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## ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate, Time, Conversions

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### ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate, Time, Conversions Prof. Jeffrey S. Parker University of Colorado Boulder Lecture 7: Coordinate, Time, Conversions * – PowerPoint PPT presentation

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Title: ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate, Time, Conversions

1
ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate, Time,
Conversions
• Prof. Jeffrey S. Parker

2
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.

3
Quiz 5
Nobody selected these. Good!
4
Quiz 5
Only ½ of the class got the right answer.
5
Quiz 5
6
Quiz 5
Z ambiguity!
Z ambiguity!
7
Quiz 5
8
Quiz 5
9
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?

10
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!
11
Do they ever get close to colliding?
12
Do they ever get close to colliding?
13
Neptunes and Plutos Orbit
• Do the orbits intersect?

Plutos Orbit
Neptunes Orbit
14
Neptune and Plutos Closest Approach
15
ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate and
Time Systems
• Prof. Jeffrey S. Parker
• University of Colorado - Boulder

16
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.

17
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).

18
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
• The angle between the Earths mean equator and
the ecliptic is called the obliquity of the
ecliptic, e23.5?.

19
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

20
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?

21
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

22
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

23
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

24
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.

25
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.

26
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

27
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

28
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

29
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.

30
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.

31
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

32
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

33
Time Systems
• Question How do you quantify the passage of
time?

34
Time Systems
• Question How do you quantify the passage of
time?
• Year
• Month
• Day
• Second
• Pendulums
• Atoms

35
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
36
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.

37
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?
38
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

39
Time Systems The Month
• Same variations in definitions exist for the
month, but the variations are more significant.

40
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

41
Time Systems The Second
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.

42
Time Systems Time Scales
43
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
44
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
45
Time Systems UTC
• UTC Coordinated Universal Time
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
46
Time Systems UTC
47
Time Systems UTC
What causes these variations?
48
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
49
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
50
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.

51
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.