University of ColoradoBoulder

ASEN 5070: Statistical Orbit Determination I

Fall 2014

Professor Brandon A. Jones

Lecture 3: Basics of Orbit Propagation

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Monday is Labor Day!

Homework 0 & 1 Due September 5

I am out of town Sept. 9-12

◦ Would anyone be interested in attending the recording of a lecture?

Announcements

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Orbital elements – Notes on Implementation

Perturbing Forces – Wrap-up

Coordinate and Time Systems

Flat Earth Problem

Today’s Lecture

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Orbit Elements – Review and Implementation

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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/angle: tp: time of perifocus (or mean anomaly at specified time) v,M: True or mean anomaly

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a – Size e – Shape v – Position

Orbit Size and Position

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

Orbit Orientation

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Will get an imaginary number from cos-1(a) if a=1+1e-16 (for example)

The 1e-16 is a result of finite point arithmetic

You may need to use something akin to the pseudocode:

Numeric Issues

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Inverse tangent has an angle ambiguity

Better to use atan2() when possible:

atan() versus atan2()

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Perturbing Forces – Wrap-up

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“Potential Energy is energy associated with the relative positions of two or more interacting particles.”

It is a function of the relative position◦ Should it be positive or negative?

Potential Energy

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For a conservative system:

Potential Energy

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Coordinate and Time Frames

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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

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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…

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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.

Time Systems

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Time Systems: Time Scales

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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?

GravityEarthquakesErrant 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)

◦ Full Moon Cycle, Lunar Year, Vague Year, Heliacal Year, Sothic Year,

Gaussian Year, Besselian Year

Time Systems: The Year

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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)

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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.

Rotating◦ Constant angular velocity: mean spin motion of a

planet◦ Osculating angular velocity: accurate spin motion

of a planet

Coordinate Frames

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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 non-rotating coordinate systems.

Coordinate Systems

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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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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?

Time and Coordinates

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Flat Earth Problem

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Flat Earth Problem

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Assume linear motion:

Flat Earth Problem

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Given an error-free state at a time t, we can solve for the state at t0

What about when we have a different observation type?

Flat Earth Problem

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Relationship between the estimated state and the observations is no longer linear

For our purposes, let’s assume the station coordinates are known.

You will solve one case of this problem for HW 1, Prob. 6

Flat-Earth Problem