# Temporal and Latitudinal Variation in Penumbra-Umbra Ratios of the Sunspots:
Analyses of RGO, Kodaikanal and Debrecen Databases

Jouni Takalo 1 Space Physics and Astronomy Research Unit, University of Oulu,
POB 3000, FIN-90014, Oulu, Finland
11email<EMAIL_ADDRESS>
###### keywords:

Sun: Sunspot cycles, Sun: Sunspot area, Sun: Penumbra-umbra ratio, Method:
Distribution analysis, Method: Statistical analysis

##  1 Introduction

It has been known almost for two hundred years that the occurrence of sunspots
is cyclic, although not strictly periodic. The length of the sunspot cycle
(SC) has varied between 9.0 and 13.7 years. The shape of the sunspot cycle has
also changed somewhat. Waldmeier noticed the asymmetry of sunspot cycles, with
the ascending phase being typically shorter than the declining phase, and that
there is an anti-correlation between cycle amplitude and the length of the
ascending phase of the cycle (Waldmeier, 1935, 1939). The number of sunspots
has been observed, at least, since the 17th century, although the measurements
in the early days were scarce and somewhat inaccurate. These early
observations include also hand-drawn figures, but these are only for short
periods (Neuhäuser, Arlt, and Richter, 2018; Carrasco et al., 2021, 2022). The
positions and areas of the sunspots have been systematically recorded by the
Royal Greenwich Observatory (RGO) since 1874 (Hathaway, 2013; Mandal et al.,
2020). Recently, efforts have been done to estimate the sunspot areas and
sometimes even trying to separate umbra and penumbra structures from the early
drawings (see the excellent review by Artl and Vaquero, 2020). Since the late
19th century photographic recordings have been done in many observatories. As
stated earlier the Royal Greenwich Observatory (RGO) started recordings in
1874 and continued these until 1976 (Hathaway, 2013), Kodaikanal observatory
in India has taken white-light images of the sunspots since 1904 (Ravindra et
al., 2013; Mandal et al., 2017; Jha et al., 2022), Mt. Wilson has collections
from 1917 up to 1982 (Howard, Gilman, and Gilman, 1984; Bogdan et al., 1988),
and Debrecen continued the recordings of RGO from 1974 until today (Baranyi et
al., 2001; Győri, Ludmány, and Baranyi, 2017).

There have been some studies of penumbra-umbra ratio ($q$) of the sunspots
during the 20th century. Nicholson (1933) and Waldmeier (1939) found
consistent results with each other that $q$ decreased as the sunspot size
increased. Jensen, Nordø, and Ringnes (1955) reported also a slight decrease
of $q$-values as a function of sunspot size but that $q$ was higher during
maxima than minima of the cycles. During cycle minima the variation of the
$q$-values was also smaller with increasing sunspot size. Antalová (1971)
noticed that $q$ is an increasing function of the sunspot size and in double
maxima cycles the ratio is higher in the first maximum than in the second
maximum. She also stated that $q$-values do not change with heliographic
latitude.

Hathaway (2013) studied the daily records of sunspot group areas compiled by
the Royal Observatory, Greenwich from May of 1874 through 1976. He found that,
on average, $q$ increases from about 5 to 6 with the group area increasing
from 100 to 2000$\,\mu$Hem, and does not change with latitude or phase of the
cycle. He, however, found a peculiar change of $q$ with time such that it
decreased smoothly from more than 7 in 1905 to lower than 3 by 1930 and then
increased again to over 7 in 1961. Carrasco et al. (2018) studied sunspot data
from the catalogue published by the Coimbra Astronomical Observatory (CAO) for
the period 1929 – 1941. They could not find such kind of change during the
analyzed period. There were also significant differences in $q$-values for the
smaller groups between the RGO and CAO analyses. They state that the main
differences are in the measurements of the umbra area, and while CAO
instrument and methods did not change during the period 1929 – 1941, there
were changes in RGO arrangements, which could have affected the discrepancy of
the results.

Jha, Mandal, and Banerjee (2019) and Jha et al. (2022) have studied $q$-values
from the recently published Kodaikanal digitized white-light images from 1904
to 2017. They have also compared their results to the $q$-values of RGO and
Debrecen datasets for the same period. They found that $q$ increases from 5.5
to 6 as the sunspot size increases from 100 to 2000 $\mu$Hem. They did not
find any systematic trend in smaller ($<$100 $\mu$Hem) sunspots as was
reported earlier in the results from RGO database by Hathaway (2013).
Furthermore, they found that the average $q$ does not show variation as a
function of latitude. They compare $q$ time-series calculated from Kodaikanal
dataset to the $q$ time-series calculated from RGO and Debrecen sunspot
datasets. It seems that for the sunspots $<$100 $\mu$ Hem, $q$-values for
Kodaikanal data fluctuate between 4 to 5 all the time and for sunspots
$\geq$100 $\mu$Hem between 5 to 6, while for the other two datasets (RGO and
Debrecen) the variation is much larger. So Kodaikanal sunspot dataset seems to
be more homogeneous than the other two datasets.

Recently Hou et al. (2022) published an article where they analyze the
recently digitized sunspot drawings observed from Yunnan Observatories (1957 –
2021) in order to study $q$ for the Solar Cycles 19 – 24. They found that $q$
is 6.63 $\pm$ 0.98, and its probability distribution fits very well to the
log-normal distribution function. They did not notice any dependence of the
$q$-value on the latitude or the phase of the cycle.

In this study we analyze the latitudinal distribution and temporal evolution
of sunspot penumbra-umbra ratio ($q$) of SC12 – SC23 (occasionally also SC24,
which is, however, only until 2017). To get more data for solid statistics, we
superimpose separately the even and odd cycles. To this end we harmonize the
duration of the cycles such that all have the same length 128 months, which is
about the average for the Solar Cycles 16 – 23, i.e. the interval for the
Kodaikanal dataset. This paper is organized as follows. Section 2 presents the
databases used in this study. In Section 3 we study the $q$-values of Debrecen
dataset for Cycles 21 – 24. Section 4 deals with the $q$-values of RGO Cycles
12 – 20 and Section 5 the $q$-values of Kodaikanal Cycles 16 – 24. In Section
6 we compare the results for the overlapping intervals of the aforementioned
datasets and give our conclusions in Section 7.

##  2 Sunspot Area Databases

In the analyses of sunspot area we use three databases. The longest dataset
for umbral and total sunspot areas is recently published Kodaikanal database
(Mandal et al., 2017; Jha et al., 2022), which is reconstructed from the
white-light images of the same observatory since 1904. Because there are gaps
before 1921, and Cycle 24 is incomplete, we use this dataset mainly for the
Solar Cycles 16 – 23. Notice that this data contains areas of separate
sunspots. Another database is that of the Royal Observatory, Greenwich-
USAF/NOAA Sunspot Data for the years 1874 – 2016. This database contains,
among others, time, latitude and whole area size (in millionths of solar
hemisphere, $\mu$Hem) for individual sunspots for SC12 – SC23, and also umbra
sizes of the sunspots for SC12 – SC20 (Hathaway, 2013). We use here the data
for Solar Cycles 12 – 20, i.e the period which contains the whole and umbra
areas of the sunspot group data. The third dataset is the Debrecen
Photoheliographic Data (DPD), which consist of daily, group and sunspot data
(Baranyi et al., 2001; Győri, Ludmány, and Baranyi, 2017). Here we use whole
areas and umbra areas of the sunspot group data data, marked in the database
with “g”. Because Debrecen database seems to be most precise for the recent
Solar Cycles 21 – 24, we use it as a preliminary database. The minima and
lengths for the Solar Cycles 12 – 24 used in this study are listed in Table 1.

Table 1.: Sunspot-cycle lengths and dates [fractional years, and year and month] of (starting) sunspot minima for Solar Cycles 12 – 24 (NGDC, 2013). Sunspot cycle | Fractional | Year and month | Cycle length
---|---|---|---
number | year of minimum | of minimum | [years]
12 | 1879.0 | 1878 December | 10.6
13 | 1889.6 | 1889 August | 12.1
14 | 1901.7 | 1901 September | 11.8
15 | 1913.5 | 1913 July | 10.1
16 | 1923.6 | 1923 August | 10.1
17 | 1933.7 | 1933 September | 10.4
18 | 1944.1 | 1944 February | 10.2
19 | 1954.3 | 1954 April | 10.5
20 | 1964.8 | 1964 October | 11.7
21 | 1976.5 | 1976 June | 10.2
22 | 1986.7 | 1986 September | 10.1
23 | 1996.8 | 1996 October | 12.2
24 | 2009.0 | 2008 December | 10.9
25 | 2020 | 2019 December |

In this study we define the penumbra-umbra ratio ($q$) as (Antalová, 1971;
Hathaway, 2013)

$q=\left(A_{\rm Sp}-A_{\rm Um}\right)/A_{\rm Um},$ (1)

where $A_{\rm Sp}$ is the sunspot total area and $A_{\rm Um}$ the umbral area
of the sunspot.

##  3 Debrecen Sunspot Group Dataset

Figure 1 shows the temporal distribution of penumbra-umbra ratio, q, for Solar
Cycles 21 – 24 (Cycle 24 only until 19th June, 2018) of Debrecen sunspot group
data. In order to get more data to a more solid statistics, we harmonize the
solar cycles having each 128 months, which is about their average cycle-length
during the longest database in this study, i.e. Kodaikanal Solar Cycles 16 –
24. We then study separately the superimposed cycles of the aforementioned
period for Debrecen data data. The blue vertical errorbars show the standard
deviation at each point of the monthly measurements. Interestingly there is a
huge valley at the maximum region of the cycles, i.e. around 40-50 months. To
analyze this valley in more detail we plot separately the temporal
distributions for areas smaller or equal to 100 $\mu$Hem (small) and over 100
$\mu$Hem (large). Figure 2 shows $q$ for small and large category sunspot
groups as blue (red errorbars) and black (magenta errorbars) curves,
respectively. The valley exists in both categories but are somewhat deeper in
the small category groups (when studying in more detail, it turned out that
the decrease is deepest for groups between 50 – 100 $\mu$Hem). We believe that
this decrease in the $q$ is related to the Gnevyshev gap (GG) (Gnevyshev,
1967) at about 35 – 40 % from the start of the cycles (Takalo and Mursula,
2018). Figure 3a shows the number of small (blue) and large (red) sunspot
groups during Cycles 21 – 23 (Cycle 24 is omitted, because it is incomplete).
Note that the number of large groups decreases more than the number of small
groups during the GG. This result is consistent with the result by Takalo
(2020a) that GG is more visible in large than in small sunspots. It is then
understandable that when the relative number of smaller sunspot groups
increases, the $q$-values are lower at the GG-region. There is another drop
after 60 months but it is caused mainly by Cycle 24, when large groups
disappear abruptly after the cycle maximum on 2014. Although the errorbars are
quite wide, the shape of the penumbra-umbra curves are very obvious regarding
that small groups dominate at the beginning and at the end of the cycles and
large groups are relatively more abundant during the maximum of the cycles
(see the inset of Figure 3a). The mean $q$-value for small groups ($\leq$100
$\mu$Hem) is 5.44 (standard deviation, std=0.90), for large groups ($>$100
$\mu$Hem) 6.09 (std=0.82) and for all groups (Fig. 1) 5.74 (std=0.82). Note
that the standard deviation in the brackets here and later is different from
the errorbars in the figures, i.e. it is calculated from the average monthly
(or average latitudinal) $q$-values during the cycles, i.e. the larger this
std is the more variation (trends) there is in the distribution in the
$q$-values.

Figure 1.: Debrecen temporal penumbra-umbra ratio of all sunspots for Cycles
SC21 – 24 (black curve). The blue errorbars show the standard deviations at
each monthly point of the ratio. Figure 2.: Debrecen temporal penumbra-umbra
ratio of the sunspot groups for Cycles SC21 – 24 separately for areas
$\leq$100 $\mu$Hem and $>$100 $\mu$Hem as blue and black curves, respectively.
The errorbars are shown with red and magenta colors, respectively. Figure 3.:
a) Number of small (blue) and large (red) sunspot groups as a function of time
for Debrecen Solar Cycles 21 – 23. The inset shows the ratio of small/large
groups along the cycles. b) Number of small (blue) and large (red) sunspot
groups as a function of latitude for Debrecen Solar Cycles 21 – 23. The inset
shows the ratio of small/large groups along the latitudes.

Figure 4a shows the Debrecen latitudinal distribution for the even and odd
cycles SC21 – SC24. The $q$-graph for the even cycles has a valley around zero
latitude, except that of different values at -3 degrees of latitude. The
maximum values seem to be between 10 to 20 degrees at both hemispheres. This
is also the region, where the larger sunspot ($>$100 $\mu$Hem) are most
abundant (see Figure 4b). It is, however, clear that the penumbra-umbra ratio
is smaller near the Equator of the Sun. This is due to the smaller size of
sunspots dominating near the Equator, which leads also to a smaller $q$-value.
Note from the inset of Figure 3b that the ratio of small/large sunspots is
about four between -2 – 2 degrees while is is under 2 from 5 to 25 degrees in
both hemispheres. It is also evident that the odd cycles have higher
$q$-values than the even cycles. This is because the odd cycles (21, 23) have
more larger category sunspot groups than the even cycles (22, 24). Figure 4b
shows the latitudinal number of large sunspot group for even (blue) and odd
(black) cycles. Note that the difference of $q$-values is largest at the
latitudes in which the difference in number of groups is also largest. Figure
4b shows also that the number of large groups has a local minimum at or near
15 degrees of latitude. This is especially clear for even cycles. This is
consistent with the results of Takalo (2020a, b), that large sunspots and
sunspot groups have local minima at the time when the average latitude of
sunspots crosses 15 degrees of latitude (Takalo, 2020b). We believe that this
in turn coincides with the Gnevyshev gap in the SSN and sunspot areas. The
mean $q$-values for the sunspot groups of even and odd Cycles 21 – 24 between
latitudes -35 – 35 are 5.53 (std=0.56) and 6.33 (std=0.55), respectively.

Figure 4.: a) Debrecen latitudinal $q$ for all sunspots of the even (blue with
magenta errorbars) and odd (black with red errorbars) Cycles 21 – 24. b)
Latitudinal distribution for the number of sunspot groups of the even (blue)
and odd (black) Cycles 21 – 24.

##  4 Kodaikanal Sunspot Dataset

We start the study of the Kodaikanal dataset with a plot of the hemispheric
umbral areas in Figure 5 for Solar Cycles 18 – 24 (note that Cycle 24 is
incomplete). The reason why we concentrate in this figure on this period is
the base-map, which shows the so-called Homogeneous Coronal Data Set (HCDS).
The HCDS is the irradiance of the Sun as a star in the coronal green line (Fe
XIV, 530.3 nm). It is derived from ground-based observations of the green
corona made by the network of coronal stations (Kislovodsk, Lomnický Štít,
Norikura, and Sacramento Peak). The coronal intensities have been measured at
72 points at five-degree separation starting from North Pole counterclockwise
around the Sun at a height around 50 arcsec. The values are calibrated to the
center of the solar disk to get absolute values of intensity, i.e. absolute
coronal units (ACU). One ACU represents the intensity of the continuous
spectrum of the center of the solar disk in the width of one Ångström at the
same wavelength as the observed coronal spectral line (1 ACU = 3.89 Wm-2 sr-1
at 530.3 nm) (Minarovjech, Rušin, and Saniga, 2011; Takalo, 2022). This
database exists only for SC18 – 24 and the temporal and spatial intensities of
this corona are shown as a colorbar on the right-side of the figure. The green
and red dots are sunspots with umbral area 75 to 150$\,\mu$Hem and greater or
equal than 150$\,\mu$Hem, respectively (we do not show the smaller sunspots
here for clarity such that the corona is also visible). The white curves show
the (relative) hemispheric total umbral areas, and the vertical black lines
the official maxima of the solar cycles. Significant information can be
extracted from Figure 5. It is evident that the corona and stronger sunspots
are located simultaneously, although corona covers larger latitudinal region.
It is seen that the maximal corona starts at latitudes 35 – 40 degrees
similarly to sunspots, but migrates both towards the Equator and poleward
during cycle evolution, although as fainter toward the Poles than equatorside.
It is also seen that the strongest Solar Cycle 19 is most symmetric around the
cycle in corona and sunspot in the northern hemisphere, while most of the
largest sunspots in the southern hemisphere are located in the ascending phase
of the cycle. The other cycles, except Cycle 24, have a prolonged tail such
that the majority of the large sunspots are after the cycle maximum. Solar
Cycle 24 has hemispheric asymmetry such that the largest northern hemisphere
sunspots exist before the cycle maximum and southern hemisphere sunspots
locate around the cycle maximum. Note also, that Solar Cycles 23 and 24 have
much fainter HCDS corona than other cycles.

Figure 5.: The butterfly diagram of Kodaikanal dataset umbral areas for Solar
Cycles 18 – 24. The green and red dots are sunspots with umbral area 75 to
150$\,\mu$Hem and greater or equal than 150$\,\mu$Hem, respectively. The
slightly smoothed white curves show total umbral area for northern and
southern hemisphere of the Sun. The colors on the base of the figure show HCDS
corona, whose intensities are shown on the right. The black vertical lines
show the sites of the maxima of the cycles. Figure 6.: a) Kodaikanal temporal
penumbra-umbra ratio ($q$) for all sunspots of even Cycles 16 – 23. b) $q$ for
all sunspots of odd Cycles 16 – 23. Figure 7.: a) Kodaikanal latitudinal
penumbra-umbra ratio ($q$) of all sunspots smaller than 100 $\mu$Hem for even
and odd Cycles 16 – 23. b) Latitudinal $q$ for sunspots larger than 100
$\mu$Hem for even and odd Cycles 16 – 23.

The recently published Kodaikanal sunspot dataset is the longest sunspot set
containing also areas of sunspot umbras recorded on a single station. We again
harmonize Solar Cycles 16 – 23 having each 128 months. Figures 6a and b show
the penumbra umbra ratio ($q$) for the even and odd cycles for all sunspots of
SC16 – SC23. These figures show that the odd cycles have moderately higher
ratio than the even cycles in the ascending phase of the average cycle. This
means that odd cycles have considerable more large sunspots in the ascending
phase than after the maximum. On the other hand the shape of the $q$-graph for
the even cycles reminds more the shape of the sunspot cycle itself, i.e,
largest sunspots are in the middle of the cycle. The error limits are smaller
than for Debrecen data, especially for the even cycles. The mean $q$ for even
and odd cycles are 5.27 and 5.43 with standard deviations 0.36 and 0.47
calculated from the monthly $q$-values, respectively.

Figures 7a and b show the latitudinal penumbra-umbra ratio for sunspots
smaller than 100 $\mu$Hem and for sunspots larger than 100 $\mu$Hem of even
(blue with red errorbars) and odd (black with magenta errorbars) cycles. We
restrict again the latitudes between -35 and 35 degrees, because the small
number of umbral data (the divisor in the calculation of penumbra-umbra ratio)
causes quite wild behavior at larger latitudes. Note that the error limits are
nevertheless large around zero latitude and at latitudes higher than 25
degrees. The latitudinal distribution of $q$-values is quite flat for small
sunspots but have more variation for large sunspots. Note that for large
category odd cycles there is again a valley around zero latitude, but there
are bad values (very wide errorbars) on both sides of the zero latitude for
the even cycles. The mean values for $q$ between -35 to 35 degrees of smaller
category and larger category sunspots are for the even cycles 4.79 (std=0.32)
and 6.02 (0.37), and for the odd cycles 5.00 (std=0.48), 6.29 (0.36),
respectively.

##  5 RGO Sunspot Group Dataset

The Royal Greenwich Observatory (RGO) compiled sunspot group observations from
a small network of observatories starting in May of 1874. These observations
lasted until 1976 after which the measurement have been done in Debrecen. The
RGO sunspot group dataset contains also umbra areas and that is why we can
study penumbra-umbra ratios for the total Solar Cycles 12 – 20 using this
dataset. We use the same common length for the cycles in order to compare the
results with Kodaikanal distributions. Figures 8a and b show the temporal
penumbra-umbra ratios of the RGO dataset for the even and odd cycles,
respectively. In this case $q$ for the even cycles seems to be somewhat higher
than for the odd cycles. The monthly $q$-graph is flatter for the odd cycles
than for the even cycles, but both have slightly higher values in the
descending part of the cycles. The mean values for $q$ are 5.20 and 4.75 with
standard deviations 0.46 and 0.41 for the even and odd cycles, respectively.
Note that now (Cycles 12 – 20) even cycles have higher average $q$-value than
odd cycles, while for Kodaikanal (Cycles 16 – 23) even cycles have lower
average $q$-value than odd cycles.

Figure 8.: a) RGO temporal penumbra-umbra ratio ($q$) of all sunspots for the
even Cycles 12 – 20. b) RGO temporal $q$ of all sunspots for the odd Cycles 12
– 20. Figure 9.: a) RGO latitudinal penumbra-umbra ratio ($q$) of sunspots
smaller than 100 $\mu$Hem for even and odd Cycles 12 – 20. b) RGO latitudinal
$q$ for sunspots larger than 100 $\mu$Hem for even and odd Cycles 12 – 20.

Figures 9a and b show the latitudinal penumbra-umbra ratios for sunspot groups
smaller than 100 $\mu$Hem and larger than 100 $\mu$Hem, respectively. The
overall shape of the ratio for small groups is flat, if not slightly convex
upwards with no valleys or humps. There are, however, huge errorbars in the
small category plots. These are due to the large variation in the RGO
penumbra/umbra ratio, especially for small groups (Hathaway, 2013; Jha et al.,
2022). As stated earlier, Carrasco et al. (2018) did not find such large
variation in the study of Coimbra Astronomical Observatory measurements for
the period 1929 – 1941. For larger sunspot groups there is a shallow valley
around zero degrees, except a couple of anomalous values at negative low
latitudes. The maxima are between 5 to 20 degrees of latitude for the even
cycles, but are wider and flatter for the odd cycles. Note that, although
there are valleys around zero latitude, they have just minor significance,
because the errorbars for large group $q$-values are also quite wide. The
$q$-values between -35 to 35 degrees for small and large groups are 4.97
(std=0.56), 4.47 (0.58) and 5.61 (0.33), 5.13 (0.45) for the even and odd
cycles, respectively.

##  6 Comparison of Kodaikanal, RGO and Debrecen Sunspot Datasets

There are overlapping Cycles 16 – 20 in RGO and Kodaikanal datasets and Cycles
21 – 23 in Kodaikanal and Debrecen dataset. That is why it is appropriate to
compare these intervals between those data. Figure 10 shows the penumbra-umbra
ratios for RGO and Kodaikanal Solar Cycles 16 – 20. It is clear that
Kodaikanal $q$ is moderately higher than RGO $q$, except and the end of the
average cycle. The shape of the $q$-graph for Kodaikanal is quite similar to
Kodaikanal even cycles (Figure 6) but the $q$-graph of the RGO has no trends.
This is because the $q$-graphs for separate cycles have opposite trends (C16
decreasing, C17 increasing etc., see also Jha et al., 2022), which cancel each
other when they are overlaid such that the sum is quite flat. The mean values
for the average Cycle between 16 – 20 are 5.35 and 4.75 with standard
deviations 0.33 and 0.26 for Kodaikanal and RGO cycles, respectively.

Figure 11 shows the $q$-graphs for Kodaikanal and Debrecen sunspot datasets of
Solar Cycles 21 – 24. We show only the errorbars of Kodaikanal $q$-values,
because the graph for Debrecen is same as in Figure 1. It is evident that
Debrecen has higher values except at the very beginning and at the end of the
cycles. Otherwise the $q$-graph of Debrecen data has much more structure than
the $q$-graph of Kodaikanal data. The most interesting difference between
these $q$-graphs is the GG-related decrease around 40-50 months in Debrecen
$q$, which does not exist in Kodaikanal $q$. Note that Kodaikanal $q$ is
highest in the ascending phase of the cycle and has also a shallow valley in
$q$-values somewhat later than Debrecen, i.e. between 50 – 60 months. The mean
$q$-value for Debrecen is 5.90 (std=0.76), and mean $q$-value for Kodaikanal
is 5.34 (std=0.56).

Figures 12a and b show the total and umbral areas for Debrecen and Kodaikanal
data, respectively. This figure may explain why Kodaikanal $q$ does not show
the GG-related valley, which is so distinct in the Debrecen temporal $q$.
Debrecen data have a deep decrease in both total sunspot and umbral area
between 42 – 51 months, which is related to GG-phenomenon. On the other hand,
Kodaikanal data have two drops, first at 35 – 42 and second very short at
Debrecen GG region (47 months). It turns out that the first is related to odd
cycles and the second to even cycles. The latter drop is narrow, because odd
cycles have maxima at the time of the even cycles minima. Neither of these
seem, however, to cause the $q$-value to decrease during the GG-regions. Note
that Kodaikanal areas are calculated from separate sunspots and Debrecen areas
from sunspot groups. We, however, have done similar a analysis also to
Debrecen sunspot data (shown with “s” in the database), and it gives results
consistent with the Debrecen $q$-values of the groups.

Figure 10.: Comparison of Kodaikanal and RGO temporal penumbra-umbra ratios
for Solar Cycles 16 – 20. Figure 11.: Comparison of Kodaikanal and Debrecen
temporal penumbra-umbra ratios for Solar Cycles 21 – 24. Figure 12.: a) Total
sunspot group areas and group umbral areas of Debrecen data for Solar Cycles
21 – 24. b) Total sunspot areas and sunspot umbral areas of Kodaikanal data
for Solar Cycles 21 – 24. (Sunspot/group areas as blue color with axis on the
left side and umbral areas as red with axis on the right side)

Figure 13a shows the comparison of RGO and Kodaikanal latutudinal $q$-values
for Cycles 16 – 20 with areas larger than 100 $\mu$Hem. Here we also show a
13-point trapezoidal smoothing of the $q$-values. Trapezoidal smoothing is a
common moving average smoothing with end points of the window having half of
the weight of the inner points. The smoothed shapes of the $q$-graphs seem to
be very similar with a shallow valley at the Equator. The average of the RGO
$q$ is again smaller than the average of the Kodaikanal $q$, i.e. mean values
(standard deviations) are 5.11 (0.15) and 5.35 (0.15) for RGO and Kodaikanal,
respectively. Figure 13b shows the comparison of Kodaikanal and Debrecen
latutudinal $q$-values for Cycles 21 – 24 with areas larger than 100 $\mu$Hem.
It is evident that the shape of the Kodaikanal $q$-graph is very similar to
that of the Kodaikanal $q$-graph in Figure 13a with slightly deeper valley at
the Equator in the smoothed curve. Note that the errorbars are behaving very
wildly, especially for Debrecen, because the $q$-values of the odd cycles are
much higher than the $q$-values of the even cycles. The Debrecen $q$-graph is,
however, still deeper at the Equator having clear maxima between 10 – 20
degres at both hemispheres. The mean $q$-values (standard deviations) are 6.22
(0.34) and 6.36 (0.41) for Kodaikanal and Debrecen, respectively. It should,
however, be noted that while it is understandable that the $q$-values are
smaller near the Equator, because sunspots and sunspot groups are smaller
there than around 10 to 25 degrees of latitude, the error limits also so wide,
that the confidence of this result is vague.

Figure 13.: a) Comparison of the latitudinal penumbra-umbra ratios ($q$) for
Kodaikanal and RGO Cycles 16 – 20 with areas larger than 100 $\mu$Hem. b)
Comparison of the latitudinal $q$-values for Debrecen and Kodaikanal Cycles 21
– 24 with areas larger than 100 $\mu$Hem.

##  7 Conclusions

We have analyzed the penumbra-umbra ratio ($q$) of sunspot latitudinal and
temporal distributions for RGO SC12 – SC20, Kodaikanal SC16 – SC23 (also
incomplete cycle 24 until june 2018) and Debrecen dataset SC21 – SC24 (SC24
until the end of 2017), in order to get enough data for a solid statistics.

The analysis of $q$-values for the Kodaikanal Cycles 16 – 24 shows that the
odd cycles have considerable more large sunspots in the ascending phase than
after the maximum. On the other hand, for the even cycles the shape of the
$q$-graph reminds more the shape of the sunspot cycle itself, i.e, largest
sunspots are in the middle of the cycle. The mean $q$-value for even and odd
cycles are 5.27 and 5.43, respectively. On the contrary, our analysis shows
that the sunspots under 100 $\mu$Hem are quite evenly distributed throughout
the cycle for both the odd and even sunspot cycles. The latitudinal $q$ for
large category sunspots and all sunspots seems to change such that it is
lowest around the Equator and increases slightly towards higher latitudes. The
latitudinal change is, however, inside the errobars and we can say that it is
insignificant. Our analysis shows again that for the small sunspots ($<$100
$\mu$Hem) the latitudinal distribution of $q$ is flat. The latitudinal average
$q$ for all sunspots is same size as the temporal average q, i.e. 5.41 and
5.65 for the even and odd cycles, respectively.

The analysis of $q$ for the RGO Cycles 12 – 20 shows quite similar results,
but now the $q$-values for the even cycles are higher than for the odd cycles.
The mean values for $q$ are now 5.20 and 4.75 for the even and odd cycles,
respectively. Note, that the $q$-values are moderately smaller for the RGO
dataset than Kodaikanal dataset. The latitudinal analysis for RGO sunspot
groups data confirms the aforementioned result, that $q$ is at lowest around
Equator and increases towards higher latitudes for the large ($>$100 $\mu$Hem)
groups. Note that the errorbars for RGO data are largest, because the level of
the $q$-values changes so much between cycles, especially for small category
sunspot groups.

For Debrecen sunspot data we superimposed the Cycles 21 – 24 in the case of
temporal evolution of the $q$-values. This is because of the shorter interval,
i.e less cycles in the Debrecen database than for the other datasets. In this
case we find an interesting phenomenon. The shape of the $q$-graph has two
clear maxima and a deep valley between them for both small and large category
sunspot groups. It turns out that the decrease in the $q$-values is
simultaneous to the drop of the total and umbral area of the Debrecen sunspot
groups. We believe that this is related to Gnevyshev gap as the deepest point
of the total and umbral area locates 36 % from the start of the cycle (Takalo
and Mursula, 2018). The average $q$-value for the Debrecen cycles is 5.74. The
latitudinal $q$-values for Debrecen sunspot groups have largest variation such
that they are clearly lowest (except some bad values) at the Equator of the
Sun. The decrease as calculated separately for the even and odd cycles seems
to be deeper than the errorbars of one standard deviation. The mean values for
the even and odd Cycles 21 – 24 between latitudes -35 – 35 are 5.53 and 6.33.

Figure 14.: The relative densities of $q$-values for RGO, Kodaikanal and
Debrecen databases.

We have also compared the overlapping intervals of the RGO and Kodaikanal
datasets for Cycles 16 – 20 and Kodaikanal and Debrecen datasets for Cycles 21
– 23. The results are consistent with the separately made analyses such that
RGO has smallest $q$-values and Debrecen highest $q$-values. The reason for
the differences in $q$ is probably difficulty in separating the border between
umbra and penumbra, especially for sunspot groups (Steinegger, Bonet, and
Vázquez, 1997). This issue is quite complicated and is not the scope of this
study. Furthermore, Foukal (2014) states that the main reason why spot areas
recorded using photographic or CCD observations are larger than those based on
drawings seems to be that the areas of spots too small to draw are still
individually measurable on good plates and CCD images.

Debrecen dataset, however, seems to have smallest umbral areas, at least for
large sunspots. The gradient method used in the analyses of Debrecen sunspot
data is described in the article by Györi (1998). Debrecen dataset seems also
to give most precise results in the sense that it shows also the effect of
Gnevyshev gap in its temporal evolution of the $q$-values. Gnevyshev gap is
somehow related to the reversal of the magnetic field of the Sun. The basic
reason for the GG is not the scope of this study, and it is an open question
now, but some good attempts have been presented (Georgieva, 2011; Karak,
Mandal, and Banerjee, 2018). There have also been suggestions about a relic
field in the Sun (Bravo and Stewart, 1996; Mursula, Usoskin, and Kovaltsov,
2001; Song and Wang, 2005). If the relic field exists, it could maybe explain
why GG is more intense in the even cycles than in the odd cycles (Takalo and
Mursula, 2020; Takalo, 2020b, 2021). It should be noted that RGO and
Kodaikanal datasets do not show the temporal decrease of the $q$-values ,
although they both have a clear twofold GG in their total area as a function
of time for Solar Cycles 16 – 20 (see Figure 15). Note that the GG region in
this case is few months later than GG of Debrecen for Solar Cycles 21 – 24.

Figure 15.: Total areas of a) Kodaikanal and b) RGO datasets for Solar Cycles
16 – 20.

Kodaikanal research group uses their own procedure in defining the border,
which is based on the threshold method introduced by Otsu (1979). Kodaikanal
data seem to be most compactly located around the mean $q$-value. This is seen
in Figure 14, where we show the relative densities of RGO, Kodaikanal and
Debrecen databases. Kodaikanal has by far the smallest standard deviation of
the $q$-distributions. Note that this standard deviation is related to the
errorbars in the $q$-graphs shown earlier for each databases. From these three
datasets RGO seems to be most heterogeneous, which is seen also as a curious
drop of $q$-values in the first third of the 20th century (Hathaway, 2013, see
also solarscience.msfc.nasa.gov/greenwch.shtml). Although the RGO data is one
of the best sunspot area datasets, the drop in $q$-values for spots smaller
than 100 $\mu$Hem has not been seen in any other datasets so far. Thus, one
must proceed with caution when comparing $q$-values between different
observatories. It should also be noted, that corrections to the RGO database
have been going on recently (Erwin et al., 2013).

### Acknowledgements

The RGO sunspot area data are from RGO-USAF/
NOAA (solarscience.msfc.nasa.gov/greenwch.shtml), Kodaikanal database
from github.com/bibhuraushan/KoSoDigitalArchive and Debrecen sunspot
datasets from fenyi.solarobs.csfk.mta.hu/DPD/. The dates of the cycle maxima
were obtained from the National Geophysical Data Center (NGDC), Boulder,
Colorado, USA (ftp.ngdc.noaa.gov), and the corona indices are retrieved from
www.ngdc.noaa.gov/stp/solar/corona.html

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