#  Solar sail with inflatable toroidal shell

V. Ya. Kezerashvili1, R. Ya. Kezerashvili1,2, O. L. Starinova3 1Physics
Department, New York City College of Technology, The City University of New
York,
Brooklyn, NY 11201, USA
2The Graduate School and University Center, The City University of New York,
New York, NY 10016, USA
3Samara National Research University, Russian Federation, Samara, Russia

###### Abstract

In the framework of a strict mathematical approach based on classical theory
of elasticity we present an idea of the deployment and stretching of the
circular solar sail attached to the inflatable toroidal shell. It is predicted
that by introducing the gas into the inflatable toroidal shell one can deploy
and stretch a large size circular solar sail membrane. The formulas for the
toroidal shell and sail membrane stresses and strains caused by the gas
pressure in the shell are derived. The analytical expressions can be applied
to a wide range of solar sail sizes. Numerical calculations for the sail of
radii up to 100 m made of CP1 membrane and attached to the toroidal shell with
the varied cross-section radius are presented. The normal transverse vibration
modes of the sail membrane under tension caused by gas pressure in the shell
are calculated. The feasibility of deployment and stretching of a solar sail
with a large size circular membrane attached to the inflatable toroidal shell
is demonstrated.

Solar sail, Current current loop

##  I Introduction

Currently the main space exploration vehicle relies on chemical propulsion
system and most missions provide energy by means of prelaunch onboard fuel. A
solar sail is propellant-less propulsion system for space exploration that
uses the Sun radiation as a propulsion mechanism. A solar sail is a large
sheet of low areal density material that gains an acceleration due to the
reflection and absorption of the Sun electromagnetic flux pol ; Colin ;
Matloff3 ; MatloffValpetti . The proposals to use solar sails cover almost the
whole spectrum of space missions, from an expedition to Mars to scientific
probes, from continuous planetary polar observation to mining exploitation of
the asteroids, even implementations for deep space exploration and
interstellar travel are considered. The successful deployment of the world’s
first interplanetary solar sail IKAROS, NASA’s first solar NanoSail-D and the
Interplanetary Society solar sail LightSail-2 demonstrates the feasibility of
exploration of the Solar system by means of the electromagnetic radiation
pressure.

Studies on a solar sail have four primary foci: i. finding low areal density
material that allows to utilize the maximum acceleration due to the solar
radiation pressure; ii. missions design for exploration of the Solar System
and beyond using the solar sail; iii. maximizing the solar thrust through the
increase of area of the solar sail made of a low areal density material; iv.
the development of the mechanism for the deployment and stretching the large
size of solar sail membrane. In this work we concentrated on the last two
objectives.

The sail’s membrane deployment strategies has attracted considerable
attention. Many different systems have been previously considered for the
sails opening. The deployment is usually performed: by uniaxial mechanisms,
such as the telescopic, deployable and inflatable booms; the extendable masts;
the centrifugal force that renders a spin-type deployment mechanism; by the
presence of guide rollers; electromechanical actuation devices, or composite
booms; the solar sail self-deployment based on shape memory alloys, (see Refs.
Genta1999 ; 9 ; Deployment4 ; 8 ; 91 ; MemorySail ; MemorySail2 ; Deployment1
; Deployment2 ; Deployment3 ; Dep4 ; Dep5  and references therein) and most
recently the deployment of the large size solar sail was considered using the
superconducting current-carrying wire VKRK20221 ; VKRK20222 . We cite these
works, but the recent literature on the subject is not limited by them. It is
worth mentioning that in the actual deployment technology, the main limit is
still the high weight of the system and the complexity of the deployment
mechanism for solar sail surface.

The concept of using inflatable structures for a spacecraft has been
extensively discussed during the past six decades. In the 1960s NASA launched
the Echo balloons that were some of the first inflatable structures to be
utilized in space 5 . Once such a structure is inflated and deployed, it
operates by similar to rigid structures principles and allows to obtain
comparable or greater performance. The advantages of inflatable structures are
a simple deployment mechanism for large space structures, the small mass and
space savings in the launch configuration 4 .

Inflatable structures have the characteristics that are particularly
advantageous for a solar sail: i. extremely lightweight that is critical for a
solar sail; ii. a simple mechanism to deploy it in orbit.

The use of inflatable structures for solar sailing goes back to 1989 when
Strobl Strobl1  proposed a hydrogen inflated hollow disk-shaped sail with a
molybdenum reflector. Later the idea of the inflatable ring sail was presented
in Ref. Hayn . Inflatable structures including the solar sail have been the
subject of interest in the past and recent years and investigated in Refs.
Strobl2 ; Genta1999 ; Matloff1 ; Kezmetloff1 ; Kezmetloff2 ; 8 ; Leigh ;
Tinker ; KezASR2021 . Most recently a sail consisting of a reflective membrane
attached to an inflatable torus-shaped shell was suggested KezASR2021 . The
sail deployment from its stowed configuration is initiated by introduction of
the inflation pressure into the toroidal shell. However, the study KezASR2021
, as well as the previous investigations did not address important issues
related to the elastic properties of the sail membrane and torus-shaped shell.
The toroidal shell is kept pressurized by the gas after deployment, withholds
all the stresses, and allows a very simple straightforward deployment
procedure of a large area circular configuration.

In this work we consider the toroidal shell that is attached to a circular
solar sail membrane to aid the membrane deployment and stretching. The sail’s
deployment from its stowed configuration is initiated by introducing inflation
pressure into the toroidal shell. We provide the rigorous consideration of
this system within the framework of the classical theory of elasticity
Timoshenko51 ; Landau7 . An important question that arises in the context of
deployable toroidal-shaped solar sail is its mass and stability. We consider
statics of a deployed torus-shaped sail, and address the structural strength
of the sail membrane and inflated toroidal shell to support the flat surface
of a circular membrane. The governing equation for a gas in an inflated
toroidal shell is assumed to be the equation for a real gas. Within such
approach we investigate the effects of both the enclosed gas pressure and
structure stiffness and the stability of the solar sail. It is demonstrated
that the effect of the enclosed gas must be considered in the analysis of the
inflatable torus with the membrane.

This article is organized in the following way. In Sec. II we consider the
solar sail configuration and within the framework of the classical theory of
elasticity derive the forces that lead to the deployment of a circular sail
membrane attached to the toroidal shell. The stress and strain of the toroidal
shell, the circular membrane and the toroidal shell-membrane structure are
addressed. The vibration of the deployed and stretched membrane are considered
in Sec. III. Results of the numerical calculations and discussion are
presented in Sec. IV. The conclusions follow in Sec. V.

##  II Solar Sail as a Circular Membrane Attached to a Toroidal Shell

A schematic of the circular solar sail attached to a torus of radius $R$ and a
circular cross-section of radius $r$ inflated by a gas is presented in Fig. 1.
The radius of the toroidal shell is significantly greater than the radius
cross-section: $R\gg r$. The solar sail membrane is stretched by forces
generated by pressure of the gas introduced into the toroidal shell. In this
Section we consider the strain in a toroidal shell due to the gas pressure and
provide the detailed consideration of the stress and strain in the circular
membrane resulting from uniformly distributed force applied to the membrane
edge and caused by the gas pressure. Finally, the system of the toroidal shell
filled by the gas attached to the sail’s circular membrane is considered.

###  II.1 Toroidal shell filled with gas

A key component and basic structural element of the torus-shaped solar sail is
a toroidal shell. The gas filled toroidal shell attached to the circular
membrane (Fig. 1$a$) provides structural support to the sail. Here we consider
a static behavior of an inflated toroidal shell following Ref. Kraus1967 . We
assume large aspect ratio, $\frac{R}{r}\gg 1$. Therefore, $R\gg r$ and it is a
very reasonable approximation to consider that the radius of the circular sail
membrane $R-r\simeq R$ is the radius of the torus. Also the thickness of the
torus shell $t_{t}$ is negligible compared to the radii of curvature.

Figure 1: (Color online) ($a$) Torus-shaped solar sail and its geometric
characteristics. ($b$) $R$ and $r$ are the radii of the torus and the toroidal
shell cross-section, respectively. Throughout of this work we assume that the
radius of the torus is much greater than the radius of the toroidal shell:
$R\gg r$. $t_{m}$ and $t_{t}$ are the thicknesses of the sail membrane and
toroidal shell, respectively.

An implicit equation for a torus with a large aspect ratio that is azimuthally
symmetric about the $z-$axis in Cartesian coordinates is

$(R-\sqrt{x^{2}+y^{2}})^{2}+z^{2}=r^{2},$ (1)

where $R$ is the distance from the center of the toroidal shell to the center
of the torus and $r$ is the radius of the toroidal shell as is shown in Fig.
1b. This torus is generated by revolving the circle $(x-R)^{2}+z^{2}=r^{2}$ of
radius $r$ in the $x-z$ $-$ plane about the $z$ axis at the distance $R$ from
the center of the circle.

In our consideration it is convenient to use toroidal and poloidal coordinates
for description of the torus geometry: the polar coordinate $(r,\theta)$ for
each cross-section and toroidal coordinate $\varphi$ around the toroidal ring.
The three coordinate $r,\theta,\varphi$ ($\theta\in\left[0,2\pi\right],$
$\varphi\in\left[0,2\pi\right]$) of curved coordinate system are orthogonal to
each other. The toroidal coordinate is used to describe those quantities along
the direction normal to the cross-section, while the poloidal coordinates are
used to describe an interior of the toroidal shell. The toroidal and poloidal
coordinate system relates to standard Cartesian coordinates as follows

$\displaystyle x$ $\displaystyle=$ $\displaystyle(R+r\cos\theta)\cos\varphi,$
$\displaystyle y$ $\displaystyle=$ $\displaystyle(R+r\cos\theta)\sin\varphi,$
(2) $\displaystyle z$ $\displaystyle=$ $\displaystyle r\sin\theta.$

Figure 2: (Color online) A schematic of the element of the toroidal shell
inflated by the gas. The force $df_{t}$ acts on the surface element of the
shell at the angle $\theta$ per unit of the length of toroidal ring at the
angle $\varphi=0$. The radius $R$ is much greater than the cross-section
radius of the torus $r$: $R\gg r$. The figure is not to scale.

Of our special interest are the forces that act on the surface area of the
torus. The total surface area of the torus can be considered as a sum of the
outer and inner surfaces as shown in Fig. 1$a$. The outer surface is a
positively curved in a contrast of the inner surface, which is negatively
curved. It is obvious that the outer surface is larger than the inner one and
the difference in surface area is $\Delta A=8\pi r^{2}$. Therefore, the
pressure of the gas $P$ in toroidal shell filled with the gas produces the
larger force on the outer surface of the shell than on the inner one.

Figure 3: (Color online) ($a$) Schematics for the action of uniform radial
forces $F_{t}$ per unit of circumferential length on the toroidal shell. ($b$)
The area of the toroidal shell cross-section is $A=2\pi rt_{t}$. The figure is
not to scale: $R\gg r$ and $r\gg t_{t}$

Let us find the net force due to the gas pressure $P$ in the toroidal shell
that keeps the torus inflated. We consider the torus circular cross-section at
the angle $\varphi=0$ as it is depicted in Fig. 2. The differential of the
force $d\mathbf{f}_{t}$ acting on the surface element $dA$ at angle $\theta$
per unit of the length of toroidal ring $Rd\varphi$ is
$d\mathbf{f_{t}=}\frac{PdA\widehat{\mathbf{n}}}{Rd\varphi}$, where
$dA=(R+r\cos\theta)d\varphi rd\theta$ and
$\widehat{\mathbf{n}}=\cos\theta\mathbf{i+\sin}\theta\mathbf{k}$ is a unit
vector normal to the surface element $dA$ with the unit vectors $\mathbf{i}$
and $\mathbf{k}$ along $x$ and $y$ axis, respectively. Therefore, the net
force that acting on the unit length of the toroidal shell at $\varphi=0$ is

$\mathbf{F}_{t}=\int_{0}^{2\pi}\frac{P}{R}(R+r\cos\theta)\left(\cos\theta\mathbf{i+\sin}\theta\mathbf{k}\right)rd\theta=\frac{\pi
Pr^{2}}{R}\mathbf{i}.$ (3)

The force $\mathbf{F}_{t}$ is directed along $x$ axis radially out of the
torus center, which is expected due to the mirror symmetry of the torus with
respect of the $x-y$ plane. The magnitude of the force $\mathbf{F}_{t}$ at
$\varphi=0$ acting on the unit length of the toroidal shell is

$F_{t}=\frac{\pi Pr^{2}}{R}.$ (4)

Due to the fact that the $z-$axis is the axis of rotational symmetry of
infinite order the magnitude of the force $F_{t}$ for any toroidal angle
$\varphi\in\left[0,2\pi\right]$ is the same as (4) for $\varphi=0$ and is
directed radially out as shown in Fig. 3. This force is responsible for the
tensile stress $\sigma_{t}$ of the toroidal shell resulting in tensile strain
$\Delta R_{t}/R$.

The net force acting on the toroidal shell cross-section of the aria $2\pi
rt_{t}$ is $\frac{1}{2}\int_{0}^{\pi}F_{t}\sin\varphi Rd\varphi=F_{t}R$. In
the limit of the linear response one can apply the Hook’s law and obtains the
tensile stress in the toroidal shell Timoshenko51

$\sigma_{t}=\frac{F_{t}R}{2\pi rt_{t}}=E_{t}\frac{\Delta R_{t}}{R}.$ (5)

In Eq. (5) $E_{t}$ and $t_{t}$ are Young’s modulus and the thickness of the
toroidal shell material, respectively. From Eq. (5) we get for the strain and
$\Delta R_{t}$ the following expression:

$\frac{\Delta R_{t}}{R}=\frac{F_{t}R}{2\pi E_{t}rt_{t}},$ (6) $\Delta
R_{t}=\frac{F_{t}R^{2}}{2\pi E_{t}rt_{t}}.$ (7)

This $\Delta R_{t}$ is for a stand-alone torus. In the equilibrium of the
stand-alone gas filled torus under the force $F_{t}$ is balanced by the
opposite elastic forces.

The toroidal shell is made from a material with density $\rho_{t}$ and the
density of the material of the membrane is $\rho_{m}$. The thicknesses of the
foil for the toroidal shell and membrane are $t_{t}$ and $t_{m}$,
respectively. Thus, the mass of the sail is

$M=\pi\rho_{m}R^{2}t_{m}+4\pi^{2}\rho_{t}rRt_{t}+M_{g},$ (8)

where ${M_{g}}$ is the mass of the gas.

###  II.2 Stress and strain of circular membrane

Ideally the torus-shaped solar sail has the attached membrane of the radius
$R-r$. However, as we mentioned before for the torus-shaped sail with large
aspect ratio when $R\gg r$, the radius of the membrane can be considered as
$R$. Consider a thin circular membrane of the radius $R$ and thickness $t_{m}$
($t_{m}\ll R$) under a uniform distributed force $F_{m}$ per unit length of a
circumference acting at the membrane edge in the radial direction as shown in
Fig. 4. If the membrane is sufficiently thin, the deformation can be treated
as uniform over its thickness and we have to deal with longitudinal
deformations of the membrane and not with any membrane bending. For a two-
dimensional case the strain tensor is a function of $x$ and $y$ coordinates
and is independent on $z$. The boundary conditions for the stress tensor on
both surfaces of the membrane are Landau7

Figure 4: (Color online) Schematics for the action of the uniformly
distributed radial force $F_{m}$ per unit of circumferential length of the
circular membrane: ($a$) top view; ($b$) side view. The figure is not to
scale: $R\gg t_{m}$

$\sigma_{ik}n_{k}=0,n_{k}=n_{z},i=x,y,z,$ (9)

where $\mathbf{n}_{z}$ is the normal vector parallel to $z-$axis, and lead to

$\sigma_{xz}=\sigma_{yz}=\sigma_{zz}=0$ (10)

in the entire volume of the membrane when $t_{m}\ll R$ Landau7 .

The equation of equilibrium in the absence of the body forces in the two-
dimensional vector form is Landau7 :

$grad\text{ }div\text{ }\mathbf{u-}\frac{1}{2}(1-\nu_{m})curl\text{
}curl\text{ }\mathbf{u}=0,$ (11)

where $\mathbf{u}$ is the displacement vector, $\nu_{m}$ is the membrane
Poisson ratio and the vector operators $grad$, $div$ and $curl$ are two-
dimensional. Due to the axial symmetry $\mathbf{u}$ is directed along the
radius and is a function of $\mathfrak{r}$ only, so $curl$ $\mathbf{u}=0$ and
Eq. (11) in polar coordinates becomes:

$div\text{
}\mathbf{u=}\frac{1}{\mathfrak{r}}\frac{d(\mathfrak{r}u)}{d\mathfrak{r}}=const\equiv
2c$ (12)

and gives

$u=c\mathfrak{r}+\frac{d}{\mathfrak{r}}.$ (13)

Therefore, the stress tensor radial and angular components are

$u_{\mathfrak{rr}}=\frac{du}{d\mathfrak{r}}=c-\frac{d}{\mathfrak{r}^{2}},\text{
}u_{\varphi\varphi}=\frac{u}{\mathfrak{r}}=c+\frac{d}{\mathfrak{r}^{2}},$ (14)

where in (14) $c$ and $d$ are some constants Handbook . In polar coordinates
at $\varphi=0$ the stress $\sigma_{ik}$ and strain $u_{ik}$ tensor components
are:

$\displaystyle\sigma_{\mathfrak{rr}}$ $\displaystyle\mathbf{=}$
$\displaystyle\sigma_{xx},\text{
}\sigma_{\varphi\varphi}\mathbf{=}\sigma_{yy},\text{ }$ (15) $\displaystyle
u_{\mathfrak{rr}}$ $\displaystyle\mathbf{=}$ $\displaystyle u_{xx},\text{
}u_{\varphi\varphi}\mathbf{=}u_{yy}.$ (16)

The general equations relating the strain tensor components to the stress
tensor components Landau7  in our case when $\sigma_{zz}=0$ become:

$\displaystyle\sigma_{xx}$ $\displaystyle=$
$\displaystyle\frac{E_{m}}{1-\nu_{m}^{2}}\left(u_{xx}+\nu_{m}u_{yy}\right),$
(17) $\displaystyle\sigma_{yy}$ $\displaystyle=$
$\displaystyle\frac{E_{m}}{1-\nu_{m}^{2}}\left(u_{yy}+\nu_{m}u_{xx}\right),$
(18)

where $E_{m}$ and $\nu_{m}$ are the Young modulus of elasticity and Poisson
ratio of the membrane, respectively. By substituting (15) and (16) at
$\varphi=0$ into (17) and (18) the stress tensor radial and angular components
become:

$\displaystyle\sigma_{\mathfrak{rr}}$ $\displaystyle\mathbf{=}$
$\displaystyle\frac{E_{m}}{1-\nu_{m}^{2}}\left(u_{\mathfrak{rr}}+\nu_{m}u_{\varphi\varphi}\right),$
(19) $\displaystyle\sigma_{\varphi\varphi}$ $\displaystyle\mathbf{=}$
$\displaystyle\frac{E_{m}}{1-\nu_{m}^{2}}\left(u_{\varphi\varphi}+\nu_{m}u_{\mathfrak{rr}}\right).$
(20)

The requirement for the deformation $u$ (14) to be finite at the membrane
center and the boundary condition for $\sigma_{\mathfrak{rr}}$ (19) combined
with $u_{\mathfrak{rr}}$ and $u_{\varphi\varphi}$ (14) at the membrane edge:

$u\left(0\right)=0\text{ and
}\sigma_{\mathfrak{rr}}(\mathfrak{r}=R)=\frac{F_{m}}{t_{m}}$ (21)

determine the values of the constants $c$ and $d$:

$c=\frac{F_{m}}{t_{m}}\frac{(1-\nu_{m})}{E_{m}},\text{ }d=0.$ (22)

The use of the constants (22) allows to obtain the expressions for the
deformation vector $\mathbf{u}$ (13) and the strain (14), as well as the
stress tensor components (19), (20):

$\displaystyle\mathbf{u}$ $\displaystyle\mathbf{=}$
$\displaystyle\frac{F_{m}}{t_{m}}\frac{(1-\nu_{m})}{E_{m}}\mathfrak{r},$ (23)
$\displaystyle u_{\mathfrak{rr}}$ $\displaystyle=$ $\displaystyle
u_{\varphi\varphi}=\frac{F_{m}}{t_{m}}\frac{(1-\nu_{m})}{E_{m}},$ (24)
$\displaystyle\sigma_{\mathfrak{rr}}$ $\displaystyle=$
$\displaystyle\sigma_{\varphi\varphi}=\frac{F_{m}}{t_{m}}.$ (25)

As expected the stress distribution (25) in the membrane deformed by the
forces acting at the edge of the membrane does not depend on the elasticity
constants of the membrane media Landau7 . Finally, both the radial deformation
$\Delta R_{m}=u(\mathfrak{r}=R)$, the strain $\frac{\Delta
R_{m}}{R}(\mathfrak{r}=R)$ and the stress
$\sigma_{\mathfrak{r}m}=\sigma_{\mathfrak{rr}}(\mathfrak{r}=R)$ of the
membrane edge are:

$\displaystyle\Delta R_{m}$ $\displaystyle\mathbf{=}$
$\displaystyle\frac{F_{m}}{t_{m}}\frac{(1-\nu_{m})}{E_{m}}R,$ (26)
$\displaystyle\frac{\Delta R_{m}}{R}$ $\displaystyle=$
$\displaystyle\frac{F_{m}}{t_{m}}\frac{(1-\nu_{m})}{E_{m}},$ (27)
$\displaystyle\sigma_{\mathfrak{r}m}$ $\displaystyle=$
$\displaystyle\frac{F_{m}}{t_{m}}.$ (28)

It is important to mention that all expressions in SubSection II.B are valid
for a membrane of thickness $t_{m}$ in the shape of a ring with concentric
internal and external radii, subject to internal membrane clamping. This is
still the same case of two-dimensional uniform expansion as considered above.

Figure 5: (Color online) Schematics for the action of uniform radial forces
$F_{t}$ and $F_{m}$ per unit of circumferential length on the toroidal shell.
The force opposite in direction and equal in magnitude to $F_{m}$ acts on the
circular membrane and is not shown. For visibility, the cross-section of the
toroidal shell is given in 3D format. The figure is not to scale: $R\gg r$ and
$r\gg t_{t}$.

###  II.3 Gas in toroidal shell

Consider the pressure of the gas in the toroidal shell. We describe the gas by
the equation for the real gas. Therefore, we are not considering atoms or
molecules of the gas as point particles that interact with toroidal shell
only. We are taking into account the volume that a real gas atoms or molecules
take up and gas atoms or molecules interaction with each other that are
experiencing an attraction at low pressure and repulsion at high pressure. The
corresponding van der Waals equation of state reads

$\left(P+a\frac{n^{2}}{V^{2}}\right)\left(V-nb\right)=nR_{g}T.$ (29)

In Eq. (29) $a$ and $b$ are the gas dependent constants, $n=\frac{M_{g}}{\mu}$
is the number of moles of the gas, where $M_{g}$ is the mass of the gas not
considering diffusion losses and $\mu$ is the molar weight of the gas, $V$ and
$T$ are volume and temperature of the gas, respectively, and $R_{g}=8.31$
J$\cdot$K-1mol-1is a universal gas constant. We assume that at equilibrium the
pressure of the gas in the toroidal shell obeys to the isochoric process
because it is confined in the interior volume $V=2\pi^{2}r^{2}R$, which only
slightly changes because of the increase of the torus radius by $\Delta R_{t}$
(7) due to the gas pressure. For the torus with large aspect ratio this
increase of the volume can be neglected. The internal pressure produces the
forces that remain normal to the surface. As it follows from Eq. (29) for the
isochoric process the pressure

$P=n\frac{R_{g}T}{2\pi^{2}r^{2}R-nb}-n^{2}\frac{a}{4\pi^{4}r^{4}R^{2}}.$ (30)

As it follows from Eq. (30) the pressure of the confined gas is defined by its
mass, temperature, the gas parameters $a$ and $b$, and the characteristic size
of the torus radii $R$ and $r$.

Using Eqs. (4) and (30) one can find the in-plane tensile force per unit
length in the membrane produced by the inflatable torus shell filled with a
gas

$F_{t}=n\frac{\pi
r^{2}R_{g}T}{2\pi^{2}r^{2}R^{2}-nbR}-n^{2}\frac{a}{4\pi^{3}r^{2}R^{3}}.$ (31)

###  II.4 Circular membrane attached to toroidal shell

In the absence of the gas, there are no stresses along the line of the
attachment of the membrane to the toroidal shell. The gas in the toroidal
shell causes the force $F_{t}$ acting on the toroidal shell radially out and
$F_{m}$ acting on the toroidal shell from the membrane radially in as shown in
Fig. 5. The latter force is in return to the stress caused by the force of the
toroidal shell on the membrane equal in magnitude to $F_{m}$. In the state of
equilibrium we have

$\Delta R_{t}=\Delta R_{m}.$ (32)

The radial forces $F_{t}$ and $F_{m}$ result in total radial force
$F_{t}-F_{m}$, which has to be balanced by the elastic forces in the toroidal
shell causing the shell radius change $\Delta R_{t}$ according to (7).
However, the force $F_{t}$ in (7) for stand-alone toroidal shell should be
replaced by the force $F_{t}-F_{m}$ for the toroidal shell and membrane
combination. Using (7) in place of $\Delta R_{t}$ with $F_{t}$ replaced to
$F_{t}-F_{m}$, and using (26) for the $\Delta R_{m}$ of membrane leads (32) to

$\frac{(F_{t}-F_{m})R^{2}}{2\pi
rt_{t}E_{s}}=\frac{F_{m}}{t_{m}}\frac{(1-\nu_{m})}{E_{m}}R,$ (33)

and

$F_{m}=F_{t}\left(1+2\pi\frac{r}{R}\frac{t_{t}}{t_{m}}\frac{E_{t}}{E_{m}}(1-\nu_{m})\right)^{-1}.$
(34)

Using (4) for $F_{t}$ Eq. (34) can be rewritten as

$F_{m}=\pi
P\frac{r^{2}}{R}\left(1+2\pi\frac{r}{R}\frac{t_{t}}{t_{m}}\frac{E_{t}}{E_{m}}(1-\nu_{m})\right)^{-1},$
(35)

where $P$ is defined by (30).

In conclusion, (27) with (35) for $F_{m}$ allow the determination of the
strain of both the toroidal shell and the membrane. Equation (28) determines
the radial pressure on the membrane edge, and (5) with $F_{t}$ replaced by
$F_{t}-F_{m}$:

$\sigma_{t}=\frac{\left(F_{t}-F_{m}\right)R}{2\pi rt_{t}}$ (36)

provides for the tensile stress of the toroidal shell for the system of the
circular membrane attached to the toroidal shell.

##  III Sail membrane

The sail surface is a circular membrane of radius $R$ represents the ”still”
drum head shape under the tensile stress produced by the inflated toroidal
shell. The boundary of the membrane is a circle of radius $R$ centered at the
origin and represents the rigid frame to which the membrane is attached. In
the equilibrium position, the membrane is stretched and fixed along its entire
boundary by the toroidal shell in the $x-y$ plane. The tension per unit length
$F_{m}$ caused by stretching the membrane is the same at all points and in all
directions and does not change during the motion.

When the deployment of the membrane ends it can experience the vibration. The
mathematical equation that governs the normal transverse vibrations of the
membrane is the wave equation with zero boundary conditions. The wave equation
has been widely studied in the literature and is as follows:

$\frac{\partial^{2}\mathfrak{u}}{\partial
t^{2}}=c_{s}^{2}\left(\frac{\partial^{2}\mathfrak{u}}{\partial
x^{2}}+\frac{\partial^{2}\mathfrak{u}}{\partial y^{2}}\right),\text{ \
}c_{s}^{2}=\frac{F_{m}}{\varkappa},$ (37)

where $\mathfrak{u}(x,y)=0$ on the boundary of the membrane, $\varkappa$ is
the mass per area of the membrane, and $c_{s}=\sqrt{\frac{F_{m}}{\varkappa}}$
is the sound speed. Equation (37) is the two-dimensional wave equation, which
is a second order partial differential equation. Due to the circular geometry
of the membrane, it is convenient to rewrite Eq. (37) using the cylindrical
coordinates

$\frac{\partial^{2}\mathfrak{u}}{\partial
t^{2}}=c_{s}^{2}\left(\frac{\partial^{2}\mathfrak{u}}{\partial\mathfrak{r}^{2}}+\frac{1}{\mathfrak{r}}\frac{\partial\mathfrak{u}}{\partial\mathfrak{r}}+\frac{1}{\mathfrak{r}^{2}}\frac{\partial\mathfrak{u}}{\partial\varphi}\right),\text{
\ for }0\leq\mathfrak{r}<R,\text{ and \ }0\leq\varphi\leq 2\pi$ (38)

with the boundary condition $\mathfrak{u}(\mathfrak{r},\varphi,t)=0$ for
$\mathfrak{r}=R,$ which means that the membrane is fixed along the boundary
circle of the radius $R$ in the $x-y-$plane for all times $t\geq 0$.

To determine solutions $\mathfrak{u}(\mathfrak{r},\varphi,t)$ that are
radially symmetric, we solve Eq. (38) following the three standard steps: i.
using the method of separation of variables, we first determine solutions as
$\mathfrak{u}(\mathfrak{r},\varphi,t)=\mathfrak{U}(\mathfrak{r},\varphi)V(t)$
and obtain two independent differential equations for
$\mathfrak{U}(\mathfrak{r},\varphi)$ and $V(t)$ functions; ii. from the
solutions of those ordinary differential equations we determine solution
(eigenfunctions) $\mathfrak{U}(r,\varphi)$ which satisfy the boundary
condition $\mathfrak{U}(\mathfrak{r},\varphi)=0$ for $\mathfrak{r}=R$ and find
the corresponding eigenvalues. Then find the periodical solution $V(t)$ by
solving the second differential equation; iii. we compose these solutions and
obtain the radially symmetric solution $\mathfrak{u}(\mathfrak{r},t)$ that
satisfy the conditions $\mathfrak{u}(\mathfrak{r},0)$ and
$\frac{\partial\mathfrak{u}(\mathfrak{r},t)}{\partial t}$ depend only on
$\mathfrak{r}.$ The corresponding method of a solution is given in Ref.
KezASR2021 . In-plane vibrations and dumping of the transfer vibrations in
particular due to the solar radiation pressure are out of the score of this
study.

##  IV Results and discussion

The theory presented in the previous sections provides expressions to
calculate different quantities related to a torus-shaped solar sail such as
stresses in the toroidal shell and membrane, the radial force that acts on the
membrane and their dependence on the gas mass. The general assumptions of
applicability of these expressions are the following:

i. both strain and stress for the membrane and toroidal shell are small and
within the limit of elasticity;

ii. the radius of the membrane is much greater than the radius of the toroidal
shell: $R\gg r$.

iii. the material thickness of the membrane and toroidal shell are
significantly smaller than the radius of the toroidal shell: $t_{m}\ll r$ and
$t_{t}\ll r$.

The first condition insures the mechanical stability of the sail and requires
that the stress in the membrane and toroidal shell have to be less than the
yield strength of the corresponding material. To satisfy the second condition
the values of $\frac{R}{r}\sim 10^{2}$ are used. To meet the third requirement
the values $\frac{t_{m}}{r}\approx\frac{t_{t}}{r}\sim 10^{-5}$ are considered.

Table 1: Model parameters for the sail membrane and hydrogen gas. $\rho_{m}$, $E_{m}$, $\nu_{m}$ and $t_{m}$ are the density, Young modulus, Poisson ratio and thickness of the sail membrane, respectively. Sail’s membrane, CP1 | Gas parameters
---|---
$\rho_{m}$, kg/m2 |  $E_{m}$, Pa | $\nu_{m}$ |  $t_{m}$, m |  $\mu,$ g/mol |  $a$, L2bar/mol2 |  $b,$ L/mol
1.43$\times 10^{3}$ | 2.17$\times 10^{9}$ | 3.40$\times 10^{-1}$ | 3.5$\times 10^{-6}$ | 2.0159 | 0.2476 | 0.02661

In our calculations we consider CP1 Polyimide film CP1  as an example of the
material of the membrane and toroidal shell (CP1 is planned to be used for the
Solar Cruiser LesJ ) and the hydrogen as the filling gas. However, to meet the
first condition the thickness of the toroidal shell is considered twofold as
for the membrane. As it follows from Eqs. (4), (34), and (36) this leads
almost twofold decrease of the shell stress. The properties of CP1,
corresponding parameters and gas constants are listed in Table 1. The yield
strength of CP1 is not available to the authors. In general, it is much less
than the Young modulus. Nonetheless, one can estimate the yield strength to be
greater than $\sim$10-2 part of the Young modulus, keeping in mind that the
real design decisions have to be based on experimentally measured values.

Figure 6: (Color online) The dependence of the areal density the solar sail on
the membrane radius for different radii of the toroidal shell. Figure 7:
(Color online) The dependence of the tensile stress of the toroidal shell
$\sigma_{t}$ and the membrane radial stress $\sigma_{\mathfrak{r}m}$ on the
membrane radius for different radii of the toroidal shell. The horizontal line
corresponds to 10${}^{-2}E_{m}$ MPa. Calculations performed for the gas mass
$M_{g}=0.750$ kg.  Figure 8: (Color online) The dependence of the membrane
radial stress $\sigma_{\mathfrak{r}m}$ on the membrane radius for different
masses of the gas in the toroidal shell. The horizontal line corresponds to
10${}^{-2}E_{m}$ MPa. Calculations performed for $r=0.1$ m.

Figure 9: (Color online) The dependence of the ratio $a_{c}/g$ on the radius
of the solar sail membrane for ($a$) different radii of the toroidal shell for
$M_{g}$ = 1.0 kg and ($b$) different masses of the gas in the toroidal shell
for $r=0.1$ m. The horizontal line for the ratio $a_{c}/g$ is obtained based
on the result from Ref. Deployment4 . Figure 10: (Color online) Limitations
for the radius of the sail. The colored area shows the allowed radii of the
sail’s membrane for the radius of the toroidal shell $r=0.1$ m.

One of the key metrics related to the performance of solar sail is the
characteristic acceleration Colin ; Matloff3 ; MatloffValpetti . The
characteristic acceleration is defined as $a_{0}=2\eta P_{0}/\chi$, where
$0.5\leq\eta\leq 1$, $P_{0}$ is the solar radiation pressure near the Earth
and $\chi=M/A$ is the areal density. In the latter expression, $M$ is the
total mass of the torus-shaped solar sail defined by Eq. (8) and $A$ is the
sail membrane area. It is worth mentioning that we neglect the projectional
area of the toroidal shell: $4\pi r(R+r)<<\pi R{{}^{2}}$. However,
consideration of this area only decreases the areal density. While the actual
sail acceleration is a function of heliocentric distance and its orientation,
the characteristic acceleration allows a comparison of solar sail design
concepts on an equal footing. The areal density $\chi$ is an important
parameter determining the sail acceleration due to electromagnetic radiation
pressure. It is easy the calculate $\chi$ for known geometry and densities of
sail materials. Less $\chi$ leads to the greater sail acceleration. The
dependence of the areal density of the solar sail on the membrane radius for
different radii of the toroidal shell is shown in Fig. 6. As is expected the
areal density increases with the increase of the toroidal shell radius and
decrease of the sail membrane radius. When the membrane radius increases from
30 m to 100 m the areal density decreases form $\sim$ 6.4 g/m2 to $\sim$ 5.2
g/m2. While the dependence of $\chi$ on the toroidal shell radius is
important, the mass of the gas has negligible contribution.

The expressions (28) and (36) are used to calculate the radial stress of the
membrane and tensile stress of the toroidal shell. Analysis of these
expressions shows the strong dependence on the membrane radius and on the
pressure of the gas, but negligible dependence on the radius of the toroidal
shell. Results of calculations performed for the different fixed radii of the
toroidal shell are presented in Fig. 7. The both stresses decrease with the
increase of membrane radius and the results show that for any given radius the
membrane stress is greater than respective toroidal shell stress. Thus, the
stability of the membrane will insure the stability of the entire shell-
membrane structure. The membrane stress stays within 10${}^{-2}E_{m}$. The
crossover points radii represent the lower limits of the sail radii.

In Eq. (30) the second term is small and one can say that the gas pressure in
the toroidal shell is approximately directly proportional to the gas mass.
Therefore, analysis of Eqs. (31) and (34) shows that the force $F_{t}$ per
unit length of the toroidal shell and the force $F_{m}$ per unit length of the
membrane also are approximately directly proportional to the gas mass. The
same strong dependence on the gas mass along with the strong dependence on the
membrane radius is reflected on both the toroidal shell and membrane stresses.
The results presented in Fig. 8 show the dependence of the membrane stress on
the membrane radius for different masses of the gas. The strong dependence of
the radii of the crossover points on the gas mass is obvious. The stable sails
need to have radii greater than the radii of crossover points in Fig. 8.

The logical question which arises now is the following: which value of the
sail radius should be considered to deploy the sail successfully? Although a
certain answer can be obtained only in the experiment it is possible to make
some estimations by introducing the ratio $a_{c}/g$ given as

$\frac{a_{c}}{g}=\frac{\left(F_{t}-F_{m}\right)}{g(2\pi
rt_{t}\rho_{t}+M_{g}/2\pi R)},$ (39)

where $g=9.8$ m/s2 is the acceleration due to gravity at the Earth surface.
The quantity $a_{c}$ is effective acceleration equivalent to centripetal
acceleration of the shell would it be spinning around its center with
frequency

$f=\frac{1}{2\pi}\left(\frac{a_{c}}{R}\right)^{1/2}.$ (40)

The greater $a_{c}/g$ ratio the greater the chance of successful deployment of
initially folded toroidal shell and sail membrane to the open state of
circular shape. A simple analysis of the partial derivatives shows that
$\sigma_{t}$, $\sigma_{rm}$, and $a_{c}/g$ are the monotonically decreasing
functions of the membrane radius $R$, radius $r$ of the toroidal shell cross-
section and increasing functions of the gas mass.

We compare the calculated values of $a_{c}/g$ ratio with the equivalent
estimate based on the successful ground demonstration of the spinning sail
deployment by Jet Propulsion Laboratory Deployment4 . The data provided in
Ref. Deployment4  for the spinning frequency 200 RPM or more of the sail of
0.8 m in diameter gives the estimate of the ratio of centripetal acceleration
over the acceleration due to gravity equal to 18 or more. The calculated
$a_{c}/g$ ratio is presented in Fig. 9. It is evident from Fig. 9$a$ that the
dependence of the ratio on the toroidal shell radius $r$ is fairy weak and
similar to the dependence of the stresses on $r$ \- see Fig. 7. Thus, the
choice of lower $r$ is preferable for the sail due to the less arial density.
Our choice for toroidal radius $r$ is 0.1 m. Results presented in Fig. 9$b$
show strong dependence of the $a_{c}/g$ ratio on the gas mass. The radii of
the crossover points of the $a_{c}/g$ versus $R$ curves with the horizontal
line at $a_{c}/g$ = 18 represent the upper limits of the sail radius. The sail
with radius less than these upper limits is expected to successfully deploy
into a circular shape.

Figure 11: (Color online) Dynamics of the vibrating membrane. Maximum membrane
deviations for the first (0,1), second (0,2) and third (0,3) modes of normal
oscillations.

In summary, there are two limits for the radii of the sail: i. the lower limit
is due to the sail stability requirement demanding the stresses to be within
limits of elasticity; ii. the upper limit is due to requirement of the
successful deployment. The radius of the sail has to simultaneously satisfy
both requirements being greater than the lower limit and less than the upper
limit. This means that the lower limit has to be less than the upper limit.
The dependence of the limits of the toroidal shell-membrane radii on the gas
mass for choice of $r=0.1$ m is presented in Fig. 10. This is kind of a solar
sail phase diagram. The coordinates of the crossover point are: $R=54.5$ m and
$M_{g}=0.870$ kg. The sails with the radius and the gas mass within the area
laying to the left of the crossover point and defined at the top by the upper
limit and at the bottom by the lower limit curves are deployable and stable.
The sails with the radius and gas mass to the right of the crossover point are
either unstable and/or not deployable. For example, in the case of CP1
Polyimide film the sail membrane of $R=51$ m attached to the toroidal shell of
$r=0.1$ m inflated by 0.75 kg of the hydrogen is the optimal fit. The same
kind of the diagram can be calculated for other choices of toroidal shell
radius $r$. The location shift of the crossover point with the variation of
$r$ from 0.1 m to 0.25 m is quit small, within 1.5 m and 60 g.

For a sail with $R=51$ m and inflatable toroidal shell with $r=0.1$ m,
$M_{g}=0.750$ kg of the gas in the toroidal shell and for the tension of the
membrane given by Eq. (35) we calculated the first three normal transverse
vibration modes of the deployed and stretched circular sail. For this tension
and the corresponding geometric characteristics of the membrane, the maximum
deviation of the solar sail membrane was found. The results of these
calculations are shown in Fig. 11. Each graph corresponds to the instant of
time when the maximum deviation of the membrane from its equilibrium position
is reached. Our results show that the total deviations of the membrane from
the equilibrium position with free modes of oscillations will not exceed a few
millimeters and the maximal deviation is observed for the first mode. However,
one should mention that these vibrations are related to the static deployed
sail and only are transverse or out of plane without taking into account any
damping in particular the dumping due to radiation pressure. These results
could dramatically change when one considers the vibration during the dynamics
of opening the sail when the gas is inserting into the toroidal shell. This
aspect along with the in-plane vibrations and dumping is out of the scope of
this paper and will be considered somewhere later.

##  V Conclusions

In this article we present the theoretical study of the circular solar sail
with inflatable torus-shaped shell. In the framework of a strict mathematical
approach based on classical theory of elasticity we justify an idea of the
deployment and stretching of the solar sail membrane attached to the
inflatable toroidal shell. We obtained the analytical expressions that can be
applied to a wide range of materials for both the shell and sail membrane. The
presented numerical example demonstrates the ability of the developed theory
to provide sound estimates of the stresses of the inflatable shell and the
solar sail membrane solely based on the geometry, gas mass and elastic
properties of the materials. Analysis of these expressions shows the strong
dependence of stresses and strains on the membrane radius and on the pressure
of the filled gas, but negligible dependence on the radius of the toroidal
shell. We performed numerical calculations for the sail of radius of 30 to 100
m made of CP1 membrane and attached to the toroidal shell of the same material
with the varied cross-section radius and gas mass. We predict that by
introducing the gas into the inflatable toroidal rim one can deploy and
stretch circular solar sail membrane up to 54 m of radius. It is worth noting
that the same calculations can be easily performed for the other suggested
materials for a solar sail such as Kapton or aluminized Mylar or others
MatlKez2008 . It is shown that the normal out of plane vibrations of the sail
membrane under tension caused by gas pressure are negligible.

To conclude, our theoretical approach and calculations demonstrate the
feasibility of deployment and stretching of the solar sail constructed as a
thin circular membrane attached to the inflatable toroidal shell. At this
point, an experimental research is necessary to find out the level of
feasibility of such a solar sail for future exploration of the solar system
and beyond by mean of the electromagnetic pressure propellant.

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# On a class of algebro-geometric solutions to the Ernst equation

Eddy B. de Leon<EMAIL_ADDRESS>Institut de
Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary,
21078 Dijon Cedex, France

###### Abstract.

We discuss a class of solutions to the Ernst equation in terms of theta
functions with characteristics. We show that it is necessary to take into
account a phase factor, which arises from a shift by a lattice vector, and
impose conditions on the characteristics of the theta functions in order for
the presented function to solve the Ernst equation for all the considered
parameters.

This work was partially supported by the EIPHI Graduate School (contract
ANR-17-EURE-0002)the Bourgogne Franche-Comté Region, the European fund FEDER,
and by the European Union Horizon 2020 research and innovation program under
the Marie Sklodowska-Curie RISE 2017 grant agreement no. 778010 IPaDEGAN. The
author thanks D. Korotkin for helpful discussions and hints.

##  1\. Introduction

The theory of multi-dimensional theta functions has proven to be useful in
many domains of mathematics and physics. In the theory of integrable systems,
they provide quasi-periodic solutions to nonlinear evolution equations such as
the Korteweg-de Vries (KdV) and nonlinear Schrödinger equations, see [1] for a
historic account and many references. Similar applications in general
relativity have different properties. Although the Einstein field equations
are in general non-integrable, they are if the spacetime under study has at
least two commuting Killing vectors (symmetries). In particular, the vacuum
stationary axisymmetric Einstein equations in Weyl coordinates can be reduced
to the integrable Ernst equation [2], which is

(1) $\Re(\mathcal{E})\Delta\mathcal{E}=(\nabla\mathcal{E})^{2},$

where the Laplacian $\Delta$ and the gradient $\nabla$ are the usual operators
in cylindrical coordinates $(\rho,\zeta,\varphi)$ and since we consider
axisymmetric spacetimes, the Ernst potential $\mathcal{E}$ does not depend on
the azimuthal coordinate $\varphi$. This means that solutions of the Einstein
equations with the aforementioned symmetries can be constructed via
quadratures of the solutions of the Ernst equation. This approach allows the
description of various rotating spacetimes, the most prominent of which is the
Kerr spacetime which is interpreted as a rotating black hole (see [5] for
further discussions), as well as counter-rotating dust disks [8, 5].

A large class of solutions to the Ernst equation was originally given by
Korotkin [6] in terms of multi-dimensional Riemann theta functions.
Alternatively, these solutions can be expressed in terms of theta functions
with characteristics, see [4]. These solutions are constructed over a family
of genus-$g$ hyperelliptic curves (the precise definitions are given in
Section 2). Namely, the dependence of these solutions on the physical
coordinates is via the modular dependence of the periods of the Riemann
surface. Thus, the physical coordinates do not enter directly the argument of
the theta functions as is the case for evolution equations, such as the KdV
and Kadomtsev-Petviashvili equations which are given on a Riemann surface
independent of the physical coordinates. In the latter case, the solutions
show quasi-periodicity properties, whereas this is not the case for solutions
to the Ernst equation. Both the dust disk solution and the Kerr solution are
constructed on a family of hyperelliptic curves of genus $2$, but the Kerr
solution is obtained as a limiting case, the so-called solitonic limit [5].

It is the purpose of this note to show that in terms of the complex variable
$\xi=\zeta+\mathrm{i}\rho$ (where $\zeta$ and $\rho$ are the physical
coordinates), the potential

(2)
$\mathcal{E}(\xi,\bar{\xi})=e^{-\pi\mathrm{i}\sum_{j}p_{j}}\frac{\Theta_{\mathbf{pq}}\left(\textstyle{\int}_{\xi}^{\infty^{+}},\mathbb{B}_{\xi}\right)}{\Theta_{\mathbf{pq}}\left(\textstyle{\int}_{\xi}^{\infty^{-}},\mathbb{B}_{\xi}\right)},$

solves the Ernst equation (thus an Ernst potential), where
$\textstyle{\int}_{\xi}^{\infty^{\pm}}$ and $\mathbb{B}_{\xi}$ are quantities
parametrized by $\xi$ and $\Theta_{\mathbf{pq}}$ is the multi-dimensional
theta function (10) with fixed arbitrary characteristics
$\mathbf{p}\in\mathbb{R}^{g}$ and $\mathbf{q}\in\mathbb{C}^{g}$, whose
components satisfy the reality conditions

(3) $\Re(q_{j})=\left\\{\begin{array}[]{ll}\frac{1}{2}\sum_{k\neq
j}p_{k},&\text{if }E_{j}=\bar{F}_{j},\\\
-\frac{1}{4}+\frac{1}{2}\sum_{k}p_{k},&\text{if
}E_{j},F_{j}\in\mathbb{R},\end{array}\right.$

where $E_{j}$, $F_{j}$ are the branch points of the defining hyperelliptic
curves (7). These terms are properly defined in Section 2. There are two
differences with respect to the potential presented in [4]. First, it is
necessary to include a phase factor, which arises upon shifting the argument
of the theta functions by a lattice vector when the characteristic
$\mathbf{p}$ is different from zero. Second, the reality conditions on the
characteristics had to be modified to (3). However, the class of solutions (2)
coincides with the one presented in [4] when $\sum_{j}p_{j}$ is an integer and
all the branch cuts are of the form $E_{j}=\bar{F}_{j}$.

This note is organized as follows. In Section 2, we recall some basic facts
regarding the Ernst equation, hyperelliptic curves and theta functions, as
well as Fay’s identity. In Section 3, we show that the potential (2) solves
the Ernst equation by using the approach presented in [4].

##  2\. Preliminaries

###  2.1. Ernst equation

The line element of a stationary axisymmetric vacuum spacetime in the Weyl-
Lewis-Papapetrou form reads

(4)
$ds^{2}=-f(dt+Ad\varphi)^{2}+f^{-1}\left[e^{2k}(d\zeta^{2}+d\rho^{2})+\rho^{2}d\varphi^{2}\right],$

where $k=k(\rho,\zeta)$, $A=A(\rho,\zeta)$, $f=f(\rho,\zeta)$. The Einstein
equations in these coordinates are equivalent to the Ernst equation together
with the relation $f=\Re(\mathcal{E})$ and the quadratures of

(5) $\displaystyle
A_{\xi}=2\rho\frac{(\mathcal{E}-\overline{\mathcal{E}})_{\xi}}{(\mathcal{E}+\overline{\mathcal{E}})^{2}},\quad
k_{\xi}=2\mathrm{i}\rho\frac{\mathcal{E}_{\xi}\mathcal{E}_{\bar{\xi}}}{(\mathcal{E}+\overline{\mathcal{E}})^{2}}.$

It is convenient to consider the Ernst equation in the complex coordinates, in
which it takes the form

(6)
$\left(\mathcal{E}+\overline{\mathcal{E}}\right)\left(\mathcal{E}_{\xi\bar{\xi}}-\frac{1}{2(\bar{\xi}-\xi)}\left(\mathcal{E}_{\bar{\xi}}-\mathcal{E}_{\xi}\right)\right)=2\mathcal{E}_{\xi}\mathcal{E}_{\bar{\xi}},$

###  2.2. Hyperelliptic curves

Recall that the period matrix $\mathbb{B}$ of any smooth algebraic curve
$\mathcal{L}$ is defined as the matrix with components
$\mathbb{B}_{jk}=\textstyle{\int}_{b_{j}}\omega_{j}$, where
$\\{a_{1},b_{1},...,a_{g},b_{g}\\}$ is a canonical basis of the first homology
group and $\\{\omega_{1},...,\omega_{g}\\}$ is a basis of holomorphic
differentials normalized with respect to the $a_{j}$ cycles, i.e.,
$\textstyle{\int}_{a_{j}}\omega_{k}=\delta_{jk}$. The period matrix
$\mathbb{B}$ is known to be complex symmetric and with positive definite
imaginary part. This matrix defines the full-rank lattice
$\Lambda_{\mathbb{B}}=\mathbb{Z}^{g}+\mathbb{B}\mathbb{Z}^{g}$ in
$\mathbb{C}^{g}$.

An algebraic curve $\mathcal{L}$ defines the Abel map
$\mathcal{A}:p\mapsto\textstyle{\int}_{p_{0}}^{p}\omega$, for a base point
$p_{0}\in\mathcal{L}$. Although $\mathcal{A}(p)$ depends on the path, it is
unique in $\mathbb{C}^{g}$ modulo $\Lambda_{\mathbb{B}}$. For simplicity, we
use the notation
$\textstyle{\int}_{p_{1}}^{p_{2}}:=\mathcal{A}(p_{2})-\mathcal{A}(p_{1})$.
Notice that the difference is independent of the base point $p_{0}$.

Consider the family of hyperelliptic curves

(7)
$\mathcal{L}_{\xi}=\\{(x,y)\in\mathbb{C}^{2}|y^{2}=(x-\xi)(x-\bar{\xi})\prod_{j=1}^{g}(x-E_{j})(x-F_{j})\\},$

parametrized by $\xi\in\mathbb{C}$, with distinct branch points
$E_{j},F_{j}\in\mathbb{C}$ and the pairwise condition that either
$E_{j}=\bar{F}_{j}$ or $E_{j},F_{j}\in\mathbb{R}$.

The basis cycles $\\{a_{j},b_{j}\\}$ of the first homology group are chosen
such that the action of the holomorphic involution
$\tau(x,y)=(\bar{x},\bar{y})$ on them is

(8) $\begin{split}\tau(a_{j})&=-a_{j},\quad\text{for }j=1,\ldots,g,\\\
\tau(b_{j})&=\left\\{\begin{array}[]{ll}b_{j}+\sum_{k\neq j}a_{k},&\text{if
}E_{j}=\bar{F}_{j},\\\ b_{j}+\sum_{k}a_{k},&\text{if
}E_{j},F_{j}\in\mathbb{R}.\end{array}\right.\end{split}$

This guarantees that the real part of the period matrix $\mathbb{B}_{\xi}$ of
$\mathcal{L}_{\xi}$ will have half-integer coefficients independent of $\xi$,
as shown in Appendix B. Figure 1 shows cycles satisfying such conditions,
where $a_{j}$ are the cycles encircling the branch cuts $[E_{j},F_{j}]$ in
counterclockwise direction and $b_{j}$ are those going from $[\xi,\bar{\xi}]$
to $[E_{j},F_{j}]$ on the $+$-sheet. In the following, the $\pm$ scripts
indicate whether we are considering the $+$ or $-$ covering sheet of
$\mathcal{L}_{\xi}$.

Figure 1. Choice of cycles.

The path $\gamma_{\infty}$ from the point $\infty^{-}$ to $\infty^{+}$ is
chosen such that

(9) $\tau(\gamma_{\infty})=\gamma_{\infty}-\sum_{k}a_{k}.$

###  2.3. Theta functions

The multi-dimensional theta function with characteristics
$\mathbf{p}\in\mathbb{R}^{g}$ and $\mathbf{q}\in\mathbb{C}^{g}$ is defined by
the series

(10)
$\Theta_{\mathbf{pq}}(\mathbf{z},\mathbb{B}):=\sum_{\mathbf{n}\in\mathbb{Z}^{g}}\exp\left(\pi\mathrm{i}\langle\mathbf{n}+\mathbf{p},\mathbb{B}(\mathbf{n}+\mathbf{p})\rangle+2\pi\mathrm{i}\langle\mathbf{n}+\mathbf{p},\mathbf{z}+\mathbf{q}\rangle\right),$

for any $\mathbf{z}\in\mathbb{C}^{g}$ and $\mathbb{B}\in\mathbb{H}^{g}$, where
$\mathbb{H}^{g}$ is the space of $g\times g$ complex symmetric matrices with
positive definite imaginary part. The latter implies that
$\Theta_{\mathbf{pq}}$ is an entire function in $\mathbb{C}^{g}$. This
function is quasi-periodic with respect to the lattice $\Lambda_{\mathbb{B}}$
and its translation by the lattice vector $\mathbf{m}\in\mathbb{Z}^{g}$ is
given by

(11)
$\Theta_{\mathbf{pq}}(\mathbf{z}+\mathbf{m},\mathbb{B})=e^{2\pi\mathrm{i}\langle\mathbf{p},\mathbf{m}\rangle}\Theta_{\mathbf{pq}}(\mathbf{z},\mathbb{B}).$

Moreover, theta functions can be defined on a curve $\mathcal{L}$ via its
period matrix $\mathbb{B}$ and the Abel map. Namely,

$\Theta:\mathcal{L}\to\mathbb{C},\quad
p\mapsto\Theta_{\mathbf{pq}}(\mathcal{A}(p)+v,\mathbb{B}),$

for any $v\in\mathbb{C}^{g}$.

Summarizing, to every point $\xi\in\mathbb{C}$ we associate the period matrix
$\mathbb{B}_{\xi}$ of the curve $\mathcal{L}_{\xi}$ as well as the Abel maps
$\textstyle{\int}_{\xi}^{\infty^{\pm}}$, which enter as the arguments of the
theta functions.

###  2.4. Fay identity

It is a relation between points on a compact Riemann surface $\mathcal{R}$,
e.g., the hyperelliptic curves (7). This functional relation is given in terms
of the theta functions with characteristics defined by the period matrix of
$\mathcal{R}$ and it holds on arbitrary points $a,b,c,d\in\mathcal{R}$, for
all $\mathbf{z}\in\mathbb{C}^{g}$.

(12)
$\begin{split}E(c,a)E(d,b)&\Theta_{\mathbf{pq}}(\mathbf{z}+\textstyle{\int}_{b}^{c})\Theta_{\mathbf{pq}}(\mathbf{z}+\textstyle{\int}_{a}^{d})+E(c,b)E(a,d)\Theta_{\mathbf{pq}}(\mathbf{z}+\textstyle{\int}_{a}^{c})\Theta_{\mathbf{pq}}(\mathbf{z}+\textstyle{\int}_{b}^{d})\\\
&=E(c,d)E(a,b)\Theta_{\mathbf{pq}}(\mathbf{z})\Theta_{\mathbf{pq}}(\mathbf{z}+\textstyle{\int}_{a}^{d}+\textstyle{\int}_{b}^{c}),\end{split}$

where
$E(x,y)=\Theta_{\mathbf{p}^{*}\mathbf{q}^{*}}(\textstyle{\int}_{y}^{x})/[h_{\Delta}(x)h_{\Delta}(y)]$
is the prime form, with the spinor $h_{\Delta}(x)$ satisfying

$h_{\Delta}^{2}(x)=\sum_{j=1}^{g}\frac{\partial\Theta_{\mathbf{p}^{*}\mathbf{q}^{*}}(0)}{\partial
z_{j}}\omega_{j}(x),$

see [7, 3] for further discussions. The theta function
$\Theta_{\mathbf{p}^{*}\mathbf{q}^{*}}$ is required to have a non-singular odd
half-integer characteristic, i.e.,
$2\mathbf{p}^{*},2\mathbf{q}^{*}\in\mathbb{Z}^{g}/(2\mathbb{Z}^{g})$ such that
$\Theta_{\mathbf{p}^{*}\mathbf{q}^{*}}(0)=0$ and
$\nabla\Theta_{\mathbf{p}^{*}\mathbf{q}^{*}}(0)\neq 0$.

##  3\. Solution to the Ernst equation

In this section, we show that the Ernst potential (2) solves the Ernst
equation (6) with fixed arbitrary characteristics
$\mathbf{p}\in\mathbb{R}^{g}$, $\mathbf{q}\in\mathbb{C}^{g}$ satisfying the
reality conditions (3). The proof is divided in three steps. First, we show
that the complex conjugate of (2) can be expressed in terms of theta functions
corresponding to the same period matrix $\mathbb{B}_{\xi}$. Second, we show
that the real part of the Ernst potential can be simplified via the Fay
identity. Third, we show that the proof presented in [4], which holds for
$\mathbf{p}=0$, can be extended to any $\mathbf{p}\in\mathbb{R}^{g}$.

We are interested in expressing the complex conjugate of the Ernst potential
in terms of theta functions of the same matrix $\mathbb{B}_{\xi}$, whose
arguments must be represented as the Abel maps of points on
$\mathcal{L}_{\xi}$, in order to use the Fay identity (12). This is given by
the following proposition.

######  Proposition 3.1.

Let $\mathcal{E}(\xi,\bar{\xi})$ be the potential defined by (2) with
characteristics $\mathbf{p}\in\mathbb{R}^{g}$, $\mathbf{q}\in\mathbb{C}^{g}$
satisfying the reality conditions (3). Then, its complex conjugate is

(13)
$\overline{\mathcal{E}(\xi,\bar{\xi})}=e^{-\pi\mathrm{i}\sum_{j}p_{j}}\frac{\Theta_{\mathbf{pq}}\left(\textstyle{\int}_{\bar{\xi}}^{\infty^{+}},\mathbb{B}_{\xi}\right)}{\Theta_{\mathbf{pq}}\left(\textstyle{\int}_{\bar{\xi}}^{\infty^{-}},\mathbb{B}_{\xi}\right)}.$

###### Proof.

From the definition of the multi-dimensional theta function (10), it can be
observed that

$\overline{\Theta_{\mathbf{pq}}(\mathbf{z},\mathbb{B})}=\alpha\cdot\Theta_{\mathbf{pq}}(-\bar{\mathbf{z}}-2\Re(\mathbf{q}+\mathbb{B}\mathbf{p})+{\rm
diag}(\Re(\mathbb{B})),\mathbb{B}),$

for all $\mathbf{z}\in\mathbb{C}^{g}$ and for any matrix
$\mathbb{B}\in\mathbb{H}^{g}$ satisfying the condition $2\Re(\mathbb{B})\in
M_{g\times g}(\mathbb{Z})$, where $\alpha\in\mathbb{C}$ is a constant
independent of $\mathbf{z}$ (see Appendix A). This condition is satisfied by
the period matrices $\mathbb{B}_{\xi}$ of hyperelliptic curves of the form (7)
with the choice of cycles shown in Figure 1, as shown in Appendix B.

Moreover, the conditions (3) on the $\mathbf{p}$, $\mathbf{q}$ characteristics
are equivalent to the vanishing of the term
$-2\Re(\mathbf{q}+\mathbb{B}\mathbf{p})+{\rm diag}(\Re(\mathbb{B}))$. Then,

$\overline{\Theta_{\mathbf{pq}}(\mathbf{z},\mathbb{B}_{\xi})}=\alpha\cdot\Theta_{\mathbf{pq}}(-\bar{\mathbf{z}},\mathbb{B}_{\xi}),$

which implies

(14)
$\overline{\mathcal{E}(\xi,\bar{\xi})}=e^{\pi\mathrm{i}\sum_{j}p_{j}}\frac{\Theta_{\mathbf{pq}}\left(-\overline{\textstyle{\int}_{\xi}^{\infty^{+}}},\mathbb{B}_{\xi}\right)}{\Theta_{\mathbf{pq}}\left(-\overline{\textstyle{\int}_{\xi}^{\infty^{-}}},\mathbb{B}_{\xi}\right)}.$

The next step is expressing the complex conjugates
$\overline{\textstyle{\int}_{\xi}^{\infty^{\pm}}}$ in terms of the Abel maps
of $\bar{\xi}$. Notice that $\bar{\mathbf{z}}=2\Re(\mathbf{z})-\mathbf{z}$ for
any $\mathbf{z}\in\mathbb{C}^{g}$ and in particular,

$\displaystyle\overline{\textstyle{\int}_{\xi}^{\infty^{\pm}}}=2\Re\left(\textstyle{\int}_{\xi}^{\infty^{\pm}}\right)-\textstyle{\int}_{\xi}^{\infty^{\pm}}.$

On the other hand, the integrals $\textstyle{\int}_{\xi}^{\infty^{\pm}}$ can
be written in terms of $\textstyle{\int}_{\infty^{-}}^{\infty^{+}}$, whose
real part is given explicitly by
$\Re(\textstyle{\int}_{\infty^{-}}^{\infty^{+}})=(\frac{1}{2},\ldots,\frac{1}{2})$
in Appendix B. Notice that
$\textstyle{\int}_{\xi}^{\infty^{+}}=-\textstyle{\int}_{\xi}^{\infty^{-}}$,
since the paths of the integrals $\textstyle{\int}_{\xi}^{\infty^{+}}$ and
$\textstyle{\int}_{\xi}^{\infty^{-}}$ have the same projection on
$\mathbb{C}\mathbb{P}^{1}$. Thus,

$\textstyle{\int}_{\infty^{-}}^{\infty^{+}}=\int_{\infty^{-}}^{\xi}+\int_{\xi}^{\infty^{+}}=-\int^{\infty^{-}}_{\xi}+\int_{\xi}^{\infty^{+}}=\pm
2\int_{\xi}^{\infty^{\pm}}.$

Implying,

(15)
$\overline{\textstyle{\int}_{\xi}^{\infty^{\pm}}}=-\textstyle{\int}_{\xi}^{\infty^{\pm}}\pm\Re(\int_{\infty^{-}}^{\infty^{+}}).$

The integral $\int_{\bar{\xi}}^{\xi}$ is one half of the integral along the
cycle encircling the cut $[\bar{\xi},\xi]$ in clockwise direction, which is
equivalent to $\sum_{k}a_{k}$. Then,

$\textstyle{\int}_{\bar{\xi}}^{\xi}=\frac{1}{2}\sum_{k}\int_{a_{k}}=\Re(\int_{\infty^{-}}^{\infty^{+}})=(\frac{1}{2},\ldots,\frac{1}{2}).$

Thus, substituting this value and (15) into (14) we obtain

(16)
$\overline{\mathcal{E}(\xi,\bar{\xi})}=e^{\pi\mathrm{i}\sum_{j}p_{j}}\frac{\Theta_{\mathbf{pq}}\left(\textstyle{\int}_{\bar{\xi}}^{\infty^{+}}-2\textstyle{\int}_{\bar{\xi}}^{\xi},\mathbb{B}_{\xi}\right)}{\Theta_{\mathbf{pq}}\left(\textstyle{\int}_{\bar{\xi}}^{\infty^{-}},\mathbb{B}_{\xi}\right)}.$

Finally, the form (13) of the complex conjugate of the Ernst potential follows
by considering the translation

$\Theta_{\mathbf{pq}}\left(\textstyle{\int}_{\bar{\xi}}^{\infty^{+}}-2\textstyle{\int}_{\bar{\xi}}^{\xi},\mathbb{B}_{\xi}\right)=e^{-2\pi\mathrm{i}\langle\mathbf{p},2\textstyle{\int}_{\bar{\xi}}^{\xi}\rangle}\Theta_{\mathbf{pq}}\left(\textstyle{\int}_{\bar{\xi}}^{\infty^{+}},\mathbb{B}_{\xi}\right)$

by the lattice vector $2\textstyle{\int}_{\bar{\xi}}^{\xi}=(1,\ldots,1)$,
which is obtained with the property (11) of theta functions. ∎

This proposition implies that the real part of the Ernst potential (2) can be
expressed in a simple form, since all the involved theta functions are now in
terms of the same period matrix $\mathbb{B}_{\xi}$, which allows us to use
Fay’s identity (12). For ease of readability, we omit the second argument
$\mathbb{B}_{\xi}$ in the sequel. Thus, using (13) from Proposition 3.1, we
obtain

$\displaystyle\mathcal{E}(\xi,\bar{\xi})+\overline{\mathcal{E}(\xi,\bar{\xi})}=e^{-\pi\mathrm{i}\sum_{j}p_{j}}\left[\frac{\Theta_{\mathbf{pq}}(\textstyle{\int}^{\infty^{+}}_{\xi})\Theta_{\mathbf{pq}}(\textstyle{\int}^{\infty^{-}}_{\bar{\xi}})+\Theta_{\mathbf{pq}}(\textstyle{\int}^{\infty^{+}}_{\bar{\xi}})\Theta_{\mathbf{pq}}(\textstyle{\int}^{\infty^{-}}_{\xi})}{\Theta_{\mathbf{pq}}(\textstyle{\int}^{\infty^{-}}_{\xi})\Theta_{\mathbf{pq}}(\textstyle{\int}^{\infty^{-}}_{\bar{\xi}})}\right],$

and considering Fay’s identity (12) with $\mathbf{z}=0$, $a=\xi$,
$b=\bar{\xi}$, $c=\infty^{-}$, $d=\infty^{+}$; the lemma

$\frac{E(\infty^{-},\xi)E(\infty^{+},\bar{\xi})}{E(\infty^{-},\bar{\xi})E(\xi,\infty^{+})}=1,$

which is proven in [4]; and the property $E(x,y)=-E(y,x)$ of the prime form,
we observe that

$\displaystyle\Theta_{\mathbf{pq}}(\textstyle{\int}^{\infty^{+}}_{\xi})\Theta_{\mathbf{pq}}(\textstyle{\int}^{\infty^{-}}_{\bar{\xi}})+$
$\displaystyle\Theta_{\mathbf{pq}}(\textstyle{\int}^{\infty^{+}}_{\bar{\xi}})\Theta_{\mathbf{pq}}(\textstyle{\int}^{\infty^{-}}_{\xi})=$
$\displaystyle\frac{E(\infty^{-},\infty^{+})E(\xi,\bar{\xi})}{E(\xi,\infty^{-})E(\bar{\xi},\infty^{+})}\Theta_{\mathbf{pq}}(0)\Theta_{\mathbf{pq}}(\textstyle{\int}_{\xi}^{\infty^{+}}+\textstyle{\int}_{\bar{\xi}}^{\infty^{-}}).$

Therefore, the real part of the Ernst potential is

(17) $\Re(\mathcal{E}(\xi,\bar{\xi}))=Q\cdot
e^{-\pi\mathrm{i}\sum_{j}p_{j}}\frac{\Theta_{\mathbf{pq}}(0)\Theta_{\mathbf{pq}}(\textstyle{\int}_{\bar{\xi}}^{\xi})}{\Theta_{\mathbf{pq}}(\textstyle{\int}_{\xi}^{\infty^{-}})\Theta_{\mathbf{pq}}(\textstyle{\int}_{\bar{\xi}}^{\infty^{-}})},$

where

$Q=Q(\xi,\bar{\xi})=\frac{1}{2}\frac{E(\infty^{-},\infty^{+})E(\xi,\bar{\xi})}{E(\xi,\infty^{-})E(\bar{\xi},\infty^{+})}=\frac{\Theta(\textstyle{\int}_{\xi}^{\infty^{-}})\Theta(\textstyle{\int}_{\bar{\xi}}^{\infty^{-}})}{\Theta(0)\Theta(\textstyle{\int}_{\xi}^{\bar{\xi}})}.$

The latter equality is obtained from the fact that $\mathcal{E}\equiv 1$ if
$\mathbf{p},\mathbf{q}=0$.

Finally, with formula (17), we show that the potential (2) solves the Ernst
equation.

######  Theorem 3.2.

Let $\mathcal{E}(\xi,\bar{\xi})$ be the potential defined by (2) with fixed
arbitrary characteristics $\mathbf{p}\in\mathbb{R}^{g}$,
$\mathbf{q}\in\mathbb{C}^{g}$ satisfying the reality conditions (3). Then, it
solves the Ernst equation (6) in complex coordinates, which can be written as

(18)
$(\mathcal{E}+\overline{\mathcal{E}})\Delta\mathcal{E}=8\mathcal{E}_{\xi}\mathcal{E}_{\bar{\xi}}.$

###### Proof.

Since the phase factor in (2) is independent of $\xi$ if the basis cycles of
the first homology group are chosen such that they satisfy conditions (8), the
derivatives $\mathcal{E}_{\xi}$, $\mathcal{E}_{\bar{\xi}}$ and the Laplacian
$\Delta\mathcal{E}$ are just those shown in [4] multiplied by this phase
factor. Namely,

$\displaystyle\mathcal{E}_{\xi}$
$\displaystyle=\frac{1}{2}c_{2}(\infty^{-},\xi,\infty^{+})e^{-\pi\mathrm{i}\sum_{j}p_{j}}\frac{\Theta_{\mathbf{pq}}(0)}{\Theta_{\mathbf{pq}}^{2}(\textstyle{\int}_{\xi}^{\infty^{-}})}D_{\xi}\Theta_{\mathbf{pq}}(0),$
$\displaystyle\mathcal{E}_{\bar{\xi}}$
$\displaystyle=\frac{1}{2}c_{2}(\infty^{-},\bar{\xi},\infty^{+})e^{-\pi\mathrm{i}\sum_{j}p_{j}}\frac{\Theta_{\mathbf{pq}}(\textstyle{\int}_{\bar{\xi}}^{\xi})}{\Theta_{\mathbf{pq}}^{2}(\textstyle{\int}_{\xi}^{\infty^{-}})}D_{\bar{\xi}}\Theta_{\mathbf{pq}}(\textstyle{\textstyle{\int}}^{\bar{\xi}}_{\xi}),$
$\displaystyle\Delta\mathcal{E}=\frac{1}{Q}c_{2}(\infty^{-},\xi,\infty^{+})c_{2}(\infty^{-},\bar{\xi},\infty^{+})e^{-\pi\mathrm{i}\sum_{j}p_{j}}\frac{\Theta_{\mathbf{pq}}(\textstyle{\int}_{\bar{\xi}}^{\infty^{-}})}{\Theta_{\mathbf{pq}}^{3}(\textstyle{\int}_{\xi}^{\infty^{-}})}D_{\bar{\xi}}\Theta_{\mathbf{pq}}(\textstyle{\textstyle{\int}}^{\bar{\xi}}_{\xi})D_{\xi}\Theta_{\mathbf{pq}}(0),$

where the coefficients $c_{2}(\infty^{-},\xi,\infty^{+})$,
$c_{2}(\infty^{-},\bar{\xi},\infty^{+})$ are functions defined in terms of
prime forms [4]. Therefore, with these values for $\mathcal{E}_{\xi}$,
$\mathcal{E}_{\bar{\xi}}$ and $\Delta\mathcal{E}$, and considering the form
(17) for $\Re(\mathcal{E})$, it follows that the potential (2) solves the
Ernst equation. ∎

######  Remark 3.3.

If the exponential factor in (2) is not taken into account, this class of
solutions is only valid if $\sum_{j}p_{j}$ is an integer, e.g.,
$\mathbf{p}=0$. Otherwise, formula (17) does not hold.

######  Remark 3.4.

An equivalent form of the class of solutions (2) is

(19)
$\mathcal{E}(\xi,\bar{\xi})=e^{-\pi\mathrm{i}\sum_{j}p_{j}}\frac{\Theta_{\mathbf{pq}}\left(\textstyle{\int}_{\xi}^{\infty^{+}}+\frac{1}{2}\Delta,\mathbb{B}_{\xi}\right)}{\Theta_{\mathbf{pq}}\left(\textstyle{\int}_{\xi}^{\infty^{-}}+\frac{1}{2}\Delta,\mathbb{B}_{\xi}\right)},$

with fixed arbitrary $\mathbf{p}\in\mathbb{R}^{g}$,
$\mathbf{q}\in\mathbb{C}^{g}$ subject to the reality condition
$\Re(\mathbf{q})+R\mathbf{p}=0$, where $R$ is given by (21) and $\Delta:={\rm
diag}(R)$.

##  Appendix A Complex conjugate of theta functions

######  Proposition A.1.

Let $\mathbb{B}$ be any Riemann matrix $\mathbb{B}\in\mathbb{H}^{g}$ whose
real part $R:=\Re(\mathbb{B})$ has half-integer coefficients. Then, the
complex conjugate of its associated theta function with characteristics (10)
can be written as

(20)
$\displaystyle\overline{\Theta_{\mathbf{pq}}(\mathbf{z},\mathbb{B})}=\alpha\cdot\Theta_{\mathbf{pq}}(-\bar{\mathbf{z}}-2\Re(\mathbf{q}+\mathbb{B}\mathbf{p})+{\rm
diag}(R),\mathbb{B}),$

where $\alpha\in\mathbb{C}$ is a constant that does not depend on
$\mathbf{z}$.

###### Proof.

In general, from the definition (10) of multi-dimensional theta functions with
characteristics $\mathbf{p}\in\mathbb{R}^{g}$ and
$\mathbf{q}\in\mathbb{C}^{g}$, it can be observed that their complex conjugate
can be written as

$\displaystyle\overline{\Theta_{\mathbf{pq}}(\mathbf{z},\mathbb{B})}=\Theta_{\mathbf{pq}}(-\bar{\mathbf{z}}-\bar{\mathbf{q}}-\mathbf{q},-\bar{\mathbb{B}})=\Theta_{\mathbf{pq}}(-\bar{\mathbf{z}}-2\Re(\mathbf{q}),\mathbb{B}-2R),$

for all $\mathbf{z}\in\mathbb{C}^{g}$ and $\mathbb{B}\in\mathbb{H}^{g}$.
Moreover, since $R$ is assumed to have half-integer coefficients,

$\begin{pmatrix}\mathbb{I}_{g}&-2R\\\
0&\mathbb{I}_{g}\end{pmatrix}\in\mathrm{Sp}(2g,\mathbb{Z}),$

implying that $\mathbb{B}-2R$ is symplectically equivalent to $\mathbb{B}$.
Thus, using the modular transformation of theta functions we obtain (20). ∎

##  Appendix B Choice of cycles

We choose cycles such that the action of the holomorphic involution
$\tau(x,y)=(\bar{x},\bar{y})$ on them satisfies (8). Figure 1 shows an example
of such cycles. These conditions imply that the real part of the period
matrices of $\mathcal{L}_{\xi}$ are half-integers and that they do not depend
on the parameter $\xi\in\mathbb{C}$. Indeed, due to the condition
$\tau(a_{j})=-a_{j}$, the action $\tau^{*}$ of $\tau$ on the normalized
differentials is $\tau^{*}(\omega_{k})=-\bar{\omega}_{k}$ (see [1]).
Therefore, the complex conjugate of the period matrices are of the form

$\displaystyle\bar{\mathbb{B}}_{jk}$
$\displaystyle=\int_{b_{j}}\bar{\omega}_{k}=-\int_{b_{j}}\tau^{*}(\omega_{k})=-\int_{\tau(b_{j})}\omega_{k},$
$\displaystyle=-\int_{b_{j}}\omega_{k}-\sum_{l}\int_{a_{l}}\omega_{k}+\sigma_{j}\int_{a_{j}}\omega_{k},$
$\displaystyle=-\mathbb{B}_{jk}-(1-\sigma_{j}\cdot\delta_{jk}),$

where $\sigma_{j}=1$ if $E_{j}=\bar{F}_{j}$ and $\sigma_{j}=0$ if
$E_{j},F_{j}\in\mathbb{R}$. Thus, the components of the real part of the
period matrices $\mathbb{B}_{\xi}$, which we denote $R=\Re(\mathbb{B}_{\xi})$,
are independent of $\xi$ and their explicit values are

(21) $R_{ij}=\left\\{\begin{array}[]{ll}0,&\text{if }i=j\text{ and
}E_{j}=\bar{F}_{j},\\\ -\frac{1}{2},&\text{otherwise}.\end{array}\right.$

Notice that the reality conditions (3) can be equivalently expressed as

$\Re(\mathbf{q})+R\cdot\mathbf{p}=\frac{1}{2}{\rm diag}(R).$

On the other hand, since the path $\gamma_{p}$ for the integral
$\textstyle{\int}_{\infty^{-}}^{\infty^{+}}\omega$ is chosen such that the
action of the holomorphic involution $\tau$ is given by (9), the complex
conjugate of $\textstyle{\int}_{\infty^{-}}^{\infty^{+}}\omega_{j}$ is

$\displaystyle\overline{\textstyle{\int}_{\infty^{-}}^{\infty^{+}}\omega_{j}}=\textstyle{\int}_{\gamma_{\infty}}\bar{\omega}_{j}$
$\displaystyle=-\textstyle{\int}_{\gamma_{\infty}}\tau^{*}(\omega_{j})=-\textstyle{\int}_{\tau(\gamma_{\infty})}\omega_{j},$
$\displaystyle=-\textstyle{\int}_{\gamma_{\infty}}\omega_{j}+\sum_{k}\textstyle{\int}_{a_{k}}\omega_{j}=-\textstyle{\int}_{\infty^{-}}^{\infty^{+}}\omega_{j}+1,$

which implies

(22)
$\Re\left(\textstyle{\int}_{\infty^{-}}^{\infty^{+}}\omega\right)=\frac{1}{2}\sum_{k}\textstyle{\int}_{a_{k}}\omega=\left(\frac{1}{2},\ldots,\frac{1}{2}\right).$

## References

  * [1] E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its and V.B. Matveev. Algebro-geometric approach to nonlinear integrable equations. Springer, Berlin (1994)
  * [2] F. Ernst. New formulation of the axially symmetric gravitational field problem. Phys. Rev. 167, 1175 (1968)
  * [3] J. Fay. Theta functions on Riemann surfaces. Springer-Verlag (1973)
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