2.1. Solar Sail Dynamics

To derive the dynamics of a solar sail propelled spacecraft it is convenient to define a Spacecraft Oriented Frame (SOF), as illustrated in Figure 1. The SOF’s X axis points in the radial direction of a heliocentric inertial frame. It has the same direction of the incoming sunlight, which is indicated by the unit vector u. The SOF’s Z axis is defined as having the same direction of the spacecraft’s orbital angular momentum, indicated by the unit vector h. The Y axis is defined in agreement with a coordinate system in dextrorotation. The sail’s orientation is represented by the unit vector n orthogonal to the solar sail surface. Two important angles are consequently defined, the azimuth α and the elevation δ, necessary to resolve n in the SOF. An important consideration is made: the n unit vector exists only in the opposite semi-space of the sunlight beam or, in other words, opposite to the sunlit layer of the solar sail. This limits both the azimuth and elevation angles to the interval [−90,90]^◦.

Fig. 1 Spacecraft oriented frame (SOF) referenced at the spacecraft’s barycenter, adapted from ^{Vulpetti et al. (2015)}. 

To determine the thrust T resulting from SRP, as a function of the sail’s orientation, it is furthermore useful to define the lightness vector L, conceived in the SOF as the impulsive acceleration normalized by the Sun’s local gravitational acceleration and defined as

where σ _c is a constant referred to as critical loading (equation 2), σ is the sail loading (equation 3), n _x is the SOF’s X axis n versor’s component, r _spec is the specular reflectance coefficient, r _diff is the diffuse reflectance coefficient, χ is the emission/diffusion coefficient (the subscript f refers to the front side of the solar sail), κ is a net thrust dimensionless factor that results from the absorbed and re-emitted power on both sides of the sail (equation 4), and a is the absorptivity coefficient.

The previously mentioned critical loading σ _c (equation 2), sail loading σ (equation 3) and net thrust dimensionless factor κ (equation 4), are defined as

where I _1AU = 1366W/m² is the energy flux emitted by the Sun at 1 Astronomic Unit, the average Sun-Earth distance, equal to 149597870700m, c ≈ 2,9979 × 10⁸ m/s is the speed of light in a vacuum, g _1UA ≈ 5,930 × 10⁻³ m/s² is the gravitational acceleration of the Sun at 1 Astronomical Unit, m is the spacecraft’s total mass, A is the solar sail surface area, T is the solar sail’s temperature and is the emittance coefficient as a function of T (again, the subscript f refers to the front side of the sail, while the subscript b refers to the back side). From equation 3 and LightSail-2 mission’s specifications, the sail loading is σ = 156.25g/m².

Further considerations are made in regard of the optical coefficients absorptivity α, diffuse reflectance r _diff and specular reflectance r _spec. They are all considered to be a function of the angle of the incident light beam, as portrayed in Figure 2.

Fig. 2 Absorptivity a, diffuse reflectance r _diff and specular reflectance r _spec coefficients as a function of the incident light angle for an aluminium sail with a root mean square roughness equal to 20nm (^{Vulpetti & Scaglione 1999}). The color figure can be viewed online. 

An incidence angle of 0^◦ represents the case where the unit vectors u and n are parallel. Since both the reflectance coefficients are also a function of the sail’s reflective layer roughness, the worst case scenario for the graphs of a value of 20nm root mean square roughness was considered for the simulations.

The Jet Propulsion Laboratory (JPL) published some of its solar sail models coefficients obtained from experimental studies developed in the early 2000’s, which are presented in Table 1 (^{McInnes 2004}). The values of and χ of a square solar sail were used for this study.

3.1. Strategy 1: The Planetary Society

The Planetary Society initial strategy consists of operating the solar sail at an “on-off” regime (^{Mansell et al. 2020}). It consists of maintaining the solar sail at full exposure to the sunlight whenever the spacecraft is traveling away from the Sun. In other words, the unit vector orthogonal to the sail n is parallel to u and, consequently, both the azimuth and elevation angles α and δ are equal to zero. On the other hand, the “off” regime consists of maintaining the solar sail without any exposure to the sunlight whenever the spacecraft is traveling in the direction of the Sun. This means that the unit vectors n and u are orthogonal between themselves and α and/or δ are equal to 90^◦. The schematics of this strategy is illustrated in Figure 3.

Fig. 3 LightSail-2 orientation schematics, adapted from ^{Betts et al. (2017b)}. The color figure can be viewed online. 

In order to maximize the thrust component in the SOF XY-plane, δ is always equal to zero. Consequently, only the value of α changes. The first day of simulation (with the purpose of zooming in the X axis) for this strategy is illustrated in Figure 4.

Fig. 4 Azimuth angle α as a function of time for Strategy 1. The color figure can be viewed online. 

It is possible to observe that two attitude maneuvers are necessary at every orbital revolution.

The first 200 days of the orbital dynamics of this system were simulated using Strategy 1, as well as the spacecraft’s altitude evolution, are presented in Figure 5.

Fig. 5 Spacecraft’s altitude as function of time for Strategy 1. The color figure can be viewed online. 

The blue region of the plot is in fact the altitude over time. It is seen as a blur due to the large scale of time covered in the X axis. As expected, it behaves as a sinusoidal wave enveloped by two other curves of the maximum and minimum altitude reached at every revolution, which in turn are seen as the dashed black lines. The dashed red line is a simple average of the envelope curves and is considered to be the average altitude as a function of the simulation instant in days h _av(t[days]). It starts at a value of 711.6867km and decreases 5.5204km to 706.1663km after 200 days. The dashed orange line indicates h _av(t) for a simulation with disabled solar sailing, meaning a null SRP net force, to demonstrate the importance of solar sailing for maintaining the spacecraft’s average altitude over time. After 200 days, h _av(200) = 671.6718km, having lost an extra 34.4945km, or 7.25 times more, of average altitude in comparison with the situation where the solar sailing is active.

Figure 6 presents the actual highest and lowest points of the orbit’s altitude values for the LightSail-2 mission, publicly made available and directly taken from The Planetary Society’s link for the LightSail-2 Mission Control (^{The Planetary Society 2020}). A similar chart can be seen in the same link. There are clear differences from the results seen in Figure 5, despite the fact that the general behavior of these figures are the same. This happens due to the simplified orbital model used in the simulation and to limited access to initial conditions information for the spacecraft. In addition, the model considered overestimates the solar radiation pressure net thrust in comparison to the drag force. Therefore, in order to approximate the simulation results to the LightSail-2’s mission published data, the SRP net thrust had to be scaled down by a factor of 7×10⁻².

Fig. 6 LightSail-2’s apogee, perigee and average altitude curves as a function of time from LightSail-2 Mission Control (https://www.planetary.org/explore/projects/lightsail-solar-sailing/lightsail-mission-control.html). The color figure can be viewed online. 

The initial average altitude of LightSail-2 mission’s at sail’s deployment, on 23 July 2019, is 717.4105km. On 3 February 2020, 200 days after sail deployment, the average altitude is 711.6255km, in contrast to the simulated value of 706.1663km. Although a divergence from the simulated and actual trajectory data is not ideal, it is not a vital requirement for this study that they match in value. As is the case, the main result is a comparison from all the attitude strategies, and since they are all simulated using the same orbital models, the results from Strategy 1 serve mainly as a comparison reference for other strategies.

3.2. Strategy 2: Maximum Ɛ˙

Solar radiation pressure propulsion is a specific type of low-thrust. Therefore, the thrust is a function of the spacecraft’s orientation, but for solar sails, its magnitude is also a function of the spacecraft’s position. Therefore, an analysis of its trajectory as a function of its orientation is possible from a traditional approach (^{Keaton 1986}) given this extra constraint |T| = f(n , r), where r is the spacecraft’s position in the heliocentric inertial frame (HIF).

From equation 5, it is known that the specific orbital energy’s rate of change over time is maximized when the net thrust component in the direction of the spacecraft’s velocity vector has its greatest magnitude. This does not necessarily mean (and often it is the case) that the sail and/or the net thrust vector should be pointed in the direction of travel. In equation 5 ṙ is the spacecraft’s velocity vector in the HIF

To determine this optimal strategy, it is necessary to search for the orientation which gives the maximum component of T at the ṙ direction. Considering one point of the simulation at a time, that is to say, having a fixed position r and velocity ṙ, this is a well bound one dimensional convex problem, with α constrained in the interval [−90,90]^◦. A simple hill climbing search algorithm is sufficient, and was used in this study to find the solutions. The first day of simulation (again, with the purpose of zooming in the X axis) for this strategy is illustrated in Figure 7.

Fig. 7 Azimuth angle α as a function of time for Strategy 2. The color figure can be viewed online. 

In contrast to the limited number of attitude maneuvers per orbital revolution from Strategy 1, this approach requires a constant control over the sail’s orientation. Nevertheless, it is possible to observe an interesting characteristic of this strategy’s behavior: the sail completes half a rotation (the azimuth angle α rises from −90^◦ to 90^◦) for every orbital revolution. This is a crucial information considered in the development of Strategy 3. It is also important to note that, from the definition of unit vector n (it exists in the semi-space opposite to the incoming sunbeam and, put differently, it always points away from the sail’s side not exposed to the sunlight), the transition of value from α = 90^◦ to α = −90^◦ does not mean a change of 180^◦ in the sail’s orientation. In this situation, the unit vector n merely changes its side back to the one in the shadow. Furthermore, for this consideration to work with the simulations in this study, it is necessary for the solar sail to have equally reflective sides.

Figure 8 displays the same quantities presented in Figure 5, but this time, with the employment of Strategy 2.

Fig. 8 Spacecraft’s altitude as function of time for Strategy 2. The color figure can be viewed online. 

There is a noticeable quantitative difference between the results of Strategies 1 and 2. The latter presents an average altitude after 200 days of 719.9000km, which is 13.7337km, or 1.94%, higher. But there is also a remarkable contrast between both simulations. While Strategy 1 decreases its average altitude over time, Strategy 2 actually increases its value. Starting from 711.7190km, it gains 8.1810km over the course of 200 days. This result is of great value given the objective of LightSail-2 mission in demonstrating the hidden potential of SRP in maneuvering a spacecraft. The more the solar sail is capable of increasing the average altitude of the spacecraft, the more the concept of solar sailing gains relevance.

3.3. Strategy 3: Constant α˙

From the desire of maintaining a performance in altitude gain close to the one obtained from Strategy 2 while trying to reduce the number of attitude maneuvers, in order to maintain a similarity to LightSail-2 mission’s approach, a third strategy was analyzed. It consists of keeping the sail at a constant rate of change of its azimuth angle α˙, with a value that guarantees it completes half a rotation for every orbital revolution, as observed in Strategy 2. But there is a minor problem with this consideration. Since the goal is to increase the spacecraft’s altitude and, consequently, its specific orbital energy, then the period of revolution will also increase. This would mean that the α˙ implemented is no longer able to maintain the half rotation per orbital revolution desired. Therefore, it is necessary to perform attitude maneuvers once in a while to correct α˙ and α. More specifically, an attitude maneuver will only be performed after a complete revolution, which means after an integer number of revolutions N _rev. As a reference, the maneuvers are performed at the point where the spacecraft is traveling directly in the direction of the Sun. In other words, when its velocity vector points in the opposite direction of its heliocentric position vector. At this instant, the desired value of α is always −90^◦.

Figure 9 illustrates some examples of the proposed strategy. Every instant of an attitude maneuver is indicated with a red cross in the graph. Once again, only the first day of simulation is presented with the intention of zooming on the X axis values and having a better view for further analysis. The top graph corresponds to an attitude maneuver once every orbital revolution (N _rev = 1). The bottom graph corresponds to an attitude maneuver once every five orbital revolutions (N _rev = 5).

Fig. 9 Azimuth angle α as a function of time for Strategy 3. N _rev = 1 for the upper graph. N _rev = 5 for the lower graph. The color figure can be viewed online. 

Figure 10 presents the azimuth angle rate of change α˙ from the examples in Figure 9, in the same time interval. Once again, the instant of the maneuver is indicated by a red cross.

Fig. 10 Azimuth angle rate of change α˙ as a function of time for Strategy 3. N _rev = 1 for the upper graph. N _rev = 5 for the lower graph. The color figure can be viewed online. 

From the data presented in Figure 10, it is interesting to observe the general behavior of α˙ throughout the simulation (Figure 11).

Fig. 11 Azimuth angle rate of change α˙ as a function of time for Strategy 3. N _rev = 1 for the upper graph. N _rev = 20 for the lower graph. The color figure can be viewed online. 

As ε increases, so does the orbital period. Therefore, α˙ is supposed to decrease in this situation. The inverse is also true. As ε oscillates, so does α˙. In the case of N _rev = 20, is it possible to observe an overall increase in the average value of α˙, which is an indication of an overall loss of ε and, consequently, average altitude. An example of N _rev = 20 was considered instead of N _rev = 5 to make this even clearer in the graphical representation of the discrete range of values of α˙. As N _rev increases, the longer the sail maintains its azimuth angle rate of change.

It became interesting to simulate a range of N _rev to evaluate for how long the spacecraft can be left without any attitude maneuver and still have a better performance than Strategy 1. The average altitude after 200 days of simulation h _av(200) and its gain compared to the initial average altitude h _gain (equation 6) are presented in Table 2, as a function of N _rev

Remarkably, only values of N _rev greater than 35 resulted in h _av(200) smaller than 706.1663km, obtained by Strategy 1. This means that over 70 times fewer maneuvers could be performed without a loss in the solar sail’s performance in keeping the spacecraft’s average altitude. An unexpected result is having a maximum value for N _rev = 4. The expected maximum was found for N _rev = 1. Despite this fact, the whole range of N _rev from 1 up to 10 presents similar results for h _av(200), all of which, as is the case of Strategy 2, show an increase of h _av over time. This indicates that other conditions are maybe more significant for this range of values. Results from Table 2 are presented graphically in Figure 12.

Fig. 12 Average altitude after 200 days as a function of N _rev for Strategy 3. The color figure can be viewed online. 

The blue dashed line at the top is the result obtained from Strategy 2. It can be considered as a maximum, yet unattainable, goal for Strategy 3. On the other hand, the red dashed line at the bottom is the result obtained from Strategy 1. Since it was already implemented in the LightSail-2 mission, it can be considered as a minimum to overcome. For values of N _rev smaller than or equal to 35, Strategy 3 proved to be a successful attempt to improve the performance of a solar sail to gain altitude.

4.1. Initial Altitude Sweep

An interesting analysis comes from the sweep of the mission’s initial altitude h(0). As indicated by Figure 5, the drag forces are greatly responsible for the spacecraft’s altitude decay. An also known fact is that, as the altitude decreases, the drag forces increase in magnitude. In turn, this would decrease the slope of the regression line. The inverse is also expected to be true. Therefore, a sweep of values in the [690,750]km interval was made (Figure 14). In addition, given the exponential nature of the atmospheric model considered, the data is fitted to the function presented in equation 7

where x is the independent variable from known data, y is the dependent variable for the fitted curve and c _n are the function’s parameters.

Fig. 14 Average altitude slope after 200 days as a function of the initial altitude h(0) for Strategy 2. The parameters of equation 7 are also displayed. The color figure can be viewed online. 

Figure 14 shows that Strategy 2 can avoid the decay of the spacecraft for initial altitudes close to 700km. Since the atmospheric model only considers that the atmosphere’s density is not null up to an altitude of approximately 865km, and therefore drag forces only exist until this altitude, it is interesting to extend the interval of analysis to verify that greater values of h(0) do not fall near the fitted curve and behave in a different manner, closer to a linear behavior (Figure 15).

Fig. 15 Average altitude after 200 days as a function of the initial altitude h(0) for Strategy 2 (extended view). Equation 7 function parameters are also displayed. The color figure can be viewed online. 

4.2. Spacecraft Mass Sweep

A fundamental parameter for solar sailing is the sail loading σ (equation 3). As the spacecraft’s mass m increases, so does σ, and consequently the sail’s capacity to maneuver the spacecraft decreases (equation 1). Given that Strategy 2 was able to increase h _av(t) over time, an effort was made to determine how much m could be increased while keeping a constant sail size (and surface area A) and still maintaining a positive slope for the average altitude regression line (Figure 16). Given the nature of equation 1, the data is fitted to the function presented in equation 8

Fig. 16 Average altitude after 200 days as a function of the spacecraft’s mass for Strategy 2. The parameters of equation 8 are also displayed. The color figure can be viewed online. 

The figure shows that, even for a massive spacecraft of nearly half a ton, the regression line’s slope still maintains a positive value. Despite being a fitted parameter, the fact that c ₃ is greater than zero indicates a limit, given this mission’s initial conditions and the employment of Strategy 2, which would guarantee no decay for the spacecraft’s average altitude. It means that, in theory, any spacecraft can use this technique.

In the search for negative ∆hav(200)∆t, the initial altitude was changed to smaller values and the same curve fit procedures were performed (Table 3). From Figure 17 it is possible to see that ∆hav(200)∆t is more sensitive to variations of h(0) for lower m, as expected. In fact, a negative ∆hav(200)∆t starts to be observed for negative values of c ₁, which happens with h(0) lower than 700km. This indicates a minimum altitude where Strategy 2 can be employed to keep a positive ∆hav(200)∆t for a wide range of spacecraft masses. The present study showed that the initial altitude of the spacecraft is the determinant variable to allow this technique to be used to keep the spacecraft in orbit, and the mass does not influence the results.

Fig. 17 Curve fit for the average altitude regression line’s slope after 200 days as a function of the spacecraft’s mass, for Strategy 2 and different initial altitudes h(0). The color figure can be viewed online. 

4.3. Orbital Plane Inclination Sweep

Solar sails are an intriguing propelling method with a unique peculiarity. Considering an heliocentric inertial frame, their resulting thrust will always have a radially positive component. As long as a sail is open, it will reflect the incident sunlight, transfer linear momentum to or from the spacecraft to produce a resulting thrust with a positive radial component, whether this is a desired effect or not. The only alternative to avoid the aforementioned limitation is to reduce the sail’s exposed surface area to zero. From a mission control perspective, this would mean increasing either the azimuth angle α or the elevation angle δ to 90^◦.

When considering a problem with the objective of increasing the average altitude over time, the desired direction in which the resulting SRP acceleration component needs to be maximized is different for various values of the orbital plane inclination. This results in slightly different desired azimuth angles over the span of an entire orbital revolution. In addition, it would be interesting to investigate different mission scenarios, considering whatever engineering limitations or obligations an organization might have in determining the orbital plane inclination of its mission. Inevitably, the variation of this orbital parameter is another interesting analysis to be made (Figure 18). Given the trigonometric relation of this parameter to the system’s dynamics, the data is fitted to the form presented in equation 9

Fig. 18 Average altitude after 200 days as a function of the orbital plane inclination i for Strategy 2. The parameters of equation 9 are also displayed. The color figure can be viewed online. 

As is well known, a smaller exposed surface area results in a lower SRP net thrust. In turn, this means lower specific orbital energy gains or losses and, theoretically, a reduced solar sailing maneuverability. But this fact could be used to the mission advantage. In the situation of higher inclinations, the use of Strategy 2 is specially capable of making the best use of the minimization of losses. In other words, whenever the spacecraft is traveling in the direction of the Sun, the energy losses can be reduced even further, compensating the inferior energy gains. This justifies the increasing average altitude regression line slope, as seen in Figure 18. It also shows that the strategy works for all ranges of inclinations studied [0,90]^o but, since ∆hav(200)∆t increases with the inclination, higher inclinations can be used if the goal is to increase the altitude of the spacecraft. This is particularly interesting if an escape from the Earth is desired.

Once again, the initial altitude was changed to lower values, followed by the same curve fit procedures, in the search for a minimum orbital plane inclination where Strategy 2 is capable of maintaining a positive ∆hav(200)∆t Table (4). These curves are displayed in Figure 19, which also indicates that, for each one, there is an interception point with the ∆hav(200)∆t=0 axis, as well as an inclination value with a maximum ∆hav(200)∆t. In turn, these particular points are presented in Figure 20.

Fig. 19 Curve fit for the average altitude regression line’s slope after 200 days as a function of the orbital plane inclination i for Strategy 2 and different initial altitudes h(0). The color figure can be viewed online. 

Fig. 20 Average altitude regression line’s slope after 200 days maximum and null values as a function of the orbital plane inclination i and the initial altitude h(0). The color figure can be viewed online. 

Figure 20 also displays a dashed red line that represents the mean value of i for max ∆hav(200)∆t from the fitted curves, which is approximately 79.8^◦. In spite of this value, the verified best case scenario of i for the employment of Strategy 2 is an inclination of 90^◦. In the bottom graph, the blue area represents regions of h(0) and i where Strategy 2 is able to maintain a positive ∆hav(200)∆t, while the red area represents the opposite region, of negative ∆hav(200)∆t. Two remarks have to be made in this case scenario. First, for h(0) > 720km, Strategy 2 is able to maintain an average altitude gain for any i, including equatorial orbits. Second, for lower h(0), the drag forces are greatly superior, reducing the linear behavior of h _av(t) and making it very difficult to fit parameters into the same function used for the other h(0) cases. This can already be verified with the parameter values of h(0) = 660km in Table 4, which are different from the rest by an order of magnitude. Nevertheless, it was possible to verify that for h(0) < 650km there are no positive ∆hav(200)∆t for any i, which indicates that this is a limiting h(0) where Strategy 2 cannot maintain an average altitude gain, even for the best value of the inclination.