ASEN 5050
SPACEFLIGHT DYNAMICS
General Perturbations
“It causeth my head to ache” - Isaac Newton
Alan Smith
University of Colorado – Boulder
Lecture 25: Perturbations
1
Announcements
• Homework #7 is out now! Due Monday morning.
– Clarification for Problem 3: you do not have to implement
BOTH a variable time-step integrator and a fixed time-step
integrator. Pick one. Then fill in that half of the table.
• Reading: Chapters 8 and 9
Lecture 24: General Perturbations
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Schedule from here out
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10/27: Three-Body Orbits
10/29: General Perturbations (Alan)
10/31: General Perturbations part 2
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11/3: Mission Orbits / Designing with perturbations
11/5: Interplanetary 1
11/7: Interplanetary 2
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11/10: Entry, Descent, and Landing
11/12: Low-Energy Mission Design
11/14: STK Lab 3
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11/17: Low-Thrust Mission Design (Jon Herman)
11/19: Finite Burn Design
11/21: STK Lab 4
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Fall Break
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12/1: Constellation Design, GPS
12/3: Spacecraft Navigation
12/5: TBD
• 12/8: TBD
• 12/10: TBD
Lecture
24: General
Perturbations
• 12/12:
Final
Review
3
Schedule from here out
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Our last lecture will be Friday 12/12.
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Final review and final Q&A.
Showcase your final projects – at least any that are finished!
Final Exam
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Handed out on 12/12
Due Dec 18 at 1:00 pm – either into D2L’s DropBox or under my door.
•
I heartily encourage you to complete your final project website by Dec 12th so you can focus
on your finals. However, if you need more time you can have until Dec 18th. As such the
official due date is Dec 18th.
•
The final due date for everything in the class is Dec 18th - no exceptions unless you have a
very real reason (medical or otherwise - see CU's policies here:
http://www.colorado.edu/engineering/academics/policies/grading). Of course we will
accommodate real reasons.
•
If you are a CAETE student, please let me know if you expect an issue with this timeframe.
We normally give CAETE students an additional week to complete everything, but the grades
are due shortly after the 18th for everyone. So please see if you can meet these due dates.
Lecture 19: Perturbations
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Final Project
• Get started on it!
• Worth 20% of your grade, equivalent to 6-7 homework assignments.
• Find an interesting problem and investigate it – anything related to
spaceflight mechanics (maybe even loosely, but check with me).
• Requirements: Introduction, Background, Description of
investigation, Methods, Results and Conclusions, References.
• You will be graded on quality of work, scope of the investigation,
and quality of the presentation. The project will be built as a
webpage, so take advantage of web design as much as you can
and/or are interested and/or will help the presentation.
Lecture 24: General Perturbations
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Final Project
•
Instructions for delivery of the final project:
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Build your webpage with every required file inside of a directory.
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Name your main web page “index.html”
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Name the directory “<LastName_FirstName>”
there are a lot of duplicate last names in this class!
You can link to external sites as needed.
i.e., the one that you want everyone to look at first
Make every link in the website a relative link, relative to the directory structure
within your named directory.
–
We will move this directory around, and the links have to work!
•
Test your webpage! Change the location of the page on your computer and make
sure it still works!
•
Zip everything up into a single file and upload that to the D2L dropbox.
Lecture 24: General Perturbations
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Space News
• Rockets are tricky.
• Sad day for Orbital and its customers yesterday.
Hopefully we will pick up all the pieces and come out
of it even stronger.
– No injuries reported, thankfully. And the ISS is not low on
any critical supplies.
Lecture 25: Perturbations
7
ASEN 5050
SPACEFLIGHT DYNAMICS
General Perturbations
“It causeth my head to ache” - Isaac Newton
Alan Smith
University of Colorado – Boulder
Lecture 25: Perturbations
8
Perturbations
Special Perturbation Techniques – Numerical integration.
Straightforward – however obtaining a good understanding of the
effects on the orbit is difficult.
General Perturbations – Use approximations to obtain analytical
descriptions of the effects of the perturbations on the orbit.
Assumes perturbing forces are small
Early work used general perturbations because of a lack of
computational power. Modern work uses special perturbations
(numerical integration) because of the wide availability of
computers. GP still useful for increasing your understanding.
Still used by AF for maintaining space object catalog (> 7000
objects).
Lecture 25: Perturbations
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Trend Analyses
• While it’s useful to be able to numerically integrate high-fidelity
equations of motion, it’s also useful to have an expectation of
possible effects for mission design and analysis
• Drag
– Clear reduction in a and e; can also impact i and the others slightly.
• SRP
– Depends, but in some circumstances it can increase e
• Earth’s oblateness
– Dramatic change in Ω, ω, and M
• Third-body effects
– Precession of the nodes, same effect as oblateness.
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Trend Analyses
• Secular Trends
• Long periodic effects
• Short periodic effects
• Mean elements
• Osculating elements
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General Perturbation Techniques
Perturbations can be categorized as secular, short period, long
period.
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Method of Perturbations
• What is a small parameter?
–
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–
–
–
J2
Cl,m, Sl,m
a / a3
Thrust / acc2B
Etc
• We need a way to generate the approximation. We’ll
use the method of Variation of Parameters (VOP)
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Variation of Parameters
• VOP was developed by Euler and improved by
Lagrange (1873)
• Describes the variations in the orbital elements over
time to first order constants.
• Need:
– Unperturbed system as a reference (2-body solution).
– A way to generalize the constants in the system to be timevarying parameters.
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Variation of Parameters
• Consider a system of six first-order differential
equations:
• These c parameters are osculating elements since
they’re no longer constant.
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Variation of Parameters
• Any six elements may be used, including the
conventional Keplerian orbital elements. But this
isn’t necessary.
• These parameters are a solution to:
– If no perturbation, then we’d have conic sections.
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Variation of Parameters
• Any six elements may be used, including the
conventional Keplerian orbital elements. But this
isn’t necessary.
• These parameters are a solution to:
– In the presence of perturbations, what is this?
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Variation of Parameters
• Unperturbed and perturbed relationships for the
velocity:
• We impose a constraint:
the condition of osculation
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Variation of Parameters
• If we take the derivative of the perturbed velocity
using the condition of osculation, we obtain:
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Variation of Parameters
• We substitute this equation into our earlier:
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Variation of Parameters
• We have:
• We can simplify this using our reference – that our
unperturbed equations of motion hold for any given
instant in time:
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Variation of Parameters
• We have:
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Variation of Parameters
• We have:
• This is a system of three equations of six variables.
We need three more equations!
• Condition of osculation:
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Variation of Parameters
• Now we have 6 equations with 6 variables
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Variation of Parameters
• Now we have 6 equations with 6 variables
• Not well-suited for computation. We need this to be
of the form:
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Variation of Parameters
• Two well-known ways to convert
to
• Lagrangian VOP and Gaussian VOP
– Gauss’ is easier to present first, though Vallado presents
Lagrange’s first. We’ll do Gauss then Lagrange
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Gaussian VOP
• Start by taking the dot product of the first equation with
and the dot product of the second with
and add them together:
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Gaussian VOP
• The parameters are mutually independent, so the term inside of the
parentheses simplifies to the Kronecker delta function (1 for i=j, 0 else)
• Hence:
Lecture 25: Perturbations
Or...
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Gaussian VOP
• The hard part!
• Gauss chose to perform these partial derivatives in the RSW
coordinate frame. The disturbing force is thus:
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Gaussian VOP
^
K
Cross-track
^
W
^
W
^
N
Figure 2.
v
^
S Along-track
^
I
^
v, T, in-track
^
J
^
R
Radial
Satellite Coordinate Systems, RSW and NTW. These coordinate systems move with the satellite. The
R axis points to the satellite, the W axis is normal to the orbital plane (and not usually aligned with the K
axis), and the S axis is normal to the position vector and positive in the direction of the velocity vector.
The S axis is aligned with the velocity vector only for circular orbits. In the NTW system, the T axis is
always parallel to the velocity vector. The N axis is normal to the velocity vector and is not aligned with
the radius vector, except for circular orbits, and at apogee and perigee in elliptical orbits.
We define in-track or tangential displacements as deviations along the T axis. In-track30
errors are not the same as along-track variations in the RSW system. One way to remember the distinction is
Lecture 25:
Perturbations
the
RSW system).
Gaussian VOP
• Vallado provides page after page of derivations of these
partials. The result:
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Gaussian VOP
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Gaussian VOP
• Note a few limitations:
• e must be < 1.0
• i and e can’t be 0
• Hence, this is limited to
moderately elliptical, nonequatorial orbits.
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Lagrangian VOP
• Let’s restart and derive the Lagrange planetary equations of
motion (LPEs), or simply the Lagrangian VOP
• Lagrange was the first person to perform this transformation.
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Lagrangian VOP
• Take the dot product of the top vector with
• And the dot product of the bottom vector with
• Yields:
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Comparison
• Gauss:
• Lagrange:
Disturbing potential in
6 parameters
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Lagrangian VOP
• We have:
• Lagrange brackets:
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Lagrangian VOP
• We have:
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Lagrangian VOP
• Some characteristics of Lagrange brackets:
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Lagrangian VOP
• Thus many of the values are easy. The remainder are tricky.
– Diagonal terms = 0
– L = Skew-symmetric
• Since the brackets are independent of time, we can evaluate
them anywhere along the orbit.
– Convenient to evaluate them at periapse!
– Convert state to PQW (perifocal) frame.
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Lagrangian VOP
• Convert to PQW:
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Lagrangian VOP
• Convert to PQW:
Note!
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Lagrangian VOP
• Convert to PQW:
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Lagrangian VOP
• Convenient independence:
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Lagrangian VOP
• We can also determine the PQW position and velocity vectors
at periapse:
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Lagrangian VOP
• Three cases arise:
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Lagrangian VOP
• After a lot of math:
Lecture 25: Perturbations
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Lagrangian VOP
• Note a few limitations:
• e must be < 1.0
• i and e can’t be 0
• Hence, this is limited to
moderately elliptical, nonequatorial orbits.
• That seemed a LOT like
the Gaussian VOP result
Lecture 25: Perturbations
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VOPs
• Lagrangian
Perturbing Potential Function
Lecture 25: Perturbations
Gaussian
Forcing function in RSW
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Lagrangian VOP
• Constructing perturbing potential functions
• Consider the spherical harmonic gravitational
potential.
– Take that potential function, remove the 2-body term, and
re-cast it in terms of the classical orbital elements.
– This leads to Kaula’s Solution:
– Can then evaluate it in the Lagrange planetary equations.
Lecture 25: Perturbations
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L.P.E.s & Kaula’s Solution
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Using the L.P.E.s
• Let’s use the Lagrange planetary equations (LPEs) to
evaluate the secular trends caused by a 2x2 gravity
field.
• Start with the potential function, R:
Lecture 25: Perturbations
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Using the L.P.E.s
• Remove all periodic effects
• Left with:
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Using the L.P.E.s
• Convert to orbital elements:
• We convert latitude:
• And use trig:
Lecture 25: Perturbations
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Using the L.P.E.s
• Remove periodic terms again:
• Yielding:
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Using the L.P.E.s
• The value of a/r varies over an orbit, since r varies.
• Average it over an orbit.
Lecture 25: Perturbations
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Using the L.P.E.s
• Evaluate this potential in the LPEs:
• Consider RAAN
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Using the L.P.E.s
• After simplifying, we find:
1st-order secular trend of RAAN over
time as a function of the orbit!
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Using the L.P.E.s
• We can certainly make this trend more accurate by
considering the first six zonals (S.H. order = 0):
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Using the L.P.E.s
• Similar techniques reveal other secular trends.
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Oblateness Perturbations
h
df
df
Lecture 25: Perturbations
mr
r˙˙ = - 3 + df
r
mr
˙
˙
r ´ r = -r ´ 3 + r ´ df
r
d
(r ´ r˙ ) = h˙ = r ´ df
dt
61
General Perturbation Techniques
Which is a “secularly precessing ellipse”. The equatorial bulge
introduces a force component toward the equator causing a
regression of the node (for prograde orbits) and a rotation of
periapse.
Note:
w˙ = 0 for i = 63.4° or 116.6°
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General Perturbation Techniques
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General Perturbation Techniques
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General Perturbation Techniques
Periapse also precesses.
w = 0 at the critical
inclination,
iw = 63.4  (116.6)
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General Perturbation Techniques
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General Perturbation Techniques
Application: Sun Synchronous orbits
360°
365.2421897
= 0.985647 deg/day
Can adjust a, e, i to
accomodate this.
h = 800 km, e = 0.0, i = 98.6°
W SEC desired =
Lecture 25: Perturbations
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General Perturbation Techniques
Sun Synchronous orbits:
– Orbit plane remains at a
constant angle (W’) with
respect to the Earth-Sun line.
– Orbit plane precession about
the Earth is equal to period of
Earth’s orbit about the Sun.
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Perturbation Magnitudes
ISS Orbit
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Perturbation Magnitudes
GPS Orbit
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FIN
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