Thrust into Space Maxwell W. Hunter, II. Newton’s 3rd Law of Motion Momentum is conserved,...
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Transcript of Thrust into Space Maxwell W. Hunter, II. Newton’s 3rd Law of Motion Momentum is conserved,...
Energy
• Kinetic energy, equation 1-4
•Ratio of kinetic energy of gun to bullet, equation 1-5
KE =mV2
2=
wV2
2g0
KEG
KEB
=VG
VB
=wB
wG
Fuel Consumption
• Specific impulse of engine, equation 1-8
• Effective exhaust velocity, equation 1-9
I sp =
T&w
vef =g0I sp
Power
• Power expended, equation 1-10
• Effective power, equation 1-11
Pe =
&wve2
2g0
Pef =
&wvef2
2g0
=Tvef
2=
g0I spT2
Internal Energy Release
• Exit velocity, equation 1-12
•Combustion temperature, equation 1-13
• Velocity of molecule, equation 1-14
ve = 2g0J h=224 h
Tco :
MV 2
2
V :
2Tco
M
Pump Power
• Pump power, equation 1-15
• Pump power for both propellants, equation 1-16
Ppu =
144Δp&wρ
Ppu =144ΔpTρI sp
The Rocket Equation
•Change in velocity, equation 1-17
• Impulsive velocity, equation 1-18
ΔV = vef lnwI
wF
⎛
⎝⎜⎞
⎠⎟= g0I sp ln
wI
wF
⎛
⎝⎜⎞
⎠⎟
ΔV = ′λ vet lnwI
wF
⎛
⎝⎜⎞
⎠⎟
Energy Efficiency
• Kinetic energy of useful load, equation 1-20
• Total energy expended by exhaust, equation 1-21
KEUL =wULΔV2
2g0
KEet =wprvet
2
2g0
Effect of Initial Velocity
• Increase of kinetic energy of useful load, equation 1-22
• Total kinetic energy expended, equation 1-23
KEUL =wUL VF
2 −VI2( )
2g0
=wUL ΔV2 +2ΔVVI( )
2g0
KEet =wpr vet
2 +VI2( )
2g0
Ballistics
• Flat earth, no drag
• From Newton’s Laws of Motion, equations in 2-1
•Range vs. velocity, equation 2-2
s =Vhtf tf =2Vv
g h=
Vv2
2g
s =V2
g
Energy
• Potential energy, equation 2-3
•Ratio of kinetic energy increase to initial kinetic energy, equation 2-4
PE =mgh=wh
ΔKE
ΔKE0=1+ 2
VI
ΔV
Forces During Motor Burning
• Velocity loss due to gravity, equation 2-5
• Figure 2-1
ΔVg = −gtb sinγ
Airplane Lift/Drag Ratio
• Airplane energy, equation 2-6
•Cruising efficiency, equation 2-7
• Velocity equivalent of energy used, equation 2-8
EA =Ds
EA =ws
L / D
ΔVEN2 =
2g0s
L /D=2g0tbV
L /D
Theoretical Propellant
PerformanceVacuum ε = 40Vacuum ε = 40 Sea Sea
LevelLevel
OxidizerOxidizer FuelFuel Mixture Mixture RatioRatio
Specific Specific GravityGravity IIspsp (sec) (sec) IIspsp (sec) (sec)
NH4ClO4 20% Al 1.74 314 266
H2O2 N2H4 2.09 1.26 325 287
N2O4 N2H4 1.40 1.22 324 292
O2 (cyro)Kerosen
e2.67 1.02 324 300
O2 (cyro) N2H4 0.95 1.07 343 313
Circular Orbits•Gravity as a
function of distance, equation 3-1
• Velocity of satellite, equation 3-2
• Period, equation 3-3
• Period, equation 3-4
g =g0
Rr
⎛⎝⎜
⎞⎠⎟2
Ve = gr =g0R
2
r
Pe =2πrVe
=2π rg
Pe =2πr3/2
g0R2
Potential Energy
• Potential energy, equation 3-5
•Maximum potential energy, equation 3-6
PE =wR 1−Rr
⎛⎝⎜
⎞⎠⎟=wR
hR+h
⎛⎝⎜
⎞⎠⎟
PEmax =wR, r→ ∞
The Vis-Vita Law• Kinetic and potential
energy, equation 3-8
• Conservation of angular momentum, equation 3-9
• Perigee velocity vs. escape velocity at perigee, equation 3-10
• Velocity, equation 3-11
KE +PE =wV2
2g0
+wR 1−Rr
⎛⎝⎜
⎞⎠⎟
Vara =Vprp
Vp2
VEp2 =
rara + rp
=2raa
V 2 =g0R2 2
r−1a
⎛⎝⎜
⎞⎠⎟
The Vis-Vita Law
• Velocity and circular velocity, equation 3-12
•Orbital period, equation 3-13
V
Vc0
⎛
⎝⎜
⎞
⎠⎟
2
=2
r / R−
1a / R
⎛⎝⎜
⎞⎠⎟
Por =2πa3/2
g0R2
Velocities Required to Establish Orbit
• Figure 3-5
• Potential energy and kinetic energy, equation 3-14
PE +KE =R 1−R2r
⎛⎝⎜
⎞⎠⎟
Planet Escape Velocities and Radii
PlanetPlanetEscape Escape VelocityVelocity
(feet/sec)(feet/sec)
Radius Radius (Earth = 1.0)(Earth = 1.0)
Earth 36,700 1.0
Venus 33,600 0.97
Pluto 32,700 1.1
Mars 16,400 0.53
Mercury 13,700 0.38
Satellite Escape Velocities and Radii
Satellite Satellite (Planet)(Planet)
Escape Escape VelocityVelocity
(feet/sec)(feet/sec)
RadiusRadius(Earth = (Earth =
1.0)1.0)Triton (Neptune) 10,400 0.31Ganymede (Jupiter)
9,430 0.39
Titan (Saturn) 8,900 0.39Io (Jupiter) 8,250 0.26Moon (Earth) 7,800 0.272Callisto (Jupiter) 7,450 0.37Europa (Jupiter) 6,900 0.23
The Planets Orbital Data
PlanetPlanet Semi-Major Semi-Major Axis AUAxis AU
PerihelioPerihelion AUn AU
Aphelion Aphelion AUAU
Mercury 0.387 0.308 0.467
Venus 0.723 0.718 0.728
Earth 1.000 0.983 1.017
Mars 1.524 1.381 1.666
Jupiter 5.203 4.951 5.455
Saturn 9.539 9.008 10.070
Uranus 19.182 18.277 20.087
Neptune 30.058 29.800 30.315
The Planets Orbital Data
Mean Celestial LongitudeMean Celestial Longitude
PlanetPlanetOff Off
Ascending Ascending NodeNode
of of PerihelioPerihelio
nn
Epoch, Epoch, 1/1/19961/1/1996
Mercury 47.93° 76.93° 210.29°
Venus 76.38 131.1° 84.87°
Earth 102.12° 98.89°
Mars 49.3° 335.44° 324.31°
Jupiter 100.11° 13.5° 87.32°
Saturn 113.42° 91.5° 347.57°
Uranus 73.9° 168.65° 166.43°
Neptune 131.4° 53° 230.02°
The Planets Orbital Data
InclinationInclination
PlanetPlanet Orbital to Orbital to EclipticEcliptic
Equatorial to Equatorial to OrbitOrbit
Mercury 7.00
Venus 3.39
Earth 23.45
Mars 1.85 25.20
Jupiter 1.31 3.12
Saturn 2.49 26.75
Uranus 0.77 97.98
Neptune 1.77 29
The Planets Orbital Data
PlanetPlanetOrbital Velocity Orbital Velocity
About Sun About Sun (ft/sec)(ft/sec)
Period of Period of Revolution Revolution
(years)(years)Mercury 157,000 0.240
Venus 114,800 0.615
Earth 97,600 1.0
Mars 79,100 1.881
Jupiter 42,800 11.86
Saturn 31,600 29.46
Uranus 22,200 84.02
Neptune 17,800 164.78
Solar System Data
Jupiter’Jupiter’s s
MoonsMoons
DiametDiameter er
(miles)(miles)
SurfaSurface ce
GravitGravityy
PerioPeriod d
(days(days))
Escape Escape Velocity Velocity
(fps)(fps)
Io 2,060 0.195 1.77 8,250Europa 1,790 0.156 3.55 6,900Ganymede
3,070 0.170 7.15 9,430
Callisto 2,910 0.112 16.69 7,450
Hyperbolic Excess Velocity
• Vis-Viva Law, hyperbolic excess velocity, equation 4-1
• Equation 4-2
• Equation 4-3
V∞2 =−
g0R2
a
V 2 =VE2 +V∞
2
V∞ = V2 −VE2
Use of Planetary Energy
•Weight of vehicle, equation 4-5
• Equation 4-6
wV V∞ +VPI( )+wV V∞ −VPF( ) =wP VPI−VPF( )
wV 2V∞( ) =wP ΔVP( )
Selected Comets
CometComet PeriheliPerihelion (AU)on (AU)
AphelioAphelion (AU)n (AU)
PerioPeriod d
(year(years)s)
PerihelioPerihelion Timen Time
Encke 0.339 4.09 3.301967-9-
12
Forbes 1.545 5.36 6.421967-12-
21
D’Arrest 1.378 5.73 6.701967-6-
17
Faye 1.652 5.95 7.411969-12-
29
Halley 0.587 35.0 76.031910-4-
20
Theoretical Liquid Propellant Performance
Equilibrium FlowVacuumVacuum Sea Sea
LevelLevel
OxidizerOxidizer FuelFuel Mixture Mixture RatioRatio
Specific Specific GravityGravity IIspsp IIspsp
Oxygen Hydrogen 4.5 0.31 456 391
Fluorine Hydrogen 9.0 0.50 475 411
Fluorine Ammonia 3.31 1.12 416 360O2-Difluoride
Kerosene 3.8 1.28 396 341
Hydrazine Diborate 1.16 0.63 401 339
HydrazinePentaborane 1.26 0.79 390 328
New Types of Engines
•Wall stress, equation 4-7
• Engine chamber weight, equation 4-8
σ =pr
2t
wco : Acot :
Acopr
2σ
Isotopic Heat Sources
Parent Parent IsotopeIsotope
Half-Half-Life Life
(year(years)s)
Type Type of of
DecaDecayy
Specific PowerSpecific Power(watts/gm)(watts/gm)
ShieldiShieldingngPurePure Fuel Fuel
CompoundCompound
Cesium-137 30 β/γ 0.42 0.067 Heavy
Plutonium-238
89 α 0.56 0.39 Minor
Curium-244 18 α 2.8 2.49Moderat
e
Polonium-210
0.38 α 141 134 Minor
Cobalt-60 5.3 β/γ 17.4 1.7 Heavy
Heavy Velocity Rockets and Gravity
Fields• Travel time, equation 5-1
•Minimum travel time in terms of inner and outer distance, equation 5-2
•Maximum travel time, equation 5-3
t f =57AUΔV
100,000⎛⎝⎜
⎞⎠⎟
t f =144 AUo −AUi( )
ΔV100,000
⎛⎝⎜
⎞⎠⎟
t f =144 AUo +AUi( )
ΔV100,000
⎛⎝⎜
⎞⎠⎟
Escape with Low Acceleration
• Velocity required to escape, equation 5-4
• For launch from circular orbit, equation 5-5
ΔV =Ve +V∞
ΔV = 2Vc2 +V∞
2 −Vc
Heliocentric Velocity Requirements
• Time to generate velocity at constant acceleration, equation 5-6
• Figure 5-6
tb =0.036a / g0( )
ΔV100,000
⎛⎝⎜
⎞⎠⎟
Cooling Limitations
• Amount by which gaseous heating raises specific impulse, equation 5-8
I sp =I sps
f
Electric Rocket Performance
•Characteristic velocity, equation 5-9
• For perfect efficiency, weight of power supply relates to weight of propellant, equation 5-10
Vch =64,100tbα
wαVch2 =wpvef
2
Fusion Rockets
• Initial weight vs. final weight, equation 6-2
•Rocket braking on arrival, equation 6-3
wI
wF
=1+
ΔVc
1−ΔVc
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
c/2vef
wI
wF
=1+
ΔVc
1−ΔVc
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
c/vef
Photon Rockets
• Effective exhaust velocity, equation 6-4
•Relativistic rocket equation 6-5
• Exhaust power of photon beam, equation 6-6
vef =εc
wI
wF
=1+
ΔVc
1−ΔVc
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
1/2ε
Pef =Tc
Mass Annihilation Rockets
•Mass annihilation rocket equation 6-7
•Mass annihilation rocket braking equation 6-8
wI
wF
=1+
ΔVc
1−ΔVc
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
1/2
wI
wF
=1+
ΔVc
1−ΔVc
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥