University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 3: Time and Coordinate Systems.

University of ColoradoBoulder

ASEN 5070: Statistical Orbit Determination I

Fall 2015

Professor Brandon A. Jones

Lecture 3: Time and Coordinate Systems

University of ColoradoBoulder 2

Homework 0 - Not required

Homework 1 Due September 4

I am out of town Sept. 15-18

Announcements


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


Orbital elements – Notes on Implementation

Perturbing Forces – Wrap-up

Coordinate and Time Systems

Today’s Lecture


Orbit Elements – Notes on Implementation


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


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


Inverse tangent has an angle ambiguity

Better to use atan2() when possible:

atan() versus atan2()

(1,1)

(-1,-1)

Same value for atan


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 function such that the gravitational acceleration is:


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


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)


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


New Integration Constant


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


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


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


Time Systems: Time Scales


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


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


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


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 Frames

In this class, we will keep the transformation simple:

In reality, this is a poor model!


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…


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


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


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


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


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

University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 3: Time and Coordinate Systems.

Documents

implementation slide

longitude slide

atan slide

class hw2 slide

oblateness j

orbit elements notes

time of perifocus

pear shape j