SOLAR SURFING THE EARTH SUN SYSTEM. Departament de Matema tica Aplicada i Ana lisi Universitat de Barcelona

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1 SOLAR SURFING THE EARTH SUN SYSTEM A. Farre s A. Jorba Departament de Matema tica Aplicada i Ana lisi Universitat de Barcelona

IKAROS: in June 21, JAXA managed to deploy

NanoSail-D2: in January 211, NASA deployed

launch SunJammer a 38m 38m solar sail in

2 IKAROS: in June 21, JAXA managed to deploy the first solar sail in space. NanoSail-D2: in January 211, NASA deployed the first solar sail that would orbit around the Earth. SunJammer: in January 215, NASA plans to launch SunJammer a 38m 38m solar sail in the vicinity of L 1. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 2 / 28

3 Background Sailing Strategies Test Mission Conclusions The Planetary Society recently deployed LightSail-1, and is planying to launch LightSail-2 in 216. A. Farre s & A. Jorba (UB) Solar Surfing the Earth - Sun RTBP 3 / 28

4 Wind Sailing or Solar Sailing A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 4 / 28

5 Solar Sailing vs Wind Sailing A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 5 / 28

6 Solar Sailing vs Wind Sailing DIFFERENT LAWS OF PHYSICS A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 5 / 28

7 Equations of Motion We use the Restricted Three Body Problem (RTBP) taking the Sun and Earth as primaries and including the solar radiation pressure as a model. Z n Sail FSun FEarth Earth 1 µ µ Y Sun X ẍ 2ẏ = Ω x + ax, Ω ÿ + 2ẋ = y + ay, Ω z = z + az, where Ω(x, y, z) = 1 2 (x 2 + y 2 ) + 1 µ + µ, and a sail = (a x, a y, a z ) is the r ps r pe acceleration given by the solar sail. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 6 / 28

8 Solar Sail Model incoming radiation Sail normal reflected radiation Sail aref The normal direction to the solar sail, n, is defined as: We consider the solar sail to be flat and perfectly reflecting (simplified model): a sail = β 1 µ rps 2 r s, n 2 n where r s = (x µ, y, z)/r ps is the Sun-sail direction, and n is the normal direction to the Sun. incoming radiation Sail n x = cos(φ(x, y) + α) cos(ψ(x, y, z) + δ), n y = sin(φ(x, y, z) + α) cos(ψ(x, y, z) + δ), aabs n z = sin(ψ(x, y, z) + δ), where φ(x, y) and ψ(x, y, z) are define r s in spherical coordinates. The angles α, δ define the sail orientation, and the parameter β is the sail lightness number which measures the sail performance. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 7 / 28

9 Family of Equilibrium Points β = 1-6 β = 5*1-6 β = T2 T1 1 T2 T1 1 T2 T β =.2 β =.5 β =.1 1 T2 T1 1 T2 T1 1 T2 T XY -plane. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 8 / 28

10 Background Sailing Strategies Test Mission Conclusions Family of Equilibrium Points Equilibrium points in the XY plane T2 T Y (AU) Y (AU) T2 T1.4 Sun Earth X (AU) X (AU) Equilibrium points in the XZ plane.4 T Z (AU) Z (AU) T2.2.3 Sun -.1 Earth A. Farre s & A. Jorba (UB) X (AU) Solar Surfing the Earth - Sun RTBP X (AU) 9 / 28

11 Family of Equilibrium Points Surface of equilibria for β =.51689, 32kg of payload mass and a sail area of 38 38m 2, i.e. the sail lightness number for the SunJammer mission. unstable stable Z (AU) Y (AU) X(AU).2 Z (AU) X(AU) Y (AU) A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 1 / 28

12 Sunjammer Mission z y x CME Sun L1 ACE Earth Sail.1 AU.2 AU Polar Observer N Sail Sail N z z x Earth x Earth Sun Summer Solstice L1 S Sun Winter Solstice L1 S A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 11 / 28

13 THE GOAL We want to design Station Keeping Strategies and Surfing Strategies to navigate along the family of equilibria in a controlled way. We will use Dynamical System Tools instead of Control Theory Algorithms. The main ideas are... To focus on the linear dynamics around an equilibrium point and study how this one varies when we change the sail orientation. Find changes in the sail orientation (i.e. the phase space) to make the system act in our favour: keep the trajectory close to a given equilibrium point and/or go from one point to another. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 12 / 28

14 StK FixPoint Surf FixPoint Linear Dynamics around Unstable Equilibria We focus on the equilibrium points that are unstable: with two real eigenvalues, λ 1 >, λ 2 <, and two pair of complex eigenvalues, ν 1,2 ± iω 1,2, with ν 1,2 << λ 1,2. The linear dynamics around these points is close to saddle centre centre. v1 p v3 p v5 p v2 v4 v6 For a fixed sail orientation α = α and δ = δ the trajectory will escape along the unstable direction and rotates around the equilibrium point in the centre projections. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 13 / 28

15 StK FixPoint Surf FixPoint Linear Dynamics around Unstable Equilibria When we change the sail orientation α = α 1 and δ = δ 1 the position of the equilibrium point is shifted. Now the trajectory will escape along the new unstable direction and rotates around the equilibrium point in the new centre projections. p2 p1 p2 p1 p p \partial p Emax A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 14 / 28

16 StK FixPoint Surf FixPoint Finding α new, δ new We do NOT know explicitly the position of the equilibrium points p(α, δ). But we can compute the linear approximation of this function: p(α, δ) = p(α, δ ) + Dp(α, δ ) (α α, δ δ ) T. There are some restrictions on the position of the new equilibria when we change α and δ. We have 2 unknowns and at least 6 conditions that must be satisfied. We will change the sail orientation so that the position of the new equilibrium point is as close as possible to the desired new equilibrium point and in the correct side in the saddle projection. To decide the new sail orientation we will assume that the eigenvalues and eigendirections do not vary when the sail orientation is changed. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 15 / 28

17 StK FixPoint Surf FixPoint Small comments on p(α, δ) close to SL 1 and SL 2 We look at the sail trajectory in the ref. system {p ; v 1, v 2, v 3, v 4, v 5, v 6 }, hence z(t) = p ini + s 1 (t) v 1 + s 2 (t) v 2 + s 3 (t) v 3 + s 4 (t) v 4 + s 5 (t) v 5 + s 6 (t) v 6 v1 p v3 p v5 p v2 v4 v6 A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 16 / 28

18 StK FixPoint Surf FixPoint Small comments on p(α, δ) close to SL 1 and SL 2 We look at the sail trajectory in the ref. system {p ; v 1, v 2, v 3, v 4, v 5, v 6 }, hence z(t) = p ini + s 1 (t) v 1 + s 2 (t) v 2 + s 3 (t) v 3 + s 4 (t) v 4 + s 5 (t) v 5 + s 6 (t) v 6 p p = (,,,,, ) α δ = (,,,,, ) A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 16 / 28

19 StK FixPoint Surf FixPoint Schematic idea: Station Keeping Strategy q1 q1 v1 pini pini Emax v2 Ideal positions for the new equilibrium point if we want to remain close to p ini. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 17 / 28

20 StK FixPoint Surf FixPoint Schematic idea: Station Keeping Strategy We look at the sails trajectory in the reference system {p ini ; v 1, v 2, v 3, v 4, v 5, v 6 }, z(t) = p ini + s 1 (t) v 1 + s 2 (t) v 2 + s 3 (t) v 3 + s 4 (t) v 4 + s 5 (t) v 5 + s 6 (t) v 6 During the station keeping algorithm: 1 when α = α, δ = δ : if s 1 (t) ε max choose new sail orientation α = α 1, δ = δ 1. 2 when α = α 1, δ = δ 1 : if s 1 (t) small restore the sail orientation: α = α, δ = δ. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 18 / 28

21 StK FixPoint Surf FixPoint Station Keeping: close to p 2.2 traj cnt 2 traj cnt 2 8e-5 traj cnt 2.2 traj cnt e-5.1 Z (AU) Y (AU) v e-5 1.e-5 1.5e-5 v 1 v 4-4e-5-8e-5-6e-5-2e-5 2e-5 6e-5 v 3 v v 5 Projections of the trajectory: from left to right YZ-plane, saddle projection, first centre projection, second centre projection. α (deg) δ (deg) Sail orientation, α and δ variation. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 19 / 28

22 StK FixPoint Surf FixPoint Schematic idea: Surfing Strategy q1 q2 pend q1 q2 pend v1 pini pini Emax v2 Ideal positions for the new equilibrium point if we want to go from p ini to p end. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 2 / 28

23 StK FixPoint Surf FixPoint Schematic idea: Surfing Strategy We look at the sails trajectory in the reference system {p ini ; v 1, v 2, v 3, v 4, v 5, v 6 }, z(t) = p ini + s 1 (t) v 1 + s 2 (t) v 2 + s 3 (t) v 3 + s 4 (t) v 4 + s 5 (t) v 5 + s 6 (t) v 6 During the surfing algorithm we start with α = α, δ = δ : 1 when s 1 (t) ε max choose new sail orientation α = α 1, δ = δ 1. 2 recompute the reference system centered around p 1, the new equilibrium point. 3 check where the p end is in this new reference frame. 4 Back to 1. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 21 / 28

24 StK FixPoint Surf FixPoint Surfing Stage 1: from p to p 1 Z (AU) Y (AU) v p end Eq. Point p ini v 1 v p ini Eq. Point p end -5e-5 5e-5 1e-4 1.5e-4 2e-4 v 3 v Eq. Point p end p ini e-4-5e-5 5e-5 1.5e-4 v 5 Projections of the trajectory: from left to right YZ-plane, saddle projection, first centre projection, second centre projection. α (deg) Sail orientation, α and δ variation. δ (deg) A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 22 / 28

25 Test Mission Description As a test mission we propose a round tour visiting 4 equilibrium points displaced 5 from the Earth-Sun line. The solar sail performance is β =.51689, which corresponds to the sail lightness number for the SunJammer mission ( 32kg of payload mass and a sail area of 38 38m 2 ) x y z α(deg) δ(deg) p e e-3.e p e-1.e e p e e-3.e+.74. p e-1.e e Table: Coordinates of the equilibrium points to visit in the test mission and their corresponding sail orientation. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 23 / 28

26 Results for the Test Mission.6 stage 1 stage 2 stage 3 stage 4.2 Z (AU) X (AU) Y (AU) Figure: XYZ projection of the solar sail trajectory. Stage 1, Stage 2, Stage 3, Stage 4 A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 24 / 28

27 Results for the Test Mission Surfing Stage 1: from p to p 1 Z (AU) Y (AU) v p end Eq. Point p ini v 1 v p ini Eq. Point p end -5e-5 5e-5 1e-4 1.5e-4 2e-4 v 3 v Eq. Point p end p ini e-4-5e-5 5e-5 1.5e-4 v 5 Surfing Stage 2: from p 1 to p 2 Z (AU) traj arc Y (AU) v p end traj arc 2 Eq. Point p ini v 1 v p ini traj arc 2 Eq. Point p end 5e v 3 v traj arc 2 Eq. Point p ini p end v 5 Projections of the trajectory: from left to right YZ-plane, saddle projection, first centre projection, second centre projection. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 25 / 28

28 Results for the Test Mission Surfing Stage 3: from p 3 to p 4.5 traj arc 3.16 p ini traj arc 3 Eq. Point traj arc 3 Eq. Point p end traj arc 3 Eq. p Point ini Z (AU) v v v Y (AU).1 p end v p ini 8e v p end v 5 Surfing Stage 4: from p 4 to p Z (AU) traj arc Y (AU) v p ini traj arc 4 Eq. Point v 1 p end v p end traj arc 4 Eq. Point p ini v 3 v traj arc 4 Eq. p Point end p ini v 5 Projections of the trajectory: from left to right YZ-plane, saddle projection, first centre projection, second centre projection. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 25 / 28

29 Results for the Test Mission α (deg) α (deg) α (deg) α (deg) δ (deg) δ (deg) δ (deg) δ (deg) Sail orientation, α and δ variation. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 25 / 28

30 Results for the Test Mission Station Keeping: close to p 1.2 8e e Z (AU).17 v v 4 v e e e-5-2e-5 2e-5 6e Y (AU) v 1 v 3 v 5 Projections of the trajectory: from left to right YZ-plane, saddle projection, first centre projection, second centre projection. α (deg) Sail orientation, α and δ variation. δ (deg) A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 26 / 28

31 Results for the Test Mission Station Keeping: close to p 2.2 traj cnt 2 traj cnt 2 8e-5 traj cnt 2.2 traj cnt e-5.1 Z (AU) Y (AU) v e-5 1.e-5 1.5e-5 v 1 v 4-4e-5-8e-5-6e-5-2e-5 2e-5 6e-5 v 3 v v 5 Projections of the trajectory: from left to right YZ-plane, saddle projection, first centre projection, second centre projection. α (deg) δ (deg) Sail orientation, α and δ variation. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 26 / 28

32 Results for the Test Mission Station Keeping: close to p traj cnt 3.5 traj cnt 3 8e-5 traj cnt 3 traj cnt 3 Z (AU) v v 4 4e-5-4e-5 v 6.1 5e-5-5e e e-5-1.e-5 -.8e-5 -.6e-5-6e-5-2e-5 2e-5 6e e-5 5e-5.15 v 1 v 3 v 5 Y (AU) Projections of the trajectory: from left to right YZ-plane, saddle projection, first centre projection, second centre projection. α (deg).2.1 δ (deg) Sail orientation, α and δ variation. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 26 / 28

33 Results for the Test Mission Station Keeping: close to p 4 Z (AU).2 traj cnt Y (AU) v 2.4 traj cnt e-5-1.e-5 -.8e-5 -.4e-5 v 1 v 4 8e-5 traj cnt 4 4e-5-4e-5-8e-5-6e-5-2e-5 2e-5 6e-5 v 3 v 6 traj cnt v 5 Projections of the trajectory: from left to right YZ-plane, saddle projection, first centre projection, second centre projection. α (deg) δ (deg) Sail orientation, α and δ variation. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 26 / 28

34 Conclusion: We have derived strategies to move along the family of equilibrium points in a controlled way only using the information of the linear dynamics. We have tested them for an example test mission to visit 4 equilibrium points. Work in Progress: Understand the relation between the parameters of these strategies and their controllability. Test the robustness of these strategies when difference sources of error are included. A. Farrés & À. Jorba (UB) Solar Surfing the Earth - Sun RTBP 27 / 28

35 apple

IAC-07-C SOLAR SAIL SURFING ALONG FAMILIES OF EQUILIBRIUM POINTS

IAC-07-C SOLAR SAIL SURFING ALONG FAMILIES OF EQUILIBRIUM POINTS 58th International Astronautical Congress, Hyderabad, India, 24-28 September 2007. Copyright IAF/IAA. All rights reserved. IAC-07-C1.4.04 SOLAR SAIL SURFING ALONG FAMILIES OF EQUILIBRIUM POINTS Ariadna