Mg II h + k Flux—Rotational Period Correlation for G-type Stars

The rate at which main sequence stars rotate is certainly amongst the most fundamental stellar parameters. It has proven to be a fundamental tool in a variety of stellar astrophysical phenomena, in particular as a test of the dynamo theories. In the time domain, rotational periods are fundamental to understanding the evolution of angular momentum in low mass stars, from the T-Tauri stage to mature objects as our Sun. Additionally, rotational velocity has become a leading tool for determining stellar ages of main sequence stars in the field, for which other methods, such as the isochrone fitting, are difficult to apply (Soderblom 2010).

Direct measurements of the stellar rotational periods rely on the analysis of brightness variations on the stellar surface due to the presence of spots. Nevertheless, aside from highly magnetic objects of early-type where the spot coverage can be large, this method represents, in many instances, a difficult task for cooler stars. In G-type stars the flux fluctuation is rather weak and requires very long and precise photometric and spectroscopic observational programs. In order to cope with these potential constraints, other indirect methods have been implemented, mainly through the analysis of the rotationally driven chromospheric activity.

The correlation between the stellar chromospheric activity and rotation was first identified in the pioneering work of Kraft (1967). This relationship is generally explained by the balance between an enhanced magnetic field induced by rotation and the breaking due to loss of angular momentum by magnetized stellar winds (Mestel 1968). An additional part of the picture is given by the internal magnetic fields generated in the interiors of convective stars (T_eff < 8000 K) through a solar-like dynamo mechanism.

Magnetic activity has been usually measured by the emission excess in the strong Ca II H and K features. The cardinal work initiated at Mount Wilson in the early sixties (Wilson 1963) was later continued through a number of extensive surveys (see, for instance, Duncan et al. 1991; Henry et al. 1996; Wright et al. 2004; Jenkins et al. 2008; Zhao et al. 2013).

Whilst the Ca II H and K lines have long been the workhorse for stellar activity studies, in particular due to their accessibility from ground based telescopes, the Mg II h and k emission lines in the mid-UV (2800 Å) has provided additional valuable diagnostics. These lines, with similar energetics to those of Ca II, present the advantage that their shallower absorption wings make the emission line easier to measure and less contaminated by the photosphere.

Early works on the Mg II h and k resonance lines by Hartmann et al. (1984) demonstrated that this chromospheric proxy strongly correlates with rotational period, P_rot. Their study was based on a relatively small sample (31 objects) covering a wide range of mass and evolutionary status, and also included numerous binaries. More recently, Cardini & Cassatella (2007, hereafter CC07) analyzed a larger stellar sample and identified a loose correlation between Mg II flux and rotational period, a situation that is improved if age instead of rotational period is used.

In this paper, we revisit the Mg II h and k doublet as a powerful diagnostics of the rotational period of G-type stars by analyzing a stellar sample of main sequence and subgiant stars observed by International Ultraviolet Explorer (IUE), complemented with high resolution observations conducted by the Space Telescope Imaging Spectrograph (STIS) and the Goddard High Resolution Spectra (GHRS) on board the Hubble Space Telescope (HST). We calibrate the Mg II h and k absolute fluxes with measured rotational periods and apply the relation to determine new rotational periods for 15 stars with high quality UV spectra.

The stellar sample upon which our analysis is based has been constructed from two different databases. The first is that of the IUE-Newly Extracted Spectra (INES¹): We searched for all high resolution data available for objects in class 44 (G-type stars of luminosity classes IV and V). This search delivered nearly 500 spectra that were subsequently visually inspected to verify their quality. A total of 57 objects have been selected from this database. We have also searched available data on G-type stars from HST and found 19 objects: 17 are from STIS, 1 from GHRS, and 1 object has 1 image from both instruments. These data were taken from the HST-MAST archive² and StarCat (Ayres 2010). For the solar spectra, we extracted a high resolution spectrum of the Moon listed in class 02 of the INES archive.

The full sample consists of 75 object plus the Sun.³ The stellar set is divided into three working subsamples. The first subsample consist of 37 stars (and the Sun) with rotational periods, that we call primary, and will be used to establish the calibration of the Mg II UV fluxes versus rotational period. For these objects, the rotational periods have been measured either through the analysis of the photometric modulation of the visible flux due to the uneven distribution of stellar spots on the stellar surface, or through variability of the Ca II emission. The data for these stars are given in Table 1 where columns 1–7 provide, respectively, the star name, the spectral type, the color index B - V, the effective temperature mainly obtained from the compilation of Soubiran et al. (2010), the calculated Mg II absolute flux (see next section), the rotational period, and a label of the reference for the period. The list of references is given at the bottom of the table. The second subsample is composed of 23 objects with secondary rotational periods, that is, periods determined through a calibration of Ca II emission and measured rotational periods. This dataset will serve as a test to compare rotational periods derived with our UV calibration with those computed from Ca II. Table 2 lists the objects included in this set. Column 1–5 are as in Table 1, while the last 3 columns provide the secondary rotational period, the reference for this period, and the rotational period estimated in this work, respectively. Finally, the third collection corresponds to 15 stars with no available information on their rotational periods, and for which we derive the first estimate of this parameter. This latter set is presented in Table 3, where we also provide our rotational periods and their estimated errors.

For the full stellar sample, we have adopted the process described in Hartmann et al. (1984) to measure the observed Mg II h and k fluxes f_h+k as the flux integral between the two minima in the h (h1v, h1r) and k (k1v, k1r) lines from the zero flux level (see Fig. 1). This process was conducted automatically, but verified (and corrected if needed) visually, in particular for stars with lower signal-to-noise ratios in these minima. This method differs from those implemented by Blanco et al. (1974), Linsky & Ayres (1978), and CC07 in that these works either include a correction for the photospheric contribution or perform the integration by considering a local continuum defined by the k1 and h1 minima. An alternative way to carry out the integration was provided by Buccino Mauas (2008), who considered a fixed 1.7 Å-width windows to calculate the flux in the k and h lines. In an ideal scenario one should eliminate the photospheric contribution by properly modeling the atmospheric absorption. Nevertheless, the space ultraviolet and in particular the Mg II features in the mid-UV still represent a challenge when modeling stellar atmospheres (see, e.g., Chavez et al. 2007). Additionally, it was argued by Hartmann et al. (1984) that the photospheric contribution in the Mg II lines is much less than for Ca II. Many objects have numerous spectra available, as many as 120 in the case of the star HD2151. For these objects we obtained the mean spectrum by weighting the available spectra by their quoted flux errors.

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Observed integrated fluxes were then converted to absolute fluxes, F_h+k, following the relation of Oranje et al. (1982):

For each object we collected the V magnitudes from SIMBAD database, while the bolometric corrections were taken from Flower (1996). Effective temperatures are mainly from Soubiran et al. (2010) and, for a few objects, from Ammons et al. (2006). In the case of Sun, we have used the IUE spectrum of the Moon LWR09968HS, which was selected among the 21 available spectra, because of its high quality. For the purposes of treating this spectrum in a similar way as the rest of the stars, we have calculated F_h+k by adopting the V magnitude of the solar analog HD102365, a G2V star with stellar parameters very close to solar: (T_eff/ log g/[Fe/H]) = (5637/4.45/-0.08) (Gratton et al. 1996). To obtain the error in the surface flux, we applied a Gaussian error propagation through equation (1). For the effective temperature we have used the errors provided in the Soubiran et al. (2010) and Ammons et al. (2006) catalogs; for stars with no quoted error in the catalogs, we adopted the mean uncertainty (95 K) of stars with temperature errors available. We assumed for V and BC the conservative values of 1% and 10% relative errors, respectively. The uncertainty on the observed flux f_h+k is obtained from the IUE and HST error vectors as a function of the wavelength; we added in quadrature the mean error in the intervals used to integrate each pair of Mg II lines. The latter is the main source of error on F_h+k.

In order to test the consistency of our derived fluxes, in Figure 2 we compare our measurements of the Mg II k line flux with those reported by Cardini (2005) (for the 23 objects in common). Whilst the correlation is clear, our fluxes are higher by approximately 25% (about 0.1 dex) on average. This excess can be plausibly explained by the different integration processes. Since Cardini (2005) did not report the IUE images that were used, it is possible that they not coincide with ours, as may be the case of HD2151 for which our flux is 80% larger.

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In the upper panel of Figure 3 we plot F_h+k versus the rotational periods for the 38 objects (including the Sun) which have primary values for their rotational periods (Table 1). In this plot the different symbols stand for the origin of the data as indicated in the panel. All stars correspond to luminosity classes objects V except for the squared symbol that shows the location of the only luminosity class IV star. The starred symbol indicates the position of the Sun. We have attempted several functional forms to calculate the best fit and found that the function:

where the coefficients A = 7.125, B = -0.162, C = 0.674, provided the lowest χ². The function is compatible with the exponential used in Noyes et al. (1984). However, as in Noyes et al. (1984), the adopted functional form does not have physical implications.

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There is a relatively tight correlation for our sample of G-type stars, with much less dispersion than that observed in other works (e.g., Hartmann et al. 1984; Cardini & Cassatella 2007) that can be explained by the exclusion of the more active cool K and M-type stars. The lines in the plot correspond to the best fitted exponential function (solid line) and the ± 1σ levels (dashed lines).

Whilst at low rotational periods (P_rot < 4 d) the diagram is less populated, the correlation indicates that for P_rot = 0 the flux appears to reach a maximum of log F_h+k ∼ 6.8. For periods larger than 4 days there is a decline of more than one decade. The three stars indicated with diamonds were not taken into account in the fitting calculation. The two with the lowest rotational periods correspond to pre-main sequence objects, whose inclusion would produce, in any case, negligible changes in the fitted function. The other object, which shows the largest deviation from the correlation depicted in Figure 3, is the metal-rich star HD182572. This a G8IV variable star whose rotational period of 41 days (log P_rot = 1.61; Baliunas et al. 1996) appears to be twice as large as predicted by its Mg II flux. Its measured rotational period is compatible with the period of ∼37 days estimated from the average rotational velocity (v sin i) of 1.9 km s^-1 in Glebocki & Gnacinski (2005) and the stellar radius of 1.38 R_⊙ (Boyajian et al. 2013). Variability can be a possible explanation for the discrepancy; however, Lockwood et al. (1997) found the root mean square of the brightness variation in Strömgren b and y bands to be just 0.0016 mag during a 12-year long interval (1984–1995). It is possible that an enhanced activity epoch around the IUE observation date (1979 December) produced such a strong UV flux, although this scenario might not be supported by the Ca II S-index monitoring by Duncan et al. (1991), unless the IUE observation actually coincided with a transient event.

An important aspect of the observed correlation of Mg II versus P_rot is the range of variation of log F_h+k. The difference between the highest and lowest values is 1.60 dex. In order to compare this difference with that to be expected from Ca II measurements, we have used the available R_H+K fluxes for the calcium line from Henry et al. (1996) and Wright et al. (2004) for the stars of our sample and transformed them into F_H+K by multiplying R_H+K by the bolometric luminosity σT⁴, according to the definition of Noyes et al. (1984). We fit a similar function to equation (2) to the F_H+K - P_rot values; this is shown in the lower panel of Figure 3, where the dotted line is the fit for Mg II case. Note that the variation of F_H+K is 1.20 dex, i.e., about 0.4 dex smaller than that for Mg II. We would like to remark that, had we included the star with the lowest rotational period (HD142361), the Ca II flux variation would have decreased to 1.09 dex due to the steeper correlation in the low period edge.

Another interesting feature of the panels depicted in Figure 3 is that the average standard deviation for both correlations are quite similar, with that of Mg II (σ_h+k ∼ 0.11 dex) slightly larger than for Ca II (∼0.09 dex). In this regard, we have conducted a simple test to understand the extent to which the standard deviation can be ascribed to the F_h+k intrinsic variability of each star. For this purpose, we have compared the amplitude of the variation of the flux in the Mg II lines, Δ log F_h+k, with σ_h+k, for the 17 stars with multiple observations. The results of such a comparison indicate that Δ log F_h+k < σ_h+k in 15 out of 17 cases. The main results of the analysis presented above demonstrates that the magnesium index is about 2.5 times more sensitive to rotational period than that of calcium.

A direct application of the exponential fit obtained in the previous section is the determination of new rotational periods (P_rot3) for the objects in Table 3. It is, however, important to compare UV derived periods with those determined from secondary methods, mainly through the calibration of the Ca II H and K lines. For this purpose we use the second stellar set in Table 2, for which we have collected calculated rotational periods from the literature: these secondary rotational periods (P_rot2) are mainly from Soderblom (1985), Wright et al. (2004), and a few from Baliunas et al. (1996) and Saar & Osten (1997). In Figure 4 we illustrate the comparison of the Mg II vs Ca II rotational periods. The dashed line correspond to slope unity.

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In general the points cluster around the one-to-one correlation with less agreement at large rotations periods (P_rot > 20 d), hence for more mature stars with less intense chromospheric emission. The point that most deviates from the correlation is that for star HD188512 (β Aql, for which derived P_rot = 35.1 d is significantly smaller than the Ca II derived periods in excess of 50 days). This star is a subgiant star which has 15 determinations of atmospheric parameters in the compilation of Soubiran et al. (2010). The effective temperatures for this object range from 4373 K to 5478 K, surface gravities from 1.3 dex to 3.79 dex, and a metallicity whose determinations vary nearly 0.6 dex. Even if we exclude the lowest gravity and highest temperature values (Luck & Heiter 2006; Luck & Lambert 1981),⁴ this star has averaged parameters that are discrepant with the rest of the luminosity class IV stars in our sample, in particular for gravity. Being a more evolved star, the correlation given by equation (2) is most probably not applicable to this object.

For the stars in Table 3 we provide for the first time an estimation of their rotational period (column 6) and their uncertainties (column 7). Errors have been obtained through Gaussian propagation of errors of the fitting parameters of equation (2). Rotational periods for this sample span a wide range of values, from P_rot = 2.7 d for HD212697 to 35.8d for HD212330. There are several objects with P_rot very similar to that of the Sun.

We have measured the flux of the mid-UV emission lines Mg II h and k for 76 stars G-type stars (including the Sun) in the main sequence or in the subgiant branch. We used the 38 objects with a rotational period obtained from the periodic variability of their light curves to derive an analytical calibration of F_h+k vs. P_rot. We found a tight correlation, which benefits from the inclusion of HST observations as they have a significantly better signal-to-noise ratio compared to the IUE spectra. This is the first time that HST UV data are used in the analysis of this relation, which we used to obtain the first estimates of the rotational periods for 15 stars with high quality mid-UV data.

Our results indicate that the Mg II h and k lines are about 2.5 times more sensitive to the rotational period than the frequently used chromospheric proxy of Ca II. The comparison of our UV-derived rotational periods with those obtained from similar correlations for Ca II shows a significant scatter for slow rotator, where perhaps the very faint Ca II H and K emission might be prone to large uncertainties.

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