# Optimal solar sail trajectories for fast deep space missions

Alesia Herasimenka, Lamberto Dell'Elce, Andreas Hein

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Alesia Herasimenka, Lamberto Dell'Elce, Andreas Hein. Optimal solar sail trajectories for fast deep space missions. 2024 AAS/AIAA Astrodynamics Specialist Conference, AAS; AIAA, Aug 2024, Broomfield (Colorado), United States. hal-04872680v2

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OPTIMAL SOLAR SAIL TRAJECTORIES FOR FAST DEEP SPACE
MISSIONS

Alesia Herasimenka*, Lamberto Dell'Elce,† and Andreas Hein‡

This paper investigates optimal trajectories for solar sails in fast deep-space
missions. By utilizing solar flybys, these missions can leverage on solar radiation
pressure assist to accelerate the solar sail away from the Sun. It enables the space-
craft to reach distant regions of the solar system or travel beyond it. To determine
feasible trajectories, we solve an optimal control problem, taking into account the
temperature constraints of the sail during its close passage near the Sun.

INTRODUCTION

Solar sail missions are promising for expeditions to the outer reaches of the solar system and interstellar travel due to their propellantless nature. One approach to achieving high velocities involves utilizing a close approach to the Sun,¹ which allows to perform a solar radiation pressure (SRP) assist, to accelerate the sail away from the solar system. This method enables speeds of up to 7 astronomical units per year (approximately 33 km/s)² using materials that are already available. A crucial determinant of solar sail performance is the area-to-mass ratio, and ongoing efforts are aimed at maximizing this parameter. For instance, previous solar sail missions had a ratio smaller than 7. The future Solar Cruiser mission aims for a ratio of around 16, while the Sundiver concept² seeks a ratio of 50 or more. Despite these advancements, achieving even higher ratios remains desirable, especially for interstellar missions.

Recent attention has focused on aerographene, an ultra-lightweight material with a density of 0.16 kg/m³,³ offering significantly improved performance potential compared to conventional materials like Mylar or Kapton. However, aerographene's completely absorptive nature poses challenges for the controllability of the sail, such as the ability to arbitrarily change its trajectory. Solar sails have inherent constraints on the thrust directions they can provide.⁴ The reflectivity of the sail directly influences its control capabilities. A highly reflective sail offers a larger range of thrust directions. Conversely, for a perfectly absorptive sail, the resulting solar radiation pressure force is always aligned with the incident solar rays, regardless of the sail's orientation relative to the Sun. For the heliocentric orbits, the controllability analysis done in⁴ does not provide any additional information. Thus, even a perfect absorptivity doesn't prevent the sail from being capable of escaping from the solar system. Nevertheless, another limitation of the aerographene material is its higher sensitivity

<footnote>*Research Associate, Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, 29, Avenue J.F Kennedy L-1855 Luxembourg.</footnote>

<footnote>†Researcher, Inria Center at Université Côte d'Azur, 06902 Valbonne, France.</footnote>

<footnote>‡Associate Professor, Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, 29, Avenue J.F Kennedy L-1855 Luxembourg.</footnote>
to the temperature, not allowing such a sail to reach a very close distance to the Sun. For these reasons, another idea would be to consider a double sail, with a perfectly reflective sail allowing to perform a close flyby of the Sun. Once the sail is at the safe distance, the satellite could drop it to allow further acceleration with the aerographene sail.

In this study, our goal is to give a preliminary analysis of possible trajectories allowing to achieve hyperbolic orbits by leveraging on the close approach to the Sun. We propose an optimal control methodology to find a trajectory that maximizes the orbital energy of a solar sail toward the farthest regions of the solar system, leveraging a solar flyby. We address the challenge of temperature constraints, which are crucial for preventing material degradation near the Sun. This involves incorporating constraints on both the state variable (distance from the Sun) and the control variable (orientation of the sail) to manage surface temperature fluctuations. Specifically, while proximity to the Sun may be necessary for gravitational and photonic assists, it also risks excessive heating, necessitating adjustments to the sail's orientation to mitigate temperature effects.

To solve this optimal control problem with combined constraints, we employ a direct approach that allows us to optimize the trajectory under the given mixed constraints. We derive optimality conditions using Pontryagin's principle to formulate an optimal control problem tailored to ideal, i.e., perfectly reflective, solar sails. We use locally optimal control laws, well-known in the literature, to find a suitable initial guess for the optimization.

## EQUATIONS OF MOTION

Let us consider a classical perturbed planar Kepler problem in Cartesian coordinates within the ecliptic plane, where the state is given by the position and velocity of the satellite:

$$x = (r, v) \tag{1}$$

The dynamics of the problem can be written by separating the drift $F_0$ and the perturbed motion $F_1$ depending on the control angle $\beta$:

$$\dot{x} = F_0(x) + F_1(x, \beta), \tag{2}$$

where

$$F_0(x) = \begin{pmatrix} v \\ -\frac{\mu}{\|r\|^3}r \end{pmatrix}, \quad F_1(x) = \begin{pmatrix} 0 \\ \mathcal{R} \cdot f_{\text{SRP}} \end{pmatrix} \tag{3}$$

In the Eq. (3), $f_{\text{SRP}}$ is the SRP force detailed in the next section, and $\mathcal{R}$ is the rotation matrix from local vertical local horizontal to the heliocentric inertia frame defined with the true anomaly $f$:

$$\mathcal{R} = \begin{pmatrix} \cos f & -\sin f \\ \sin f & \cos f \end{pmatrix} \tag{4}$$

## Solar radiation pressure

Solar sails are satellites that utilize SRP for orbital maneuvers. SRP results from the interaction between photons and the sail's surface, with its magnitude depending on the distance from the Sun
<figure><img src="image_2.png" /><figcaption>(a) Schematic representation.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>(b) Force sets for different optical properties.</figcaption></figure>

**Figure 1:** Left: Different components of the solar radiation force. Right: control sets for different reflectivity coefficients and $s = 1$. The blue sets corresponds to the perfectly reflective sail, referring to the ideal sail model, when the red set illustrates forces generated by a perfectly absorptive sail, often called a cannonball model.

to the sail, $||r||$. Let $\Phi_{SR} = 1367 \text{ W m}^{-2}$ represent the solar flux at a distance of $r_{\oplus} = 1 \text{ AU}$, and let $c$ be the speed of light. Then, a simple model for the SRP is given by: [5, Chap. 3]

$$P_{SR} = \frac{\Phi_{SR}}{c} \left(\frac{r_{\oplus}}{r}\right)^2 \qquad (5)$$

In this study, similar to,⁴ a flat sail with surface area $A$ and mass $m$ is considered. Various optical and geometric properties influence the resulting SRP force, as explained in details in,⁶ which comprises components of absorbed, reflected specularly and diffusely, and thermal radiation, denoted as $f_a$, $f_{rs}$, $f_{ru}$, $f_e$, respectively. We denote the Sun-sail direction $\hat{s}$, corresponding to $\hat{r}$, and the unit vector normal to the sail $\hat{n}$.

As illustrated in Fig. 1a, we introduce the direction of specular reflection, denoted by $\hat{\xi}$, and the unit tangent vector $\hat{t}$, which lies in the plane formed by $\hat{s}$ and $\hat{n}$. We follow [7, Chap. 2] and express the unit vectors $\hat{s}$ and $\hat{\xi}$ in terms of $\hat{n}$ and $\hat{t}$,

$$ \begin{aligned} \hat{s} &= \cos \beta \hat{n} + \sin \beta \hat{t}, \\ \hat{\xi} &= \cos \beta \hat{n} - \sin \beta \hat{t}, \end{aligned} \qquad (6) $$

so the forces presented above can be expressed as:⁸

$$ \begin{aligned} f_a &= \kappa \cos \beta \hat{s} = \kappa \cos \beta(\cos \beta \hat{n} + \sin \beta \hat{t}), \\ f_{rs} &= \kappa \rho s \cos \beta \hat{\xi} = \kappa \rho s \cos \beta(\cos \beta \hat{n} - \sin \beta \hat{t}), \\ f_{ru} &= \kappa B_f \rho (1 - s) \cos \beta \hat{n}, \\ f_e &= \kappa (1 - \rho) \frac{\varepsilon_f B_f - \varepsilon_b B_b}{\varepsilon_b + \varepsilon_f} \cos \beta \hat{n}. \end{aligned} \qquad (7) $$

In (7), $\kappa$ is a constant parameter equal to the area-to-mass ratio of the sail multiplied by the SRP: $AP_{SR} m^{-1}$. The parameter $\varepsilon$ has small magnitude, stressing the fact that the overall force is small;
$\rho \in [0, 1]$ is the fraction of reflected radiation to total amount of radiation illuminating the sail; $s \in [0, 1]$ is the fraction of specularly reflected radiation to total reflected radiation; $\varepsilon_b$ and $\varepsilon_f$ are the back and front surface emissivity coefficients, respectively, and $B_b$ and $B_f$ are back and front non-Lambertian coefficients, respectively. The SRP force is the sum of these four contributions:

$$f_{\text{SRP}} = f_a + f_{rs} + f_{ru} + f_e \qquad (8)$$

Let us use the notations used in:⁹

$$b_1 = 1 - \rho s, \quad b_2 = 2\rho s, \quad b_3 = B_f \rho(1 - s) + (1 - \rho) \frac{\varepsilon_f B_f - \varepsilon_b B_b}{\varepsilon_f + \varepsilon_b}$$

Let us introduce a vector $\hat{e}_1$ orthogonal to $\hat{s}$ such that $(\hat{s}, \hat{e}_1)$ form a frame $\mathcal{S}$. With these, in $\mathcal{S}$, the SRP force reads

$$f_{\text{SRP}} = \kappa \cos \beta \left[ \begin{array}{l} (b_1 + b_2) \cos^2 \beta + b_3 \cos \beta + b_1 \sin^2 \beta \\ b_2 \sin \beta \cos \beta + b_3 \sin \beta \end{array} \right] \qquad (9)$$

Figure 1b shows how the resulting force sets depend on the optical parameters of the sail, and in particular, its reflectivity.

## Temperature constraint

Going closer to the Sun can provide better acceleration and, thus, a more efficient solution for exiting the solar system. However, the sail's material must withstand the extreme heat near the Sun. Kapton, for example, is known to remain stable under temperatures from -269° C to 400° C. Therefore, we should impose a temperature constraint in the model.

We assume that the sail has no thermal inertia due to its very small thickness, meaning that it heats up and cools down instantaneously. We consider only the radiative heat transfer. Furthermore, we do not take into account the temperature of the satellite's body, restricting the model to the temperature of the sail alone.

$$T_{\text{sat}} = \left( T_{\text{DS}}^4 + \frac{(1 - \rho) \cdot \Phi_{\text{SR}} \cdot \cos \beta}{\|r\|^2 \sigma (\varepsilon_b + \varepsilon_f)} \right)^{\frac{1}{4}} \qquad (10)$$

where $\sigma = 5.67051 \cdot 10^{-8}$ is the Stefan-Boltzmann constant, and $T_{\text{DS}} = 3\text{K}$ refers to the temperature of the deep space.

Given that the optical properties of the sail are fixed in time ($\rho, \varepsilon_b, \varepsilon_f$), the constraint (10) is a mixed constraint, since it depends both on the state and the control variables. Thus, the closer is the sail to the Sun and the bigger is the projection of its surface facing the solar rays, the bigger is the sail's temperature. This mixed constraint is different from fixing a minimum distance from the Sun, what was previously done in the literature.¹⁰ Indeed, there exists a one-parameter family of solutions satisfying $T_{\text{sat}}(r, \beta) = T_{\text{max}}$:

$$\cos \beta = \gamma \|r\|^2, \qquad (11)$$

where $\gamma = \frac{(T_{\text{max}}^4 - T_{\text{DS}}^4) \sigma (\varepsilon_b + \varepsilon_f)}{(1 - \rho) \cdot \Phi_{\text{SR}}}$.
Note that for an ideal solar sail $\rho = 1$, and $T_{sat} = T_{DS}$ is always satisfied, since all the photons are perfectly reflected from the surface, without heating it. In reality, no surface is perfectly reflecting, and even if we use a perfectly reflective model for a simulation, we enforce a penalty on the reflectivity coefficient in the temperature equation, so that $\rho < 1$. In the following example we use $\rho = 0.88$ for the temperature constraint.⁷

## OPTIMAL CONTROL

In order to derive the necessary conditions for the optimality, we should first write the Hamiltonian of the system. We augment the system with a co-state $p = (p_r, p_v)$. The Hamiltonian is

$$H = p_r \cdot v + p_v \cdot \left( -\frac{\mu}{\|r\|^3} r + \mathcal{R} \cdot f_{\text{SRP}} \right) \quad (12)$$

The Pontryagin's maximum principle states that the optimal control $\beta^*$ is given by the maximisation of the Hamiltonian:

$$\beta^* = \arg \max_{\beta \in [-\frac{\pi}{2}, \frac{\pi}{2}]} H \quad (13)$$

If we denote $\psi = p_v \cdot \mathcal{R} \cdot f_{\text{SRP}}$, as done in,⁶ then there exists an implicit function relating $\beta, \psi$, and the optical properties of the sail given by $b_i, i = 1, 2, 3$:

$$\frac{\sin \beta(b_1 + 3b_2 \cos^2 \beta + 2b_3 \cos \beta)}{(\cos^2 \beta(b_2 \cos \beta + b_3) - \sin^2 \beta(2b_2 \cos \beta + b_3))} = \frac{\psi_2}{\psi_1} \quad (14)$$

For more details on the geometrical interpretation of the Pontryagin's maximum principle (PMP) we invite the reader to refer to.⁶ For a perfectly reflective solar sail, the control law is well-known:⁹

$$\beta = \tan^{-1} \left( \frac{-3 + \sqrt{9 + \frac{\psi_2^2}{\psi_1^2}}}{4 \frac{\psi_2^2}{\psi_1^2}} \right) \quad (15)$$

In optimal control theory, managing mixed constraints can be approached using different methods: either by incorporating the constraints into the Hamiltonian of the system and deriving the necessary optimality conditions or by employing a multi-arc solution using the multiple shooting numerical method. To determine the most suitable approach, it is essential first to analyze the structure of the solution. Therefore, we propose to solve this optimal control problem (OCP) using direct methods. Our goal is to maximize the final orbital energy of the satellite

$$\varepsilon(t_f) = \frac{-\mu}{\sqrt{\|r(t_f)\|}} + \frac{1}{2} \|v(t_f)\|^2 \quad (16)$$

under the temperature constraint. The resulting OCP is:

$$
\begin{array}{ll}
\max & \varepsilon(t_f) \quad \text{subject to} \\
& \dot{x} = F_0(x) + F_1(x, \beta), \\
& T_{\text{sat}} \le T_{\text{max}} \\
& x(0) = x_0, \quad t \in [0, t_f]
\end{array}
\quad (17)
$$
<figure><img src="image_4.png" /><figcaption>**Figure 2:** Excess velocity for hyperbolic orbits achieved after one year of simulations using locally optimal control laws with a perfectly reflective solar sail. The white space corresponds to the case where the hyperbolic orbit was not achieved after one year.</figcaption></figure>

for the given initial position $x_0$ and the final time $t_f$. The following section gives more details on the choice of the initial conditions. This problem can initially be solved for a perfectly reflective solar sail. Subsequently, by employing differential continuation, we can obtain the optimal solution for a non-ideal sail by gradually deforming the control set. This approach can also be applied to other parameters of the problem, such as the area-to-mass ratio and temperature modeling.

# NUMERICAL SIMULATIONS

## Case study

We consider a scenario where the sail is first brought to a close distance to the sun, either by the SRP propulsion, or directly by a launcher. To simplify the case study, we consider that the satellite follows an elliptic orbit with an apoapsis at 1 astronomical unit (AU), connecting the departure from Earth to the initial point, from which the trajectory is optimized onwards. The closer is this initial point, the faster can the sail achieve a hyperbolic orbit.

Besides the initial distance from the Sun, the area-to-mass ratio of the sail has also a major impact on the final velocity of the sail. Figure 2 shows achievable hyperbolic excess velocities by a solar sail depending on its closest approach to the Sun and the area-to-mass ratio after a single passage through the perihelion using local-optimal control laws described in the next section. The white zones corresponds to the simulations where the hyperbolic orbit was not achieved after one year of simulation. For comparison, aerographene solar sail could achieve area-to-mass ratio surpassing 1000, but these results don't take into account the mass of the payload and remain theoretical until now, as the aerographene is still under development and is not yet produced industrially.
<figure><img src="image_5.png" /><figcaption>**Figure 3:** Trajectory achieved with locally optimal control laws and corresponding orbital energy.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4:** Trajectories for different initial distances from the Sun and their energies.</figcaption></figure>

## Initial guess provided by local-optimal feedback control law

Using local-optimal control law given by Eq. (15), it is possible to obtain an initial guess on the optimized trajectory. Figure 3 shows a trajectory achieved by using the local-optimal control law for a perfectly reflective solar sail with an area-to-mass ratio of 45 m/kg, allowing to instantaneously maximize the semi-major axis. The initial position is fixed at 0.15 AU from the Sun. The red arc shows an elliptic orbit connecting the Earth orbit to the initial position. The blue trajectory corresponds to the elliptic trajectory of the sail, while the hyperbolic arc is illustrated in purple. Yellow arrows show the orientation of the SRP. Hyperbolic character of the orbit can be verified by calculating the orbital energy, represented in the Fig. 3. Hyperbolic orbit is achieved as soon as $ε > 0$.

Figure 4 shows the results in terms of the trajectory and the orbital energy, respectively, for different initial distances from the Sun (0.10, 0.15 and 0.30 AU). The yellow trajectory corresponds to the previous example starting at 0.15 AU, when the blue and the red start from 0.10 AU and 0.30 AU from the Sun, respectively. Bringing sail to 0.10 AU enables to achieve a hyperbolic orbit directly, without requiring an additional passage through the perihelion, but can pose problems in terms of the excessive temperature of the materials. The yellow trajectory starting at 0.30 AU, on the contrary, requires 2 additional flybys to achieve a hyperbolic orbit. Figure 4 shows the corresponding energy for all three trajectories, showing that the increases occur exactly at the perihelion. The energy plot further shows that the final energy of the trajectory starting at 0.30 AU, after 8 years of simulation, is actually higher than that of the trajectory starting at 0.10 AU, which achieved a hyperbolic velocity without additional flybys. This example highlights that maximizing energy might not always be the most suitable objective function, depending on the mission scenario. It may
<figure><img src="image_7.png" /><figcaption>Figure 5: Optimized trajectory arc and the corresponding orbital energy.</figcaption></figure>

be more advantageous to perform additional flybys to achieve a higher velocity, even if it means spending more time within the solar system. For very long missions, such as interstellar travel, this approach can ultimately be preferable.

## Optimized trajectories achieved using direct methods

Now, using the locally optimal trajectories, it is possible to initialize a shooting method for solving the OCP. In order to do that, we use the *OptimalControl.jl* toolbox available in Julia language and allowing to solve different OCP using direct and indirect methods.

*Trivial solution without the constraints on the orientation of the sail.* The very first simulation shows an obvious solution of the problem, which is launching the satellite into the Sun. Indeed, passing through the Sun numerically allows to maximize the energy, since $\epsilon \to \infty$ as $\|r\| \to 0$. At the same time the temperature constraint is not violated, since for the $\beta = \pm \frac{\pi}{2}$ the satellite stays “invisible” for the Sun as $\cos \beta = 0$, and its temperature is equal to the temperature of the deep space. This result is due to the fact that the satellite is approximated as a thin sail, and no surface is exposed to the Sun when its is aligned with the solar rays. In reality, the body of the satellite will still receive the light from the Sun and heat up. A possible solution to this issue without complicating the model could be either to add a penalty in the temperature constraint, such that the term $\cos \beta$ becomes $\cos(\beta) + k$, where k is a small constant; another solution is to constraint the orientation of the sail such that $\|\beta\| \le \frac{\pi}{2} - k$. In the following simulations, we impose an additional constraint on the control

$$ \|\beta\| \le 0.9 \cdot \frac{\pi}{2}, \qquad (18) $$

which actually has a physical meaning since it's impossible to maintain the sail perfectly oriented to the solar rays because of the attitude limitations related to power generation.¹¹

<footnote>*https://control-toolbox.org/OptimalControl.jl</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 6:** Globally optimized trajectory and the corresponding orbital energy.</figcaption></figure>

*Trajectories obtained by continuing the initial time.* After adding the control constraint (18) to the OCP (17), we use the local-optimal trajectory as an initial guess. Given that the globally optimized trajectory is very different from the initial guess, to allow relatively fast convergence of the solver, we first solve the problem for a small arc $[t_{\text{init}}, t_f]$, $t_{\text{init}} \gg t_0$ instead of the whole trajectory. Figure 5 shows the optimized trajectory arc and its energy respectively. It is clear from the resulting energy that the new trajectory has a better final cost.

Then, continuation on the initial conditions allows to gradually retrieve the whole optimized trajectory. Figure 6 shows the achieved trajectory as well as the resulting orbital energy. Comparison of both trajectories in Figure 6 shows that the optimized trajectory achieved with the direct methods allows to achieve a hyperbolic orbit faster, and the resulting final energy is higher that for the local optimal solution: $1.2 > 0.3$. A sudden decrease in the energy of the optimized trajectory around $t \approx 2.5$ years corresponds to the perihelion passage, where the satellite approaches the Sun for further acceleration.

Figure 7 shows the solution of the problem in terms of the state $(r_1, r_2, v_1, v_2)$ and the costate $(p_{r1}, p_{r2}, p_{v1}, p_{v2})$ variables. The costate variables have discontinuities ("jumps") which can be explained by the constraint violation. Indeed, Figure 8 shows the temperature of the sail along its trajectory, and we can see a peak in the temperature which happens at the perihelion, when the satellite is close to the Sun. Nevertheless, the constraint set to $T_{\text{max}} = 600$ K is satisfied. We suppose that there exists a *touch point*, where the constraint is activated, what corresponds to the observed peak, and to the discontinuities in the costate variables. These discontinuities have to be taken into account for the indirect resolution.

Finally, Figure 9 illustrated the solution of the problem in terms of the control angle $\beta$. The saturation of the control at $t \approx 2.5$, visible on the zoomed picture, corresponds to the passage at perihelion, where the sails also flips around, what explains the change of the sign of the resulting angle. This is due to the fact that decreasing the exposed area to the sunlight allows to satisfy the imposed temperature constraint, and thus, to approach the Sun closer. The value of 81 deg
<figure><img src="image_9.png" /><figcaption>**Figure 7:** Solution of the optimization.</figcaption></figure>

corresponds to the imposed constraint on the control given by Eq. (18). The control has an entire bang structure: the thrust is always maximum, because the control set has no interior for the solar sails, only the orientation changes. Of course, the presented solution makes an assumption that the orientation of the sail can be changed instantaneously, which is not feasible in practice. Additional constraints on the control of the solar sail can improve the feasibility of the concept.

## CONCLUSION AND WAY FORWARD

We performed a preliminary study of possible trajectories for deep space missions using solar sails. Even though the local optimal control law gives a fast and good approximation of the trajectory, the resulting solution is far from the optimal one. Using direct methods to solve an OCP combined with continuation on the parameters of the model allows to retrieve the whole optimized trajectory for such maneuvers, with a much better result in terms of the final cost. Using direct ap-
<figure><img src="image_10.png" /><figcaption>**Figure 8:** Temperature of the sail along its trajectory.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 9:** The resulting control angle $\beta$.</figcaption></figure>

proach allows to find an approximation of the costate variables. In the given simulation we suppose a *touch point* which occurs on the temperature constraint, resulting in the jumps of the costate variables. These discontinuities have to be accounted for the further computations implying shooting methods.

The next step is to solve the problem with an indirect method, and then to use differential continuation to find trajectories feasible by a non-ideal solar sail.

## ACKNOWLEDGMENT

This work was partially supported by the Prix Coup de Cœur from the President of Université Côte d'Azur, within the framework of the IdEx UCAJEDI program promoting research excellence.

The authors would like to thank Prof. Jean-Baptiste Caillau, Université Côte d'Azur, the developer of the *OptimalControl.jl* toolbox, for his help with the use of the software.
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