NESSI: Empirical Sun-as-a-Star Spectrum Tool

• NESSI is a public code that numerically integrates high-resolution solar observations using empirical center-to-limb variation to produce a disk-integrated spectrum.
• It translates resolved spectral lines into a Sun-as-a-star spectrum by incorporating differential rotation, Doppler shifts, and options for active region substitution.
• The method outperforms continuum limb-darkening approximations, offering improved accuracy for exoplanet transit modeling and stellar activity diagnostics.

The Numerical Empirical Sun-as-a-Star Integrator (NESSI) is a public code for turning high-resolution solar observations of only part of the solar disk into a disk-integrated, Sun-as-a-star (SAAS) spectrum (Pietrow et al., 2023). It was developed to bridge a specific observational gap: solar telescopes and spectrographs often resolve small-scale structure with excellent spatial and spectral detail, but their field of view is usually too small to compare directly with stellar-style full-disk spectra. NESSI performs that translation numerically by using empirical center-to-limb variation (CLV) information rather than relying only on continuum limb-darkening approximations, with the broader aim of testing common assumptions in stellar physics and in unresolved exoplanet and activity diagnostics (Pietrow et al., 2023).

1. Scientific rationale and problem setting

NESSI was developed because spectral line CLV matters for stellar characterization, not just continuum brightness variation. Many existing transit and stellar models use continuum limb-darkening laws from atmosphere codes like PHOENIX or ATLAS, which can be adequate for broad photometric bands but are not necessarily valid for spectral lines. The distinction is consequential for exoplanet transits, where incorrect limb-darkening prescriptions can bias inferred planet radii, and for radial velocities and activity studies, where active regions such as spots and plage alter line profiles in a position-dependent way and their CLV differs from quiet-Sun CLV (Pietrow et al., 2023).

The central physical point is that line CLV is much more complicated than continuum limb darkening. Continuum CLV can often be approximated by smooth analytic laws from atmosphere models, but spectral lines form over a range of atmospheric depths and are sensitive to changing thermodynamic and velocity conditions across the disk. As a result, line depth, equivalent width, asymmetry, and line width can all change with viewing angle. Broadband photometry may tolerate continuum-only CLV reasonably well; spectroscopic observables, especially narrow lines and activity-sensitive diagnostics, do not (Pietrow et al., 2023).

The Sun is the ideal testbed because its surface structure can be observed in detail, but those observations are often restricted to a small field of view. NESSI addresses that restriction by allowing local observations to be turned into a Sun-as-a-star spectrum for direct comparison to full-disk measurements. The paper explicitly frames NESSI as an analogue of stellar surface-integration tools such as SOAP, but adapted to the Sun and built around observed CLV curves rather than continuum-only approximations (Pietrow et al., 2023).

2. Empirical disk integration and formal structure

NESSI starts from a disk-center quiet-Sun spectrum and a set of CLV curves at each wavelength point. It then evaluates how the spectrum changes as a function of position across the solar disk, applies rotational Doppler shifts, and integrates the contribution from all surface elements into one synthetic full-disk spectrum. The required ingredients are a spectral profile or wavelength range of the quiet Sun at disk center, CLV curves for each wavelength point, a differential rotation velocity profile, and the orientation of the rotation axis relative to the observer. Surface heterogeneity can be incorporated by replacing some grid elements with CLV appropriate to sunspots, plage, flares, and related structures (Pietrow et al., 2023).

A compact expression of the intended procedure is

$$F_\lambda^{\mathrm{SAAS}} \approx \sum_i I_\lambda(\mu_i)\,W_i,$$

where $I_\lambda(\mu_i)$ is the specific intensity or line profile at limb angle $\mu_i=\cos\theta_i$, $W_i$ is the geometric weight of the surface element including projected area and visibility factors, and each contribution is Doppler shifted by the local rotation velocity before summation. In continuous form, the construction is conceptually

$$F_\lambda^{\mathrm{SAAS}}=\int_{\mathrm{visible\ disk}} I_\lambda(\mu,\phi)\,\mu\,d\Omega,$$

with the local line profile modified by the CLV, the rotation-induced wavelength shift, and, where relevant, the replacement of quiet-Sun intensities by those of active regions (Pietrow et al., 2023).

The methodological distinction is not merely numerical. NESSI is not simply applying continuum limb darkening to a line profile; it uses wavelength-dependent empirical CLV so that the line shape itself varies correctly from disk center to limb. This is the point at which NESSI departs from simplified Sun-as-a-star constructions that treat angular dependence as a continuum weighting problem only (Pietrow et al., 2023).

3. Inputs, workflow, and operational assumptions

The workflow described for NESSI is sequential. One chooses a line or wavelength range at disk center, attaches CLV profiles for that line as a function of $\mu$, discretizes the solar disk into grid points, determines $\mu$ at each point, assigns the corresponding local line profile from the CLV set, applies the rotational Doppler shift, optionally replaces the quiet-Sun profile with a spot, plage, or flaring profile, and then sums all weighted contributions to produce the synthetic disk-integrated spectrum. The result is then compared to observed full-disk spectra (Pietrow et al., 2023).

Input or component	Function in NESSI	Stated source or form
Quiet-Sun spectrum at disk center	Base spectral profile	$\mu=1$ profile
CLV curves for each wavelength point	Angular dependence of line shape	Empirical CLV
Differential rotation velocity profile	Rotational Doppler shifts	Balthasar et al. (1986) in the demonstrations
Rotation-axis orientation	Observer-relative geometry	Required input
Optional active-region replacement	Surface heterogeneity	Sunspots, plage, flares, etc.

The demonstration cases in the original NESSI paper adopt several simplifying assumptions. The disk is treated as quiet Sun only, with no active regions included. The $\mu=1$ line profile represents the disk-center intensity. Rotation is described by a prescribed differential rotation law. The CLV is assumed to be sufficiently sampled by the available empirical observations. The line CLV observations are taken to encode the relevant physics even for lines that are difficult to model in 1D LTE, such as those requiring 3D NLTE treatment (Pietrow et al., 2023).

These assumptions delimit the code’s baseline use case. A plausible implication is that NESSI functions both as a forward model and as a diagnostic framework for asking which discrepancies arise from inadequate CLV treatment, which from missing surface heterogeneity, and which from the observational sampling itself.

4. Demonstration on H$\alpha$ and O I 7772 Å

The paper demonstrates NESSI using empirical CLV observations for two lines: H$I_\lambda(\mu_i)$0 from the Institut für Astrophysik Göttingen (IAG) spectral atlas and O I 7772 Å from Swedish 1-m Solar Telescope (SST) line CLV measurements. In both tests, the $I_\lambda(\mu_i)$1 profile was used as the disk-center input, the rotation field used the differential rotation law of Balthasar et al. (1986), and the Sun was treated as a quiet disk with no active regions. The synthetic spectra were compared against continuum limb-darkening curves from Neckel & Labs (1994) and against the full-disk IAG flux atlas of Reiners et al. (2016). For the oxygen line, the atlas was convolved down to the SST resolution using STIC (Pietrow et al., 2023).

For each case, the comparison involved three constructions: a spectrum synthesized using line CLV, a spectrum synthesized using continuum limb darkening, and a spectrum using only the disk-center profile. The line-CLV-based synthesis matches the observed full-disk atlas much better, with residuals at about the percent level. The continuum-only approach performs worse, which the paper uses to show that continuum limb darkening is not enough for detailed line-profile work (Pietrow et al., 2023).

The paper also notes residual structure associated with telluric-line regions in the atlas and with interpolation over removed spectral intervals. Remaining discrepancies are suggested to arise in part because the IAG full-disk atlas averages over weeks of observations, likely including some contribution from untracked surface activity. This does not invalidate the method; rather, it identifies the level at which atlas construction, tellurics, and temporal averaging enter the comparison (Pietrow et al., 2023).

A common misconception addressed by these demonstrations is that continuum limb-darkening prescriptions are an adequate surrogate for spectral-line angular behavior. NESSI’s two-line tests show that this is not generally true when the observable of interest is the detailed line profile rather than a broadband flux.

5. Extension to flare synthesis and pseudo-Sun-as-a-star spectroscopy

A later application uses NESSI to synthesize Sun-as-a-star flare spectra from 19 small area field-of-view optical observations of solar flares acquired by the Swedish 1-m Solar Telescope between 2011 and 2024, corresponding to 20 flare events because one SST dataset contains a double flare. These observations span X9.3 to C1.2 and were obtained with CRISP and CHROMIS. The stated motivation is that earlier pseudo-Sun-as-a-star methods integrated only a small region and normalized it by a quiet region, but did not fully account for the rest of the solar disk; NESSI adds the missing full-disk quiet-Sun construction, empirical CLV, differential rotation, and flare insertion at the correct disk location (Wilde et al., 10 Jul 2025).

In that flare framework, NESSI builds the solar disk on a polar grid, assigns each position a line profile based on measured CLV, shifts each profile according to a differential rotation law citing Balthasar et al. (1986), and integrates over the disk to obtain a full-disk quiet-Sun spectrum. The flare is then injected by taking the average flare spectra measured in the SST field of view, dividing by the corresponding subfield of the NESSI quiet-Sun disk at the same location and same size as the SST field of view, and multiplying the resulting flare factor back onto the full quiet disk. Before comparison, each dataset is normalized using a manually selected quietest Sun reference region inside the field of view, with the scaling constant $I_\lambda(\mu_i)$2 obtained by minimizing

$I_\lambda(\mu_i)$3

The disk-integrated contrast profile is then defined as

$I_\lambda(\mu_i)$4

The paper describes this as intensity-preserving disk integration rather than simple area scaling (Wilde et al., 10 Jul 2025).

The flare study uses the synthetic spectra to analyze contrast profiles, differenced equivalent widths, and Voigt residuals. It reports a major validation against the true Sun-as-a-star flare observation from HARPS-N of the X9.3 flare: the synthetic SST-derived contrast profiles were degraded to 5-minute cadence and HARPS-N-like noise, while leaving spectral resolution essentially unchanged because SST/CRISP and HARPS-N have similar resolving power ($I_\lambda(\mu_i)$5 versus $I_\lambda(\mu_i)$6). For that event, the SST field of view captured only about 40% of the flare, so the synthetic profile was multiplied by 2.5 to approximate the full flare, and the degraded synthetic profiles resembled the HARPS-N observations well in both shape and intensity for the interval where the flare was well covered (Wilde et al., 10 Jul 2025).

The same study identifies physical features in pseudo-Sun-as-a-star flare spectra, including core intensity modulation, redshifted emission, filament and flare-loop imprints, and offsets between $I_\lambda(\mu_i)$7 and GOES peaks, while also identifying spurious or misleading features such as false quasi-periodic oscillations and false profile modulation or abrupt jumps caused by seeing variations, tracking drift, normalization choices, or telescope repointing. It further concludes that CME detection from unresolved stellar line profiles is highly ambiguous: blue-shifted absorption alone is not a reliable CME diagnostic, flare presence does not imply a CME, and overlying filaments, loops, and coronal rain can generate false positives (Wilde et al., 10 Jul 2025).

6. Place within empirical Sun-as-a-star modeling and stated limitations

NESSI belongs to a broader class of empirical Sun-as-a-star modeling efforts that seek to reconstruct unresolved observables from spatially structured solar information. A relevant precedent is the empirical modeling of radiative versus magnetic flux for the Sun-as-a-star, which studies the relationship between full-disk solar radiative flux at different wavelengths and average solar photospheric magnetic-flux density using daily measurements from the Kitt Peak magnetograph and other instruments over most of Solar Cycles 21–23. That work finds that total solar irradiance and red continuum variability are not monotonic functions of disk-averaged magnetic flux, whereas chromospheric and coronal channels are approximately linear, and it models the Sun as a linear time-invariant system with a finite impulse response kernel. For most chromospheric and coronal channels, the reconstructions account for 85–90% of the variability (Preminger et al., 2010).

That precedent is methodologically distinct from NESSI, but the conceptual continuity is direct. Both approaches operate in the Sun-as-a-star regime, both are empirical rather than purely theoretical atmosphere syntheses, and both treat unresolved observables as the output of structured solar physics that must be integrated rather than assumed. This suggests that NESSI occupies the spectral-line and spatial-integration side of the same broader program in which full-disk observables are reconstructed from physically informative, but incomplete, solar measurements.

The limitations stated for NESSI and its applications are explicit. In the original line-CLV demonstrations, the disk is quiet Sun only, no active regions are included, the $I_\lambda(\mu_i)$8 profile is taken as the disk-center intensity, and the empirical CLV sampling is assumed to be adequate (Pietrow et al., 2023). In the flare application, additional caveats include limited field of view, variable flare coverage fractions, the assumption of a linear scaling between observed coverage and emitted intensity, quiet reference regions that are not always truly quiet, seeing variations, tracking jumps, drifts, MOMFBD reconstruction artifacts, incomplete line datasets, and the fact that some lines lack limb-darkening or CLV data, so not all observations can be processed equally well (Wilde et al., 10 Jul 2025).

Within those limits, the scientific value of NESSI is specific and well defined. It makes spectral-line CLV explicit in full-disk integration, enables direct comparison between spatially resolved solar spectroscopy and unresolved stellar-style spectra, and provides a framework in which spots, plage, and flares can be inserted into a disk-integrated synthesis. Its broader significance lies in showing that detailed angular dependence of line formation is not a secondary correction but a determining factor in Sun-as-a-star and stellar-spectroscopic inference (Pietrow et al., 2023).