ASEN 5050
SPACEFLIGHT DYNAMICS
Coordinate, Time, Conversions
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 7: Coordinate, Time,
Conversions
1
Announcements
• Office hours today, cancelled (PhD prelim exam).
Let me know if you need to chat and can’t make it to
any other office hours.
• Homework #3 is due Friday 9/19 at 9:00 am
• Concept Quiz #6 will be available at 10:00 am, due
Wednesday morning at 8:00 am.
• Reading: Chapter 3
Lecture 7: Coordinate, Time,
Conversions
2
Quiz 5
Nobody selected these. Good!
Lecture 7: Coordinate, Time,
Conversions
3
Quiz 5
Only ½ of the class got the
right answer.
Please convince your
neighbor that you know the
correct answer!
Lecture 7: Coordinate, Time,
Conversions
4
Quiz 5
Lecture 7: Coordinate, Time,
Conversions
5
Quiz 5
Z ambiguity!
Z ambiguity!
Lecture 7: Coordinate, Time,
Conversions
6
Quiz 5
Lecture 7: Coordinate, Time,
Conversions
7
Quiz 5
Lecture 7: Coordinate, Time,
Conversions
8
Challenge #3
• We examined Pluto’s and Neptune’s orbits last time.
• Question: since Pluto sometimes travels interior to
Neptune’s 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?
Lecture 7: Coordinate, Time,
Conversions
9
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
–
–
–
–
Pluto’s inclination is ~17 deg
Neptune’s inclination is ~2 deg
Pluto’s Longitude of Ascending Node is ~110 deg
Neptune’s Longitude of Ascending Node is ~131 deg
• Pluto and Neptune are in resonance
– Neptune orbits the Sun 3x when Pluto orbits 2x.
Lecture 7: Coordinate, Time,
Conversions
8 people got a point!
10
Do they ever get close to colliding?
Lecture 7: Coordinate, Time,
Conversions
11
Do they ever get close to colliding?
Lecture 7: Coordinate, Time,
Conversions
12
Neptune’s and Pluto’s Orbit
• Do the orbits intersect?
Pluto’s Orbit
Neptune’s
Orbit
Lecture 7: Coordinate, Time,
Conversions
13
Neptune and Pluto’s Closest Approach
Lecture 7: Coordinate, Time,
Conversions
14
ASEN 5050
SPACEFLIGHT DYNAMICS
Coordinate and Time Systems
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 7: Coordinate, Time,
Conversions
15
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 nonrotating coordinate
system!
• Earth J2000 or ecliptic
J2000 or Mars, etc.
Lecture 7: Coordinate, Time,
Conversions
16
Coordinate Systems
Celestial Sphere
– Celestial poles intersect
Earth’s rotation axis.
– Celestial equator extends
Earth equator.
– Direction of objects
measured with right
ascension (a) and
declination (d).
Lecture 7: Coordinate, Time,
Conversions
17
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 Earth’s orbit about the Sun.
The angle between the Earth’s mean
equator and the ecliptic is called the
obliquity of the ecliptic, e~23.5.
Lecture 7: Coordinate, Time,
Conversions
18
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
Lecture 7: Coordinate, Time,
Conversions
19
Coordinate Systems
• Coordinate Systems = Frame + Origin
– Inertial coordinate systems require that the system be nonaccelerating.
• Inertial frame + non-accelerating origin
– “Inertial” coordinate systems are usually just non-rotating
coordinate systems.
• Is the Earth-centered J2000 coordinate system
inertial?
Lecture 7: Coordinate, Time,
Conversions
20
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
Lecture 7: Coordinate, Time,
Conversions
21
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
Lecture 7: Coordinate, Time,
Conversions
22
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 12:00:00 TT (Terrestrial Time)
• Z = Spin axis of Earth at same time
• Y = Completes right-handed coordinate frame
Lecture 7: Coordinate, Time,
Conversions
23
Useful Coordinate Systems
• EMO2000
• Earth-centered Mean Orbit and Equinox of J2000
– Center = Earth
– Frame = Inertial
• X = Vernal Equinox at 1/1/2000 12:00:00 TT (Terrestrial Time)
• Z = Orbit normal vector at same time
• Y = Completes right-handed coordinate frame
– This differs from EME2000 by ~23.4393 degrees.
Lecture 7: Coordinate, Time,
Conversions
24
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
• JPL’s 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.
Lecture 7: Coordinate, Time,
Conversions
25
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
Lecture 7: Coordinate, Time,
Conversions
26
Useful Coordinate Systems
• Earth Rotation

The angular velocity vector ωE 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)

Lecture 7: Coordinate, Time,
Conversions
Components of ωE depend on
observations; difficult to predict
over long periods
27
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
Lecture 7: Coordinate, Time,
Conversions
28
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 Earth’s 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.
Lecture 7: Coordinate, Time,
Conversions
29
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.
Lecture 7: Coordinate, Time,
Conversions
30
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
Lecture 7: Coordinate, Time,
Conversions
31
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
Lecture 7: Coordinate, Time,
Conversions
32
Time Systems
• Question: How do you quantify the passage of time?
Lecture 7: Coordinate, Time,
Conversions
33
Time Systems
• Question: How do you quantify the passage of time?
•
•
•
•
•
•
Year
Month
Day
Second
Pendulums
Atoms
Lecture 7: Coordinate, Time,
Conversions
34
Time Systems
• Question: How do you quantify the passage of time?
•
•
•
•
•
•
Year
Month
Day
Second
Pendulums
Atoms
Lecture 7: Coordinate, Time,
Conversions
What are some issues with
each of these?
Gravity
Earthquakes
Snooze alarms
35
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 Earth’s rotation relative to distant stars.
– Dynamical Time
• Unit of duration is defined based on the orbital motion of the Solar System.
Lecture 7: Coordinate, Time,
Conversions
36
Time Systems: The Year
• The duration of time required to traverse from one
perihelion to the next.
(exaggerated)
• The duration of time it takes for the Sun to occult a
very distant object twice.
These vary from
year to year.
Why?
Lecture 7: Coordinate, Time,
Conversions
37
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 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
Lecture 7: Coordinate, Time,
Conversions
38
Time Systems: The Month
• Same variations in definitions exist for the month, but
the variations are more significant.
Lecture 7: Coordinate, Time,
Conversions
39
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
Lecture 7: Coordinate, Time,
Conversions
40
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.
Lecture 7: Coordinate, Time,
Conversions
41
Time Systems: Time Scales
Lecture 7: Coordinate, Time,
Conversions
42
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
Lecture 7: Coordinate, Time,
Conversions
43
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
Lecture 7: Coordinate, Time,
Conversions
44
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-UTC=10 sec
Since 1972, leap seconds keep |UT1-UTC| < 0.9 sec
In June, 2012, the 25th leap second was added such that TAI-UTC=35 sec
UTC
Lecture 7: Coordinate, Time,
Conversions
45
Time Systems: UTC
Lecture 7: Coordinate, Time,
Conversions
46
Time Systems: UTC
What causes these
variations?
Lecture 7: Coordinate, Time,
Conversions
47
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
Lecture 7: Coordinate, Time,
Conversions
TT
48
Time Systems: TDB
• TDB = Barycentric Dynamical Time
• JPL’s “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
Lecture 7: Coordinate, Time,
Conversions
TDB
49
Time Systems: Summary
• Long story short
• In astrodynamics, when we integrate the equations of motion of a satellite,
we’re 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 won’t worry about the fine details in this class, but in reality spacecraft
navigators do need to worry about the details.
– Fortunately, most navigators don’t; rather, they permit one or two specialists to
worry about the details.
– Whew.
Lecture 7: Coordinate, Time,
Conversions
50
Announcements
• Office hours today, cancelled (PhD prelim exam).
Let me know if you need to chat and can’t make it to
any other office hours.
• Homework #3 is due Friday 9/19 at 9:00 am
• Concept Quiz #6 will be available at 10:00 am, due
Wednesday morning at 8:00 am.
• Reading: Chapter 3
Lecture 7: Coordinate, Time,
Conversions
51