Presentation Information
[O12-P58]Observational studies on the wavelength dependence of limb darkening
*Yuki Sakurai1, *Mayumi Eguchi1, *Haruka Ogura1, *Ren Kurosawa1, *Shoma Koizumi1, *Kento Sasaki1, *Naoki Terahara1, *Haruki Nakane1, *Fuka Watabe1 (1. Kawaguchi Municipal High School)
Keywords:
Sun,limb darkening,spectroscopy
1.Abstract and Research Background
We are conducting research using a spectrograph owned by our school. In this study, we investigated the solar limb darkening. Solar limb darkening is a phenomenon in which the Sun gradually becomes dimmer from the center toward the limb. As shown in Figure 1, in the central region of the Sun, we can see through from the photosphere to deeper layers, allowing us to observe light from higher-temperature layers. In contrast, in the limb region, we can only see through to shallow layers, so we can only observe light from relatively lower-temperature layers. Therefore, we conducted slit-scan spectroscopic observations and confirmed that the limb darkening exhibits wavelength dependence in the visible light region. Furthermore, based on these results, we calculated the temperature of the Sun’s central region at each wavelength using Planck’s law and theoretically examined the observable depth from the visible to the near-infrared region.
2. Observation Method
Location: Rooftop of Kawaguchi Municipal High School, Kawaguchi City, Saitama Prefecture
Date and Time: March 24, 2026, 12:27–14:05 (JST)
Telescope: Refractor (60 mm aperture, 330 mm focal length, Sharpstar ED60)
Attenuation Filters: ND1000 + ND8
R1 (attached during near-infrared observations to exclude higher-order spectra in the visible light region)
Spectrometer: Low-dispersion spectrometer VEGA (Showa Machinery Co., Ltd.)
Camera: ZWO ASI 178MM
We divided the observations into the visible light range and near-infrared region (normal exposure) and the near-infrared region (overexposure), and performed five slit-scan observations for each, utilizing the Sun’s diurnal motion. Slit-scan observations have the advantage of not requiring primary processing of dark flats, as they use a fixed sensor position. Furthermore, because the camera’s sensitivity is low in the near-infrared region, resulting in significant variations in observed values across wavelengths and making it difficult to obtain sufficient data, observations were conducted separately for normal exposure and overexposure.
3. Analysis Method
① The spectral average intensity of the data obtained was calculated by averaging the data in 50-nm intervals across the ranges shown in Figure 2: from 350 nm to 700 nm in the visible light region and from 650 nm to 1000 nm in the near-infrared region. Subsequently, the ratio of edge dimming was plotted on a graph, with the maximum spectral intensity at each wavelength set to 1 (Figs. 5 and 6). We also presented the standard errors of the 95% confidence intervals for the 500 nm–550 nm and 800 nm–850 nm bands (Figs. 7 and 8).
② The intensity ratio between the central and peripheral regions of the Sun was calculated using the spectral data derived in step ①. Subsequently, using Planck’s formula [1] (Fig. 3), the spectral intensity I at each wavelength was considered equivalent to the radiant intensity Bλ (T ), and the temperature T of the Sun’s central region was calculated for each wavelength. Furthermore, using a previous study [2] (Fig. 4) that describes the relationship between solar temperature and depth, we derived the observable depth into the Sun’s core for each observable wavelength (Figs. 9 and 10).
4. Results
① As shown in Figures 5 and 6, the intensity decreases across all wavelength bands as one approaches the solar limb.
Figures 5 and 6 indicate that, compared to longer wavelengths, spectral intensity tends to decrease more significantly for shorter wavelengths in the 850 nm to 900 nm range.
② As shown in Figure 9, in the visible light region, as the wavelength becomes shorter, the temperature increases and the depth of penetration increases.
In the near-infrared region, as the wavelength becomes longer, the temperature increases and the depth of penetration increases.
5. Discussion
・The attenuation of spectral intensity from the center of the Sun toward the periphery in all wavelength bands is thought to be due to the Stefan-Boltzmann law.
・The wavelength dependence observed in the range from visible light to 900 nm, where the limb dimming increases as the wavelength becomes shorter, is thought to be due to Wien’s displacement law.
・The opposite trend observed for temperature and depth at wavelengths of 650 nm to 700 nm and beyond (Fig. 10) is thought to be due to the influence of bound-to-free (b-f) transitions of negative hydrogen ions (H-) [3] (Fig. 11).
・Regarding the wavelength dependence, we speculate that, in addition to Wien’s displacement law, absorption by negative hydrogen ions may also be influencing the results.
We are conducting research using a spectrograph owned by our school. In this study, we investigated the solar limb darkening. Solar limb darkening is a phenomenon in which the Sun gradually becomes dimmer from the center toward the limb. As shown in Figure 1, in the central region of the Sun, we can see through from the photosphere to deeper layers, allowing us to observe light from higher-temperature layers. In contrast, in the limb region, we can only see through to shallow layers, so we can only observe light from relatively lower-temperature layers. Therefore, we conducted slit-scan spectroscopic observations and confirmed that the limb darkening exhibits wavelength dependence in the visible light region. Furthermore, based on these results, we calculated the temperature of the Sun’s central region at each wavelength using Planck’s law and theoretically examined the observable depth from the visible to the near-infrared region.
2. Observation Method
Location: Rooftop of Kawaguchi Municipal High School, Kawaguchi City, Saitama Prefecture
Date and Time: March 24, 2026, 12:27–14:05 (JST)
Telescope: Refractor (60 mm aperture, 330 mm focal length, Sharpstar ED60)
Attenuation Filters: ND1000 + ND8
R1 (attached during near-infrared observations to exclude higher-order spectra in the visible light region)
Spectrometer: Low-dispersion spectrometer VEGA (Showa Machinery Co., Ltd.)
Camera: ZWO ASI 178MM
We divided the observations into the visible light range and near-infrared region (normal exposure) and the near-infrared region (overexposure), and performed five slit-scan observations for each, utilizing the Sun’s diurnal motion. Slit-scan observations have the advantage of not requiring primary processing of dark flats, as they use a fixed sensor position. Furthermore, because the camera’s sensitivity is low in the near-infrared region, resulting in significant variations in observed values across wavelengths and making it difficult to obtain sufficient data, observations were conducted separately for normal exposure and overexposure.
3. Analysis Method
① The spectral average intensity of the data obtained was calculated by averaging the data in 50-nm intervals across the ranges shown in Figure 2: from 350 nm to 700 nm in the visible light region and from 650 nm to 1000 nm in the near-infrared region. Subsequently, the ratio of edge dimming was plotted on a graph, with the maximum spectral intensity at each wavelength set to 1 (Figs. 5 and 6). We also presented the standard errors of the 95% confidence intervals for the 500 nm–550 nm and 800 nm–850 nm bands (Figs. 7 and 8).
② The intensity ratio between the central and peripheral regions of the Sun was calculated using the spectral data derived in step ①. Subsequently, using Planck’s formula [1] (Fig. 3), the spectral intensity I at each wavelength was considered equivalent to the radiant intensity Bλ (T ), and the temperature T of the Sun’s central region was calculated for each wavelength. Furthermore, using a previous study [2] (Fig. 4) that describes the relationship between solar temperature and depth, we derived the observable depth into the Sun’s core for each observable wavelength (Figs. 9 and 10).
4. Results
① As shown in Figures 5 and 6, the intensity decreases across all wavelength bands as one approaches the solar limb.
Figures 5 and 6 indicate that, compared to longer wavelengths, spectral intensity tends to decrease more significantly for shorter wavelengths in the 850 nm to 900 nm range.
② As shown in Figure 9, in the visible light region, as the wavelength becomes shorter, the temperature increases and the depth of penetration increases.
In the near-infrared region, as the wavelength becomes longer, the temperature increases and the depth of penetration increases.
5. Discussion
・The attenuation of spectral intensity from the center of the Sun toward the periphery in all wavelength bands is thought to be due to the Stefan-Boltzmann law.
・The wavelength dependence observed in the range from visible light to 900 nm, where the limb dimming increases as the wavelength becomes shorter, is thought to be due to Wien’s displacement law.
・The opposite trend observed for temperature and depth at wavelengths of 650 nm to 700 nm and beyond (Fig. 10) is thought to be due to the influence of bound-to-free (b-f) transitions of negative hydrogen ions (H-) [3] (Fig. 11).
・Regarding the wavelength dependence, we speculate that, in addition to Wien’s displacement law, absorption by negative hydrogen ions may also be influencing the results.