© ESO, 2011

1. Introduction

The age of a star is one of the most difficult stellar parameters to determine. Although several age-dating methods are becoming more precise, stellar chronology still holds many uncertainties, especially when the objects are in the field and moderately old. For ages above 0.7 Gyr, open clusters and moving groups are, in general, not very useful as age calibrators, since they are typically at large distances, making their individual component stars difficult to study in detail, particularly their activity characteristics. This is clearly illustrated in recently proposed calibrations (Mamajek & Hillenbrand 2008) that, while providing good performance for young ages (15–20% uncertainty), do not yield accurate results beyond 0.5 Gyr because of the lack of calibrators and the increasing unreliability of rotation period estimates (differential rotation effects or a weak signal from low-amplitude modulations). Furthermore, the calibrations are not applicable to stars later than K2.

The determination of the ages of low-mass stars has many applications in astrophysics (Mamajek & Hillenbrand 2008; Soderblom 2010), including the study of galactic populations or any use of intermediate-age G-, K-, and M-type stars as tracers of evolution, such as calibrating the decrease in high-energy emissions. Our interest lies in this use of age calibrations for low-mass stars, because of the importance of stellar emissions, and their time variability, to the understanding of exoplanet atmospheres. The host star to a planet is, by far, its main source of energy, and many studies (e.g. Lammer et al. 2003; Penz et al. 2008) have demonstrated that stellar high-energy emissions have a strong impact on planetary atmospheres. Thus, characterizing such emissions is central to proper modelling of exoplanet properties. High-energy emissions are related to the magnetic activity of the star, and this activity decays throughout the stellar lifetime in an as yet poorly understood manner, making it difficult to calculate the accumulated effects of the UV and X-ray emission on the planetary environments. The decay in activity with stellar age is intimately linked to the rotational evolution of the stars (e.g. Skumanich 1972; Ayres 1997). Although the time evolution of high-energy emissions of solar-type stars is already well constrained from The Sun in Time project (Ribas et al. 2005), that of late GKM stars still needs much improvement. Our objective is to extend The Sun in Time project to cooler stars. This is justified by the many differences in the high-energy emissions between, e.g., G-type and M-type stars and by the interest of the latter as hosts to exoplanets.

In this work we discuss the first part of our larger plan to determine age-activity relationships for low-mass stars in general. The paper is organised as follows. In Sect. 2 we present the results for stars with stellar ages below 0.7 Gyr. The methodology of the age determination for WDs is explained in Sect. 3. In Sect. 4 we collect our sample and describe its selection criteria. In Sect. 5 we present the observations and the data reduction. Section 6 is devoted to the analysis, atmospheric parameter determination, and the subsequent age determination method. This is followed by Sect. 7 where we discuss the low-mass companions in the context of defining age sequences, and finally in Sect. 8 we elaborate on our conclusions.

2. Preliminary work

We started by collecting data on the evolution of X-ray emissions for the younger stars (<1 Gyr). Their ages were determined from cluster and moving group membership (Montes et al. 2001), and the log L_X values were obtained from a thorough list provided by Pizzolato et al. (2003). We complemented the measurements with X-ray data estimated directly from ROSAT measurements for some field stars with age estimates from different methods.

The young main sequence phase was studied by means of stars belonging to the IC 2602, IC 2391, Pleiades, α Persei, Hyades clusters, and the Ursa Majoris moving group. The early G-type star sample, as we explained in Sect. 1, was already available from The Sun in Time project. These stars have well-known rotation periods, temperatures, and metallicities. In general, we only consider stars in narrow spectral type bins (or effective temperature) to avoid contamination from the intrinsic variation in magnetic activity with stellar mass. The age range  <0.07 Gyr is covered by stars in the IC 2602, IC 2391, and α Persei clusters. We should point out that these clusters are younger than 0.07 Gyr (0.03, 0.03, and 0.05 Gyr, respectively), but since saturation is present until 0.07 Gyr or longer in the case of K and M stars (Stȩpień et al. 2001; Jeffries et al. 2011), considering data with different ages does not affect the value of log L_X.

For ages above 1 Gyr cluster or moving group membership is not a useful age determination method. Other age indicators, such as the use of rotation period, age-activity relations, asteroseismology, or theoretical isochrones, are more useful in this age domain. Some of them have been used to obtain the ages of a few GKM stars older than  ~1 Gyr. Only a handful of field stars have reliable ages in this domain. These stars are α Cen B and Proxima Cen, with ages determined from the isochrone and asteroseismologic age of their close companion α Cen A (Porto de Mello et al. 2008). HR7703 shows space motions that are typical of a thick disk star, so we can very roughly assume an age of 10 Gyr. The X-ray luminosities of these stars were determined from the ROSAT database following the calibration in Schmitt et al. (1995). For stars older than  ~6 Gyr long-term changes in high-energy emissions are, in general, difficult to distinguish from short-term stellar activity variations.

We have put together all the compilled data, and the evolution of log L_X with age for three spectral type intervals (G0-5, K0-5, and M0-5) is illustrated in Fig. 1. In the case of G type stars, the plotted values are quite reliable as they come from a thorough analysis of the The Sun in Time sample. For K- and M-type stars, however, the plotted values are very crude results corresponding to the few stars described above that just have rough age estimates. It is likely that the uncertainty of each point is at least of a few tenths of a dex. The figure shows, as expected, that M-type stars stay at saturated activity levels for a longer period of time than G-type stars. According to these results, solar-like G0-5 stars are at saturated emission levels until ages of  ~100 Myr, and their X-ray luminosity decreases rapidly. K-type stars have saturated emission levels for a little longer (~200 Myr) and then also decrease rapidly. Finally, M0-M5 stars seem to have saturated emission levels up to 0.5 Gyr or more and then decrease in an analogous way to G- and K-type stars.

Fig. 1 X-ray luminosity versus age for solar-like stars and for field and cluster K- and M-type stars. Only a few K and M stars are available beyond 0.7 Gyr.

The interval corresponding to ages youger than  ~0.7 Gyr is covered well by the used cluster and moving group stars (IC 2602, IC 2391, Pleiades, α Persei, Hyades clusters, and the Ursa Majoris moving group), but it is obvious from our preliminary analysis that a more complete sample of older stars is needed to define reliable age-activity relationships. Furthermore, it is interesting to note that an age of 0.7 Gyr is a key point for our Sun, since it represents the time at which life is supposed to have appeared on the Earth’s surface. Thus, data for older stars of different types will be very important for modelling planetary atmospheres in a regime that can be relevant to potential life on their surface, as happened to our Earth.

3. Methodology

We have developed an age determination method based on the use of wide binaries where one of the components, a WD, is used as a chronometer. The members of a wide binary are assumed to have been born simultaneously and with the same chemical composition. Since they are well separated (100–1000 AU), we can assume that no interaction has occurred between them in the past and they have evolved as single stars (Oswalt et al. 1988). We are interested in wide binaries composed by a WD and a star with GKM spectral type. The evolution of a WD can be described as a cooling process, which is relatively well understood at present (Salaris et al. 2000). The total age of the WD can be expressed as the sum of its cooling time plus the pre-WD lifetime of its progenitor. Thus, ages can be obtained from an initial-final mass relationship and stellar tracks to account for the pre-WD lifetime. This procedure is analogous to that described by Catalán et al. (2008b). It is sound to assume that the age of the WD is the same as the that of the low-mass companion, since both members of a wide binary were born simultaneously.

We selected a sample of 30 wide binaries containing a WD and a GKM star (see Table 1). In the sample we favour WD components classified as a DA, i.e., with the unique presence of Balmer lines in their spectra. As we demonstrate in Sect. 6, the fits to the spectral features of these WDs yield realistic values for the atmospheric parameters (effective temperature, T_eff and surface gravity, log g). We have collected a large amount of observational data (photometry and spectroscopy) on the WD components to implement the proposed approach. We aim at determining total system ages (i.e., cooling ages plus progenitor lifetime) with precisions of 10–20%. This is sufficient in our context since magnetic activity is an intrinsically variable phenomenon and any relationship will have some inherent dispersion no matter how accurate the ages and coronal/chromospheric fluxes.

4. Sample selection

The sample of wide binaries was compilled from the available literature. For this we considered the recent revision of the New Luyten Two Tenths (NLTT) Catalogue(Luyten 1979–1980) by Chanamé & Gould (2004), an NLTT catalogue with Hipparcos stars from Gould & Chanamé (2004) and later completed with some pairs from a selection of WD+M systems made by Silvestri et al. (2005).

Our first selection was done only considering the catalogues of Chanamé & Gould (2004) and Gould & Chanamé (2004) and following a careful evaluation procedure. First, the WD component had to be classified as a DA (i.e., with the sole presence of Balmer lines), so that the fitting procedure can be sufficiently accurate to derive atmospheric parameters. Secondly, the other component of the pair should be a star of spectral type G, K, or M. For the WD type classification we have used the McCook & Sion (1999, 2006) catalogues, and the spectral classification of the low mass stars was made by calculating their V − J index and taking the spectral classification into account for main sequence stars in Johnson et al. (1966). Information about the two proper motion components of both members of the wide binary is available in Chanamé & Gould (2004) and Gould & Chanamé (2004) catalogues for most of the members of the sample. We have completed and checked this information with data from USNO catalogue and calculated the total proper motion of each star. We only consider in our sample the wide binaries that satisfy the NLTT proper motion threshold, μ ≥ 180 mas yr^-1 and that they show the same direction in their proper motions.

The resulting sample was completed considering some wide binaries from the catalogue of Silvestri et al. (2005). They studied the relationship between age and chromospheric activity for 189 binary systems composed of a WD and an M-type star. For this purpose, the authors determined the system’s age using WDs as stellar clocks but assuming a rough value for their mass, the typical 0.6 M_⊙ (Silvestri et al. 2001). Conversely, the ages estimated are potentially unreliable since the mass of each WD is an important parameter for estimating their progenitor lifetime. With the aim of improving age estimates, we performed a revision of the sample of Silvestri et al. (2005) and completed our sample by considering the pairs containing a DA WD and a GKM star in their catalogue. The targets from Silvestri et al. (2005) were also screened following the proper motion criteria explained in the previous paragraph.

Our final sample is composed of 30 wide binaries. The sample with all relevant information, including the V − J indices of the companions and their inferred spectral types, is provided in Table 1. For NLTT55287 J photometry was not available and the spectral type was determined from the B − V index. In Table 1 we provide an approximate value for V − J corresponding to an M8 star. Although photometric information about the two members of the binary was available in the catalogues, we checked all V magnitudes in the SIMBAD database. We found some differences or confusions between the magnitudes of the two members in some cases. For these targets we made a thorough analysis to make sure that each member was correctly identified. Figure 2 shows a histogram with the spectral type distribution of the low-mass stars. As can be seen, most of the sample is composed by M-type stars plus a few K-type stars and one G-type star. As we discuss in Sect. 6, the WD sample mainly contains DA-type WDs, but also a few DC-type WDs.

Fig. 2 Histogram of the spectral type distribution of the low-mass stars in our sample, which are mostly of spectral type M.

5. Observations and data reduction

We performed optical long-slit low-resolution spectroscopic observations for our analysis, with the goal of obtaining measurements with high signal-to-noise ratio (~75–100). We were able to obtain spectroscopic data for 25 WDs in our sample during different observing campaigns between February of 2009 and February of 2010. The observing runs were carried out at Observatorio del Roque de los Muchachos (ORM) and the German-Spanish Astronomical Center at Calar Alto (CAHA). The 3.5 m telescope was used during three nights in February of 2009 at CAHA, and the William Herschel Telescope (WHT) during 4.5 nights in February and September of 2009 and February of 2010 at ORM. Table 2 provides more details about the observation setups used. We managed to observe 25 out of 30 initial targets, with the remaining five not visible at the time the observing runs were allocated.

The spectroscopic observations covered the main H Balmer lines, from Hα to Hε or H9, whenever possible. The two spectrographs used included two separate spectroscopic channels (blue and red arms) behind the common entrance slit aperture. We chose a proper dichroic position in order to observe the Hα line in the red arm and all the other visible Balmer lines in the blue arm, covering at least a spectral range of  ~3500–7500 Å. A suitable grism selection was done to place Hα centred and unvignetted for the red arm and a maximum number of unvignetted Balmer lines for the blue arm. The slit width for each observation was chosen according to the seeing (~1′′–2′′). Spectra of high quality are essential to derive the atmospheric parameters with accuracy. We performed as many exposures as necessary to guarantee a high signal-to-noise ratio (~75–100) for the final spectrum of each object (after the corresponding reduction). The exposure times and signal-to-noise ratios for all the targets are shown in Table 3. For targets with individual observations with low signal-to-noise ratio, we co-added individual 1800 s exposures to minimise the effects of cosmic ray hits on the CCD.

The WD spectra were reduced using the standard procedures within the single-slit tasks of IRAF¹. First, the images were bias-corrected and flatfield-normalised, and then, the spectra were extracted and wavelength-calibrated using arc lamp observations. Flux calibrations were not performed since they are not necessary for our subsequent analysis, and most of the nights were not photometric. We applied the heliocentric correction to each spectrum prior to coadding multiple observations obtained in different nights.

6. White dwarf analysis

After the corresponding reduction, we carried out a first inspection of the spectra. Although all the WDs we selected had previously been classified as DA, we found that four of them do not show the bluer Balmer lines in their spectra, so they are likely to be the DC type. For these four targets we checked the part of the spectrum observed with the red arm. The Hα line was not visible in any of them, and is faint in DC WDs, the resolution of the instrument and the low signal-to-noise ratio of the spectrum yields in the absence of such line for these four spectra. Table 1 lists the observed 21 DA type WDs in the first set of rows and remaining four observed DCs in the second set of rows.

Once the spectra of the DA targets were correctly selected, we proceeded to the continuum normalization procedure. The normalization has to be carried out carefully because the determination of atmospheric parameters is very sensitive to variations in the continuum. We carefully defined several clean continuum windows by considering the central part of each spectrum and avoiding any irregular feature or the wings of the spectral lines. We used Legendre polynomials of order 15 to 20 for the normalization procedure.

6.1. Atmospheric parameters

Before calculating the atmospheric parameters of the WDs (T_eff and log g), we determined the radial velocities of each star using the IRAF task fxcor. Each spectrum was cross-correlated with a reference model from a grid computed by Koester. The resulting radial velocity values can be found in Table 4. The radial velocities determined were small in a number of cases (ranging from 10 to 90 km s^-1) when compared with the resolution element of our observations. For some of the stars, however, the measured radial velocities were higher (ranging from 100 to 200 km s^-1) thus becoming relevant to the subsequent analysis. In our procedure we took all the radial velocities for consistency into account.

We derived the atmospheric parameters of the WDs by performing a fit of the observed Balmer lines using the models of Koester (priv. comm.), following a procedure based on χ² minimization. The Balmer lines in such WD models were calculated with the modified Stark broadening profiles of Tremblay & Bergeron (2009), kindly made available by the authors. The model grid used covers the range T_eff = 5000–100 000 K, in steps of 1000 K at low temperatures (T_eff < 14   000 K), steps of 2000 K (from T_eff = 16 000 to 20 000 K), steps of 5000 K (from T_eff = 20   000 to 40 000 K), and steps of 10 000 K for the higher temperatures (T_eff > 40   000 K). The range of log g is between 5.0 and 9.0 steps of 0.5 dex. The WD models had been previously normalised to the continuum and convolved with a Gaussian instrumental profile with the proper FWHM in order to have the same resolution as the observed spectra. The fit of the line profiles to synthetic models was done using a subroutine based on the simplex χ² minimization method of Press et al. (1992). We observed that the fitting procedure applied to all the observed lines (in the red and blue arms) suffered from considerable systematic effects. After running a number of tests we decided to consider only the lines observed with the blue arm for the analysis to avoid systematic errors because the Hα line in the red arm is less adequate for atmospheric parameter determination.

Fig. 3 Fits to the individual Balmer lines for the WDs observed at the WHT of ORM. The line profiles correspond to Hβ (bottom) up to Hϵ, H8, or H9 (top) depending on the object. We have applied vertical shifts for clarity. The solid lines are the observed spectra and the dashed lines the model that best fits the profile.

Both T_eff and log g were obtained simultaneously. It is well known that the changes in the profiles of the Balmer lines induce a certain degree of degeneracy to the procedure. There are often two numerically valid solutions corresponding to minima of the χ² function around the maximum strength of the Balmer lines (T_eff ≈ 12   000 K), i.e., a “cool” and a “hot” solution. In some cases these two minima are well separated and their individual determination is straightforward. But for some other objects, the parameters corresponding to the two scenarios are so close as to become harder to distinguish. We tried to obtain both solutions for the WDs by changing the initial T_eff for the analysis. We were able to distinguish between the “cool” and “hot” solutions by finding the minimum value of χ² for both cases and comparing the results with the photometric temperatures obtained as explained in Sect. 6.2.

Given that our spectra have a very high signal-to-noise ratio, the main contributor to the uncertainty in the fitting process (besides the adequacy of the models themselves) is the normalization procedure. The normalization can potentially modify the broad profiles of the spectral lines and thus introduce systematic errors in the resulting parameters. We carried out a careful analysis to quantify such an error source. Using the IRAF continuum task, we obtained the blaze functions for each individual spectrum. To do so, we devised a procedure that is based on normalizing each individual spectrum with the blaze functions corresponding to all the remaining spectra taken with the same instrument. Each of the resulting spectra was then re-normalised (to correct for obvious trends) employing a 5-order Legendre function for the observations with higher signal-to-noise and order 3 for the ones with low signal-to-noise ratio. Finally, each of the resulting spectra for every target were fitted to models as explained above to determine the atmospheric parameters. The final uncertainties were estimated following the prescription of Bergeron et al. (1992), i.e., by deriving them from the independent fits of the individual exposures for any given star before combination, but in our case by considering all the normalised individual spectra. We are aware that this procedure may overestimate the error since some of the renormalised spectra were clearly not optimal. However, we prefer to make certain that all possible systematic errors in the fitting procedure are accounted for by taking a conservative approach and considering the uncertainties derived in this way. The results of the fits and the estimated uncertainties are shown in Table 4. Figures 3 and 4 show the fits to the Balmer line profiles for the DA WDs in our sample.

Fig. 4 Same as Fig. 3 for the WDs observed at the 3.5-m telescope at CAHA.

6.2. Photometric effective temperatures

Some stars in our sample were not visible in the epochs of our observing runs. For these stars, and for the ones with no Balmer lines in their spectrum, we used the available IR photometry to obtain their photometric T_eff. The V magnitudes were obtained from the catalogues used to build the sample and from the SIMBAD database, JHK photometry was collected from the 2MASS All Sky Catalog when available. For V magnitude we assumed an uncertainty of 0.02 mag as errors are seldom available.

The T_eff for each object was obtained by adapting the method of Masana et al. (2006) to WDs. This procedure consists in calculating synthetic photometry using the WD atmospheres of Holberg & Bergeron (2006). Subsequently, we developed a fitting algorithm to compare the observational and theorical values and minimise the χ² using the Levenberg-Marquardt method, and χ² was defined from the differences between the observed and synthetic VJHK magnitudes. The uncertainties in the calculated photometric temperatures were derived from the covariance matrix. The VJHK magnitudes and photometric T_eff for the sample stars are provided in Table 5.

In the cases where both spectroscopic data and V and JHK 2MASS photometry were available, the photometric T_eff was used to evaluate the reliability of the resulting T_eff obtained by the line profile analysis. In most of the cases, both T_eff were in good agreement, as shown in Fig. 5. NLTT26379 is the object showing the largest deviation. For our subsequent analysis, we consider the T_eff value obtained from spectroscopy, since it should be more reliable. It is worth noting that the spectroscopic analysis of WDs with T_eff < 12   000 K should be done with special care, since their atmospheres could be enriched in helium while retaining their DA spectral type (Bergeron et al. 1992). In this case, photometric T_eff provide the needed cross-check to ensure that the overall atmospheric parameters are consistent and accurate. It is reassuring to confirm that our photometric and spectroscopic T_eff values are in good concordance for the cooler targets of our sample.

Fig. 5 Comparison of photometric and spectroscopic Teff derived in this work for 15 WDs in our sample. The dashed line indicates 1:1 correspondence. The χ2 of the difference between the measurements is 0.57.

Together with T_eff, our procedure requires a value for log g. Given the absence of any other possibility for the objects with photometric T_eff, we decided to adopt a “mean” log g estimated from the WDs in the sample with spectroscopic parameters. We employed all WDs with reliable log g values (i.e., the first part of Table 4). We obtained log g = 8.01    ±    0.20, in which the error bar is not the standard deviation but the overall range.

6.3. Masses, cooling times and total ages

Once we derived the T_eff and log g of each WD, its mass (M_WD) and cooling time (t_cool) can be obtained by using the appropriate cooling sequences. We have adopted the cooling tracks of Salaris et al. (2000), which consider a carbon-oxygen (C/O) core WD (with a higher abundance of O at the central core) with a thick hydrogen envelope ontop of a helium buffer, q(H) = M_H/M = 10^-4 and q(He) = M_He/M = 10^-2. These improved cooling sequences include an accurate treatment of the crystallization process of the C/O core, including phase separation upon crystallization, together with up-to-date input physics suitable for computing WD evolution. The resulting values for M_WD and t_cool are listed in Table 7.

Systematic errors related to uncertainties on the CO stratification and envelope composition may affect the cooling sequences (Salaris 2009). To check the sensitivity of our results to the adopted cooling tracks, we also used the sequences of Fontaine et al. (2001), which consider different core and envelope compositions. The composition of the cooling sequences employed can be found in Table 6. The derived masses do not change appreciably when adopting alternative cooling sequences, with differences less than  ~ 0.02 M_⊙ over the entire mass range. The average of the differences in t_cool were found to be around 9%. Thus, we adopted a systematic error of 9% in t_cool in our subsequent analysis of the cooling ages.

For three of our targets, included at the bottom of Table 4, our spectroscopic fitting procedure yielded low values of log g (<7.7) with relatively poor fits. The photometric temperatures available for two of these targets (NLTT10976 and NLTT29967) show good agreement with the spectroscopic value, thus indicating the consistency of the analysis. Thus, although the log g values that we obtained seem reliable, we could not infer reliable masses for them using the cooling sequences of Salaris et al. (2000) since they are out of the covered parameter range. The values for M_WD and t_cool obtained via extrapolation for these three targets can be found in Table 7. We obtained M_WD < 0.5   M_⊙, which yields progenitor masses lower than 1 M_⊙.

From the derived M_WD and t_cool values, the mass of the progenitor (M_prog) can be determined using an initial-final mass relationship (Catalán et al. 2008a). From the calculated M_prog, we used the stellar tracks of Domínguez et al. (1999) to derive the progenitor’s lifetime (t_prog). The total age of the WDs, and consequently that of the low-mass companion, follows directly by adding t_prog and t_cool. The final age value is provided in Table 7. In this table we compiled the final atmospheric parameters, the mass of the WD, the mass of the progenitor star, the cooling time of the WD, the main-sequence time of the progenitor star, the total age, the ratio between the cooling time and the total age, and the V − J colour of the low-mass companions. The first 18 WDs in the Table have atmospheric parameters determined from the Balmer line profile fitting. The three objects in the middle were analysed spectroscopically, but the derived log g are outside of the valid parameter range of our procedure. The nine WDs at the end of the table have temperatures determined by VJHK photometry and the mean log g value for our sample estimated from the spectroscopic determinations.

Some of the WDs in our sample have atmospheric parameters and, in some cases, masses, cooling times, and ages estimated from previous investigations, such as Bergeron et al. (2001), Silvestri et al. (2001), Gianninas et al. (2005), and Zhao et al. (2011). We can compare our results to the previous determination. Our T_eff and log g values are compared to those in the literature and are shown in Fig. 6. In general the agreement between the T_eff determinations is good within the uncertainties, but there are some discrepant results for some of the targets, such as NLTT26379 and G86-B1B. In the case of G86-B1B, our analysis shows good accord between spectroscopic and photometric determinations and thus, given the available cross-check, we prefer the value we determined. In the case of NLTT26379, our photometric and spectroscopic T_effs are discrepant at the 2.7-σ level, and the result of Silvestri et al. (2001) (using photometry) lies squarely in between. Given the better a priori reliability of the line profile analysis for T_eff above 12 000 K, we subsequently adopt the spectroscopic T_eff value. The comparison of the log g values between our values and the literature yields no very discrepant results.

Fig. 6 Comparison of Teff (top) and log g (bottom) determined by us with values obtained by Bergeron et al. (2001), Silvestri et al. (2001), Gianninas et al. (2005), and Zhao et al. (2011). The dashed line is the 1:1 correspondence.

6.4. Photometric distances

We have collected available parallax measurements for the low-mass companions and completed the information with photometric distances. The photometric distances were determined using the evolutionary models for low-mass stars described in Baraffe et al. (1997) and using the ages determined for each wide binary system. The results for the sample are collected in Table 8. For NLTT19314 and WOLF672B, we could not directly perform a photometric distance determination since their estimated ages are greater than 8 Gyr, which is the age limit of the models we used. Thus, we just adopted a value of 8 Gyr, since the evolutionary changes of these stars between 8 and 13 Gyr are negligible.

7. Discussion

The main product of this part of the work is the determination of ages for 27 low-mass stars that are members of wide binary pairs. For eighteen of these age determinations we employed WD atmospheric parameters coming from a detailled spectroscopic fit, while for nine additional systems we used photometric T_eff values and an estimated log g from the average of the rest. The final age is the sum of two contributions, namely the cooling time of the WD and the lifetime of the progenitor, mostly during the main sequence. As can be seen in Table 7, there is a clear distinction between the subsamples with spectroscopic and photometric temperatures in that the former shows relatively short cooling ages (usually  < 1 Gyr), while these are significantly longer for the latter (~1–5 Gyr). This is not a coincidence since older WDs are cooler and the Balmer lines in their spectrum are weak or nonexistent.

To further exploit our sample, it is worth evaluating the reliability of our estimated ages. The uncertainties we provide in Table 7 come from the random error estimates related to our fitting procedure, but they do not include systematic contributions potentially arising from the relationships and models used. We are not especially concerned with the contribution from models since the WD cooling process is relatively simple and main-sequence lifetime estimates should be quite reliable. However, there is some degree of concern as to the possible existence of systematic uncertainties related to the initial-final mass relationship of WDs. Many improvements have been achieved during these last years, for instance with the coverage of the low-mass domain (Catalán et al. 2008a). However, there is still a relatively large cosmic dispersion in the empirical initial-final mass relationship that could be related, e.g., to metallicity effects or the poorly-known mass-loss processes during the AGB phase. For this reason, we believe that t_cool is more reliable than t_prog.

Given the considerations above, we favour systems in which the contribution of WD cooling time to the total age is at its maximum. In addition, we aim at defining a sample of companion stars with a wide range of spectral types that cover as much of the important 1–6 Gyr age interval as possible. From these two premises, we found that a constraint of t_cool/Age > 30% gives the best compromise between age robustness and sample size, so therefore selected systems fulfilling this requirement. These objects are highlighted in boldface in Table 7. Eighteen systems in our sample fulfil this condition. Almost all of the cooler WDs analysed in this work belong to this reliable sample, which is expected because their cooling times are relatively long. The last column of Table 7 includes, for reference, the V − J index of the companion to each WD so that the proper sample selection for the next step can be made. Following the same prescription as was used in the Sun in Time project, one should define the spectral type (or mass, or V − J) intervals and select stars that cover a wide age range. Given the V − J colours of our sample stars, we grouped them into five bins, namely early K, late K, early M, early-mid M, and mid-late M. Table 9 shows these bins with the V − J and ages of the stars included in them. Each bin has at least two stars (to be added to the cluster averages presented in Sect. 2), covering roughly the interval between 1 and 5 Gyr. These are the stars for which we collect activity information so that they can be used to trace the time-evolution of their high-energy emissions from the start of the zero-age main-sequence to at least the age of our Sun. This is illustrated in Fig. 7, which is the same as Fig. 1 but with vertical lines showing the position of the ages that will be sampled with the stars we have analysed. The new ages cover the relevant area that is currently unpopulated and beyond what can be studied with open cluster data.

Fig. 7 log LX vs. age diagram for G0-G5, K0-K5, and M0-M5 spectral type bins. The vertical lines correspond to the ages of the WDs with reliable age determinations from our study. The G0-G5 spectral type interval is already fully covered from the The Sun in Time project.

8. Conclusions

We studied a sample of 30 wide binary systems composed of a WD component and a low-mass companion. We have obtained high-signal-to-noise low resolution spectroscopy for 25 of the WDs, which allowed us to carry out a full analysis of their spectra and a fit of WD models to derive atmospheric parameters, T_eff and log g. Total ages for the wide binary systems were obtained from a procedure based on the cooling sequences of Salaris et al. (2000), the initial-final mass relationship of Catalán et al. (2008a), and the stellar tracks of Domínguez et al. (1999). We estimate that the total ages for 18 of the systems are especially reliable as the relative contribution from the cooling time of the WD (which is supposed to be relatively unaffected by systematic errors) is greater than 30%. The resulting sample covers spectral types from early K to mid-late M and, very importantly, it contains stars that populate the critical age interval between 1 and 5 Gyr for which the determination of ages using other methods is difficult. Most of our targets are within a few tens of parsecs, so the study of their activity properties is much easier than for stars belonging to intermediate- or old-age clusters.

For the selected stars we are obtaining high-resolution spectroscopy to measure chromospheric fluxes and X-ray data so that we can build reliable relationships describing the time evolution of emissions related to stellar magnetic activity. As our final goal, our work should provide critical input information to any effort of understanding the long-term evolution of the atmospheres of planets around low-mass stars, including their potential habitability. It will also allow better understanding of the evolution of high-energy emissions from low-mass stars. The activity-age relationship will allow us not only to address the time evolution of high energy radiation, but also to use it as an age calibrator when the stellar high-energy emissions can be measured. This age indicator is very interesting because it will be based on the star’s own properties and will be useful in a regime (>1 Gyr) where estimating stellar ages is still very difficult.

Acknowledgments

We thank D. Koester for providing us with his white dwarf models. We acknowledge support from the Spanish MICINN grant AYA2009-06934. S.C. is supported by a Marie Curie Intra-European Fellowship within the 7th European Community Framework Programme.

References

All Tables

All Figures

	Fig. 1 X-ray luminosity versus age for solar-like stars and for field and cluster K- and M-type stars. Only a few K and M stars are available beyond 0.7 Gyr.
In the text

	Fig. 2 Histogram of the spectral type distribution of the low-mass stars in our sample, which are mostly of spectral type M.
In the text

	Fig. 3 Fits to the individual Balmer lines for the WDs observed at the WHT of ORM. The line profiles correspond to Hβ (bottom) up to Hϵ, H8, or H9 (top) depending on the object. We have applied vertical shifts for clarity. The solid lines are the observed spectra and the dashed lines the model that best fits the profile.
In the text

	Fig. 4 Same as Fig. 3 for the WDs observed at the 3.5-m telescope at CAHA.
In the text

	Fig. 5 Comparison of photometric and spectroscopic Teff derived in this work for 15 WDs in our sample. The dashed line indicates 1:1 correspondence. The χ2 of the difference between the measurements is 0.57.
In the text

	Fig. 6 Comparison of Teff (top) and log g (bottom) determined by us with values obtained by Bergeron et al. (2001), Silvestri et al. (2001), Gianninas et al. (2005), and Zhao et al. (2011). The dashed line is the 1:1 correspondence.
In the text

	Fig. 7 log LX vs. age diagram for G0-G5, K0-K5, and M0-M5 spectral type bins. The vertical lines correspond to the ages of the WDs with reliable age determinations from our study. The G0-G5 spectral type interval is already fully covered from the The Sun in Time project.
In the text