ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 2: Basics of Orbit Propagation
University of Colorado
Boulder
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Homework 0
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Homework 1 – Due September 4 by start of lecture
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Office Hours:
◦ You do not need to turn it in
◦ Intended to be a tutorial for those who have never used a
numeric integrator (ode45()/ode113()/odeint())
◦ Prof. Jones: W 3-4pm & Th 11-12noon
 ECNT 420
◦ Eduardo: W 2-3pm & Th 2:30-4:30pm
 ECAE 1B44 (not CCAR meeting room!)
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Syllabus questions?
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The orbit propagation problem (Chapter 2)
◦ Two-body problem
◦ Orbital elements
◦ Potential theory
◦ Perturbed Satellite Motion
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Two-Body Motion
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First, we start with Newton’s
second law:
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Combine that with Newton’s law
of universal gravitation
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To get the gravity acceleration
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For unperturbed dynamics about a central
body:
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What’s μ?
μ = GM
G = 6.67384 ± 0.00080 × 10-20 km3/kg/s2
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MEarth ~ 5.97219 × 1024 kg
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μEarth = 398,600.4415 ± 0.0008 km3/s2
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◦ or 5.9736 × 1024 kg
◦ or 5.9726 × 1024 kg
◦ Use a value and cite where you found it!
(Tapley, Schutz, and Born, 2004)
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How do we measure the value of μEarth?
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We express orbit energy as a specific energy
(energy per unit mass) instead of absolute
energy
For two-body motion, the specific energy is
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Center of mass of two bodies moves in straight
line with constant velocity
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Specific angular momentum
◦ Consequence: motion is planar
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Energy per unit mass (scalar) is constant
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The mass of the satellite is negligible relative to
the primary body
◦ What are some possible exceptions to this?
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The coordinate system is inertial
◦ What is the problem with this assumption?
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The bodies are spherical with uniform mass
density
◦ Is that true?
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No other forces act on the system
◦ That isn’t true either…
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Orbit Elements
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The six orbit elements (or Kepler elements) are
constant in the two-body problem
◦ 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 (different options):
 tp: time of perifocus
 Mean/true/eccentric anomaly at some time (requires two
parameters)
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Consider an ellipse
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Periapse/perifocus/periapsis
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◦ Perigee, perihelion
r = radius
rp = radius of periapse
ra = radius of apoapse
a = semi-major axis
e = eccentricity = (ra-rp)/(ra+rp)
rp = a(1-e)
ra = a(1+e)
ω = argument of periapse
f/υ = true anomaly
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i - Inclination
Ω - RAAN
ω – Arg. of Perigee
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Cartesian Coordinates
◦ x, y, z, vx, vy, vz in some coordinate frame
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Keplerian Orbital Elements
◦ a, e, i, Ω, ω, ν (or similar set)
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Topocentric Elements
◦ Right ascension, declination, radius, and time rates
of each
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When are each of these useful?
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Shape:
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Orientation:
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Position:
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What if i=0?
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If orbit is equatorial, i = 0 and Ω is undefined.
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
◦ In that case we can use the “True Longitude of Periapsis”, i.e., the angle
from the vernal equinox (inertial X-axis) to the perifocus
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Shape:
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Orientation:
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Position:
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What if e=0?
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If orbit is circular, e = 0 and ω is undefined.
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
◦ In that case we can use the “Argument of Latitude”, i.e., the angle from
ascending node to satellite
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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?
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If orbit is circular and equatorial, neither ω nor Ω are defined
◦ a = semi-major axis
◦ e = eccentricity
◦ i = inclination
◦ Ω = right ascension of ascending node
◦ ω = argument of periapse
◦ ν = true anomaly
◦ In that case we can use the “True Longitude”, i.e., the angle between the
satellite position vector and the vernal equinox
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Shape:
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Orientation:
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Position (at time t):
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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
◦ M = mean 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”
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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.
We’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!
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A Brief Introduction to Potential Theory
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Potential theory provides a means for
studying harmonic functions
Given the potential energy V for a system,
then
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In astrodynamics, we often write
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This gives us
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In HW 1, you will use potential theory to
derive the two-body equation
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Homework 1
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Homework # 1
Problem 1:
Problem 2:
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Problem 3:
Problem 4:
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Homework # 1
Problem 5:
Problem 6:
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Problem 7:
Solution method discussed
next week!
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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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l 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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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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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
From D. King-Hele, 1964, Theory of
Satellite Orbits in an Atmosphere
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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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