Modeling HMI observables for the study of solar oscillations

TL;DR

The paper addresses how solar oscillation perturbations map to Helioseismic and Magnetic Imager (HMI) observables, highlighting inaccuracies in simple intensity- and velocity-projection assumptions. It constructs a realistic background atmosphere using MPS-ATLAS patched to Model S, performs full radiative-transfer synthesis of the Fe I 6173 Å line, and computes HMI observables by convolving with the six instrument filters. A first-order perturbation theory is developed and validated, revealing substantial amplitude and phase deviations (approximately $10\%$ and $10^{\circ}$) in both hmi.V_45s and hmi.Ic_45s, dependent on the mode and disk position. The framework enables improved interpretation of mode visibility and systematic corrections in helioseismology and can be adapted to other instruments or asteroseismology applications.

Abstract

Context: Helioseismology aims to infer the properties of the solar interior by analyzing observations of acoustic oscillations. The interpretation of the helioseismic data is however complicated by the non-trivial relationship between helioseismic observables and the physical perturbations associated with acoustic modes, as well as by various instrumental effects. Aims: We aim to improve our understanding of the signature of acoustic modes measured in the Helioseismic and Magnetic Imager (HMI) continuum intensity and Doppler velocity observables by accounting for radiative transfer, solar background rotation, and spacecraft velocity. Methods: We start with a background model atmosphere that accurately reproduces solar limb darkening and the Fe I 6173Å spectral line profile. We employ first-order perturbation theory to model the effect of acoustic oscillations on inferred intensity and velocity. By solving the radiative transfer equation in the atmosphere, we synthesize the spectral line, convolve it with the six HMI spectral windows, and deduce continuum intensity (hmi.Ic_45s) and Doppler velocity (hmi.V_45s) according to the HMI algorithm. Results: We analytically derive the relationship between mode displacement in the atmosphere and the HMI observables, and show that both intensity and velocity deviate significantly from simple approximations. Specifically, the continuum intensity does not simply reflect the true continuum value, while the line-of-sight velocity does not correspond to a straightforward projection of the velocity at a fixed height in the atmosphere. Our results indicate that these deviations are substantial, with amplitudes of approximately 10% and phase shifts of around 10 degrees across the detector for both observables. Moreover, these effects are highly dependent on the acoustic mode under consideration and the position on the solar disk.

Figures (12)

• Figure 1: Background sound speed (red) and density (blue) from Model S (dashed), MPS-ATLAS (solid), and the patched model used in this study (dots). The gray region represents the domain where the patching between Model S and MPS-ATLAS is done. The zero level of height in this context is defined at one solar radius.
• Figure 2: Background intensity computed from the MPS-ATLAS solar model atmosphere (solid), the standard solar Model S (dashed), and from the observations (dots). Left: limb darkening ($I_c(\mu) / I_c(\mu=1)$) in the continuum around HMI line compared with the observations from Neckel1994. Right: synthesized $\mathrm{Fe I}$ spectral line background intensity ($\tilde{I}(\lambda,\mu) / I_c(\mu=1)$) compared with the observations from IAG spectral atlas Ellwarth2023 at $\mu = 1$ (blue) and $\mu=0.5$ (red). The wavelength offset is with respect to the center of the HMI line $\lambda_{\rm HMI} = 6173.33$ Å. The black dashed curves are the six HMI filtergrams.
• Figure 3: Response function for velocity $K(s,\mu)$ (in Mm$^{-1}$) defined by Eq. \ref{['eq:response_function_V']} as a function of height $s$ and limb angle $\mu$. The black and white lines represent, respectively, the maximum and center of gravity of the response function for each value of $\mu$.
• Figure 4: Test of the direct computation of the perturbed intensity $(I_\lambda-I_\lambda^0)/I_\lambda^0$ (dots) for $\mu=0.5$ and comparison with first-order computations $\delta I_\lambda / I_\lambda^0$ (solid line). The thermodynamic, geometrical, and line contributions are represented separately in blue, green, and red, respectively, while the full intensity is in black. The wavelength offset is with respect to the center of the HMI line $\lambda_{\rm HMI} = 6173.33$ Å.
• Figure 5: Difference between the velocity perturbation caused by an example radial p-mode ($l =0, n=20, \omega_{nl}/2\pi = 2.90$ mHz) of amplitude 1.0 m/s (at the surface at the disk center) computed from the HMI algorithm (without background velocity) and a simple line-of-sight approximation evaluated at the formation height at each position on the disk $h_\mu$ or at disk center $h_{\mu=1}$. Note that $\delta \mathrm{v}_{\rm los}$ is purely imaginary so that Re[$\delta \mathrm{v}_{\rm th}$] and Re[$\delta \mathrm{v}_{\rm geom}$] are a deviation from the simple line-of-sight approximation. A center-to-limb effect is observed in the imaginary part and in the thermodynamic contribution. For this radial mode, the geometrical component is small.