Transcript ASEN 5050 SPACEFLIGHT DYNAMICS

ASEN 5050
SPACEFLIGHT DYNAMICS
Time Systems, Conversions, f & g
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 8: Time, Conversions
1
Announcements
• Homework #3 is due Friday 9/19 at 9:00 am
– You must write your own code.
– For this HW, please turn in your code (preferably in one
text/Word/PDF document)
– After this assignment, you may use Vallado’s code, but if you do
you must give him credit for work done using his code. If you
don’t, it’s plagiarism.
• Concept Quiz 7 active and due Friday at 8:00 am.
• I’ll be at the career fair Monday, so I’m delaying
Monday’s office hours to 2:00.
• Reading: Chapter 3
Lecture 8: Time, Conversions
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Concept Quiz 6
p
a(1- e2 )
r=
=
1+ ecos n 1+ ecos n
Lecture 8: Time, Conversions
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Concept Quiz 6
Scheduling spacecraft observations requires complete knowledge of time!
UT1 and UTC are unpredictable.
Lecture 8: Time, Conversions
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Concept Quiz 6
y
x
Lecture 8: Time, Conversions
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Space News
• NASA just announced which companies will be used
to launch our astronauts into orbit!
• Boeing
– CST-100
– $4.2 Billion
• SpaceX
– Dragon
– $2.6 Billion
Lecture 8: Time, Conversions
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Final Project
• Reminder to think about your final project, even now.
• Objective: go beyond the scope of this class in some way.
Build an informative website describing your project.
Gloat to your friends.
• I have an opportunity for several people to work on the
mission design for a mission to Mars. If you’re
interested, email me or come by office hours.
– Today and any Wednesday 2-4
– Next Monday at 2:00 (future Mondays at 11)
Lecture 8: Time, Conversions
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ASEN 5050
SPACEFLIGHT DYNAMICS
Time Systems
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 8: Time, Conversions
8
Time Systems
• Time is important
• Signal travel time of electromagnetic waves
– Altimetry, GPS, SLR, VLBI
• For positioning
– Orbit determination
– One nanosecond (10–9 second) is 30 cm of distance
– Relative motion of celestial bodies
• Scheduling maneuvers
Lecture 8: Time, Conversions
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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 8: Time, Conversions
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Time Systems: Time Scales
Lecture 8: Time, Conversions
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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)
• Atomic clocks drift 1 second in about 20 million years.
TAI
Lecture 8: Time, Conversions
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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 8: Time, Conversions
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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 8: Time, Conversions
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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 8: Time, Conversions
TT
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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 8: Time, Conversions
TDB
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Present time differences
• As of 17 Sept 2014,
– TAI is ahead of UTC by 35 seconds.
– TAI is ahead of GPS by 19 seconds.
– GPS is ahead of UTC by 16 seconds.
• The Global Positioning System (GPS) epoch is January 6, 1980 and
is synchronized to UTC.
Lecture 8: Time, Conversions
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Fundamentals of Time
Julian Date (JD) – defines the number of mean solar days since
4713 B.C., January 1, 0.5 (noon).
Modified Julian Date (MJD) – obtained by subtracting 2400000.5
days from JD. Thus, MJD commences at midnight instead of
noon.
Civilian Date
JD
1980 Jan 6 midnight
2444244.5
GPS Standard Epoch
2000 Jan 1 noon
2451545.0
J2000 Epoch
Algorithm 14 in book.
Lecture 8: Time, Conversions
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Time Systems: Summary
• 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 8: Time, Conversions
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ASEN 5050
SPACEFLIGHT DYNAMICS
Coordinate Systems
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 8: Time, Conversions
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Coordinate Systems
• An interesting scenario that involves two coordinate
frames playing together:
Lecture 8: Time, Conversions
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The Moon’s Librations
• The librations can be explained via three facts:
1. The Moon spins about its axis at a very consistent rate
• And it is tidally locked to the Earth
2. The Moon’s orbit is not circular.
3. The Moon’s spin axis is not aligned with its orbital axis
Lecture 8: Time, Conversions
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The Moon’s Librations
• The librations can be explained via three facts:
1. The Moon spins about its axis at a very consistent rate
• And it is tidally locked to the Earth
2. The Moon’s orbit is not circular.
M = 270°
Periapse
M = 0°
Apoapse
M = 180°
Lon = 0°
Lecture 8: Time, Conversions
M = 90°
Moon’s orbit
(exaggerated)
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Coordinate Systems
• An interesting scenario that involves two coordinate
frames playing together:
So this image may be interpreted as being
a view of the Moon in the Earth-Moon
rotating frame, where the Moon’s surface
rotates according to the “Moon Fixed”
coordinate system.
Lecture 8: Time, Conversions
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Coordinate Systems
Geocentric Coordinate System (IJK)
- aka: Earth Centered Inertial (ECI), or the Conventional
Inertial System (CIS)
- J2000 – Vernal equinox on Jan 1, 2000 at noon
- non-rotating
Intersection of ecliptic and celestial eq
Lecture 8: Time, Conversions
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Coordinate Systems
Earth-Centered Earth-Fixed Coordinates (ECEF)
Topocentric Horizon Coordinate System (SEZ)
Lecture 8: Time, Conversions
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Coordinate Systems
Perifocal Coordinate System (PQW)
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Coordinate Systems
Satellite Coordinate Systems:
RSW – Radial-Transverse-Normal
NTW – Normal-Tangent-Normal; VNC is a rotated version
Lecture 8: Time, Conversions
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Coordinate Systems
Satellite Coordinate Systems:
RSW – Radial-Transverse-Normal
NTW – Normal-Tangent-Normal; VNC is a rotated version
V
R
C
S
Lecture 8: Time, Conversions
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Coordinate Transformations
Coordinate rotations can be accomplished through rotations about
the principal axes.
0 ù
sin a ú
ú
cos a úû
- sin a ù
0 ú
ú
cos a ûú
0
é1
cos a
ROT 1( a ) = ê0
ê
- sin a
êë0
0
écos a
1
ROT 2( a ) = ê 0
ê
0
ëê sin a
é cos a
ROT 3( a ) = ê- sin a
ê
êë 0
Lecture 8: Time, Conversions
sin a
cos a
0
0ù
0ú
ú
1úû
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Coordinate Transformations
To convert from the ECI (IJK) system to ECEF, we simply rotate
around Z by the GHA:
rECEF = ROT 3( q GST )rIJK
GST = Greenwich Sidereal Time
or rIJK = ROT 3( -q GST )rECEF
ignoring precession, nutation, polar motion, motion of equinoxes.
Lecture 8: Time, Conversions
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Coordinate Transformations
To convert from ECEF to SEZ:
rSEZ = ROT 2( 90° - f )ROT 3( l )rECEF
= ROT 2( 90° - f )ROT 3( q LST = l + q GST )rIJK
Lecture 8: Time, Conversions
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Coordinate Transformations
• One of the coolest shortcuts for building
transformations from one system to any other,
without building tons of rotation matrices:
The unit vector in the S-direction,
expressed in I,J,K coordinates
Lecture 8: Time, Conversions
(sometimes this is
easier, sometimes not)
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Coordinate Transformations
• You can check Vallado, or some of the appendix
slides of this presentation for additional
transformations.
• I’d like to provide some conceptual purpose for
considering different coordinate systems!
Lecture 8: Time, Conversions
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Scenario: Tracking Stations
• Consider a satellite in orbit.
• How long is the satellite overhead, as viewed by a ground
station in Goldstone, California?
– What’s the elevation/azimuth time profile of the pass?
• Need: elevation (and azimuth) angles of satellite as
viewed by station.
– Need: satellite’s states represented in SEZ coordinates
• Transform satellite from IJK to ECEF
• Transform satellite from ECEF to SEZ
• Compute elevation and azimuth angles
Lecture 8: Time, Conversions
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Scenario: Solar Power
• A satellite is nadir-pointed with body-fixed solar
panels pointed 90 deg away from nadir. How should
the satellite rotate to maximize the energy output of
the panels? What is the incidence angle of the Sun
over time?
• Need: satellite state represented as RSW
• Compute angles to the Sun in that frame
Lecture 8: Time, Conversions
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Brainteaser
• If you were to plot the position and velocity of a
satellite over time using RSW coordinates, what
would you find?
– Say, an elliptical orbit
R
S
Lecture 8: Time, Conversions
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Challenge #4
• If you were to plot the position and velocity of a
satellite over time using VNC (Velocity-NormalConormal) coordinates, what would you find?
– Say, an elliptical orbit
V
Lecture 8: Time, Conversions
C
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Latitude/Longitude
Geocentric latitude
Lecture 8: Time, Conversions
(Vallado, 1997)
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Latitude/Longitude
For geodetic latitude use:
tan f gc
tan f gd =
1 - eÅ2
where e=0.081819221456
(Vallado, 1997)
Lecture 8: Time, Conversions
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Announcements
• Homework #3 is due Friday 9/19 at 9:00 am
– You must write your own code.
– For this HW, please turn in your code (preferably in one
text/Word/PDF document)
– After this assignment, you may use Vallado’s code, but if you do
you must give him credit for work done using his code. If you
don’t, it’s plagiarism.
• Concept Quiz 7 due Friday at 8:00 am.
• I’ll be at the career fair Monday, so I’m delaying
Monday’s office hours to 2:00.
• Reading: Chapter 3
Lecture 8: Time, Conversions
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Coordinate Transformations
To convert between IJK and PQW:
rIJK = ROT 3( -W )ROT 1( -i )ROT 3( -w )rPQW
rPQW = ROT 3( w )ROT 1( i )ROT 3( W )rIJK
To convert between PQW and RSW:
R
S
rRSW = ROT 3(n )rPQW
rPQW = ROT 3( -n )rRSW
Thus, RSW  IJK is:
rIJK = ROT 3( -W )ROT 1( -i )ROT 3( -u )rRSW
where
Lecture 8: Time, Conversions
P
u =n + w
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Latitude/Longitude
Rotate into ECEF
rECEF = ROT 3( q GST )rIJK
æ ry ö
l = tan çç ÷÷
è rx ø
-1 æ rz ö
f gc = sin ç ÷
èrø
-1
rECEF
ér cos f cos lù érx ù
ê
ú ê ú
= ê r cos f sin l ú = êry ú
êë r sin f úû êërz úû
r = rx2 + ry2 + rz2
Lecture 8: Time, Conversions
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Right Ascension/Declination
rIJK
ér cos d cos aù é rI ù
ê
ú ê ú
= ê r cos d sin a ú = ê rJ ú
êë r sin d úû êërK úû
r = rI2 + rJ2 + rK2
thus,
æ rK ö
d = sin ç ÷
èrø
-1
æ rJ ö
a = tan ç ÷
è rI ø
-1
Lecture 8: Time, Conversions
( l = a - q GST )
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Azimuth-Elevation
Compute slant-range vector from site to satellite:
r IJK = rIJK - rsiteIJK
Rotate into SEZ
r SEZ = ROT 2( 90° - f )ROT 3( l )ROT 3( q GST )r IJK
é- r cos( el ) cos( b )ù
since r SEZ = ê r cos( el ) sin( b ) ú
ê
ú
r sin( el )
êë
úû
r S2 + r E2
cos(el) =
r
r
sin( el ) = Z
r
sin( b ) =
Lecture 8: Time, Conversions
rE
r S2 + r E2
- rS
cos( b ) =
r S2 + r E2
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Topocentric Horizon System (SEZ)
Lecture 8: Time, Conversions
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Azimuth-Elevation
Alternatively:
rECEF = ROT 3( q GST )rIJK
r ECEF = rECEF - rsiteECEF
r SEZ = ROT 2( 90° - f )ROT 3( l )r ECEF
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