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International Journal of Application or Innovation in Engineering & Management (IJAIEM

Most interplanetary spacecraft have used finite burns to escape Earth's orbit and be captured by their destination's gravity. Solar sail is an attractive technology to potentially lessen propellent expenditure. Optimal control of the sail
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  International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org  Volume 7, Issue 8, August 2018 ISSN 2319 - 4847   Volume 7, Issue 8, August 2018 Page 93 A BSTRACT    Most interplanetary spacecraft have used finite burns to escape Earth’s orbit and be captured by their destination’s gravity. Solar sail is an attractive technology to potentially lessen propellent expenditure. Optimal control of the sail coning angle to the Sun can both reduce transit time and the mission’s propulsive velocity change. This research sought to determine the savings in hyperbolic excess velocity at departure and arrival for a baseline spacecraft and steering angle. Keywords:  Interplanetary, Solar Sail, Hyperbolic Excess Velocity, Lagrangian Variation of Parameters (VOP)   1.   INTRODUCTION Most interplanetary spacecraft have used finite propulsive burns to escape Earth’s orbit and be captured by their destination’s gravity. The propellent expended for these maneuvers is a large part of the mass and cost of these missions. If the hyperbolic excess velocity at departure and/or arrival could be reduced, propellant mass could be saved. Solar sail is an attractive technology to potentially optimize propellant expenditure. It converts solar photon momentum to spacecraft acceleration, providing a free, continuous propulsion source. This propulsion system is only limited by sail size, reflective parameters, and orientation to the sun. Methods to optimize transit time and propellant expenditure with solar sails could directly translate to reduced mission cost and/or more scientific exploration of the target. The methodology of low thrust presented by Vallado details orbit-raising. [1] Raising is accomplished by determining the required control angle between the thrust and velocity vectors. Unfortunately, it requires a multiple-revolution scenario for a closed-form solution, which may be less desirable for a Mars mission. McInnes presents a method using Lagrangian variation of parameters (VOP). [3] Additionally, optimization of the sail angle to provide the maximum force in the desired direction. [3] However, this technique means there will be means that there will usually be a transverse force to that in the desired direction. Stevens et al. were able to find Earth-Mars rendezvous trajectories, but did not keep the coning angle optimal. [4] This research sought to optimally control the sail coning angle to the Sun to reduce transit time and the mission’s propulsive velocity change, thus ensuring optimal savings in hyperbolic excess velocity at departure and arrival of the baseline spacecraft and steering angle. 2.   METHODOLOGY 2.1   Solar Sail Solar sail uses solar photon momentum to accelerate the spacecraft. The solar flux has an inverse-square relationship to the distance from the sun, just like gravitation. Therefore, the maximum acceleration provided by the sail and the acceleration due to solar gravity can form the lightness value, the ratio b   [4] b   = a srp a grav  (1) The relationship presented in above equation provides the mission designer the ability to design trajectories without prior knowledge of spacecraft mass, sail area, and sail reflective parameters. It allows the acceleration due to solar radiation pressure be expressed in terms of gravitational acceleration. [3] The angle between the sun vector and the sail-normal vector is called the coning angle, a  . Figure 1 represents two-dimensional representation of the system. Optimal Interplanetary Trajectories for Solar Sail Deepak Gaur 1 , M. S. Prasad 2 1 M. Tech. (Avionics), Amity Institute of Space Science and Technology, Amity University, Noida, U.P., India 2 Director, Amity Institute of Space Science and Technology, Amity University, Noida, U.P., India  International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org  Volume 7, Issue 8, August 2018 ISSN 2319 - 4847   Volume 7, Issue 8, August 2018 Page 94 Figure 1 Sail craft coning angle. [5] This parameter is steered to produce the sail force in the desired direction. However, it is not optimal to just steer to the coning angle between 0  and p  2 . Considering the sail force direction: n = cos a  ˆ r   + sin a  ˆ  p ´ ˆ  p  (2) And the vector that maximizes sail force to be applied: q = cos a  ˆ r   + sin a  ˆ  p ´ ˆ  p  (3) The force magnitude  f  q  is [3]  f  q  = 2 PA n × ˆ r  ( ) 2 n × q ( )  (4) By combining equations (2)-(4) and setting the result’s derivative equal to zero, the optimal coning angle is found by the following expression: tan a  opt   = - 3 + 9 + 8tan 2  ⌢ a  4tan  ⌢ a   (5) The coning angle is shown against the desired force angle in Figure 2. Due to the transverse force component being limited by the product of sin a  and cos a  , one can see that maximizing the force in the transverse direction (with respect to the Sun) requires a coning angle of 35.25°. The two-dimensional polar equations of motion for a large gravitational body and solar radiation pressure given as: [3] (6) (7)  International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org  Volume 7, Issue 8, August 2018 ISSN 2319 - 4847   Volume 7, Issue 8, August 2018 Page 95 Figure 2 Optimal coning angle. From equations (2) and (3), it is evident that a  provides no sail force at p  2 . 2.2   Optimal Trajectories Optimal trajectories can be found with Lagrangian VOP technique. To optimally increase an orbital parameter, one may take the VOP for that equation and solve for a  2 by means of  f  l   = 2 PA n × ˆ r  ( ) 2 n × l  ( )  (8) where l  represents VOP components in the RSW frame. For the coplanar problem, commanded a  is a   = arctan l   R l  S  æ è ç ö ø÷  (9) For the orbit semi-major axis to increase, the expression for a  is found to be tan  ⌢ a   = 1 +  e cos  f e sin  f   (10) Force along this vector can be minimized by the use of equation (5). 3.   SIMULATION AND RESULTS 3.1   Raising Semi-Major Axis  Solar sail trajectories for various values of b   are generated to optimally increase the spacecraft semi-major axis to that of Mars and Jupiter. Planetary orbits are assured to be co-planar and circular. Perturbing accelerations are not taken into consideration, just those of solar radiation pressure and gravity of sun. Trajectories are assured to start outside of  International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org  Volume 7, Issue 8, August 2018 ISSN 2319 - 4847   Volume 7, Issue 8, August 2018 Page 96 Earth’s Sphere of Influence (SOI), where V  inf   inity is zero. Thus, the only D V  needed in these trajectories is to inject the spacecraft in the target planet’s orbit. Resultant trajectories can be seen for Mars in Figure 3-5. Figure 3 Mars, b  = 0.05 Figure 4 Mars, b  = 0.10 Figure 5 Mars, b  = 0.15 Table 1 shows the performance of these trajectories compared to Hohmann transfer.  International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org  Volume 7, Issue 8, August 2018 ISSN 2319 - 4847   Volume 7, Issue 8, August 2018 Page 97 Table 1: Mars Transfer with Optimal Raise of Semi-Major Axis b    D V  total  (km/sec) Transfer Time (days) Phase Angle (degrees) Intercept Angle (degrees) Hohmann 5.594 259 180 0 0.05 2.376 747 530 6 0.10 4.410 251 191 10 0.15 7.552 186 146 18 Trajectories where the semi-major axis is optimally increased provided advantages in D V  savings over pure Hohmann transfers. For the lowest lightness value simulated, more than 50% of D V  is saved at the cost of almost three times the transfer time. For b  = 0.10, the transfer time is similar, but the continuous sail thrust saves over 1.184 km/sec. This is not as attractive as it seems, however, due to the greater sail area required and the challenges in implementing it. b  = 0.15 actually saved more than 2 months of transfer time, but the D V   exceeded the Hohmann transfer by 1.958 km/sec. The cone angles used in each trajectory is presented in Figure 6. Figure 6 Mars Cone Angles The cone angles were capped at 35.25°. Higher lightness values allowed for lower cone angles, which generated higher forces. Resultant trajectories for Jupiter are presented in Figure 7-9. Figure 7 Jupiter, b  = 0.05
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