Introduction
The position, speed and trajectory of an object in space need to be described in a certain coordinate system. Depending on the selected coordinate system, the difficulty of describing the above motion states will also vary greatly. The coordinate system is defined theoretically by a series of principles and regulations, and its specific implementation is called a coordinate frame or reference frame. It should be noted that when the focus of our discussion is not on the issue of theoretical regulation or concrete realization, sometimes they are not strictly distinguished. In order to have a deep understanding of various celestial coordinate systems, it is first necessary to introduce the basic concepts of precession and nutation.
1. Celestial coordinate system
1.1 precession
The precession of the vernal equinox (an intersection of the celestial equator and ecliptic) due to the long-term motion of the celestial equator and celestial ecliptic is called precession. Among them, the precession caused by the long-term movement of the celestial equator is called equatorial precession; the precession caused by the long-term movement of the celestial ecliptic is called ecliptic precession. The equatorial precession was originally called the diurnal and lunar precession, and the ecliptic precession was called the planetary precession. Since the terms "solar and lunar precession" and "planetary precession" that have been used for more than one hundred years are inaccurate and prone to misunderstanding, the 26th IAU General Assembly decided to adopt Fukushima's suggestion and rename them as equatorial precession and ecliptic precession respectively .
(1) Equatorial precession
The precession of the celestial equator is caused by the torque of the sun, moon, and planets on the equatorial bulge on the earth, and finally the phenomenon that the vernal equinox moves westward on the ecliptic by about 50.39" every year is called equatorial precession. (
2 ) Ecliptic precession
Due to the gravitational force of the planets, the center of mass of the earth-moon system revolves around the sun (ecliptic plane) changes, thus causing the spring equinox to move eastward about 0. 1 ′ ′ 0.1^{\prime \prime} every year on the celestial equator0.1'' The phenomenon is called ecliptic precession. The change of the plane of the ecliptic will also reduce the angle between the yellow and the red by about0.4 7 ′ ′ 0.47^{\prime \prime}0.47'' .
(3) Total precession and precession model
The sum of the components of equatorial precession and ecliptic precession on the ecliptic is called total precession. In other words, under the joint action of equatorial precession and ecliptic precession, the celestial longitude will change, and the amount of change islll equivalent:
l = Ψ ′ − λ ′ cos ε l=\Psi^{\prime}-\lambda^{\prime} \cos \varepsilonl=Ps′−l′cosε
式中,Ψ ′ \Psi^{\prime}Ps′ is the amount of the vernal equinox moving westward on the ecliptic due to the equatorial precession;λ ′ \lambda^{\prime}l′ is the amount of spring equinox moving eastward on the equator due to ecliptic precession;ε \varepsilonε is the intersection angle between yellow and red.
So far, several precession models have been successively established in the world, such as the IAU 1976 precession model (L77 model), the IAU 2000 precession model, the IAU 2006 precession model (P03) model, and the B03 model established by Bretagnon et al. F03 model established by Fukushima, etc. At the 26th IAU General Assembly in 2006, it was decided to officially adopt the IAU 2006 precession model from January 1, 2009.
(4) Precession correction
If we use the following method to form an instantaneous celestial coordinate system: take the center of the celestial sphere as the coordinate origin,XXThe X- axis points to the instantaneous equinox,ZZThe Z axis points to the instantaneous flat north celestial pole,YYY axis is perpendicular toXXX axis andZZThe Z axis forms a right-handed rectangular coordinate system, so due to precession, the three coordinate axes of these instantaneous celestial coordinate systems will have different orientations. A fixed object in space, such as a star without proper motion, has different coordinates in different instantaneous celestial coordinate systems, and cannot be compared with each other. To this end, we need to choose a fixed celestial coordinate system as a reference, and use different observation timesti t_itiThe measured celestial coordinates are all reduced to the fixed celestial coordinate system for mutual comparison and ephemeris of celestial bodies. This fixed celestial coordinate system is called the protocol celestial coordinate system. At present, we choose the flat celestial coordinate system at J2000.0 as the protocol celestial coordinate system. O − γ 0 y 0 p 0 O-\gamma_0 y_0 p_0 in the figureO−c0y0p0It is the protocol celestial coordinate system, where XXThe X- axis points to the equinox at J2000.0γ 0 , Z \gamma_0, Zc0,The Z axis points toJ 2000.0 \mathrm{J} 2000.0The flat north celestial pole p 0 , Y p_0, Yat J 2000.0p0,Y axis is perpendicular toX , ZX , ZThe X and Z axes form a right-handed coordinate system. Schematic diagram of precession correction:
For any moment ti t_itiThere are many ways to reduce the observed value of the celestial coordinate system to the protocol, and the simplest method is to use the method of coordinate system rotation. It can be seen from the figure that to put ti t_itiInstantaneous celestial coordinate system O − γ yp O-\gamma ypO−γ y p transforms tot 0 t_0t0The protocol celestial coordinate system at time O − γ 0 y 0 P 0 O-\gamma_0 y_0 P_0O−c0y0P0, only three coordinate rotations are required. The first is around ZZZ axis rotationζ \zetaζ angle, so thatXXX axis fromγ \gammaγ points toBBB ; followed by aroundYYY- axis rotationθ \thetaθ angle, so thatZZZ axis fromO p O_pOpConvert to O p 0 , XO p_0, XOp0,X axis fromBBB turns to point toAAA ; wind aroundZZZ axis rotationη 0 \eta_0the0Angle, make xxX axis fromAAA turns to point toγ 0 \gamma_0c0. F is there:
( XYZ ) t 0 = RZ ( η 0 ) RY ( − θ ) RZ ( ζ ) ( XYZ ) ti = ( cos η 0 sin η 0 0 − sin η 0 cos η 0 0 0 0 1 ) ( cos θ 0 sin θ 0 1 0 − sin θ 0 cos θ ) ( cos ζ sin ζ 0 − sin ζ cos ζ 0 0 0 1 ) ( XYZ ) ti = ( p 11 p 12 p 13 p 21 p 22 p 23 p 31 p 32 p 33 ) ( XYZ ) L i X = [ p ] ( YYZ ) is \begin{aligned} &\left(\begin{array}{l} X \\ Y \ \Z\end{array}\right)_{t_0}=\ballsymbol{R}_Z\left(\eta_0\right) \ballsymbol{R}_Y(-\theta)\ballsymbol{R}_Z(\zeta) \left(\begin{array}{l}X\\Y\\Z\end{array}\right)_{t_i}\\&=\left(\begin{array}{ccc}\cos \eta_0& \without \eta_0&0\\ -\without\eta_0&\cos \eta_0&0\\0&0&1 \end{array}\right)\left(\begin{array}{ccc}\cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 &\cos \theta \end{array}\right)\left(\begin{array}{ccc} \cos \zeta & \sin \zeta & 0 \\ -\sin \zeta & \cos \zeta & 0 \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{l} X \\ Y \\ Z \end{array}\right)_{t_i} \\ &=\left(\begin{array}{lll} p_{11} & p_{12} & p_{13} \\ p_{21} & p_{22} & p_{23} \\ p_{31} & p_{32} & p_{33} \end{array}\right)\left(\begin{array}{l} X \\ Y \\ Z \end{array}\right)_{L_i}^X=[p]\left(\begin{array}{c} Y \\ Y \\ Z \end{array}\right)_{t_i} \end{aligned}p_{33} \end{array}\right)\left(\begin{array}{l} X \\ Y \\ Z \end{array}\right)_{L_i}^X=[p]\left(\begin{array}{c} Y \\ Y \\ Z \end{array}\right)_{t_i} \end{aligned}p_{33} \end{array}\right)\left(\begin{array}{l} X \\ Y \\ Z \end{array}\right)_{L_i}^X=[p]\left(\begin{array}{c} Y \\ Y \\ Z \end{array}\right)_{t_i} \end{aligned}⎝
⎛XYZ⎠
⎞t0=RZ( h0)RY( − θ ) RZ( g )⎝
⎛XYZ⎠
⎞ti=⎝
⎛costhe0−sinthe00sinthe0costhe00001⎠
⎞⎝
⎛cosi0−sini010sini0cosi⎠
⎞⎝
⎛cosg−sing0singcosg0001⎠
⎞⎝
⎛XYZ⎠
⎞ti=⎝
⎛p11p21p31p12p22p32p13p23p33⎠
⎞⎝
⎛XYZ⎠
⎞LiX=[p]⎝
⎛YYZ⎠
⎞ti
where, [ p ] [\boldsymbol{p}][ p ] is called the precession matrix, and its nine elements are:
{ p 11 = cos η 0 cos θ cos ζ − sin η 0 sin ζ p 12 = cos η 0 cos θ sin ζ + sin η 0 cos ζ p 13 = cos η sin θ p 21 = − sin η 0 cos θ cos ζ − cos η 0 sin ζ p 22 = − sin η 0 cos θ sin ζ + cos η 0 cos ζ p 23 = − sin η 0 sin θ p 31 = − sin θ cos ζ p 32 = − sin θ sin ζ p 33 = cos θ \left\{\begin{array}{l} p_{11} =\cos \eta_0 \cos \theta \cos \zeta-\sin \eta_0 \sin \zeta \\ p_{12}=\cos \eta_0 \cos \theta \sin \zeta+\sin \zeta_0 \cos \zeta \ \ p_{13}=\cos \eta_0 \sin \theta \\ p_{21}=-\sin \eta_0 \cos \theta \cos \zeta-\cos \eta_0 \sin \zeta \\ p_{22}= -\sin \eta_0 \cos \theta \sin \zeta+\cos \eta_0 \cos \zeta \\ p_{23}=-\sin \eta_0 \sin \theta \\ p_{31}=-\sin \theta\ cos \zeta \\ p_{32}=-\sin \theta \sin \zeta \\p_{33}=\cos \theta \end{array}\right.⎩
⎨
⎧p11=costhe0cosicosg−sinthe0singp12=costhe0cosising+sinthe0cosgp13=costhe0sinip21=−sinthe0cosicosg−costhe0singp22=−sinthe0cosising+costhe0cosgp23=−sinthe0sinip31=−sinicosgp32=−sinisingp33=cosi
Conversely, transform from the protocol celestial coordinate system to any time ti t_itiFor the quantitative equations, the equivalent equation:
( XYZ ) yi = [ p ] − 1 ( XYZ ) t 0 [ p ] − 1 = RZ ( − ζ ) R γ ( θ ) RZ ( − η 0 ) = ( . p 11 ′ p 12 ′ p 13 ′ p 21 ′ p 22 ′ p 23 ′ p 31 ′ p 32 ′ p 33 ′ ) \begin{gathered} \left(\begin{array}{c}X\\Y\\ Z\end{array}\right)_{y_i}=[\bold symbol{p}]^{-1}\left(\begin{array}{l}X\\Y\\Z\end{array}\ right)_{t_0} \\ {[\ballsymbol{p}]^{-1}=\ballsymbol{R}_Z(-\zeta) \ballsymbol{R}_\gamma(\theta) \ballsymbol{R} _Z\left(-\eta_0\right)=\left(\begin {array}{lll} p_{11}^{\prime} & p_{12}^{\prime} & p_{13}^{\prime } \\ p_{21}^{\prime} & p_{22}^{\prime} & p_{23}^{\prime} \\ p_{31}^{\prime} & p_{32}^{ \prime} & p_{33}^{\prime}\end{array}\right)}\end{gathered}⎝
⎛XYZ⎠
⎞yi=[p]−1⎝
⎛XYZ⎠
⎞t0[p]−1=RZ( − g ) Rc( i ) RZ( − h0)=⎝
⎛p11′p21′p31′p12′p22′p32′p13′p23′p33′⎠
⎞
In the formula,
{ p 11 ′ = cos η 0 cos θ cos ζ − sin η 0 sin ζ p 12 ′ = − sin η 0 cos θ cos ζ − cos η 0 sin ζ p 13 ′ = − sin θ cos ζ p 21 ′ = cos η 0 cos θ sin ζ + sin η 0 cos ζ p 22 ′ = − sin η 0 cos θ sin ζ + cos η 0 cos ζ p 23 ′ = − sin θ sin ζ p 31 ′ = cos η 0 sin θ p 32 ′ = − sin η 0 sin θ p 33 ′ = cos θ \left\{\begin{ array}{l} p_{11}^{\prime}=\cos \theta_0 \cos \theta \cos \zeta-\sin \eta_0 \sin \zeta \\ p_{12}^{\prime}=-\ sin \theta_0 \cos \theta \cos \zeta-\cos \eta_0 \sin \zeta \\p_{13}^{\prime}=-\sin \theta \cos \zeta \\p_{21}^{\ prime}=\cos \theta_0 \cos \theta \sin \zeta+\sin \zeta_0 \cos \zeta \\ p_{22}^{\prime}=-\sin \eta_0 \cos \theta \sin \zeta+\cos \eta_0 \cos \zeta \\ p_{23}^{\prime}=-\sin \theta \sin \zeta \\ p_{31}^{\prime}=\cos \eta_0 \sin \theta \\ p_ {32}^{\prime}=-\sin \theta_0 \sin \theta\\ p_{33}^{\prime}=\cos \theta \end{array}\right.⎩
⎨
⎧p11′=costhe0cosicosg−sinthe0singp12′=−sinthe0cosicosg−costhe0singp13′=−sinicosgp21′=costhe0cosising+sinthe0cosgp22′=−sinthe0cosising+costhe0cosgp23′=−sinisingp31′=costhe0sinip32′=−sinthe0sinip33′=cosi
岁差 parameterη 0、ζ、θ \eta_0、\zeta、\thetathe0, ζ , θ can be obtained by the precession model. The calculation formula given by the IAU 2006 precession model is as follows:
η 0 = 2.65054 5 ′ ′ + 2306.08322 7 ′ ′ t + 0.298849 9 ′ ′ t 2 + 0.01801828 t 3 − 5.97 1 ′ ′ × 1 0 − 6 t 4 − 3.17 3 ′ ′ × 1 0 − 7 t 5 ζ = − 2.65054 5 ′ ′ + 2306.07718 1 ′ ′ t + 1.0927348 t 2 + 0.01826837 t 3 + 2.859 6 ′ ′ × 1 0 − 5 t 4 − 2.90 4 ′ ′ × 1 0 − 7 t 5 θ = 2004.19190 3 ′ ′ t − 0.429493 4 ′ ′ t 2 − 0.0418226 4 ′ ′ t 3 − 7.08 9 ′ ′ × 1 0 − 6 t 4 − 1.27 4 ′ ′ × 1 0 − 7 t 5 \begin{aligned} \eta_0=& 2.650545^{\prime \prime}+2306.083227^{\prime \prime} t+0.2988499^{\prime \prime} t^2+0.01801828 t^3 \\ &-5.971^{\prime \prime} \times 10^{-6} t^4-3.173^{\prime \prime} \times 10^{-7} t^5 \\ \zeta=&-2.650545^{\prime \prime}+2306.077181^{\prime \prime} t+1.0927348 t^2+0.01826837 t^3 \\ &+2.8596^{\prime \prime} \times 10^{-5} t^4-2.904^{\prime \prime} \times 10^{-7} t^5 \\ \theta=& 2004.191903^{\prime \prime} t-0.4294934^{\prime \prime} t^2-0.04182264^{\prime \prime} t^3-7.089^{\prime \prime} \times 10^{-6} t^4 \\ &-1.274^{\prime \prime} \times 10^{-7} t^5 \end{aligned} the0=g=i=2.650545′′+2306.083227′′t+0.2988499′′t2+0.01801828t3−5.971′′×10- 6 t4−3.173′′×10− 7 t5−2.650545′′+2306.077181′′t+1.0927348t _2+0.01826837t _3+2.8596′′×10−5t4−2.904′′×10− 7 t52004.191903′′t−0.4294934′′t2−0.04182264′′t3−7.089′′×10- 6 t4−1.274′′×10− 7 t5
where, ttt is J 2000.0 \mathrm{J} 2000.0from the reference momentJ 2000.0 Julian century number. Theoretically, TDB time should be used when calculating multidifference, but in practice TT time is always used, because the maximum difference between the two time systems is only1.7 ms 1.7 \mathrm{~ms}1.7 ms , the influence on precession is only2. 7 ′ ′ × 1 0 − 9 2.7^{\prime \prime} \times 10^{-9}2.7′′×10− 9 , which can be ignored.
1.2 Nutation
(1) The basic concept of nutation
Since the relative positions of the moon, the sun and the major planets and the earth have periodic changes, the moment acting on the equatorial bulge of the earth is also changing, and the mass center of the earth-moon system revolves around the sun. Periodic perturbations also exist on the orbital surface, therefore, on the basis of precession, there are various small periodic changes of different sizes and periods—nutation. The most important one is the amplitude of 9. 2 ′ ′ 9.2^{\prime \prime}9.2′′ (angular nutation), the period is18.6 18.618.6 -year period term. This is because the orbital plane of the moon's revolution around the earth---the intersecting angle between the white equator plane and the earth's equator plane will be18.6 18.6The cycle of 18.6 years is at1 8 ∘ 1 7 ′ 18^{\circ} 17^{\prime}18∘17′ to2 8 ∘ 3 5 ′ 28^{\circ} 35^{\prime}28∘35' caused by the back and forth changes.
(2) Nutation model
Many nutation models have been established so far, such as the IAU 1980 nutation model and the IAU 2000 nutation model. Currently, the IAU 2000 nutation model is widely used. This model is composed of two parts: daily and lunar nutation and planetary nutation. The daily and lunar nutation is composed of 678 periodic items with different amplitudes and different periods. , while the planetary nutation is composed of 687 periodic items with different amplitudes and different periods.
- 日、月章动
{ Δ Ψ = ∑ i = 1 678 [ ( A i + A i ′ t ) sin f i + ( A i ′ ′ + A i ′ ′ ′ t ) cos f i ] Δ ε = ∑ i = 1 678 [ ( B i + B i ′ t ) cos f i + ( B i ′ ′ + B i ′ ′ ′ t ) sin f i ] \left\{\begin{array}{l} \Delta \Psi=\sum_{i=1}^{678}\left[\left(A_i+A_i^{\prime} t\right) \sin f_i+\left(A_i^{\prime \prime}+A_i^{\prime \prime \prime} t\right) \cos f_i\right] \\ \Delta \varepsilon=\sum_{i=1}^{678}\left[\left(B_i+B_i^{\prime} t\right) \cos f_i+\left(B_i^{\prime \prime}+B_i^{\prime \prime \prime} t\right) \sin f_i\right] \end{array}\right. { DPS=∑i=1678[(Ai+Ai′t)sinfi+(Ai′′+Ai′′′t)cosfi]D e=∑i=1678[(Bi+Bi′t)cosfi+(Bi′′+Bi′′′t)sinfi]
式中, Δ Ψ \Delta \PsiΔΨ is the nutation of the celestial meridian, which is the change of the celestial meridian due to the nutation;Δ ε \Delta \varepsilonΔ ε is the angle nutation, which is the common eccentric angle ε \varepsiloncaused by the nutationε -variation;Ai、Ai′、Ai′′、Ai′′′ 以及 B i 、 B i ′ 、 B i ′ ′ 、 B i ′ ′ B_i 、 B_i^{\prime} 、 B_i^{\prime \prime} 、 B_i^{\prime \prime} Bi、Bi′、Bi′′、Bi′′given by the table; ttt is J 2000.0 \mathrm{J} 2000.0from the reference momentJ 2000.0 Julian century number.
fi = N 1 I + N 2 I ′ + N 3 F + N 4 D + N 5 Ω f_i=N_1 I+N_2 I^{\prime}+N_3 F+N_4 D+N_5 \Omegafi=N1I+N2I′+N3F+N4D+N5
In the Ω formula, N 1 , N 2 , N 3 , N 4 , N 5 N_1, N_2, N_3, N_4, N_5N1,N2,N3,N4,N5The value of is also given by the table; I , I ′ , F , D , Ω I, I^{\prime}, F, D, \OmegaI,I′,F,D,Ω is some parameters related to the positions of the sun and the moon, and there are fixed calculation formulas for calculation. - 行星章动
{ Δ Ψ = ∑ i = 1 687 ( A i sin f i + A i ′ cos f i ) Δ ε = ∑ i = 1 667 ( B i cos f i + B i ′ sin f i ) \left\{\begin{array}{l}\Delta \Psi=\sum_{i=1}^{687}\left(A_i \sin f_i+A_i^{\prime} \cos f_i\right) \\ \Delta \varepsilon=\sum_{i=1}^{667}\left(B_i \cos f_i+B_i^{\prime} \sin f_i\right)\end{array}\right. { DPS=∑i=1687(Aisinfi+Ai′cosfi)D e=∑i=1667(Bicosfi+Bi′sinfi)It is a parameter related to the position of the major planets, which is calculated by a fixed formula.
The accuracy of the above nutation model is better than 0.2 mas 0.2 \mathrm{mas}0.2 mas , for precision requirement only1 mas 1 \mathrm{mas}For users of 1 mas , you can use the simplified formula to calculate, the precise model is called IAU2000 A 2000 \mathrm{~A}2000 A nutation model, and the simplified model is called the IAU 2000B model. inBBIn model B , there are only 77 diurnal and lunar nutation items and 1 planetary nutation deviation item. For GPS satellites,1 mas 1 \mathrm{mas}1 mas will cause about13 cm 13 \mathrm{~cm}13 cm satellite position error.
(3) The infinitesimal equivalent of the equation
is given by:
[ N ] = R γ ( − ε − Δ ε ) ⋅ RZ ( − Δ ψ ) ⋅ R x ( ε ) = ( n 11). n 12 n 13 n 21 n 22 n 23 n 31 n 32 n 33 ) [N]=\ballsymbol{R}_\gamma(-\varepsilon-\Delta \varepsilon) \cdot \ballsymbol{R}_Z(-\; Delta \psi) \cdot \ball symbol{R}_x(\itempsilon)=\left(\begin{array}{lll}n_{11} & n_{12} & n_{13}\\n_{21} & n_ {22} &n_{23}\\n_{31}&n_{32}&n_{33}\end{array}\right)[N]=Rc( − e−D e )⋅RZ( − Δ ψ )⋅Rx( e )=⎝
⎛n11n21n31n12n22n32n13n23n33⎠
⎞
In the formula,
{ n 11 = cos Δ ψ n 12 = − sin Δ ψ cos ε n 13 = − sin Δ ψ sin ε n 21 = sin Δ ψ cos ( ε + Δ ε ) n 22 = cos Δ ψ cos ε cos ( ε + Δ ε ) + sin ε sin ( ε + Δ ε ) n 23 = cos Δ ψ sin ε cos ( ε + Δ ε ) − cos ε sin ( ε + Δ ε ) n 31 = sin Δ ψ sin ( ε + Δ ε ) n 32 = cos Δ ψ cos ε sin ( ε + Δ ε ) − sin ε cos ( ε + Δ ε ) n 33 = cos Δ ψ sin ε sin ( ε + Δ ε ) + cos ε cos ( ε + Δ ε ) \left\{\begin{array}{l}n_{11}=\cos \ Delta \psi \\ n_{12}=-\sin \Delta \psi \cos \varepsilon \\ n_{13}=-\sin \Delta \psi \sin \varepsilon \\ n_{21}=\sin \Delta \psi \cos (\varepsilon+\Delta \varepsilon) \\ n_{22}=\cos \Delta \psi \cos \varepsilon \cos (\varepsilon+\Delta \varepsilon)+\sin \varepsilon \sin (\varepsilon+\ Delta \varepsilon) \\ n_{23}=\cos \Delta \psi \sin \varepsilon \cos (\varepsilon+\Delta \varepsilon)-\cos\varepsilon \sin (\varepsilon+\Delta \varepsilon) \\ n_{31}=\sin \Delta \psi \sin (\varepsilon+\Delta \varepsilon) \\ n_{32}=\cos \Delta \psi \cos \varepsilon \sin (\varepsilon+\Delta \varepsilon)-\sin \varepsilon \cos (\varepsilon+\Delta \varepsilon) \\ n_{33}=\cos \Delta \psi \sin \varepsilon \sin (\varepsilon+\ Delta \varepsilon)+\cos \varepsilon \cos (\varepsilon+\Delta \varepsilon)\end{array}\right.⎩
⎨
⎧n11=cosD psn12=−sinD pscosen13=−sinD pssinen21=sinD pscos ( e+D e )n22=cosD pscosecos ( e+D e )+sinesin ( e+D e )n23=cosD pssinecos ( e+D e )−cosesin ( e+D e )n31=sinD pssin ( e+D e )n32=cosD pscosesin ( e+D e )−sinecos ( e+D e )n33=cosD pssinesin ( e+D e )+cosecos ( e+D e ).
1.3 Celestial coordinate system
The celestial coordinate system is a coordinate system used to describe the position or direction of natural and artificial celestial bodies in space. According to the different origins of the selected coordinates, it can be divided into the station-centered celestial coordinate system, the earth-centered celestial coordinate system and the solar system barycentric celestial coordinate system. In classical astronomy, since the distance to most natural celestial bodies cannot be accurately measured, but only their directions can be accurately measured, the celestial bodies are always projected onto the celestial sphere, and then the two spherical angles ( θ , λ ) are used to (\theta, \lambda)( i ,λ ) to represent its direction. But for the satellite and its distance can often be measured accurately, so the spherical coordinates( θ , λ , r ) (\theta, \lambda, r)( i ,l ,r ) to represent its position, and space Cartesian coordinates can also be used to( X , Y , Z ) (X, Y, Z)(X,Y,Z ) indicates its position. The geocentric celestial equatorial coordinate system is widely used in GPS measurement, and the origin of this coordinate system is at the center of mass of the earth,XXX- axis points to the equinox,ZZZ axis coincides with the earth's rotation axis, pointing to the north celestial pole,YYY axis is perpendicular toXXX axis andZZThe Z axis forms a right-handed Cartesian coordinate system. Due to the existence of precession and nutation, the north celestial pole and vernal equinox also have "true" and "flat". We call the North Celestial Pole and Vernal Equinox when only the precession is taken into account but not the nutation, and we call the North Celestial Pole and the Vernal Equinox which take into account both the Lunar Aberration and Nutation and can reflect their true positions as the True North Celestial Pole and True North Celestial Pole. Equinox.
(1) True geocentric celestial equatorial coordinate system (instantaneous geocentric celestial equatorial coordinate system)
we put the coordinate origin at the center of the earth, XXX- axis points to true equinox,ZZZ axis points to true north celestial pole,YYY axis is perpendicular toXXX axis andZZThe right-handed coordinate system composed of the Z axis is called the true geocentric celestial equatorial coordinate system or the instantaneous geocentric celestial equatorial coordinate system. Astronomical observations are always carried out in the true celestial coordinate system, and the obtained observation values also belong to this coordinate system. However, due to the influence of age and nutation, the orientation of the three coordinate axes in the true celestial coordinate system is constantly changing. The celestial body coordinates ( α , δ ) (\alpha, \delta)obtained after observing a fixed celestial body (such as a star without proper motion) at different times( a ,δ ) are not the same, so it is not suitable to use this coordinate system to compile star catalogs, indicating the position and direction of celestial bodies.
(2) Flat geocentric celestial equatorial coordinate system
We put the origin of the coordinates at the center of the earth, XXThe X- axis points to the equinox,ZZZ axis points to the flat north celestial pole,YYY axis is perpendicular toXXX axis andZZThe right-handed coordinate system composed of the Z axis is called the flat earth-centered celestial equatorial coordinate system. Of course, in fact, precession and nutation are superimposed together. The reason why we artificially separate the long-term average motion (precession) from many small periodic changes (nutation) on this basis is that In order to make the concept and steps of coordinate transformation clearer. They can also be combined and calculated at the same time during calculation.
The directions of the three coordinate axes of the flat-celestial spherical coordinate system are still not fixed, but their changing law is very simple, and can be calculated conveniently. Obviously, we should not use the flat celestial coordinate system to describe the position and direction of celestial bodies.
(3) Agreement on the geocentric celestial equatorial coordinate
system The position of celestial bodies needs to be described in a fixed coordinate system. Theoretically speaking, this kind of sky-fixed coordinate system can be chosen arbitrarily, as long as the direction of the coordinate axis remains unchanged. However, in order to avoid each country going its own way, in fact, it is always stipulated by the international authoritative unit through consultation and used uniformly . The currently widely used protocol celestial coordinate system is the international celestial coordinate system GCRS and BCRS stipulated by the IAU. The origin of the coordinates of the former is located at the center of the earth, which is used to calculate satellite orbits and compile satellite ephemeris; the origin of coordinates of the latter is located at the barycenter of the solar system. It is used to calculate the orbits of planets and compile star catalogs. XX of the international celestial coordinate systemX axis points toJ 2000.0 J 2000.0J2000.0 (JD = 2451545.0 ) =2451545.0) =Equinox at 2451545.0 ) , ZZWhen the Z axis points to the flat north celestial pole at J2000.0,YYY axis is perpendicular toXXX axis andZZThe Z axis forms a right-handed coordinate system. Obviously, this is only a theoretical regulation and definition, and the concrete realization of the international celestial coordinate system is called the international celestial reference frame. The international celestial reference frame is realized through the directions of a group of extragalactic radio sources determined by the International Earth Rotation and Reference Frame Service IERS using VLBI observations.
The three coordinate axes in GCRS point to three fixed directions in space. Although the origin of coordinates is revolving around the sun, it is still a fairly good inertial coordinate system. We usually call it a quasi-inertial coordinate system. The orbital motion equation of GPS satellites is usually established and solved in GCRS, and then transformed into ITRS through coordinate conversion.
2. Earth coordinate system
The earth coordinate system is also called the earth coordinate system. Since this coordinate system will rotate with the earth, it is also called a ground-fixed coordinate system. The main task of the earth coordinate system is to describe the position of objects on the earth or in near-Earth space.
According to the position of the origin of coordinates, the earth coordinate system can be divided into a reference coordinate system and a geocentric coordinate system. The coordinates directly obtained in GPS measurement are generally geocentric coordinates, and space geodetic coordinates B, L, HB, L, H can be used.Expressed in the form of B , L , H , or space Cartesian coordinates X, Y, ZX, Y, ZX , Y , Z form to represent. In order to clarify the earth coordinate system, it is necessary to first introduce the basic concept of pole shift.
2.1 Pole shift
The intersection of the Earth's axis of rotation and the ground is called the pole. Due to the movement of matter on the earth's surface (such as ocean currents, ocean tides, etc.) and the movement of matter inside the earth (such as the movement of the mantle), the position of the pole (strictly speaking, it should be the pole of the celestial almanac) will change. This phenomenon is called pole. shift. The position of the instantaneous pole at any moment can be expressed in a specific coordinate system by the coordinate components X p X_pXp和Y p Y_pYpTo represent, as shown in the figure below, the origin of the coordinate system is located at the origin of the international agreement CIO \mathrm{CIO}CIO (strictly speaking, the reference pole of IERS),XXX- axis is the starting meridian,YYThe Y axis is longitude27 0 ∘ 270^{\circ}270∘ 's meridian. Theoretically, the coordinate system is a spherical coordinate system, but due to the small value of the polar shift( < 1 ′ ) \left(<1^{\prime \prime}\right)(<1′′ ), so it can be regarded as a plane coordinate system. Currently, the pole shift value( X p , Y p ) \left(X_p, Y_p\right)(Xp,Yp) is accurately measured and published by IERS through VLBI, SLR, GPS, DORIS and other space geodetic methods (see Table 2-1). In newL 2 C \mathrm{L}_2 \mathrm{C}L2C andL 5 \mathrm{L}_5L5The corresponding short-term forecast value is given in the navigation message for users to use.
2.2 Instantaneous (true) earth coordinate system
For the convenience of application, when establishing the earth coordinate system, we need to make the coordinate axes be (or parallel) to some important points, lines, and planes on the earth. For example, let ZZThe Z axis is coincident with (or parallel to) the earth's rotation axis, so thatXXThe X- axis is located on the intersection line of the origin meridian plane and the equatorial plane (or parallel to this line), etc. However, due to the existence of pole shift, the directions of the three coordinate axes in the instantaneous earth coordinate system are constantly changing within the earth body, so the coordinates of fixed points on the ground will also change continuously. Obviously, the instantaneous earth coordinate system is not suitable for expressing point location. Instantaneous pole coordinates:
2.3 Protocol Earth Coordinate System
In order to keep the coordinates of fixed points on the ground fixed, it is necessary to establish a coordinate system that is completely fixed with the earth itself. Theoretically speaking, there are many ways to choose this coordinate system. Also, in order to prevent various coordinate systems from appearing, it still needs to be negotiated and uniformly stipulated by an international authoritative organization. This is the resolution of the International Earth Reference System ITRS IUGG, ITRS is defined by IERS, and its specific regulations are as follows:
(1) The origin of the coordinates is located at the center of mass of the entire earth including the ocean and atmosphere;
(2) The scale is the scale within the local earth frame in the sense of general relativity;
(3) The direction of the coordinate axis is determined by BIH 1984.0 1984.01984.0 ;
(4) The change of the orientation of the coordinate axis with time should satisfy the condition that "the crust has no overall rotation".
ITRS is implemented and maintained by IERS using space geodetic technologies such as VLBI, SLR, GPS, and DORIS. The specific implementation of ITRS is called the International Terrestrial Reference Frame ITRF. The coordinate frame usually adopts the spatial Cartesian coordinate system ( X , Y , Z ) (X, Y, Z)(X,Y,Z ) form to represent. If you need to use space geodetic coordinates( B , L , H ) (B, L, H)(B,L,H ) , it is recommended to use GRS 80 ellipsoid (a = 6378137.0 m , e 2 = 0.0069438003 ) \left.a=6378137.0 \mathrm{~m}, e^2=0.0069438003\right)a=6378137.0 m,e2=0.0069438003 ) . ITRF is a set of IERS station coordinates( X , Y , Z ) (X, Y, Z)(X,Y,Z ) , the annual rate of change of station coordinates( Δ X / (\Delta X /( Δ X /年,Δ Y / \Delta Y /ΔY/ 年, Δ Z / \Delta Z / ΔZ / year) and the corresponding earth orientation parameter EOP, ITRF is currently the most accurate earth reference frame recognized internationally . The precise ephemeris of the IGS is based on this framework.
With the increase in the number of stations, the improvement of observation accuracy, the accumulation of observation data, and the improvement of data processing methods, IERS is also constantly improving and perfecting the framework. So far, IERS has established and announced 12 different ITRF versions. These versions use ITRF yy { }_{yy}yyIn the form of , where yy \mathrm{yy}yy indicates the last year of the data used to build this version. For example, ITRF97 { }_{97}97Indicates that this version was established by IERS using various relevant materials obtained up to the end of 1997. Of course, the time of publication and use is after 1997. The 12 different ITRF versions are ITRF 88 , ITRF 89 ITRF_{88}, \mathrm{ITRF}_{89}ITRF88,ITRF89 、 I T R F 90 \mathrm{ITRF}_{90} ITRF90、 I T R F 91 \mathrm{ITRF}_{91} ITRF91 、 I T R F 92 \mathrm{ITRF}_{92} ITRF92 、 I T R F 93 \mathrm{ITRF}_{93} ITRF93、 I T R F 94 \mathrm{ITRF}_{94} ITRF94、 I T R F 96 \mathrm{ITRF}_{96} ITRF96 、 I T R F 97 \mathrm{ITRF}_{97} ITRF97、 I T R F 2006 \mathrm{ITRF}_{2006} ITRF2006、 I T R F 2005 \mathrm{ITRF}_{2005} ITRF2005and ITRF 2008 \mathrm{ITRF}_{2008}ITRF2008It is not difficult to see that before 1997, ITRF was updated almost every year. Afterwards, with the improvement of the accuracy of the framework, it gradually stabilized, and the update cycle of the version gradually increased.
2.4 World Geodetic Coordinate System
The world geodetic coordinate system is a global geocentric coordinate system established by the United States. It has successively launched different versions such as WGS60, WGS66, WGS72 and WGS84. Among them, WGS 84 replaced WGS 72 in 1987 and became the coordinate system used by the global positioning system broadcast ephemeris, and has been widely used by countries all over the world with the popularization of GPS navigation and positioning technology.
According to different angles and occasions of discussing issues, WGS 84 is sometimes regarded as a coordinate system and sometimes as a reference frame, unlike ITRS and ITRF, which can be clearly distinguished. As a coordinate system, WGS 84 also meets the four requirements proposed by IERS when establishing ITRS, that is to say, theoretically speaking, WGS 84 should be consistent with ITRS. But it's different from ITRF. WGS 84 uses space geodetic coordinates ( B , L , H ) (B, L, H) in many occasions(B,L,H ) form to indicate the position of a point, because ITRS and ITRF are mainly used in geodesy and geodynamics research fields, while WGS 84 is mostly used in navigation and positioning fields, users prefer to use( B , L , H ) (B, L, H)(B,L,H ) to represent the position of the point, at this time the WGS 84 ellipsoid (a = 6378137.0 m , f = 1 / 298.257223563 a=6378137.0 \mathrm{~m}, f = 1 / 298.257223563a=6378137.0 m,f=1/298.257223563 )。
In order to improve the accuracy of the WGS 84 framework, the U.S. Defense Mapping Agency (DMA) used the observation data of the Global Positioning System and the GPS satellite tracking stations of the U.S. Air Force, as well as the GPS observation data of some IGS stations to conduct a joint solution. During the calculation, the station coordinates of the IGS station in the ITRF frame were taken as fixed values, and the coordinates of other stations were recalculated, thus obtaining a more accurate WGS 84 frame. This improved frame is called WGS 84 (G730), where the G in brackets means that the frame is determined using GPS data, and 730 means that the frame is used from the 730th week of GPS time (that is, January 2, 1994). day). WGS 84(G730) and ITRF 92 \mathrm{ITRF}_{92}ITRF92The degree of agreement is up to 10 cm 10 \mathrm{~cm}10 cm level. Since then, the United States has refined the WGS84 framework twice, one in 1996, and the refined framework is called WGS84 (G873). This frame is used from week 873 of GPS time (September 29, 1996 0h). On October 1, 1996, the US Defense Mapping Agency DMA merged into the newly established US National Imagery and Mapping Agency NIMA (National Imagery and Mapping Agency). Since then, NIMA has used WGS 84 (G873) to calculate precise ephemeris. The systematic error between this ephemeris and the precise ephemeris of IGS (using ITRF 9 framework) is less than or equal to2 cm 2 2001 2 \mathrm{~cm}^2 20012 cm2 In 2001, the United States carried out the third refinement of WGS 84 and obtained the WGS 84 (G1150) framework. The frame is used from week 1150 of GPS time (January 20, 20020 h 0 \mathrm{~h}0 h ), 与 I T R R F 2000 ITRRF_{2000} ITRRF2000are in good agreement, the average difference in each component is less than 1 cm 2 1 \mathrm{~cm}^21 cm 2。
From: Chapter Two of "GPS Measurement and Data Processing".