Transcript ASEN 5050 SPACEFLIGHT DYNAMICS

ASEN 5050
SPACEFLIGHT DYNAMICS
General Perturbations
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 26: General Perturbations
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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.
• I’ll be in my office from 10-1 and 2-3 for office
hours.
• Reading: Chapters 8 and 9
Lecture 26: 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
26: General
Perturbations
• 12/12:
Final
Review
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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.
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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.
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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 26: General 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.
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Final Project
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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.
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We will move this directory around, and the links have to work!
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Test your webpage! Change the location of the page on your computer and make
sure it still works!
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Zip everything up into a single file and upload that to the D2L dropbox.
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Space News
Reflections of sunlight off of seas in Titan’s northern latitudes. Methane and ethane.
Lecture 26: General Perturbations
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Implementing Perturbations
• Any force model may be integrated using a numerical
integrator, such as ode45.
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Two-body equations of motion
n-body equations of motion
Accelerations caused by solar radiation pressure
Accelerations caused by atmospheric drag
Accelerations caused by any other effect
– Put them all together and integrate that collective force
model to build the spacecraft’s trajectory.
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Implementing Perturbations
• An example of the derivative function for an
integrator such as ode45.
function dX = deriv( t, X )
x = X(1);
…
vz = X(6);
a_sun = -GM_sun * r_sun/norm(r_sun)^3
a_mercury = -GM_mercury * r_mercury/norm(r_mercury)^3
…
a_neptune = -GM_neptune * r_neptune/norm(r_neptune)^3
a_earth_harmonics = blah blah blah
a_SRP = blah blah blah
a_earth_albedo = blah blah blah
a_drag = blah blah blah
a_outgassing = blah blah blah
a_thermal_emmission = blah blah blah
a = a_sun + a_mercury + … + a_thermal_emmission
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Perturbation Magnitudes
ISS Orbit
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Perturbation Magnitudes
GPS Orbit
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Perturbation Magnitudes
Earth – Mars
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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 perturbative 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).
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General Perturbation Techniques
Perturbations can be categorized as secular, short period, long
period.
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Perturbations
c = c0 + c˙1 (t - t0 )+ K1 cos(2w )+ K2 sin(2n + w)+ K 3 cos(2n )
secular
long-periodic
mixed-periodic short-periodic
This equation is known as a “Poisson Series”
Ki = sums and products of polynomials in a, e, i
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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
• Different derivation, similar results:
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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.
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VOPs
• Lagrangian
Perturbing Potential Function
Lecture 26: General 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.
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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:
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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:
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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.
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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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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 =
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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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How does this compare to reality?
• 28.5 deg inclined, somewhat
eccentric orbit:
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Conclusions
• Consider a perturbation’s trends (secular drift) and
periodic effects (fast and slow)
• What steps are taken to estimate these?
• Gaussian VOP
– Rate of change of orbital elements evaluated using a perturbing
potential field
• Lagrangian VOP
– Rate of change of orbital elements evaluated using a forcing
function
• J2 is famous for its secular effects on the node, the
argument of periapse, and the mean anomaly.
Lecture 26: General Perturbations
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ASEN 5050
SPACEFLIGHT DYNAMICS
Mid-Term Exam
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 26: General Perturbations
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Mid-Term Exam
• Problem 1
• Common errors:
– Since this isn’t a circular orbit, the DT is not half of the
orbital period. Though the Delta-nu is indeed 180 deg.
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Mid-Term Exam
• Hard way: generate the two-body orbit and map the
latitude:
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Mid-Term Exam
• Problem 1 solution
• Easy way (and precise):
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Argument of latitude: u = ω + ν
At ascending node: u = 0 deg
At descending node: u = 180 deg
ω = 310 deg
At ascending node: ν = -310 deg = 50 deg
At descending node: ν =-130 deg = 230 deg
– Compute time past periapse for both
– Delta-t = 1.138 hours
Lecture 17: Perturbations
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Mid-Term Exam
• Problem 2
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Mid-Term Exam
• Problem 2
Periapsis Range: 5473.0 km
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Mid-Term Exam
• Problem 2
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Mid-Term Exam
• Problem 2
Use eccentricity and semi-major axis, and find the eccentric anomaly
needed to make radius = 6378.1363 km.
Convert to TimePastPeriapse.
And also compute TimePastPeriapse of initial state. We find:
Time past periapse of initial state: -2049.99566 sec
Time past periapse of impact:
-676.70889 sec
Duration of time to impact:
1373.28677 sec = 0.38147 hours
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Mid-Term Exam
• Problem 2
Lecture 26: General Perturbations
The missile will strike the surface traveling
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8811.91852 m/s in velocity
Mid-Term Exam
• Problem 3
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Mid-Term Exam
• Problem 3
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Mid-Term Exam
C/W Equations: Algorithm 48
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Mid-Term Exam
• Problem 3
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Mid-Term Exam
• Problem 3
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omega: 0.00114400182253 rad/s
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Satellite A will be located at a position of
[0.0, 0.0, 80.8050388] in meters, after 7 minutes
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Satellite B will be located at a position of
[79.17972504, 142.440310957, 0.0] in meters, after 7 minutes
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Satellite A will have a velocity relative to the shuttle of
[0.0, 0.0, 0.177354562] in m/s, after 7 minutes
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Satellite B will have a velocity relative to the shuttle of
[0.369764446, 0.2188365,
0.0] in m/s, after 7 minutes
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Mid-Term Exam
• Problem 3
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Mid-Term Exam
• Problem 3
• Satellite A's state relative to Satellite B at t=7
minutes, in meters and m/s, before executing any
maneuver (A-B):
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x0: -79.179725045 m
y0: -142.440310958 m
z0: 80.8050388697 m
vx0: -0.369764446945 m/s
vy0: -0.218836500483 m/s
vz0: 0.17735456256 m/s
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Mid-Term Exam
• Problem 3
• What velocity is NEEDED? Equation 6-66
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Mid-Term Exam
• Problem 3
• What velocity is NEEDED? Equation 6-66
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The velocity that Satellite A needs to obtain (m/s):
vx: 0.00953422198546
vy: 0.247527429082
vz: -0.0745206696574
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Mid-Term Exam
• Problem 3
• What CHANGE in velocity is needed?
• The Delta-V impulse that A has to perform, in m/s
relative to B:
• Delta-Vx: 0.37929866893
• Delta-Vy: 0.466363929565
• Delta-Vz: -0.251875232218
• Delta-V magnitude: 0.651769842549
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Mid-Term Exam
• Problem 3
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Mid-Term Exam
• Problem 3
• Use Algorithm 48 again to propagate state of A relative to
B. You should see the position go to zero!!!
• The rendezvous Delta-V impulse that A has to perform, in
m/s relative to B:
• Delta-Vx: -0.17983714361
• Delta-Vy: -0.066363929565
• Delta-Vz: 0.11873790189
• Delta-V magnitude: 0.225486715161
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Mid-Term Exam
• Problem 3
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Mid-Term Exam
• Problem 3
• Answer: yes!
• If satellite A does not do anything, it will collide with
the shuttle in about 45 minutes. Half an orbit later.
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Mid-Term Exam
• Problem 4
• Inclination = 90+14 = 104 deg
• P = 30/360*86164.09056 seconds = 7180.341 sec =
1.995 hours
• P = 2*pi*sqrt(a^3/mu)  a = 8044.32 km
• Altitude = 1666.18 km
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Mid-Term Exam
• Problem 5
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Mid-Term Exam
• Problem 5
Earth
Moon
Umbra
Penumbra
Total Eclipse
Sunlight
380,000 km
Moon
Umbra
Penumbra
Annular Eclipse
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Mid-Term Exam
• Problem 5
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Mid-Term Exam
• Problem 5
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mu = muEarth + muMoon = 403503.2405 km3/s2
rp = 362600 km
ra = 405400 km
a = 384,000 km
e = 0.0557292
Period of lunar orbit: 2353711.295 sec = 39228.5216 min
= 653.80869 hours = 27.242029 days
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Mid-Term Exam
• Problem 5
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Mid-Term Exam
• Problem 5
• use equation: r = a(1-e cosE)
• r = 380000 km + rEarth = 386378.1363 km
• E = 1.68215420 rad = 96.380336 deg
• Time past periapse: 609395.909 sec = 10156.5985 min = 169.27664 hours
= 7.053193 days
• Total Duration within 386378 km: 1218791.818 sec = 20313.1970 min =
338.55328 hours = 14.106387 days
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Mid-Term Exam
• Problem 5
Percentage of time the Earth and Moon are
within 386378 km: 51.782%
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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.
• I’ll be in my office from 10-1 and 2-3 for office
hours.
• Reading: Chapters 8 and 9
Lecture 26: General Perturbations
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