, different member stars have different mass and rotation rates. Stellar mass and rotation rate can influence CA level (Noyes et al. 1984; Mamajek & Hillenbrand 2008). Stellar cycle modulations also change CA level (Baliunas et al. 1995; Lorenzo-Oliveira et al. 2018). Some stars in our sample may have flare or starspots, which affects CA level. Binaries and ISM can also influence CA level. Appendix B simply discusses the impact of binaries and ISM on log R'. In our sample, some stars may not belong to open clusters and they affect the mean values. Besides, our data-processing method also contributes to the scatter. For those stars whose EW are very close to the basal line, a small difference in EW between two stars can cause a large difference in log R' (see Table 4).

In order to decrease the influence of stellar mass on CA, we divide T_eff into three equal bins and plot the mean log R' versus age log t again, as shown in Figure 7. In all T_eff ranges, we can see the trend that as age increases the mean value decreases slowly or remains unchanged, and then decreases rapidly. The scatter is smaller at low T_eff range than at high T_eff range. That means log R' is more sensitive to stellar age at lower T_eff, which is consistent with Figure 2 of Zhao et al. (2011). In their figure, the quantity log S_HK used to indicate CA level discerned more from each other at redder color. The trend that CA shows a slow decline and then a rapid decline is more evident for cooler stars. This phenomena may be related to stellar inner structure. Those stars at low T_eff range have thicker convective zone than those at high T_eff range. So those stars at a low T_eff range can maintain a strong surface magnetic field on a longer timescale than at a high T_eff range (Fang et al. 2018; West et al. 2008).

Zoom In Zoom Out Reset image size

There are some open clusters that deviate from locations of expectation, or have a large error bar. Many of them are discussed above. In Figure 7(c), Melotte 25 has a low mean value and a large error bar. The reason is that this cluster has only three member stars at 6000 K < T_eff < 7000 K, of which one member star has a low $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ value ($\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }=-5.81$), pulling down the mean value. In Figure 7(d), Alessi 20 has a larger mean value. With 4000 K < T_eff < 5000 K, this open cluster has only two member stars, whose [Fe/H] are poorer compared to the other member stars. One of these two stars shows very strong Hα emission line. Maybe these two stars are not member stars of the cluster. Roslund 6 has only two member stars with 4000 K < T_eff < 5000 K. The ${\mathrm{EW}}_{{\rm{H}}\alpha }$ of these two stars are very large. One is 15.13 Å, the other is 3.90 Å. Their spectra show very strong H α emission line. This is not just Hα, there are other emission lines in these two spectra such as: Hβ, Ca ii HK, N ii, and so on. Maybe the two stars are in a special term. For example, they have large spots on stellar surface.

4.2.2. log R' versus log t in a Narrow [Fe/H] Range

[Fe/H] has more of an effect on $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ than $\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }$ (Rocha-Pinto & Maciel 1998; Rocha-Pinto et al. 2000; Lorenzo-Oliveira et al. 2016). Maybe there is a negative correlation between $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ and [Fe/H]. We narrow the [Fe/H] range to −0.2 < [Fe/H] < 0.2 and plot the mean log R' versus age log t again. The sample is split on a star-by-star basis. Figure 8 shows log R' versus log t with 4000 K < T_eff < 7000 K and −0.2 < [Fe/H] < 0.2. By comparing Figures 8 and 6, we find that there is no obvious difference. Figures 9(a) and (c) show log R' versus log t with 4000 K < T_eff < 5500 K and without [Fe/H] limit. Figures 9(b) and (d) show log R' versus log t with 4000 K < T_eff < 5500 K and −0.2 < [Fe/H] < 0.2. By comparison, no obvious difference is formed. We also see no large difference of log R' versus log t relation when narrowing the [Fe/H] range to −0.1 < [Fe/H] < 0.1.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Quadratic function is used to fit data points with 4000 K < T_eff < 5500 K in two [Fe/H] ranges. Figure 9 shows fitting curves and relationships. For Hα, two stars of Alessi 20 and two stars of Roslund 6 mentioned in Section 4.2.1 are removed. The relationships are also listed in Equations (5)–(8). Age for field stars can be approximately estimated by solving these quadratic equations. Via Monte Carlo simulation, a distribution of log t can be obtained with a log R' value and its error. For Equations (5) and (6), we calculate two distribution of log t at two $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ values ($\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }=-4.90,-5.23$). The error of $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ is set to 0.15 dex. The errors of log t are about 0.40 dex, 0.28 dex at log t = 8.75, 9.44 corresponding to $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }=-4.90,-5.23$. The error of log t at log t = 9.44 is smaller than at log t = 8.75. This is because the fitting curve gets steeper when log t increases and $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ is projected to a smaller range of log t. For Equations (7) and (8), the errors of log t are about 0.40 dex, 0.28 dex at log t = 8.60, 9.40 corresponding to log ${R}_{{\rm{H}}\alpha }^{{\prime} }=-4.90,5.40$. The error of $\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }$ is set to 0.20 dex.

Equations (5) and (7) are used to estimate ages of corresponding clusters whose log t > 8.00. The results and relative error are shown in Table 3. The accuracy of Equation (5) is about 40%, while the accuracy of Equation (7) is about 60%. The ages of NGC 1647 and Ascc 10 cannot be estimated by Equation (5) because the two clusters have very large $\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }$ mean values that exceed the maximum value of Equation (5).

Equations (5) and (7) are also used to estimate ages of open clusters whose ages are not found in literatures. However, only the open cluster RSG 1 is available to estimate age. The cluster has log t = 8.69 estimated by Equation (5) and log t = 8.52 estimated by Equation (7). In a following paper, we will use Equations (5)–(8) to roughly estimate ages of field stars.

$$\begin{eqnarray}&&\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} }=-12.45+2.10\mathrm{log}t-0.14{\left(\mathrm{log}t\right)}^{2},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}{R}_{\mathrm{CaK}}^{{\prime} } & = & -11.22+1.82\mathrm{log}t-0.13{\left(\mathrm{log}t\right)}^{2},\\ & & -0.2\lt [\mathrm{Fe}/{\rm{H}}]\lt 0.2\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} }=-12.16+2.22\mathrm{log}t-0.16{\left(\mathrm{log}t\right)}^{2},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}{R}_{{\rm{H}}\alpha }^{{\prime} } & = & -9.84+1.66\mathrm{log}t-0.13{\left(\mathrm{log}t\right)}^{2},\\ & & -0.2\lt [\mathrm{Fe}/{\rm{H}}]\lt 0.2.\end{array}\end{eqnarray} \tag{ 8 }$$