Does currently available technology have the capacity to facilitate a manned mission to Mars?

Transcription

1 Furze Platt Senior School Does currently available technology have the capacity to facilitate a manned mission to Mars? Daniel Messias Date: 8/03/2015 Candidate Number: 7158 Centre Number: 51519

2 Contents Introduction... 2 Launch to LEO... 2 LEO to Earth-Escape... 3 Earth-Mars Hohmann Transfer... 3 Mars Aero-capture... 4 Landing on Mars Surface... 5 Returning to LMO... 5 LMO to Mars Escape... 6 Mars-Earth Hohmann Transfer... 6 Earth Landing... 6 Delta-V Total and Mission Feasibility... 7 Works Cited Daniel Messias

3 Introduction Since Yuri Gagarin became the first man to reach outer Space humankind has strived to explore further away from Earth. The next big target for the world's space agencies is a Mars manned mission. In this piece I am going to calculate a delta-v and time cost to see if the rockets of today could be used for such a mission. My mission can be broken down into a number of orbital manoeuvres, each resulting in an amount of delta-v needed to execute them. The total delta -v can then be calculated for the entire mission. The plan is launch to LEO (Low Earth Orbit), then accelerate until escape velocity is achieved leaving the spacecraft in a solar orbit of 1AU, i.e. the same as the Earth's (Assuming that the SOI (Sphere[s] of Influence) of the Sun and Earth have definite edges). From there I am going to use a Hohmann transfer to reach Mars SOI and perform an aero-capture to enter LMO (Low Mars Orbit). Next the craft will land, where the crew will stay until the next return window opens, upon which they will return similarly to how they arrived. In detail, the calculations look as follows. Launch to LEO The initial stage consists of launching the rocket to LEO. The orbital velocity needed to reach a circular parking orbit of 200km can be calculated as follows (Orbits, 2010), (Williams, 2013): v GM r μ Earth v o = = 7.79km s 1 R Earth + 200km μ Earth = Gravitational Constant Mass of Earth = km 3 s 2 R Earth = Equatorial radius of Earth = 6 371km Although we need 7.79km s -1 to reach a stable parking orbit of 200km, we need to account for the resistance we will encounter when launching from Earth. The exact amount of drag will depend on the aerodynamic properties of the rocket and the air density from where we launch. For the purposes of this example we will assume that to reach LEO the drag the rocket would experience will be 1.75km s -1 (Tajmar). This brings the total delta-v needed for this stage to approximately 9.54km s Daniel Messias

4 LEO to Earth-Escape Escape velocity is defined as the speed at which the kinetic energy plus the gravitational potential energy of an object is zero. In other words it is the speed at which an object will no longer eventually be pulled into a massive body. Escape velocity at a given altitude is 2 times the velocity of a circular orbit at the same given height. This means it can be calculated as follows (Nave): v e 2GM r This equation tells us the velocity needed to escape, however we want to know the change in velocity needed to escape, so the following equation should be used. v e 2GM r v o v e = 2μ Earth R Earth 7.79km s 1 = 3.22km s 1 μ Earth = Gravitational Constant Mass of Earth = km 3 s 2 R Earth = Equatorial radius of Earth = 6 371km After this manoeuvre the spacecraft will be on what could be described as a solar orbit equivocal to that of Earth's. As the spacecraft only just reaches escape velocity, the difference between its orbital speed and that of Earth's will be negligible. This means for the next section we can consider the spacecraft to be in an orbit around the sun of radius 1AU. The total delta-v for this section is 3.22km s -1. Earth-Mars Hohmann Transfer The next section consists of transferring the spacecraft from just outside Earth's SOI to inside Mars'. There are a number of ways to perform an interplanetary transfer. The most obvious idea might be to wait for Earth and Mars to be at closest approach, point the craft at Mars and burn towards it. Providing you could provide enough thrust in a short enough period of time this would technically work, however as Earth is moving around the Sun faster than mars, you would have to be fast enough to prevent missing Mars completely once your burn is finished. Unsurprisingly this is a very inefficient and impractical method of transfer, however there are two more realistic alternatives, namely the Hohmann transfer (named after German scientist Walter Hohmann) and the Bi-elliptic transfer (first published by Ary Sternfield). For simplicities sake I am going to just model a Hohmann transfer, however both have their merits. In brief, despite a bi-elliptic transfer requiring one more engine burn, there are situations when it is more efficient that the Hohmann transfer, however in general it takes more travel time to complete, which is an issue in the viability of sending a manned mission to mars. 3 Daniel Messias

5 A Hohmann transfer consists of two burns, the first to move the spacecraft onto its transfer orbit, and the second to circularize the spacecraft at its new orbit height. The basic principle is to provide an impulse that expands the current circular orbit into an elliptical one whereby the periapsis remains at the starting orbital height and the apoapsis is at the target orbital height. The spacecraft then coasts until it reaches apoapsis and provides a second impulse expanding the periapsis to the same orbital height as the new apoapsis (Braeunig, 2013). In the case of this Mars mission the starting height will be 1AU, i.e. at Earth, and the target height will be the orbital height of Mars from the Sun. Using a table of Mars' closest approaches to Earth, I am going to assume a mission set for the intercept to be in October This would mean the launch would have to be approximately half the duration of the Hohmann transfer orbit prior to this date, as the transfer needs to end at closest approach. At closest approach the minimum distance from Earth would be 0.415AU (Mars Close Approaches). I am going to use only the first half of the Hohmann transfer in the mission to raise my spacecraft's aphelion to AU, and assumed there would be a negligible mid-course correction to place the apoapsis within Mars atmosphere. The second half of my Hohmann transfer, i.e. circularizing around LMO will be achieved through aero-capture. This is where Mars atmosphere is used to slow down the spacecraft, reducing its orbital velocity and effectively gaining a free circularisation. The calculations for the delta-v needed to perform the Hohmann transfer are as follows (Williams, Sun Fact Sheet, 2013). v = GM ( 2r 2 1) v r 1 r 1 + r insert = u Sun 2 1AU ( AU 1AU AU μ Sun = Gravitational Constant Mass of Sun = km 3 s 2 1AU = km 1.415AU = km 1) = 2.46km s 1 After this stage the spacecraft will be heading on a hyperbolic trajectory towards Mars' atmosphere, ready for aero-capture. The total delta-v for this stage is 2.46km s -1. Mars Aero-capture Now the spacecraft is within Mars' SOI and is heading on a hyperbolic trajectory deep into the atmosphere. The drag created as the spacecraft passes through the Martian atmosphere will create a large amount of resistance, dramatically reducing the velocity of the spacecraft, performing a free equivalent of a burn at periapsis, lowering the height of the apoapsis to a point where it is within Mars' SOI, in other words exiting the spacecraft from its hyperbolic trajectory and ideally placing it into an elliptical orbit of eccentricity of less than 1. It might be more practical to use aero-braking instead of aero-capture, which consists of multiple passes through the atmosphere, the first of which needs only exit the spacecraft from its hyperbolic trajectory, achieving an eccentricity of less than 1 can be achieved in further passes (Johnston, Esposito, Alwar, Demcak, Graat, & Mase). Once the apoapsis is lowered to a reasonable height in LMO, a small burn must be made at apoapsis to raise the periapsis out of the atmosphere and a stable orbit is achieved. For the purposes of this mission, that burn size will be assumed to be negligible, and the final LMO altitude will be assumed to be around 200km. 4 Daniel Messias

6 Using the orbital speed equation from earlier it is possible to calculate the speed the spacecraft will be travelling at this time (Orbits, 2010), (Williams, Mars Fact Sheet, 2013). v GM r μ Mars v o = = 3.45km s 1 R Mars μ Mars = Gravitational Constant Mass of Mars = km 3 s 2 R Mars = Equatorial radius of Mars = 3 396km Landing on Mars Surface Unlike the Moon, Mars has an atmosphere, which makes landing on its surface a lot easier. A small retrograde burn from LMO will knock the spacecraft out of a stable orbit and onto a trajectory towards the planet. It will now be necessary to reduce the velocity of the spacecraft from 3.45 km s -1 to zero before it hits the ground. When NASA launched the Curiosity rover to Mars it (along with the sky-crane) entered Mars atmosphere at over 5.8 km s -1, and the atmosphere reduced its velocity pre-parachute-deployment to 0.45 km s -1. After parachute deployment the rover slowed to 80 m s -1. The sky-crane device that safely landed Curiosity was equipped with 8 thrusters that could at full power exert 3100N of thrust each, which removed the final 80 m s -1 of velocity needed for landing. This means the total delta-v for the landing stage, like the aero-braking, is negligible (Coulter, 2013). After landing, the crew would conduct all the research and general tasks needed for survival on Mars, and wait until the next transfer window opened up, which would be one synodic period later, in December 2022 (Mars Close Approaches). Like during the Earth-Mars transfer, I would need to launch half the transfer orbital period prior to this date in order that the orbits coincide at closest approach. Returning to LMO The spacecraft now needs to return to LMO, which will mean a burn through the Martian atmosphere. Like in the calculations for the Earth launch, we will need to know the velocity of a stable orbit at 200km around Mars, which we calculated earlier to be 3.45 km s -1. In the earth launch we added 1.75 km s -1 of delta-v to our total to account for drag. Mars has an average atmosphere of 0.6% of Earth s mean sea level pressure, so we will only add 0.6% of 1.75 km s -1 to the 3.45 km s -1 needed for the LMO orbit (Lunine, 1999). In other words, the total delta-v to launch from Mars surface to LMO is 3.50 km s Daniel Messias

7 LMO to Mars Escape The return process is going to be very similar to the arrival, however in this case we need to escape the gravitational pull of Mars instead of Earth. Using the same equation from the LEO to Earth Escape calculation we get the following (Nave). v e 2GM r v o v e = 2μ Mars R Mars 3.45km s 1 = 1.43km s 1 μ Mars = Gravitational Constant Mass of Earth = km 3 s 2 R Earth = Equatorial radius of Earth = 3 396km Therefore the total delta-v for this section is 1.43 km s -1. Mars-Earth Hohmann Transfer Again we are going to use the Hohmann equation. The difference is at this point Mars will be further away from Earth than their last closest approach. This time around the distance between them will be 0.545AU. Substituting this new value into the first part of the Hohmann Transfer Equation yields the following results (Braeunig, 2013). v = GM ( 2r 2 1) v r 1 r 1 + r insert = u Sun 2 1AU ( AU 1) = 3.03km s 1 1AU U μ Sun = Gravitational Constant Mass of Sun = km 3 s 2 1AU = km 1.545AU = km After this section the spacecraft will be travelling on a hyperbolic trajectory towards Earth. The total delta-v for this manoeuvre is 3.03 km s -1. Earth Landing Earth orbits the sun at an average speed of km s -1 (Williams, 2013), so it safe to assume this will be the minimum speed that the spacecraft would enter our atmosphere with. The more direct the trajectory towards the surface the greater the entrance speed. As the aim of the piece is to consider the viability of a mars mission with regards to the rocketry we have available at the moment, I am not going to consider the effects of the g-force on the human body entering at such a high velocity, however it would definitely be viable to do multiple passes through Earth s atmosphere where it is thinner to reduce the force the passengers would experience, and slow them to a safe descent over a longer duration. Either way, once a hyperbolic trajectory towards Earth s atmosphere is achieved, no matter what altitude the periapsis may be, fuel will no longer be required as Earth s atmosphere and parachutes can reduce the velocity of the spacecraft to zero as it hits the surface. 6 Daniel Messias

8 Delta-V Total and Mission Feasibility Totalling up the delta-v for each of the mission stages yields the following number: = km s 1 In order to determine whether or not the mission is viable I need to know the delta-v of a modern day rocket. I am going to use the specification of the upcoming NASA rocket named the Space Launch System, and calculate its approximate delta-v capacity. Using the ideal rocket equation and totalling up the available delta v gives the following (Ideal Rocket Equation). v = g 0 I sp ln m full m empty g 0 = Acceleration due to gravity on Earth ' surface = 9.81 m s 2 I sp = The specific impulse of the engine m full = Mass of rocket with propellant m empty = Mass of rocket after propellant/stage is burned Total Mass = (2 751) = 2838t There is a slight problem with using the rocket equation for the launch as there are a pair of boosters and main engine set each with a different I sp firing at the same time. The following equation will calculate the net I sp of the first stage (Specific Impulses). I sp = i F i F i i I spi I splaunch = ( ) ( ) = 254 s Launch: g ln 2838 = 5.74 km s Upper Stage: g ln 268 = 9.96 km s 1 31 Orion v = 1.50 km s 1 (NASA) Total v = km s 1 From these calculations it would appear that that another 5.98km s -1 would need to be found in order for the mission to be viable with the currently available rocketry. There are a few immediately available options to reduce the delta-v absence, however there is always the option to await the design of more efficient and powerful rockets. 7 Daniel Messias

9 One option is to plan a more efficient flight path. The one I proposed was in no way optimised; it was merely a reasonable route. Improvements such as waiting for a closer approach of the planets or increasing the burn size at perigee to maximise use of the Oberth effect would reduce the delta-v cost of the mission. Another option would be to use on-orbit refuelling; that could mean construction the interplanetary stage in LEO before sending up a crew to go to Mars, or sending a mission to Mars, then sending a refuelling vehicle when the crew is ready to depart again. It might even be possible to create fuel using resources available on Mars, however this would then increase the weight of the mission and add huge complexities. Either way, there are plenty of viable ways to gain more delta-v for the mission, and as the gap between currently available and required velocity is so low, I believe it can be concluded that using modern day rocketry a mission to mars is definitely viable. 8 Daniel Messias

10 Works Cited Braeunig, R. A. (2013). Orbital Mechanics. Retrieved March 07, 2014, from Rocket & Space Technology: Coulter, D. (2013, July 30). Strange but True: Curiosity's Sky Crane. Retrieved March 07, 2014, from NASA: Ideal Rocket Equation. (n.d.). Retrieved March 07, 2014, from NASA: Johnston, M. D., Esposito, P. B., Alwar, V., Demcak, S. W., Graat, E. J., & Mase, R. A. (n.d.). Mars Global Surveyor Aerobraking at Mars. Retrieved March 07, 2014, from NASA: Lunine, J. I. (1999). Earth: Evolution of a Habitable World. Cambridge University Press. Mars Close Approaches. (n.d.). Retrieved March 07, 2014, from Richard's Astronomical Observatory: NASA. (n.d.). Orion Quick Facts. Retrieved March 07, 2014, from Lunar & Planetary Institute: Nave, R. (n.d.). Escape Velocity. Retrieved March 07, 2014, from Hyper Physics: Orbits. (2010, May 18). Retrieved March 07, 2014, from Astronomy Notes: Specific Impulses. (n.d.). Retrieved March 07, 2014, from Kerbal Space Program Wiki: Tajmar, D. M. (n.d.). Advanced Space Propulsion Systems. Retrieved March 07, 2014, from Institute of Lightweight Design and Structural Biomechanics: Williams, D. D. (2013, July 1). Earth Fact Sheet. Retrieved March 07, 2014, from NASA: Williams, D. D. (2013, July 01). Mars Fact Sheet. Retrieved March 07, 2014, from NASA: Williams, D. D. (2013, July 01). Sun Fact Sheet. Retrieved March 07, 2014, from NASA: 9 Daniel Messias