ASEN 5070
Statistical Orbit determination I
Fall 2012
Professor George H. Born
Professor Jeffrey S. Parker
Lecture 3: Astro and Coding
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Homework 1
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Another change in office hours
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Yep, D2L
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HW 2 will be posted after this class
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79% of responses were
correct.
Answer in mid-lecture.
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98% of responses were
correct.
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70% of responses were
correct.
Answer in mid-lecture.
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Answer in mid-lecture.
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95% of responses were
correct.
Answer in mid-lecture.
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Review solutions for HW1
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Show HW2
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Coordinate Frames and Time Systems
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Homework details
◦ Cartesian to Keplerian conversions
◦ When elements aren’t well-defined.
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Integrators
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Coding hints and tricks
◦ LaTex: intro
◦ MATLAB: ways to speed up your code
◦ Python: intro
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Inertial: fixed orientation in space
◦ Inertial coordinate frames are typically tied to
hundreds of observations of quasars and other very
distant near-fixed objects in the sky.
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Rotating
◦ Constant angular velocity: mean spin motion of a
planet
◦ Osculating angular velocity: accurate spin motion of
a planet
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Coordinate Systems = Frame + Origin
◦ Inertial coordinate systems require that the system
be non-accelerating.
 Inertial frame + non-accelerating origin
◦ “Inertial” coordinate systems are usually just nonrotating coordinate systems.
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Is the Earth-centered J2000 coordinate
system inertial?
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ICRF
International Celestial Reference Frame, a realization of the ICR
System.
Defined by IAU (International Astronomical Union)
Tied to the observations of a selection of 212 well-known
quasars and other distant bright radio objects.
◦ Each is known to within 0.5 milliarcsec
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Fixed as well as possible to the observable universe.
Motion of quasars is averaged out.
◦ Coordinate axes known to within 0.02 milliarcsec
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Quasi-inertial reference frame (rotates a little)
Center: Barycenter of the Solar System
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ICRF2
Second International Celestial Reference Frame, consistent with
the first but with better observational data.
Defined by IAU in 2009.
Tied to the observations of a selection of 295 well-known
quasars and other distant bright radio objects (97 of which are in
ICRF1).
◦ Each is known to within 0.1 milliarcsec
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Fixed as well as possible to the observable universe.
Motion of quasars is averaged out.
◦ Coordinate axes known to within 0.01 milliarcsec
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Quasi-inertial reference frame (rotates a little)
Center: Barycenter of the Solar System
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EME2000 / J2000 / ECI
Earth-centered Mean Equator and Equinox of
J2000
◦ Center = Earth
◦ Frame = Inertial (very similar to ICRF)
 X = Vernal Equinox at 1/1/2000 12:00:00 TT
(Terrestrial Time)
 Z = Spin axis of Earth at same time
 Y = Completes right-handed coordinate frame
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EMO2000
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Earth-centered Mean Orbit and Equinox of
J2000
◦ Center = Earth
◦ Frame = Inertial
 X = Vernal Equinox at 1/1/2000 12:00:00 TT
(Terrestrial Time)
 Z = Orbit normal vector at same time
 Y = Completes right-handed coordinate frame
◦ This differs from EME2000 by ~23.4393 degrees.
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Note that J2000 is very similar to ICRF and ICRF2
◦ The pole of the J2000 frame differs from the ICRF pole by ~18
milliarcsec
◦ The right ascension of the J2000 x-axis differs from the ICRF by
78 milliarcsec
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JPL’s DE405 / DE421 ephemerides are defined to be
consistent with the ICRF, but are usually referred to as
“EME2000.” They are very similar, but not actually the same.
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ECF / ECEF / Earth Fixed / International
Terrestrial Reference Frame (ITRF)
Earth-centered Earth Fixed
◦ Center = Earth
◦ Frame = Rotating and osculating (including
precession, nutation, etc)
 X = Osculating vector from center of Earth toward the
equator along the Prime Meridian
 Z = Osculating spin-axis vector
 Y = Completes right-handed coordinate frame
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Earth Rotation
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The angular velocity vector
ωE is not constant in
direction or magnitude
◦ Direction: polar motion
 Chandler period: 430 days
 Solar period: 365 days
◦ Magnitude: related to length
of day (LOD)
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Components of ωE depend
on observations; difficult to
predict over long periods
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Principal Axis Frames
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Planet-centered Rotating System
◦ Center = Planet
◦ Frame:
 X = Points in the direction of the minimum moment of
inertia, i.e., the prime meridian principal axis.
 Z = Points in the direction of maximum moment of
inertia (for Earth and Moon, this is the North Pole
principal axis).
 Y = Completes right-handed coordinate frame
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IAU Systems
Center: Planet
Frame: Either inertial or fixed
 Z = Points in the direction of the spin axis of the
body.
 Note: by convention, all z-axes point in the solar system
North direction (same hemisphere as Earth’s North).
 Low-degree polynomial approximations are used to
compute the pole vector for most planets wrt ICRF.
 Longitude defined relative to a fixed surface feature
for rigid bodies.
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Example:
◦ Lat and Lon of Greenwich, England, shown in EME2000.
◦ Greenwich defined in IAU Earth frame to be at a
constant lat and lon at the J2000 epoch.
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Synodic Coordinate Systems
Earth-Moon, Sun-Earth/Moon, JupiterEuropa, etc
◦ Center = Barycenter of two masses
◦ Frame:
 X = Points from larger mass to the smaller mass.
 Z = Points in the direction of angular momentum.
 Y = Completes right-handed coordinate frame
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Converting from ECI to ECF
P is the precession matrix (~50
arcsec/yr)
N is the nutation matrix (main
term is 9 arcsec with 18.6 yr
period)
S’ is sidereal rotation (depends on
changes in angular velocity
magnitude; UT1)
W is polar motion
◦ Earth Orientation Parameters
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Caution: small effects may be
important in particular application
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Question: How do you quantify the passage
of time?
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Question: How do you quantify the passage
of time?
Year
Month
Day
Second
Pendulums
Atoms
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Question: How do you quantify the passage
of time?
Year
Month
Day
Second
Pendulums
Atoms
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What are some issues
with each of these?
Gravity
Earthquakes
Snooze alarms
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Countless systems exist to measure the passage of time. To
varying degrees, each of the following types is important to the
mission analyst:
◦ Atomic Time
 Unit of duration is defined based on an atomic clock.
◦ Universal Time
 Unit of duration is designed to represent a mean solar day as uniformly as
possible.
◦ Sidereal Time
 Unit of duration is defined based on Earth’s rotation relative to distant stars.
◦ Dynamical Time
 Unit of duration is defined based on the orbital motion of the Solar System.
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The duration of time required to traverse
from one perihelion to the next.
(exaggerated)
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The duration of time it takes for the Sun to
occult a very distant object twice.
These vary from
year to year.
Why?
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Definitions of a Year
◦ Julian Year: 365.25 days, where an SI “day” = 86400 SI “seconds”.
◦ Sidereal Year: 365.256 363 004 mean solar days
 Duration of time required for Earth to traverse one revolution about the
sun, measured via distant star.
◦ Tropical Year: 365.242 19 days
 Duration of time for Sun’s ecliptic longitude to advance 360 deg.
Shorter on account of Earth’s axial precession.
◦ Anomalistic Year: 365.259 636 days
 Perihelion to perihelion.
◦ Draconic Year: 365.620 075 883 days
 One ascending lunar node to the next (two lunar eclipse seasons)
◦ Full Moon Cycle, Lunar Year, Vague Year, Heliacal Year, Sothic
Year, Gaussian Year, Besselian Year
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Same variations in definitions exist for the
month, but the variations are more
significant.
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Civil day: 86400 SI seconds (+/- 1 for leap second on
UTC time system)
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Mean Solar Day: 86400 mean solar seconds
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Sidereal Day: 86164.1 SI seconds
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Stellar Day: 0.008 seconds longer than the Sidereal Day
◦ Average time it takes for the Sun-Earth line to rotate 360 degrees
◦ True Solar Days vary by up to 30 seconds, depending on where
the Earth is in its orbit.
◦ Time it takes the Earth to rotate 360 degrees relative to the
(precessing) Vernal Equinox
◦ Time it takes the Earth to rotate 360 degrees relative to distant
stars
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From 1000 AD to 1960 AD, the “second” was defined to be
1/86400 of a mean solar day.
Now it is defined using atomic transitions – some of the most
consistent measurable durations of time available.
◦ One SI second = the duration of 9,192,631,770 periods of the
radiation corresponding to the transition between the two
hyperfine levels of the ground state of the Cesium 133 atom.
◦ The atom should be at rest at 0K.
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TAI = The Temps Atomique International
◦ International Atomic Time
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Continuous time scale resulting from the statistical analysis
of a large number of atomic clocks operating around the
world.
◦ Performed by the Bureau International des Poids et Mesures (BIPM)
TAI
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UT1 = Universal Time
Represents the daily rotation of the Earth
Independent of the observing site (its longitude, etc)
Continuous time scale, but unpredictable
Computed using a combination of VLBI, quasars, lunar laser
ranging, satellite laser ranging, GPS, others
UT1
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UTC = Coordinated Universal Time
Civil timekeeping, available from radio broadcast signals.
Equal to TAI in 1958, reset in 1972 such that TAI-UTC=10 sec
Since 1972, leap seconds keep |UT1-UTC| < 0.9 sec
In June, 2012, the 25th leap second was added such that TAIUTC=35 sec
UTC
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What causes
these variations?
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TT = Terrestrial Time
Described as the proper time of a clock located on the geoid.
Actually defined as a coordinate time scale.
In effect, TT describes the geoid (mean sea level) in terms of
a particular level of gravitational time dilation relative to a
notional observer located at infinitely high altitude.
TT-TAI=
~32.184 sec
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TT
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TDB = Barycentric Dynamical Time
JPL’s “ET” = TDB. Also known as Teph. There are other
definitions of “Ephemeris Time” (complicated history)
Independent variable in the equations of motion governing
the motion of bodies in the solar system.
TDB-TAI=
~32.184 sec+
relativistic
TDB
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Long story short
In astrodynamics, when we integrate the equations of motion
of a satellite, we’re using the time system “TDB” or ~”ET”.
Clocks run at different rates, based on relativity.
The civil system is not a continuous time system.
We won’t worry about the fine details in this class, but in
reality spacecraft navigators do need to worry about the
details.
◦ Fortunately, most navigators don’t; rather, they permit one or two
specialists to worry about the details.
◦ Whew.
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Questions on Coordinate or Time Systems?
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Quick Break
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Next topics:
◦ Cartesian to Keplerian conversions.
◦ Integration
◦ Coding Tips and Tricks
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Shape:
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Orientation:
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Position:
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
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Shape:
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Orientation:
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Position:
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What if i=0?
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
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Shape:
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Orientation:
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Position:
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What if i=0?
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
◦ If orbit is equatorial, i = 0 and Ω is undefined.
 In that case we can use the “True Longitude of Periapsis”
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Shape:
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Orientation:
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Position:
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What if e=0?
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
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Shape:
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Orientation:
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Position:
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What if e=0?
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
◦ If orbit is circular, e = 0 and ω is undefined.
 In that case we can use the “Argument of Latitude”
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Shape:
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Orientation:
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Position:
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What if i=0 and e=0?
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
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Shape:
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Orientation:
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Position:
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What if i=0 and e=0?
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
◦ If orbit is circular and equatorial, neither ω nor Ω are defined
 In that case we can use the “True Longitude”
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Shape:
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Orientation:
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Position:
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Special Cases:
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
◦ If orbit is circular, e = 0 and ω is undefined.
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In that case we can use the “Argument of Latitude” ( u = ω+ν )
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In that case we can use the “True Longitude of Periapsis”
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In that case we can use the “True Longitude”
◦ If orbit is equatorial, i = 0 and Ω is undefined.
◦ If orbit is circular and equatorial, neither ω nor Ω are defined
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Handout offers one conversion.
I’ve coded up Vallado’s conversions
◦ ASEN 5050 implements these
◦ Check out the code RVtoKepler.m
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Check errors and/or special cases when i or e are very small.
Also good to check the angular momentum vector.
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An “Integrator” in the context of astrodynamics is something that
propagates the state of an object forward (or backward) in time.
Say we have a satellite in a realistic force model (not just 2-body)
at some state “X”.
We have models to describe the accelerations on it.
Where will the satellite go in the future?
Earth
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We know the current state and time.
We know the derivative of the state.
Earth
é
ù
ê R ú
X = ê V ú,
ê
ú
êë C úû
é
ù
ê V ú
X =ê A ú
ê
ú
êë 0 úû
Accelerations are the equations of
motion (two-body, J2, Drag, etc)
2nd order ODE may be integrated
as system of 1st order ODEs.
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Many techniques to predict the motion of an object.
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Simplest: 1st order Taylor expansion: Euler’s Method
é
ù
R
0
ê
ú
X(0) = ê V0 ú
ê
ú
ê C ú
ë
û
X(10)
X(0)
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X(20)
é
ù
R
+10V
0
0
ê
ú
X(10) = ê V0 +10A0 ú
ê
ú
ê
ú
C
ë
û
é
ê R10 +10V10
X(20) = ê V10 +10A10
ê
ê
C
ë
ù
ú
ú
ú
ú
û
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Many techniques to predict the motion of an object.
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Euler’s Method with predictor/corrector using trapezoidal rule
X(10)
X(0)
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X(20)
é
ê
X(0) = ê
ê
ê
ë
é
ê
X(10) = ê
ê
ê
ë
é
ê
ê
X(10) = ê
ê
ê
ë
ù
R0 ú
V0 ú
ú
C ú
û
ù
R0 +10V0 ú
V0 +10A0 ú
ú
ú
C
û
(
)
V0 + 5 ( A0 + A10 )
R0 + 5 V0 + V10
C
ù
ú
ú
ú
ú
ú
û
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Many techniques to predict the motion of an object.
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Runge-Kutta (RK4)
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Many techniques to predict the motion of an object.
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Varying time-step / adaptive time-step methods
◦ Perform a 4th and a 5th-order approximation. Check the difference.
◦ If smaller than tolerance, keep the 5th order state and move on.
◦ If not smaller than tolerance, reduce the time-step and repeat.
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Small time-steps when trajectory and/or force model changes
rapidly.
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Many techniques to predict the motion of an object.
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Symplectic integrators
◦ Rather than focusing on maintaining position accuracy, these integrators
focus on conserving energy.
◦ Tend to drift along-track
◦ In a conservative force-field, a satellite’s specific energy should be
constant.
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(Show course website’s
MATLAB integrator handout)
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Questions on Integrators?
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Next topics:
◦ Coding Tips and Tricks
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MATLAB!
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