Velocity field in the solar granulation from two-vantage points

(Solar Orbiter Nugget #49 by Takayoshi Oba (1,*), Luis R. Bellot Rubio (2), Daniele Calchetti (1), Johann Hirzberger (1), Sami K. Solanki (1), Yukio Katsukawa (3))

1. Introduction

The quiet solar photosphere exhibits a cellular pattern based on bright granules which are surrounded by dark channels called intergranular lanes [1]. A granule is the manifestation of an overshooting hot upflow from the convective unstable subsurface layers into the stable photosphere. This vigorous convective process gives rise to dynamical phenomena such as exploding granules [2], supersonic flows [3], vortex tubes [4], and reversed granulation [5]. In addition, the gas convection plays a major role in structuring the chromosphere and the corona, by converting kinetic energy into magnetic energy through flux tubes that extend into these upper atmospheric layers. To understand these phenomena, numerous studies have been conducted to characterize the gas convection in the solar surface. 
One of the key physical quantities for understanding these phenomena is the horizontal gas flow on the solar surface, as all the aforementioned phenomena are closely associated with it. However, deriving this physical quantity is quite challenging. Since Doppler shift measurements provide the velocity of gas motion along the line-of-sight (LOS), a straightforward approach might use observations near the solar limb, where the LOS reflects horizontal motion. However, foreshortening effects considerably degrades the effective spatial resolution. A promising approach is to take advantage of the stereoscopic observation. The joint use of the Solar Orbiter / Polarimetric Helioseismic Imager (SO/PHI) [6] and Hinode/Solar Optical Telescope-Spectropolarimeter (SOT/SP) [7] is particularly well-suited for studying the small-scale feature, such as the granulation, thanks to their stable observations from space and their comparable high spatial resolution. This article introduces how these instruments, observing from different viewing angles, provide a novel approach to study the solar granulation and its related velocity fields. 

2. Data and coalignment

Both Hinode/SP and SO/PHI provide spectropolarimetric observations of photospheric absorption lines. While Hinode/SP employs a slit-based system, SO/PHI is based on a tunable filtergraph. Both instruments are able to retrieve the LOS velocity by Doppler shifts from their spectral line. The target is a quiet region observed on 10th April 2023. During the observational campaign, SO/PHI-HRT (high resolution telescope) took spectral scans over a period of 6 hours with a 1-minute cadence, while Hinode/SP repeatedly rastered a large field-of-view. The observation location was near the solar equator, positioned off disk-center for both instruments. The heliocentric angles, expressed as cosθ (where θ is the deviation of the LOS of the observer from the surface normal at the observation point), were 0.8 for Hinode and 0.9 for Solar Orbiter (Fig. 1). The separation angle between the Sun-Hinode and the Sun-SO lines was 63 degrees. 
Since the granulation pattern is an extremely small-scale phenomenon, precise coalignment is essential. Several challenges must be addressed, including differences in pixel sampling, image distortion, viewing angles, and even observation systems (i.e., slit-based vs. filtergraph). In our coalignment approach, scaling and offsets in the X- and Y-directions are determined through the affine transformation to maximize the correlation [8]. In addition, Hinode/SP’s slit-scan system rasters the spatial region step by step, while the granulation evolves on relatively short timescales. To address this, temporal interpolation is applied to a time series of the SO/PHI snapshots. This ensures that any pixel in the resulting SO/PHI composite map corresponds to the same observation time as Hinode/SP at each slit position. 

Figure 1. Spatial relation among Earth (Hinode), the Sun, and Solar Orbiter in the Geocentric Solar Ecliptic coordinate system at the observation time of 10th April 2023. θHinode/SP is the heliocentric angle for the Hinode/SP observation, whereas θSO/PHI is the heliocentric angle for the SO/PHI observation. 

3. Results

Figure 2 shows the intensity maps from Hinode/SP (left panel) and SO/PHI (right panel). It should be noted that the two maps are composed of the same number of spatial pixels in the X and Y directions, enabling a direct one-to-one pixel comparison. The two intensity images exhibit a strong correspondence with a correlation coefficient of 0.91. While the intensity contrasts are comparable, the Hinode/SP map appears slightly sharper. 

Figure 2. Continuum intensity map from Hinode/SP. Right: Composite continuum intensity map derived from a time series of the SO/PHI observations.

Figure 3 (a) and (b) provide enlarged views for a detailed comparison between the two images. While the overall correspondence is good, residual difference can be seen, particularly in regions with large granules. This discrepancy is more discernible in Fig.3 (c), which illustrates the intensity profiles along a slice cutting through a large granule. These observed discrepancies are likely due to the opacity effect: in the photosphere, opacity is primarily temperature-dependent [9], i.e., a granule with hot-material creates a hill whereas an intergranular lane with cool-material creates a valley [10], resulting in the different heights or locations being sampled by the two instruments with different viewing angles. Further investigation is needed to better understand the origins of these discrepancies. 

Figure 3. Panel (a): Intensity map of Hinode/SP. Panel (b): Composite intensity map derived from the SO/PHI observations. Panel (c): Intensity profile from Hinode/SP (black) and SO/PHI (red), at the central position marked with the vertical white lines in panels (a) and (b).

Figure 4 provides another closer view of the intensity maps (top row) and the LOS velocity (bottom row). It is evident from the two LOS velocity maps that SO/PHI sees smaller velocities than Hinode/SP, as the same color scale is applied to both maps. Several factors contribute to this difference. First, SO/PHI has slightly lower spatial resolution compared to Hinode/SP, as indicated by a bit sharper intensity image of Hinode/SP. Second, the different sensitivity to vertical and horizontal flow fields according to their respective heliocentric angle. With cosθHinode/SP=0.8 for Hinode/SP, it is more sensitive to the horizontal flow than cosθSO/PHI=0.9 for SO/PHI, resulting in faster LOS velocity since the horizontal flow is expected to be approximately twice as fast as vertical flow [11][12]. Additionally, instrumental differences, such as the limited number of wavelength samples in SO/PHI, further contribute to the discrepancy in the amplitude of the LOS velocity. 

Figure 4. Top row: Intensity maps from Hinode/SP and SO/PHI, along with their intensity profiles at the central location, from left to right, respectively. Bottom raw: Same as the top row, but for LOS velocity field. Note that the left side is the viewing direction from Hinode/SP whereas the right side is that from SO/PHI, as illustrated by the right most cartoon that describes the geometric configuration of Hinode/SP and SO/PHI with respect to the observational target region.

Although several issues still remain to be addressed for a direct comparison of the two LOS velocity maps, one implication regarding the general properties of the gas convection can be readily drawn. An illustrative example is highlightened by the magenta dashed rectangular boxes in the right column. The LOS velocity map of Hinode/SP shows a strong blue shift on the left side of the granule (toward the disk center), whereas the right side (toward the limb) exhibits a red shift. In contrast, the LOS velocity map of SO/PHI shows that the left side (toward the limb for SO/PHI) displays a red shift or weak blue shift, whereas the right side (toward the disk center for SO/PHI) shows strong blue shift. The right most cartoon should serve as a helpful reference for understanding the geometric configuration. These findings suggest that the granule exhibits a fountain-like divergent flow. This behavior is naturally expected from convection theory, but this result represents the first ever direct observational confirmation using Doppler measurements. 

4. Conclusions

Hitherto, only one of the three components of the photospheric velocity vector could be obtained. Deriving the full velocity vector is essential to characterize the dynamical behaviors seen in the solar atmospheres. Our coalignment approach allows a direct comparison of both instruments with different viewing angles. The comparison of the coaligned LOS velocity maps clearly reveals the presence of a divergent flow in a granule, as predicted from convection theory. 
The next step is to derive 2 of the 3 components of the velocity field in the photosphere. This process requires decomposition of the two LOS velocity fields into the vertical and horizontal flows, taking into account of the heliocentric angle for both instruments. The spatial distribution of the velocity field vectors can characterize an act of divergence/compression and shear/rotation in the gas convection. This valuable information offers insights into how convection drives dynamical phenomena, acts on magnetic flux tubes, and connects to the upper solar atmospheres. 

Affiliations

(1) Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany. *email: oba@mps.mpg.de
(2) Instituto de Astrofísica de Andalucía (IAA-CSIC), Apartado de Correos 3004, E-18080 Granada, Spain
(3) National Astronomical observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

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