kepler Universidad Polit cnica de Madrid by MikeJenny

VIEWS: 17 PAGES: 33

									                        Kepler's Problem
                   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
                           Universidad Politécnica de Madrid




OCTOBER 2007
Kepler's problem
Kepler's problem



                   P
Some vector algebra...



                     P
  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

								
To top