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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
  • University of Colorado Boulder

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.
  • Reading Chapter 3

3
Quiz 5
Nobody selected these. Good!
4
Quiz 5
Only ½ of the class got the right answer.
Please convince your neighbor that you know the
correct 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
    Earths orbit about the Sun.
  • 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
  • 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.

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
  • 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
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.
  • Reading Chapter 3
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