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ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate, Time,

Conversions

- Prof. Jeffrey S. Parker
- University of Colorado Boulder

Announcements

- Office hours today, cancelled (PhD prelim exam).

Let me know if you need to chat and cant make it

to any other office hours. - Homework 3 is due Friday 9/19 at 900 am
- Concept Quiz 6 will be available at 1000 am,

due Wednesday morning at 800 am. - Reading Chapter 3

Quiz 5

Nobody selected these. Good!

Quiz 5

Only ½ of the class got the right answer.

Please convince your neighbor that you know the

correct answer!

Quiz 5

Quiz 5

Z ambiguity!

Z ambiguity!

Quiz 5

Quiz 5

Challenge 3

- We examined Plutos and Neptunes orbits last

time. - Question since Pluto sometimes travels interior

to Neptunes orbit, could they ever collide? - If so, what sort of order of duration do we need

to wait until it may statistically happen?

Years? Millennia? Eons?

Challenge 3

- They are statistically never going to collide!

(unless something crazy happens, like we

encounter another star) - Pluto and Neptune are quite far non coplanar
- Plutos inclination is 17 deg
- Neptunes inclination is 2 deg
- Plutos Longitude of Ascending Node is 110 deg
- Neptunes Longitude of Ascending Node is 131 deg
- Pluto and Neptune are in resonance
- Neptune orbits the Sun 3x when Pluto orbits 2x.

8 people got a point!

Do they ever get close to colliding?

Do they ever get close to colliding?

Neptunes and Plutos Orbit

- Do the orbits intersect?

Plutos Orbit

Neptunes Orbit

Neptune and Plutos Closest Approach

ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate and

Time Systems

- Prof. Jeffrey S. Parker
- University of Colorado - Boulder

Coordinate Systems

- Given a full state, with position and velocity

known. - Or, given the full set of coordinate elements.

- What coordinate system is this state represented

in? - Could be any non-rotating coordinate system!
- Earth J2000 or ecliptic J2000 or Mars, etc.

Coordinate Systems

- Celestial Sphere
- Celestial poles intersect Earths rotation axis.
- Celestial equator extends Earth equator.
- Direction of objects measured with right

ascension (a) and declination (d).

Coordinate Systems

- The Vernal Equinox defines the reference

direction. A.k.a. The Line of Aries - The ecliptic is defined as the mean plane of the

Earths orbit about the Sun. - The angle between the Earths mean equator and

the ecliptic is called the obliquity of the

ecliptic, e23.5?.

Coordinate Frames

- 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 Systems

- 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. - Is the Earth-centered J2000 coordinate system

inertial?

Useful Coordinate Systems

- ICRF
- International Celestial Reference Frame, a

realization of the ICR System. - Defined by IAU (International Astronomical Union)
- Tied to the observations of a selection of 212

well-known quasars and other distant bright radio

objects. - Each is known to within 0.5 milliarcsec
- Fixed as well as possible to the observable

universe. - Motion of quasars is averaged out.
- Coordinate axes known to within 0.02 milliarcsec
- Quasi-inertial reference frame (rotates a little)
- Center Barycenter of the Solar System

Useful Coordinate Systems

- ICRF2
- Second International Celestial Reference Frame,

consistent with the first but with better

observational data. - Defined by IAU in 2009.
- Tied to the observations of a selection of 295

well-known quasars and other distant bright radio

objects (97 of which are in ICRF1). - Each is known to within 0.1 milliarcsec
- Fixed as well as possible to the observable

universe. - Motion of quasars is averaged out.
- Coordinate axes known to within 0.01 milliarcsec
- Quasi-inertial reference frame (rotates a little)
- Center Barycenter of the Solar System

Useful Coordinate Systems

- EME2000 / J2000 / ECI
- Earth-centered Mean Equator and Equinox of J2000
- Center Earth
- Frame Inertial (very similar to ICRF)
- X Vernal Equinox at 1/1/2000 120000 TT

(Terrestrial Time) - Z Spin axis of Earth at same time
- Y Completes right-handed coordinate frame

Useful Coordinate Systems

- EMO2000
- Earth-centered Mean Orbit and Equinox of J2000
- Center Earth
- Frame Inertial
- X Vernal Equinox at 1/1/2000 120000 TT

(Terrestrial Time) - Z Orbit normal vector at same time
- Y Completes right-handed coordinate frame
- This differs from EME2000 by 23.4393 degrees.

Useful Coordinate Systems

- Note that J2000 is very similar to ICRF and ICRF2
- The pole of the J2000 frame differs from the ICRF

pole by 18 milliarcsec - The right ascension of the J2000 x-axis differs

from the ICRF by 78 milliarcsec - JPLs DE405 / DE421 ephemerides are defined to be

consistent with the ICRF, but are usually

referred to as EME2000. They are very similar,

but not actually the same.

Useful Coordinate Systems

- ECF / ECEF / Earth Fixed / International

Terrestrial Reference Frame (ITRF) - Earth-centered Earth Fixed
- Center Earth
- Frame Rotating and osculating (including

precession, nutation, etc) - X Osculating vector from center of Earth toward

the equator along the Prime Meridian - Z Osculating spin-axis vector
- Y Completes right-handed coordinate frame

Useful Coordinate Systems

- Earth Rotation

- The angular velocity vector ?E is not constant in

direction or magnitude - Direction polar motion
- Chandler period 430 days
- Solar period 365 days
- Magnitude related to length of day (LOD)
- Components of ?E depend on observations

difficult to predict over long periods

Useful Coordinate Systems

- Principal Axis Frames
- Planet-centered Rotating System
- Center Planet
- Frame
- X Points in the direction of the minimum moment

of inertia, i.e., the prime meridian principal

axis. - Z Points in the direction of maximum moment of

inertia (for Earth and Moon, this is the North

Pole principal axis). - Y Completes right-handed coordinate frame

Useful Coordinate Systems

- IAU Systems
- Center Planet
- Frame Either inertial or fixed
- Z Points in the direction of the spin axis of

the body. - Note by convention, all z-axes point in the

solar system North direction (same hemisphere as

Earths North). - Low-degree polynomial approximations are used to

compute the pole vector for most planets wrt

ICRF. - Longitude defined relative to a fixed surface

feature for rigid bodies.

Useful Coordinate Systems

- Example
- Lat and Lon of Greenwich, England, shown in

EME2000. - Greenwich defined in IAU Earth frame to be at a

constant lat and lon at the J2000 epoch.

Useful Coordinate Systems

- Synodic Coordinate Systems
- Earth-Moon, Sun-Earth/Moon, Jupiter-Europa, etc
- Center Barycenter of two masses
- Frame
- X Points from larger mass to the smaller mass.
- Z Points in the direction of angular momentum.
- Y Completes right-handed coordinate frame

Coordinate System Transformations

- Converting from ECI to ECF

- 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

Time Systems

- Question How do you quantify the passage of

time?

Time Systems

- Question How do you quantify the passage of

time? - Year
- Month
- Day
- Second
- Pendulums
- Atoms

Time Systems

- Question How do you quantify the passage of

time? - Year
- Month
- Day
- Second
- Pendulums
- Atoms

What are some issues with each of

these? Gravity Earthquakes Snooze alarms

Time Systems

- 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 Earths

rotation relative to distant stars. - Dynamical Time
- Unit of duration is defined based on the orbital

motion of the Solar System.

Time Systems The Year

- The duration of time required to traverse from

one perihelion to the next. - The duration of time it takes for the Sun to

occult a very distant object twice.

(exaggerated)

These vary from year to year. Why?

Time Systems The Year

- 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 Suns ecliptic longitude to

advance 360 deg. Shorter on account of Earths

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 Month

- Same variations in definitions exist for the

month, but the variations are more significant.

Time Systems The Day

- Civil day 86400 SI seconds (/- 1 for leap

second on UTC time system) - Mean Solar Day 86400 mean solar seconds
- Average time it takes for the Sun-Earth line to

rotate 360 degrees - True Solar Days vary by up to 30 seconds,

depending on where the Earth is in its orbit. - Sidereal Day 86164.1 SI seconds
- Time it takes the Earth to rotate 360 degrees

relative to the (precessing) Vernal Equinox - Stellar Day 0.008 seconds longer than the

Sidereal Day - Time it takes the Earth to rotate 360 degrees

relative to distant stars

Time Systems The Second

- From 1000 AD to 1960 AD, the second was defined

to be 1/86400 of a mean solar day. - Now it is defined using atomic transitions some

of the most consistent measurable durations of

time available. - One SI second the duration of 9,192,631,770

periods of the radiation corresponding to the

transition between the two hyperfine levels of

the ground state of the Cesium 133 atom. - The atom should be at rest at 0K.

Time Systems Time Scales

Time Systems TAI

- TAI The Temps Atomique International
- International Atomic Time
- Continuous time scale resulting from the

statistical analysis of a large number of atomic

clocks operating around the world. - Performed by the Bureau International des Poids

et Mesures (BIPM)

TAI

Time Systems UT1

- UT1 Universal Time
- Represents the daily rotation of the Earth
- Independent of the observing site (its longitude,

etc) - Continuous time scale, but unpredictable
- Computed using a combination of VLBI, quasars,

lunar laser ranging, satellite laser ranging,

GPS, others

UT1

Time Systems UTC

- UTC Coordinated Universal Time
- Civil timekeeping, available from radio broadcast

signals. - Equal to TAI in 1958, reset in 1972 such that

TAI-UTC10 sec - Since 1972, leap seconds keep UT1-UTC lt 0.9 sec
- In June, 2012, the 25th leap second was added

such that TAI-UTC35 sec

UTC

Time Systems UTC

Time Systems UTC

What causes these variations?

Time Systems TT

- TT Terrestrial Time
- Described as the proper time of a clock located

on the geoid. - Actually defined as a coordinate time scale.
- In effect, TT describes the geoid (mean sea

level) in terms of a particular level of

gravitational time dilation relative to a

notional observer located at infinitely high

altitude.

- TT-TAI 32.184 sec

TT

Time Systems TDB

- TDB Barycentric Dynamical Time
- JPLs ET TDB. Also known as Teph. There are

other definitions of Ephemeris Time

(complicated history) - Independent variable in the equations of motion

governing the motion of bodies in the solar

system.

- TDB-TAI 32.184 sec relativistic

TDB

Time Systems Summary

- Long story short
- In astrodynamics, when we integrate the equations

of motion of a satellite, were using the time

system TDB or ET. - Clocks run at different rates, based on

relativity. - The civil system is not a continuous time system.
- We wont worry about the fine details in this

class, but in reality spacecraft navigators do

need to worry about the details. - Fortunately, most navigators dont rather, they

permit one or two specialists to worry about the

details. - Whew.

Announcements

- Office hours today, cancelled (PhD prelim exam).

Let me know if you need to chat and cant make it

to any other office hours. - Homework 3 is due Friday 9/19 at 900 am
- Concept Quiz 6 will be available at 1000 am,

due Wednesday morning at 800 am. - Reading Chapter 3