Download Early History & Fiction; Orbital Motion

Seminar 2!
Early History & Fiction!
Orbital Motion!
FRS 112, Princeton University!
Robert Stengel"
Decades of the Great Dreams"
From the Earth to the Moon, Jules Verne"
Equations of Motion"
Momentum and Energy"
"
Rockets, Missiles, and Men in Space, Ch 2!
Round the Moon, excerpts!
Understanding Space, Ch 4!
Copyright 2015 by Robert Stengel. All rights reserved. For educational use only.!
http://www.princeton.edu/~stengel/FRS.html!
1!
A Timeline of Events Critical to Space Travel – V!
Date
17th-18th
cent
Science Fact
Gottfried Leibniz (1646-1716):
Calculus (1675), founder of
"dynamics". Isaac Newton
(1642-1727): Laws of Motion,
Universal Gravitation, Principia
(1687).
Science Fiction
Jonathan Swift,
Gulliver's Travels
(1726),
18th-19th
cent
Pierre-Simon Laplace (1749Voltaire, Micromégas
(1752)
1827): Solar system
cosmology and dynamics. Carl
Friedrich Gauss (1777-1855):
Orbit calculation, shape of the
Earth. "Balloonatics":
Montgolfier brothers &
deRozier (1783), 1st manned
flight; Madame Blanchard;
Major Turner; Glaisher &
Coxwell to 35K ft
World Scene
1689: Peter the Great,
Czar of Russia. 1690:
Battle of the Boyne.
1729: Bach's St.
Matthew Passion.
1746: PRINCETON IS
FOUNDED!
AMERICAN REVOLUTION;
FRENCH REVOLUTION;
AGE OF THE
ENLIGHTENMENT;
INDUSTRIAL
REVOLUTION. 1781:
Herschel discovers
Uranus. 1800s:
Exploits of Napoleon.
Steamships, railroads,
Babbage's Analytical
Engine.
2!
Gottfreid Leibniz!
(1646-1716)!
!" German mathematician and philosopher"
!" Along with Newton, invented “infinitesimal calculus”"
!" Established notation for derivatives, integrals, and “chain
rule”, e.g.,"
dy d !" f ( x ) #$
=
;
dx
dx
y = f ( x );
! y dx = ! f ( x ) dx;
d 2 y d %' d !" f ( x ) #$ )'
= &
*
dx 2 dx (' dx +'
! y dx = ! y
dx
du
du
dy dy du1
du
=
i
i!i n
dx du1 du2
dx
3!
Pierre Simon Laplace!
(1749-1827)!
!"
!"
!"
!"
!"
!"
Mathematical astronomy and statstics"
The “French Newton”"
Mécanique Céleste (1799-1825), 5 volumes"
Laplace transform!
Laplace’s equation!
The “figure” of the Earth and its potential
(TBD); spherical harmonics"
"
L [ x(t)] = x(s) = # x(t)e! st dt
0
s = $ + j% , ( j = i = !1)
! 2" = 0
4!
Karl Friedrich Gauss!
(1777-1855)!
•" German mathematician who contributed to
many fields"
•" Determination of conic section orbits from
observations"
•" Method of “least squares”"
•" “Gaussian distribution”"
•" Discovered “Fast Fourier
Transform” (Cooley, Tukey, 1965) before
there was a Fourier Transform (1807)!
•" Circuit laws, electromagnetics"
5!
Gaussian probability density function of random variable, x
2
( )
#
1
2
pr ( x ) =
e 2"
2!"
µ : Mean value of x; " : Standard deviation of x
x# µ
!" Describes variability of many physical symbols "
!" Also called a “Normal Distribution”"
!" Classic “bell-shaped curve”"
6!
Charles Babbage!
(1791-1871)!
!" Conceived of Difference Engine to tabulate
polynomials (1822)"
!" Analytical Engine with programmable arithmetic
logic (1837)"
!" Neither machine was built in his lifetime"
7!
Decades of the
Great Dreams!
8!
Near and Far Sides
of the Moon"
Clementine mission, 1994"
Albedo!
Height above Reference!
9!
A Timeline of Events Critical to Space Travel – VI!
Date
Early 19th cent
Late 19th cent
Science Fact
Karl Jacobi (18041851): mathematics of
dynamical systems.
William Congreve:
Military & rescue
rockets.
Science Fiction
Edgar Allan Poe, The
Unparalleled
Adventure of One
Hans Pfaall (1835),
The moon hoax (False
account of Herschel's
research, 1835)
Henri Poincaré (1854- 1865: Jules Verne,
1912): Restricted 3From the Earth to the
body problem. Kirchoff Moon, Alexandre
& Bunsen:
Dumas, Voyage a la
Spectroscope (1859)
lune, Henri de Parville,
Un Habitant de la
planete Mars, Anon,
Voyage a la lune,
English clergyman,
Chrysostom Trueman
(nom de plume), The
History of a Voyage to
the Moon, Camille
Flammarion, Mondes
imaginaires et mondes
reels
World Scene
Bombardment of
Copenhagen (1807);
WAR OF 1812 (StarSpangled Banner);
Battle of Waterloo
AMERICAN CIVIL WAR;
FRANCO-PRUSSIAN
WAR; VICTORIAN ERA.
1866: Nobel invents
dynamite. 1876: Bell
invents telephone.
1895: Roentgen
discovers X-rays.
10!
1865!
Jules Verne
(1828-1905)!
11!
Orbital Motion!
12!
Orbits 101"
Satellites!
Escape and Capture
(Comets, Meteorites)!
13!
Two-Body Orbits are Conic Sections "
14!
Orbits 102"
(2-Body Problem) !
•" e.g., "
–" Sun and Earth or"
–" Earth and Moon or"
–" Earth and Satellite"
•" Circular orbit: radius and velocity
are constant"
•"
Low Earth orbit: 17,000 mph = 24,000
ft/s = 7.3 km/s"
•" Super-circular velocities"
–" Earth to Moon: 24,550 mph = 36,000
ft/s = 11.1 km/s"
–" Escape: 25,000 mph = 36,600 ft/s =
11.3 km/s"
•" Near escape velocity, small changes
have huge influence on apogee"
15!
Newton’s 2nd Law"
!" Particle of fixed mass (also called a point mass)
acted upon by a force changes velocity with "
!" acceleration proportional to and in direction of force "
!" Inertial reference frame "
!" Ratio of force to acceleration is the mass of the
particle: F = m a"
dv ( t )
d
= ma ( t ) = F
!" mv ( t ) #$ = m
dt
dt
! fx #
&
%
F = % fy & = force vector
% f &
%" z &$
! v (t )
x
#
d
m # vy ( t )
dt #
#" vz ( t )
$ ! f $
& # x &
& = # fy &
& #
&
f
z
&% #"
&%
16!
Equations of Motion for a Particle "
Integrating the acceleration (Newton’s 2nd Law)
allows us to solve for the velocity of the particle"
! fx m $ ! a $
& # x &
#
dv ( t )
1
= v! ( t ) = F = # fy m & = # ay &
m
dt
# f m & #
&
&% #" az &%
#" z
v (T ) = !
T
0
T
T 1
dv ( t )
dt + v ( 0 ) = ! a ( t ) dt + v ( 0 ) = !
F dt + v ( 0 )
0
0 m
dt
3 components of velocity"
! v (T )
# x
# vy ( T )
#
#" vz (T )
$
! a (t )
x
&
T#
& = ' # ay ( t )
& 0#
&%
#" az ( t )
$
! v (0) $
! f (t ) m $
! v (0) $
x
x
&
&
&
&
#
# x
T#
& dt + # vy ( 0 ) & = ' # fy ( t ) m & dt + # vy ( 0 ) &
&
& 0#
&
&
#
#
&%
#" vz ( 0 ) &%
#" fz ( t ) m &%
#" vz ( 0 ) &%
Equations of Motion for a Particle "
Integrating the velocity allows us to solve for the
position of the particle"
! x! ( t )
#
dr ( t )
= r! ( t ) = v ( t ) = # y! ( t )
dt
#
#" z! ( t )
r (T ) = !
T
0
$ ! v (t )
& # x
& = # vy ( t )
& #
&% #" vz ( t )
$
&
&
&
&%
T
dr ( t )
dt + r ( 0 ) = ! v dt + r ( 0 )
0
dt
3 components of position"
! x (T )
#
# y (T )
#
#" z (T )
! v (t )
$
x
&
T#
& = ' # vy ( t )
& 0#
#" vz ( t )
&%
$
! x (0) $
&
&
#
& dt + # y ( 0 ) &
&
&
#
&%
#" z ( 0 ) &%
18!
17!
Trajectories Calculated
with Flat-Earth Model"
!" Constant gravity, g, is the only force in the
model, i.e., no aerodynamic or thrust force"
!" Can neglect motions in the y direction"
Dynamic Equations"
v x ( t ) = v x0
v!z ( t ) = !g
x! ( t ) = vx ( t )
( z positive up )
Analytic (Closed-Form) Solution"
v x ( T ) = v x0
T
z! ( t ) = vz ( t )
Initial Conditions"
vz (T ) = vz0 ! " g dt = vz0 ! gT
v x ( 0 ) = v x0
0
x ( T ) = x 0 + v x0 T
T
z (T ) = z0 + vz0 T ! " gt dt = z0 + vz0 T ! gT 2 2
vz ( 0 ) = vz0
x ( 0 ) = x0
0
z ( 0 ) = z0
19!
Trajectories Calculated
with Flat-Earth Model"
vx ( 0 ) = 10m / s
vz ( 0 ) = 100, 150, 200m / s
x (0) = 0
z (0) = 0
20!
Nike-Tomahawk 2-Stage
Sounding Rocket"
Nike-Ajax AntiAircraft Missile"
45-kg payload to
370-km altitude"
Trajectory well approximated using flat-Earth
model from 2nd stage burnout"
21!
MATLAB Code for FlatEarth Trajectories"
Script for "
Analytic Solution"
!
g
t
!
vx0
vz0
x0
z0
!
vx1
vz1
x1
z1
Script for "
Numerical Solution"
tspan
= 40; % Time span, s!
xo
= [10;100;0;0]; % Init. Cond.!
[t1,x1] = ode45('FlatEarth',tspan,xo);!
=
=
9.8;
!
0:0.1:40;!
=
=
=
=
10;!
100;!
0;!
0;!
=
=
=
=
vx0;!
vz0 – g*t;!
x0 + vx0*t;!
z0 + vz0*t - 0.5*g*t.* t;!
Function for "
Numerical Solution"
function xdot = FlatEarth(t,x) !
%
x(1)
=
vx!
%
x(2)
=
vz!
%
x(3)
=
x!
%
x(4)
=
z!
g
!
=
9.8;!
xdot(1) =
0;!
xdot(2) =
-g;!
xdot(3) =
x(1);!
xdot(4) =
x(2);!
xdot
=
xdot';!
22!
end!
Sub-Orbital (Sounding) Rockets
1945 - Present!
NASA Wallops Island
Control Center"
Canadian!
Black Brant XII!
LTV!
Scout!
23!
Spherical Model of the
Rotating Earth"
Spherical model of earth
s surface,
earth-fixed (rotating) coordinates"
! xo $
! cos LE cos ' E
&
#
#
R E = # yo & = # cos LE sin ' E
# z &
#
sin LE
" o %E "
$
&
&R
&
%
LE : Latitude (from Equator), deg
! E : Longitude (from Prime Meridian), deg
R : Radius (from Earth's center), deg
Earth's rotation rate, !, is 15.04 deg/hr
24!
Non-Rotating (Inertial) Reference
Frame for the Earth"
Celestial longitude, "C, measured
from First Point of Aries on the
Celestial Sphere at Vernal Equinox"
!C = !E + "( t # t epoch ) = !E + " $t
25!
Transformation Effects of Rotation"
Transformation from inertial frame to rotating frame"
$ cos !"t
R E = & # sin !"t
&
0
&%
sin !"t
cos !"t
0
0
0
1
$ cos !"t
'
) R = & # sin !"t
) I &
0
&%
)(
sin !"t
cos !"t
0
0
0
1
' $ xo '
)
)&
y
)& o )
)( & zo )
(I
%
Location of satellite, rotating and inertial frames"
" cos LE cos ! E
$
rE = $ cos LE sin ! E
$
sin LE
#
%
" cos LE cos !C
'
$
' ( R + Altitude ); rI = $ cos LE sin !C
'
$
sin LE
&
#
%
'
' ( R + Altitude )
'
&
Orbital calculations generally are made in an
inertial frame of reference"
26!
Gravity Force Between Two
Point Masses"
Magnitude of gravitational attraction"
Gm1m2
F=
r2
G : Gravitational constant = 6.67 ! 10 "11 Nm 2 / kg 2
m1 : Mass of 1st body = 5.98 ! 10 24 kg for Earth
m2 : Mass of 2 nd body = 7.35 ! 10 22 kg for Moon
r : Distance between centers of mass of m1 and m2 , m
Force between Earth and Moon = 1.98 x 1020N!
27!
Acceleration Due To Gravity"
Gm1m2
r2
Gm µ
= 2 1 ! 21
r
r
F2 = m2 a1on2 =
a1on2
“Inverse-square
Law”!
µ1 = Gm1 : Gravitational parameter of 1st mass
At Earth’s surface, acceleration due to gravity is"
ag ! goEarth =
µE
2
Rsurface
3.98 ! 1014 m 3 s 2
2
=
2 = 9.798 m s
( 6, 378,137m )
What is the acceleration due to gravity at the Moon’s
surface (R = 1,375,000 m)?"
28!
Gravitational Force Vector
of the Earth"
Force always directed toward the Earth’s center"
( x +
µ E " rI %
µE *
Fg = !m 2 $ ' = !m 3 * y - (vector), as rI = rI
rI # rI &
rI
*) z -,
I
(x, y, z) establishes the direction of
the local vertical"
rI
=
rI
! x $
&
#
# y &
! cos LE cos ' I
#" z &%
#
I
=
# cos LE sin ' I
x I2 + yI2 + zI2 #
sin LE
"
$
&
&
&
%
29!
Equations of Motion for a Particle
in an Inverse-Square-Law Field "
Integrating the acceleration (Newton’s 2nd Law)
allows us to solve for the velocity of the particle"
( x +
dv ( t )
µ E " rI %
µE *
1
= v! ( t ) = Fg = ! 2 $ ' = ! 3 * y m
dt
rI # rI &
rI
*) z -,
v (T ) = !
T
0
T
T 1
dv ( t )
dt + v ( 0 ) = ! a ( t ) dt + v ( 0 ) = !
F dt + v ( 0 )
0
0 m
dt
3 components of velocity"
! v (T )
# x
# vy ( T )
#
#" vz (T )
$
! x r3
I
&
T#
& = ' µ E ( # y rI 3
0
&
#
3
&%
#" z rI
! v (0) $
$
&
# x
&
& dt + # vy ( 0 ) &
&
#
&
#" vz ( 0 ) &%
&%
30!
Equations of Motion for a Particle
in an Inverse-Square-Law Field "
Same as before. Integrating the velocity allows
us to solve for the position of the particle"
! x! ( t )
#
dr ( t )
= r! ( t ) = v ( t ) = # y! ( t )
dt
#
#" z! ( t )
r (T ) = !
T
0
$ ! v (t )
& # x
& = # vy ( t )
& #
&% #" vz ( t )
$
&
&
&
&%
T
dr ( t )
dt + r ( 0 ) = ! v dt + r ( 0 )
0
dt
3 components of position"
! x (T )
#
# y (T )
#
#" z (T )
! v (t )
$
x
&
T#
& = ' # vy ( t )
& 0#
#" vz ( t )
&%
$
! x (0) $
&
&
#
& dt + # y ( 0 ) &
&
&
#
&%
#" z ( 0 ) &%
31!
Dynamic Model with InverseSquare-Law Gravity"
!" No aerodynamic or thrust force"
!" Neglect motions in the z direction"
Dynamic Equations"
v!x ( t ) = ! µ E x I ( t ) rI 3 ( t )
Initial Conditions at Equator"
vx ( 0 ) = 7.5, 8, 8.5 km / s
v!y ( t ) = ! µ E yI ( t ) rI 3 ( t )
vy ( 0 ) = 0
y! I ( t ) = vy ( t )
y ( 0 ) = 6, 378 km = R
x! I ( t ) = vx ( t )
where rI ( t ) = x I2 ( t ) + yI2 ( t )
x (0) = 0
32!
Trajectories Calculated with
Inverse-Square-Law Model"
33!
Equatorial Orbits Calculated with
Inverse-Square-Law Model"
34!
MATLAB Code for Spherical-Earth
Trajectories"
Script for Numerical Solution"
R
=
6378; %
Earth Surface Radius, km!
tspan
=
6000; %
seconds!
options
=
odeset('MaxStep', 10)!
xo
=
[7.5;0;0;R];!
[t1,x1]
=
ode15s('RoundEarth',tspan,xo,options);!
for i = 1:length(t1)!
v1(i)
=
sqrt(x1(i,1)*x1(i,1) + x1(i,2)*x1(i,2));!
r1(i)
=
sqrt(x1(i,3)*x1(i,3) + x1(i,4)*x1(i,4));!
end!
Function for "
Numerical Solution"
function xdot = RoundEarth(t,x)!
%
x(1)
=
vx!
%
x(2)
=
vy!
%
x(3)
=
x!
%
x(4)
=
y!
mu =
3.98*10^5; %
km^2/s^2!
r
=
sqrt(x(3)^2 + x(4)^2);!
!
xdot(1) =
-mu * x(3) / r^3;!
xdot(2) =
-mu * x(4) / r^3;!
xdot(3) =
x(1);!
xdot(4) =
x(2);!
xdot
=
xdot';!
end!
35!
Work"
!" “Work” is a scalar measure of change in energy"
!" With constant force,"
In one dimension, W12 = F ( r2 ! r1 ) = F"r
In three dimensions, W12 = FT ( r2 ! r1 ) = FT "r
!" With varying force,"
" dx %
'
$
W12 = ! F dr, dr = $ dy '
r1
$# dz '&
r2
r2
(
T
= ! f x dx + fy dy + fz dz
r1
)
36!
Conservative Force"
!" Assume that the 3-D force
field is a function of position"
Iron Filings!
Around Magnet!
F = F(r)
!" The force field is conservative if"
r2
r1
Force Emanating
from Source!
T
F
r
dr
+
F
(
)
!
! ( r ) dr = 0
T
r1
r2
… for any path between
r1 and r2 and back"
37!
Gravitational Force Field"
!" Gravitational force field"
Fg = !m
µE
rI
rI 3
!" Gravitational force field is
conservative because"
r2
r
1
µE
µ
! " m 3 rI drI ! " m E3 rI drI = 0
rI
rI
r1
r2
… for any path between
r1 and r2 and back"
38!
How Do We Know that Gravitational
Force is Conservative?"
Because the force is the derivative (with
respect to r) of a scalar function of r called
the potential, V(r):"
V ( r ) = !m
µ
µ
+ Vo = !m
1 2 + Vo
T
r
(r r)
" !V !x %
" x %
'
!V ( r ) $
µ$
'
= $ !V !y ' = m 3 $ y ' = (Fg
!r
r
$ !V !z '
$# z '&
&
#
This derivative is also called the gradient
of V with respect to r"
39!
Potential Energy"
Potential energy is defined with respect to a
reference radius, r1"
µ
µ
+m
r2
r1
(The term Vo cancels)
!PE ! V ( r2 ) " V ( r1 ) = "m
!PE = 0 when r2 = r1
Example:"
m = 10kg
µ = 3.98 ! 1014 m 3 s 2
r1 = 6, 378,000 m
r2 = 10,000,000 m
1
$ #1
'
!PE = 3.98 " 1015 & 7 +
% 10 6.378 " 10 6 )(
= 2.26 " 10 8 kg i m 2 / s 2 = 2.26 " 10 8 joules
40!
Kinetic Energy"
Apply Newton’s 2nd Law to the definition of Work"
dr
= v; dr = vdt
dt
r2
t2
T
" dv %
W12 = ! F dr = m ! $ ' v dt
# dt &
r
t
1
t2
T
1
t
1
d T
1 2 d 2
= m ! ( v v ) dt = m ! ( v ) dt
2 t1 dt
2 t1 dt
F=m
dv
dt
1 2 2 1
= mv = m "# v 2 ( t 2 ) ! v 2 ( t1 ) $%
2
2
t1
t
=
1
m "# v22 ! v12 $% ! T2 ! T1 ! &KE
2
Work evaluated as the integral from 1st to 2nd
time"
T is the kinetic energy of the point mass, m!
41!
Total Energy"
!" Potential energy of mass,
m, depends only on the
gravitational force field "
!" Kinetic energy of mass, m,
depends only on the velocity
magnitude measured in an
inertial frame of reference "
!" Total energy is the sum
of the two:"
PE = !m
T ! KE =
µ
r
1 2
mv
2
E = PE + KE
µ 1
= !m + mv 2
r 2
42!
Energy in a Conservative System"
Energy is constant on the trajectory"
E = PE + KE
= !m
µ 1 2
+ mv = Constant
r 2
E2 ! E1 = 0
"
µ 1 2% "
µ 1 2%
$# !m r + 2 mv2 '& ! $# !m r + 2 mv1 '& = 0
2
1
"
µ
µ% " 1 2 1 2%
!m
+
m
= $ mv2 ! mv1 '
$#
&
2
r2
r1 '& # 2
P1!
PE2 ! PE1 = KE2 ! KE1
In a potential force field (e.g.,
inverse-square-law gravity),
change in potential energy
between any 2 points on the orbit
equals change in kinetic energy"
P2!
43!
Conservation of Energy"
Energy is conserved in an elastic collision, i.e. no
losses due to friction, air drag, etc."
“Newton’s Cradle” illustrates interchange of
potential and kinetic energy in a gravitational field"
44!
Ellipse"
As Conic Section!
As Tilted Circle!
45!
Elliptical Planetary Orbits"
!" Assume satellite mass is
negligible compared to Earth’s
mass"
!" Then "
!" Center of mass of the 2 bodies is
at Earth’s center of mass"
!" Center of mass is at one of
ellipse’s focal points"
!" Other focal point is “vacant”"
r=
p
, m or km
1+ e cos!
! : True Anomaly
Angle from perigee direction, deg or rad
46!
Properties of Elliptical Planetary Orbits"
!" Eccentricity can be determined from
apogee and perigee radii"
e=
ra ! rp ra ! rp
=
ra + rp
2a
rp = a (1! e )
ra = a (1+ e)
!" Semi-major axis is the average of the
two"
a=
ra + rp
2
47!
Properties of Elliptical Planetary Orbits"
!" Semi-latus rectum, p, can be
expressed as a function of a and e!
p = a (1! e2 )
!" Semi-minor axis, b, can be
expressed as a function of ra and rp!
b = ra rp
!" Area of the ellipse, A, is!
A=! ab
= ! a 2 (1" e) , m 2
48!
Next Time:!
Precursors to Space Flight: #Rocket, Missiles,
and Men in Space, Prophets with Some Honor,
Ch 4 (ER)!
… the Heavens and the Earth, Introduction, Ch 1!
Orbital Motion: #Understanding Space, Ch 5!
!
49!
!upplemental Ma"ria#l
50!
Equations that Describe Ellipses"
x 2 y2
+ =1
a2 b2
a : Semi - major axis, m or km
b : Semi - minor axis, m or km
a!
b!
y!
x!
o!
x (! ) = a cos (! )
y (! ) = b sin (! )
! : Angle from x - axis (origin at center) rad
51!
Constructing Ellipses"
F1P1 + F2 P1 = F1P2 + F2 P2 = 2a
Foci (from center),
! xf
#
# yf
"
$
!
2
2
& = # ' a 'b
&
0
%1,2 #"
String, Tacks, and Pen!
$ !
&, #
&% #"
$
a2 ' b2 &
0
&%
Archimedes Trammel!
Hypotrochoid!
52!
Ellipses"
Semi - latus rectum ("The Parameter"),
b2
p= , m
a
b2
Eccentricity, e = 1! 2
a
2
b
= 1! e; b = a 1! e
a2
53!