# Trajectory design and optimization of a solar sail probe

Jeric Garrido<EMAIL_ADDRESS>Jose Perico Esguerra11footnotemark: 1
<EMAIL_ADDRESS>National Institute of Physics, University of the
Philippines, Diliman, Quezon City 1101, Philippines

###### Abstract

There is a desire to observe the sun’s poles to further deepen our
understanding of solar activity. However, because of the large speeds needed
to perform out-of-ecliptic plane maneuvers, chemical and electric rocket
propulsion mechanisms have been proven to be costly and impractical, leaving
alternative space technology systems like solar sails to be considered for
these applications. In this paper, we study the possibility of using a solar
sail as a probe observing the sun. We design and optimize the trajectories of
the solar sail probe through the surface constraint approach, with the
assumption that the sail moves on a displaced spherical surface. We first
review the surface constraint approach, focusing on its important assumptions
and limitations. Then, we solve and obtain a family of radial and azimuthal
trajectory equations by choosing the correct constraint equation. We
characterize the trajectories based on the functional dependence of the sail’s
orientation with the polar angle. Finally, we determine the trajectories of
the probe that will give us the minimum flight time. Results show that
increasing the number of mission stages decreases the total flight time, with
minimal changes in the sail’s radial and polar velocities. Furthermore,
changing the functional dependence of the clock angle resets the azimuthal
velocity, making the sail not reverse its direction and directly approach the
sun along the spherical surface.

###### keywords:

solar sail sun probe , displaced sphere , surface constraint approach ,
Laplace-Runge Lenz vector

††journal: Advances in Space Research

##  1 Introduction

Solar sailing is a propulsion technology which uses the radiation pressure of
the sun for space travel (Garrido & Esguerra, 2023a; McInnes, 2004; Gong &
Macdonald, 2019). Through the momentum transfer of solar photons to its
surface, a solar sail is capable of attaining its mission objectives without
the need for a conventional chemical thruster. While this novel mission
concept was developed by Soviet scientists Tsiolkovsky and Tsander in the
1920s, it took almost a century for solar sailing to be successfully
demonstrated with the launch of IKAROS in 2010 (Vulpetti et al., 2014; Fu et
al., 2016; McInnes, 2004; Tsuda et al., 2011). Since then, additional sails
like NanoSail D2, LightSail 1, and LightSail 2 have been successfully
deployed, showing the feasibility of using solar sailing technology in various
mission applications (Alhorn et al., 2011; Betts et al., 2017; Spencer et al.,
2021). The future of solar sailing is promising with the recently-launched NEA
Scout mission by the NASA and the upcoming OKEANOS mission by JAXA (Pezent et
al., 2018; Mori et al., 2020), as well as other proposed missions like the
Sundiver concept which may enable solar sails to reach their target
destinations with ultrafast speeds (Turyshev et al., 2023).

Solar sails have potential applications in some missions where the use of
conventional rocket propulsion mechanisms is impractical. For example, solar
sails can be used in deep space exploration and missions towards Neptune and
beyond (Vulpetti, 2012; Lingam & Loeb, 2020; Dachwald, 2005; Macdonald et al.,
2010), in generation of non-Keplerian orbits (McKay et al., 2011; Gong et al.,
2011; McInnes & Simmons, 1992; McInnes, 1998), and in planetary and asteroid
observations (Lappas et al., 2009; Ozimek et al., 2009; Bookless & McInnes,
2008; Baoyin & McInnes, 2006; Waters & McInnes, 2007; Peloni et al., 2016;
Song & Gong, 2019; Morrow et al., 2001; Macdonald et al., 2007; Zeng et al.,
2015, 2016, 2019; Heiligers et al., 2019). For this paper though, we focus on
one particular application which is on monitoring solar activity. There is a
desire to study and perform measurements at the polar regions of the sun,
which necessitates off-ecliptic plane orbits or trajectories with high
inclination (Turyshev et al., 2023). Consequently, a large $\Delta v$ is
necessary for these missions, which is costly and difficult for typical
chemical and electric propulsion mechanisms (Liewer et al., 2013; Turyshev et
al., 2023). The continuously-applied nature of solar radiation force allows
for these large changes in velocity, making it viable for these mission
applications. For example, the GeoStorm mission concept was envisioned to
provide real-time monitoring of Sun activity (West, 2004; Garrido & Esguerra,
2023a). On the other hand, the Solar Polar Imager (SPI) mission concept was
proposed to place a light sail in a highly-inclined circular orbit of radius
0.48 AU and at least $0.75^{\circ}$ inclination to provide measurements from
high altitudes (Thomas et al., 2020). The proponents of the SPI mission claim
that by observing the sun’s poles, our understanding of the mechanism of solar
activity cycles, polar magnetic field reversals, and the internal structure
and dynamics of the sun will be revolutionized (Liewer et al., 2013). The SPI
mission concept has been referenced in other mission proposals, such as the
High Inclination Solar Mission (HISM) concept which aims to observe the sun
and the heliosphere using remote, in situ and plasma wave instruments
(Kobayashi et al., 2020). On the other hand, the POLARIS mission concept was
proposed to address the limitation of the SPI mission, which is to determine
the relationship between the magnetism and dynamics of the sun’s polar regions
(Appourchaux et al., 2009).

Two aspects of solar sailing research are trajectory design and optimization.
Similar to any spacecraft in continuous propulsion, the trajectory equation of
a solar sail can be obtained by solving its equation of motion, yielding
numerical solutions. The nonlinear nature of the solar sail equation of motion
makes obtaining optimal analytic trajectories difficult (Conway, 2010).
However, for very special cases, semi-analytic solutions can also be
determined, which can be used as preliminary guesses for more robust numerical
solutions. A set of good initial guesses is necessary since the accuracy of
the final trajectories mainly depend on these solutions. Some semi-analytic
solutions for continuously-propelled systems such as the logarithmic spiral
trajectory and the exponential sinusoid have been studied in Refs.
(Petropoulos & Sims, 2002; McKay et al., 2011).

Different approaches have been developed to obtain these semi-analytic
trajectories. For instance, the shape-based approach developed in Petropoulos
(2003) and Petropoulos & Longuski (2004) a priori assumes a form of the
trajectory equation with unknown parameter values. These parameters are then
adjusted to match with the mission requirements. These shaping algorithms have
been used in solving the Lambert’s problem for exponential sinusoids (Izzo,
2006), as well as in three-dimensional rendezvous trajectories (Vasile et al.,
2007). Other shaping methods include using Fourier series (Taheri &
Abdelkhalik, 2012) and Bezier curves (Fan et al., 2020) in trajectory design
and optimization.

Previously, we proposed another approach of solar sail trajectory design and
optimization, which is the surface constraint approach (Garrido & Esguerra,
2023a). Unlike shape-based algorithms which assume a form of the orbit or the
trajectory equation, in this approach, we directly solve the solar sail
equation of motion by deriving a generalized Laplace-Runge-Lenz vector, and
then assume a surface where the sail is expected to move. The method was found
to be useful in obtaining an optimal transfer towards the orbit of an asteroid
with a highly-inclined orbit (Garrido & Esguerra, 2023a, b).

In this paper, we explore the possibility of using a solar sail as a probe
approaching the sun. To do this, we use the surface constraint approach in the
design and flight time optimization of a solar sailing spacecraft for a sun
probe mission. Unlike in our previous study where we applied the method to a
solar sail on a cylindrical surface of revolution, we want to further show the
method’s versatility by solving a different surface, in this case a sphere
displaced from the ecliptic plane. After obtaining a family of trajectories,
we can then choose the one that gives us the minimum flight time.

The paper is organized as follows: We first review the important aspects of
the surface constraint approach (Section 2). We then use this approach in
obtaining the families of trajectories of a solar sail moving on a displaced
spherical surface (Section 3). It turns out that the form of the trajectories
depend on how the clock angle varies with the polar angle. Finally, we apply
our approach in the design and optimization of a solar probe trajectory
(Section 4).

##  2 A Review of the Surface Constraint Approach

We use the surface constraint approach to design the trajectories of a solar
sail probe that is constrained to move on a displaced sphere (Garrido &
Esguerra, 2023a). Let us briefly discuss this approach in this section.

Consider a solar sail that is in a heliocentric inertial reference frame
represented using spherical coordinates $(r,\theta,\varphi)$. This sail
follows ideal force model such that it is rigid and flat, its surface is
perfectly-reflecting, and the sun is an ideal blackbody and a point source of
radiation. Then, if $\hat{\mathbf{n}}$ is the unit normal to the sail’s
surface, the equation of motion in coordinate-free form, using a heliocentric
inertial reference frame, is given by

$\ddot{\mathbf{r}}=-\frac{\mu}{r^{2}}\hat{\mathbf{r}}+\frac{\beta\mu}{r^{2}}(\hat{\mathbf{n}}\cdot\hat{\mathbf{r}})^{2}\hat{\mathbf{r}}$
(1)

where $\beta$ is the sail’s lightness number or the ratio of the solar
radiation pressure force and the gravitational force, $\mu=GM_{\odot}$ is the
universal gravitational parameter, $G$ is the universal gravitational constant
and $M_{\odot}$ is the solar mass. We solve this equation of motion, yielding
solutions of the form $r(\theta,\varphi)$. (See Figure 1)

Fig. 1: A solar sail on a three-dimensional heliocentric inertial reference
frame.

Representing the unit normal vector in this coordinate system, two angles
define the unit vector: the cone angle which measures the angle between the
radial unit vector $\hat{\mathbf{r}}$ and $\hat{\mathbf{n}}$, and the clock
angle which measures the projection of $\hat{\mathbf{n}}$ with the local
horizontal plane (Garrido & Esguerra, 2023a). Consequently, the unit normal
vector can be decomposed into its radial, azimuthal and polar components as
follows:

$\begin{split}\hat{\mathbf{n}}&=\hat{\mathbf{r}}\cos\alpha+\hat{\boldsymbol{\theta}}\sin\alpha\cos\delta+\hat{\boldsymbol{\varphi}}\sin\alpha\sin\delta\\\
&=\hat{\mathbf{r}}n_{r}+\hat{\boldsymbol{\theta}}n_{\theta}+\hat{\boldsymbol{\varphi}}n_{\varphi}\end{split}$
(2)

We have shown in Garrido & Esguerra (2023a, b) that if the radial and polar
components of the velocity are related by a function $G(\theta)$, which we
define as the constraint equation, that is, if

$\dot{r}=rG(\theta)\dot{\theta},$ (3)

then one can determine a generalized Laplace-Runge-Lenz (LRL) vector, that in
return can be used to generate orbit and/or trajectory equations. This is
possible provided that the cone angle, and hence the radial component of the
unit normal vector, is constant throughout the mission. However, there is no
limit on the form of the polar and azimuthal components, except the fact that
the unit vector must be normalized.

Consequently, the trajectory of the sail can be specified using two equations:
the radial equation which relates the radial and polar components, and the
azimuthal equation which relates the azimuthal and polar coordinates. In our
approach, the polar angle $\theta$ is the independent parameter. If $r_{0}$ is
the initial position of the sail, $J$ is the squared magnitude of the LRL
vector, then the radial equation takes the form

$r(\theta)=r_{0}\exp\left(\int_{\theta_{0}}^{\theta}\mathrm{d}\nu\
G(\nu)\right)=\frac{h^{2}}{\mu+J-\beta\mu
n_{r}^{2}}\exp\left(\int_{\theta_{0}}^{\theta}\mathrm{d}\nu\ G(\nu)\right).$
(4)

On the other hand, the azimuthal equation is given by

$\varphi(\theta)=\varphi(\theta_{0})+\int_{\theta_{0}}^{\theta}\mathrm{d}\nu\frac{2n_{\varphi}(\nu)+A(\nu)}{\sin\nu(2n_{\theta}(\nu)+B(\nu))}$
(5)

where

$A(\nu)=\int_{\theta_{0}}^{\nu}\mathrm{d}\eta\ n_{\varphi}(\eta)G(\eta)$ (6)

and

$B(\nu)=\int_{\theta}^{\nu}\mathrm{d}\eta\
n_{\theta}(\eta)G(\eta)\cos(\nu-\eta).$ (7)

These two equations define the sail’s trajectory in space. The radial equation
specifies the surface where the sail is expected to move (Garrido & Esguerra,
2023a). While the radial equation is given a priori by the constraint
equation, the surface constraint approach assures that the trajectory of the
sail satisfies the solar sail equation of motion. In fact, it was shown that
for the solution not to be extraneous, the following inequality, known as the
complete trajectory characterization condition, must be satisfied:

$J=-\mu+\beta\mu
n_{r}^{3}+\frac{2\beta\mu}{G(\theta_{0})}\left(\frac{n_{r}^{2}(1-n_{r}^{2})}{n_{\theta}(\theta_{0})}\right)\geq
0.$ (8)

On the other hand, the azimuthal equation specifies the path taken by the
light sail in space. While the radial equation mainly depends on the cone
angle, the azimuthal equation depends only on the sail’s clock angle, a
consequence of the condition that the cone angle is always constant throughout
the mission. In return, the azimuthal equation is scale-independent relative
to the lightness number.

Other physical quantities from the constraint, radial and azimuthal equations.
For example, the time derivatives $\dot{r}$, $\dot{\theta}$ and
$\dot{\varphi}$ can be determined directly from equations (3), (4) and (5):

$\displaystyle{\dot{\theta}=\pm\left[\frac{\beta\mu
n_{r}^{2}}{r^{3}G(\theta)}\left(2n_{\theta}(\theta)+\int_{\theta_{0}}^{\theta}\mathrm{d}\eta\
n_{\theta}(\eta)G(\eta)\cos(\theta-\eta)\right)\right]^{1/2}}.$ (9)
$\dot{r}=\pm\left(\frac{\beta\mu}{r}n_{r}^{2}G(\theta)\left[2n_{\theta}(\theta)+\int_{\theta_{0}}^{\theta}\mathrm{d}\eta\
n_{\theta}(\eta)G(\eta)\cos(\theta-\eta)\right]\right)^{1/2},$ (10)

and

$\dot{\varphi}=\pm\frac{\mathrm{d}\varphi}{\mathrm{d}\theta}\frac{\mathrm{d}\theta}{\mathrm{d}t}=\pm\left[\frac{\beta\mu
n_{r}^{2}}{r^{3}G(\theta)}\right]^{1/2}\frac{2n_{\varphi}(\theta)+\int_{\theta_{0}}^{\theta}\mathrm{d}\eta\
n_{\varphi}(\eta)G(\eta)}{\sin\theta\left[2n_{\theta}(\theta)+\int_{\theta_{0}}^{\theta}\mathrm{d}\eta\
n_{\theta}(\eta)G(\eta)\cos(\theta-\eta)\right]^{1/2}}.$ (11)

Finally, from the polar time derivative $\dot{\theta}$, we can also determine
the total flight time of the sail from its initial position
$(r_{0},\theta_{0},\varphi_{0})$ to its final position
$(r_{f},\theta_{f},\varphi_{f})$:

$\begin{split}T(\theta_{0}|\theta_{f})=\displaystyle{\int_{\theta_{0}}^{\theta_{f}}\mathrm{d}\theta\left(\frac{r^{3}G(\theta)/\beta\mu
n_{r}^{2}}{2n_{\theta}(\theta)+\int_{\theta_{0}}^{\theta}\mathrm{d}\eta\
n_{\theta}(\eta)\ G(\eta)\cos(\theta-\eta)}\right)^{1/2}}.\end{split}$ (12)

In summary, the surface constraint approach consists of the following steps:

  * 1.

Determine the most appropriate geometry of the surface and the form of
$n_{\theta}$ (or $n_{\varphi}$) based on the mission requirements

  * 2.

From the chosen surface geometry, choose the correct surface constraint
equation (3).

  * 3.

Equation (3) immediately gives us the correct radial and azimuthal equations
depending on the initial position $(r_{0},\theta_{0},\varphi_{0})$ and
velocity $(\dot{r}_{0},\dot{\theta}_{0},\dot{\varphi}_{0})$.

  * 4.

From the radial and azimuthal equations, other physical quantities such as
velocities and flight time can then be determined.

##  3 Trajectories on a Displaced Sphere

###  3.1 Constraint Equation

As an application of the surface constraint approach, let us consider a light
sail that is constrained to move on a sphere of radius $R$ and is centered at
$(0,0,a)$ of our heliocentric coordinate system. Note that $a$ here can be
positive or negative, where $a>0$ ($a<0$) means that the center of the
displaced sphere is above (below) the ecliptic plane.

Fig. 2: A sail’s trajectory constrained on a displaced sphere. The shaded
region is the surface where the sail is constrained to move.

We first obtain the constraint equation appropriate for this system. In
rectangular coordinates, the equation of the sphere is given by

$x^{2}+y^{2}+(z-a)^{2}=R^{2}.$ (13)

This can be re-written in spherical coordinates using the transformation
$x=r\sin\theta\cos\varphi$, $y=r\sin\theta\sin\varphi$, and $z=r\cos\theta$.
The resulting equation is then

$r=a\cos\theta\pm\sqrt{R^{2}-a^{2}\sin^{2}\theta}.$ (14)

To determine $G(\theta)$, we differentiate $r$ with respect to $\theta$,
giving us

$\frac{\mathrm{d}r}{\mathrm{d}\theta}=\mp\frac{a\sin\theta}{\sqrt{R^{2}-a^{2}\sin^{2}\theta}}\left(a\cos\theta\pm\sqrt{R^{2}-a^{2}\sin^{2}\theta}\right)=\mp
r\frac{a\sin\theta}{\sqrt{R^{2}-a^{2}\sin^{2}\theta}}$ (15)

Comparing this equation with
$\frac{\mathrm{d}r}{\mathrm{d}\theta}=rG(\theta)$, the constraint equation is
thus given by

$G(\theta)=\mp\frac{a\sin\theta}{\sqrt{R^{2}-a^{2}\sin^{2}\theta}}.$ (16)

The positive root takes the upper half of the sphere from $\theta\in[0,\pi/2)$
while the negative root takes the lower half of the spherical surface i.e.
$\theta\in[\pi/2,\pi)$. While in this paper, we focus on the positive root, it
is possible to generate trajectories using the negative root of the radial
equation. Using the positive root, the radial equation takes the form

$r(\theta)=a\cos\theta+\sqrt{R^{2}-a^{2}\sin^{2}\theta}$ (17)

while the azimuthal equation depends on the functional dependence of the clock
angle on $\theta$.

###  3.2 Constant Clock Angles

The path of the sail depends mainly on the choice of $\delta(\theta)$. Let us
consider the case when both the cone and clock angles are constant throughout
the mission such that $\delta(\theta)=\delta_{0}$. Then, the azimuthal
equation is given by

$\varphi(\theta)=\varphi(\theta_{0})+\tan\delta_{0}\int_{\theta_{0}}^{\theta}\mathrm{d}\nu\dfrac{2-\int_{\theta_{0}}^{\nu}\mathrm{d}\eta\
\dfrac{a\sin\eta}{\sqrt{R^{2}-a^{2}\sin^{2}\eta}}}{\sin\nu\left[2-\int_{\theta_{0}}^{\nu}\mathrm{d}\eta\
\dfrac{a\cos(\eta-\nu)\sin\eta}{\sqrt{R^{2}-a^{2}\sin^{2}\eta}}\right]}.$ (18)

The above equation can be evaluated numerically, which together with the
radial equation, gives us the trajectory of the sail.

Figure 3 shows the trajectories of the sail for different values of the clock
angle $\delta_{0}$. In our simulations, we assume that $a=1.00$ AU, $R=1.5$
AU, and the cone angle is at $-35.26^{\circ}$, which is the optimal value for
$\alpha$. As the clock angle increases, there is also an increase in the
number of turns in the trajectory. This is signified by the large azimuthal
displacement between the sail at $\theta=0^{\circ}$ and at
$\theta=180^{\circ}$.

(a) $\delta=30^{\circ}$

(b) $\delta=45^{\circ}$

(c) $\delta=60^{\circ}$

(d) $\delta=75^{\circ}$

Fig. 3: Trajectories constrained on a displaced sphere for different constant
clock angles.

These results are consistent with what is shown in Garrido & Esguerra (2023a),
which is unsurprising because in both cases, the azimuthal displacement varies
linearly with $\tan\delta_{0}$. At a specific value of $\theta$ the integral
reduces to a constant, and thus is independent of the form of $\delta_{0}$. In
fact. one can show that this is true for any non-trivial form of $G(\theta)$.
That is, if the clock angle is constant with respect to the polar angle, then

$\varphi(\theta)=\varphi(\theta_{0})+(\tan\delta_{0})\tilde{\varphi}(G,\theta).$
(19)

where $\tilde{\varphi}(G,\theta)$ is the integral term in Eq. (5).

###  3.3 Periodic $n_{\theta}$ and $n_{\varphi}$

It is possible to use a different form of the clock angle, as long as the
normalization condition for $\hat{\mathbf{n}}$ is satisfied. Let us consider
the following functional dependence of $n_{\theta}$ and $n_{\varphi}$, for
some constant $k$:

$n_{\theta}=\sqrt{1-n_{r}^{2}}\sin k\theta;\quad\quad
n_{\varphi}=\sqrt{1-n_{r}^{2}}\cos k\theta$ (20)

The constant $\sqrt{1-n_{r}^{2}}$ assures that the unit vector is normalized.
Then, the azimuthal orbit equation takes the form

$\varphi(\theta)=\varphi(\theta_{0})+\int_{\theta_{0}}^{\theta}\mathrm{d}\nu\dfrac{2\cos
k\nu+\int_{\theta_{0}}^{\nu}\mathrm{d}\eta\ \dfrac{a\cos
k\eta\sin\eta}{\sqrt{R^{2}-a^{2}\sin^{2}\eta}}}{\sin\nu\left(2\sin
k\nu+\int_{\theta}^{\nu}\mathrm{d}\eta\ \dfrac{a\sin
k\eta\sin\eta\cos(\eta-\nu)}{\sqrt{R^{2}-a^{2}\sin^{2}\eta}}\right)}.$ (21)

The trajectories are shown in Figure 4 for different values of $k$. The
trajectories exhibit different characteristics for $k<1$ and $k\geq 1$. For
$k$ values smaller than 1 shown in Figures 4(a) to 4(c), the azimuthal
position of the sail increases monotonically as $\theta$ increases. As $k$
decreases, the total number of revolutions increases. This is due to the
relationship between $\delta$ and $\theta$. For smaller values of $k$, it will
take a larger value of $\theta$ for $\delta$ to reach $90^{\circ}$, hence we
see larger azimuthal displacements, and more revolutions for these values of
$k$.

(a) $k=1/3$

(b) $k=1/2$

(c) $k=2/3$

(d) $k=1$

(e) $k=3/2$

(f) $k=2$

Fig. 4: Trajectories constrained on a displaced sphere for different values of
$k$.

An interesting trend occurs when $k\geq 1$. For this set of $k$ values, the
clock angle will reach $90^{\circ}$ even with a polar angle smaller than
$90^{\circ}$, changing the sign of $n_{\theta}$. Consequently, the azimuthal
displacement becomes negative for angles greater than $90^{\circ}$, reversing
the trajectory in the azimuthal direction. This reversal is evident in the
trajectories shown in Figures 4(d) to 4(f).

By definition, $\delta$ only takes values from $-180^{\circ}$ to
$180^{\circ}$. Hence, for values of $k$ larger than 1, there is an critical
value of $\theta$ for which the upper limit of the clock angle is reached.
Beyond this value, the trajectory becomes ill-defined so that a re-definition
of the clock angle is necessary. We note that this upper limit of $\delta$ is
reached at about $\theta=82.5^{\circ}$ for $k=2$.

Finally, we compare our results to those written in the literature. The
increasing number of revolutions for $0<k<1$, and the apparent reversal of the
trajectories for larger values of $k$ are also observed in previous cases,
such as trajectories that are constrained on a cylinder, which we previously
studied in Garrido & Esguerra (2023a). However, one striking difference
between trajectories on a displaced sphere and on a cylinder is on the
behavior near $\theta=0$. For trajectories constrained on a displaced sphere,
more revolutions are observed near the ecliptic pole than near the ecliptic
plane (Garrido & Esguerra, 2023a). This is different from trajectories
constrained on a cylinder where the paths cluster near $\theta=90^{\circ}$.

##  4 Mission application: a solar probe

A possible mission application of orbits constrained on displaced spheres is
on observing solar activity. It is possible to use the surface constraint
approach to determine the optimal transfer towards a terminal location close
to the sun, making the sail a solar probe. We consider the range of distances
when the sail still follows nearly ideal behavior and its film’s structure
stays intact and does not deteriorate as the sail approaches the sun.
Consequently, we consider the case when the light sail’s terminal position is
at $\theta=80^{\circ}$.

We first assume that the sail is located at a distance of $1\text{ AU}$ above
the ecliptic plane of the sun. Then, the sail has a lightness number of
$\beta=0.15$ and follows the radial equation (14), where $R=a=0.5\text{ AU}$.
Further, we consider the case when the polar and azimuthal components are
periodic in $\theta$ such that the sail’s control follows equation (20).

We determine the trajectories that will make the flight time a minimum by
splitting the trajectories into different sub-intervals such that each stage
has a unique value of $k$. Assuming that the control switch is done
instantaneously, for $n$ sub-intervals the total flight time is given by

$T(\theta_{0}|\theta_{f})=\sum_{i=0}^{N}T_{k_{i}}(\theta_{i}|\theta_{i+1}).$
(22)

For the single stage trajectory, we determine the correct value of $k$ that
will make the flight time a minimum. For piecewise continuous trajectories, we
determine the transition points and the value of $k$ in each sub-trajectory.
We consider a single optimal trajectory, a trajectory with two stages, a
trajectory with three stages, and a trajectory with four stages in this paper.
We then use Brent’s method for the single-stage trajectory optimization while
we use Trust Region constrained optimization for other three trajectories
(Brent, 2013; Forsythe et al., 1977; Conn et al., 2000).

The results of the optimization process are shown in Figures 5 and 6, as well
as in Table 1. We observe that the flight time decreases upon addition of
stages. It should be noted, however, that there is hardly any change in $k$
values from the first two sub-intervals of the four-stage trajectory.
Interestingly, this slight change in the $k$ values of the first two parts,
making the number of stages four from three, improved the total flight time by
a few days.

(a) Single trajectory

(b) Two-stage trajectory

Fig. 5: Optimal single and two-stage trajectories where $n_{\theta}$ is
periodic with respect to $\theta$.

(a) Three-stage trajectory

(b) Four-stage trajectory

Fig. 6: Optimal three-stage and four-stage trajectories where $n_{\theta}$ is
periodic with respect to $\theta$. Table 1: Details for the solar probe
mission. The cone angle is set at the optimal cone angle of $-36.25^{\circ}$.

|  $k$ Values | Transition Points | Flight Time, d
---|---|---|---
Single | 1.50968650 | None | 157.83
Two-Stage |  1.72659908 1.24968798 | $(0.5217\text{ AU},58.556^{\circ},1282.01^{\circ})$ | 144.34
Three-Stage |  1.72659913 1.53541026 1.20115984 |  $(0.6480\text{ AU},49.610^{\circ},1284.34^{\circ})$ $(0.4022\text{ AU},66.286^{\circ},1284.49^{\circ})$ | 142.18
Four-Stage |  1.72659977 1.72659709 1.42118131 1.18060364 |  $(0.7559\text{ AU},40.897^{\circ},1282.74^{\circ})$ $(0.5537\text{ AU},56.377^{\circ},1284.94^{\circ})$ $(0.3508\text{ AU},69.463^{\circ},1284.96^{\circ})$ | 140.95

We can gain more understanding about these trajectories by looking at the
radial, polar, and azimuthal velocities as functions of the polar angle (See
Figure 7.) We note that the velocities here are measured with respect to the
reference point which is the sun. Obtaining the velocities relative to the
earth is straightforward though. The radial velocity is negative, indicating
that the sail points towards the direction of the sun, while the polar
velocity is positive, consistent with our convention. Because in all of the
optimal trajectories, $k>1$, it is expected that the sail reverses its
azimuthal direction, indicating a change in sign in the azimuthal velocity
plot. The sail starts with a smaller magnitude of $\dot{r}$ but a large value
of azimuthal velocity, implying that at smaller values of $\theta$, the sail’s
motion is mainly helicoidal. As $\theta$ increases, the azimuthal velocity
decreases, making the trajectory follow a more directly downward curve.
Consequently, the sail approaches the sun, increasing its speed, consistent
with the increase in the magnitude of the radial velocity in the
$\dot{r}$-vs-$\theta$ plot.

(a) Single stage

(b) Two-stage

(c) Three-stage

(d) Four-stage

Fig. 7: Velocity profiles for an time optimal transfer for a solar sail probe
with different number of stages. The vertical dashed lines indicate the
transition points.

We can also compare the velocity components for different number of stages. We
re-arranged the plots in Figure 7 to compare the velocity component for
different number of stages (See Figure 8). We observed that while there is a
noticeable change in the velocity profile from the single stage to two-stage
trajectories, there is a small variation among the velocities in the two-
stage, three-stage, and four-stage optimal trajectories. Hence, using a four-
stage optimal trajectory gives us the most optimal flight time without
expending additional effort in changing the radial and polar valocities of the
sail.

(a) Radial Velocity

(b) Polar Velocity

(c) Azimuthal Velocity

Fig. 8: Components of the velocity for different number of stages.

A different case is observed in the azimuthal component of the velocity. There
is a jump among the different parts of the
$r\dot{\varphi}\sin\theta$-vs-$\theta$ for multiple stage optimal
trajectories. These jumps occur at the transition points and are common for
the two-stage, three-stage, and four-stage transfers. Hence, for the optimal
trajectory to be followed, the sail must drastically change its azimuthal
speed near the transition point.

The transition points are located at $\theta>\pi/4$ where the trajectory is
not helicoidal anymore. At these regions, since $k>1$, the sail will reverse
its trajectory in the azimuthal direction. By changing its $k$ value at these
points, the sail corrects its orientation for it not to immediately reverse
its direction, but for it to traverse directly towards the direction of the
sun along the spherical surface. This ‘self-correcting’ mechanism makes the
total flight time a minimum.

##  5 Conclusions and recommendations

In this paper, we used the surface constraint approach in designing and
optimizing the trajectories of a solar sailing spacecraft that is constrained
on a displaced spherical surface. We showed that assuming that the cone angle
of the sail is constant, the generalized Laplace-Runge-Lenz vector is
conserved, hence both the radial and azimuthal equations can be determined. By
choosing the appropriate constraint equation, we were successful in obtaining
a family of trajectories that are highly-dependent on the form of
$\delta(\theta)$.

For the case when both the cone and clock angles are constant throughout the
mission, we found out that the trajectory becomes more helicoidal as the clock
angle increases, which is consistent with the observations in our previous
study (Garrido & Esguerra, 2023a). This result was also supported by the fact
that the azimuthal force increases as the clock angle increases.

Interesting results were also obtained for the case when the component of the
unit normal vector followed equation (20). For $k<1$, the sail traverses a
spiral with increasing number of turns as $k$ decreases. On the other hand, a
reversal of the trajectory in the azimuthal direction was observed for $k\geq
1$, which became more pronounced for larger values of $k$. These
characteristics were also observed in our previous study on a sail on a
cylindrical surface, even if they differ in the form of the constraint
equation.

So far, only the constant and periodic components of $n_{\theta}$ have been
studied in the literature. However, the surface constraint approach allows for
any form of the functional dependence of $\delta$, provided that the
normalization condition for the unit normal vector is satisfied. Studying
other forms of $n_{\theta}$ can be a topic of future research. By changing the
form of $n_{\theta}$, we can get a family of trajectories different from what
we currently have. Obtaining these families of trajectories will enrich our
library of controls that we can use for optimization. As a consequence, we can
formulate an optimization problem in $n_{\theta}$, that is, given same value
of the lightness number, the surface constraint equation, and the initial and
terminal conditions, what functional dependence of $\delta$ and/or
$n_{\theta}$ will give us the minimum flight time?

A common characteristic of the surfaces we considered for the surface
constraint approach is that they are all symmetric with respect to the $z$
axis. We can extend our analysis to those surfaces of revolution with the same
property. Furthermore, we have not yet considered the case when the constraint
equation is also a function of the azimuthal angle $\varphi$. What will happen
to the trajectory equations if that is indeed the case?

Finally, we obtained an optimal transfer trajectory for a solar probe mission.
Similar to what was done in Garrido & Esguerra (2023b), we considered split-
continuous trajectories, with each stage having a unique $k$ value. We
observed that by increasing the number of stages, the flight time becomes
smaller. There is a great improvement in the flight time from the single stage
to the two-stage optimal trajectories. Interestingly, the number of stages can
thus be increased without major changes in the sail’s velocity. At the
transition points, the sail’s azimuthal velocity resets, allowing for the sail
to correct its path and approach the sun directly. The velocities we have
obtained are comparable to those desired in the literature, for example in
Turyshev et al. (2023).

The results we have shown here can be used in the preliminary design and
optimization of solar sail missions. We hope that our results contribute in
planning future missions concerning solar observations.

## References

  * Alhorn et al. (2011) Alhorn, D., Casas, J., Agasid, E. et al. (2011).  Nanosail-D: The small satellite that could!  In Proceedings of the 25th Annual AIAA/USU Conference on Small Satellites Small but Mighty, SSC11-VI-1.  https://digitalcommons.usu.edu/smallsar/2011/all2012/1/.
  * Appourchaux et al. (2009) Appourchaux, T., Liewer, P., Watt, M. et al. (2009).  Polar investigation of the sun—polaris.  Experimental Astronomy, 23, 1079–1117. doi:10.1007/s10686-008-9107-8.
  * Baoyin & McInnes (2006) Baoyin, H., & McInnes, C. R. (2006).  Solar sail equilibria in the elliptical restricted three-body problem.  J Guid Control Dynam, 29(3), 538–543. doi:10.2514/1.15596.
  * Betts et al. (2017) Betts, B., Nye, B., Vaughn, J. et al. (2017).  Lightsail 1 mission results and public outreach strategies.  In Fourth International Symposium on Solar Sailing.
  * Bookless & McInnes (2008) Bookless, J., & McInnes, C. (2008).  Control of Lagrange point orbits using solar sail propulsion.  Acta Astronaut, 62(2-3), 159–176. doi:10.1016/j.actaastro.2006.12.051.
  * Brent (2013) Brent, R. P. (2013).  Algorithms for minimization without derivatives.  Courier Corporation.
  * Conn et al. (2000) Conn, A. R., Gould, N. I., & Toint, P. L. (2000).  Trust region methods.  SIAM.
  * Conway (2010) Conway, B. (2010).  Spacecraft trajectory optimization.  Cambridge University Press volume 29 of Cambridge Aerospace Series.
  * Dachwald (2005) Dachwald, B. (2005).  Optimal solar sail trajectories for missions to the outer solar system.  J Guid Control Dynam, 28(6), 1187–1193. doi:10.2514/1.13301.
  * Fan et al. (2020) Fan, Z., Huo, M., Qi, N. et al. (2020).  Initial design of low-thrust trajectories based on the bezier curve-based shaping approach.  Proceedings of the Institution of Mechanical Engineers, Part G: J Aerospace Eng, 234(11), 1825–1835. doi:10.1177/0954410020920040.
  * Forsythe et al. (1977) Forsythe, G. E. et al. (1977).  Computer methods for mathematical computations.  Prentice-hall.
  * Fu et al. (2016) Fu, B., Sperber, E., & Eke, F. (2016).  Solar sail technology-a state of the art review.  Prog Aerosp Sci, 86, 1–19. doi:10.1016/j.paerosci.2016.07.001.
  * Garrido & Esguerra (2023a) Garrido, J., & Esguerra, J. P. (2023a).  A surface constraint approach for solar sail trajectories.  Advances in Space Research, 71(10), 4256–4275. doi:10.1016/j.asr.2022.12.041.
  * Garrido & Esguerra (2023b) Garrido, J. V., & Esguerra, J. P. H. (2023b).  Flight time optimization of cylindrical constrained solar sail trajectories.  In Proceedings of the Samahang Pisika ng Pilipinas (pp. SPP–2023–PB–26).  volume 41.  URL: https://proceedings.spp-online.org/article/view/SPP-2023-PB-26.
  * Gong & Macdonald (2019) Gong, S., & Macdonald, M. (2019).  Review on solar sail technology.  Astrodynam, 3, 93–125. doi:10.1007/s42064-019-0038-x.
  * Gong et al. (2011) Gong, S.-P., Li, J.-F., & Zeng, X.-Y. (2011).  Utilization of an H-reversal trajectory of a solar sail for asteroid deflection.  Res Astron Astrophys, 11(10), 1123. doi:10.1088/1674-4527/11/10/001.
  * Heiligers et al. (2019) Heiligers, J., Fernandez, J. M., Stohlman, O. R. et al. (2019).  Trajectory design for a solar-sail mission to asteroid 2016 HO3.  Astrodynam, 3(3), 231–246.
  * Izzo (2006) Izzo, D. (2006).  Lambert’s problem for exponential sinusoids.  J Guid Control Dynam, 29, 1242–1245. doi:10.2514/1.21796.
  * Kobayashi et al. (2020) Kobayashi, K., Johnson, L., Thomas, H. D. et al. (2020).  The high inclination solar mission (hism).  In AGU Fall Meeting Abstracts (pp. SH011–0004).  volume 2020.
  * Lappas et al. (2009) Lappas, V., Mengali, G., Quarta, A. et al. (2009).  Practical systems design for an Earth-magnetotail-monitoring solar sail mission.  J Spacecraft Rockets, 46, 381–399. doi:10.2514/1.32040.
  * Liewer et al. (2013) Liewer, P., Alexander, D., Appourchaux, T. et al. (2013).  A solar polar imager concept: Observing solar activity from a new perspective.  NASA, Tech. Rep, .
  * Lingam & Loeb (2020) Lingam, M., & Loeb, A. (2020).  Propulsion of spacecraft to relativistic speeds using natural astrophysical sources.  Astrophys J, 894(1), 36. doi:10.3847/1538-4357/ab7dc7.
  * Macdonald et al. (2007) Macdonald, M., Hughes, G., Falkner, P. et al. (2007).  GeoSail: an elegant solar sail demonstration mission.  J Spacecraft Rockets, 44, 784–796. doi:10.2514/1.22867.
  * Macdonald et al. (2010) Macdonald, M., McInnes, C., & Hughes, G. (2010).  Technology requirements of exploration beyond Neptune by solar sail propulsion.  J Spacecraft Rockets, 47(3), 472–483. doi:10.2514/1.46657.
  * McInnes (1998) McInnes, C. (1998).  Dynamics, stability, and control of displaced non-Keplerian orbits.  J Guid Control Dynam, 21, 799–805. doi:10.2514/2.4309.
  * McInnes (2004) McInnes, C. (2004).  Solar sailing: technology, dynamics and mission applications.  Springer Science & Business Media.
  * McInnes & Simmons (1992) McInnes, C. R., & Simmons, J. F. (1992).  Solar sail halo orbits part II-Geocentric case.  J Spacecraft Rockets, 29(4), 472–479. doi:10.2514/3.55639.
  * McKay et al. (2011) McKay, R., Macdonald, M., Biggs, J. et al. (2011).  Survey of highly non-Keplerian orbits with low-thrust propulsion.  J Guid Control Dynam, 34, 645–666. doi:10.2514/1.52133.
  * Mori et al. (2020) Mori, O., Matsumoto, J., Chujo, T. et al. (2020).  Solar power sail mission of OKEANOS.  Astrodynam, 4(3), 233–248. doi:doi.org/10.1007/s42064-019-0067-8.
  * Morrow et al. (2001) Morrow, E., Acheeres, D., & Lubin, D. (2001).  Solar sail orbit as asteroids.  J Spacecraft Rockets, 38, 279–286. doi:10.2514/2.3682.
  * Ozimek et al. (2009) Ozimek, M., Grebow, D., & Howell, K. (2009).  Design of solar sail trajectories with applications to Lunar South Pole convergence.  J Guid Control Dynam, 32, 1884–1897. doi:10.2514/1.41963.
  * Peloni et al. (2016) Peloni, A., Ceriotti, M., & Dachwald, B. (2016).  Solar-sail trajectory design for a multiple near-earth-asteroid rendezvous mission.  J Guid Control Dynam, 39, 2712–2724. doi:10.2514/1.G000470.
  * Petropoulos (2003) Petropoulos, A. (2003).  A shape-based approach to automated, low-thrust, gravity-assist trajectory design..  Ph.D. thesis School of Aeronautics and Astronautics, Purdue University West Lafayette, IN.
  * Petropoulos & Longuski (2004) Petropoulos, A., & Longuski, J. (2004).  Shape-based algorithm for the automated design of low-thrust, gravity assist trajectories.  J. Spacecraft Rockets, 41, 787–796. doi:10.2514/1.13095.
  * Petropoulos & Sims (2002) Petropoulos, A., & Sims, J. (2002).  A review of some exact solutions to the planar equations of motion of a thrusting spacecraft.  In Proceedings of the the 2nd International Symposium on Low Thrust Trajectories.  Toulouse, France.
  * Pezent et al. (2018) Pezent, J. B., Sood, R., & Heaton, A. (2018).  Near Earth Asteroid (NEA) Scout solar sail contingency trajectory design and analysis.  In 2018 Space Flight Mechanics Meeting (p. 0199).  doi:10.2514/6.2018-0199.
  * Song & Gong (2019) Song, Y., & Gong, S. (2019).  Solar-sail trajectory design for multiple near-earth asteroid exploration based on deep neural networks.  Aerosp Sci Technol, 91, 28–40. doi:10.1016/j.ast.2019.04.056.
  * Spencer et al. (2021) Spencer, D. A., Betts, B., Bellardo, J. M. et al. (2021).  The Lightsail 2 solar sailing technology demonstration.  Adv Space Res, 67(9), 2878–2889. doi:10.1016/j.asr.2020.06.029.
  * Taheri & Abdelkhalik (2012) Taheri, E., & Abdelkhalik, O. (2012).  Shape based approximation of constrained low-thrust space trajectories using Fourier series.  J Spacecraft Rockets, 49, 535–546. doi:10.2514/1.58789.
  * Thomas et al. (2020) Thomas, D., Kobayashi, K., Mike, B. et al. (2020).  Solar polar imager concept.  In ASCEND 2020 (p. 4060).  doi:10.2514/6.2020-4060.
  * Tsuda et al. (2011) Tsuda, Y., Mori, O., Funase, R. et al. (2011).  Flight status of IKAROS deep space solar sail demonstrator.  Acta Astronaut, 69, 833–840. doi:10.1016/j.actaastro.2011.06.005.
  * Turyshev et al. (2023) Turyshev, S. G., Garber, D., Friedman, L. D. et al. (2023).  Science opportunities with solar sailing smallsats.  Planetary and Space Science, (p. 105744). doi:10.1016/j.pss.2023.105744.
  * Vasile et al. (2007) Vasile, M., De Pascale, P., & Casotto, S. (2007).  On the optimality of a shape-based approach based on pseudo-equinoctial elements.  Acta Astronaut, 61, 286–297. doi:10.1016/j.actaastro.2007.01.017.
  * Vulpetti (2012) Vulpetti, G. (2012).  Fast solar sailing: astrodynamics of special sailcraft trajectories.  Springer Science & Business Media volume 30 of Space Technology Library.
  * Vulpetti et al. (2014) Vulpetti, G., Johnson, L., & Matloff, G. (2014).  Solar sails: a novel approach to interplanetary travel.  Springer.
  * Waters & McInnes (2007) Waters, T. J., & McInnes, C. R. (2007).  Periodic orbits above the ecliptic in the solar-sail restricted three-body problem.  J Guid Control Dynam, 30(3), 687–693. doi:10.2514/1.26232.
  * West (2004) West, J. (2004).  The Geostorm Warning Mission: enhanced opportunities on new technology.  In 14th AAS/AIAA Space Flight Mechanics Conference (pp. 1–14).
  * Zeng et al. (2016) Zeng, X., Gong, S., Li, J. et al. (2016).  Solar sail body-fixed hovering over elongated asteroids.  J Guid Control Dynam, 39(6), 1223–1231. doi:10.2514/1.G001061.
  * Zeng et al. (2019) Zeng, X., Vulpetti, G., & Circi, C. (2019).  Solar sail H-reversal trajectory: A review of its advances and applications.  Astrodynam, 3(1), 1–15. doi:10.1007/s42064-018-0032-y.
  * Zeng et al. (2015) Zeng, X.-Y., Jiang, F.-H., & Li, J.-F. (2015).  Asteroid body-fixed hovering using nonideal solar sails.  Res Astron Astrophys, 15(4), 597. doi:10.1088/1674-4527/15/4/011.