Another Look at NonRotating Origins

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Another Look at Non-Rotating Origins

• George Kaplan
• Astronomical Applications Department
• U.S. Naval Observatory
• gkaplan_at_usno.navy.mil

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Another Look at Non-Rotating OriginsAbstract
Two non-rotating origins were defined by
the IAU in 2000 for the measurement of Earth
rotation the Celestial Ephemeris Origin (CEO)
in the ICRS and the Terrestrial Ephemeris Origin
(TEO) in the ITRS. Universal Time (UT1) is now
defined by an expression based on the angle ?
between the CEO and TEO. Many previous papers,
e.g., Capitaine et al. (2000), developed the
position of the CEO in terms of a quantity s, the
difference between two arcs on the celestial
sphere. A similar quantity s? was defined for
the TEO. continued
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Abstract (cont.)
As an alternative, a simple vector
differential equation for the position of a
non-rotating origin on its reference sphere is
developed here. The equation can be easily
numerically integrated to high precision. This
scheme directly yields the ICRS right ascension
and declination of the CEO, or the ITRF longitude
and latitude of the TEO, as a function of time.
This simplifies the derivation of the main
transformation matrix between the ITRF and the
ICRS. This approach also yields a simple vector
expression for apparent sidereal time. The
directness of the development may have
pedagogical and practical advantages for the vast
majority of astronomers who are unfamiliar with
the history of this topic.
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Objectives

• Develop alternative mathematical description of
  non-rotating origins, precise enough for
  practical computations
• Avoid spherical trig, composite arcs, etc.
• Understandable to the average astronomer
• Directly yield RA Dec of CEO, lat and lon of
  TEO
• Plot out path of CEO on sky and TEO on Earth
• Develop corresponding rotation matrices for ITRS
  ? ICRS transformation and test
• The NRO for Dummies

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NRO Definition

• Standard
• Given a pole P(t) and a point of origin ? on the
  equator. If O is the geocenter, then the
  orientation of the cartesian axes OP, O?, O?
  (where O? is orthogonal to both OP and O?) is
  determined by the condition that in any
  infinitesimal displacement of P there is no
  instantaneous rotation around OP. Then ? (also
  ?) is a non-rotating origin on the moving
  equator.
• Alternative
• A non-rotating origin, ?, is a point on the
  moving equator whose instantaneous motion is
  always orthogonal to the equator.

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ecliptic
?1
?1
equator of t1
?2
equator of t2
?2
equator of t3
?3
?3
Dec
ICRF
RA
Motion of a non-rotating origin, ?, compared to
that of the true equinox, ?.
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Two equators, two poles, two NROs, ?t apart ?t ? 0
? n
n0
n1
e1
x1
?2
e0
O
?x
x0
?1
n0, n1, x0, x1 are unit vectors
x1 x0 ?x ?x ?z n0
where ?z is a scalar to be determined
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Differential equation for NRO position

• If n(t) is a unit vector toward the
  instantaneous pole and x(t) is a unit vector
  toward an instantaneous non-rotating origin, then
  in a few steps we get
  
• x(t) ? x(t) n(t)
  n(t) (1)
• We want to obtain the path of the NRO, x(t).
• Note that if P, N, and F are the matrices for
  precession, nutation, and frame bias, then
• n(t) F Pt(t) Nt(t) is the path of
  the pole in the ICRS

.
.
0 0 1
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Numerical Integration of NRO Position, x(t)

• Conceptually and practically, it is simple to
  integrate x(t) ?
  x(t) n(t) n(t)
• Can use, for example, standard 4th-order Runge
  Kutta integrator
• Fixed step sizes ? 1 day OK
• Really a 1-D problem played out in 3-D, so can
  apply constraints at each step x 1 and x
  n 0

?
?
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Applications of x(t)

• x(t) is the path of the CEO if n(t) is the path
  of the CIP (celestial pole) and a proper initial
  point ? x(J2000) ? is selected. The path of the
  CIP is defined by the IAU2000A precession-nutation
  model, as implemented by the algorithms in the
  IERS Conventions (2000). The initial point,
  x(J2000), should be chosen to provide continuity
  of Earth orientation measurements on 2003 Jan 1.
• x(t) is the path of the TEO if n(t) is path of
  terrestrial pole (xp, yp) as defined by
  observations and x(J2000) 0.

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Transformations Made Easy (1 of 3)

• If n(t) represents the path of the CIP (celestial
  pole) and x(t) represents the path of the CEO
  (celestial non-rotating origin), then let y n
  ? x. The orthonormal triad x y n then
  represents the axes of the intermediate frame
  (equator-of-date frame with CEO as azimuthal
  origin) expressed ICRS coordinates
• Full terrestrial-to-celestial transformation then
  becomes
• rc C R3(-?) W? rt where C ( x y n )
  
• (W? is the polar motion matrix and R3 is a
  simple rotation about the z axis, which at
  that step is n(t).)
• continued...

x1 y1 n1 x2 y2 n2 x3 y3 n3

(2)
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Transformations Made Easy (2 of 3)

• If p represents the apparent direction of a
  celestial object in ICRS coordinates (its virtual
  place), then its azimuthal coordinate, A, in the
  equator-of-date frame, measured eastward from the
  CEO, is
• A arctan ( p y / p x )
  (3)
• Its Greenwich Hour Angle (GHA), is simply GHA
  ? ? A.
• (Here, the Greenwich meridian is defined to be
  the plane that passes through the TEO and the CIP
  axis.)
• continued...

?
?

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Transformations Made Easy (3 of 3)

• We know the position of the instantaneous (true)
  equinox in ICRS coordinates
• ?(t) F Pt(t) Nt(t)
• So the GHA of the true equinox apparent
  sidereal time must be simply
• GHA ? GAST ? ? arctan ( ? ? y /
  ? ? x ) (4)
• Need s? The node of the instantaneous equator
  on the ICRS equator is just
• N ? n(t) Then s can be
  easily computed as the difference
• between the arcs x(t)N and ?0 N
• (?0 is the ICRS origin of RA)

1 0 0

0 0 1
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Does All This Work?

• Yes!
• Compare the quantity s, computed using the CEO
  integration scheme, with that computed using the
  IERS analytical formula (series).
• In the upper plot, the integrated and IERS
  curves completely overlap the bottom plot shows
  the difference ? only a few microarcseconds.

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Long-Term CEO Position

• This plot shows the output of the integrator
  from a 50,000-year integration, using a very
  simple precession algorithm and no nutation. It
  shows the changing position of the CEO on the sky
  in a fixed (kinematically non-rotating) frame
  such as the ICRS.

Start at J2000
approx. ecliptic
This plot is similar to those previously obtained
by Fukushima and Guinot from analytic
developments. Note that, unlike the equinox, the
CEO does not return to the same point on the sky
after each precession cycle.
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Earth Orientation

• These plots show the difference in Earth
  orientation computed using the integrated CEO
  scheme (eq. 2) and that computed using a
  conventional equinox-based sidereal-time method.
  For sidereal time, the new IERS formula was used.
  The difference is only a few micro- arcseconds
  within two centuries of 2000.

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Earth Orientation

• These plots show the same difference as on the
  previous slide, but only IERS algorithms and
  parameters were used.
• The overall systematic pattern is obviously the
  same as when using the integrated CEO scheme.

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Apparent Sidereal Time

• This plot shows the difference between apparent
  sidereal time computed from the integrated CEO
  scheme, eq. (4), and that computed using a
  conventional sidereal-time formula. The new
  IERS formula for sidereal time was used as the
  conventional formula. Units are
  microarcseconds.

The systematic pattern is the same as that seen
previously in earth orientation RA, as one would
expect.
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Earth Orientation Revisited

• These plots show the difference in Earth
  orientation computed using the integrated CEO
  scheme (eq. 2) and that computed using a
  conventional equinox-based sidereal-time method.
  However, for this comparison, the new Capitaine
  et al. (2003) formulas for precession and
  sidereal time were used. These appear to be more
  self-consistent than the IERS formulas.

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An Interesting Corollary

• V. Slabinski (2002, private communication)
  derived the NRO equation of motion (eq. 1)
  independently from the first definition of the
  NRO. He also showed that the equation implies
  that all NROs on a common equator must maintain
  constant arc distances between them. This makes
  sense, because we require the same rotation
  measurement to be obtained for all NROs.

The above plot shows how well the integrator
satisfies this condition. Four NROs were
integrated independently, starting at year 1700
at RA0, 1?, 95?, and ?160?, respectively. The
six curves show how well the pairwise
combinations maintain constant arc separations.
No combination exceeds 0.5 µas over 6 centuries.
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Conclusions

• It is possible to provide a description of the
  non-rotating origin (NRO) concept that is both
  intuitively simple and mathematically precise
• might be attractive to the broader astronomical
  community
• The development is based on a numerical
  integration of an NRO position as a function of
  time, using an equation of motion that is simple
  to derive and elegant
• Applied to the celestial system (ICRS), this
  scheme directly yields the RA and Dec of the CEO
  as a function of time
• Terrestrial-celestial frame transformations and
  related algorithms become almost trivial
• The results are good ? the numerical precision
  seems to be at least as good as that from the
  IERS algorithms

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This poster is available on the web
at http//aa.usno.navy.mil/kaplan/NROs.pps
or http//aa.usno.navy.mil/kaplan/NROs.pdf The
text version (abridged content) submitted to the
IAU Joint Discussion 16 proceedings is at
http//aa.usno.navy.mil/kaplan/NROsJD16proc.pd
f George Kaplan can be contacted at the U.S.
Naval Observatory at gkaplan_at_usno.navy.mil