Statistical analysis of the Si I 6560.58 Å line observed by CHASE

The equivalent width (EW) of an absorption line is defined as the integration of the normalized line depth profile over wavelength:

For all these three observations, we calculate the values of EW for each pixel on the solar disk. Data pixels at the solar limb where $\mu<0.15$ are discarded since this line becomes relatively weak. We also calculate the EWs from the VALC model for $\mu=1.0$ until $\mu=0.05$.

The probability density functions of the EW as function of $\mu$ in the three observations are shown in Fig. 4 as gray shades. The averaged value and 1$\sigma$ range at each $\mu$ are overplotted as red and blue curves, while the calculated values from the VALC model are shown in green diamonds. These values are also listed in Table 2 for comparison.

It is clear that the center-to-limb variation of EW as revealed from the three observations are quite similar, and agrees well with model calculations. Generally speaking, the EW increases from center to limb, and reaches its peak value near $\mu=0.2$, and then begins to decrease. The variation of the EW could be interpreted in the following way. As $\mu$ increases, both the Si I line and the continuum are formed in higher layers with a lower local temperature, leading to a decrease in both the line and the continuum intensity. A smaller line intensity tends to decrease the EW, while a smaller continuum intensity tends to increase the EW. Since the line and continuum are formed in different heights, the decreasing percentage of their intensity at a certain $\mu$ could vary. The competition of these two factors leads to the final results: when $\mu$ decreases from 1.0 to 0.2, the line intensity decreases more sharply than continuum; while for $\mu<0.2$ the decrease in continuum dominates. The differences of the averaged values from observations and calculated values from models are less than 1 mÅ, which is within the 1$\sigma$ range.

The only recorded EW value of this line in literature is from the revised Rowland Table by Moore et al. (1966), which reads 22 mÅ at the disk center. We also measure the EW from the Jungfraujoch atlas (BASS2000, Delbouille et al. 1973) and the Kitt Peak atlas (Kurucz et al. 1984), and they give values of 18.52 and 17.27 mÅ at the disk center. All these values are far larger than the observed values from CHASE, even outside the 1$\sigma$ range, which is mostly due to the fact that the Si I line is blended with a telluric absorption line. Although the Kitt Peak atlas has corrected the atmospheric transmission, it seems that their evaluation is still not accurate.

The full width at half maximum (FWHM) of an absorption profile is determined by various line broadening mechanisms, and is close to the FWHM of the line profile in the optically thin regime. For optically thick lines, the FWHMs of the line profile and the absorption profile are not necessarily the same due to the so-called opacity broadening (Rathore & Carlsson 2015), while they are still positively related. Here, we measure the FWHM of the line depth profile $R_{\lambda}$ for each pixel on the solar disk where $\mu\geq 0.15$. Note that the observed profiles are not corrected with a deconvolution of the instrumental profile. Thus, the measured FWHM has included the instrumental FWHM of 72.6 mÅ (Li et al. 2022).

In Fig. 5 we show the probability density functions of the FWHM as function of $\mu$ in the three observations. Despite of the spikes and sub-structures, there is still an increasing trend for the FWHM from the disk center to the solar limb. However, the calculated FWHM from the VALC model after convolution with the CHASE instrumental profile, as denoted by green diamonds and also listed in Table 2, is quite different from the observed values. Without any turbulent velocities, the calculated FWHM decreases towards the solar limb, since for a smaller $\mu$, the line forms higher in the photosphere where the lower temperature narrows the thermal width. For comparison, the measured FWHM is 166 mÅ from the Jungfraujoch atlas (BASS2000, Delbouille et al. 1973) and 158 mÅ from the Kitt Peak atlas (Kurucz et al. 1984), after convolution with the CHASE instrumental profile. The discrepancy of observed and calculated FWHM values are explained below.

Traditionally, non-thermal turbulent velocities are introduced to explain the excess width in the absorption profiles and line profiles. A dichotomy of microturbulence and macroturbulence is employed from the geometric scale of the motions. Microturbulence occurs within the mean free path of photons, so the line absorption is changed, resulting in a larger EW and FWHM. However, macroturbulence does not influence the line absorption and only broadens the line profile.

We recalculate the EW of the Si I line assuming different values of microturbulent velocity in the VALC model. The results for the $\mu=0.6$ case are shown in Fig. 6. The value of $\mu=0.6$ is chosen arbitrarily here for illustration, and the increasing trend is similar for other values of $\mu$. One can see that the EW increases by 1.1 mÅ for a microturbulent velocity of 4 km s^-1. Given the large uncertainties in the EW measurement from the observations, it would be unreliable to derive the microturbulent velocities.

As stated above, the FWHM of an absorption line is influenced by both micro- and macroturbulent velocities, and Fig. 6 provides a schematic view of their contributions as calculated from the VALC model. The line profile has been convolved with the CHASE instrumental profile in order to compare with observations. The macroturbulent velocity is then added by convolving the line depth profile with a Gaussian velocity distribution:

As shown in Fig. 6, both turbulent velocities could effectively increase the FWHM. However, the increase of the FWHM is not obvious for small macroturbulent velocities, under the spectral resolution of CHASE.

In order to separate the contributions to the FWHM from micro- and macroturbulent velocities, we fix the values for microturbulent velocities at different positions on the solar disk. We take the empirical formula of Takeda (2022), which shows an increasing trend of the microturbulent velocities towards the solar limb, with 1 km s^-1 at the disk center and 1.97 km s^-1 at $\mu=0.2$. The macroturbulent velocities are then derived from a similar relation as in Fig. 6 after inclusion of the microturbulent velocities. The results from the three observations are shown in Fig. 7. There is also an increasing trend towards the solar limb, with 2.80 km s^-1 at the disk center and 3.52 km s^-1 at $\mu=0.2$. The center-to-limb variation of the macroturbulent velocity can be interpreted as the intrinsic anisotropy of photospheric motions. If we consider a radial turbulent velocity $\xi_{r}$ and a tangential turbulent velocity $\xi_{t}$, then the total macroturbulent velocity would be $\xi_{\mathrm{macro}}^{2}=\xi_{r}^{2}\mu^{2}+\xi_{t}^{2}(1-\mu^{2})$ (Takeda & UeNo 2017). A least-square fit to the averaged values in our observations (Fig. 7) gives $\xi_{r}=2.85$ km s^-1 and $\xi_{t}=3.58$ km s^-1. These values are slightly larger than previous ones derived from the Fe I lines (Takeda & UeNo 2017; Sheminova 2022), since the Si I line forms deeper in the photosphere where the unresolved turbulent motions are faster, as revealed from observations and simulations (Gray 1977, 1978; Takeda 1995; Beeck et al. 2012; Takeda & UeNo 2017).