ASEN 5050 SPACEFLIGHT DYNAMICS Intro to Perturbations

ASEN 5050SPACEFLIGHT DYNAMICS

Intro to Perturbations

Prof. Jeffrey S. Parker

University of Colorado – Boulder

Lecture 17: Intro to Perturbations 1

Announcements


Concept Quiz 11


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• Earth phasing orbits, followed by lunar phasing orbits

Lecture 17: Intro to Perturbations

Credit: NASA/Goddard

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LADEE’s Mission to the Moon• Lunar Orbit


Credit: NASA/Ames / ADS

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LADEE’s Mission to the Moon• Lunar orbit perturbations


Credit: NASA/Ames / ADS

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ASEN 5050SPACEFLIGHT DYNAMICS

Perturbations

Prof. Jeffrey S. Parker

University of Colorado – Boulder


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

• 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

• 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 Problem


3-Body Problem

• We want to know the position vector of the satellite relative to the Earth over time


3-Body Problem


Lecture 17: Intro to Perturbations 17Be cognizant of the signs – the signs are defined according to how the vectors are drawn!

3-Body Problem



Direct Effect Indirect Effect

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

• 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 Analysis

SystemTwo-Body Problem

Full Three-Body

RTBP: 3rd Body Massless

CRTBP:Circular orbits

PCRTBP (Synodic Frame)

Degrees of Freedom 12 18 18 18 12

Integrals of Motion 12 10 12 13 9

Linear Momentum 6 6 6 6 4

Angular Momentum 3 3 3 3 1

Energy 1 1 1 1 1

Kepler’s Laws 2 0 2 2 2

Jacobi Integral N/A N/A N/A 1 1

x y

M2

M1

z

Dynamical Analysis


Full Three-Body








Energy 1 1 1 1 1



x y

M3

M2

M1

z

Dynamical Analysis


Full Three-Body








Energy 1 1 1 1 1



x y

M3 ~ 0

M2

M1

z

M1 and M2 follow conic trajectories about their COM

Dynamical Analysis


Full Three-Body








Energy 1 1 1 1 1



y

x

M3 ~ 0

M2

M1

z

M1 and M2 follow circular orbits about their COM

Synodic Frame

Dynamical Analysis


Full Three-Body



PCRTBP: Planar





Energy 1 1 1 1 1



x

M3 ~ 0

M2M1

y

Planar motion

Synodic Frame

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

• Say we have a state (pos, vel) and some equations of motion.


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


• 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!



• Euler integration


Actual Trajectory


• How do we improve it?• Take smaller time-steps• Take smarter steps


Actual Trajectory

Euler Integration Example


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 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:

Lecture 17: Intro to Perturbations 38Actual Trajectory

Midpoint Integration Example


EulerMidpoint

Midpoint Integration Example


EulerMidpoint

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 integration• 4th order Runge-Kutta “RK4” or “The Runge-Kutta

method”


Weighted average correction system, related to Simpson’s Rule

Example RK4


Example RK4 Shorter Time Interval




Oversampled

Undersampled

Use a variable time-step!Use a variable time-step!

Variable Time-Steps

• These may be implemented in many ways.• Method 1.

1. Compute Xi+1 using one full step

2. Compute Xi+1 using two half-steps

3. Compare them.a. If they are similar to within a tolerance, then use the second state

and move on.

b. If they are similar to within some fraction of the tolerance, then increase the time-step.

c. 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-Steps

• These may be implemented in many ways.• Method 2.

1. Compute Xi+1 using a 4th order scheme

2. Compute Xi+1 using a 5th order scheme

3. Compare them.a. If they are similar to within a tolerance, then use the second state

and move on.

b. If they are similar to within some fraction of the tolerance, then increase the time-step.

c. 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-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

• 4th order Runge-Kutta “RK4” or “The Runge-Kutta method”


Runge-Kutta Take 2

• The weights may be placed into a table, called a Butcher Table


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 2

• The general explicit Runge-Kutta scheme:


The method is consistent if:

Butcher Tables


RK4 Euler Midpoint

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 Tables

• Runge-Kutta-Fehlberg method consists of a 5th order method and a 4th order check. Hence:


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

• Similar form:


A may be a full matrix!


• 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.


• 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.


Announcements


ASEN 5050 SPACEFLIGHT DYNAMICS Intro to Perturbations

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