Thrust into Space Maxwell W. Hunter, II. Newton’s 3rd Law of Motion Momentum is conserved, equation 1- 1.

Thrust into SpaceMaxwell W. Hunter, II

Newton’s 3rd Law of Motion

•Momentum is conserved, equation 1-1

mGVG =mBVB

Force

• Force, equation 1-2

•Weight, equation 1-3

F =ma

w =mg0

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

Guns as Rockets

• Paris Gun, WW I

•Change in velocity, equation 1-6

ΔV = ve 1−wF

wI

⎛

⎝⎜⎞

⎠⎟

Rocket Engines

• Thrust, equation 1-7

T =

&wve

g0

+ pe−pat( )Ae

Rocket Nomenclature

• Figure 1-1

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

Rocket Energy Efficiency

• Figure 1-2

Nozzle Altitude Effect

• Figure 1-3

Nozzle Altitude Performance

• Figure 1-4

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

⎛

⎝⎜⎞

⎠⎟

The Rocket Equation

• Figure 1-5

Useful Load

•Useful load, equation 1-19 wI =wUL +

wpr

′λ

The Rocket Equation

• Figure 1-6

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

External Energy Efficiency

• Figure 1-7

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

External Energy Efficiency

• Figure 1-8

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

Airplane Lift/Drag Ratio

• Figure 2-2

Automobile Lift/Drag Ratio

• Figure 2-3

Ship Lift/Drag Ratio

• Figure 2-4

Solid-Propellant Rockets

• Figure 2-5

Solid Rockets

• Acceleration of guns or rockets, equation 2-9

•Honest John Missile

a =ΔV2

2g0s

Required Acceleration

• Figure 2-6

Four Decades of Development

• Figure 2-7

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

Elliptical Orbit Nomenclature

• Figure 3-1

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→ ∞

Escape Velocity

• Escape velocity, equation 3-7 VE = 2gr =

2g0R2

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

Optimum Ballistic Missile Trajectories

• Figure 3-2

Global Rocket Velocities

• Figure 3-3

Hohmann Transfer

• Figure 3-4

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

Gravity Losses

• Effective gravity, equation 3-15

gef =g−Vh

2

r=

Vc2 −Vh

2

r

Large, Solid Propellant Motors

• Figure 3-6

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

The Outer Solar System

• Figure 4-1

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

Hyperbolic Excess Velocity

• Figure 4-2

Solar System Hyperbolic Excess

Velocity

• Figure 4-3

Hohmann Transfer Velocities

• Figure 4-4

Hohmann Transfer Travel Time

• Figure 4-5

Synodic Period of Planets

• Synodic period, equation 4-4

• Figure 4-6

PS =1

1P1

−1P2

Solar Probe Type Missions with Two Impulse Transfers

• Figure 4-7

Elastic Impact Analogy for the Use of

Planetary Energy

• Figure 4-8

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( )

Planetary Swing-Around Angle

• Figure 4-9

Distance from Center of Sun (Astronomical Units) Solar

Probe Velocity Requirements

• Figure 4-10

Out-of-Ecliptic Velocity

Requirements

• Figure 4-11

Solar System Travel Times

• Figure 4-12

Planetary Arrival Velocities

• Figure 4-13

Planetary Capture Velocities

• Figure 4-14

Payload Velocity Requirements

• Figure 4-15

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

Earth-Mars Launch Windows

• Figure 4-16

Earth-Mars Launch Windows

• Figure 4-17

Round Trip Synodic Period Effects

• Figure 4-18

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

High-Performance Chemical Rockets

• Figure 4-19

New Types of Engines

•Wall stress, equation 4-7

• Engine chamber weight, equation 4-8

σ =pr

2t

wco : Acot :

Acopr

2σ

New Engine Types

• Figure 4-20

Nuclear Thermal Rockets

• Einstein’s famous equation 4-9

• Kiwi-A rocket engine

E =mc2

Graphite Solid-Core Engine

• Figure 4-21

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

Nuclear Vehicle Shielding

Comparison

• Figure 4-22

Required Fuel Weights for Single-Stage Space

Launch Vehicles

• Figure 4-23

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

⎛⎝⎜

⎞⎠⎟

Minimum Travel Times from Earth Including

Braking Requirements

• Figure 5-1

Average Travel Times from Earth Including

Braking Requirements

• Figure 5-2

Solar System Synodic Periods

• Figure 5-3

Travel Times Between Planets

• Figure 5-4

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

Total Velocity to Escape

• Figure 5-5

Heliocentric Velocity Requirements

• Time to generate velocity at constant acceleration, equation 5-6

• Figure 5-6

tb =0.036a / g0( )

ΔV100,000

⎛⎝⎜

⎞⎠⎟

Specific Impulse From Nuclear

Reactions

• Figure 5-7

Typical Gaseous Core

Engines

• Figure 5-8

• Power output, equation 5-7

Pr =0.501T

1,000⎛⎝⎜

⎞⎠⎟

4

Cost of Nuclear Fission Fuel and

Propellant

• Figure 5-9

Cooling Limitations

• Amount by which gaseous heating raises specific impulse, equation 5-8

I sp =I sps

f

Thrust/Weight Ratio of Gaseous Fission

Engines

• Figure 5-10

Types of Electrical Rocket

Thrusters

• Figure 5-11

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

Electrical Rocket Performance

• Figure 5-12

Single-Stage Spaceship Fuel and

Propellant Costs

• Figure 5-13

Transportation vs. Ammunition Re-Use

Assumptions

• Figure 5-14

Spaceship Payload Capability

• Figure 5-15

Single-Stage Spaceship Fuel, Propellant, and

Structure Costs

• Figure 5-16

Single-Stage Spaceship Fuel, Propellant, and

Structure Costs

• Figure 5-17

Dose to Ground Observer from

Gaseous Core Rockets

• Figure 5-18

Gaseous Fission Powered Spaceship

• Figure 5-19

Acceleration Distance

• Figure 5-20

The Near Stars

• Figure 6-1

The Galaxy

• Figure 6-2

Hypothetical Galactic Community

• Figure 6-3

Time Dilation

• Ship time, equation 6-1 ts =tEA 1− V / c( )

2

Interstellar Travel Time Dilation

Effects

• Figure 6-4

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

Fusion Starship Weight Ratio

• Figure 6-5

Fusion Starship Power

• Figure 6-6

Cost of Nuclear Rocket Fuel and

Propellant

• Figure 6-7

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

Starship Weight Ratio

• Figure 6-8

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

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Starship Power

• Figure 6-9

Mass Annihilation Rockets

•Overall time dilation effect, equation 6-9

•Relation between time dilation achieved and rocket weight, equation 6-10

• Equation 6-11

ts =tEA 1+Vc

⎛⎝⎜

⎞⎠⎟

1−Vc

⎛⎝⎜

⎞⎠⎟

=tEA 2 1−Vc

⎛⎝⎜

⎞⎠⎟

wI

wF

=2

1−ΔVc

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

wI

wF

=2tEA

ts

⎛

⎝⎜⎞

⎠⎟

2