Q2.pdf

Solar Sailing To Mars Kishan Sankharva (160332)

Outline 1. 2. 3. 4. 5. 6.

Introduction Problem Statement Literature Review Technique Results and Discussion Conclusion

1. 2. 3. 4. 5. 6.

Introduction Problem Statement Literature Review Technique Results and Discussion Conclusion

Introduction • 1957: Sputnik I launch – Dawn of the space age • Tons of improvement in efficiency of propellants and thrusters since then • 2020s: First manned mission to Mars (probably?) MAVEN (NASA, 2013)

MOM (ISRO, 2013)

Wet Mass

2454 kg

1337 kg

Dry Mass

809 kg

483 kg

Mass Ratio

3.0

2.8

Payload

65 kg

13.4 kg

• Yet Inefficient !!

Introduction (contd.) • Chemical propellant rockets are complex to design • Rocket engine + fuel occupy a large proportion of mass • Interstellar travel not feasible • Solar energy to propel spacecrafts? • Solar sails first proposed in 1925, courtesy Friedrich Zander • Not much research on the subject • IKAROS: only spacecraft to be propelled primarily by solar sails • Hypothesis: Should provide better mass ratio, payload capacity

1. 2. 3. 4. 5. 6.

Introduction Problem Statement Literature Review Technique Results and Discussion Conclusion

Problem Statement • Optimize trajectory of a spacecraft from Earth to Mars. • Use only solar sails to propel the spacecraft. • Constraints: – Total mass (sail + payload) = 2000 kg – Surface mass density of sail = 7 g m-2 – Relative velocity at the time of entry into Martian atmosphere < 9 km s-1

• Assume sail is made up of highly reflective material • Initial condition: Rocket launches the spacecraft in GEO at zero relative velocity w.r.t. Earth. • Start the mission when Mars is at closest approach to Earth

1. 2. 3. 4. 5. 6.

Introduction Problem Statement Literature Review Technique Results and Discussion Conclusion

Radiation Pressure

• Pressure on a mirror due to radiation is given by

2 $ cos ( ) != * • If Sun emits a total power !+ then

!+ $ , = 4., ( • Thus, force on the spacecraft is

2!+ 1 cos ( ) 0⃗ = 32 ( 4.* ,

Equations of Motion • Total force on the spacecraft is given by

2%& * cos . / 45⨀7 ⃗ "= 21 − 0̂ 4() 0 . 0. • Here, 21 = cos / 0̂ + sin / =< is normal to the sail • Radial component:

7 0̈ − 0=̇ .

2%& * cos @ / 45⨀7 = − 4() 0 . 0.

• Tangential component:

2%& * sin / cos . / 7 0=̈ + 20̇ =̇ = 4() 0.

1. 2. 3. 4. 5. 6.

Introduction Problem Statement Literature Review Technique Results and Discussion Conclusion

Nature of the Problem • Coupled second order differential equations in two variables – difficult to solve analytically • Solve numerically • Choose origin to be the center of Sun. • Four initial conditions needed: – "# = 1 &. (., *# = 0, "#̇ = 0, *̇ # = Ω • We have two parameters: ., & • . is very crucial – can accelerate and decelerate the spacecraft just by changing the value of .

Solution • To keep things simple, keep angle constant while accelerating, until we reach certain radial distance • Then, change the angle so as to decelerate until we reach Mars. • Another parameter – the radial distance r’ at which we would be changing the angle. We define r’ as

!"

!$% + '!( = 1+'

where, !$% is radius of Mars orbit, !( is radius of Earth’s orbit and ' is a dimensionless parameter • Need to iterate over different values of A, * and ' to find best trajectories in terms of payload capacity and time required to reach Mars.

Solution (contd.) • Decouple the equations as follows 1. $̇ = &' 2. )̇ = &*

3. &'̇ =

$&*,

4. &*̇ = −

+

,./ 4 5678 9

,@A @B '

+

<=⨀

': ,./ 4 7CD 9 567: 9 0123 ':

0123

':

−

• Can solve these using fourth order RK method to obtain different possible combinations of A, E and F.

1. 2. 3. 4. 5. 6.

Introduction Problem Statement Literature Review Technique Results and Discussion Conclusion

! = 200000 %& , ( = 0.34, , =

,/ = 0 10

! = 200000 %& , ( = 0.34, , =

, / = 100 0123 10

! = 200000 %& , ( = 0.34, , =

, / = 200 0123 10

! = 200000 %& , ( = 0.34, , =

, / = 300 0123 10

! = 200000 %& , ( = 0.34, , =

, / ~ 419 2345 10

Plot of journey time and velocity of approach at Mars vs parameter ! for area of sails = 200,000 m2

Results • Velocity of approach varies around 7000-8000 m s-1 for each of A = {1.9, 2.0, 2.1, 2.2} x 105 m2. • ! is also almost same for all A values. • However, there seems to be linear relation between area of sails, time taken and payload capacity. Area (m2)

! tmin (in units of ") (days)

Relative velocity (m s-1)

Payload Capacity (kg)

190,000

0.115

440

7184

670

200,000

0.115

417

7142

600

210,000

0.105

399

7113

530

220,000

0.105

381

7248

460

Table 2: Values of various parameters needed for minimum journey time for different values of A

Plot of t vs A

Discussion • For A = 200,000 m2, we can send payload mass ratio of 0.3 to Mars in 417 days. • MAVEN took 282 days for payload mass ratio of 0.026 • MOM took 298 days for payload mass ratio of 0.01 • Reasons: – Propulsion systems are complex and massive. – the fuel takes up more than 60% of the launch mass

1. 2. 3. 4. 5. 6.

Introduction Problem Statement Literature Review Technique Results and Discussion Conclusion

Conclusion • Although solar sails provide very small thrust, over time it adds up, making its performance comparable to rockets. • Thus, solar sails, being environment friendly, are good substitute for rockets.

THANK YOU