USNO Circular 179 - U.S. Naval Observatory

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44 PRECESSION & NUTATION 2. A rotation around the new z-axis (the direction toward the ecliptic pole of J2000.0) by the angle −ψA, the amount of precession of the equator from J2000.0 to t. 3. A rotation around the new x-axis (the direction along the intersection of the mean equator of t with the ecliptic of J2000.0) by the angle −ωA, the obliquity of the mean equator of t with respect to the ecliptic of J2000.0. After the rotation, the fundamental plane is the mean equator of t. 4. A rotation around the new z-axis (the direction toward the mean celestial pole of t) by the angle χA, accounting for the precession of the ecliptic along the mean equator of t. After the rotation, the new x-axis is in the direction of the mean equinox of date. If we let S1 = sin (ɛ0) S2 = sin (−ψA) S3 = sin (−ωA) S4 = sin (χA) then the precession matrix can also be written: P(t) = ⎛ ⎜ ⎝ C1 = cos (ɛ0) C2 = cos (−ψA) (5.9) C3 = cos (−ωA) C4 = cos (χA) C4C2 − S2S4C3 C4S2C1 + S4C3C2C1 − S1S4S3 C4S2S1 + S4C3C2S1 + C1S4S3 −S4C2 − S2C4C3 −S4S2C1 + C4C3C2C1 − S1C4S3 −S4S2S1 + C4C3C2S1 + C1C4S3 S2S3 −S3C2C1 − S1C3 −S3C2S1 + C3C1 (5.10) Existing applications that use the 3-angle precession formulation of Newcomb and Lieske can be easily modified for the IAU 2000A precession, by replacing the current polynomials for the angles ζA, zA, and θA with the following: ζA = 2.650545 + 2306.083227 T + 0.2988499 T 2 + 0.01801828 T 3 − 0.000005971 T 4 − 0.0000003173 T 5 zA = −2.650545 + 2306.077181 T + 1.0927348 T 2 + 0.01826837 T 3 − 0.000028596 T 4 − 0.0000002904 T 5 θA = 2004.191903 T − 0.4294934 T 2 − 0.04182264 T 3 − 0.000007089 T 4 − 0.0000001274 T 5 (5.11) The 3-angle precession matrix is P(t) = R3(−zA) R2(θA) R3(−ζA), but any existing correct construction of P using these three angles can still be used. The expression for the mean obliquity of the ecliptic (the angle between the mean equator and ecliptic, or, equivalently, between the ecliptic pole and mean celestial pole of date) is: ɛ = ɛ0 − 46.836769 T − 0.0001831 T 2 + 0.00200340 T 3 − 0.000000576 T 4 − 0.0000000434 T 5 (5.12) where, as stated above, ɛ0 = 84381.406 arcseconds. This expression arises from the precession formulation but is actually used only for nutation. (Almost all of the obliquity rate — the term linear in T — is due to the precession of the ecliptic.) ⎞ ⎟ ⎠
PRECESSION & NUTATION 45 Where high accuracy is not required, the precession between two dates, t1 and t2, not too far apart (i.e., where |t2 − t1| ≪ 1 century), can be approximated using the rates of change of right ascension and declination with respect to the mean equator and equinox of date. These rates are respectively m ≈ 4612.16 + 2.78 T n ≈ 2004.19 − 0.86 T (5.13) where the values are in arcseconds per century and T is the number of centuries between J2000.0 and the midpoint of t1 and t2. If the dates are expressed as Julian dates, T = ((t1 + t2)/2 − 2451545.0)/36525. Then, denoting the celestial coordinates at t1 by (α1, δ1) and those at t2 by (α2, δ2), α2 ≈ α1 + τ (m + n sin α1 tan δ1) δ2 ≈ δ1 + τ (n cos α1) (5.14) where τ = t2 − t1, expressed in centuries. These formulas deteriorate in accuracy at high (or low) declinations and should not be used at all for coordinates close to the celestial poles (how close depends on the accuracy requirement and the value of τ) . 5.4.2 Formulas for Nutation Nutation is conventionally expressed as two small angles, ∆ψ, the nutation in longitude, and ∆ɛ, the nutation in obliquity. These angles are measured in the ecliptic system of date, which is developed as part of the precession formulation. The angle ∆ψ is the small change in the position of the equinox along the ecliptic due to nutation, so the effect of nutation on the ecliptic coordinates of a fixed point in the sky is simply to add ∆ψ to its ecliptic longitude. The angle ∆ɛ is the small change in the obliquity of the ecliptic due to nutation. The true obliquity of date is ɛ ′ = ɛ + ∆ɛ. Nutation in obliquity reflects the orientation of the equator in space and does not affect the ecliptic coordinates of a fixed point on the sky. The angles ∆ψ and ∆ɛ can also be thought of as small shifts in the position of the celestial pole (CIP) with respect to the ecliptic and mean equinox of date. In that coordinate system, and assuming positive values for ∆ψ and ∆ɛ, the nutation in longitude shifts the celestial pole westward on the sky by the angle ∆ψ sin ɛ, decreasing the pole’s mean ecliptic longitude by ∆ψ. Nutation in obliquity moves the celestial pole further from the ecliptic pole, i.e., southward in ecliptic coordinates, by ∆ɛ. (Negative values of ∆ψ and ∆ɛ move the pole eastward and northward in ecliptic coordinates.) The effect of nutation on the equatorial coordinates (α,δ) of a fixed point in the sky is more complex and is best dealt with through the action of the nutation matrix, N(t), on the equatorial position vector, r mean(t). Where high accuracy is not required, formulas that directly give the changes to α and δ as a function of ∆ψ and ∆ɛ are available and are given at the end of this subsection. The values of ∆ψ and ∆ɛ are obtained by evaluating rather lengthy trigonometric series, of the general form ∆ψ = N i=1 (Si + ˙ SiT ) sin Φi + C ′ i cos Φi
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UNITED STATES NAVAL OBSERVATORY CIR
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ii 3.4.4 Relationship to Other Syst
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iv INTRODUCTION recommendations. Ma
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Page 10 and 11: A Few Words about Constants This ci
Page 12 and 13: Abbreviations and Symbols Frequentl
Page 14 and 15: xii ABBREVIATIONS & SYMBOLS s CIO l
Page 16 and 17: 2 RELATIVITY In 1991, the IAU made
Page 18 and 19: 4 RELATIVITY Geocentric Celestial R
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Page 30 and 31: 16 TIME SCALES method and the time
Page 32 and 33: 18 TIME SCALES through the local ge
Page 34 and 35: 20 CELESTIAL REFERENCE SYSTEM 3.1 T
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Page 44 and 45: 30 EPHEMERIDES DE200 nor DE405 have
Page 46 and 47: 32 EPHEMERIDES scaling. LB = 1.550
Page 48 and 49: 34 PRECESSION & NUTATION 5.1 Aspect
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Page 98 and 99: 84 REFERENCES Høg, E., Fabricius,
Page 100 and 101: 86 REFERENCES Nelson, R. A., McCart
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