ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones

University of ColoradoBoulder

ASEN 5070: Statistical Orbit Determination IFall 2014

Professor Brandon A. Jones

Lecture 2: Basics of Orbit Propagation

University of ColoradoBoulder 2

Monday is Labor Day!

Homework 0 – You could already have it finished

Homework 1 – Due September 5

Syllabus questions?

Announcements


The orbit propagation problem

Orbital elements

Perturbed Satellite Motion

Coordinate and Time Systems

Today’s Lecture


Brief Review of Astrodynamics


What’s μ (other than a greek letter)?

Review of Astrodynamics


What’s μ (other than a greek letter)?

μ is the gravitational parameter of a massive body

μ = GM



What’s μ?


μ = GM

What’s G? What’s M?



What’s μ?


μ = GM

What’s G? Universal Gravitational Constant What’s M? The mass of the body



What’s μ? μ = GM G = 6.67384 ± 0.00080 × 10-20 km3/kg/s2

MEarth ~ 5.97219 × 1024 kg ◦ or 5.9736 × 1024 kg◦ or 5.9726 × 1024 kg◦ Use a value and cite where you found it!

μEarth = 398,600.4415 ± 0.0008 km3/s2 (Tapley, Schutz, and Born, 2004)

How do we measure the value of μEarth?




Problem of Two Bodies

XYZ is nonrotating, with zero acceleration; an inertial reference frame

µ = G(M1 + M2)


Center of mass of two bodies moves in straight line with constant velocity

Angular momentum per unit mass (h) is constant, h = r x V = constant, where V is velocity of M2 with respect to M1, V= dr/dt◦ Consequence: motion is planar

Energy per unit mass (scalar) is constant

Integrals of Motion


Orbit Plane in Space

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Equations of Motion in the Orbit Plane

The uθ component yields:

which is simply h = constant


Solution of ur Equations of Motion

The solution of the ur equation is (as function of θ instead of t):

where e and ω are constants of integration.


The Conic Equation Constants of integration: e and ω

◦ e = ( 1 + 2 ξ h2/µ2 )1/2

◦ ω corresponds to θ where r is minima

Let f = θ – ω, then

r = p/(1 + e cos f) which is “conic equation” fromanalytical geometry (e is conic “eccentricity”,

p is “semi-latus rectum” or “semi-parameter”, and f is the “true anomaly”)

Conclude that motion of M2 with respect to M1 is a “conic section”◦ Circle (e=0), ellipse (0<e<1), parabola (e=1), hyperbola (e>1)

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Types of Orbital Motion


Orbit Elements


Six Orbit Elements

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: tp: time of perifocus (or mean anomaly at specified time)

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i - Inclination Ω - RAAN ω – Arg. of Perigee

Orbit Orientation


Keplerian Orbital Elements

Consider an ellipse

Periapse/perifocus/periapsis◦ 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

Astrodynamics Background


Cartesian Coordinates◦ x, y, z, vx, vy, vz in some coordinate frame

Keplerian Orbital Elements◦ a, e, i, Ω, ω, ν (or similar set)

Topocentric Elements◦ Right ascension, declination, radius, and time rates

of each

When are each of these useful?

Satellite State Representations


Shape:◦ a = semi-major axis◦ e = eccentricity

Orientation:◦ i = inclination◦ Ω = right ascension of ascending node◦ ω = argument of periapse

Position:◦ ν = true anomaly

What if i=0?

◦ If orbit is equatorial, i = 0 and Ω is undefined. In that case we can use the “True Longitude of Periapsis”






What if e=0?

◦ If orbit is circular, e = 0 and ω is undefined. In that case we can use the “Argument of Latitude”






What if i=0 and e=0?

◦ If orbit is circular and equatorial, neither ω nor Ω are defined In that case we can use the “True Longitude”





Position:◦ ν = true anomaly◦ M = mean anomaly

Special Cases:◦ If orbit is circular, e = 0 and ω is undefined.

In that case we can use the “Argument of Latitude” ( u = ω+ν )◦ If orbit is equatorial, i = 0 and Ω is undefined.

In that case we can use the “True Longitude of Periapsis”◦ If orbit is circular and equatorial, neither ω nor Ω are defined

In that case we can use the “True Longitude”



Handout offers one conversion. We’ve coded up Vallado’s conversions

◦ ASEN 5050 implements these◦ Check out the code RVtoKepler.m

Check errors and/or special cases when i or e are very small!

Cartesian to Keplerian Conversion


Perturbed Satellite Motion


Perturbed Motion

The 2-body problem provides us with a foundation of orbital motion

In reality, other forces exist which arise from gravitational and nongravitational sources

In the general equation of satellite motion, a is the perturbing force (causes the actual motion to deviate from exact 2-body)


Perturbed Motion: Planetary Mass Distribution

Sphere of constant mass density is not an accurate representation for planets

Define gravitational potential, U, such that the gravitational force is


Gravitational Potential The commonly used

expression for the gravitational potential is given in terms of mass distribution coefficients Jn, Cnm, Snm

n is degree, m is order

Coordinates of evaluation point are given in spherical coordinates: r, geocentric latitude φ, longitude


Gravity Coefficients

The gravity coefficients (Jn, Cnm, Snm) are also known as Stokes Coefficients and Spherical Harmonic Coefficients

Jn, Cn0:◦ Gravitational potential represented in zones of latitude; referred to

as zonal coefficients

Cnm, Snm:◦ If n=m, referred to as sectoral coefficients◦ If n≠m, referred to as tesseral coefficients

These parameters may be used for orbit design (take ASEN 5050 for more details!)


Shape of Earth: J2, J3

U.S. Vanguard satellite launched in 1958, used to determine J2 and J3

J2 represents most of the oblateness; J3 represents a pear shape

J2 = 1.08264 x 10-3

J3 = - 2.5324 x 10-6


Atmospheric Drag

Atmospheric drag is the dominant nongravitational force at low altitudes if the celestial body has an atmosphere

Depending on nature of the satellite, lift force may exist

Drag removes energy from the orbit and results in da/dt < 0, de/dt < 0

Orbital lifetime of satellite strongly influenced by drag

From D. King-Hele, 1964, Theory of Satellite Orbits in an Atmosphere


Other Forces What are the other forces that can perturb a satellite’s

motion?

◦ 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


Coordinate and Time Frames


Coordinate Frames

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° N or

φ = -90° S


Coordinate Systems and Time

The transformation between ECI and ECF is required in the equations of motion

◦ Depends on the current time!◦ Thanks to Einstein, we know that time is not simple…


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.

Time Systems


Time Systems: Time Scales


Question: How do you quantify the passage of time?

Year Month Day Second Pendulums Atoms

Time Systems

What are some issues with each of these?

GravityEarthquakesSnooze alarms


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

Time Systems: The Year


Coordinate Systems and Time

Equinox location is function of time◦ Sun and Moon interact with

Earth J2 to produce Precession of equinox (ψ) Nutation (ε)

Newtonian time (independent variable of equations of motion) is represented by atomic time scales (dependent on Cesium Clock)


Precession / Nutation

Precession

Nutation (main term):


Sidereal rate of rotation: ~2π/86164 rad/day Variations exist in magnitude of ωE, from upper

atmospheric winds, tides, etc. UT1 is used to represent such variations

◦ UTC is kept within 0.9 sec of UT1 (leap second) Polar motion and UT1 observed quantities Different time scales: GPS-Time, TAI, UTC, TDT Time is independent variable in satellite equations of

motion; relates observations to equations of motion (TDT is usually taken to represent independent variable in equations of motion)

Earth Rotation and Time


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.

Rotating◦ Constant angular velocity: mean spin motion of a

planet◦ Osculating angular velocity: accurate spin motion

of a planet

Coordinate Frames


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 non-rotating coordinate systems.

Coordinate Systems


Converting from ECI to ECF

Coordinate System Transformations

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

Caution: small effects may be important in particular application

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ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones

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