Analysis of Solar Diameter Measurements Made at the Basilica of San Petronio during and after the Maunder Minimum

The Sun is an essential source of energy for the natural phenomena taking place on the Earth. Moreover, it constitutes our closest laboratory of stellar physics. Therefore, it is important to know how the physical parameters of the Sun evolve. One of the fundamental Sun parameters for astrophysics and geophysics is the solar diameter, as stellar physical models are adjusted with the value of this parameter and changes in it could affect Earth's climate (Rozelot 2001). However, the relation between solar activity and solar diameter is still a matter of debate. Observations show that the variations of the solar diameter during a solar cycle are very small (Bush et al. 2010; Meftah et al. 2015; Kosovichev & Rozelot 2018). The link between these variations and solar activity is also not clear. Some studies state an anticorrelation between the solar diameter and the number of sunspots (Laclare et al. 1996; Egidi et al. 2006), others find a correlation (Sveshnikov 2002; Noël 2004), and others indicate that the relationship between these two variables is not significant (Kuhn et al. 2004; Lefebvre et al. 2006).

Furthermore, the long-term behavior of the solar diameter is also a controversial issue. Studies of solar diameter measurements taken at the Royal Greenwich Observatory during the years 1836–1953 show a 225 secular decrease in the value of the solar diameter (Eddy & Boornazian 1979). Moreover, Gilliland (1981) found a probable secular decrease of 01 after analyzing five different sets of solar observations covering 265 yr. In contrast, other authors state that there are no long-term trends in solar diameter. For instance, a later analysis concludes that the secular decrease found by Eddy & Boornazian (1979) is due to errors of interpretation in the measurements (Parkinson et al. 1980). Toulmonde (1997) compares 30 series of measurements over 300 yr and concludes that there are not any significant secular variations. No significant trends in solar diameter using data from the Royal Observatory of the Spanish Navy from 1773 to 2006 were found by Vaquero et al. (2016) either. In addition, a recent analysis of the observations taken at the Kodaikanal Observatory over nearly a century indicates that the solar diameter remains almost constant, showing a very feeble decrease over the century studied (Hiremath et al. 2020).

One of the most interesting periods to assess the value of the solar diameter is the MM, as it is the only "grand minimum" period recorded during the telescopic era. This period covers the years 1645–1715, when sunspots were rarely observed according to the observations of the astronomers of that time (Eddy 1976; Usoskin et al. 2015). Analyzing data from the observations made by Picard and la Hire over the period of 1660–1719, Ribes et al. (1987) concluded that the diameter of the Sun was 4'' larger during the MM. However, this result was criticized by O'dell & van Helden (1987) who claim that Ribes et al. (1987) had misinterpreted the error in the measurements. In addition, Morrison et al. (1988) also rejected the hypothesis of a larger Sun during the MM by studying observations of a total solar eclipse that took place in 1715. Nevertheless, Pap et al. (2001) upgraded the data set of Toulmonde (1997), who had compared 30 series of measurements of the solar diameter spanning a period from the seventeenth century to the end of the twentieth century, showing a larger Sun during the MM again.

Ground-based observations of the solar diameter are far less precise than those made from space (Meftah et al. 2015; Kosovichev & Rozelot 2016). For example, the accuracy of the measurements made at the Basilica of San Petronio studied in this work is 1000 mas, whereas the accuracy of the space measurements is around 10 mas (Damé et al. 2017). Furthermore, the latter were carried for the first time in the 1990s and therefore only a few decades of data are available (Rozelot et al. 2018). Although the differences between the solar diameter measurements from the ground are still controversial, some studies are finding the causes. For instance, some works on variations in solar diameter measurements made with astrolabes indicate that the variations are due to the mechanisms dwelling in the interface zone between the lower stratosphere and the upper troposphere (Badache-Damiani & Rozelot 2006; Badache-Damiani et al. 2007).

The aim of this paper is to analyze a series of measurements of the solar diameter taken between 1655 and 1736 at the meridian line of the Basilica of San Petronio at Bologna (Figure 1). These measurements are particularly interesting because they cover the MM (1655–1715) and the following two decades (1716–1736). Moreover, since they were taken with the same instrument and methodology, their time distribution allows us to compare the solar diameter value during the MM and the value after this period. Although our data set consists of ground-based measurements (which are less precise than those made from space), they are accurate enough to detect big changes in solar diameter, which some authors state took place between the MM and the subsequent period (Ribes et al. 1987; Pap et al. 2001). These changes in the solar diameter could be relevant for studies of the evolution of our star, especially if they were connected with the anomalies in the rate of solar rotation during the MM that have been suggested by several authors. In particular, Ribes & Nesme-Ribes (1993) indicated that the Sun rotates slowly at the equator but showing a higher level of differential rotation with latitude during the MM. These findings were confirmed by Casas et al. (2006). Hence, the purpose of this work is to bring a new result to the controversy about the size of the Sun during the MM by studying a series of measurements that had not been analyzed before. In Section 2, we describe the data, history, and features of the meridian line of the Basilica of San Petronio as well as the methodology used to obtain the solar diameter. The methodology we have followed to analyze the measurements is explained in Section 3. Section 4 contains the results and the discussion of the analysis. Finally, the main conclusions are presented in Section 5.

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The solar diameter measurements taken at the meridian of the Basilica of San Petronio (44^◦ 29' N, 11^◦ 20' E, 54 m) between 1655 and 1736 were published by Manfredi (1736). From this book, 4033 observations of the solar diameter have been retrieved. These observations are exhibited in tables (Figure 2) indicating the date, the meteorological conditions under which the observation was made, the measurements of the tangents of the solar limb after removing the penumbra, the distance between the solar limbs and the meridian's vertex after removing the penumbra, the apparent solar diameter and the distance from the center of the solar image to the meridian's vertex. In this work, the data of the apparent solar diameter are analyzed (sixth column of Figure 2). These data are given in arcminutes and arcseconds. The measurements are not distributed homogeneously over time. In fact, as can be seen in Figure 3, there are periods when very few (or even no) observations were made. These periods cover the years 1661–1666, 1677–1689, 1691–1695, and 1716–1718. The reason behind the scarcity of measurements in these periods is unknown. Figure 3 displays the whole daily time series of solar diameter measurements and indicates the number of observations made in each year. In that figure, some outliers can be seen. They may be the result of personal biases. In Section 4, after statistical analyses, they are removed.

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The history and features of the meridian line of the Basilica of San Petronio and the method of measurement are reviewed in the following subsection.

2.1. The Meridian Line of Cassini

The most accurate instrument to determine the position of the Sun during the year in the 15th century was the meridian line (Heilbron 1999). For that reason, a meridian line was built in 1575 by E. Danti on the floor of the Basilica of San Petronio to check the dates of the Gregorian calendar. In 1655, G. D. Cassini rebuilt the meridian line because it was no longer useful due to alterations that had been made to the building. The stated aim of this new meridian line was to continue checking the dates of the Gregorian calendar, but Cassini also used it to verify Kepler's second law (by observing if the apparent diameter of the Sun decreased in the same way as its apparent speed). Thus, he constructed the longest meridian line of that time making the most of the size of the nave and avoiding the pillars that supported the basilica (Figure 4). The hole of the meridian, with a diameter of 2.7 cm, was located at a height of 27.07 m from the meridian's vertex. A scale in hundredths of the height of the hole center was marked along the meridian line, which was 66.72 m long (Heilbron 1999).

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There were many observers at the Basilica of San Petronio. The first regular observers were Cassini, Grimaldi, and Riccioli, who were in charge of leading the observations and recruiting observers. The author that published the measurements we analyze in the present work (Manfredi) began as an observer under the guidance of Guglielmini, who was a professor of mathematics at the University of Bologna. In 1699, he succeeded Guglielmini as professor and was left in charge of directing the observations that he later published together with those made by his predecessors (Heilbron 1999). The measurements on the meridian were not easy to do, as the image fluctuated and moved. Therefore, two observers were needed: a senior one tried to find the solar limbs by observing points equally illuminated as the center of the Sun crossed the meridian line and indicated to another observer where to place marks. That was complicated due to the trembling of the image and the uncertainty to identifying the appropriate points. For that reason, Cassini gave the following guideline: "If you always take the same limit of the light and the edge of the fluctuation furthest from the center, at least you will have the same proportion in comparing the apparent diameter at different times" (Cassini 1695; Heilbron 1999). Once the marks were placed, they obtained the distance from each mark (of the solar limbs) to the meridian's vertex using the meridian scale and increased the accuracy of this measurement a thousand times by taking the distance of each mark from the nearest scale division by a pair of dividers and measuring the openings against a graduated plate. The size of the Sun's image marked on the scale mainly depended on the altitude of the Sun (although it also depended on the Sun–Earth distance). Since the bigger the Sun's image, the higher the accuracy, this method produced measurements with an accuracy that depended on the time of the year. The maximum accuracy took place during the winter solstice with a value of 03 (image of 168 cm on the meridian line) and the minimum one during the summer solstice with a value of 2'' (image of 26 cm on the meridian line).

The measurements were also affected by the ring of the penumbra, which had to be subtracted from each image to obtain the apparent solar diameter. Furthermore, the eye cannot detect the rim of the penumbra, but an apparent boundary within it. For this reason, the astronomers made corrections to the measurements considering this effect (Heilbron 1999). Refraction was another effect affecting the images of the Sun. It was corrected using a model developed by Cassini that is highly exact for the solar altitudes associated to our data (Young 2004). Despite the fact that we know that the astronomers of the Basilica of San Petronio made the above corrections, the extent of the errors associated to personal biases is unknown (i.e., determining when the center of the Sun image crossed the meridian line or detecting the limbs of the images). Moreover, the uncertainty related to the aforementioned corrections is unknown too (the data presented here include the corrections since they are the only data available in Manfredi 1736). Nevertheless, these measurements are useful to make comparisons between them because they were taken with the same instrument and methodology (including the corrections).

The meridian had to be restored in 1695 and 1722 because the distance between the hole and the floor had decreased and the meridian line had lost stability. These deteriorations were due to the fact that the Basilica of San Petronio was sinking into the ground. On both occasions the measurements were corrected to take into account the displacement of the hole. Finally, in 1776 the meridian was completely renewed and since then no modifications have been made to it (Heilbron 1999).

A statistical analysis comparing the measurements made in the MM and the ones made after that period is carried out to analyze the value of the solar diameter during the MM. For this purpose, the identification and removal of outliers is needed in a previous step. The result of a Kolmogorov–Smirnov test performed to the measurements shows that the data do not follow a normal distribution. Therefore, a box plot diagram is used in order to display the outliers. After removing them the new data set seems to contain more outliers. Then, a linear regression analysis is performed to the data after a linearization in order to detect outliers through a study of standardized residuals. As the time evolution of the data follows a cosine distribution, the function used to linearize it was:

$$\begin{eqnarray}&&\cos \left(\displaystyle \frac{2\pi \left(d-p\right)}{365.25}\right),\end{eqnarray} \tag{ 1 }$$

where p is the most representative perihelion day (using the "NASA JPL Horizons ephemeris system") and d is the day of the year.

Residuals must follow a normal distribution so that the analysis of standardized residuals is correct. The result of a Kolmogorov–Smirnov test suggests that the residuals do not follow a normal distribution. Therefore, to obtain a normal distribution, a transformation of Johnson is applied to the residuals (Chou et al. 1998). Then, the standardized residuals can be computed. Those greater than 2 or smaller than −2 are considered outliers and their corresponding values of solar diameter are filtered out.

Once the outliers have been eliminated, the data set covering the years 1655–1736 is separated into two periods: one during the MM spanning 1655–1715 and another one after the MM covering 1716–1736. In a first analysis, three Mann–Whitney U tests and three Student's t-tests with different alternative hypotheses are performed to compare the median and the average of the solar diameter in the two periods, respectively. The differences in the medians and averages and their corresponding confidence intervals calculated for several confidence levels are also computed with each test. A general description of these statistical methods can be found in García (1992).

In a second analysis, the influence of the distribution of the days of the year in which the measurements were made is removed. For that purpose, the solar diameter average value (with its standard deviation) for each day of the year of each period is calculated. Three Mann–Whitney U tests (Student's t-tests) comparing the median (average) value of the set of daily averages for the period 1655–1715 and that for the period 1716–1736 is performed. Different alternative hypotheses are considered for each test (the same hypotheses as those considered in the first analyses). Also, the differences and their corresponding confidence intervals calculated for several confidence levels are computed using each test. Note that the original data set is affected by errors of different origins whose real impact on the data is unknown. So that the difference in solar diameter between the periods calculated in this study can be compared with those of other studies, we multiply the value of the difference and their corresponding confidence interval by a constant K defined as:

$$\begin{eqnarray}&&K=\displaystyle \frac{{D}_{\max }-{D}_{\min }}{{D}_{\max }^{{\prime} }-{D}_{\min }^{{\prime} }},\end{eqnarray} \tag{ 2 }$$

where D_max and D_min are the solar diameter values in the perihelion and aphelion, respectively, obtained from the software "NASA JPL Horizons ephemeris system." ${D}_{\max }^{{\prime} }$ and ${D}_{\min }^{{\prime} }$ are the solar diameter values in the perihelion and aphelion, respectively, from a polynomial fit of degree six in which the independent variable is the day of the year and the dependent variable is the diameter using the data collected in the Basilica of San Petronio.

The cleaned data (after the two removals of outliers) are displayed in Figure 5. The box plot analysis detected 64 outliers (1.58% of the measurements) and the analysis of the standardized residuals revealed 188 outliers (4.66%). These outliers were uniformly distributed over both periods.

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The distribution of measurements over both periods, along with the median and the average solar diameter for each period is shown in Table 1. The median after the MM is 0107 (64) higher than that for the MM. The same applies to the average solar diameter, which is around 0.06 (38) higher for the second period. However, the errors associated with these values are unknown, and therefore, at this point of the study, firm conclusions cannot been drawn.

Table 1. Number of Measurements, Median, and Average of the Two Periods Considered in the Present Study (After the Elimination of Outliers)

Period	Number of Measurements	Median	Average
1655–1715	2849	31368	31410
1716–1736	932	31475	31474

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The Mann–Whitney U tests applied to the periods 1655–1715 and 1716–1736 reveal that solar diameter median was smaller during the MM. Taking into account that the null hypothesis considers that the medians of the periods are equal, p-values are less than 0.05 (namely less than 0.001 in both cases) when the alternative hypotheses state that the medians of the periods are different or the median of the first period is lower. In these cases, the null hypotheses are rejected for a 95% confidence level and statistical differences in the value of the solar diameter during 1655–1715 and 1716–1736 are found. Likewise, the p-value is greater than 0.05 (namely more than 0.999) when the alternative hypothesis states that the median of the first period is greater. Then, the null hypothesis is accepted for a 95% confidence level and it is preferable to consider the same solar diameter median in both periods rather than a bigger one during the MM. Similar results are found for the solar diameter average with the Student's t-tests. In addition, Table 2 supports the idea of a smaller solar diameter during the MM since the differences (in medians and in averages) and their confidence intervals have negative values. The above results are not in accordance with Ribes et al. (1987) and Pap et al. (2001). Nevertheless, confidence intervals for the difference in Table 2 show that, with a 99% probability, the difference in the medians and the averages in the two periods is 01167 and 01089 at the most in absolute value, respectively. These values are equivalent to approximately 7'' in both cases. They are small considering the features of the measurement method and also compared with the solar diameter variation during the year (see Figure 5). Therefore, after this analysis we conclude that the solar diameter during the MM was slightly (but not significantly) smaller than that for the subsequent period 1716–1736.

Table 2. Estimation of the Difference Between the Median and the Average of the Solar Diameter Measurements Taken in the Period 1655–1715 and Those for the Period 1716–1736

	Difference	Confidence Interval for the Difference	Confidence Level
Median	−006667	(−011667, −001667)	99%
Average	−006391	(−010893, −001902)	99%

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Note that the above results can be influenced by the distribution of the measurements over the year, i.e., the days of the year in which the measurements were made. After an evaluation of the seasonal distribution of the dates when the measurements were made, a different distribution in the periods was found. Therefore, a second analysis taking into account this effect is required.

Solar diameter daily averages with their corresponding confidence intervals of two standard deviations for each period are exhibited in Figure 6. There are no data outside the confidence intervals. However, there are 29 days without records during 1716–1736. Therefore, there are no averages for those days. Consequently, to avoid that the distribution of days of observation over the year influences the results, the solar diameter averages of the above 29 days of the year for the period 1655–1715 are removed in this second analysis. Mann–Whitney U tests and Student's t-tests applied to the solar diameter daily averages of the periods 1655–1715 and 1716–1736 yield p-values greater than 0.05 in all the tests (0.701, 0.350, and 0.650 for Mann–Whitney U tests, and 0.672, 0.336, and 0.664 for Student's t-tests). Therefore, the null hypothesis is accepted in all the tests for a 95% confidence level. This implies that the medians (and the averages) of the two periods are statistically equal. The difference and the confidence interval for the difference for each test with a confidence level of 99% are also computed. The difference multiplied by the constant K and the confidence interval for the difference multiplied by K for each test are displayed in Table 3 (K = 0.806). The fact that both confidence intervals for the difference are almost centered on zero indicates that there is no difference in the medians and the averages of the solar diameter between 1655–1715 and 1716–1736. In addition, the estimated differences in median and average are −001072 and −001104, respectively. These values are equivalent to approximately 06 in both cases. Therefore, the differences between the periods are not significant because they are smaller than the mean accuracy of the meridian of the Basilica of San Petronio.

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Table 3. Estimation of the Difference Between the Median and the Average of the Solar Diameter Daily Averages During the Period 1655–1715 and Those for the Period 1716–1736 Multiplied by a Constant K = 0.806

	Difference	Confidence Interval for the Difference	Confidence Level
Median	−001072	(−008698, 005776)	99%
Average	−001104	(−007850, 005634)	99%

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Finally, after the above statistical analyses, we conclude that there is no difference between the solar diameter during the MM (1655–1715) and that for the subsequent period 1716–1736. This result disagrees with that obtained by Ribes et al. (1987) who state that the solar diameter was 4'' larger during the MM. Also, it differs from Pap et al. (2001), who found a larger Sun during the MM. On the other hand, our result supports those of Morrison et al. (1988) and O'dell & van Helden (1987).

Solar diameter measurements taken at the Basilica of San Petronio (Bologna, Italy) during the years 1655–1736 are analyzed in this work. The extent of the errors associated with the measurements is not completely known because the metadata about the conditions of the observations are limited. In addition, personal biases must be taken into consideration. However, these measurements are useful to make comparisons between them because they were taken with the same instrument and methodology. For this reason, the goal of this work is to study if there is a change in the value of the solar diameter between the MM (1665–1715) and the subsequent period (1716–1736).

The results obtained in the present study indicate that there is no statistically significant difference in the solar diameter medians or averages of the periods. Likewise, the difference between the medians and between the averages are estimated around 06 in both cases. These differences are below the mean accuracy of the meridian of the Basilica of San Petronio. Therefore, we conclude that there is not a significant difference between the solar diameter value measured during the MM (1655–1715) and that for the subsequent period 1716–1736. Furthermore, although there are some errors associated with the measurements that are not completely known, our analysis clearly indicates that the major changes in the solar diameter outlined by Ribes et al. (1987), Toulmonde (1997), and Pap et al. (2001) disagree with the observations taken in the Basilica of San Petronio.

It should be pointed out that the series of measurements presented in this study had not been analyzed before. Therefore, since there is controversy about the value of the solar diameter during the MM, the results of the present study can be of interest to the scientific community.

This research was supported by the Economy and Infrastructure Counselling of the Junta of Extremadura through grant GR18097 (co-financed by the European Regional Development Fund) and by the Ministerio de Economìa y Competitividad of the Spanish Government (CGL2017-87917-P). The authors are grateful to Paco Bellido (https://pacobellido.naukas.com/) for providing the image shown in Figure 1.