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1ME 461 / 561 - Chapter 1
ME 461 / 561 Orbital Mechanics
Chapter 1 - The Two-Body Problem
Professor Christopher D. Hall
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Keplers Laws:
First Law (1609):
The orbit of each planet is
an ellipse with the sun at a focus.
Second Law (1609):
The line joining the planet to the sun
sweeps out equal areas in equal time.
Third Law (1619):
The square of the period of a planet is
proportional to the cube of its mean
distance from the sun.
These laws were formed for motion of planets about the sun, butalso apply to artificial satellites orbiting the sun, planets, or moons
Keplers Laws:
First Law (1609):
The orbit of each planet is
an ellipse with the sun at a focus.
Second Law (1609):
The line joining the planet to the sun
sweeps out equal areas in equal time.
Third Law (1619):
The square of the period of a planet is
proportional to the cube of its mean
distance from the sun.
These laws were formed for motion of planets about the sun, butalso apply to artificial satellites orbiting the sun, planets, or moons
FocusFocus
P2 a3
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Newtons Laws (Principia, 1687):
First Law:
Every body continues in its state of rest or of uniform motion in a
straight line unless it is compelled to change that state by forcesimpressed upon it.
Second Law:
Third Law:
Universal Gravitational Law:
Newtons Laws (Principia, 1687):
First Law:
Every body continues in its state of rest or of uniform motion in a
straight line unless it is compelled to change that state by forcesimpressed upon it.
Second Law:
Third Law:
Universal Gravitational Law:
- ~F
m1 m2
~F
~F id(m~iv)dt
m
M
~r~Fgm=
GMm
r2~r
r
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The N-Body Problem
~Fother =
~FDrag+
~FThrust+
~Fsolarpressure +
~Fperturbe+ . . .
Applying Newtons 2nd Law for the jth particle: Applying Newtons 2nd Law for the jth particle:
~rjn,
~rn ~rjStart
point
Start
pointEnd
point
End
point
Z
Y
X
O ~Fother
~rj
mj
~Fgj
mn
P~F =
id(mij~vj)
dt
~Fgj = GmjnX
k=1
mkr3kj
~rkj
~rn
m2
m1 ~rjn
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The N-Body Problem
~Fother| {z}0
+~Fgj =mj
id(i~vj)
dt =mj
id2(i~rj)
dt2
Gravitational forcebetween earthand satellite
Gravitational forcebetween earthand satellite
Perturbing effects ofsun, moon,
Perturbing effects ofsun, moon,
~rj =
G
nXk=1
mk
r
3
kj
~rkj
~r12 = G(m1+ m2)
r312~r12
| {z }
nXk=3
Gmk~rk2
r3k2 ~rk1
r3k1
| {z }
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The Two-Body Problem
Assumptions
The bodies are spherically
symmetric (we can assumeeach as a point mass).
The only force is thegravitational force, alongthe line joining the centers
of two bodies.
Assumptions
The bodies are spherically
symmetric (we can assumeeach as a point mass).
The only force is thegravitational force, alongthe line joining the centers
of two bodies.
Z
X0
m
X
O0
Y0
Z0
Y
~rm
~rM
~r
M
~r=~rm ~rM
Differential Eq. of therelative motion
Differential Eq. of therelative motion
Nonrotating FrameNonrotating Frame
Inertial FrameInertial Frame
m~rm =
GMm
r3 ~r
~rm =
GM
r3 ~r
M~rM= GMm
r3 ~r~rM= Gmr3 ~r~r= G(M+m)r3 ~r
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G(M+ m) GM
When ( M>>m) i.e. , artificial satellites, space probes, ballistic missiles, When ( M>>m) i.e. , artificial satellites, space probes, ballistic missiles,
Constants (Integrals) of the Motion:
Conservation of mechanical energy
Conservation of angular momentum
Constants (Integrals) of the Motion:
Conservation of mechanical energy
Conservation of angular momentum
Equation of motion of TBP
E=T+ V
PotentialenergyPotentialenergyKinetic
energyKineticenergy
MechanicalenergyMechanicalenergy
~h= ~r ~v
~r+ r3 ~r=~0
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Conservation of Mechanical Energy
Lets start with the EOM:Lets start with the EOM:
Using the fact thatUsing the fact that
Then:Then:
vv+ r2 r= 0
which can be written as:which can be written as:
ddt
v2
2
+ ddt
c r
= 0
E= v2
2 +
c r
~r+
r3~r=~0
~r ~r+ r3 ~r ~r= 0
~r ~r= rr
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Conservation of Mechanical Energy
Kinetic energy perunit mass of satelliteKinetic energy perunit mass of satellite
Potential energy per
unit mass of satellite
Potential energy per
unit mass of satellite
Specific mechanical
energy
Specific mechanical
energy
If:If:
c= 0 limrV(r) = 0c= rM V(rM) = 0
E= v2
2 r
If:If:
E= v2
2 +
c r
| {z }
Vis-viva (living force) Eq.:Vis-viva (living force) Eq.:
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Matlab Function: rv2E.m
functionE=rv2E(R,V,mu)
%RV2E FunctiontocalculatethespecificmechanicalenergyofaKeplerian orbit.
% Theinputsofrv2Earethepositionvectorofthesatellite,R,thevelocity
% vectorofthesatellite,V,andthegravitationalparameter ofthecentral
% body,mu. Theoutputisthespecificmechanicalenergy,E.
%
%
Theposition
and
velocity
vectors
must
be
either
3x1
or
1x3 matrices,
and
% thefunctionwillresizebothofthemsothattheyare3x1. Ifavalue
% formuisnotinputed,thedefaultvalueisthatforEarth(3.986e5km^3/s^2)
%Determineifinputvectorsarethecorrectsizeandresizethem
iflength(R)~=3||length(V)~=3
error('Inputvectorsarenotthecorrectsize. Pleaseinputnewvectors.')
end
R=[R
(1);
R
(2);
R
(3)];
V=[V(1);V(2);V(3)];
%Ifnovalueofmuisgiven,usethedefaultvalueforEarth
ifnargin
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Matlab Program using rv2E.m
%Thisprogram(anmfile)calculatestheenergywhilevaryingthemagnitudeofthevelocityvector%foragivenradiusvector,thenmakesaplotofEvs |V|
%Setparametersmu=3.986e5;Npoints =100;
%InitializevariablesR=[7000;0;0];Vrange =linspace(1,15,Npoints);%eachofthesevelocitymagnitudeswillbeusedtocomputeEErange =zeros(1,Npoints);
fori=1:Npoints
V=[0;
Vrange(i);
0];
%
set
V
perpendicular
to
R,
so
either
periapsis or
apoapsis
Erange(i)=rv2E(R,V,mu);end
figurehg=plot(Vrange,Erange); %usehandlegraphics tomakeprofessionalgraphset(hg,'linewidth',2)
xlabel('Velocity magnitude(km/s)','fontsize',12)
ylabel('Energy (km^2/s^2)','fontsize',12)
%Thisprogram(anmfile)calculatestheenergywhilevaryingthemagnitudeofthevelocityvector%foragivenradiusvector,thenmakesaplotofEvs |V|
%Setparametersmu=3.986e5;Npoints =100;
%InitializevariablesR=[7000;0;0];Vrange =linspace(1,15,Npoints);%eachofthesevelocitymagnitudeswillbeusedtocomputeEErange =zeros(1,Npoints);
fori=1:Npoints
V=[0;
Vrange(i);
0];
%
set
V
perpendicular
to
R,
so
either
periapsis or
apoapsis
Erange(i)=rv2E(R,V,mu);end
figurehg=plot(Vrange,Erange); %usehandlegraphics tomakeprofessionalgraphset(hg,'linewidth',2)
xlabel('Velocity magnitude(km/s)','fontsize',12)ylabel('Energy (km^2/s^2)','fontsize',12)
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Matlab Program using rv2E.m
Thisplotistheresultofrunningthe
Matlab mfile that
callstheMatlabfunctionrv2E.m
Notethatforsmallvelocitiestheenergyisnegative,andforlargevelocitiestheenergyispositive
Thinkaboutthat
pointwhere
the
energyiszeroandaskwhatitmightmean
Thisplotistheresultofrunningthe
Matlab mfile that
callstheMatlabfunctionrv2E.m
Notethatforsmallvelocitiestheenergy
isnegative,
and
for
largevelocitiestheenergyispositive
Thinkaboutthat
pointwhere
the
energyiszeroandaskwhatitmightmean
0 5 10 15-60
-40
-20
0
20
40
60
Velocity magnitude (km/s)
Energy
(km
2/s2)
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Conservation of Angular Momentum
We start with the EOM:We start with the EOM:
Cross product withCross product with ~r
Using identity:Using identity:
Then:Then:
~r ~v = ~h=const.
~r+ r3~r=~0
~r ~r+
r3~r ~r
| {z }=0= ~0
ddt
~r ~r
=~r ~r= ~0
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Conservation of Angular Momentum
Local
vertic
al
Local
vertic
al
local
loca
l
horizontal
horizo
ntal
~v
M
m~r
is the flight-path angleis the flight-path angle
is the zenith angleis the zenith angle
The motion takes place in a plane that isfixed in inertial space.
This plane is called the orbital plane
The motion takes place in a plane that isfixed in inertial space.
This plane is called the orbital plane
andandto the plane ofto the plane of
oror h=r v cos
~h ~r ~v
~vr
~v
tan = vrv
h=rv sin
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The Trajectory Equation
Using the identity:Using the identity:
LHS of Eq. (a)LHS of Eq. (a)
RHS of Eq. (a) :RHS of Eq. (a) :
= r ~v rr2 ~r= ddt~rr
(c)
( ~A ~B) ~C= ~B( ~A ~C) ~C( ~A ~B)
r3~h ~r) = r3 (~r ~v) ~r= r3 ~v(~r ~r) ~r(~r ~v)
~r= r3~r~r ~h= r3 (~h ~r) (a)
~r ~h= ddt(~r ~h) (b)
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The Trajectory EquationCombining Eqs. (b) & (c)Combining Eqs. (b) & (c)
After integrationAfter integration
Constant vectorConstant vector
Using the identity:Using the identity:
h2 = r+rBcos r= h2/
1+(B/) cos < ~r,~B >
~A ( ~B ~C) = ( ~A ~B) ~C
d
dt(
~r ~
h) =
d
dt~r
r ~r ~h= ~rr +
~B
~r ~r ~h= ~r~rr + ~r ~B
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The Trajectory Equation
Conic Section Eq.Conic Section Eq. Trajectory Eq. for TBPTrajectory Eq. for TBP
r= p1+e cos r= h
2
/1+(B/) cos
The eccentricity:The eccentricity:
The parameter orsemi-latus rectum:
The parameter orsemi-latus rectum:
p= h2
e= B
p
F
~r
CircleCircle
EllipseEllipse
ParabolaParabola
HyperbolaHyperbola
EccentricityEccentricity Type of orbit or TrajectoryType of orbit or Trajectory
e= 0
0< e 1
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Some terminology[2] Satellite: any object which travels in an elliptical orbit around a
planet is called a satellite of that planet.
Space probe: any object which travels in an open trajectory; i.e.,a hyperbolic trajectory in a vicinity of a planet and in ellipticalorbit around the sun is called a space probe.
Interstellar probe: any object with a velocity greater thanheliocentric escape velocity is called an interstellar probe.
Orbitor Trajectory: path of a spacecraft or natural body in
space. We use orbit for closed and trajectory for open paths.
Barycenter: the location of the center of mass of two bodies.
Satellite: any object which travels in an elliptical orbit around aplanet is called a satellite of that planet.
Space probe: any object which travels in an open trajectory; i.e.,a hyperbolic trajectory in a vicinity of a planet and in ellipticalorbit around the sun is called a space probe.
Interstellar probe: any object with a velocity greater thanheliocentric escape velocity is called an interstellar probe.
Orbitor Trajectory: path of a spacecraft or natural body in
space. We use orbit for closed and trajectory for open paths.
Barycenter: the location of the center of mass of two bodies.
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Some important orbit terminology
Peri means the closest point. Perigee means the closest point to theearth. Perihelion means the closest point to the sun, and so forth.
Apo means the farthest point. Apogee means the farthest pointfrom the earth. Aphelion means the farthest point from the sun.
Keplerian orbit:is an orbit in which
1. The only force is gravity2. The central body is spherically symmetric
3. The primary mass is much greater than secondary mass
4. There are no other masses in the system
Also called two-body problem orbit or unperturbed orbit
Peri means the closest point. Perigee means the closest point to theearth. Perihelion means the closest point to the sun, and so forth.
Apo means the farthest point. Apogee means the farthest pointfrom the earth. Aphelion means the farthest point from the sun.
Keplerian orbit:is an orbit in which
1. The only force is gravity2. The central body is spherically symmetric
3. The primary mass is much greater than secondary mass
4. There are no other masses in the system
Also called two-body problem orbit or unperturbed orbit
A memory aid:a b c d e f g h I j k l m n o p q r s t u v w x y zmemory aid: a b c d e f g h I j k l m n o p q r s t u v w x y z
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Planet Symbols
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Conic Sections
A conic section is the locus of points such that the ratio of the absolutedistance from a given point (a focus) to the absolute distance from agiven line (a directrix) is a positive constant e.
A conic section is the locus of points such that the ratio of the absolutedistance from a given point (a focus) to the absolute distance from agiven line (a directrix) is a positive constant e.
Hyperbola
branches
Hyperbola
branches
CircleCircle
EllipseEllipse
ParabolaParabola
FFFF
AA
DirectrixDirectrix
rr
r/er/e
HH
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Conic Sections
CircleCircle
EllipseEllipse
ParabolaParabola
a=
c=
2c
2pF
2c
FF0
2p
2p
2a
F0F,
2a
2a
F F0 2p
HyperbolaHyperbola
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Conic Sections
= p1+e =a(1 e)
=
p
1e =a(1 + e)
e= caEccentricity:Eccentricity:
The parameter:The parameter:
Periapsis radius:Periapsis radius:
Apoapsis radius:Apoapsis radius:
The eccentricity vector:The eccentricity vector: ~e= ~v~h
~rr
(Except for parabola)(Except for parabola)
(Except for parabola)(Except for parabola)
p= h2
p=a(1 e2
)
ra=
p
1+e cos 180o
rp = p
1+e cos0o
~e= (v2 r )~r (~r ~v)~v
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Conic Sections Relating Relating E andand
Circle and Ellipse:Circle and Ellipse:
Parabola:Parabola:
Hyperbola:Hyperbola: E>0
E= 0
E
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The Circular Orbit
For Circular orbit we have:For Circular orbit we have: e= 0
Circular velocity:Circular velocity:
The period is:The period is:
v2cs2
a =
2a vcs =
p
a
T Pcs = 2
= 2 avcsTPcs=
2
a(3/2)
r= p1+0cos = h2
=a
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Pop Quiz (doesnt count)
An earth satellite is in a circular orbit with
altitude 400 km. What are the satellitesspeed and orbital period?
Useful information: = 3.986 10
5 km3/s2
R = 6378 km
Answer:
An earth satellite is in a circular orbit with
altitude 400 km. What are the satellitesspeed and orbital period?
Useful information: = 3.986 10
5 km3/s2
R = 6378 km
Answer:
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The Elliptic Orbit
bSemi-minorAxisSemi-minor
Axis
FocusFocus
Empty
Focus
Empty
Focus
c
a
aSemi-majorAxis
Semi-major
Axis
Center of mass\
Barycenter
Center of mass\
Barycenter
a2 =b2 + c2
rp+ ra = 2a
e= rarp
ra+rp
The eccentricity: The eccentricity:
Periapsis
(Perifocal)
Periapsis
(Perifocal)
Apoapsis
(Apofocal)
Apoapsis
(Apofocal)
The period is: The period is:
from calculus:from calculus:
h= r2ddt dt= r2
hd (a)
dA= 1
2r2
d (b)Eliminating between (a) & (b):Eliminating between (a) & (b):r
2d dt= 2hdA
For one period:For one period: T P= 2abh T P= 2
a3/2
&&
Line of
apses
Line of
apses
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Pop Quiz (doesnt count) An earth satellite is in an elliptical orbit with
perigee altitude 400 km and apogee altitude
900 km. What are the satellites speed at
perigee and apogee, and orbital period?
Useful information:
= 3.986 105 km3/s2
R
= 6378 km
Answer:
An earth satellite is in an elliptical orbit with
perigee altitude 400 km and apogee altitude
900 km. What are the satellites speed atperigee and apogee, and orbital period?
Useful information:
= 3.986 105 km3/s2
R = 6378 km
Answer:
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The Parabolic Trajectory
For a parabolic trajectory we have:For a parabolic trajectory we have:
Escape velocity :Escape velocity :
e= 1
rp= p1+cos 0 = p2
vesc=q
2r
Note:Note: vesc=
2vcs
v2esc2
r =
v22|{z}0
r|{z}0
= 0
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Pop Quiz (doesnt count) An earth satellite is in a circular orbit with
altitude 400 km. What is the satellites speed?
What is the satellites escape speed? Suppose
the satellites speed is changed to the escape
speed, then what is the satellites speed when
the radius reaches infinity?
Useful information:
= 3.986 105 km3/s2
R = 6378 km
An earth satellite is in a circular orbit with
altitude 400 km. What is the satellites speed?
What is the satellites escape speed? Supposethe satellites speed is changed to the escape
speed, then what is the satellites speed when
the radius reaches infinity?
Useful information:
= 3.986 105 km3/s2
R = 6378 km
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The Hyperbolic Orbit Turning angle: Turning angle:
Hyperbolic excess velocity: Hyperbolic excess velocity:
E= v2bo2
rbo=
v2
2
r
v2 =v2bo 2rbo =v2bo v2esc
F F0
c b
c2 =a2 + b2
a
Note: is denoted byand it is called departure energy.
Note: is denoted byand it is called departure energy.v
2 C3
vbovbo
rborbo
rrv
v
sin 2 = ac =
1e
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Canonical Units and The Reference Orbit In order to simplify the arithmetic, we use the following canonical
units in Astrodynamics.
Mass: We assume the mass of the central body is 1, i.e. 1 mass
unit. Distance: Mean distance from earth to sun is called Astronomical
unit, AU.
Distance: For orbits about a planet, typically use radius of planetas 1 Distance unit, or D.U.
Time: We choose the time unit so that the period of the orbit isTP = 2 T.U.
In order to simplify the arithmetic, we use the following canonicalunits in Astrodynamics.
Mass:We assume the mass of the central body is 1, i.e. 1 mass
unit. Distance: Mean distance from earth to sun is called Astronomical
unit, AU.
Distance: For orbits about a planet, typically use radius of planetas 1 Distance unit, or D.U.
Time: We choose the time unit so that the period of the orbit isTP = 2 T.U.
1AU/TU1AU
SunSunPlanet
1 DU
1 DU/TU1 DU/TUExercise:
What is ?
Exercise:
What is ?
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Extrasolar Planets
More than 700 planets orbitingother stars have been discovered
since the first was detected in 1995
Most of these are large planets
(Jupiter-sized and larger)
Detection is mainly by wobble of
the stars motion
At least one planet has been
detected by observing its transit
across the star
Planet Period (days) Eccentricity,e
Mass, m(Jupiter masses)
Semi-majoraxis, a(AU)
B 4.617 0.02 0.69 0.059
C 241.3 0.24 2.05 0.828
D 1308 0.31 4.29 2.556
Our sun wobbles 12 m/s due to Jupiter
Upsilon Andromedaehas 3 planets!
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References
[1] http://solarsystem.nasa.gov
[2] Mission Geometry; Orbit Constellation
Design and Management, J. Wertz, 2001.
[1] http://solarsystem.nasa.gov
[2] Mission Geometry; Orbit Constellation
Design and Management, J. Wertz, 2001.