Invariant manifolds for a Solar sail es ∗ , ` Angel Jorba † , Marc Jorba-Cusc´ o † Ariadna Farr´ ∗ University of Maryland Baltimore County & NASA Goddard Space Flight Center † Universitat de Barcelona M 3 ES 2 , Rome, March 22 2019 1 / 70
Outline 1 Background 2 Station Keeping around Equilibria 3 Dynamics near an asteroid 4 Periodic time-dependent effects 2 / 70
Background What is a Solar Sail ? It is a new concept of spacecraft propulsion that takes advantage of the Solar radiation pressure to propel a satellite. The impact of the photons emitted by the Sun on the surface of the sail and its further reflection produce momentum on it. Solar Sails open a new range of applications not accessible by a tradi- tional spacecraft. 3 / 70
Background Several solar sails have already been in space: IKAROS: (Interplanetary Kite-craft Accelerated by Radiation Of the Sun). It is a Japan Aerospace Exploration Agency experimental space- craft with a 14 × 14 m 2 . sail. The spacecraft was launched on May 21st 2010, together with Akatsuki (Venus Climate Orbiter). On December 8th 2010, IKAROS passed by Venus at about 80,800 km distance. NanoSail-D2: On January 2011 NASA deployed a small solar sail (10 m 2 , 4kg.) in a low Earth orbit. It reentered the atmosphere on Septem- ber 17th 2011. LightSail-A: This is a small test spacecraft (32 m 2 ) of the Planetary Society. It has been launched on May 20th 2015 and it deployed its solar sail on June 7th 2015. It has reentered the atmosphere on June 14th 2015. 4 / 70
Background The Restricted Three-Body Problem 1 × L 5 0.5 L 2 L 1 L 3 × × × ◦ � 0 E S -0.5 × L 4 -1 -1.5 -1 -0.5 0 0.5 1 1.5 5 / 70
Background The Restricted Three-Body Problem Defining momenta as P X = ˙ X − Y , P Y = ˙ Y + X and P Z = ˙ Z , the equations of motion can be written in Hamiltonian form. The corresponding Hamiltonian function is H = 1 Z ) + YP X − XP Y − 1 − µ − µ 2( P 2 X + P 2 Y + P 2 , r 1 r 2 1 = ( X − µ ) 2 + Y 2 + Z 2 and r 2 2 = ( X − µ + 1) 2 + Y 2 + Z 2 . being r 2 6 / 70
Background The Solar Sail As a first model, we consider a flat and perfectly reflecting Solar Sail: the force due to the solar radiation pressure is normal to the surface of the sail ( � n ), and it is defined by the sail orientation and the sail lightness number . The sail orientation is given by the normal vector to the surface of the sail, � n . It is parametrised by two angles, α and δ . The sail lightness number is given in terms of the dimensionless parameter β . It measures the effectiveness of the sail. The acceleration of the sail due to the radiation pressure is given by: a sail = β m s n � 2 � � � � r s ,� n . r 2 ps 7 / 70
Background The Sail Effectiveness The parameter β is defined as the ratio of the solar radiation pressure in terms of the solar gravitational attraction. With nowadays technology, it is considered reasonable to take β ≈ 0 . 05. This means that a spacecraft of 100 kg has a sail of 58 × 58 m 2 . 8 / 70
Background A Dynamical Model We use the Restricted Three Body Problem (RTBP) taking the Sun and Earth as primaries and including the solar radiation pressure. Z Sail n � � F Sun � F Earth X 1 − µ µ Earth Sun Y 9 / 70
Background Equations of Motion The equations of motion are: y + x − (1 − µ ) x − µ − µ x + 1 − µ + β 1 − µ n � 2 n x , � � r s ,� x ¨ = 2˙ r 3 r 3 r 2 ps pe ps � 1 − µ + µ � y + β 1 − µ n � 2 n y , � � r s ,� y ¨ = − 2˙ x + y − r 3 r 3 r 2 ps pe ps � 1 − µ + µ � z + β 1 − µ n � 2 n z , z ¨ = − � � r s ,� r 3 r 3 r 2 ps pe ps where � n = ( n x , n y , n z ) is the normal to the surface of the sail with 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 ) , n z = sin( ψ ( x , y , z ) + δ ) , and � r s = ( x − µ, y , z ) / r ps is the Sun - sail direction. 10 / 70
Background Equilibrium Points The RTBP has 5 equilibrium points ( L i , i = 1 , . . . , 5). For small β , these 5 points are replaced by 5 continuous families of equilibria, parametrised by α and δ . For a small value of β , we have 5 disconnected families of equilibria near the classical L i . For a fixed and larger β , these families merge into each other. We end up having two disconnected surfaces, S 1 and S 2 , where S 1 is like a sphere and S 2 is like a torus around the Sun. All these families can be computed numerically by means of a contin- uation method. 11 / 70
Background Interesting Missions Applications Observations of the Sun provide information of the geomagnetic storms, as in the Geostorm Warning Mission. z y CME x ACE Sun Earth L 1 0 . 01 AU Sail 0 . 02 AU Observations of the Earth’s poles, as in the Polar Observer. N N Sail Sail z z x Earth Earth x L 1 L 1 Sun Sun Winter Solstice Summer Solstice S S 12 / 70
Background Interesting Missions Applications To ensure reliable radio communication between Mars and Earth even when the planets are lined up at opposite sides of the Sun. Sun − Mars L1/L2 Hover Sun − Earth L1/L2 Hover Mars Sun Earth 5º 2.5º 1au 1.52 au 13 / 70
Background Periodic Motion Around Equilibria We must add a constrain on the sail orientation to find bounded motion. One can see that when α = 0 and δ ∈ [ − π/ 2 , π/ 2] (i.e. only move the sail vertically w.r.t. the Sun - sail line): The system is time reversible ∀ δ by z , − t ) and Hamiltonian only for R : ( x , y , z , ˙ x , ˙ y , ˙ z , t ) → ( x , − y , z , − ˙ x , ˙ y , − ˙ δ = 0 , ± π/ 2. There are 5 disconnected families of equilibrium points parametrised by δ , we call them FL 1 ,..., 5 (each one related to one of the Lagrangian points L 1 ,..., 5 ). Three of these families ( FL 1 , 2 , 3 ) lie on the Y = 0 plane, and the linear behaviour around them is of the type saddle × centre × centre. The other two families ( FL 4 , 5 ) are close to L 4 , 5 , and the linear behaviour around them is of the type sink × sink × source or sink × source × source. 14 / 70
Background We focus on ... We focus on the motion around the equilibrium on the FL 1 family close to SL 1 (they correspond to α = 0 and δ ≈ 0). We fix β = 0 . 051689. We consider the sail orientation to be fixed along time. Sun Earth L2 L3 L1 SL1 SL3 SL2 (Schematic representation of the equilibrium points on Y = 0 ) Let us see the periodic motion around these points for a fixed sail orientation and show how it varies when we change, slightly, the sail orientation. 15 / 70
Background P -Family of Periodic Orbits Periodic Orbits for δ = 0. 0.03 0.015 0.015 P - Family Halo Halo 0.02 0.01 0.01 0.01 0.005 0.005 0 0 0 Y Z Z -0.01 -0.005 -0.005 -0.02 -0.01 -0.01 -0.03 -0.015 -0.015 -0.995 -0.99 -0.985 -0.98 -0.975 -0.97 -0.965 -0.995 -0.99 -0.985 -0.98 -0.975 -0.97 -0.02 -0.01 0 0.01 0.02 X X Y Halo 1 Halo 1 Halo 2 Halo 2 Planar Planar 0.01 0.01 0.005 0.005 Z Z Z Z 0 0 -0.005 -0.005 0.02 -0.01 0 0.02 -0.01 Y 0 -0.02 -0.99 -0.99 -0.98 -0.97 Y X -0.98 -0.02 X -0.97 16 / 70
Background P -Family of Periodic Orbits Periodic Orbits for δ = 0 . 01. Main family of periodic orbits for δ = 0 . 01 0.025 0.015 P Family B P Family B P Family B 0.02 0.01 0.015 0.01 0.005 0.005 0 0 0 Y Z Y -0.005 -0.005 -0.01 -0.015 -0.01 -0.02 -0.025 -0.015 -0.995 -0.99 -0.985 -0.98 -0.975 -0.97 -0.995 -0.99 -0.985 -0.98 -0.975 -0.97 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 X X Z Secondary family of periodic orbits for δ = 0 . 01 0.03 0.015 0.015 P Family A P Family A P Family A 0.02 0.01 0.01 0.01 0.005 0.005 0 0 0 Y Z Z -0.01 -0.005 -0.005 -0.02 -0.01 -0.01 -0.03 -0.015 -0.015 -0.995 -0.99 -0.985 -0.98 -0.975 -0.97 -0.965 -0.995 -0.99 -0.985 -0.98 -0.975 -0.97 -0.965 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 X X Y 17 / 70
Background P -Family of Periodic Orbits Periodic Orbits for δ = 0 . 01. Fami A Fami A Fami B Fami B 0.01 0.01 0.005 0.005 Z Z Z Z 0 0 -0.005 -0.005 0.02 -0.02 -0.01 -0.01 0 -0.99 0 -0.97 Y Y -0.98 0.02 X -0.99 -0.98 -0.02 X -0.97 Fami A Fami B 0.01 0.005 0 Z Z -0.005 -0.01 0.02 -0.97 0 Y -0.98 -0.02 X -0.99 18 / 70
Background V - Family of Periodic Orbits δ = 0 δ = 0.001 0.03 0.03 0.02 0.02 0.01 0.01 Z Z Z Z 0 0 -0.01 -0.01 -0.02 0.0006 -0.02 0.0006 0.0003 0.0003 -0.03 -0.03 -0.9816 -0.9812 -0.9808 -0.9804 0 -0.9816 -0.9812 -0.9808 -0.9804 0 Y Y -0.0003 -0.0003 X X -0.0006 -0.0006 -0.98 -0.98 δ = 0.005 δ = 0.01 0.03 0.03 0.02 0.02 0.01 0.01 Z Z Z Z 0 0 -0.01 -0.01 -0.02 0.0006 -0.02 0.0006 0.0003 0.0003 -0.03 -0.03 0 0 -0.9816 -0.9812 -0.9808 -0.9804 -0.9816-0.9812-0.9808-0.9804 -0.98 -0.0006 Y Y -0.0003 -0.0003 X -0.0006 X -0.98 19 / 70
Station Keeping around Equilibria Station Keeping around Equilibria 20 / 70
Station Keeping around Equilibria Station Keeping around equilibria Goal: Design station keeping strategy to maintain the trajectory of a solar sail close to an unstable equilibrium point. We want to use Dynamical Systems Tools to find a station keeping algorithm for a Solar Sail. 21 / 70
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