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ASEN 5050 SPACEFLIGHT DYNAMICS Intro to Perturbations. Prof. Jeffrey S. Parker University of Colorado – Boulder. Announcements. STK LAB 2 Alan will be in ITLL 2B10 Fri 2-3 STK Lab 2 will be due 10/17, right when the mid-term exam starts. Homework #5 is due right now! CAETE by Friday 10/17

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asen 5050 spaceflight dynamics intro to perturbations

ASEN 5050SPACEFLIGHT DYNAMICSIntro to Perturbations

Prof. Jeffrey S. Parker

University of Colorado – Boulder

announcements
Announcements
  • STK LAB 2
    • Alan will be in ITLL 2B10 Fri 2-3
    • STK Lab 2 will be due 10/17, right when the mid-term exam starts.
  • Homework #5 is due right now!
    • CAETE by Friday 10/17
  • Homework #6 will be due Friday 10/17 10/24
    • CAETE by Friday 10/24 10/31
  • Mid-term Exam will be handed out Friday, 10/17 and will be due Wed 10/22. (CAETE 10/29)
    • Take-home. Open book, open notes.
    • Once you start the exam you have to be finished within 24 hours.
    • It should take 2-3 hours.
space news
Space News
  • Anyone watch the ISS?
  • Anyone see the lunar eclipse?
  • There’s an event in 1.5 weeks that is directly related to the lunar eclipse we just had. Anyone have an idea what the event is?
ladee s mission to the moon
LADEE’s Mission to the Moon
  • Earth phasing orbits, followed by lunar phasing orbits

Credit: NASA/Goddard

ladee s mission to the moon1
LADEE’s Mission to the Moon
  • Lunar Orbit

Credit: NASA/Ames / ADS

ladee s mission to the moon2
LADEE’s Mission to the Moon
  • Lunar orbit perturbations

Credit: NASA/Ames / ADS

asen 5050 spaceflight dynamics perturbations

ASEN 5050SPACEFLIGHT DYNAMICSPerturbations

Prof. Jeffrey S. Parker

University of Colorado – Boulder

orbital perturbations
Orbital Perturbations
  • You’ll notice that LADEE’s orbit is not strictly conical.
  • So far, we’ve only considered orbital solutions to the two-body problem
    • Point-masses
  • In reality, nothing is ever in orbit about a point-mass without any other perturbations
    • (even in an orbit about a black hole!)
  • The two-body relationship is typically the dominant orbital dynamic. Everything else is a small perturbation
    • Realistic gravitational masses
    • Other gravitating bodies
    • Atmospheric drag
    • Solar radiation pressure
    • Spacecraft effects
    • Even relativity and other subtle effects.
perturbation discussion strategy
Perturbation Discussion Strategy
  • We know the 2-body problem *really well!*
  • Introduce the 3-body and n-body problems
    • We’ll cover halo orbits and low-energy transfers later
  • Numerical Integration
  • Introduce aspherical gravity fields
    • J2 effect, sun-synchronous orbits
  • Introduce atmospheric drag
    • Atmospheric entries
  • General perturbation techniques
  • Further discussions on mean motion vs. osculating motion.
gravitational perturbations
Gravitational Perturbations
  • Start by considering the effects of other gravitating bodies.
  • Recall the two-body equation of motion:

which is a differential equation describing the motion of msat WRT m.

  • How would this change if we had multiple gravitating bodies?
3 body problem1
3-Body Problem
  • We want to know the position vector of the satellite relative to the Earth over time
3 body problem2
3-Body Problem
  • We want to know the position vector of the satellite relative to the Earth over time

Be cognizant of the signs – the signs are defined according to how the vectors are drawn!

3 body problem3
3-Body Problem
  • We want to know the position vector of the satellite relative to the Earth over time

Indirect Effect

Direct Effect

n body problem
n-Body Problem
  • The equation of motion for the position vector of a satellite in the presence of n bodies.
  • … relative to Body “1” (Earth?)
full 2 body problem
Full 2-Body Problem
  • How about the perturbations that result in being in orbit about a non-spherical body?

Images from Park, Werner, and Bhaskaran, “Estimating Small-Body Gravity Field from Shape Model and Navigation Data”, Journal of Guidance, Control, and Dynamics, Vol. 33, No. 1, Jan – Feb 2010.

dynamical analysis2
Dynamical Analysis

z

M3 ~ 0

M1 and M2 follow conic trajectories about their COM

M1

M2

x

y

dynamical analysis3
Dynamical Analysis

z

M3 ~ 0

y

M1 and M2 follow circular orbits about their COM

M1

M2

x

Synodic Frame

dynamical analysis4
Dynamical Analysis

y

M3 ~ 0

M2

Planar motion

M1

x

Synodic Frame

building solutions to the n body problem
Building Solutions to the n-Body Problem
  • We have more degrees of freedom than we have integrals of motion!
  • Conic sections are no longer solutions.
  • Most common method used to build solutions to the n-Body problem is to take initial conditions and integrate them forward in time.
    • Build a trajectory using knowledge of the equations of motion.
numerical integration
Numerical Integration
  • Say we have a state (pos, vel) and some equations of motion.

Accelerations due to 2-body, n-body, etc.

numerical integration1
Numerical Integration
  • We want to recover the spacecraft’s trajectory using knowledge of the derivative of its state over time.
  • If we were to accurately integrate the derivative function over time, using the spacecraft’s initial state as the constant of motion, then we could recover its trajectory.
  • Lots of ways to do this. Some are better than others!
numerical integration2
Numerical Integration
  • Euler integration

Actual Trajectory

numerical integration3
Numerical Integration
  • How do we improve it?
  • Take smaller time-steps
  • Take smarter steps

Actual Trajectory

higher order terms
Higher order terms
  • Here’s what we just tried:
  • What about this modification?:
  • That would be better!
    • But really hard to implement in a general sense.
higher order terms1
Higher order terms
  • Here’s what we just tried:
  • How about a correction term. Here’s a second-order scheme, usually referred to as a midpoint method:

Actual Trajectory

midpoint integration example1
Midpoint Integration Example

Midpoint

Euler

Note: this does take 2x as many derivative function calls, but the improvement is better than just doubling Euler’s!

runge kutta integrators
Runge-Kutta Integrators
  • Runge-Kutta integration
  • 4th order Runge-Kutta “RK4” or “The Runge-Kutta method”

Weighted average correction system, related to Simpson’s Rule

example rk4 shorter time interval3
Example RK4 Shorter Time Interval

Undersampled

Oversampled

Use a variable time-step!

variable time steps
Variable Time-Steps
  • These may be implemented in many ways.
  • Method 1.
    • Compute Xi+1 using one full step
    • Compute Xi+1 using two half-steps
    • Compare them.
      • If they are similar to within a tolerance, then use the second state and move on.
      • If they are similar to within some fraction of the tolerance, then increase the time-step.
      • If they are sufficiently dissimilar, then reduce the time-step and try again. Do not move on until they are similar within the tolerance.
variable time steps1
Variable Time-Steps
  • These may be implemented in many ways.
  • Method 2.
    • Compute Xi+1 using a 4th order scheme
    • Compute Xi+1 using a 5th order scheme
    • Compare them.
      • If they are similar to within a tolerance, then use the second state and move on.
      • If they are similar to within some fraction of the tolerance, then increase the time-step.
      • If they are sufficiently dissimilar, then reduce the time-step and try again. Do not move on until they are similar within the tolerance.
variable time steps2
Variable Time-Steps
  • These may be implemented in many ways.
  • Some algorithms change the time-steps every iteration; some either cut it in half or double it. Some use a formula.
  • When plotting the trajectories, it’s generally best to plot at a constant time-step.
    • Often the integrator takes many steps between each outputted time-step.
    • Often the integrator does not actually evaluate the trajectory AT a desired time-step.
      • Either force it to (expensive)
      • Or interpolate (less expensive)
runge kutta take 2
Runge-Kutta Take 2
  • 4th order Runge-Kutta “RK4” or “The Runge-Kutta method”
runge kutta take 21
Runge-Kutta Take 2
  • The weights may be placed into a table, called a Butcher Table
runge kutta take 22
Runge-Kutta Take 2
  • The weights may be placed into a table, called a Butcher Table
runge kutta take 23
Runge-Kutta Take 2
  • The general explicit Runge-Kutta scheme:

(oh yes, I did just copy and paste that from wikipedia. They have a great write-up.)

runge kutta take 24
Runge-Kutta Take 2
  • The general explicit Runge-Kutta scheme:

The method is consistent if:

butcher tables
Butcher Tables

RK4

Euler

Midpoint

butcher tables1
Butcher Tables
  • To build an adaptive time-step Butcher Table:
    • Assemble an order p Butcher Table
    • Append an order p-1 Butcher Table to it.
butcher tables2
Butcher Tables
  • Runge-Kutta-Fehlberg method consists of a 5th order method and a 4th order check. Hence:
drawbacks of explicit rk methods
Drawbacks of Explicit RK Methods
  • Explicit Runge-Kutta methods are easy to derive and require relatively few computations to implement.
  • They are usually quite good for astrodynamic applications.
  • Don’t be afraid to use them 
  • However, for stiff problems they are unstable. They are particularly unstable for solving partial differential equations.
  • To combat this, implicit Runge-Kutta methods have been formulated.
implicit runge kutta methods
Implicit Runge-Kutta Methods
  • Similar form:

A may be a full matrix!

implicit runge kutta methods1
Implicit Runge-Kutta Methods
  • These require an iterative scheme, since, for instance, k1 depends on itself!
  • Example: the backward Euler method:

This uses the slope of the next state to integrate the current state forward in time.

Notice that k1 depends on itself.

implicit runge kutta methods2
Implicit Runge-Kutta Methods
  • Gauss-Legendre method may be scaled to include as many stages as desired. The 4th order (2 stages) method has this Butcher table:
  • All collocation methods are implicit Runge-Kutta methods, but not all implicit Runge-Kutta methods are collocation methods.
announcements1
Announcements
  • STK LAB 2
    • Alan will be in ITLL 2B10 Fri 2-3
    • STK Lab 2 will be due 10/17, right when the mid-term exam starts.
  • Homework #5 is due right now!
    • CAETE by Friday 10/17
  • Homework #6 will be due Friday 10/17 10/24
    • CAETE by Friday 10/24 10/31
  • Mid-term Exam will be handed out Friday, 10/17 and will be due Wed 10/22. (CAETE 10/29)
    • Take-home. Open book, open notes.
    • Once you start the exam you have to be finished within 24 hours.
    • It should take 2-3 hours.