Impact of Spectral Resolution on $S$-index and Its Application to Spectroscopic Surveys

PHOENIX high-resolution synthetic spectra have a resolution of 500,000. They are modeled under the local thermodynamic equilibrium (LTE), with non-LTE corrections applied for some specific lines including the Ca II H and K lines (Husser et al., 2013). The library covers a range of effective temperatures ($T_{\rm{eff}}$) from 2300 K to 12,000 K, surface gravities (log$g$) from 0 to 6, and metallicities ([Fe/H]) from $-$4 to 1. In this work, we mainly focused on late-type dwarf and giant stars so that we only used the synthetic spectra with $T_{\rm{eff}}$ between 2500 K and 6500 K and log$g$ between 2 and 6. Meanwhile, all [Fe/H] grids have been included. No $\alpha$-enhancement was considered. We treated targets with log$g$ larger than 3.5 as dwarfs and those with log$g$ smaller than 3.5 as giants. Meanwhile, we marked targets with $6000<T_{\rm{eff}}\leq 6500$ K as F-type stars, $5300<T_{\rm{eff}}\leq 6000$ K as G-type stars, $4000<T_{\rm{eff}}\leq 5300$ K as K-type stars, and $T_{\rm{eff}}\leq 4000$ K as M-type stars.

To simulate spectra with chromospheric activity, we first constructed two normalized Gaussian profiles centered at 3934.78 Å  and 3969.59 Å. In order to derive the full width at half maximum (FWHM) corresponding to stellar activity, we first conducted an estimation of the FWHM for the emission cores of the Ca II H and K lines using HARPS spectra¹¹1http://archive.eso.org/wdb/wdb/adp/phase3_main/form for several active stars and the spectrum of solar spot region (Cretignier et al., 2024). In Figure 1 we plot two examples. The gray spectrum corresponds to HD 165341A (K0V), observed with HARPS while the red spectrum is the solar spectrum at the spot region (Cretignier et al., 2024). The results suggest that an FWHM of 0.35 Å  matches well with the observed spectra. As a result, in this work the FWHM of each Gaussian profile was set to be 0.35 Å. These Gaussian profiles were then scaled by a group of factors, ranging from 1 to 10 with steps of 1, multiplying the maximum flux within the wavelength range between 3800 Å  and 4100 Å, respectively, to represent different levels of chromospheric activity. Finally, these Gaussian profiles were added to the PHOENIX spectra. Figure 2 provides some examples.

We calculated the $S$-index following

excluding the factors of 8 and $\alpha$. H and K represent the integrated fluxes corresponding to the H and K bands, centered at 3969.59 Å  and 3934.78 Å, respectively. Note that these wavelengths mark the vacuum cores of the Ca II H and K lines. The integration window for these bands was a triangle bandpass with a FWHM of 1.09 Å. R and V refer to the integrated fluxes within two 20 Å  rectangle reference bands on the red and violet sides of H and K lines, centered at 4001 Å  and 3901 Å, respectively. In addition, considering the Wilson-Bappu effect, which indicates that the widths of the emission line cores of the Ca II H and K lines are broader in giants compared to dwarfs (Wilson & Vainu Bappu, 1957), we also used a 2.18 Å integration range for the $H$ and $K$ bands to calculate the $S$-indices for giants. It is worth noticing that a wider bandpass would wash out the weaker activity signal since more photospheric contribution from the wings would be brought in (Gomes da Silva et al., 2022; Pietrow et al., 2024).

It should be noted that we used the original PHOENIX spectra (with flux units of erg/s/Å/cm²) for the calculations, while in many observations, normalized spectra were used (e.g. Wright et al., 2004; Schröder et al., 2009; Han et al., 2024). Therefore, we first conducted a check to see whether the $S$-indices calculated from normalized and unnormalized spectra are the consistent. We randomly selected a group of templates with various $T_{\rm{eff}}$ and log$g$ values and fixed [Fe/H] values of 0.0. The comparison of the $S$-indices shows only a minor deviation, suggesting that both methods yield consistent results (Figure 3).

We first calculated the $S$-indices for the original PHOENIX spectra with a resolution of $R=500,000$ and labeled them as $S_{\rm{O}}$. Then, the spectra were convolved to various resolutions of $R\sim 1800$, 2000, 3000, 4000, 5000, 6000, 7000, 7500, 8000, 9000, 10,000, 15,000, 20,000, 30,000, 50,000, and 100,000 using a Gaussian window, and the corresponding $S$-indices were calculated, labeled as $S_{\rm{R}}$. For giants, the $S$-indices calculated using 2.18 Å  window were denoted as $S_{\rm{R,2A}}$. There are a total of 22,140 dwarf templates and 11,070 giant templates at each resolution. These calculations will be used to investigate the influence of spectral resolution on the values of $S$-indices.

We used the standard nonparametric Kolmogorov-Smirnov (K-S) test to quantitatively assess at which resolution the $S_{\rm{O}}$ and $S_{\rm R}$ samples can be treated as the same. Generally, the two samples can be regarded as different if the null hypothesis probability $P_{\rm K-S}$ is much smaller than one. To be conservative, here we treat they as the same if $P_{\rm K-S}$ $>$ 0.99.

It is obvious that at low resolutions, the $S_{\rm{O}}$ and $S_{\rm R}$ samples are significantly different (Figure 4). Moreover, at the same resolution, the difference become more notable as the Ca II H and K emission levels increase, indicating that stronger emission lines could amplify the impact of resolution. Conversely, at high resolution, the two samples are quite consistent.

For both dwarfs and giants with $S$-indices calculated using a 1.09 Å  window, the knee point, where the $S_{\rm O}$ and $S_{\rm R}$ samples can be considered consistent, is around $R\sim 30,000$. For giants with $S$-indices calculated with a 2.18 Å  window, the knee point is about $R\sim 15,000$. We proposed that for spectra with $R\gtrsim 30,000$, the effects of resolution on $S$-index estimation can be ignored (i.e., $S_{\rm O}=S_{\rm R}$). Our results are roughly consistent with Bouchy et al. (2001) who proposed a 50,000 resolution threshold. The difference in threshold may be due to the different wavelength ranges. Bouchy et al. (2001) used a wider wavelength range to investigate the influence of spectral resolution on line profile while we only focused on the narrow emission cores of Ca II H and K lines.

Considering the impact of spectral resolution on the calculation of $S$-indices, especially for low-resolution cases, one should convert the $S_{\rm R}$ values to $S_{\rm O}$ values to mimic the resolution effects. Here we provide the calibrations for some large spectroscopic sky surveys, including the LAMOST low-resolution survey with $R\sim 1800$ (Cui et al., 2012), the SEGUE survey with $R\sim 2000$ (Yanny et al., 2009), the SDSS-V/BOSS survey with $R\sim 2000$ (Kollmeier et al., 2017), the DESI survey with $R\sim 4000$ (DESI Collaboration et al., 2016), and some upcoming sky surveys like the MSE survey with $R\sim 6000$ (Hill et al., 2018) and the MUST survey with $R\sim 4000$ (Zhao et al., 2024).

Figure 5 shows the comparison of $S_{\rm O}$ and $S_{\rm R=1800}$ for different types of stars. For stellar templates with [Fe/H] $\geq-1$, the $S_{\rm O}$ and $S_{\rm R=1800}$ exhibit a linear relationship, whereas for templates with [Fe/H] $<-1$, the relationship becomes non-linear. Since the integration window includes the line wings, the slope of the wings will affect the flux within the integrated region. For metal-rich stars, their broad wings have shallower slopes, making the integration less sensitive to slope variations. As a result, changes in metallicity, which affect the slopes, only have a minor impact on the integrated flux and thus lead to tight relations between $S_{\rm{R}}$ and $S_{\rm{O}}$. In contrast, metal-poor stars exhibit narrower line wings with steeper slopes. Consequently, the integrated flux becomes more sensitive to the wings, and changes in slopes caused by variations in [Fe/H] significantly influence the integrated flux. This results in large dispersions in the $S_{\rm{R}}$-$S_{\rm{O}}$ relations.

After excluding metal-poor stars, the relations are similar across different types of stars and the correlation is tighter for late-type stars. Therefore, we will focus our analysis on the relations for stellar templates with [Fe/H] $\geq-1$. In addition, our analysis reveals that excluding A- to M-type stars with [Fe/H] $<-1$ from the LAMOST DR10 catalog reduces the target sample by merely 4$\%$, suggesting that the excluding of these stars will not significantly influence the statistical analysis. For templates with [Fe/H] $\geq-1$, although there are small differences in the $S_{\rm{R}}$-$S_{\rm O}$ relations, we treat them as a single group.

Figure 6 compares $S_{\rm O}$ and $S_{\rm R}$ for dwarfs with a 1.09 Å widow, giants with a 1.09 Å window and giants with a 2.18 Å window, respectively. Obviously, the deviation between $S_{\rm{O}}$ and $S_{\rm R}$ gradually increases as the spectral resolution decreases. Additionally, for both dwarfs and giants, when the emission levels of Ca II H and K lines are high, the relationships are very tight. However, when the emission levels are low, the relations would split into several branches in low-resolution cases (e.g., $R\leq 2000$).

For each resolution, a linear regression was applied to fit the relationship between $S_{\rm O}$ and $S_{\rm{R}}$ using templates with [Fe/H] $\geq-1$ with all the emission strengths:

Furthermore, for low-resolution spectra, we divided the sample into two groups based on activity levels: a low-activity group ($S_{\rm R}<$ 0.02 for dwarfs and $S_{\rm R}<$ 0.01 for giants) and a high-activity group ($S_{\rm R}\geq$ 0.02 for dwarfs and $S_{\rm R}\geq$ 0.05 for giants). Individual linear regression was applied to each group. The fitting results are listed in table 1. These individual fittings are recommended for $S$-index calibrations. Additionally, for the low-activity group, we performed linear fittings for different types of stars, and the fitting results were found to be very similar. Finally, since the linear relations are tighter when calculating the $S$-indices using a 2.18 Å  compared to those that correspond to 1.09 Å  window (Figure 6), as reported by Duncan et al. (1991); Schröder et al. (2018), we suggested to use a 2.18 Å  window to calculate the $S$-indices for giants.

The $S$-indices represent the total emission of the Ca II H and K lines, which includes the photospheric contribution unrelated to magnetic activity. To address this, the $R_{\rm{HK}}^{{}^{\prime}}$, which represents the “flux excess” in the Ca II H and K lines, was introduced as the standard chromospheric activity indicator. It can be derived from the $S_{\rm MWO}$ through a series of complex steps (Noyes et al., 1984; Mittag et al., 2013). In the following analysis we will provide conversions from $S_{\rm R}$ to $S_{\rm MWO}$.

In Section 3.2, we presented the conversions from $S_{\rm R}$ to $S_{\rm O}$. Establishing the relationship between $S_{\rm R}$ and $S_{\rm MWO}$ would become straightforward only if we can determine the relationship between $S_{\rm O}$ and $S_{\rm MWO}$. Since $S_{\rm O}$ is consistent with $S_{\rm R\gtrsim 30,000}$ (Section 3.1) we can use stars common to high-resolution spectroscopic surveys and the MWO observations to derive these relationship.

Fortunately, there are some common sources between the HARPS spectral observations and the MWO observations (Boro Saikia et al., 2018). The HARPS spectra, with a very high resolution of $R=$ 120,000, allow the $S_{\rm HARPS}$ to be treated as $S_{\rm O}$ without requiring further calibration for resolution effects. All these spectra were corrected for radial velocities using the values provided by the HARPS data-reduction software (Pepe et al., 2002). For each target, we normalized all the spectra and calculated the $S$-indices of HARPS spectra with $S_{\rm{HARPS}}=(H+K)/(R+V)$. The median value of these measurements for each source was adopted as the final result (i.e., $S_{\rm{HARPS}}$). We then applied a linear regression to $S_{\rm{HARPS}}$ values and $S_{\rm{MWO}}$ values from Boro Saikia et al. (2018) following

The fitting coefficients are $k_{\rm{HARPS}}=20.767\pm 0.515$ and $b_{\rm{HARPS}}=0.021\pm 0.005$. Since $S_{\rm{HARPS}}$ were not multiplied by a factor of 8$\alpha$, the fitting results are differ from those reported by Boro Saikia et al. (2018). Figure 7 shows the fitting results.

Finally, we provided the general relations between $S_{\rm R}$ and $S_{\rm MWO}$ as follows,

The results of $k^{{}^{\prime}}$ and $b^{{}^{\prime}}$ for each resolution are summarized in table 1.

The LAMOST low-resolution spectra have been widely used in the studies of stellar magnetic activity (e.g. Zhang et al., 2020a, b; Li et al., 2024; Ye et al., 2024; Zhang et al., 2021). Some studies (e.g., Karoff et al., 2016) calculated the $S$-indices as $S_{\rm LAMOST}=8\alpha(H+K)/(R+V)$ and simply treated it as $S_{\rm{MWO}}$ to derive $R_{\rm{HK}}^{{}^{\prime}}$ following the method of Mittag et al. (2013). Although the authors adopted $8\alpha$ as the calibration factor, their approach may introduce inaccuracies in the results. Furthermore, Karoff et al. (2016) used the air wavelength to define the line centers of the Ca II H and K lines, whereas the LAMOST spectra are provided in vacuum wavelengths. Given the low resolution of LAMOST spectra, this $\approx$1 Å difference could strongly affect the calculation of $S_{\rm{LAMOST}}$.

The most appropriate approach is to first convert $S_{\rm{LAMOST}}$ to $S_{\rm MWO}$ and then calculate $R_{\rm{HK}}^{{}^{\prime}}$. However, there are no common targets between the LAMOST and MWO observations. Consequently, Li et al. (2024) and Han et al. (2024) used common targets between the LAMOST and HARPS observations to do the calibration using the HARPS $S$-indices (Boro Saikia et al., 2018), subsequently converting them into $S_{\rm{MWO}}$. Similarly, Zhang et al. (2022) used common stars between the LAMOST observations and the $S_{\rm MWO}$ catalog given by Duncan et al. (1991), while Zhang et al. (2024) utilized common targets between the LAMOST observations and several catalogs with $S$-indices calibrated to the MWO measurement.

To compare our results with the conversion relations derived observationally, we plotted the relations from Zhang et al. (2022), Zhang et al. (2024), and Han et al. (2024) in Figure 8. The coefficients (i.e., slopes and intercepts) of the linear relations are 2.68 and $-0.3$ in Han et al. (2024) or 2.26 and $-$0.27 for solar-like stars (5400 K $<T_{\rm eff}<$ 6500 K) in Zhang et al. (2024). Note that the LAMOST $S$-indices calculated in these studies were multiplied by a factor of $8\times 1.8$, which explains why their coefficients differ from our results in table 1. Additionally, Zhang et al. (2022) used an exponential function to fit the relationship between $S_{\rm{LAMOST}}$ and $S_{\rm{MWO}}$ for solar-like stars. Since both the relations from Zhang et al. (2022) and Zhang et al. (2024) only include solar-like stars with $S_{\rm{LAMOST}}<0.5$, their relations differ from our results. The magenta line in Figure 8 corresponds to the model with $k^{{}^{\prime}}=50.111$ and $b^{{}^{\prime}}=-0.519$ for dwarfs with $S\geq 0.02$ and $\rm R=1800$. The fitting result is different from those from previous literature, especially at large $S_{\rm{LAMOST}}$ region.

The differences of the $S_{\rm{LAMOST}}-S_{\rm{MWO}}$ relations may be caused by the biased sample, since there are only a few targets with large $S$-indices. In previous studies, which provided calibrations from observations, they may be suffered from sample selection bias, low signal-to-noise ratio, and low resolution that could lead to the inaccuracy in the measurement of radial velocities and thus influence the determining of line centers. These effects would lead to large uncertainties while converting $S_{\rm{LAMOST}}$ to $S_{\rm{O}}$ observationally (For example the blue shaded area in Figure 8).

First, in this work we used a 0.35 Å  Gaussian profile to simulate stellar activity. However, different structures and physical processes in the chromosphere, including the velocity gradients of the fluid, the intensity of shock wave and the non-LTE radiative transfer, would influence the widths of line cores (e.g. Carlsson & Stein, 1997). To test the potential influence, in Figure 9 we plot the relations of $S_{\rm{R=1800}}$ and $S_{\rm{O}}$ corresponding to various FWHM of emission cores, i.e., 0.3 Å,   0.35 Å,   0.4 Å,   and 0.5 Å. Obviously, the relations are similar. We conclude that when the simulated emission cores are within the integrated windows, the fitting result will not change significantly.

Second, the rotational broadening may also have notable impact on the line profile. We further tested the effect using a representative stellar model with $T_{\rm{eff}}=4900$ K, log$g$ $=$ 4.5 and [Fe/H] $=$ 0.0, corresponding to a K-type dwarfs. Assuming a typical radius of 0.7$R_{\odot}$ and a very short rotation period of 1 day, we derived a FWHM of 0.5 Å  of the simulated emission core, which is still smaller than the typical width of integrated window. Thus the rotational broadening will not affect our results significantly.