# Boosting GUI Prototyping with Diffusion Models

Jialiang Wei∗, Anne-Lise Courbis∗, Thomas Lambolais∗,
Binbin Xu∗, Pierre Louis Bernard∗∗ and Gérard Dray∗
∗: EuroMov Digital Health in Motion, Univ Montpellier, IMT Mines Ales, Ales,
France
∗∗: EuroMov Digital Health in Motion, Univ Montpellier, IMT Mines Ales,
Montpellier, France
∗<EMAIL_ADDRESS>∗∗<EMAIL_ADDRESS>
###### Abstract

GUI (graphical user interface) prototyping is a widely-used technique in
requirements engineering for gathering and refining requirements, reducing
development risks and increasing stakeholder engagement. However, GUI
prototyping can be a time-consuming and costly process. In recent years, deep
learning models such as Stable Diffusion have emerged as a powerful text-to-
image tool capable of generating detailed images based on text prompts. In
this paper, we propose UI-Diffuser, an approach that leverages Stable
Diffusion to generate mobile UIs through simple textual descriptions and UI
components. Preliminary results show that UI-Diffuser provides an efficient
and cost-effective way to generate mobile GUI designs while reducing the need
for extensive prototyping efforts. This approach has the potential to
significantly improve the speed and efficiency of GUI prototyping in
requirements engineering.

##  I Introduction

The exponential growth of mobile technology and the increasing dependence on
mobile devices for various daily activities have significantly influenced the
design and development of mobile applications. The mobile GUI (graphical user
interface) plays a critical role in mobile applications since it is the
primary means of interaction between users and their devices. A well-designed
mobile UI can significantly enhance the user experience, making it simpler for
users to navigate and achieve their desired tasks, resulting in increased user
engagement and retention [1, 2]. Additionally, an engaging and user-friendly
GUI can set an application apart from its competitors and increase its chances
of success in the highly competitive mobile app market [3]. As the competition
among mobile apps intensifies, it’s critical for developers to create
innovative and user-friendly GUI to meet users’ evolving expectations.

GUI prototyping is a crucial technique that allows developers to create an
initial version of a GUI design, assess its effectiveness, and refine it based
on feedback from stakeholders. This technique is highly valuable in the
context of requirements engineering, as it can help refine requirements,
reduce development risks, and promote stakeholders’ engagement [4]. Despite
the benefits of GUI prototying, it can be a time-consuming and costly process
[5].

To enhance the GUI prototyping process, various approaches have been proposed,
including the use of established tools that provide basic components and
templates for creating GUIs. The industry has widely embraced commercial tools
like Figma [6], InVision Studio [7], Adobe XD [8], Moqups [9], Sketch [10], as
well as open source tools like Pencil Project [11] to streamline the
prototyping process. In recent literature, several GUI search and retrieval
approaches have been proposed [12, 13, 14, 15, 16, 17, 18, 19]. These
approaches aim to provide users with the ability to search for inspiration
from existing designs and reuse them to streamline the GUI prototyping
process, which can reduce the time and effort spent on creating new designs.

Recently, deep learning-based text-to-image models have emerged as a promising
approach to generate highly detailed and structured images based on text
descriptions [20, 21, 22, 23]. Among these models, Stable Diffusion [21] has
shown impressive results in generating high-quality images from textual input.
In this study, we propose a novel approach called UI-Diffuser that leverages
the Stable Diffusion model to generate mobile UI designs through simple text
prompts and UI components, as illustrated in Figure 1. The preliminary results
of UI-Diffuser show its potential to enhance the effectiveness of mobile UI
design and to decrease time consumption for prototyping, leading to cost
reductions for this phase.

Figure 1: UI image generation with UI-Diffuser

The rest of the paper is organized as follows: Section II presents the
background of this research. Section III details the framework of UI-Diffuser.
Section IV shows the samples of generated UI images and discusses their
limitations. Section V concludes the paper and outlines the future directions.

##  II Background

###  II-A GUI Prototyping

To streamline and enhance the process of GUI prototyping, numerous strategies
have been previously introduced.

Prototyping tools that provides basic components and templates are widely used
in practice, such as Figma [6], InVision Studio [7], Adobe XD [8], Moqups [9],
Sketch [10], and Pencil Project [11]. However, utilizing these tools
effectively requires users to possess design experience.

Some researchers introduced GUI retrieval approaches that take the sketch
(GUIfetch [12], SWiRE [13], Wireframe-Based UI Design Search of Chen et al.
[14]) or the screenshot (Screen2vec [15], VINS [16]) as input and find designs
that are similar to the input. Although useful, these methods require an
initial rudimentary GUI prototype. Moreover, while these strategies primarily
aim to support the GUI design’s prototyping by presenting alternative designs,
they are not ideal for interactive GUI prototyping during requirements
elicitation.

GUI search engines, such as Guigle [17], Gallery D.C. [18], and RaWi [19],
allow users to search for existing GUI designs and components using textual
queries. Wei et al. [24] have put forth an proposition of combining app
features with app UIs. While these approaches can serve as a source of
inspiration and guidance, blindly copying existing designs without proper
attribution or permission can result in legal issues such as copyright
infringement.

GUIGAN, as introduced by Zhao et al. [25], leverages previously collected GUI
components from existing mobile app to compose new designs. These new
composite designs not only comply with accepted standards of GUI structure,
but also cater to consumer aesthetics. However, it is important to note that
this approach offers limited control over the generation process.

In summary, these existing approaches fall short in providing support to
analysts during the requirements elicitation phase through GUI prototyping.

###  II-B Image synthesis

Image synthesis is a process of creating new images from diverse forms of
image descriptions, including textual descriptions, sketch images, noise, and
others.

In 2014, Goodfellow et al. [26] introduced Generative Adversarial Network
(GAN) as a means of generating realistic image. GANs consist of two deep
neural networks: a generator and a discriminator. The generator creates new
images to deceive the discriminator, which aims to distinguish real from fake
images. Both networks are trained simultaneously and this process continues
until the generator produces images that are indistinguishable from real
images. However, the lack of diversity and the challenges associated with
training GANs limit their scalability and hinder their applicability to novel
domains [27].

Diffusion models (DMs) [28] are neural networks trained to denoise images
blurred with Gaussian noise by learning to reverse the diffusion process.
Recent studies [28] have demonstrated that DMs are capable of generating high-
quality images, and possess desirable attributes such as distribution
coverage, a stable training objective, and scalability. Several companies
recently released their image synthesis tools based on DMs like DALL$\cdot$E 2
[20], Midjourney [29], Stable Diffusion [21], and DreamBooth [30]. Despite
their usefulness in generating images, existing tools for image synthesis
generally lack efficacy in generating UIs. Since 2023, some researchers
utilize DMs for the generation of UI layout [31, 32, 33, 34]. However, to the
best of our knowledge, no DM-based models have been developed specifically for
generating UIs.

##  III UI-Diffuser

UI-Diffuser is a novel approach that facilites requirements engineers in
rapidly prototyping mobile app UIs through a two-step process (cf. Figure 2).
In the first step, UI-Diffuser takes input UI components, such as text,
buttons, and images, and generates a layout that considers the arrangement of
these components (see Section III-A). In the second step, the generated layout
is used to complete a mobile UI image based on the textual description
provided by the user (see Section III-B). In the subsequent subsections, we
will describe each step of UI-Diffuser in detail.

Figure 2: Overview of UI-Diffuser

###  III-A Layout Generation

The objective of layout generation is to create realistic graphic scenes that
consist of diverse components with varying attributes, such as category, size,
position, and between-component relationships [31]. This task is critical for
simplifying graphic design tasks, especially for structured scenes like
documents and user interfaces.

To achieve this goal, we employ LayoutDM [34] in this study. LayoutDM builds
on the discrete-state space diffusion models [35, 36] and has been trained on
the Rico dataset [37] – a dataset of user interface designs for mobile
applications containing 25 categories of UI components, such as text button,
toolbar, and icon. We will detail the Rico dataset in Section III-B2. LayoutDM
allows the generation of UI layout with given conditions, such as a list of
components. Figure 2 illustrates the generation of layout with a given
component list.

###  III-B UI Generation

Given the generated layout and a textual description, UI-Diffuser is able to
generate UI images fitting the layout and the description.

####  III-B1 Architecture

To generate UI images from layout and description, we utilized Stable
Diffusion [21] augmented with ControlNet [38]. The proposed model workflow is
illustrated in Figure 3. It contains four components: Text Encoder, Image
Information Creator, Image Decoder, and ControlNet.

Figure 3: Overview of UI generation model

The Text Encoder transforms raw text inputs into numerical vectors called
token embeddings. CLIP [39] is a pre-trained model that has been trained on a
large-scale corpus of text and images using a contrastive learning approach.
CLIP is employed as text encoder due to its remarkable performance in encoding
both text and images into a shared latent space.

The Image Information Creator is responsible for generating image embeddings
based on the given token embeddings, which are then used by the Image Decoder
to produce the final image. The diffusion process occurs inside this
component, in a step-by-step fashion. Starting from a noisy image, each step
of the diffusion process adds more relevant information that aligns with the
input text. UNet [40] is used in this component. During the pre-training, a
large number of images are blurred with Gaussian noise and the UNet is trained
by denoising the image.

To support additional input conditions, such as the layout image in our case,
ControlNet [38] is integrated with UNet. ControlNet is an end-to-end neural
network architecture that enables the control of large image diffusion models,
like Stable Diffusion, to learn task-specific input conditions. ControlNet
clones the weights of the Image Information Creator into a “trainable copy”.
The original Image Information Creator preserves the network capability
learned from billions of images during the pre-training, while the “trainable
copy” is trained on the Rico datasets to learn the conditional control. The
“trainable copy” and the Image Information Creator are connected with a unique
type of convolution layer called “zero convolution”. We will train the
ControlNet during the fine-tuning.

Finally, the Image Decoder generates images using the image embeddings.
Variational Autoencoder (VAE) [41] is a type of generative model consisting of
an encoder network, which maps input images into a lower-dimensional latent
space, and a decoder network, which maps the latent space back to the original
images space. The VAE encoder-decoder pair have been pre-trained on a large
number of images to accurately reconstruct input images. The decoder network
of VAE is used as the Image Decoder.

####  III-B2 Dataset for fine-tuning

The fine-tuning of ControlNet [38] requires a dataset including input images,
conditioning images and text prompts. In the context of UI generation, this
dataset should consist of UI screenshots, wireframes that depict the page
layout, and textual descriptions of the screenshots. To this end, we leveraged
the Rico dataset [37], which provides the first two elements: the screenshots
and wireframes. And we generated the textual descriptions of these screenshots
using XUI [42].

The Rico dataset [37] is one of the largest mobile app design datasets to
date, encompassing design data from more than 9.3k Android applications across
27 categories. This dataset provides access to the visual, textual,
structural, and interactive design properties of more than 66k distinct UI
screens. In this work, we utilized the screenshots, wireframes, and
hierarchies of the Rico dataset.

Figure 4: Example of a screenshot and its wireframe from Rico dataset

To prepare the Rico dataset for fine-tuning the Stable Diffusion model, a
preprocessing pipeline was applied to its screenshots and wireframes. The
pipeline includes the following steps:

  * •

remove the screenshots and wireframes on landscape. For this preliminary work,
we focus on portrait UI design.

  * •

resize the screenshots and wireframes to 288x512. The original size of the
images is 1080x1920 or 540x960. However, the Stable Diffusion model we use
accept only images with height/width less than 512 as its input during the
training process.

TABLE I: Samples of generated UI images [1pt] Components and Descriptions |  Generated Layouts |  Generated UIs |  |  |  |  |
---|---|---|---|---|---|---|---
A login page with input fields. |  |  |  |  |  |  |
A tutorial app having text components. |  |  |  |  |  |  |
A gallery page of an app. |  |  |  |  |  |  |
A maps app. |  |  |  |  |  |  |
A mediaplayer app. |  |  |  |  |  |  |
A profile app with a big image. |  |  |  |  |  |  |
[1pt] |  |  |  |  |  |  |

The Rico dataset does not provide the textual description of the screenshots.
For this reason, we used XUI [42] model to generate the descriptions. XUI is a
tool to generate automatically informative description of a given UI
screenshot. It takes the screenshots, wireframes, and hierarchies of the UIs
as input, and generate their natural language descriptions. For instance, the
generated description of Figure 4 is “That screen maybe is a list screen. You
may see a list of elements, typically arranged in rows. You may notice a text
ubicated at the center area.”. During the training, we randomly replace 50%
text prompts with a default prompt (“A nice screenshot of a mobile app”). This
facilitates ControlNet’s capability to better recognize semantic contents from
the wireframes. This is primarily because, in cases where the prompt lacks
sufficient information for the Stable Diffusion model, the model may rely more
heavily on the semantics of the input wireframes to compensate for the lack of
meaningful guidance from the prompt.

The processed screenshots and wireframes from Rico dataset and the generated
textual descriptions of XUI are then used for the subsequent fine-tuning.

####  III-B3 Fine-tuning procedures

During the fine-tuning procedures, the parameters of Text Encoder, the Image
Information Creator, and the Image Decoder are all locked. The update is only
applied to the parameters of ControlNet.

Our model are implemented with the Diffusers
library111https://github.com/huggingface/diffusers. We employed the
“runwayml/stable-diffusion-v1-5”
checkpoint222https://huggingface.co/runwayml/stable-diffusion-v1-5 and the
“lllyasviel/sd-controlnet-seg”
checkpoint333https://huggingface.co/lllyasviel/sd-controlnet-seg from
HuggingFace. We trained the model for 1 epoch on the processed Rico dataset
with a batch size of 4, equivalent to about 16,000 steps. We use AdamW for
optimization with a learning rate of $1e^{-5}$. The training was performed
using Nvidia Tesla T4 with 16GB VRAM.

##  IV Demo and Discussion

Table I illustrates some samples of UI images generated by UI-diffuser. Given
UI components and a brief description, UI-Diffuser generates UIs with various
designs that roughly meet the requirements. UI-Diffuser can produce a UI image
within a matter of seconds. Compared to traditional prototyping, UI-Diffuser
can generate GUI prototypes at a much quicker rate.

At first glance, the UI images generated by UI-diffuser appear to be of high
quality. However, upon closer examination, some details are missing. It’s
important to note that the generated UIs may not always conform to the
components category, as illustrated by the five UIs in the fifth row of Table
I, where the ”advertisement” component at the bottom of the layout is
disregarded. Additionally, certain generated UIs may not meet aesthetic
standards.

Consequently, the current UIs produced by UI-Diffuser may be more suitable for
inspiring UI designers than serving as fully functional UI prototypes.

##  V Conclusion and Roadmap

This paper presents UI-Diffuser, a GUI prototyping tool that utilizes layout
components and simple text prompts to produce mobile UI designs. Through a
demo, we demonstrate that UI-Diffuser can generate UI images that align the
given components and textual descriptions, highlighting the potential
advantages of UI-Diffuser in GUI prototyping.

To advance this research, we intend to perform a comprehensive evaluation of
the UI-Diffuser by investigating three critical factors: the aesthetics of the
generated UIs, their compatibility with UI components, and their compatibility
with textual descriptions. The aesthetics evaluation will be carried out
manually. We shall rate the aesthetics based on a predetermined set of
criteria. As for the compatibility with UI components, we will manually assess
the number of correctly generated UI components. Finally, we will use
CLIPScore [43] to calculate the compatibility of the descriptions and their
generated UIs.

Moreover, we propose enhancing UI-Diffuser from three aspects:

Developping a dataset with high-quality screenshot descriptions: the quality
of the image descriptions within a training dataset has a significant impact
on the performance of Stable Diffusion. In this work, we used XUI [42] to
generate screenshots’ descriptions. Although XUI is a valuable tool for
generating descriptions of screenshots, it only categorizes screenshots into
roughly 20 categories, which is insufficient in depicting the numerous
functions of modern mobile apps. Moreover, the descriptions generated by XUI
lack sufficient detail in describing the UI components. To address these
limitations, we plan to investigate alternative UI image captioning tools that
can generate more comprehensive UI descriptions.

Cropping components from generated UIs: while generated UI images can inspire
requirements engineers in GUI prototyping, they are usually not editable or
directly reusable. To overcome this limitation, we take inspiration from
Kolthoff et al. [19] and propose cropping each component of the generated UI
image based on its absolute position in the generated layout image. The GUI
components may overlap with each other, leading to a blank space of the lower
component when cropping the top component. In such cases, the blank space can
be filled with the top-ranked RGB color from the color histogram of the lower
component. The cropped components can be then reused in further prototyping.

Generating code from generated UIs: the ability to generate code from UI
designs can significantly accelerate the development of application
prototypes. As the layout images generated by UI-Diffuser contain the
components’ category as well as their size and position, it is possible to
generate corresponding code [44]. We intend to develop a GUI code generator
that compromises two steps: (i) extract location and size of each components
from generated layout image, and the style from generated UI image, (ii)
generate GUI code for each kind of components according to its attributes.

We believe that exploring the potential of Diffusion Models for generating UIs
is a promising research direction, as it can significantly improve the speed
and efficiency of GUI prototyping in requirements engineering.

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# Dependence of North–South Difference in the Slope of Joy’s law on Amplitude
of Solar Cycle

J. Javaraiah Bikasipura, Bengaluru-560 111, India.
Formerly working at Indian Institute of Astrophysics, Bengaluru-560 034,
India.
email: jajj55@yahoo.co.in;jdotjavaraiah<EMAIL_ADDRESS>

###### keywords:

Sun: Dynamo – Sun: Solar activity – Sun: Sunspots

##  1 Introduction

The tilt angles of bipolar magnetic regions are one of the important
ingredients for solar-dynamo models (Bhowmik and Nandy, 2018, and references
therein). Tilt angles have a significant role in the generation of polar
magnetic fields and reverse of polarity of polar fields in a 22-year solar
magnetic cycle. The tilt angles and Joy’s law (increase of tilt angle with
latitude) depend on several properties of solar magnetic active regions
(sunspot groups), e.g. latitude, area, formation and evolution, rotation and
meridional motions, and also on solar-cycle phase (Howard, 1991; Sivaraman,
Gupta, and Howard, 1999; Muneer and Sing, 2002; Zharkova and Zharkov, 2008).
Dasi-Espuig et al. (2010) analysed Mt. Wilson Observatory (1917 – 1986) and
Kodaikanal Observatory (1906 – 1986) sunspot-group tilt-angle data and found
the existence of a reasonably good anti-correlation between the ratio of mean
tilt angle [$\langle\gamma\rangle$] over a solar cycle to mean absolute
latitude [$\langle|\lambda|\rangle$] over the same solar cycle and the
strength/amplitude of the solar cycle (note that ‘$\langle.\rangle$’ implies
the mean over a time interval, i.e. here over a solar cycle). Some authors
confirmed this result and some others criticized it for various reasons (Jiao,
Jiang, and Wang, 2021). There exists a considerable north–south
difference/asymmetry in strengths/amplitudes of many solar cycles (e.g.
Carbonell, Oliver, and Ballester, 1993; Verma, 1993; Norton and Gallagher,
2010; McIntosh et al., 2013; Javaraiah, 2019; Ravindra, Chowdhury, and
Javaraiah, 2021). North–south asymmetry also exists in the rotational and the
meridional motions of solar plasma, magnetic field, and tracers such as
sunspot groups (Hathaway and Wilson, 1990; Javaraiah and Gokhale, 1997; Haber
et al., 2002; Javaraiah, 2003; Gigolashvili et al., 2005; Javaraiah and
Ulrich, 2006; Xie, Shi, and Qu, 2018; Lekshmi, Nandy, and Antia, 2018; Wan and
Gao, 2022). It is believed that the action of Coriolis force on rising
magnetic flux tubes responsible for the tilts of bipolar active regions and
Joy’s law (see D’Silva and Choudhuri, 1993; Fisher, Fan, and Howard, 1995;
Bhowmik and Nandy, 2018; Jiao, Jiang, and Wang, 2021). Since there exists
north–south asymmetry in the solar rotation, there could be a difference in
the action of Coriolis forces on the rising magnetic-flux tubes in northern
and southern hemispheres. Therefore, north–south asymmetry may exist in the
slope of Joy’s law and it also may depend on amplitude of the solar cycle.
McClintock and Norton (2013) analysed the Mt. Wilson sunspot-group data (1917
– 1986) and found the aforementioned result (anti-correlation between
$\langle\gamma\rangle/\langle|\lambda|\rangle$ and strength of cycle) from the
southern hemisphere’s data and not found from the northern hemisphere’s data.
Dasi-Espuig et al. (2010) and McClintock and Norton (2013) used
$\langle\gamma\rangle/\langle|\lambda|\rangle$ to remove the effect of
latitudinal dependence of mean tilt angle in the relation between the mean
tilt angle and the strength of the solar cycle. However, the anti-correlation
between $\langle\gamma\rangle$ alone and strength of a solar cycle is
statistically insignificant. Therefore, one may have a doubt that the
existence of a good correlation between $\langle|\lambda|\rangle$ and the
strength/amplitude of a solar cycle have a significant influence on the
aforementioned result. That is, the significant anti-correlation between
$\langle\gamma\rangle/\langle|\lambda|\rangle$ and the strength/amplitude of a
solar cycle might be largely an artifact of the good correlation between the
denominator [$\langle|\lambda|\rangle$] of
$\langle\gamma\rangle/\langle|\lambda|\rangle$ and the strength/amplitude of
the solar cycle. It is difficult to differentiate the coefficients of Joy’s
law (namely the slope of the linear relation between tilt angle and latitude)
of different solar cycles due to large uncertainties in the derived
coefficients (Dasi-Espuig et al., 2010). Hence, Dasi-Espuig et al. (2010) and
McClintock and Norton (2013) did not calculate correlation between the slope
of Joy’s law and amplitude/strength of solar cycle. In the present analysis,
by using the same aforementioned data of tilt angles of the sunspot groups
measured in Mt. Wilson Observatory (MWOB) during the period 1917 – 1986 and
Kodaikanal Observatory (KOB) during the period 1906 – 1986, we study the
dependence of the slope of Joy’s law (coefficient of Joy’s law) on the
amplitude of the solar cycle by determining it from the whole sphere’s data
and northern and southern hemispheres’ data. We obtain the relationship
between the slope (and its north–south asymmetry) and the amplitude of the
solar cycle by determining the linear-least-square fits to the data of these
parameters taking into account the uncertainties in all these parameters.

In the next section we describe the data and analysis. In Section 3 we
describe the results, and in Section 4 we present the conclusions and briefly
discuss them.

##  2 Data Analysis

Here we use the daily data: heliographic latitude (area weighted) [$\lambda$],
area [$A$], tilt angle [$\gamma$], etc. of sunspot groups measured in Mt.
Wilson Observatory (MWOB) during the period 1917 – 1986, and Kodaikanal
Observatory (KOB) during the period 1906 – 1986. These data are available at
the website www.ngdc.noaa.gov/stp/solar/sunspotregionsdata.html. We had used
these data in an our earlier analysis of angular velocities of sunspot groups
(Javaraiah, Bertello, and Ulrich, 2005). In both the northern and southern
hemispheres a positive tilt angle implies that the leading spot is closer to
the Equator than the following spot. We use the amplitudes $175.7\pm 11.8$,
$130.2\pm 10.2$, $198.6\pm 12.6$, $218.7\pm 10.3$, $285.0\pm 11.3$, $156.6\pm
8.4$, and $232.9\pm 10.2$ (values of $R_{\rm M}$, the maximum 13-month
smoothed monthly mean values of Version-2 of the international sunspot number,
SN) of Solar Cycles 15, 16, 17, 18, 19, 20, and 21, respectively (taken from
Pesnell, 2018). Here the data reduction and analysis are similar as in Dasi-
Espuig et al. (2010) and McClintock and Norton (2013). We excluded the data
correspond to the zero values of the tilt angles, and also that correspond to
the difference between the tilt angles of a sunspot group observed in two
consecutive days is $\geq 16^{\circ}$ day-1. We binned the daily tilt-angle
data during each of the Solar Cycles 15 – 21 into seven different
$5^{\circ}$-latitude (absolute) intervals $0^{\circ}$ – $5^{\circ}$,
$5^{\circ}$ – $10^{\circ}$, $10^{\circ}$ – $15^{\circ}$, $15^{\circ}$ –
$20^{\circ}$, $20^{\circ}$ – $25^{\circ}$, $25^{\circ}$ – $30^{\circ}$, and
$30^{\circ}$ – $35^{\circ}$. We calculated the mean value, [$\bar{\gamma}$],
of the tilt angles, and the corresponding standard error in each latitude bin.
We fitted these data to Joy’s law in the form:

$\bar{\gamma}=m|\lambda|+c,$ $None$

where $\lambda$ is the mid-value of a latitude bin (in the case of area-
weighted tilt angle $\bar{\gamma}_{\rm aw}$ is used instead of
$\bar{\gamma}$). Here the slope $m$ is referred to as the coefficient of Joy’s
law. The calculations of linear-least-square fits were done by taking into
account the uncertainties in both the abscissa [$|\lambda|$] and the ordinate
[$\bar{\gamma}$], namely the standard error in the case of $\bar{\gamma}$ and
the value $2.5^{\circ}$, i.e. half of the range of a latitude interval, in the
case of $|\lambda|$. The calculations were done by using the tilt-angle data
and also the area-weighted tilt-angle data of whole sphere, and separately for
northern and southern hemispheres’ tilt-angle data. The area-weighted mean
tilt angle, $\sum(\gamma_{i}\times A_{i})/\sum A_{i}=X/Y$, where $X$ is the
mean of ($\gamma_{i}\times A_{i}$) and $Y$ is the mean of $A_{i}$,
$i=1$,…,$n$, and $n$ is the number of data points in a given latitude bin. If
$\Delta(X)$ and $\Delta(Y)$ are the uncertainties in $X$ and $Y$,
respectively, then the uncertainty in the ratio $X/Y$ will be approximately
equal to ($Y\times\Delta(X)-X\times\Delta(Y)$)/$Y^{2}$ (see also Javaraiah and
Gokhale, 1997). Note that here $\Delta(X)$ and $\Delta(Y)$ represent the
standard errors of $X$ and $Y$, respectively. The coefficients of Joy’s law
over the whole sphere and in northern and southern hemispheres were determined
from the combined data of all cycles (15 – 21), and from the data of each
individual cycle. The MWOB data of Solar Cycle 15 are incomplete (note that
the beginning of this solar cycle was 1913). In the case of this solar cycle,
the number of data points is found to be very few in the $30^{\circ}$ –
$35^{\circ}$ latitude bins of the northern hemisphere (zero in both MWOB and
KOB data) and the southern hemisphere (6 and 2 in MWOB and KOB data,
respectively). Hence, the data in this latitude bin are not considered in the
linear-least-square fits of this solar cycle’s data in all three cases: whole
sphere, northern hemisphere, and southern hemisphere. We determined
correlation between the slope $m$ and amplitude $R_{\rm M}$ during Solar
Cycles 15 – 21. We also determined the corresponding linear regression. The
separate northern and southern hemispheres’ sunspot-number data are not
available for Solar Cycles 15 – 21. Hence, we compared the northern and
southern hemispheres’ values of the slope of Joy’s law and the corresponding
north–south asymmetry, with the amplitude $R_{\rm M}$ (maximum total sunspot
number) of the solar cycle. All the linear regression analyses presented in
this article are done by using the Interactive Digital Library (IDL) software
FITEXY.PRO, which is downloaded from the website
idlastro.gsfc.nasa.gov/ftp/pro/math/. This software is very useful because the
uncertainties of both the abscissa and ordinate will be taken care in the
calculations of linear-least-square fit.

Table 1.: Values of mean tilt angle [$\bar{\gamma}$, in deg.] and of mean
area-weighted tilt angle [$\bar{\gamma}_{\rm aw}$, in deg.] of sunspot groups
in absolute latitude intervals $0^{\circ}$ – $5^{\circ}$, $5^{\circ}$ –
$10^{\circ}$, $10^{\circ}$ – $15^{\circ}$, $15^{\circ}$ – $20^{\circ}$,
$20^{\circ}$ – $25^{\circ}$, $25^{\circ}$ – $30^{\circ}$, and $30^{\circ}$ –
$35^{\circ}$ ($|\lambda|$ represents the mid-value of an absolute latitude
interval) determined from the whole-sphere data, and separately from the
northern and southern hemispheres’ data, during the entire period. $n$
represents the number of data points in each latitude interval. The
uncertainties in $\bar{\gamma}$ and $\bar{\gamma}_{\rm aw}$ are the
corresponding standard errors.

| Whole sphere | North hemisphere | South hemisphere
---|---|---|---
$|\lambda|$ | $\bar{\gamma}$ | $\bar{\gamma}_{\rm{aw}}$ | $n$ | $\bar{\gamma}$ | $\bar{\gamma}_{\rm{aw}}$ | $n$ | $\bar{\gamma}$ | $\bar{\gamma}_{\rm{aw}}$ | $n$
|  |  | Derived from Mt. Wilson data |  |  |
2.5 | $1.44\pm 0.76$ | $3.32\pm 0.82$ | 1670 | $0.79\pm 1.08$ | $4.34\pm 1.34$ | 820 | $2.06\pm 1.07$ | $2.41\pm 0.98$ | 850
7.5 | $2.43\pm 0.39$ | $3.28\pm 0.43$ | 5914 | $3.18\pm 0.56$ | $3.78\pm 0.60$ | 2845 | $1.73\pm 0.54$ | $2.80\pm 0.62$ | 3069
12.5 | $4.39\pm 0.34$ | $5.08\pm 0.35$ | 7719 | $4.40\pm 0.46$ | $5.21\pm 0.48$ | 4108 | $4.37\pm 0.50$ | $4.93\pm 0.51$ | 3611
17.5 | $4.84\pm 0.38$ | $5.01\pm 0.42$ | 6461 | $5.55\pm 0.53$ | $5.75\pm 0.61$ | 3310 | $4.10\pm 0.54$ | $4.18\pm 0.58$ | 3151
22.5 | $5.45\pm 0.48$ | $6.31\pm 0.51$ | 3881 | $6.11\pm 0.64$ | $7.69\pm 0.64$ | 2190 | $4.60\pm 0.73$ | $4.55\pm 0.81$ | 1691
27.5 | $6.56\pm 0.76$ | $7.65\pm 0.80$ | 1764 | $7.61\pm 1.02$ | $9.58\pm 1.04$ | 1051 | $5.02\pm 1.14$ | $4.36\pm 1.23$ | 713
32.5 | $6.72\pm 1.41$ | $7.46\pm 1.57$ | 544 | $7.03\pm 1.81$ | $8.68\pm 1.98$ | 318 | $6.28\pm 2.25$ | $5.38\pm 2.57$ | 226
|  |  | Derived from Kodaikanal data |  |  |
2.5 | $2.19\pm 0.74$ | $3.71\pm 0.91$ | 1583 | $1.95\pm 1.01$ | $4.84\pm 1.36$ | 842 | $2.47\pm 1.09$ | $2.23\pm 1.14$ | 741
7.5 | $2.94\pm 0.40$ | $3.81\pm 0.44$ | 5732 | $3.65\pm 0.57$ | $5.41\pm 0.62$ | 2936 | $2.20\pm 0.56$ | $1.90\pm 0.60$ | 2796
12.5 | $4.56\pm 0.33$ | $6.02\pm 0.36$ | 7857 | $5.38\pm 0.45$ | $6.25\pm 0.49$ | 4111 | $3.67\pm 0.48$ | $5.72\pm 0.53$ | 3746
17.5 | $4.74\pm 0.39$ | $4.88\pm 0.41$ | 6480 | $5.31\pm 0.53$ | $5.67\pm 0.57$ | 3226 | $4.18\pm 0.57$ | $4.01\pm 0.60$ | 3254
22.5 | $5.31\pm 0.48$ | $5.40\pm 0.55$ | 4180 | $6.38\pm 0.62$ | $6.29\pm 0.69$ | 2418 | $3.85\pm 0.76$ | $4.04\pm 0.92$ | 1762
27.5 | $7.38\pm 0.73$ | $9.37\pm 0.87$ | 1866 | $7.12\pm 0.99$ | $10.20\pm 1.30$ | 1010 | $7.69\pm 1.09$ | $8.36\pm 1.09$ | 856
32.5 | $4.56\pm 1.38$ | $6.91\pm 1.82$ | 618 | $6.68\pm 1.70$ | $8.71\pm 2.46$ | 346 | $1.85\pm 2.28$ | $4.28\pm 2.65$ | 272

table1

##  3 Results

###  3.1 Recovering Joy’s law From the Combined Data of All Solar Cycles

In Table 1 we have given the values of the mean tilt angle [$\bar{\gamma}$]
and of the mean area-weighted tilt angle [$\bar{\gamma}_{\rm aw}$] of sunspot
groups in each absolute latitude intervals $0^{\circ}$ – $5^{\circ}$,
$5^{\circ}$ – $10^{\circ}$, $10^{\circ}$ – $15^{\circ}$, $15^{\circ}$ –
$20^{\circ}$, $20^{\circ}$ – $25^{\circ}$, $25^{\circ}$ – $30^{\circ}$, and
$30^{\circ}$ – $35^{\circ}$ ($|\lambda|$ represents the mid-value of an
absolute latitude interval) separately determined from the data of sunspot
groups in northern and southern hemispheres and from the combined data (whole
sphere data) during all solar cycles (15 – 21), i.e. data of sunspot groups
measured in MWOB during the period 1917 – 1986 and in KOB during the period
1906 – 1986. The uncertainties in $\bar{\gamma}$ and $\bar{\gamma}_{\rm aw}$
are the corresponding standard errors and $n$ represents the number of data
points in a latitude interval. As we can see in this table the values of both
$\bar{\gamma}$ and $\bar{\gamma}_{\rm aw}$ are reasonably accurate (the ratio
of mean value to standard error is considerably large) except, particularly in
the case of KOB data, at $|\lambda|=32.5^{\circ}$ (i.e. in $30^{\circ}$ –
$35^{\circ}$ latitude interval) of southern hemisphere. Figure 1 shows the
Joy’s law derived from the data given in Table 1 for the whole sphere and
Figures 2 and 3 are the same as Figure 1, but the Joy’s laws are derived from
the northern and southern hemispheres’ data, respectively. In all these
figures, the values of the correlation coefficient [$r$], rms (root-mean-
square deviation), $\chi^{2}$, and the corresponding probability [$P$] are
given, and in Table 2 the details on the best-fit linear equations are given.
As we can see in these figures and in Table 2, all the linear relations (Joy’s
laws) derived from particularly MWOB data are good, i.e. the values of $r$ are
sufficiently high, the values of $\chi^{2}$ are significantly low, and the
ratios of the slopes to the corresponding standard deviations are
substantially high. Overall, the relations derived from KOB data are
relatively less reliable, especially, the relations correspond to both the
tilt-angle and the area-weighted tilt-angle data of the southern hemisphere
are statistically insignificant (see Figures 3c and 3d).

As we can see in Table 2, there exist some differences in the values of the
slopes of the linear equations obtained from MWOB and KOB data. These
differences are statistically significant in the case of area-weighted tilt-
angle data of both the northern and southern hemispheres. The values of the
slopes determined from the MWOB and KOB whole-sphere’s tilt-angle data well
match within their uncertainties limits with the value 0.2 that was obtained
by Norton and Gilman (2005) by using tilt angles of 650 active regions
observed in Michelson Doppler Imager data during the period 1996 – 2004. Both
MWOB and KOB whole sphere’s values of the slopes (determined from both the
tilt-angle and area-weighted tilt-angle data) are slightly lower than the
corresponding values $0.26\pm 0.05$ and $0.28\pm 0.06$ that were obtained from
the area-weighted tilt-angle data by Dasi-Espuig et al. (2010) by forcing the
linear fit through the origin. The values of northern and southern
hemispheres’ slopes determined from MWOB area-weighted tilt-angle data
reasonably match the corresponding values 0.26 and 0.13 determined from the
same data by McClintock and Norton (2013). In the case of MWOB area-weighted
tilt-angle data, the slope of northern hemisphere is significantly (more than
95 % confidence level) larger ($\approx$130 %) than that of the southern
hemisphere. In the case of KOB data, the slope corresponding to the area-
weighted tilt-angle data of the southern hemisphere looks to be larger than
that of the northern hemisphere, but as already mentioned above, the relations
obtained from the KOB southern hemisphere’s data are unreliable. Hence, in
this case it is not possible to draw any conclusion on the north–south
difference in the slope.

Figure 1.: Mean tilt angle [$\bar{\gamma}$] and mean area-weighted tilt angle
[$\bar{\gamma}_{\rm aw}$] versus mid-value of the corresponding $5^{\circ}$
latitude bin, determined from Mt. Wilson Observtory (MWOB) and Kodaikanal
Observatory (KOB) whole-sphere sunspot-group data. The continuous line
represents the corresponding linear-least-square best fit and the dotted line
(red) represents one-rms level. The values of the correlation coefficient
[$r$], rms, and $\chi^{2}$ and the corresponding probability [$P$] are also
given.f1

Figure 2.: Mean tilt angle [$\bar{\gamma}$] and mean area-weighted tilt angle
[$\bar{\gamma}_{\rm aw}$] versus mid-value of the corresponding $5^{\circ}$
latitude bin, determined from Mt. Wilson Observtory (MWOB) and Kodaikanal
Observatory (KOB) northern hemisphere sunspot-group data. The continuous line
represents the corresponding linear-least-square best fit and the dotted line
(red) represents one-rms level. The values of the correlation coefficient
[$r$], rms, and $\chi^{2}$ and the corresponding probability [$P$] are also
given.f2

Figure 3.: Mean tilt angle [$\bar{\gamma}$] and mean area-weighted tilt angle
[$\bar{\gamma}_{\rm aw}$] versus mid-value of the corresponding $5^{\circ}$
latitude bin, determined from Mt. Wilson Observatory (MWOB) and Kodaikanal
Observatory (KOB) southern hemisphere sunspot-group data. The continuous line
represents the corresponding linear-least-square best fit and the dotted line
(red) represents one-rms level. The values of the correlation coefficient
[$r$], rms, and $\chi^{2}$ and the corresponding probability [$P$] are also
given.f3 Table 2.: Values of the intercept [$c$] and the slope [$m$], and the
corresponding standard deviations [$\sigma_{c}$ and $\sigma_{m}$],
respectively, of the best-fit linear relationships derived from the mean
values of tilt angles and area-weighted tilt angles of sunspot groups in
different $5^{\circ}$-latitude intervals of whole-sphere, and northern and
southern hemispheres, by using the entire periods MWOB and KOB sunspot-group
data. The values of correlation coefficient [$r$], Student’s t [$\tau$] and
the corresponding probability [$P$], and the ratio $m/\sigma_{m}$ are also
given.

$Data$ | $c$ | $\sigma_{c}$ | $m$ | $\sigma_{m}$ | $r$ | $\tau$ | $P$ | $m/\sigma_{m}$
---|---|---|---|---|---|---|---|---
|  |  | Derived from tilt angles |  |
MWOB Whole | 1.26 | 0.66 | 0.20 | 0.03 | 0.97 | 8.92 | 0.0001 | 6.67
MWOB North | 1.31 | 0.86 | 0.22 | 0.05 | 0.95 | 6.80 | 0.0005 | 4.40
MWOB South | 1.23 | 0.83 | 0.17 | 0.04 | 0.93 | 5.66 | 0.0012 | 4.25
KOB Whole | 1.97 | 0.61 | 0.16 | 0.03 | 0.77 | 2.70 | 0.0214 | 5.33
KOB North | 2.41 | 0.79 | 0.18 | 0.04 | 0.93 | 5.66 | 0.0012 | 4.50
KOB South | 1.47 | 0.83 | 0.15 | 0.04 | 0.36 | 0.86 | 0.2138 | 3.75
|  |  | Derived from area-weighted tilt angles |  |
MWOB Whole | 2.34 | 0.67 | 0.18 | 0.04 | 0.97 | 8.92 | 0.0001 | 4.50
MWOB North | 2.32 | 0.89 | 0.23 | 0.05 | 0.94 | 6.16 | 0.0008 | 4.60
MWOB South | 2.57 | 0.90 | 0.10 | 0.05 | 0.82 | 3.20 | 0.0119 | 2.00
KOB Whole | 2.39 | 0.81 | 0.20 | 0.04 | 0.79 | 2.88 | 0.0173 | 5.00
KOB North | 4.15 | 1.03 | 0.13 | 0.06 | 0.84 | 3.46 | 0.0090 | 2.17
KOB South | 0.78 | 0.99 | 0.24 | 0.05 | 0.61 | 1.72 | 0.0729 | 4.80

table2

###  3.2 Recovering Joy’s law From the Data of Individual Solar Cycles

In this section we find no significant results from KOB data. Here we present
only the results derived from MWOB data. We do not know the exact reason for
the discrepancies in the results derived from KOB and MOB data, but during
some solar cycles the KOB data seem to be more inconsistent due to a large
missing observations (Ravindra, Chowdhury, and Javaraiah, 2021).

In Table 3 we have given the values of the intercept [$c$], slope [$m$], and
the corresponding standard deviation [$\sigma$] of the best-fit linear
relationships derived from the mean values of tilt angles and area-weighted
tilt angles of sunspot groups in different $5^{\circ}$ absolute latitude
intervals of the whole sphere during each solar cycle. The values of the
correlation coefficient $r$, $\chi^{2}$, and the corresponding probability
[$P$], and the ratio $m/\sigma_{m}$ are also given in this table. The
corresponding results determined from northern and southern hemispheres’ data
are given in Tables 4 and 5. The best-fit linear relations obtained from the
data of some cycles are found to be reasonably good (the value of $m$ is
statistically significant, i.e. the ratio $m/\sigma_{m}$ is $\geq 2$) and the
values of $\chi^{2}$ are reasonably small ($P$ is much larger than 0.05) and
the fits of some other cycles are found to be not good.

Figure 4 shows cycle-to-cycle variations in different parameters: whole
sphere’s slope [$m_{\rm W}$], northern hemisphere’s slope [$m_{\rm N}$],
southern hemisphere’s slope [$m_{\rm S}$], and north–south asymmetry in slope
[$m_{\rm N}-m_{\rm S}$], determined from the MWOB tilt-angle data and the
area-weighted tilt-angle data. For the sake of comparison in this figure, the
variation in the amplitude [$R_{\rm M}$] of the solar cycle is also shown.
Note that $m_{\rm W}$, $m_{\rm N}$, and $m_{\rm S}$ represent the values of
the slope $m$ of whole sphere, northern hemisphere, and southern hemisphere
that are given in Tables 3, 4, and 5, respectively. The uncertainty in the
north–south asymmetry $m_{\rm N}-m_{\rm S}$ is the square-root of the ratio of
the sum of the squares of the standard deviations of $m_{\rm N}$ and $m_{\rm
S}$ to the number of data points (number of absolute latitude bins: 7). As we
can see in this figure, the patterns of $m_{\rm W}$ and $m_{\rm N}$ are
similar and both seem to have no significant correlation with the amplitude of
the solar cycle. It looks to be there exists an anti-correlation between
$m_{\rm N}$ and $m_{\rm S}$ (mainly in Figure 4a). There is a suggestion of a
strong anti-correlation between $m_{\rm S}$ and the amplitude of solar cycle,
and a strong correlation between the north–south asymmetry in the slope and
the amplitude of the solar cycle (mainly in Figure 4b).

Figure 5 shows the relationship between slope $m_{\rm W}$ (the values of $m$
given Table 3) of Joy’s law derived from the whole-sphere area-weighted tilt-
angle data of a solar cycle and the amplitude [$R_{\rm M}$] of the solar
cycle. The correlations are determined with and without the data point of
Solar Cycle 15 because it has a high value and is an outlier (more outside the
one-rms level, see Figure 5a). As can be seen in this figure there exists a
small and insignificant anti-correlation and the corresponding best-fit linear
relations are not good. That is, the slopes of the best-fit linear equations
are found to be only 1 – 1.4 times the corresponding standard deviations,
although the values of $\chi^{2}$ are smaller than that of the 5% significant
level. In the case of tilt-angle data (without area weighting) the
correlations found to be much smaller (hence not shown). Overall, these
results suggest that there exists only a weak linear relationship between $m$
and $R_{\rm M}$ of a solar cycle in the case of the whole sphere data.
However, Jiao, Jiang, and Wang (2021) have found the existence of a
significant anti-correlation between the coefficient of Joy’s law and the
strength of the solar cycle by using the area-weighted tilt-angle data of the
sunspot groups with polarity angular separation $>2.5^{\circ}$.

Figure 6 shows the relationship between $m_{\rm N}$ (the values of $m$ given
in Table 4) obtained from tilt-angle data, and also that obtained from area-
weighted tilt-angle data, in the northern hemisphere during a solar cycle and
$R_{\rm M}$ of the solar cycle. As can be seen in this figure, there exist no
significant correlations and the corresponding best-fit linear relations are
not good. That is, the value of $r$ is statistically insignificant. The rms is
large, but it could be mainly due to the data point of Solar Cycle 15, which
is an outlier. In the case of tilt-angle data, the $\chi^{2}$ is significant
on more than 95% confidence level (probability $P$ is smaller than 0.05). In
the case of the area-weighted tilt-angle data, the value of $\chi^{2}$ is
reasonably small but the ratio $m/\sigma_{m}$ is found to be significantly
small. Overall, there exists no significant linear relationship between
$m_{\rm N}$ and $R_{\rm M}$ of a solar cycle determined from the northern
hemisphere’s data. (The values of the slopes are positive, whereas the values
of $r$ are negative, see Figure 6. This discrepancy is due to in the linear-
least-square fit calculations uncertainties in abscissae and ordinates were
taken into account, whereas in the calculations of $r$ the uncertainties are
not taken into account).

Figure 7 is the same as Figure 6, but determined from the southern
hemisphere’s data (the values of $m$ given in Table 5). As we can see in this
figure $m_{\rm S}$ is correlated with $R_{\rm M}$ reasonably well in both the
cases of the tilt-angle data and the area-weighted tilt-angle data, and we
obtained the following linear relations.

In the case of tilt-angle data:

$m_{\rm S}=0.61\pm 0.17-(0.002\pm 0.00085)R_{\rm M},\ {\rm and}$ $None$

in the case of area weighted tilt-angle data:

$m_{\rm S}=0.84\pm 0.21-(0.0035\pm 0.001)R_{\rm M}.$ $None$

These best-fit linear Equations 2 and 3 are reasonably good, i.e. the slopes
are about 2.3 and 3.5 times larger than the corresponding standard deviations,
respectively. The $P$-values of the corresponding correlations are 0.004 and
0.01, i.e. the correlations are significant on more than 95% confidence level.
The corresponding $\chi^{2}$-values are reasonably small, i.e. much lower than
the value (12.592) of the 5% level of significance. Moreover, only one data
point is only slightly outside one-rms level. Overall, there exists a
significant linear relationship between the slope [$m_{\rm S}$] determined
from the southern hemisphere’s data and $R_{\rm M}$.

Figure 8 shows the relationship between north–south asymmetry in the slope
[$m_{\rm N}-m_{\rm S}$] of a solar cycle and the amplitude [$R_{\rm M}$] of
the solar cycle, for both the cases of tilt angles and area-weighted tilt
angles (see also Figure 4). We obtained the following linear relations.

In the case of tilt angles:

$m_{\rm N}-m_{\rm S}=-0.60\pm 0.11+(0.0031\pm 0.0005)R_{\rm M},\ {\rm and}$
$None$

in the case of area-weighted tilt angles:

$m_{\rm N}-m_{\rm S}=-0.63\pm 0.11+(0.0036\pm 0.0005)R_{\rm M}.$ $None$

In the case of Equation 4, the corresponding correlation is insignificant and
$\chi^{2}$ is very large mainly because the data point of Solar Cycle 15 is
far away from the one-rms level. However, the slope is 6.2 times larger than
the corresponding standard deviation, suggesting a possibility of the
existence of the linear relation between $m_{\rm N}-m_{\rm S}$ and $R_{\rm M}$
of a solar cycle. In the case of Equation 5 the correlation is highly
significant (Student’s t is 5.9 and the corresponding $P=0.001$), the
$\chi^{2}$ is considerably small (the corresponding $P=0.61$), only one data
point is only slightly out of the one-rms, and moreover the slope is 7.2 times
larger than the corresponding standard deviation. Overall, we find that there
exists a good linear relationship between the north–south difference in the
slope and the amplitude [$R_{\rm M}$] of a solar cycle.

Table 3.: Whole sphere: Values of the intercept [$c$] and the slope [$m$], and
the corresponding standard deviations [$\sigma_{c}$ and $\sigma_{m}$],
respectively, of the best-fit linear relationships derived from the mean
values of tilt angles and area-weighted tilt angles of sunspot groups in
different $5^{\circ}$-latitude intervals of the whole-sphere during each solar
cycle (SC). The values of correlation coefficient [$r$], $\chi^{2}$ and the
corresponding probability [$P$], the ratio $m/\sigma_{m}$, and the rms are
also given. a indicates incomplete data.

SC | $c$ | $\sigma_{c}$ | $m$ | $\sigma_{m}$ | $r$ | $\chi^{2}$ | $P$ | $m/\sigma_{m}$ | rms
---|---|---|---|---|---|---|---|---|---
|  |  | Derived from tilt angles |  |  |
15a | $-3.29$ | 2.41 | 0.55 | 0.15 | 0.82 | 5.38 | 0.25 | 3.67 | 2.93
16 | 0.69 | 1.84 | 0.25 | 0.10 | 0.87 | 5.89 | 0.32 | 2.50 | 1.63
17 | 2.29 | 1.48 | 0.18 | 0.08 | 0.53 | 4.58 | 0.47 | 2.25 | 1.62
18 | 1.79 | 1.32 | 0.18 | 0.07 | 0.21 | 8.94 | 0.11 | 2.57 | 2.90
19 | 0.21 | 1.20 | 0.20 | 0.06 | 0.95 | 3.19 | 0.67 | 3.33 | 0.75
20 | 1.50 | 1.38 | 0.18 | 0.07 | 0.81 | 4.00 | 0.55 | 2.57 | 1.21
21 | 1.47 | 1.45 | 0.22 | 0.08 | 0.95 | 1.45 | 0.92 | 2.75 | 0.72
|  |  | Derived from area-weighted tilt angles |  |  |
15a | -1.36 | 2.57 | 0.53 | 0.16 | 0.81 | 7.50 | 0.11 | 3.31 | 2.79
16 | 2.65 | 1.74 | 0.17 | 0.09 | 0.82 | 4.05 | 0.54 | 1.89 | 2.02
17 | 2.39 | 1.55 | 0.24 | 0.08 | 0.55 | 9.21 | 0.10 | 3.00 | 2.60
18 | 3.13 | 1.49 | 0.16 | 0.08 | 0.01 | 10.68 | 0.06 | 2.00 | 3.51
19 | 1.35 | 1.18 | 0.15 | 0.06 | 0.84 | 6.46 | 0.26 | 2.50 | 0.98
20 | 0.18 | 1.55 | 0.32 | 0.08 | 0.92 | 2.69 | 0.75 | 4.00 | 1.24
21 | 3.00 | 1.58 | 0.16 | 0.08 | 0.54 | 6.86 | 0.23 | 2.00 | 2.21

table3

Table 4.: Northern hemisphere: Values of the intercept [$c$] and the slope
[$m$], and the corresponding standard deviations [$\sigma_{c}$ and
$\sigma_{m}$], respectively, of the best-fit linear relationships derived from
the mean values of tilt angles and area-weighted tilt angles of sunspot groups
in different $5^{\circ}$-latitude intervals of the northern hemisphere during
each solar cycle (SC). The values of correlation coefficient [$r$], $\chi^{2}$
and the corresponding probability [$P$], the ratio $m/\sigma_{m}$, and the rms
are also given. a indicates incomplete data.

SC | $c$ | $\sigma_{c}$ | $m$ | $\sigma_{m}$ | $r$ | $\chi^{2}$ | $P$ | $m/\sigma_{m}$ | rms
---|---|---|---|---|---|---|---|---|---
|  |  | Derived from tilt angles |  |  |
15a | $-5.70$ | 3.45 | 0.79 | 0.21 | 0.91 | 4.64 | 0.33 | 3.76 | 2.98
16 | 1.43 | 2.24 | 0.18 | 0.12 | 0.78 | 3.27 | 0.66 | 1.50 | 2.12
17 | 5.00 | 2.05 | 0.05 | 0.11 | 0.03 | 5.60 | 0.35 | 0.45 | 2.05
18 | 3.93 | 1.86 | 0.14 | 0.10 | 0.31 | 6.15 | 0.29 | 1.40 | 3.50
19 | 0.19 | 1.47 | 0.27 | 0.08 | 0.96 | 2.09 | 0.84 | 3.38 | 0.88
20 | 1.24 | 1.82 | 0.15 | 0.10 | $-0.13$ | 2.92 | 0.71 | 1.50 | 2.94
21 | 0.13 | 2.04 | 0.34 | 0.11 | 0.69 | 6.88 | 0.23 | 3.09 | 2.76
|  |  | Derived from area weighted-tilt angles |  |  |
15a | $-0.86$ | 3.09 | 0.49 | 0.19 | 0.82 | 6.52 | 0.16 | 2.58 | 2.72
16 | 2.20 | 2.14 | 0.15 | 0.11 | 0.48 | 6.08 | 0.30 | 1.36 | 3.23
17 | 3.68 | 2.31 | 0.25 | 0.12 | 0.14 | 10.20 | 0.07 | 2.08 | 6.92
18 | 4.84 | 1.91 | 0.13 | 0.10 | 0.06 | 9.34 | 0.10 | 1.30 | 4.30
19 | 1.24 | 1.37 | 0.25 | 0.07 | 0.94 | 1.87 | 0.87 | 3.57 | 0.88
20 | 0.96 | 1.95 | 0.25 | 0.10 | 0.14 | 2.64 | 0.75 | 2.50 | 4.23
21 | 2.57 | 2.19 | 0.25 | 0.12 | 0.20 | 12.60 | 0.03 | 2.08 | 3.68

table4

Table 5.: Southern hemisphere: Values of the intercept [$c$] and the slope
[$m$], and the corresponding standard deviations [$\sigma_{c}$ and
$\sigma_{m}$], respectively, of the best-fit linear relationships derived from
the mean values of tilt angles and area-weighted tilt angles of sunspot groups
in different $5^{\circ}$-latitude intervals of the southern hemisphere during
each solar cycle (SC). The values of the correlation coefficient [$r$],
$\chi^{2}$ and the corresponding probability [$P$], the ratio $m/\sigma_{m}$,
and the rms are also given. a indicates incomplete data.

SC | $c$ | $\sigma_{c}$ | $m$ | $\sigma_{m}$ | $r$ | $\chi^{2}$ | $P$ | $m/\sigma_{m}$ | rms
---|---|---|---|---|---|---|---|---|---
|  |  | Derived from tilt angles |  |  |
15a | $-0.33$ | 3.06 | 0.31 | 0.19 | 0.38 | 3.67 | 0.45 | 1.63 | 2.88
16 | 0.09 | 2.70 | 0.31 | 0.14 | 0.43 | 8.00 | 0.16 | 2.21 | 4.57
17 | $-0.18$ | 2.00 | 0.30 | 0.10 | 0.57 | 4.77 | 0.44 | 3.00 | 2.61
18 | 0.87 | 1.77 | 0.15 | 0.09 | $-0.03$ | 10.60 | 0.06 | 1.67 | 3.08
19 | 1.62 | 1.51 | 0.02 | 0.08 | 0.54 | 1.80 | 0.88 | 0.25 | 1.10
20 | 1.57 | 2.04 | 0.24 | 0.11 | 0.64 | 6.40 | 0.27 | 2.18 | 2.57
21 | 2.25 | 2.04 | 0.15 | 0.11 | 0.65 | 5.72 | 0.33 | 1.36 | 1.97
|  |  | Derived from area weighted-tilt angles |  |  |
15a | 0.73 | 3.81 | 0.38 | 0.24 | 0.69 | 4.54 | 0.34 | 1.58 | 2.76
16 | 2.36 | 2.67 | 0.25 | 0.14 | 0.34 | 3.84 | 0.57 | 1.79 | 3.59
17 | 1.11 | 1.79 | 0.21 | 0.09 | 0.32 | 4.04 | 0.54 | 2.33 | 2.69
18 | 4.18 | 1.87 | $-0.03$ | 0.10 | $-0.28$ | 11.47 | 0.04 | $-0.30$ | 2.82
19 | 3.82 | 1.94 | $-0.14$ | 0.10 | $-0.56$ | 5.33 | 0.38 | $-1.40$ | 1.39
20 | $-0.92$ | 2.21 | 0.40 | 0.11 | 0.87 | 4.90 | 0.43 | 3.64 | 2.02
21 | 3.64 | 2.57 | 0.09 | 0.14 | 0.24 | 15.14 | 0.01 | 0.64 | 5.06

table5

Figure 4.: Cycle-to-cycle variations in different parameters: whole sphere’s
slope $m_{\rm W}$ (yellow filled square), northern hemisphere’s slope $m_{\rm
N}$ (black filled circle), southern hemisphere’s slope $m_{\rm S}$ (blue
filled triangle), and north–south asymmetry in slope $m_{\rm N}-m_{\rm S}$
(red cross). (a) Determined from MWOB tilt-angle data and (b) determined from
MWOB area-weighted tilt-angle data. Note that the data of Solar Cycle 15 are
incomplete. The variation in $R_{\rm M}$ (green filled star) is also shown.f4

Figure 5.: The slope [$m_{\rm W}$] of Joy’s law derived from the whole sphere
area-weighted tilt-angle data in $5^{\circ}$ absolute latitude bins versus
amplitude [$R_{\rm M}$] of solar cycle, (a) for all Cycles 15 – 21 and (b) for
Cycles 16 – 21. The continuous line represents the corresponding linear-least-
square best fit and the dotted line (red) represents one-rms level. The values
of the correlation coefficient [$r$], rms, and $\chi^{2}$ and the
corresponding probability [$P$] are also given.f5

Figure 6.: The slope [$m_{\rm N}$] of Joy’s law in northern hemisphere versus
amplitude [$R_{\rm M}$] of solar cycle derived from the average values of (a)
tilt angles, and (b) area-weighted tilt angles, in $5^{\circ}$ latitude bins.
The continuous line represents the corresponding linear-least-square best fit
and the dotted line (red) represents one-rms level. The values of the
correlation coefficient [$r$], rms, and $\chi^{2}$ and the corresponding
probability [$P$] are also given.f6

Figure 7.: The slope [$m_{\rm S}$] of Joy’s law in southern hemisphere versus
amplitude [$R_{\rm M}$] of solar cycle derived from the average values of (a)
tilt angles, and (b) area-weighted tilt angles, in $5^{\circ}$ latitude bins.
The continuous line represents the corresponding linear-least-square best fit
and the dotted line (red) represents one-rms level. The values of the
correlation coefficient [$r$], rms, and $\chi^{2}$ and the corresponding
probability [$P$] are also given.f7

Figure 8.: The hemispheric difference [$m_{\rm N}-m_{\rm S}$] in the slopes of
Joy’s law (see also Figure 4), derived from (a) tilt angles and (b) area-
weighted tilt angles shown in Figures 6 and 7, versus amplitude [$R_{\rm M}$]
of solar cycle. The continuous line represents the corresponding linear-least-
square best fit and the dotted line (red) represents one-rms level. The values
of the correlation coefficient [$r$], rms, and $\chi^{2}$ and the
corresponding probability [$P$] are also given.f8

##  4 Discussion and Conclusions

Study of tilt angles of solar bipolar magnetic regions is important because
the tilt angles have an important role in the solar dynamo. We analysed the
data on tilt angles of sunspot groups measured at MWOB during the period 1917
– 1986 and at KOB during the period 1906–1986. We have used the amplitudes
(the values of $R_{\rm M}$) of Solar Cycles 15 – 21. We binned the daily tilt-
angle data during each of the Solar Cycles 15 – 21 into different
$5^{\circ}$-latitude intervals and calculated the mean value of the tilt
angles in each latitude interval and the corresponding standard error. We
fitted these binned data to Joy’s law, i.e. the linear relationship between
tilt angle and latitude of an active region. The linear-least-square fit
calculations were done by taking into account the uncertainties in both the
abscissa (latitude) and ordinate (mean tilt angle). The calculations were
carried out by using both the tilt-angle and area-weighted tilt-angle data on
the whole sphere, northern hemisphere, and southern hemisphere during the
whole period and during each individual solar cycle. We find a significant
difference (north–south asymmetry) between the slopes of Joy’s law recovered
from northern and southern hemispheres’ whole period MWOB data of area-
weighted tilt angles. The slope obtained from the whole sphere’s MWOB data of
a solar cycle is found to be weakly anti-correlated (statistically
insignificant) to the amplitude [$R_{\rm M}$] of the solar cycle. No
correlation is found between the slope obtained from the northern hemisphere’s
data and the amplitude of solar cycle, whereas the slope obtained from the
southern hemisphere’s data is found to be reasonably well anti-correlated with
the amplitude of solar cycle. In the case of area weighted tilt-angle data,
highly statistically significant correlation is found between the north–south
asymmetry in the slope of a solar cycle and the amplitude of the solar cycle.
The corresponding best-fit linear equations are found to be statistically
significant. These results are not found in KOB data.

McClintock and Norton (2013) analysed the Mt. Wilson sunspot-group data (1917
– 1986) and found that anti-correlation between
$\langle\gamma\rangle/\langle|\lambda|\rangle$ and strength of a solar cycle
is significant only in southern hemisphere. Here we analysed the same data and
find a significant anti-correlation between the slope and amplitude $R_{\rm
M}$ of solar cycle only from the southern hemisphere’s data. That is, the
behavior of slope in northern and southern hemispheres is closely similar to
that of $\langle\gamma\rangle/\langle|\lambda|\rangle$ found by McClintock and
Norton (2013). (Note that the slope $m$ represents the latitudinal gradient of
tilt angle, whereas $\langle\gamma\rangle/\langle|\lambda|\rangle$ is
considered as the latitude independent mean tilt angle.) Here we also find the
existence of a significant correlation between north–south difference in the
slope and $R_{\rm M}$ of a solar cycle. The reason behind why the correlation
between the slope and $R_{\rm M}$ is significant only in southern hemisphere
is not clear to us. We find no significant correlation between the slope (or
north–south difference $m_{\rm N}-m_{\rm S}$) and north–south difference in
the mean area of sunspot groups (taken from Javaraiah, 2022) at the epoch of
$R_{\rm M}$ of a solar cycle, indicating that no correlation exists between
the former and the north–south asymmetry in the amplitude of the solar cycle.

The $m_{\rm S}$ strongly varies and strongly anti-correlates with $R_{\rm M}$,
but $m_{\rm N}$ is almost constant (if we exclude the large value of Solar
Cycle 15, see Figures 4 and 6) and there exists no correlation between $m_{\rm
N}$ and $R_{\rm M}$. Hence, there exists only a weak anti-correlation between
$m_{\rm W}$ (derived from the combined northern and southern hemispheres’
data) and $R_{\rm M}$ and a good positive correlation exists between $m_{\rm
N}-m_{\rm S}$ and $R_{\rm M}$. These results convey that it is important to
determine the slope of the Joy’s law separately from the tilt-angle data of
the two hemispheres. However, the physical reason behind why $m_{\rm S}$
strongly anti-correlates with $R_{\rm M}$ and no correlation exists between
$m_{\rm N}$ and $R_{\rm M}$ is not known to us. According to Jiao, Jiang, and
Wang (2021) the Coriolis force involved in the formation of the tilt angle
depends on the local expansion rate of the rising flux tube and the local
rotation rate, hence the tilt angles of the two hemispheres could be
independent and uncoupled. As already mentioned in Section 1, there exist
north–south differences in the rotational and meridional motions of sunspot
groups and also there exist cycle-to-cycle variations in these differences
(e.g. Javaraiah and Ulrich, 2006). The differences in the cycle-to-cycle
variations in the rotational and meridional motions of sunspot groups in
northern and southern hemispheres may be responsible for the hemispheric
differences in the cycle-to-cycle variations in the coefficient of Joy’s law.
It needs to be investigated.

#### Acknowledgments

The author thanks the anonymous reviewer for useful comments and suggestions.
The author acknowledges the work of all the people who contribute to and
maintain the MWOB and KOB sunspot databases. We have used the maximum values
of Solar Cycles 15 – 21 determined by Pesnell (2018) from the time series of
13-month smoothed monthly mean values of version 2 of international sunspot
number (SN) available at www.sidc.be/silso/datafiles.

## Declarations

## Conflict of interest

The author declares that he has no conflicts of interest.

## Data Availability

All data generated or analysed during this study are included in this article.

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