THE INITIAL–FINAL MASS RELATION AMONG WHITE DWARFS IN WIDE BINARIES

Spectroscopic observations were made at two observatories. In the southern hemisphere, observations were conducted at Cerro Tololo Inter-American Observatory (CTIO) using the Blanco 4 m telescope. Northern hemisphere observations were conducted at Kitt Peak National Observatory (KPNO) using the Mayall 4 m telescope.

At CTIO, the Ritchey-Chrétien (RC) Cassegrain spectrograph on the 4 m Blanco telescope was used on two separate observing runs (2004 February and 2005 February) to obtain optical spectroscopy of wide pairs, as well as standard flux calibration stars. During these two observing runs the KPGL1 grating was used to obtain a scale of 0.95 Å pixel⁻¹ (R ∼ 2000). A Loral 3K CCD (L3K) was used with the RC spectrograph. It is a thinned 3K × 1K CCD with 15 μm pixels. A spectral range of approximately 3800–6700 Å was achieved.

At KPNO, the RC spectrograph with the BL450 grating set for the second order to yield a resolution of 0.70 Å pixel⁻¹ (R ∼ 2000) was used to obtain optical spectra during 2006 November and 2010 July with the Mayall 4 m telescope. The 2K × 2K T2KB CCD camera with 24 μm pixels was used to image the spectra. An 8 mm CuSO₄ order-blocking filter was added to decrease first-order overlap at the blue end of the spectrum. A spectral range of approximately 3800–5000 Å was achieved.

The data were reduced with standard IRAF⁴ reduction procedures. In all cases, program objects were reduced with calibration data (bias, flat, arc, flux standard) taken on the same night. Data were bias-subtracted and flat-fielded, and one-dimensional spectra were extracted using the standard aperture extraction method. A wavelength scale was determined for each target spectrum (including stellar flux standards) using HeNeAr arc lamp calibrations. Flux standard stars were used to place the spectra on a calibrated flux scale. We emphasize that the final flux calibrations for the targets provide only relative fluxes, as most nights were not spectrophotometric.

The radial velocity of each MS star was determined by cross-correlation between the observed spectra and a set of MS template spectra. The F, G, and K template spectra were generated from a theoretical atmosphere grid (Castelli & Kurucz 2003). The dM template spectra were compiled using observed M dwarf spectra from the Sloan Digital Sky Survey.⁵ The wavelength ranged from roughly 3900 to 9200 Å (see Bochanski et al. 2007). Our typical internal measurement uncertainties in radial velocity were $\sigma _{\rm v_{r}}$ = ± 4.6 km s⁻¹.

Our sample included nine DA WDs and two DB WDs. For DA WDs the T_eff and log g were derived via simultaneous fitting of the Hβ to H8 Balmer line profiles using the procedure outlined by Bergeron et al. (1992). The line profiles in both observed spectra and model spectra were first normalized using two points at the continuum level on either side of each absorption line. Thus, the fit should not be affected by the flux calibration. Model atmospheres used for this fitting were derived from model grids provided by Koester (2010). Details of the input physics and methods can be found in that reference. Fitting of the line profiles was carried out using the IDL package MPFIT (Markwardt 2008), which is based on χ² minimization using the Levenberg–Marquardt method. This package can be downloaded from the project Web site.⁶ Errors in T_eff and log g were calculated by stepping the parameter in question away from their optimum values and re-determining the minimum χ² until the difference between this and the true minimum χ² corresponded to 1σ for a given number of free model parameters. Our spectra of the DB WDs were not of high enough signal-to-noise ratio (S/N) to do line profile fitting on such weak lines. For these Degenerate B (DB) WDs we adopted the T_eff and log g from Voss et al. (2007).

Figure 1 shows fits of the observed Balmer lines of the nine Degenerate A (DA) WDs. The derived T_eff, log g, and uncertainties are shown in Columns 2 and 3 of Table 2. Estimated T_eff and log g for nine DAs were also available in the literature, allowing the comparisons shown in Figure 2. For most of the WDs, the difference between our T_eff value and the literature value is smaller than 1000 K and the log g difference is smaller than 0.1 dex. For WD1544−374, our log g is consistent with that of C08b and Kawka et al. (2007) but differs by 0.2 dex with that of Vauclair et al. (1997). For WD2253−08, the log g values in the literature differ substantially; however, our value is consistent with that of C08b. In Figure 2, the dotted line represents the unit slope relation. In general, our results are consistent with those in the literature.

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Reid (1996) reported that the pair BD+44 1847/G116-16 is a non-physical pair because the gravitational redshift K_RS = 113.9 km s⁻¹ corresponds to 1.1 M_☉ which is inconsistent with 0.75 M_☉ from line profile fitting by Bergeron et al. (1995). We determined a mass of 0.77 M_☉ that is almost the same as Bergeron et al. (1995). Although our spectral resolution is only ∼1.8 Å, we attempted a rough redshift measurement of this pair. The doppler shift of the WD component measured from Hβ was about 120 ± 10 km s⁻¹ which is consistent with that of Reid (1996). The parallaxes of the components from the Simbad Astronomical Database (Genova 2006) for this pair are 34.6 mas and 19.36 mas for the primary and secondary, respectively. Hence, we also conclude this pair is non-physical. It was therefore eliminated from our determination of the IFMR.

From our T_eff and log g estimation for each star, its current mass (M_f) and cooling time (t_cool) were estimated from Bergeron's cooling sequences.⁷ For the pure hydrogen model atmospheres hotter than T_eff = 30,000 K, we used the carbon-core cooling models of Wood (1995), with thick hydrogen layers of q_H = M_H/M_* = 10⁻⁴. For T_eff cooler than 30,000 K, we used cooling models similar to those described in Fontaine et al. (2001) but with carbon–oxygen cores and q_H = 10⁻⁴ (see Bergeron et al. 2001). For the pure helium atmospheres we used similar models but with q_H = 10⁻¹⁰. M_WD and t_cool were then calculated by spline interpolation based on the T_eff and log g. The final M_WD and t_cool values are shown in Columns 4 and 5 of Table 2. The uncertainties in mass are estimated based on the uncertainties in T_eff and log g when interpolating in the cooling sequences. The average mass of our 11 WDs is 0.628 M_☉. The average mass of only the nine DA WDs is 0.635 M_☉, which is consistent with Bergeron et al. (1995; 0.626 M_☉).

Age is one of the most difficult to determine properties of any star. Ages for single lower MS stars derived from isochrone fitting are especially uncertain because of the narrow vertical dispersion of isochrones within the MS in an H-R diagram. Small uncertainties in luminosity and metallicity propagate into large uncertainties in age. Lachaume et al. (1999) investigated the precision of age determinations for a sample of nearby MS stars of spectral types B9–K9. They found that the isochrone method is best for hot stars, while stellar rotation (Barnes 2007) is best for cool stars. The MS components of our wide binaries are F–K stars. Most of our wide binaries are older than 1 Gyr, so measurement of their rotation period would require a large investment of observing time and very precise photometry.

For decades, chromospheric activity (CA) has been known to inversely correlate with stellar age (Skumanich 1972). Early work by Wilson (1963, 1968) and Vaughan & Preston (1980) established Ca ii H&K emission as a useful marker of CA in lower MS stars. The most commonly used measurement index of Ca ii H&K emission is R'_HK, defined as the ratio of the emission from the chromosphere in the cores of the Ca ii H&K lines to the total bolometric emission of the star. Soderblom et al. (1991), Lachaume et al. (1999), and Mamajek & Hillenbrand (2008) presented detailed investigations of the relation between R'_HK and stellar age.

Soderblom (2010, Figure 8) compared ages of G dwarfs derived from isochrone placement and from HK activity index. It is clear that the isochrone ages were larger than activity ages. For some stars, isochrone ages were inconsistent with the age of the Galactic disk. Most stars in our sample are late MS G stars and K stars, for which the isochrone method does not work well, as discussed above.

Following Hall et al. (2007), for each star we computed the flux ratio S_HK:

$$\begin{eqnarray} S_{\rm HK}&\equiv &{\alpha \frac{H+K}{R+V}}, \end{eqnarray} \tag{ 1 }$$

where H and K are the fluxes measured in 2 Å rectangular windows centered on the line cores of Ca ii H&K; R and V are the fluxes measured in 20 Å rectangular "pseudocontinuum" windows on either side. These bands are similar to those used in the Mount Wilson chromospheric activity survey program (Baliunas et al. 1995), except that the bands centered on Ca ii H&K are wider (2 Å) than those used at Mount Wilson (1 Å) because of our instrumental resolution. Here α is 10, indicating the pseudocontinuum windows are 10 times wider than the H&K windows in wavelength coverage. Figure 3 presents our H&K measurements for these 10 MS stars.

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In order to derive a transformation relation between our instrumental systems and the Mount Wilson system we selected six stars which have stable CA from Baliunas et al. (1995) as "standard CA stars." Since S_MW values for four MS stars in our CTIO sample were found in the literature, we only observed these "CA standard stars" with the KPNO telescope. Table 3 provides data for these standard stars. Column 1 lists the stars' name; Columns 2 and 3 give our S_HK and the published S_MW, respectively. Figure 4 shows the correlation between S_HK and published S_MW; Equation (2) is the transformation equation obtained by a least-squares fit. The scatter is $\sigma _{S_{\rm MW}}$ = 0.017:

$$\begin{eqnarray} S_{\rm MW}&=-0.069+1.308S_{\rm HK}. \end{eqnarray} \tag{ 2 }$$

The range of S_HK for which Equation (2) applies is limited by the range of S_HK in "standard CA stars," which is about 0.15 < S_HK < 0.55. With Equation (2), the S_HK value we measured at KPNO can be directly transformed to S_MW. For wide binary stars observed at CTIO we adopted the S_MW values from the literature wherever possible. If more than one value was available in the literature, reference was given to those derived from higher resolution spectra. This is based on the conclusion by Jenkins et al. (2011) that CA measures from low-resolution spectra significantly increase the rms scatter when calibrating onto a common system such as the Mt. Wilson system. The S_MW values of CD-59 1275 and CD-38 10983 were obtained from Henry et al. (1996); 40 Eri A came from Baliunas et al. (1995), and CD-37 10500 was found in Jenkins et al. (2011). Only two MS stars observed at CTIO did not have S_MW values in the literature. For these, the S_HK (CTIO) were first calibrated into S_HK (KPNO) using the empirical relation between the two instruments (Zhao et al. 2011a): S_HK (CTIO) = S_HK (KPNO) + 0.095. Then, the S_MW of these two stars were computed using Equation (2). For the wide binary MS stars observed at KPNO, the S_MW were directly calculated from Equation (2).

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The S_MW index includes both photospheric and chromospheric contributions. The photospheric flux can be removed approximately using the procedure of Noyes et al. (1984), who derived a quantity R_HK ∝ F_HK/σ T⁴_eff, where F_HK is the flux per square centimeter in the H and K bandpasses. The quantity R_HK can be derived from S_MW by modeling the variation in the continuum fluxes as a function of effective temperature (using B − V as a proxy). R_HK must then be further corrected by subtracting the photospheric contribution in the cores of the H and K lines. The logarithm of the final quantity R'_HK is a useful measure of the chromospheric emission, essentially independent of the effective temperature.

We computed R'_HK for the six stars observed at CTIO and the five stars observed at KPNO. Their colors (B − V) meet the condition of Noyes et al. (1984; B − V < 1.1). Both the estimates of S_MW and log R'_HK are tabulated for our program stars in Table 4.

CA versus age relations were published by Soderblom et al. (1991), Donahue (1998), Lachaume et al. (1999), and Mamajek & Hillenbrand (2008). Rocha-Pinto & Maciel (1998) investigated the metallicity dependence of the Soderblom et al. (1991) relation. Because we intended to explore the effects of metallicities within our sample, we used their relation to estimate our ages:

$$\begin{eqnarray} \log t&=&(-1.50\pm 0.003)\log R\mbox{$^\prime $}_{\rm HK}+(2.25\pm 0.12) \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \Delta \log t&=&{ -0.193\hbox{--}1.382[{\rm Fe/H}]}\nonumber\\ &&{-0.213[{\rm Fe/H}]^{2}+0.270[{\rm Fe/H}]^{3}}. \end{eqnarray} \tag{ 4 }$$

The total age estimates and uncertainties for our program stars derived from Equations (3) and (4) are listed in Column 6 of Table 4. The age uncertainties originate mainly from the S_HK uncertainties as they affect Equation (3). The S_HK uncertainties are determined from the average of the standard deviations of measurements for stars with more than two observations (about ±4.6%). The total internal uncertainty of our age determinations is about ±0.17 dex. Independent age determinations for a couple of pairs were found in the literature. These pairs are listed in Table 5. Column 1 gives their identifications. Columns 2, 3, and 4 list the ages included in Holmberg et al. (2009), Valenti & Fischer (2005), and Barnes (2007), respectively. Our ages are younger than isochrone fitting ages in all four cases. For 40 Eri A, the isochrone fitting age is unreasonably large, while our age for this star is close to the rotation age 4.75 ± 0.75 Gyr from Barnes (2007). For CD-38 10983, our age is also consistent with the rotation age, but younger than the isochrone age, which has a large uncertainty. For CD-59 1275, our age is close to the isochrone age. For CD-37 10500, the error bar is too large for the isochrone age to be useful. Thus, even in this small sample it can be seen how difficult it is to get consistent isochrone ages for late G and K dwarfs. Small uncertainties in luminosity and metallicity cause large uncertainties in age.

Ages from isochrone fitting of clusters with well-defined turnoffs are unquestionably more accurate than those of single field stars. To estimate the accuracy of our CA age estimates, we selected 75 member stars in M67 having R'_HK and B − V data from Mamajek & Hillenbrand (2008). All the original HK observations of these objects are from Giampapa et al. (2006). The CA age of each member star was estimated with the above formula. Figure 5 is the resulting CA age distribution of this sample. The dotted line is a Gaussian fit. The average age of those member stars is 3.28 Gyr with 1σ 0.95 Gyr (1σ uncertainty of 29%). The difference between the average CA age for this cluster and its turnoff age (4.0 Gyr) is about 0.73 Gyr. This difference, about 18%, is well within the formal uncertainties of our wide binaries' age determinations.

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All are M67 stars discussed above are solar-type dwarfs, which are likely to have activity cycles, perhaps similar to the Sun's 11–22 year period. The stars with very high HK values could have been observed at maximum activity during their cycle (perhaps a few are close interacting binaries). The stars with very low HK values could be in a Maunder minimum state. Most likely these are the main sources of scatter in CA ages derived from coeval populations of stars, like clusters. We did not attempt to quantify these factors here because each of our objects has so far only been observed a small number of times. This is almost certainly the dominant source of scatter in CA ages.

As a final reality check, we randomly drew 10 stars from the data shown in Figure 5 and computed the age and 1σ estimates. The age of M67 derived from a 10-star sample is 3.3 Gyr ± 0.9, which represents a 28% uncertainty—very comparable to the 30% claimed by Soderblom (2010) and others for CA age determinations. In our 10 wide binary sample the formal age uncertainty is 0.17 dex (48%; computed from uncertainties listed in Table 1). We do not propose that the age of any single star has been determined to a precision any better than this. These uncertainties have been fully propagated into the error bars shown in the plots shown in Figures 6 and 7 below.

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The difference between the total age of an MS star and the cooling age of its WD companion provides an estimate of the latter's progenitor MS lifetime (given in Column 6 of Table 2).

The metallicity of each MS star was measured by comparing the observed spectrum to a set of template spectra. Initially, a library of low-resolution theoretical spectra was generated using the SYNTH program based on Kurucz's New Opacity Distribution Function atmospheric models (Castelli & Kurucz 2003). The details can be found in Zhao et al. (2011a, 2011b). Figure 3 of Zhao et al. (2011a) presents an example of how we determined stellar metallicity and its uncertainty in 40 Eri A. The final [Fe/H] values are given in Column 7 of Table 2. In Section 6, the metallicity is indicated by Z, where [Fe/H] = 0.0 is equivalent to Z = 0.019.