asen 5050 spaceflight dynamics time systems, conversions, f & g
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ASEN 5050SPACEFLIGHT 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 3Lecture 8: Time, Conversions 2
Concept Quiz 6
Lecture 8: Time, Conversions 3
Concept Quiz 6
Lecture 8: Time, Conversions 4
Scheduling spacecraft observations requires complete knowledge of time!UT1 and UTC are unpredictable.
Concept Quiz 6
Lecture 8: Time, Conversions 5
x
y
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 6
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 7
ASEN 5050SPACEFLIGHT 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 9
• 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.
Time Systems
Lecture 8: Time, Conversions 10
Time Systems: Time Scales
Lecture 8: Time, Conversions 11
• 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.
Time Systems: TAI
TAI
Lecture 8: Time, Conversions 12
• 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
Time Systems: UT1
UT1
Lecture 8: Time, Conversions 13
• 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
Time Systems: UTC
UTC
Lecture 8: Time, Conversions 14
• 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.
Time Systems: TT
TT
TT-TAI=~32.184 sec
Lecture 8: Time, Conversions 15
• 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.
Time Systems: TDB
TDB
TDB-TAI=~32.184 sec+relativistic
Lecture 8: Time, Conversions 16
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 17
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 18
• 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.
Time Systems: Summary
Lecture 8: Time, Conversions 19
ASEN 5050SPACEFLIGHT DYNAMICS
Coordinate Systems
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 8: Time, Conversions 20
Coordinate Systems• An interesting scenario that involves two coordinate
frames playing together:
Lecture 8: Time, Conversions 21
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 22
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.
Moon’s orbit (exaggerated)
PeriapseM = 0°
ApoapseM = 180°
M = 90°
M = 270°
Lon = 0°
Lecture 8: Time, Conversions 23
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 24
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-rotatingIntersection of ecliptic and celestial eq
Lecture 8: Time, Conversions 25
Coordinate Systems
Earth-Centered Earth-Fixed Coordinates (ECEF)
Topocentric Horizon Coordinate System (SEZ)
Lecture 8: Time, Conversions 26
Coordinate Systems
Perifocal Coordinate System (PQW)
Lecture 8: Time, Conversions 27
Coordinate SystemsSatellite Coordinate Systems:
RSW – Radial-Transverse-Normal
NTW – Normal-Tangent-Normal; VNC is a rotated version
Lecture 8: Time, Conversions 28
Coordinate SystemsSatellite Coordinate Systems:
RSW – Radial-Transverse-Normal
NTW – Normal-Tangent-Normal; VNC is a rotated version
CVR
S
Lecture 8: Time, Conversions 29
Coordinate Transformations
Coordinate rotations can be accomplished through rotations about the principal axes.
Lecture 8: Time, Conversions 30
Coordinate Transformations
To convert from the ECI (IJK) system to ECEF, we simply rotate around Z by the GHA:
ignoring precession, nutation, polar motion, motion of equinoxes.
Lecture 8: Time, Conversions 31
Coordinate Transformations
To convert from ECEF to SEZ:
Lecture 8: Time, Conversions 32
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
(sometimes this is easier, sometimes not)
Lecture 8: Time, Conversions 33
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 34
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 35
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 36
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
Lecture 8: Time, Conversions 37
R
S
Challenge #4
• If you were to plot the position and velocity of a satellite over time using VNC (Velocity-Normal-Conormal) coordinates, what would you find?– Say, an elliptical orbit
Lecture 8: Time, Conversions 38
CV
(Vallado, 1997)
Latitude/Longitude
Geocentric latitude
Lecture 8: Time, Conversions 39
(Vallado, 1997)
Latitude/Longitude
For geodetic latitude use:
where e=0.081819221456
Lecture 8: Time, Conversions 40
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 3Lecture 8: Time, Conversions 41
Coordinate Transformations
To convert between IJK and PQW:
To convert between PQW and RSW:
Thus, RSW IJK is:
R
S
P
Lecture 8: Time, Conversions 42
Latitude/Longitude
Rotate into ECEF
Lecture 8: Time, Conversions 43
Right Ascension/Declination
thus,
Lecture 8: Time, Conversions 44
Azimuth-Elevation
Compute slant-range vector from site to satellite:
Rotate into SEZ
Lecture 8: Time, Conversions 45
Topocentric Horizon System (SEZ)
Lecture 8: Time, Conversions 46
Azimuth-Elevation
Alternatively:
Lecture 8: Time, Conversions 47
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