Intensities of the hydrogen Balmer lines of solar-like stars revealed by the LAMOST spectroscopic surveys

The activity indexes are evaluated for the first four Balmer lines (H$\alpha$, H$\beta$, H$\gamma$, and H$\delta$ lines) in the LRS data as explained in Section 1. For each of the lines, a central band at the line center and two continuum bands on the two sides of the line are specified to derive the activity index of the line. Table 1 tabulates the center wavelengths of the four Balmer lines (denoted by $\lambda_{\mathrm{cen}}$), the center wavelengths of the two continuum bands on the violet side and the red side of the lines (denoted by $\lambda_{\mathrm{cont1}}$ and $\lambda_{\mathrm{cont2}}$, respectively), and the widths of the central and continuum bands adopted in this work. All the wavelength values displayed in Table 1 are in vacuum as adopted by the LAMOST. The center wavelengths of the four Balmer lines shown in Table 1 are from the paper by Stoughton et al. [31], as used by the Sloan Digital Sky Survey (SDSS). The locations of the continuum bands are determined by referring to the continuum profile of the LRS spectra of solar-like stars. The widths of the central and continuum bands are determined by ensuring that each central band contains at lease one data point and each continuum band contains about 20 data points; this choice of the band widths is necessary to keep the uncertainty of the derived activity indexes at an acceptable level. Figures 2a–2d give a diagram illustration of the central and continuum bands adopted in this work for the H$\alpha$, H$\beta$, H$\gamma$, and H$\delta$ lines, respectively, as listed in Table 1.

For each of the four Balmer lines in the LRS data, the mean flux in the central band (denoted by $\overline{F}_{\mathrm{cen}}$; indicated by the circle symbol in Fig. 2) and the mean fluxes in the two continuum bands on the violet side and red side of the line (denoted by $\overline{F}_{\mathrm{cont1}}$ and $\overline{F}_{\mathrm{cont2}}$, respectively; indicated by the filled hexagons in Fig. 2) are measured; then the activity index of the Balmer line (denoted by $L$, where the letter L indicates that the activity index is derived from LRS data) is calculated by using the following formula:

$$L=\frac{\overline{F}_{\mathrm{cen}}}{a\cdot\overline{F}_{\mathrm{cont1}}+b\cdot\overline{F}_{\mathrm{cont2}}}.$$

(1)

The coefficients $a$ and $b$ in Equation (1) are the weight factors for $\overline{F}_{\mathrm{cont1}}$ and $\overline{F}_{\mathrm{cont2}}$, respectively, whose values are inversely proportional to the distances of the continuum bands from the center of the line, that is,

$$a=\frac{\lambda_{\mathrm{cont2}}-\lambda_{\mathrm{cen}}}{\lambda_{\mathrm{cont2}}-\lambda_{\mathrm{cont1}}},$$

(2)

$$b=\frac{\lambda_{\mathrm{cen}}-\lambda_{\mathrm{cont1}}}{\lambda_{\mathrm{cont2}}-\lambda_{\mathrm{cont1}}},$$

(3)

and the sum of $a$ and $b$ equals unity. The specific values of $a$ and $b$ of the four Balmer lines are given in Table 1. If the two continuum bands are symmetrical with respect to the center of the line, both $a$ and $b$ are $0.5$ (i.e., equally weighted). If the two continuum bands are asymmetrical relative to the line center, such as for the H$\gamma$ line as illustrated in Fig. 2c, the values of $a$ and $b$ are different (see Table 1). The $a\cdot\overline{F}_{\mathrm{cont1}}+b\cdot\overline{F}_{\mathrm{cont2}}$ term in the denominator of Equation (1) can also be understood as the virtual continuum flux at the center wavelength of the Balmer line (indicated by the diamond symbol in Fig. 2), which is linearly interpolated from the values of $\overline{F}_{\mathrm{cont1}}$ and $\overline{F}_{\mathrm{cont2}}$ as implied in Equations (2) and (3) and illustrated by the dashed line in Fig. 2.

The $L$-index values of the four Balmer lines (denoted by $L_{\mathrm{H\alpha}}$, $L_{\mathrm{H\beta}}$, $L_{\mathrm{H\gamma}}$, and $L_{\mathrm{H\delta}}$, respectively) are derived for the LRS spectral sample of solar-like stars obtained in Section 2. The wavelength shift in the original LRS data caused by the stellar radial velocity is corrected by using the rv parameter in the LRS catalog before the activity index evaluation. The uncertainties of the activity index values are also estimated by considering the flux uncertainty, the radial velocity uncertainty, and the uncertainty due to the discrete data points [18].

As explained in Section 2, a stellar object may be observed multiple times at different dates by the LRS. Then, for a same stellar source, there might be multiple measurements of the $L$-index for each of the Balmer lines. These multiple $L$-index measurements of each Balmer line probably correspond to different activity levels of the source and present a certain degree of fluctuations around a mid value. For this reason, the median of the multiple measurements of each Balmer line is used as the representative $L$-index value of the line of the same source. This operation is performed for each stellar object in the stellar source sample of solar-like stars obtained in Section 2.

The data of the $L_{\mathrm{H\alpha}}$, $L_{\mathrm{H\beta}}$, $L_{\mathrm{H\gamma}}$, and $L_{\mathrm{H\delta}}$ indexes (and their uncertainties) of the four Balmer line of the LRS spectral sample and the stellar source sample are provided in the dataset of this paper (see Supplementary Note and Supplementary Tables S1 and S3). Some $L$-index and uncertainty values are unavailable for a small portion of the LRS spectral sample owing to the invalid LRS flux data around the Balmer lines; those missing values (as well as the missing values for other activity measures obtained in this work) are marked as $-9999.0$ in the dataset.

Figure 3 shows the distribution histograms of the derived $L_{\mathrm{H\alpha}}$, $L_{\mathrm{H\beta}}$, $L_{\mathrm{H\gamma}}$, and $L_{\mathrm{H\delta}}$ index values of the four Balmer lines (see Figs. 3a, 3c, 3e, and 3g) and their uncertainties $\delta L_{\mathrm{H\alpha}}$, $\delta L_{\mathrm{H\beta}}$, $\delta L_{\mathrm{H\gamma}}$, and $\delta L_{\mathrm{H\delta}}$ (see Figs. 3b, 3d, 3f, and 3h) for the stellar source sample of solar-like stars employed in this work. The medians of the $L$ and $\delta L$ distributions of the four Balmer lines shown in Fig. 3 are given in Table 2. It can be seen from Fig. 3 and Table 2 that the $L$-index values of the four Balmer lines are in the range of $0.5\sim 0.9$, and the uncertainties of the $L$-index values are on the order of magnitude of $10^{-2}$. The relatively wider $\delta L_{\mathrm{H\gamma}}$ and $\delta L_{\mathrm{H\delta}}$ distributions (Figs. 3f and 3h) than the $\delta L_{\mathrm{H\alpha}}$ and $\delta L_{\mathrm{H\beta}}$ distributions (Figs. 3b and 3d) is owing to the lower S/N values in the blue band than in the red band of LRS. The distributions of the $L_{\mathrm{H\alpha}}$, $L_{\mathrm{H\beta}}$, $L_{\mathrm{H\gamma}}$, and $L_{\mathrm{H\delta}}$ indexes of the four Balmer line shown in Fig. 3 will be analyzed in detail in Section 4.

The MRS data have a higher spectral resolution than the LRS data, but only contain the H$\alpha$ line of the Balmer series (see Fig. 1). An activity index, $I_{\mathrm{H\alpha}}$, was introduced for the H$\alpha$ line in the MRS data in the work by He et al. [18]. The formula of $I_{\mathrm{H\alpha}}$ is similar to that of the $L$-index defined for the Balmer lines in the LRS data in Equation (1), but the definition of the central and continuum bands of the $I_{\mathrm{H\alpha}}$ index is different from that of the $L_{\mathrm{H\alpha}}$ index because of the higher spectral resolution of the MRS data. In the definition of the $I_{\mathrm{H\alpha}}$ index, the width of the central band of H$\alpha$ line is 0.25 Å; the wavelength offset of the two continuum bands is $\pm 25$ Å relative to the H$\alpha$ line center; and the width of each continuum band is 5 Å [18].

Because the MRS and LRS data are collected in different spectral resolutions, for a same stellar source, the $I_{\mathrm{H\alpha}}$ value derived from the MRS data and the $L_{\mathrm{H\alpha}}$ value from the LRS data are generally different. In order to obtain a MRS H$\alpha$ activity index that is compatible with the LRS $L_{\mathrm{H\alpha}}$ index, the spectral resolution of the original MRS data is first reduced to the same resolution as the LRS data by Gauss smoothing, then the H$\alpha$ activity index is calculated from the smoothed MRS data using the same approach as for the $L_{\mathrm{H\alpha}}$ index of the LRS data. The obtained H$\alpha$ activity index based on the smoothed MRS data is denoted by $\widetilde{I}_{\mathrm{H\alpha}}$, and the formula for calculating $\widetilde{I}_{\mathrm{H\alpha}}$ is

$$\widetilde{I}_{\mathrm{H\alpha}}=\frac{\overline{\mathcal{F}}_{\mathrm{cen,H\alpha}}}{a_{\mathrm{H\alpha}}\cdot\overline{\mathcal{F}}_{\mathrm{cont1,H\alpha}}+b_{\mathrm{H\alpha}}\cdot\overline{\mathcal{F}}_{\mathrm{cont2,H\alpha}}},$$

(4)

where $\mathcal{F}$ represents the smoothed MRS spectral flux. The definitions for the central and continuum bands in Equation (4) and the weight factors $a_{\mathrm{H\alpha}}$ and $b_{\mathrm{H\alpha}}$ for the mean fluxes of the two continuum bands are the same as listed in Table 1 (note that both $a_{\mathrm{H\alpha}}$ and $b_{\mathrm{H\alpha}}$ equal to $0.5$).

The values of the $\widetilde{I}_{\mathrm{H\alpha}}$ index are derived for the MRS spectral sample of solar-like stars obtained in Section 2. The wavelength shift in the original MRS data caused by the stellar radial velocity is corrected by using the rv_r0 parameter in the MRS catalog before the $\widetilde{I}_{\mathrm{H\alpha}}$ index evaluation. The uncertainties of the $\widetilde{I}_{\mathrm{H\alpha}}$ values are also estimated using the similar approach as for the $L$-index of the LRS spectra.

A stellar object may be observed multiple times at different dates by the MRS (see Section 2), and hence there might be multiple measurements of the $\widetilde{I}_{\mathrm{H\alpha}}$ index for a same stellar source. As the operation for the multiple measurements of the $L$-index of the LRS data, for each stellar object in the stellar source sample of solar-like stars obtained in Section 2, the multiple measurements of the $\widetilde{I}_{\mathrm{H\alpha}}$ index of the object are aggregated by the median of the multiple $\widetilde{I}_{\mathrm{H\alpha}}$ values.

Figure 4 shows the distribution histograms of the $\widetilde{I}_{\mathrm{H\alpha}}$ index values (filled histogram in Fig. 4a) and their uncertainties $\delta\widetilde{I}_{\mathrm{H\alpha}}$ (filled histogram in Fig. 4b) derived from the MRS data for the stellar source sample of solar-like stars employed in this work. The histograms of the $L_{\mathrm{H\alpha}}$ index and its uncertainty $\delta L_{\mathrm{H\alpha}}$ of the LRS data for the stellar source sample (as having been shown in Figs. 3a and 3b) are also plotted in Fig. 4 (line histograms) for reference. It can be seen from Fig. 4a that the magnitudes of the $\widetilde{I}_{\mathrm{H\alpha}}$ index from the MRS data and the $L_{\mathrm{H\alpha}}$ index from the LRS data are similar. Figure 4b shows that the uncertainty of $\widetilde{I}_{\mathrm{H\alpha}}$ is about one order of magnitude smaller than the uncertainty of $L_{\mathrm{H\alpha}}$, which can be interpreted as the effect of smoothing on the MRS data, that is, both the spectral flux uncertainty and the uncertainty owing to discrete data points are dramatically reduced by the smoothing procedure.

In order to determine the quantitative relationship between the $\widetilde{I}_{\mathrm{H\alpha}}$ index and the $L_{\mathrm{H\alpha}}$ index, the distribution of the difference between the $\widetilde{I}_{\mathrm{H\alpha}}$ and $L_{\mathrm{H\alpha}}$ values (i.e., $\widetilde{I}_{\mathrm{H\alpha}}-L_{\mathrm{H\alpha}}$) is investigated for the stellar source sample of solar-like stars employed in this work, which is shown in Fig. 5a. It can be seen from Fig. 5a that there is a small offset between the values of $\widetilde{I}_{\mathrm{H\alpha}}$ and $L_{\mathrm{H\alpha}}$. To find out the exact value of the offset, the envelope of the $\widetilde{I}_{\mathrm{H\alpha}}-L_{\mathrm{H\alpha}}$ distribution histogram shown in Fig. 5a is fitted by a normal distribution model. The fitted probability density function (PDF) of the normal distribution is also plotted in Fig. 5a as a solid black curve. The center location of the fitted normal distribution (denoted by $C=0.01357$ and indicated by a vertical dashed line in Fig. 5a) reflects the offset between $\widetilde{I}_{\mathrm{H\alpha}}$ and $L_{\mathrm{H\alpha}}$, and the standard deviation of the fitted normal distribution (equaling to $0.01093$) has the similar scale as the $L_{\mathrm{H\alpha}}$ uncertainty (about $10^{-2}$ with a median of 0.011; see Fig. 3b and Table 2), which implies that the scatter of the $\widetilde{I}_{\mathrm{H\alpha}}-L_{\mathrm{H\alpha}}$ distribution is mainly caused by the uncertainty of $L_{\mathrm{H\alpha}}$, since the uncertainty of $\widetilde{I}_{\mathrm{H\alpha}}$ is about one order of magnitude smaller than that of $L_{\mathrm{H\alpha}}$ (see Fig. 4b).

To compensate for the offset between $\widetilde{I}_{\mathrm{H\alpha}}$ and $L_{\mathrm{H\alpha}}$, a new activity index (denoted by $M_{\mathrm{H\alpha}}$, where the letter M indicates that the index is derived from the MRS data) is introduced based on the $\widetilde{I}_{\mathrm{H\alpha}}$ index, which is calculated with the formula

$$M_{\mathrm{H\alpha}}=\widetilde{I}_{\mathrm{H\alpha}}-0.01357,$$

(5)

where the constant number $0.01357$ reflects the offset between $\widetilde{I}_{\mathrm{H\alpha}}$ and $L_{\mathrm{H\alpha}}$ (see Fig. 5a). The uncertainty of the $M_{\mathrm{H\alpha}}$ index ($\delta M_{\mathrm{H\alpha}}$) is the same as that of the $\widetilde{I}_{\mathrm{H\alpha}}$ index (i.e., $\delta M_{\mathrm{H\alpha}}=\delta\widetilde{I}_{\mathrm{H\alpha}}$) according to the error propagation rules.

The values of the $M_{\mathrm{H\alpha}}$ index and their uncertainties are derived from the values of the $\widetilde{I}_{\mathrm{H\alpha}}$ index for the MRS spectral sample and the stellar source sample of solar-like stars obtained in Section 2. The data of the $M_{\mathrm{H\alpha}}$ index (and its uncertainty) are provided in the dataset of this paper (see Supplementary Note and Supplementary Tables S2 and S3). Considering that the $\widetilde{I}_{\mathrm{H\alpha}}$ index is an intermediate parameter, it is not included in the dataset; if desired, it can be derived from the data of $M_{\mathrm{H\alpha}}$ by using Equation (5).

Figure 5b shows the distribution histogram of the derived $M_{\mathrm{H\alpha}}$ index values (filled histogram) for the stellar source sample of solar-like stars employed in this work. The histogram of the $L_{\mathrm{H\alpha}}$ index for the stellar source sample (as having been shown in Fig. 3a) is also plotted in Fig. 5b (line histogram) for reference. The median of the $M_{\mathrm{H\alpha}}$ distribution shown in Fig. 5b, as well as the median of the $\delta M_{\mathrm{H\alpha}}$ distribution (same as the $\delta\widetilde{I}_{\mathrm{H\alpha}}$ distribution shown in Fig. 4b), is given in Table 2. It can be seen from Fig. 5b that the distribution of the $M_{\mathrm{H\alpha}}$ index derived from the MRS data is consistent with the distribution of the $L_{\mathrm{H\alpha}}$ index from the co-source LRS data. The wider span of the $L_{\mathrm{H\alpha}}$ index distribution is due to the larger uncertainty of the $L_{\mathrm{H\alpha}}$ index values. Detailed analysis on the distributions of the activity indexes will be performed in Section 4.

The $L$-index defined for the Balmer lines in the LRS data (see Equation (1)) can also be expressed with the stellar absolute flux:

$$L=\frac{\overline{f}_{\mathrm{cen}}}{a\cdot\overline{f}_{\mathrm{cont1}}+b\cdot\overline{f}_{\mathrm{cont2}}},$$

(6)

where $f$ represents the absolute spectral flux on the stellar surface (in unit of erg s^-1 cm^-2 Å^-1) and $\overline{f}$ denotes the mean flux of $f$ in a given band. In Section 3.1, the $L$-index values have been obtained from the observed LRS spectral data, then the absolute flux in the central band of a Balmer line, $\overline{f}_{\mathrm{cen}}$, can be calculated from the $L$-index by using the following equation:

$$\overline{f}_{\mathrm{cen}}=(a\cdot\overline{f}_{\mathrm{cont1}}+b\cdot\overline{f}_{\mathrm{cont2}})\cdot L.$$

(7)

By expressing the absolute flux $\overline{f}_{\mathrm{cen}}$ as the ratio to the stellar bolometric flux $\sigma T_{\mathrm{eff}}^{4}$ (where $\sigma$ is the Stefan-Boltzmann constant), the absolute flux index (denoted by $\ell$) is defined for the Balmer lines in the LRS data as

$$\ell=\frac{\overline{f}_{\mathrm{cen}}}{\sigma T_{\mathrm{eff}}^{4}}=\frac{a\cdot\overline{f}_{\mathrm{cont1}}+b\cdot\overline{f}_{\mathrm{cont2}}}{\sigma T_{\mathrm{eff}}^{4}}\cdot L.$$

(8)

Because $f$ in the numerator of Equation (8) is the spectral flux (flux per unit wavelength) and the $\sigma T_{\mathrm{eff}}^{4}$ term (stellar bolometric flux) in the denominator of Equation (8) is the integral flux of all wavelengths, the unit of the absolute flux index $\ell$ is Å^-1.

Equation (8) is used to transform the $L$-index of the Balmer lines in the LRS data to the absolute flux index $\ell$. The $\overline{f}_{\mathrm{cont1}}$ and $\overline{f}_{\mathrm{cont2}}$ (absolute fluxes of the two continuum bands) and then the $(a\cdot\overline{f}_{\mathrm{cont1}}+b\cdot\overline{f}_{\mathrm{cont2}})/(\sigma T_{\mathrm{eff}}^{4})$ term in Equation (8) are calculated by using the synthetic spectra of the PHOENIX stellar atmosphere model [32] published in the work by Husser et al. [33]. A comparison between the PHOENIX synthetic spectra and the observed spectra in the literature [24] showed that the PHOENIX model can provide a satisfactory representation of the continuum flux of solar-like stars. The stellar atmospheric parameters of the PHOENIX grid of synthetic spectra [33] utilized in this work are $T_{\mathrm{eff}}=5700,5800,5900$ K, $\log g=3.5,4.0,4.5,5.0$ dex, and $\mathrm{[Fe/H]}=-1.5,-1.0,-0.5,0.0,0.5,1.0$ dex, which cover the stellar atmospheric parameter ranges of the LAMOST spectral samples of solar-like stars employed in this work (see Section 2). The spectral resolution of the original PHOENIX synthetic spectra is reduced to the same resolution as the LRS data by Gauss smoothing. The values of the $(a\cdot\overline{f}_{\mathrm{cont1}}+b\cdot\overline{f}_{\mathrm{cont2}})/(\sigma T_{\mathrm{eff}}^{4})$ term in Equation (8) on the PHOENIX grid are first calculated based on the smoothed PHOENIX synthetic spectra; the values of $(a\cdot\overline{f}_{\mathrm{cont1}}+b\cdot\overline{f}_{\mathrm{cont2}})/(\sigma T_{\mathrm{eff}}^{4})$ between the PHOENIX grid points are then calculated using linear interpolation from the values on the grid.

The $\ell$-index values of the four Balmer lines (denoted by $\ell_{\mathrm{H\alpha}}$, $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$, respectively) are calculated from the values of the $L_{\mathrm{H\alpha}}$, $L_{\mathrm{H\beta}}$, $L_{\mathrm{H\gamma}}$, and $L_{\mathrm{H\delta}}$ indexes obtained in Section 3.1. The uncertainties of the $\ell$-index values are also estimated, which originate from the uncertainties of the $L$-index values and the uncertainties of the stellar atmospheric parameters determined from the LRS data. In order to be consistent with the absolute flux index defined for the MRS data (see the following subsection), the $\ell$-index values of the four Balmer lines and their uncertainties are only evaluated for the stellar source sample of solar-like stars obtained in Section 2. The data of the $\ell_{\mathrm{H\alpha}}$, $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ indexes (and their uncertainties) of the stellar source sample are provided in the dataset of this paper (see Supplementary Note and Supplementary Table S3).

Figure 6 shows the distribution histograms of the derived $\ell_{\mathrm{H\alpha}}$, $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ index values of the four Balmer lines (see Figs. 6a, 6c, 6e, and 6g) and their uncertainties $\delta\ell_{\mathrm{H\alpha}}$, $\delta\ell_{\mathrm{H\beta}}$, $\delta\ell_{\mathrm{H\gamma}}$, and $\delta\ell_{\mathrm{H\delta}}$ (see Figs. 6b, 6d, 6f, and 6h) for the stellar source sample of solar-like stars employed in this work. The medians of the $\ell$ and $\delta\ell$ distributions of the four Balmer lines shown in Fig. 6 are given in Table 2. It can be seen from Fig. 6 and Table 2 that the $\ell$-index values of the four Balmer lines are in the range of $0.6\times 10^{-4}\sim 1.1\times 10^{-4}$ Å^-1, and the uncertainties of the $\ell$-index values are on the order of magnitude of $10^{-6}$ Å^-1. The distributions of the $\ell_{\mathrm{H\alpha}}$, $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ indexes of the four Balmer line shown in Fig. 6 will be analyzed in detail in Section 4.

In Section 3.1, the activity index $M_{\mathrm{H\alpha}}$ have been obtained from the smoothed MRS data, which is introduced to be quantitatively consistent with the $L_{\mathrm{H\alpha}}$ index of the LRS data. Similar to the absolute flux index $\ell$ defined for the Balmer lines in the LRS data, an absolute flux index, $m_{\mathrm{H\alpha}}$, can be defined based on the $M_{\mathrm{H\alpha}}$ index for the H$\alpha$ line in the MRS data as

$$m_{\mathrm{H\alpha}}=\frac{\overline{f}_{\mathrm{cen,H\alpha}}}{\sigma T_{\mathrm{eff}}^{4}}=\frac{a_{\mathrm{H\alpha}}\cdot\overline{f}_{\mathrm{cont1,H\alpha}}+b_{\mathrm{H\alpha}}\cdot\overline{f}_{\mathrm{cont2,H\alpha}}}{\sigma T_{\mathrm{eff}}^{4}}\cdot M_{\mathrm{H\alpha}},$$

(9)

where $a_{\mathrm{H\alpha}}=b_{\mathrm{H\alpha}}=0.5$ as usual (see Table 1).

In order to be quantitatively consistent with the $\ell_{\mathrm{H\alpha}}$ index of the LRS data, the stellar atmospheric parameters determined from the LRS spectra are used to evaluate the values of the $(a_{\mathrm{H\alpha}}\cdot\overline{f}_{\mathrm{cont1,H\alpha}}+b_{\mathrm{H\alpha}}\cdot\overline{f}_{\mathrm{cont2,H\alpha}})/(\sigma T_{\mathrm{eff}}^{4})$ term in Equation (9); another consideration is that, although the uncertainties of the stellar atmospheric parameters of the LRS and MRS data are at the similar level, the trailing of the stellar parameter uncertainty distributions of the MRS data is longer than that of the LRS data (see Supplementary Methods and Supplementary Fig. S2). For this reason, the $m_{\mathrm{H\alpha}}$ index and its uncertainty are only evaluated for the stellar source sample of solar-like stars obtained in Section 2. The data of the $m_{\mathrm{H\alpha}}$ index (and its uncertainty) of the stellar source sample are provided in the dataset of this paper (see Supplementary Note and Supplementary Table S3).

Figure 7 shows the distribution histograms of the derived $m_{\mathrm{H\alpha}}$ index values (filled histogram in Fig. 7a) and their uncertainties $\delta m_{\mathrm{H\alpha}}$ (filled histogram in Fig. 7b) for the stellar source sample of solar-like stars employed in this work. The histograms of the $\ell_{\mathrm{H\alpha}}$ index values and their uncertainty for the stellar source sample (as having been shown in Figs. 6a and 6b) are also plotted in Fig. 7 (line histograms) for reference. The medians of the $m_{\mathrm{H\alpha}}$ and $\delta m_{\mathrm{H\alpha}}$ distributions shown in Fig. 7 are given in Table 2. It can be seen from Fig. 7a that the distribution of the $m_{\mathrm{H\alpha}}$ index derived from the MRS data is consistent with the distribution of the $\ell_{\mathrm{H\alpha}}$ index from the co-source LRS data. Figure 7b shows that the uncertainty of $m_{\mathrm{H\alpha}}$ is about one order of magnitude smaller than that of $\ell_{\mathrm{H\alpha}}$. The wider span of the $\ell_{\mathrm{H\alpha}}$ index distribution than the $m_{\mathrm{H\alpha}}$ index distribution (see Fig. 7a) is owing to the larger uncertainty of the $\ell_{\mathrm{H\alpha}}$ index values. Detailed analysis on the distribution of the absolute flux indexes will be performed in Section 4.

The absolute flux indexes defined in Section 3.2 express the absolute fluxes at the centers of the Balmer lines with the ratios to the stellar bolometric flux, which counteracts the dependence of the activity indexes with $T_{\mathrm{eff}}$ shown in Fig. 8, and hence are more suitable for comparing activity properties of stars with different $T_{\mathrm{eff}}$ and intensity magnitudes of different Balmer lines. The absolute activity indexes obtained in Section 3.2 include the $\ell_{\mathrm{H\alpha}}$, $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ indexes from the LRS data and the $m_{\mathrm{H\alpha}}$ index from the MRS data. The distributions of these absolute flux indexes with $T_{\mathrm{eff}}$ for the stellar source sample of solar-like stars employed in this work are shown in Fig. 9. The $T_{\mathrm{eff}}$ parameter used in Fig. 9 is determined from the LRS data. The medians of the absolute flux index distributions, as given in Table 2, are indicated by the horizontal dashed lines in Fig. 9.

It can be seen from Fig. 9 that the trend of the activity indexes with $T_{\mathrm{eff}}$ shown in Fig. 8 has been largely weakened in the distribution plots of the absolute flux indexes with $T_{\mathrm{eff}}$. The magnitudes of the $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ indexes (represented by the medians of the index distributions; see the horizontal dashed lines in Fig. 9) are at a similar level with small fluctuations among the lines (see Figs. 9a–9c), while the magnitude of the $\ell_{\mathrm{H\alpha}}$ index (see Fig. 9d) is lower than that of the other three lines. The consistency between the $\ell_{\mathrm{H\alpha}}$ index (obtained from the LRS data) and the $m_{\mathrm{H\alpha}}$ index (obtained from the MRS data), as well as the smaller uncertainty of the $m_{\mathrm{H\alpha}}$ data than the $\ell_{\mathrm{H\alpha}}$ data (as having been shown in Section 3.2), is demonstrated in Figs. 9d and 9e.

To investigate the dependence of the absolute flux indexes of the Balmer lines on stellar surface gravity ($\log g$), in Fig. 10, the distributions of the absolute flux indexes $\ell_{\mathrm{H\alpha}}$, $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, $\ell_{\mathrm{H\delta}}$, and $m_{\mathrm{H\alpha}}$ with $\log g$ are shown for the stellar source sample of solar-like stars employed in this work. The $\log g$ parameter used in Fig. 10 is determined from the LRS data.

It can be see from Figs. 10d and 10e that, for the H$\alpha$ line, there is a trend of the $\ell_{\mathrm{H\alpha}}$ or $m_{\mathrm{H\alpha}}$ index with the stellar surface gravity; that is, the bottom envelope of the $\ell_{\mathrm{H\alpha}}$ or $m_{\mathrm{H\alpha}}$ distribution tends to be higher with the increase of stellar surface gravity for $\log g$ values greater than about $4.2$ dex, and there is an abrupt increase of $\ell_{\mathrm{H\alpha}}$ or $m_{\mathrm{H\alpha}}$ at $\log g$ values around about $4.5$ dex. This trend can also be seen in the distribution plots of the absolute flux indexes of the other three Balmer lines ($\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$; see Figs. 10a–10c), but is not as clear as that of the H$\alpha$ line.

The distributions of the $\ell_{\mathrm{H\alpha}}$, $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, $\ell_{\mathrm{H\delta}}$, and $m_{\mathrm{H\alpha}}$ indexes with stellar metallicity ($\mathrm{[Fe/H]}$) are displayed in Fig. 11 to show the dependence of the absolute flux indexes of the Balmer lines on metallicity of stars. The $\mathrm{[Fe/H]}$ parameter used in Fig. 11 is determined from the LRS data.

It can be seen from Figs. 11d and 11e that, for the H$\alpha$ line, there is a positive correlation trend between the $\ell_{\mathrm{H\alpha}}$ or $m_{\mathrm{H\alpha}}$ index and the stellar metallicity for the sample with lower $\ell_{\mathrm{H\alpha}}$ or $m_{\mathrm{H\alpha}}$ values (less than about $0.77\times 10^{-4}$ Å^-1), and the sample with higher $\ell_{\mathrm{H\alpha}}$ or $m_{\mathrm{H\alpha}}$ values (greater than about $0.77\times 10^{-4}$ Å^-1) distributes around $\mathrm{[Fe/H]}\sim 0.1$ dex and forms a spike-shaped structure. For the other three Balmer lines (see Figs. 11a–11c), the trends of the relations between the absolute flux indexes ($\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$) and the stellar metallicity are not monotonic, and the shapes of the trends of the three Balmer lines are different from each other as exhibited in Figs. 11a–11c.

The distributions of the absolute flux indexes of the four Balmer lines with stellar effective temperature, surface gravity, and metallicity described above show that the behavior of the H$\alpha$ line in response to stellar activity is very different from that of the H$\beta$, H$\gamma$, and H$\delta$ lines. In order to find out the relationship between the absolute flux indexes of the four Balmer lines, in Fig. 12, the scatter plots of $\ell_{\mathrm{H\alpha}}$, $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ indexes vs. $m_{\mathrm{H\alpha}}$ index are shown for the stellar source sample of solar-like stars employed in this work. Because the uncertainty of the $m_{\mathrm{H\alpha}}$ index from MRS data is smaller than that of the $\ell_{\mathrm{H\alpha}}$ index from LRS data (see Section 3.2 and Table 2), the $m_{\mathrm{H\alpha}}$ index is used as the representative index of the H$\alpha$ line (see the horizontal axis of the plots in Fig. 12), and its relations with the $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ indexes of the other three Balmer lines are investigated (see Figs. 12a–12c). The scatter plot of $\ell_{\mathrm{H\alpha}}$ vs. $m_{\mathrm{H\alpha}}$ is shown in Fig. 12d (as well as in other panels of Fig. 12 in gray) for comparison.

It can be seen from Fig. 12d that the data points of $\ell_{\mathrm{H\alpha}}$ vs. $m_{\mathrm{H\alpha}}$ are distributed mainly along the $1:1$ line (with a correlation coefficient of 0.86), demonstrating the consistency between the $\ell_{\mathrm{H\alpha}}$ index (from LRS data) and the $m_{\mathrm{H\alpha}}$ index (from MRS data). The scatter plots of $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ vs. $m_{\mathrm{H\alpha}}$ in Figs. 12a–12c show that the relations between the absolute flux indexes of the $H\beta$, $H\gamma$, and $H\delta$ lines and the H$\alpha$ line are not monotonic. With the increase of the $m_{\mathrm{H\alpha}}$ index, the $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ indexes first present trend of increasing and then deceasing, and finally increase synchronously with the $m_{\mathrm{H\alpha}}$ index (see the short arrows in Figs. 12a–12c for the turning positions of the upper envelopes of the data point distributions). The deviation from the positive correlation between the $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ indexes and the $m_{\mathrm{H\alpha}}$ index becomes greater from the lower-order to higher-order Balmer lines (with the correlation coefficients of $\ell_{\mathrm{H\beta}}$, $\ell_{\mathrm{H\gamma}}$, and $\ell_{\mathrm{H\delta}}$ vs. $m_{\mathrm{H\alpha}}$ being 0.65, 0.37, and 0.22, respectively). The different behaviors of the Balmer lines in response to stellar activity imply their different forming mechanisms. This issue will be discussed in Section 5.1.