Center-to-Limb Variations in Solar Plage

• Center-to-limb variations (CLVs) are systematic changes in spectral properties across the solar disk, defined by the heliocentric angle (μ = cosθ).
• Empirical linear fits of the Si IV 1402.77 Å line in plage quantify rising intensity, decreasing Doppler redshift, and increasing non-thermal velocity from disk center to limb.
• These measurements differentiate solar regimes and underpin forward modeling of unresolved plasma dynamics and magnetic structuring in the solar atmosphere.

Center-to-limb variations (CLVs) are the systematic changes in radiative and kinematic observables across the apparent surface of the Sun (or other stars) as a function of the heliocentric angle, parameterized by μ = cos θ (with θ the angle between the local surface normal and the observer’s line of sight). In the context of the solar transition region, CLVs provide key diagnostic constraints on plasma dynamics, radiative transfer effects, and the structuring of solar magnetic fields. Recent high-resolution IRIS observations have, for the first time, quantified the CLVs of the Si IV 1402.77 Å line in active region plage, enabling precise comparisons to quiet-Sun and coronal-hole regimes and illuminating the combined roles of line-of-sight geometry, unresolved motions, and magnetic field strength (Kayshap et al., 4 Nov 2024).

1. Analytical Parameterization of CLV in Solar Plage

Empirical CLVs for Si IV 1402.77 Å in solar plage are well described by linear functions of μ for three key spectroscopic quantities:

Quantity	Linear Fit Y(μ) = A + B μ	Units
log₁₀ I(μ)	(1.67 ± 0.03) – (0.56 ± 0.06) μ	dex
v_D(μ)	(0.05 ± 0.55) + (8.89 ± 0.96) μ	km/s
v_nt(μ)	(30.84 ± 0.49) – (7.22 ± 0.87) μ	km/s

Here, I(μ) is the Si IV spectral intensity, v_D(μ) is the Doppler velocity, and v_nt(μ) is the non-thermal velocity derived from Gaussian linewidth fits. These relations capture the key features of CLV in plage:

• Intensity rises linearly from disk center (low) to limb (high): log₁₀ I(μ) decreases with μ, i.e.,

$I(\mu) = 10^{1.67 - 0.56\mu}$

• Doppler velocity (redshift) decreases linearly from ~8.9 km s⁻¹ at μ = 1 (disk center) to ≈ 0 at μ = 0 (limb).
• Non-thermal velocity increases linearly from ~23.6 km s⁻¹ (disk center) to ~30.9 km s⁻¹ (limb).

This analytic form accurately describes the observed binned means across the solar disk (see summary table above).

2. Physical Interpretation of CLV Behavior

The observed CLV trends in plage are governed by the interplay of line-of-sight geometry, unresolved dynamics, and magnetic structuring:

• Line-of-sight column depth: Moving from disk center to limb (μ ↓), the geometric LOS through the thin transition region increases as 1/μ, leading to a larger emission measure and hence higher line intensity toward the limb.
• Non-thermal broadening: At the limb, the LOS integrates through more spatially unresolved, predominantly transverse (to the radius) motions, leading to an increase in v_nt (non-thermal velocity). At disk center, the sightline samples a smaller physical depth and thus less unresolved transverse motion, minimizing v_nt.
• Doppler redshift: For predominantly radial downflows, the LOS velocity component is v₀ μ. Thus, the net redshift is maximized at disk center (where LOS aligns with flow) and diminishes to zero at the limb. This is the classical “center-to-disk” redshift pattern and reflects the geometry of the projected bulk flows in plage.
• Magnetic-field dependence: At disk center (μ ≈ 1), intensity and v_nt both scale with local photospheric field strength (|B_LOS|), indicating denser, more energetic plasma and more vigorous unresolved dynamics within the small-scale flux tube population. However, v_D shows negligible dependence on B, implying that the net redshift is a global phenomenon, not a local magnetically-induced flow.

3. Direct Quantitative Comparison: Plage, Quiet Sun, and Coronal Hole

CLV amplitudes and absolute parameter values in plage are systematically higher than in other solar regimes:

Parameter	Plage (μ=1)	QS (μ=1)	CH (μ=1)
Intensity I	~12.9	≈6.4	<6.4
v_D [km/s]	8.9	5.7	4.9
v_nt [km/s]	23.6	~14.6	~9.6

These plage enhancements are attributed to higher plasma density and stronger, multipolar magnetic structuring, which increase the emission measure and the amplitude of unresolved small-scale motions. The linear μ-dependence of intensity and v_nt persists in all regions but is offset vertically by absolute field and density differences.

4. Supporting Formulations and Diagnostic Equations

Interpretation of CLVs in terms of line formation is anchored by the decomposition of the observed Gaussian line width:

• Total spectral width (w):

$w^2 = w_{\rm I}^2 + w_{\rm T}^2 + w_{\rm NT}^2$

• instrumental: $w_{\rm I}=26$ mÅ,
• thermal: $$w_{\rm T}^2 = 4\ln2 \left(\frac{\lambda_0}{c}\right)^2 \frac{2kT_i}{m}$$,
• non-thermal: $w_{\rm NT}$ corresponding to $v_{\rm nt}$.
  • Non-thermal speed:

$$v_{\rm nt} = \xi = \sqrt{\frac{w_{\rm NT}^2}{4\ln 2}} \frac{c}{\lambda_0}$$

The rest-wavelength for Doppler correction is set by a linear fit of centroid vs. μ, with the limb value adopted as λ₀ = (1402.7668) Å.

5. Observational Constraints and Broader Implications

Empirical CLVs from IRIS Si IV 1402.77 Å establish several key constraints:

• The close match in CLV functional form between plage and quiet Sun (with only a vertical offset) demonstrates that geometric and dynamical factors dominate the variation, while absolute values reflect magnetic/thermodynamic structuring.
• The linear μ-dependence of both intensity and v_nt quantifies the column depth and unresolved “microturbulence” scaling, providing a direct input for forward modeling.
• The invariance of the global Doppler redshift with local B-field suggests that in plage, mass flows are organized on larger spatial scales than individual flux tubes.
• The decoupling of v_nt and intensity from v_D as a function of |B_LOS| supports the use of CLVs as diagnostics of unresolved dynamics and magnetic mass loading.

These trends must be reproduced by any physical model seeking to capture the chromospheric and transition region structure of active regions.

6. Connection to Instrumental/Analytic Approaches and Future Uses

The observed linear CLVs provide robust benchmarks for modeling strategies:

• Line-of-sight and unresolved motions are separable via the quantitative scaling laws, offering calibration for transfer codes and forward modeling.
• Plage, quiet Sun, and coronal hole can be distinguished by vertical offsets in these CLV laws, allowing for systematic mapping of solar atmosphere dynamics across activity regimes.
• In stellar and exoplanet atmospheric studies, these empirical solar CLVs enable accurate construction of spatially inhomogeneous radiative-transfer models and correction of activity-driven biases in transmission or emission spectroscopy.

A plausible implication is that extending this methodology to other transition region lines and to multi-line studies will enable a more unified and detailed description of atmospheric structuring across the full solar disk, essential for both solar–stellar comparative studies and precision exoplanet atmosphere modeling.