Credit most likely to Gary Larson's Far Side  image from Mike Gruntman's page

Propulsion System Engineering EG 101 Mars:  A 21st Century Frontier J. M. Powers

• Engineering: Designing and implementing technology to some end
• Engineering closely wed to science, uses science, and even becomes science, but goals are different
• Engineering encompasses global system design (e.g. mission feasability) all the way down to small scale component design (e.g. Mars rover)
• Rational engineering incorporates scientific principles and empirical knowledge in system design and development
• For Mars Pathfinder: scientific achievement is the end
• Other ends include economic, political, military etc.
• Dilbert recruiting engineers and some more
• Underlying Scientific Principles in Propulsion
• Newton's Laws of Motion
• In the absence of a net force, bodies in motion stay in motion; bodies at rest stay at rest.
• The product of the mass of a body and its acceleration is equal to the net force applied to the body: F = m a
• For every action, there is an equal and opposite reaction.
• Newton's Law of Gravitational Attraction: F = G m₁ m₂/r²
• Total energy is conserved: Kinetic + Potential + Thermal + Chemical + ... = constant
• Application of Scientific Principles to Propulsion Engineering
• m_total d²x/dt² = m_dot v_e - m_totalg: important in getting off the ground
• 1/2 m_pf v² = m_sp q_chem: kinetic energy balanced with chemical energy, important in interplanetary travel
• x = v t: relation between distance and time for constant velocity motion
• Classes of space propulsion systems
• Chemical rocket (on board fuel AND oxidizer)
• Solid Propellants
• homogeneous: single molecule has both fuel AND oxidizer--unstable, explosive
• heterogeneous: ground slurry of metallic (often Aluminum) fuel and nitramine oxidizer stirred together in polymeric mixture--safer
• High energy density (q_chem~5 MJ/kg)
• Difficult to control burning rates
• Liquid Propellants
• monopropellants: single molecule which breaks down exothermically on ignition, eg. hydrazine N₂ H₄
• bipropellants: separate liquid fuel and liquid oxidizer, e.g.. O₂ and H₂
• Lower energy density (q_chem~1 MJ/kg)
• Easier to control with valves and pumps
• Cryogenic storage necessary to maintain liquid state
• Electric
• on board low energy inert gas
• energy typically from solar panels or on-board nuclear reactor input to accelerate and expel gas
• long duration (10,000 hours)
• low thrust (220 mN)
• Important parameters for Mars Pathfinder Flight
• Payload mass: m_pf=552 kg
• Distance from Earth to Mars: ~7.81 x 10¹⁰ m
• Propellant energy density: ~q_chem=5 MJ/kg
• Available economic resources: $250,000,000
• Calculations for Mars flight
• Launch 4 December 1996; 1:58:07 EST
• Launch Site: Kennedy Space Center, Florida
• Launch Vehicle: McDonnell-Douglas (now Boeing!)Delta II (7925) rocket
• Can lift 2036 kg into geostationary transfer orbit
• rocket mass =m_total =230,000 kg (hence gravitational force on earth surface = m_total g = (230,000 kg)(9.81 m/s²) = 2,256,000 N
• rocket height = 38.1 m
• thrust at liftoff =F_thrust = 2,630,000 N
• Given: rocket mass, gravitational acceleration, thrust available
• Find: estimates for initial acceleration, velocity after 60 s, distance after 60 s
• Assume: mass constant (very conservative!)
• Analysis:

(mass)*(acceleration)=net force
m_totala = -m_totalg + F_thrust
a = -g + F_thrust/m_total
a = -9.81 m/s² + 2,630,000 N/230,000 kg
a = -9.81 m/s² + 13.37 m/s²
a = 1.62 m/s²
dv/dt = a
v = dy/dt = at + v_o; (v_o = 0 here)
y = 1/2 a t^2 + v_ot + y_o (y_o = 0 here)

v = dy/dt = at = (1.62 m/s²) t
y = 1/2 a t² = (1/2)(1.62 m/s²) t² = 0.81 m/s² t²

After 1 minute of thrust (t = 60 s)
v = (1.62 m/s²) (60 s) = 97.2 m/s
y = (0.81 m/s²) (60 s)² = 2,916 m

• Cost of Delta 7925 rocket ~ $61,000,000
• First achieve Earth orbit and discard big booster rocket
• After orbit achieved, final boost out of orbit from small stage III solid rocket
• Chemical rockets can ultimately achieve the velocity of the expelled gases
• Light product gases can reach up to 7 km/s = 7 x 10³ m/s
• Earth's mean solar radius = 1.50 x 10⁸ km = 1 Astronomical Unit (AU)
• Mars's mean solar radius = 2.28 x 10⁸ km = 1.52 AU
• Earth-Mars distance ~ 2.28 x 10⁸ km - 1.50 x 10⁸ km = 7.81 x10⁷ km = 7.81 x 10¹⁰ m
• Time to reach Mars ~ distance/speed = (7.81x 10¹⁰ m)/(7 10³ m/s) = 1.11 10⁷ s = 4 months: close!
• Similarly,
• Earth-moon @ 3,900 m/s -> 7 days
• Earth-Neptune @ 13,400 m/s -> 29.9 years
• Earth-Alpha-Centauri @ 30,000,000 m/s -> 50 years
• Seven month, zero thrust, constant velocity cruise time to Mars; estimate how much solid propellant needed to get the Pathfinder to Mars
• Given: m_pf = 552 kg, v_final = 7 x 10³ m/s, v_initial ~ 0 m/s, q_chem = 5 x10⁶ J/kg
• Find: mass of solid propellant (m_sp) necessary to achieve final velocity
• Assume: no initial kinetic energy,  100% conversion of chemical->kinetic energy (liberal! assumption due to 2nd Law of Thermo),  constant mass system (overly-conservative!)
• Analysis: kinetic energy_initial + chemical energy_initial = kinetic energy_final + chemical energy_final
• 0 + m_sp q = 1/2 m_pf v_final² + 0
• m_sp = 1/2 m_pf v_final²/q_chem
• m_sp = (1/2)(552 kg)(7 x 10³ m/s)²/(5 x10⁶ J/kg) = 2700 kg of propellant!
• Total cost of operations during cruise, landing, and surface operations: $14,000,000
• Capture into Martian orbit
• Entry to Martian Surface (4 July 1997)
• Entry mass = 552 kg
• Surface area = 5.52 m²
• Drag coefficient = C_D = F_D/(rho v² A) = 1.7
• F_D = rho v² A C_D = (0.0012 kg/m³)(7500 m/s)²(5.52 m²)(1.7)=633,000 N
• ma = -mg + F_D
• a = -g + F_D/m
• v = -g t + (F_D/m) t + v_o
• v_land = -g t + (F_D/m) t + v_o
• t = (v_o-v_land)/(g-F_D/m)
• t = (-7500 m/s-(-100 m/s))/(3.69 m/s² - (633,000 N)/(552 kg)) = ~ (552 kg)(7500 m/s)/(633,000 N) = 6.5 s!
• Very short re-entry time using "aerodynamic braking"
• Analysis very deficient!
• Some tests and analysis indicated some small susceptibility to instability in Martian entry
•  Sources and Links
• Braun, R. D., Powell, R. W., Engelund, W. C., Gnoffo, R. A., Weilmuenster, K. J., and Mitcheltree, R. A., 1995, "Mars Pathfinder Six-Degree-of-Freedom Entry Analysis," Journal of Spacecraft and Rockets, Vol. 32, No. 6, pp. 993-1000.
• Chen, Y.-K., Henline, W. D., and Tauber, M. E., 1995, "Mars Pathfinder Trajectory Based Heating and Ablation Calculations," Journal of Spacecraft and Rockets, Vol. 32, No. 2, pp. 225-230.
• Cook, R. A., and McNamee, J. B., 1993, " Mission Design for the Mars Environmental Survey, AAS 93-563, pp. 967-981.
• Cornelisse, J. W., Schoyer, H. F. R, and Wakker, K. F., 1979, Rocket Propulsion and Spaceflight Dynamics, Pitman: London.
• Halliday, D., and Resnick, R., 1978, Physics, Wiley: New York.
• Hill, P., and Peterson, C., 1992, Mechanics and Thermodynamics of Propulsion, Second Edition, Addison-Wesley:  Reading, MA.
• Loh, W. H. T., 1968, Jet, Rocket, Nuclear, Ion and Electric Propulsion: Theory and Design, Springer-Verlag: New York.
• Micheltree, R. A., and Gnoffo, P. A., 1995, "Wake Flow About the Mars Pathfinder Entry Vehicle," Journal of Spacecraft and Rockets, Vol. 32, No. 5, pp. 771-776.
• Reiber, D. B., 1988, The NASA Mars Conference, Vol. 71 American Astronautical Society Science and Technology Series, Univelt: San Diego.
• Shirley, D. L., 1996, "`Mars on $300K a Day':The Mars Exploration Program," Acta Astronautica, Vol. 39, No. 1-4, pp. 101-110.
• Spear, A. J., 1996, "Mars Pathfinder Project Progress Report," Acta Astronautica, Vol. 39, No. 1-4, pp. 91-99.
• Stoker, C., 1989, The Case for Mars III:  Strategies for Exploration-Technical, Vol. 75 American Astronautical Society Science and Technology Series, Univelt: San Diego.
• Mars Pathfinder Launch Vehicle Homepage
• Mars Fact Sheet
• U.S. Air Force Phillips Laboratory Propulsion Directorate

Mars Pathfinder Launch and Cruise Homepage

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