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Seminar 3!Precursors to Space Flight!
Orbital Motion!FRS 112, Princeton University!
Robert Stengel"
Copyright 2015 by Robert Stengel. All rights reserved. For educational use only.!http://www.princeton.edu/~stengel/FRS.html!
Prophets with Some Honor"The Human Seed and Social Soil: Rocketry and Revolution"
Orbital Motion"Energy and Momentum"
Orbital Elements"
1!
Precursors to Space Flight: !Rocket, Missiles, and Men in Space, Ch 4 (ER) "
… the Heavens and the Earth, Introduction, Ch 1"Orbital Motion: !Understanding Space, Ch 5"
Introduction, …the Heavens and the Earth"
2!
Introduction, …the Heavens and the Earth"
3!
Introduction, …the Heavens and the Earth"
4!
Ch. 1, The Human Seed and Social Soil: Rocketry and Revolution!
5!
6!
Ch. 1, The Human Seed and Social Soil: Rocketry and Revolution!
7!
Ch. 1, The Human Seed and Social Soil: Rocketry and Revolution!
The Human Seed and Social Soil: Rocketry and Revolution!
8!
Early 20th Century Rocket Vehicles"
Opel RAK.1 Airplane"https://www.youtube.com/watch?
v=vsqg28y_s3s"
Valier Rocket Car"
Opel RAK.3 Car"
Opel RAK.6 Automobile"https://www.youtube.com/watch?
v=LmQIWpTW-W8"
Opel RAK.1 Plane"
9!
10!
Konstantin Tsiolkovsky (1857-1935)"
•# Russian “father of spaceflight”"•# High school teacher who wrote of
rocket-propelled vehicles"•# “The rocket equation” (1897):
vehicle speed change depends on"–# Rocket exhaust velocity"–# Ratio of vehicle’s full-to-empty mass"
!v = vexhaust lnminitial
m final 11!
Orbital Motion!
12!
Specific Energy…"… is the energy per unit of the
satellite’s mass"
KES =12mmv2 = 1
2v2
!# Specific total energy:"
!# Specific kinetic energy:"
PES = ! mm
µr= ! µ
r
ES = PES + KES = ! µr+ 12v2
!# Specific potential energy:"P1"
P2"
13!
!# Specific total energy is inversely proportional to the semi-major axis (see App. C.4, Sellers)"
12v2 = µ
r+ E
v = µ 2r! 1a
"#$
%&'
!# Velocity is a function of radius and specific energy (drop subscript)"
E = ! µ2a
“Vis Viva (Living Force) Integral”"
!# Velocity is a function of radius and semi-major axis (see http://en.wikipedia.org/wiki/Vis-viva_equation)"
v = 2 µr+ E!
"#$%&
14!
Orbital Period"
•# From Kepler’s 3rd Law , Period of the Orbit, P, is (see App. C.6, Sellers)"
P = 2! a3
µ, min
•# Thus, the orbital period is related to the specific total energy as"
P = ! "µ2
2E 3 , min
where E < 0 for an ellipse15!
Examples of Circular Orbit Periods for Earth and Moon"
Period, min"Altitude above "
Surface, km" Earth" Moon"0" 84.5" 108.5"
100" 86.5" 118"1000" 105.1" 214.6"
10000" 347.7" 1905"
16!
Angular Momentum of a Particle (Point Mass)!
h = r !mv( ) = m r ! v( ) = m r ! !r( )17!
Angular Momentum of a Particle!
•# Moment of linear momentum of a particle"–# Mass times components of the velocity
that are perpendicular to the moment arm"
•# Cross Product: Evaluation of a determinant with unit vectors (i, j, k) along axes, (x, y, z) and (vx, vy, vz) projections on to axes"
r ! v =i j kx y zvx vy vz
= yvz " zvy( ) i + zvx " xvz( ) j + xvy " yvx( )k
h = r !mv( ) = m r ! v( )
18!
Dimension of energy?!
Dimension of linear momentum?!
Dimension of angular momentum?!
Scalar (1 x 1)!
Vector (3 x 1)!
Vector (3 x 1)!19!
Cross Product in Column Notation!
r ! v =i j kx y zvx vy vz
= yvz " zvy( )i + zvx " xvz( ) j + xvy " yvx( )k
r ! v =
yvz " zvy( )zvx " xvz( )xvy " yvx( )
#
$
%%%%%
&
'
(((((
Column notation"
Cross product identifies perpendicular components of r and v"
20!
Can We Define a Cross-Product-Equivalent Matrix?"
r ! v ==
yvz " zvy( )zvx " xvz( )xvy " yvx( )
#
$
%%%%%
&
'
(((((
=? ? ?? ? ?? ? ?
#
$
%%%
&
'
(((
vxvyvz
#
$
%%%%
&
'
((((
Cross-product-equivalent matrix"
!r "0 !z yz 0 !x!y x 0
"
#
$$$
%
&
'''
Cross product"
21!
Angular Momentum Vector is Perpendicular to Both Moment
Arm and Velocity!
h = mr ! v = m
yvz " zvy( )zvx " xvz( )xvy " yvx( )
#
$
%%%%%
&
'
(((((
= m0 "z yz 0 "x"y x 0
#
$
%%%
&
'
(((
vxvyvz
#
$
%%%%
&
'
((((
= m!rv
22!
Specific Angular Momentum Vector of a Satellite!
hS =
mmr ! v = r ! v = r ! !r
… is the angular momentum per unit of the satellite’s mass, referenced to the center of attraction"
It is constant and perpendicular to the orbital plane"
How do we know that?"
23!
Recall"
a t( ) = !v t( ) = !!r t( ) = ! µ
r2 t( )rI t( )r t( )
"
#$%
&'= ! µ
r3 t( ) r t( )
… or"
!!r + µ
r3r = 0
Equations of Motion for a Particle in an Inverse-Square-Law Field "
24!
Cross Products of Radius and Radius Rate!
Then"
… because they are parallel"
r ! !!r + µ
r3r"
#$%&'= 0
r ! r = 0 !r ! !r = 0
ddtr ! !r( ) = !r ! !r( ) + r ! !!r( ) = r ! !!r( )Chain Rule for Differentiation"
25!
Specific Angular Momentum!
Consequently"
r ! !!r + µ
r3r"
#$%&'= r ! !!r( ) + µ
r3r ! r( )
hS = Constant
!hS " r # !r( ) (Perpendicular to the plane of motion)
Orbital plane is fixed in inertial space"
= ddtr ! !r( ) = dhS
dt= 0
26!
Eccentricity Vector!
With triple vector product identity (see Supplement)"
!!r + µ
r3r!
"#$%&' h = !!r ' h+ µ
r3r ' h = 0
e = Eccentricity vector Constant of integration( )
!!r ! h = " µ
r3r ! h = " µ
r3r ! r ! !r( )
!!r ! h = " µ
r3r ! r ! !r( ) = " µ
r2!rr " r!r( ) = µ d
dtrr
#$%
&'(
Integrating"
!!r ! h( )dt" = !r ! h = µ r
r+ e#
$%&'(
27!
Significance of Eccentricity Vector!
!r ! h" µ r
r+ e#
$%&'(
)*+
,-.
T
h = 0
! !r " h( )T h# µrTh
r# µeTh = 0
!"µeTh = 0
0" 0"
!# e is perpendicular to angular momentum,"
!# which means it lies in the orbital plane"!# Its angle provides a reference direction
for the perigee" 28!
Classical Orbital Elements"Dimension and Time"
Orientation"
a : Semi-major axise : Eccentricityt p: Time of perigee passage
! :Longitude of the Ascending/Descending Nodei : Inclination of the Orbital Plane" : Argument of Perigee
29!
In-plane Parameters of an Elliptical Orbit"
Dimensions of the orbit"
p = h2
µ = Semi-latus rectum
h = Magnitude of angular momentum
E = ! µr+ 1
2v2 = ! µ
2a= Specific total energy
e = 1+ 2Ep µ = Magnitude of eccentricity vector
a = p1! e2 = Semi-major axis
ra = a 1+ e( ) = Apogee radiusrp = a 1! e( ) = Perigee radius
µE = 3.98 !105 km3 s2
RE = 6,378 km
30!
In-plane Parameters of an Elliptical Orbit"
Position and velocity of the satellite"
r = p1+ ecos!
= h2 µ1+ ecos!
! = True anomaly
v = µar
2a " r( )
Period of the orbit"
P = 2! a3µ
31!
Orientation of an Elliptical Orbit"
32!
Position and Velocity Following Launch Determine Orbital Elements"
Identical major axes = "Identical orbital periods"
33!
FPS-16 Radar"
Planar Orbit Establishment from Measured Radius, Angular Rate, Velocity, and Time"
!o = cos"1 1e
pro"1
#$%
&'(
)
*+
,
-.
/ o = cos"1 a " roae
#$%
&'( : Eccentric Anomaly
t p =a3
µ/ o " esin/ o( )
rp = a 1" e( )
h = ro2 !!o
p= h2 µ
E = vo2
2" µro
a = µro2µ " rovo
2
e = 1" p a2
Given ro, !!o,vo,to!!o = vo cos" o ro( )
" o : Flight Path Angle from Local Horizontal
34!
Effect of Launch Latitude on Orbital Parameters"
•# Launch latitude establishes minimum orbital inclination"•# Time of launch establishes line of nodes"•# Argument of perigee established by"
–# Launch trajectory"–# On-orbit adjustment"
Typical launch inclinations from
Wallops Island"(Latitude = 38°)"
35!
Typical Satellite Orbits "
Sun-Synchronous
Orbit"
GPS Constellation"26,600 km"
36!
Geo-Synchronous Ground Track "
Geo-Synchronous Ground Track"
42, 164 km"
Marco Polo-1 & 2"
37!
Orbital Lifetime of a Satellite"•# Aerodynamic drag causes orbit to decay"
dVdt
= !CD"V2S /2
m# !B* "V 2S /2
B* = CDS /m
•# Drag is highest at perigee"–# Air drag circularizes the orbit"
•# Large change in apogee"•# Small change in perigee"•# Until orbit is ~circular"•# Final trajectory is a spiral"
•# Aerodynamic drag causes orbit to decay"
•# Air density decreases exponentially with altitude"
! = !SLe"h/hscale
!SL = air density at sea level; hscale = atmospheric scale height
38!
Orbital Lifetime of a Satellite"
dadt
= ! µaB* "SLe! a!R( ) / hscale
•# Time, tdecay, to reach earth s surface (a = R) from starting altitude, h0"
•# Aerodynamic drag causes energy loss, reducing semi-major axis, a"
•# Variation of a over time"
e! a!R( ) / hs
aa0
a
" da = ! µB* #SL dt0
t
"
tdecay = hscaleµRB* !SL
eh0 / hscale "1( )39!
NRL Starshine 1 Orbital Decay (2003)"
http://www.azinet.com/starshine/descript.htm!
ISS Altitude = 330-435 km"
40!
Next Time:!Early Space Age !"
…the Heavens and the Earth, Ch 2 to 5; "!
Launch Dynamics & Staging !"Understanding Space, Sec 9.1, 14.1 (pp.
535-542), 14.3"
41!
SSuupppplleemmeennttaall MMaatteerriiaall
42!
Prophets with Some Honor"
43!
Long-Distance Communication"
44!
Industrial Revolution"
45!
Electric Power Storage, Generation, and Transmission"
46!
Pangaea and Eusthenopteron"
47!
Industrial Revolution and Government Science"
48!
Government, Technology, and War"
49!
Background Math!
a ! b ! c( ) " a i c( )b # a ib( )c= aTc( )b # aTb( )c
Triple Vector Product Identity!
r i !r = rT !r = r dr
dt
Dot Product of Radius and Rate !
50!
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