University of ColoradoBoulder

ASEN 5070: Statistical Orbit Determination I

Fall 2015

Professor Brandon A. Jones

Lecture 3: Time and Coordinate Systems

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Homework 0 - Not required

Homework 1 Due September 4

I am out of town Sept. 15-18

Announcements

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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

As stated in the syllabus, legible hand-generated derivations are okay as an image in the PDF

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!

Homework Format/Requirements

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Orbital elements – Notes on Implementation

Perturbing Forces – Wrap-up

Coordinate and Time Systems

Today’s Lecture

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Orbit Elements – Notes on 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

6

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You 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 this pseudocode:

Numeric Issues

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Inverse tangent has an angle ambiguity

Better to use atan2() when possible:

atan() versus atan2()

(1,1)

(-1,-1)

Same value for atan

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Perturbed Satellite Motion

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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)

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Perturbed Motion: Planetary Mass Distribution

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

Define gravitational potential function such that the gravitational acceleration is:

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Gravitational Potential Function

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

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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

You will only need to implement a J2 model for this class (HW2)

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Our new orbit energy is

Is this constant over time? Why or why not?

What if we only include J2 in U’ and pure rotation about Z-axis for Earth?

Orbit Energy

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New Integration Constant

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Atmospheric Drag

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

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

Orbital lifetime of satellite strongly influenced by drag

You will use a simple exponential model for the atmospheric density (HW2)

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

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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

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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.

◦ 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 Sundial

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)

Main idea: When we talk about time, we need to be precise with our statements!

Time Systems: The Year

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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

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.

◦ Underlying problem: How do we estimate the inertial frame from a rotating one?

Coordinate 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 Frames

In this class, we will keep the transformation simple:

In reality, this is a poor model!

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Coordinate Systems and Time

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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Coordinate Systems and Time

Accurate representations must account for precession, nutation, and other effects

Classic definition of ECF and ECI transformation based on an `Equinox’

Modern definitions instead use the “Celestial Intermediate Origin” (CIO)Animations/Images courtesy of WikiCommons

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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. Why is a frame at the center of the Earth not a true

inertial frame?

Coordinate Systems

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Converting from ECR to ECI

Coordinate System Transformations

BPN accounts for nutation, precision and a bias term

R is the Earth’s rotation, which is not constant! (In this class, we only include this component)

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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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?

Make sure your propagator is working by looking at the constants of motion◦ Specific energy◦ Specific angular momentum

Once you complete HW 1, you are ready to start HW 2!

Homework 2