Presentation Information
[O12-P52]Generalization of correction values for sunspot relative number
*Shuya SUZUKI1, *Yuta KANEKO1, Seiya MORITA1, Kenta ISHIDA1, Juri TAKANO1, Minato TAKEYAMA1, Uiko DEWA1, Yuna NISHIZAWA1 (1. Shizuoka Prefectural Iwata Minami High School)
Keywords:
Correction value for relative sunspot number,National Astronomical Observatory,aberration,Sunspot observation using projection methods
Since 1945, Iwata Minami High School has maintained a continuous record of solar activity through sunspot observations using the projection method and calculated relative sunspot numbers. However, the replacement of our primary telescope in 2023 introduced a significant challenge: the performance difference between the old and new instruments rendered our longitudinal data inconsistent. While our institution historically assumed a correction value of k=1.0, we hypothesized that this discrepancy could be resolved by deriving a telescope-specific k-value calibrated against the National Astronomical Observatory of Japan (NAOJ). Since the conventional empirical method for calculating k requires years of comparative data, we developed a methodology to estimate an average correction value based on the physical and optical specifications of the telescope.
The relative sunspot number depends on the observation threshold—the minimum sunspot area resolvable on a projection board. It is established that the distribution of sunspot areas follows a log-normal distribution (Bogdan et al., 1988). Therefore, the correction value k can be modeled as the ratio of the areas under the probability density function (PDF) that exceed the respective thresholds of the NAOJ and the target telescope. In this study, the mean value of the PDF was determined to be 2.7212 (n=346566) based on NAOJ data, with a standard deviation of 1.13 based on previous research (Ibid.). The PDF graph is analyzed by shifting it horizontally by the value of the NAOJ's observation threshold to standardize the comparison.
This observation threshold is primarily governed by the telescope's ability to represent a sunspot on the projection board. Optical aberrations, specifically chromatic and spherical aberrations, significantly impact the resolution of sunspot images. Since aberrations are constant across the projection field, we focused on the values at the edge of the solar disk. By incorporating the factors that a sunspot must retain approximately 20% of its original darkness to be visible (Blackwell, 1946) and that actual lenses reduce the aberrations of a standard equi-convex lens by approximately 95% (Smith, 2007), the threshold was expressed as a telescope-specific constant. These values were converted from m2 to Millionths of a Solar Hemisphere (MSH).
Furthermore, due to the Sun's spherical geometry, sunspots appear foreshortened on a 2D projection. The observed area on the projection board is related to the true area by the factor of cosα, where α [rad] is the angular distance from the solar center. To represent k in a single formula, we derived a representative mean value for cosα. Following Maunder’s (1904) finding that sunspots appear with nearly uniform probability within the 0° to 30° latitude zone over a solar cycle, we assumed a uniform distribution in this region. Considering the 7.25° tilt of the Sun’s axis, the mean value of cosα was calculated to be approximately 0.8206. This allows the correction value to be expressed solely through constants determined by the telescope's performance.
The NAOJ utilizes high-precision image analysis, theoretically capable of detecting spots smaller than 1 MSH. However, to account for atmospheric turbulence, we defined the NAOJ’s reliable threshold as 3 MSH. Based on these parameters, the calculated correction factor for the new telescope at Iwata Minami High School is k=1.89.
Future work will involve verifying the practical utility of this model by applying it to historical observation data and calculating short-term correction factors.
We express our deepest gratitude to Mr. Hisayuki Tori, Mr. Hiroyuki Kurematsu, and Mr. Akira Aoshima for their invaluable guidance.
The relative sunspot number depends on the observation threshold—the minimum sunspot area resolvable on a projection board. It is established that the distribution of sunspot areas follows a log-normal distribution (Bogdan et al., 1988). Therefore, the correction value k can be modeled as the ratio of the areas under the probability density function (PDF) that exceed the respective thresholds of the NAOJ and the target telescope. In this study, the mean value of the PDF was determined to be 2.7212 (n=346566) based on NAOJ data, with a standard deviation of 1.13 based on previous research (Ibid.). The PDF graph is analyzed by shifting it horizontally by the value of the NAOJ's observation threshold to standardize the comparison.
This observation threshold is primarily governed by the telescope's ability to represent a sunspot on the projection board. Optical aberrations, specifically chromatic and spherical aberrations, significantly impact the resolution of sunspot images. Since aberrations are constant across the projection field, we focused on the values at the edge of the solar disk. By incorporating the factors that a sunspot must retain approximately 20% of its original darkness to be visible (Blackwell, 1946) and that actual lenses reduce the aberrations of a standard equi-convex lens by approximately 95% (Smith, 2007), the threshold was expressed as a telescope-specific constant. These values were converted from m2 to Millionths of a Solar Hemisphere (MSH).
Furthermore, due to the Sun's spherical geometry, sunspots appear foreshortened on a 2D projection. The observed area on the projection board is related to the true area by the factor of cosα, where α [rad] is the angular distance from the solar center. To represent k in a single formula, we derived a representative mean value for cosα. Following Maunder’s (1904) finding that sunspots appear with nearly uniform probability within the 0° to 30° latitude zone over a solar cycle, we assumed a uniform distribution in this region. Considering the 7.25° tilt of the Sun’s axis, the mean value of cosα was calculated to be approximately 0.8206. This allows the correction value to be expressed solely through constants determined by the telescope's performance.
The NAOJ utilizes high-precision image analysis, theoretically capable of detecting spots smaller than 1 MSH. However, to account for atmospheric turbulence, we defined the NAOJ’s reliable threshold as 3 MSH. Based on these parameters, the calculated correction factor for the new telescope at Iwata Minami High School is k=1.89.
Future work will involve verifying the practical utility of this model by applying it to historical observation data and calculating short-term correction factors.
We express our deepest gratitude to Mr. Hisayuki Tori, Mr. Hiroyuki Kurematsu, and Mr. Akira Aoshima for their invaluable guidance.