ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 3: Time and Coordinate Systems
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Boulder
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Homework 0 - Not required
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Homework 1 Due September 4
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I am out of town Sept. 15-18
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Provide enough detail to answer the problem and
allow for grading
◦ For a figure, we need labeled axes, fonts big enough to
read, etc.
◦ We do not need a detailed description unless it is
requested in the problem set
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As stated in the syllabus, legible hand-generated
derivations are okay as an image in the PDF
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If you are spending more than 20 minutes on the
write-up, you may be including too much detail
◦ This will not be true for the project write-up!
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Orbital elements – Notes on Implementation
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Perturbing Forces – Wrap-up
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Coordinate and Time Systems
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Orbit Elements – Notes on Implementation
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The six orbit elements (or Kepler elements) are
constant in the problem of two bodies (two
gravitationally attracting spheres, or point masses)
◦ Define shape of the orbit
 a: semimajor axis
 e: eccentricity
◦ Define the orientation of the orbit in space
 i: inclination
 Ω: angle defining location of ascending node (AN)
 : angle from AN to perifocus; argument of perifocus
◦ Reference time/angle:
 tp: time of perifocus (or mean anomaly at specified time)
 v,M: True or mean anomaly
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You will get an imaginary number from cos-1(a) if
a=1+1e-16 (for example)
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The 1e-16 is a result of finite point arithmetic
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You may need to use something akin to this
pseudocode:
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Inverse tangent has
an angle ambiguity
Better to use atan2()
when possible:
(1,1)
Same value
for atan
(-1,-1)
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Perturbed Satellite Motion
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The 2-body problem
provides us with a
foundation of orbital motion
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In reality, other forces exist
which arise from
gravitational and
nongravitational sources
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In the general equation of
satellite motion, a is the
perturbing force (causes the
actual motion to deviate
from exact 2-body)
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Sphere of constant
mass density is not an
accurate representation
for planets
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Define gravitational
potential function such
that the gravitational
acceleration is:
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The commonly used expression for the gravitational potential is
given in terms of mass distribution coefficients Jn, Cnm, Snm
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n is degree, m is order
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Coordinates of evaluation point are given in spherical coordinates:
r, geocentric latitude φ, longitude 
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U.S. Vanguard satellite
launched in 1958, used
to determine J2 and J3
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J2 represents most of the
oblateness; J3 represents
a pear shape
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J2 ≈ 1.08264 x 10-3
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J3 ≈ - 2.5324 x 10-6
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You will only need to
implement a J2 model for
this class (HW2)
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Our new orbit energy is
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Is this constant over time? Why or why not?
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What if we only include J2 in U’ and pure
rotation about Z-axis for Earth?
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From D. King-Hele, 1964, Theory of
Satellite Orbits in an Atmosphere
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Atmospheric drag is the
dominant nongravitational
force at low altitudes if the
celestial body has an
atmosphere
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Drag removes energy from
the orbit and results in
da/dt < 0, de/dt < 0
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Orbital lifetime of satellite
strongly influenced by drag
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You will use a simple
exponential model for the
atmospheric density (HW2)
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What are the other forces that can perturb a satellite’s
motion?
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Solar Radiation Pressure (SRP)
Thrusters
N-body gravitation (Sun, Moon, etc.)
Electromagnetic
Solid and liquid body tides
Relativistic Effects
Reflected radiation (e.g., ERP)
Coordinate system errors
Spacecraft radiation
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Coordinate and Time Frames
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Countless systems exist to measure the passage of time. To
varying degrees, each of the following types is important in
astrodynamics:
◦ Atomic Time
 Unit of duration is defined based on an atomic clock.
◦ Sidereal Time
 Unit of duration is defined based on Earth’s rotation relative to distant stars.
◦ Universal Time
 Unit of duration is designed to represent a mean solar day as uniformly as
possible.
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Dynamical Time
 Unit of duration is defined based on the orbital motion of the Solar System.
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Question: How do you quantify the passage
of time?
Year
Month
Day
Second
Pendulums
Atoms
Sundial
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What are some issues
with each of these?
Gravity
Earthquakes
Errant elbows
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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)
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Main idea: When we talk about time, we need to be precise with
our statements!
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Rotating: changing orientation in space
◦ The Earth is a rotating bodyThink about the motion of a
top. The Earth has similar changes in the rotation axis
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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 nearfixed objects in the sky.
◦ Underlying problem: How do we estimate the inertial
frame from a rotating one?
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Define xyz reference frame
(Earth centered, Earth fixed;
ECEF or ECF), fixed in the
solid (and rigid) Earth and
rotates with it
Longitude λ measured from
Greenwich Meridian
0≤ λ < 360° E; or measure λ
East (+) or West (-)
Latitude (geocentric latitude)
measured from equator (φ is
North (+) or South (-))
◦ At the poles,
φ = -90° S
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φ = + 90° N or
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In this class, we will keep the
transformation simple:
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In reality, this is a poor model!
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The transformation between ECI and ECF is
required in the equations of motion
◦ Why?
◦ Depends on the current time!
◦ Thanks to Einstein, we know that time is not simple…
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Animations/Images courtesy of WikiCommons
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Accurate representations
must account for
precession, nutation, and
other effects
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Classic definition of ECF
and ECI transformation
based on an `Equinox’
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Modern definitions instead
use the “Celestial
Intermediate Origin” (CIO)
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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.
 Why is a frame at the center of the Earth not a true
inertial frame?
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Converting from ECR to ECI
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BPN accounts for nutation, precision and a bias term
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R is the Earth’s rotation, which is not constant! (In this class, we
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W is polar motion
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Caution: small effects may be important in particular application
only include this component)
◦ Earth Orientation Parameters
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We did not spend a lot of time on this
subject, but it is very, very important to orbit
determination!
What impact can the coordinates and time
have on propagation and observing a
spacecraft?
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Propagate a spacecraft where the model includes
the two-body, J2, and drag forces
Observe the change in the orbital elements over
time as a result of these forces
◦ Why would they change?
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Make sure your propagator is working by looking
at the constants of motion
◦ Specific energy
◦ Specific angular momentum
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Once you complete HW 1, you are ready to start
HW 2!
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