Another Look at Non-Rotating Origins George Kaplan Astronomical Applications Department U.S. Naval Observatory

2003 JulyIAU JD16 & Div. 12 Another Look at Non-Rotating Origins Abstract 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…

2003 JulyIAU JD16 & Div. 13 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.

2003 JulyIAU JD16 & Div. 14 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”

2003 JulyIAU JD16 & Div. 15 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.

2003 JulyIAU JD16 & Div. 16 Motion of a non-rotating origin, , compared to that of the true equinox, . equator of t 3 ecliptic 11 11 equator of t 2 equator of t 1 22 33 22 33 RA Dec ICRF

2003 JulyIAU JD16 & Div. 17 O n0n0 n1n1  n n e0e0 e1e1 x0x0 ΔxΔx x1x1 11 x 1 = x 0 + Δx Δx = Δz n 0 where Δ z is a scalar to be determined n 0, n 1, x 0, x 1 are unit vectors Two equators, two poles, two NROs,  t apart  t  0 22

2003 JulyIAU JD16 & Div. 18 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 P t (t) N t (t) is the path of the pole in the ICRS

2003 JulyIAU JD16 & Div. 19 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 

2003 JulyIAU JD16 & Div. 110 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 (x p, y p ) as defined by observations and x(J2000) = 0.

2003 JulyIAU JD16 & Div. 111 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 r c = C R 3 (-  ) W r t where C = ( x y n ) = (W is the polar motion matrix and R 3 is a simple rotation about the z axis, which at that step is n(t).) continued... x 1 y 1 n 1 x 2 y 2 n 2 x 3 y 3 n 3 (2)

2003 JulyIAU JD16 & Div. 112 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... 

2003 JulyIAU JD16 & Div. 113 Transformations Made Easy (3 of 3) We know the position of the instantaneous (true) equinox in ICRS coordinates:  (t) = F P t (t) N t (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)

2003 JulyIAU JD16 & Div. 114 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.

2003 JulyIAU JD16 & Div. 115 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. 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. Start at J2000

2003 JulyIAU JD16 & Div. 116 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.

2003 JulyIAU JD16 & Div. 117 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.

2003 JulyIAU JD16 & Div. 118 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.

2003 JulyIAU JD16 & Div. 119 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.

2003 JulyIAU JD16 & Div. 120 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 RA=0, 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.

2003 JulyIAU JD16 & Div. 121 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

2003 JulyIAU JD16 & Div. 122 This poster is available on the web at or The text version (abridged content) submitted to the IAU Joint Discussion 16 proceedings is at George Kaplan can be contacted at the U.S. Naval Observatory at