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Orbital Dynamics and Attitude Control
EuMAS-European Masters Course in Aeronautics and Space Technology

Dr. Rafael Ramis Abril
Departamento de Física Aplicada a la Ingeniería Aeronáutica

OCTOBER 2007
Kepler's problem
Kepler's problem

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Some vector algebra...

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Kepler's problem
“First integrals” or “conservation laws”

Angular momentum is constant
Kepler's problem
“First integrals” or “conservation laws”

“Eccentricity vector” is constant

P
Kepler's problem
“First integrals” or “conservation laws”

Energy (kinetic + potential) is constant

P

Can be derived from previous expressions
Kepler's problem
The trajectory (orbit) is contained in a plane (orbital plane)
Eccentricity vector is inside the orbital plane
Kepler's problem
Angular momentum and eccentricity vector define the “perifocal frame”
Orientation defined by 3 angles: i=inclination, =longitude (right
ascension) of node, and =argument of periapse
Kepler's problem
Conversion formulas between angles and perifocal frame
Kepler's problem

Trajectory (in polar coordinates)

f=”true anomaly”
p=”parameter” or “semilatus rectum”
=”argument of latitude”
Kepler's Problem

Shape of orbits. For fixed “p”

e=0 --> circular
0<e<1 --> elliptic
e=1 --> parabolic
e>1 --> hiperbolic
Kepler's Problem

Velocity vector
Kepler's Problem
Velocity components
Kepler's Problem
Kepler's Problem
Kepler's Problem
Kepler's Problem

Kepler's equation
Kepler's Problem
Kepler's equation can be solved iteratively
M=E-e sin E
Kepler's Problem
Kepler's equation can be solved by Fourier Series

M=E-e sin E

R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Chapter 5

D. Brouwer & G. M. Clemence, Methods of Celestial Mechanics, Chapter II
Kepler's Problem

Solution of the initial value problem, given position ri
and velocity vi at time ti, compute values at time t

●From ri and vi compute vectors h and e
●Compute h, p, a, e, and n
●Obtain Euler angles of perifocal frame i, , and 
●From ri compute (i, fi,) Ei, and Mi
●From t i compute 
Kepler's Problem

Solution of the initial value problem, given position r   i
and velocity v i at time t i, compute values at time t

●From  i, , and determine ue and up
●From a and e compute n
●From t and  compute M
●Solve Kepler equation to determine E
●Obtain dE/dt
●Obtain x, y, vx and vy (in perifocal frame)
●Compute r and v
Kepler's Problem
Regular equations allow to overcome all these difficulties !!

R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Chapter 5

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