2. Line-strength variability during the solar cycle

Past efforts to identify solar-cycle variability of photospheric lines in the spectrum of the Sun-as-a-star included 35 years of monitoring from Kitt Peak (Livingston et al. Reference Livingston, Wallace and White2007; Livingston et al. Reference Livingston, White and Wallace2010), but the results were rather inconclusive, except for chromospherically influenced lines. This acknowledges the challenges in maintaining stable instrumentation and observational conditions over longer epochs with spectrometers operated in air and readjusted between different observing programs. A review of these and other earlier efforts is in Dravins & Ludwig (Reference Dravins and Ludwig2024).

These limitations are now circumvented by Sun-as-a-star telescopes feeding extreme-precision radial-velocity spectrometers operated in a vacuum and (almost) without any readjustments. Their time series permits not only radial-velocity monitoring but also precise stellar spectroscopy.

2.2. Photospheric line changes during successive years

Figure 1 shows systematic patterns in measured line-strength variations. With increasing chromospheric activity, Fe absorption gradually weakens, showing the greatest amplitude for Fe II at shorter wavelengths, while diminishing for Fe I and at longer wavelengths.

Figure 1.

Left: Observed relative changes of Fe I and Fe II absorption line equivalent widths in the spectrum of the Sun-as-a-star, measured with HARPS-N, between summer seasons in 2016 (red), 2017 (cyan), and 2018 (gray), as a function of the Mt.Wilson Ca II H&K chromospheric activity S-index. The variability largely shadows the Ca II H&K activity-cycle signal, but with much smaller amplitudes on sub-percent levels. Right: Relative and systematic changes in the absorption equivalent widths between the three strong lines of the Mg I green triplet, and the semiforbidden Mg I $\lambda$ 457 nm line.

Figure 1 Long description

Panel A: A scatter plot shows the relationship between Fe II λ435-475 percent equivalent width and Mt. Wilson Ca II H&K S-index. Data points are color-coded in red, cyan, and gray. Panel B: A scatter plot shows the relationship between Mg I b2 percent absorption and Mg I b1 percent absorption. Data points are color-coded in red, cyan, and gray. Panel C: A scatter plot shows the relationship between Fe I λ430-445 percent equivalent width and Mt. Wilson Ca II H&K S-index. Data points are color-coded in red, cyan, and gray. Panel D: A scatter plot shows the relationship between Mg I b3 percent absorption and Mg I b1 percent absorption. Data points are color-coded in red, cyan, and gray. Panel E: A scatter plot shows the relationship between Fe I λ520-535 percent equivalent width and Mt. Wilson Ca II H&K S-index. Data points are color-coded in red, cyan, and gray. Panel F: A scatter plot shows the relationship between Mg I λ457 percent absorption and Mg I b1 percent absorption. Data points are color-coded in red, cyan, and gray.

For stronger, chromospherically influenced lines such as the green triplet of Mg I b $_1$ 518.3, b $_2$ 517.2, and b $_3$ 516.7 nm, the behavior is more complex, and the changes are systematically different among the lines in this triplet. Their atomic energy levels are coupled by transitions between a common upper level and three different lower levels, a configuration that makes non-LTE effects very likely. Somewhat similar relations are observed among the H $\alpha$ , H $\beta$ , and H $\gamma$ Balmer lines of hydrogen. By contrast, the formation of the semi-forbidden Mg I 457.1 nm intercombination line is expected to be relatively simple, being dominated by atomic collisions. Indeed, its behavior is very different, showing no significant variability during these years.

Such patterns of photospheric line variability demonstrate that they contain nontrivial information and measure something different from chromospheric indices. However, detailed studies of such signals at their sub-percent levels remain challenging. With equivalent-width measures formed over many tens of pixels in the spectrometer, each with a nominal S/N $\sim$ 350, and after averaging over several similar lines, one would expect the photometric noise contribution to be less than 0.001, raising the question of what part of the scatter seen in Fig. 1 is physical and what is instrumental?

It would certainly be interesting to follow the relative radial-velocity and line-shape changes between groups of different classes of lines during the solar activity cycle; however, that remains beyond the capabilities of current radial-velocity instruments. Their wavelength calibration is stable but not absolute between different grating orders. They reach sub-m s $^{-1}$ precision, but only after averaging over complete stellar spectra, not for limited groups of selected lines. Although their spectral resolutions are classed as ‘high’, those are lower than in synthetic spectra or in spectrometers at solar telescopes, smearing out signatures such as the exact bisector shapes. For HARPS-N and similar instruments, such limitations are discussed by Hanassi-Savari et al. (Reference Hanassi-Savari, Pietrow and Druett2025). For such reasons, the current studies concern equivalent widths only, which – at least to a first approximation – should be insensitive to such issues.

Several precision radial-velocity instruments are used to monitor the Sun-as-a-star, and comparisons among them confirm that their deduced radial-velocity fluctuations largely agree Zhao et al. (Reference Zhao, Dumusque and Ford2023). With time series now reaching a full solar cycle, it should be feasible to examine whether the deduced line-strength variations also agree among the instruments, thereby isolating what is solar and what is instrumental.

3. Theoretical modeling of radial-velocity jittering

Quiet granulation covers most of the solar surface and provides most of the solar flux. For its modeling, surface convection is simulated with time-dependent 3D hydrodynamics, along with the concurrent computation of complete spectra from ultraviolet to infrared. With hyper-high spectral resolution, the changing shapes and wavelength shifts of representative Fe I and Fe II lines are followed during the simulation sequences, seeking systematic patterns of variability, particularly those that correlate with excursions in apparent radial velocity.

Solar surface volumes that realistically can be simulated cover only a small surface fraction: in the present modeling with the CO $^5$ BOLD code this is a 3D Cartesian “local-box” model of 5.6 $\times$ 5.6 Mm $^2$ ; for modeling methodology, see Freytag et al. (Reference Freytag, Steffen and Ludwig2012). From this simulation volume, spatially averaged line profiles are computed in LTE at various emergent angles, and corresponding full-disk profiles synthesized for a non-rotating Sun. The radial-velocity jittering for integrated sunlight, built up from such a small simulation area, typically amounts to $\pm$ 150 m s $^{-1}$ , scaling to $\sim$ 2 m s $^{-1}$ for the real full solar disk. Most photospheric lines vary in phase but differences in radial-velocity excursions between various line-groups are found to be about 10% of their values. Amplitudes are greater for stronger and for ionized lines, decreasing at longer wavelengths. An example is in Fig. 2 with details in Dravins & Ludwig (Reference Dravins and Ludwig2023).

Figure 2.

Synthetic spectral lines for the spectrum of a full solar disk, built up by spectral-line components from a small simulation area of non-magnetic granulation, sampled at 20 instances during the 3D hydrodynamic simulation. Top: A strong Fe I line in the blue, $\lambda$ 436.59 nm, $\chi$ = 2.99 eV. Bottom: A weak Fe I line in the longward red, $\lambda$ 675.27 nm, $\chi$ = 4.64 eV. Right: zoomed-up line centers on a uniform wavelength scale in units of radial velocity (top); tick marks on the vertical axes correspond to those for the full profiles. Hyper-high spectral resolution, $\lambda$ / $\Delta\lambda$ $\sim$ 900,000.

Current computational constraints do not permit properly conducting detailed simulations of granulation coupled with large-scale features such as supergranular and meridional flows, p-mode oscillations, and others, which certainly may contribute significant radial-velocity signals but would require much larger simulation volumes.

4. Magnetic granulation

The second largest contribution to solar irradiance comes from the plage and faculae areas of magnetic granulation. Magnetic fields disturb and dampen the convective velocity patterns, causing different asymmetries in the line profiles and resulting in much smaller convective blueshifts. The varying area coverage of magnetic granulation during the solar cycle has been identified as the main driver of long-term changes in solar apparent radial velocity (Meunier et al. Reference Meunier, Desort and Lagrange2010a, b). While contributions from sunspots also affect spectral lines (Komori et al. Reference Komori, Brewer and Zhao2025), their fraction of the solar photospheric flux is modest.

While non-magnetic granulation in full-disk spectra, in principle, can be modeled from fundamental physical parameters of stellar temperature, gravity, and chemical composition, there are more degrees of freedom for magnetic granulation. Not only can their area coverage and latitudinal and longitudinal surface distributions be different (and unknown for other stars), but also the degree and distribution of magnetic flux concentrations. For the spectral synthesis, an additional complication (or opportunity!) arises with the potential of handling the components of magnetically sensitive lines in a situation characterized by a sometimes uneven precision of laboratory data.

Nevertheless, line-profile variability in magnetic areas could reveal signatures that might discriminate against other effects. A sequence of magnetohydrodynamic CO $^5$ BOLD solar models was computed as above, representing a range of magnetic flux densities. The vertical magnetic flux through the atmosphere was fixed by initial conditions and preserved during the modeling sequence. Field lines were constrained to pass vertically through the top and bottom boundaries of the simulation volume. For details, including synthetic surface images for different magnetic flux levels and at different limb angles, see Ludwig et al. (Reference Ludwig, Steffen and Freytag2023). Synthetic LTE spectra for different flux densities were computed at numerous instances throughout the simulations, and Fig. 3 provides some examples for 2400 G (240 mT), compared to the non-magnetic case.

Figure 3.

Synthetic solar profiles for an ordinary Fe II line ( $\lambda$ = 465.698 nm, $\chi$ = 2.89 eV) in non-magnetic and magnetic (2400 G, 240 mT) granulation. Thin curves are spatially averaged profiles across the simulation area, sampled at 20 temporal instances; black for non-magnetic; blue for magnetic. Left-hand groups show profiles at solar disk center, at right integrated for a full (non-rotating) solar disk, using center-to-limb profiles from the small simulation area. Spatially and temporally averaged profiles are at bottom left. The non-magnetic convective blueshift is greater (by $\sim$ 400 m s $^{-1}$ ), and furthermore is slightly larger at disk center than in integrated sunlight. The opposite holds for magnetic granulation, apparently caused by a more ‘corrugated’ stellar surface, an effect also causing increased plage and faculae brightness toward the limb (Ludwig et al. Reference Ludwig, Steffen and Freytag2023). Bottom right: Hyper-high spectral resolution ( $\lambda$ / $\Delta\lambda$ $\sim$ 900,000) enables to study variations inside line profiles. The temporal standard deviations show the longward line flanks to be the most variable in magnetic granulation versus the line core in the non-magnetic case.

5. Outlook: possibilities and challenges

A significant addition to Sun-as-a-star facilities will be the PoET telescope, to be installed at ESO VLT on Paranal (Santos et al. Reference Santos, Cabral and Leite2025). Besides providing spectra of the Sun-as-a-star with the ESPRESSO spectrometer’s higher resolving power of $\sim$ 200,000, it will be able to select limited portions of the solar disk to examine local radial velocities. This should help to identify locations contributing to solar velocity jittering and also enable novel tests of hydrodynamic model atmospheres. Until now, these have been verified against spatially and temporally averaged spectral line shapes, granulation morphology, and intensity modulation. Predictions such as different radial velocity jittering amplitudes among various classes of spectral lines – such as discussed above – cannot yet be tested because of lack of adequate instrumentation. For Sun-as-a-star, it is still not possible to discern differences between line-group jittering of amplitudes between, say, 2.0 and 2.2 m s $^{-1}$ , as deduced for the full solar disk. However, such extrapolated full-disk values emanate from small simulation areas where the amplitudes are much greater and should be readily measurable with ESPRESSO.

Different levels of approximation and sophistication are possible in computing synthetic spectral lines from dynamic 3-D atmospheres, eventually envisioning ‘full’ non-LTE treatments with Zeeman-sensitive line components, a connection to the lower chromosphere, and others. Although not all such observations are yet feasible, the types of observations foreseen with PoET may well guide us toward highly precise 3-D spectra. To realize corresponding spatially resolved spectroscopy across the surfaces of other stars is marginally possible using exoplanet transits to differentially isolate surface portions, for which theoretical line-profile predictions can be made (Dravins et al. Reference Dravins, Ludwig and Dahlén2017a,b, 2018, 2021a,b). In the nearish future, the availability of stable spectrometers at extremely large telescopes should make such observations less challenging.

The significance of non-LTE effects for also ‘ordinary’ photospheric lines measured for radial velocities is suggested by studies of how different assumed levels of chromospherically influenced temperatures in the upper photosphere affect 1-D non-LTE line formation in modeled G2 V stars (Vieytes et al. Reference Vieytes, Zhao and Bedell2025). The response to different levels of chromospheric heating is markedly different among various Fe I lines, even those with similar strengths at nearby wavelengths, carrying a promise that they may contain additional and perhaps unique signatures of solar atmospheric modulation. Such lines could be selected to observationally search for their variability, although challenging non-LTE 3D calculations might be needed for a fuller understanding.

Other relevant spectrometers for Sun-as-a-star spectroscopy include those in the near infrared, e.g., NIRPS at ESO on La Silla (Bouchy et al. Reference Bouchy, Doyon and Pepe2025). As seen in both observed and synthetic spectra, the amplitudes of physical variability decrease toward longer wavelengths, suggesting that the near infrared could be advantageous for exoEarth searches. However, there is the complexity of fewer atomic lines of significant strength in that region, as well as effects from enhanced telluric absorption.

The large ANDES spectrometer is being built for the ESO ELT (Palle et al. Reference Palle, Biazzo and Bolmont2025). While it will, of course, have outstanding light-collecting power and should provide very high S/N spectra for brighter stars, the challenges of interfacing a realistically large cross-dispersed échelle spectrometer to an extremely large telescope limit its spectral resolution to ranges similar to those of current instruments. In the longer term, one could certainly wish for hyper-high resolutions of 1,000,000, which may possibly require diffraction-limited adaptive-optics photonic instruments to limit their physical size. Already, NIRPS is taking advantage of adaptive optics to limit its size, but realizing the same for an extremely large telescope in the shorter visual wavelengths remains an interesting challenge.