Page 68 - 卫星导航2021年第1-2合期

Han et al. Satell Navig (2021) 2:18 Page 6 of 10

conception and defnition. Simply, the coordinate origin Here T ≡ TCG , called Geocentric Coordinate Time
of GCRS is defned at the mass center of the Earth, and (TCG), M E denotes the mass of the Earth, R is the radius
the coordinate axes near the Earth are orthogonal to each of the earth, P (cosθ) is Legendre expansion, S the
i
lm
E
other with no spatial rotation relative to the coordinate angular momentum of Earth rotation, a E the semi-major
axes of BCRS. Te coordinate axes of GCRS are essen- axis of the Earth’s equator, and (C lm , S lm ) the geocentric
tially defned by the coordinate relationship between gravitational potential coefcients.
GCRS and BCRS, which are only locally straight and IAU2006 Resolution B2 clearly stated that GCRS orien-
orthogonal. Terefore, if BCRS were viewed as straight tation is derived from ICRS-oriented BCRS. According to
line coordinates, GCRS would be curvilinear coordinates. the Eq. (4), if the geodesic precession is ignored, under the
According to IAU2000 Resolution B1.3, the spatial post-Newtonian approximation, the coordinate relation-
coordinate axes of GCRS are consistent with the spatial ship between GCRS and BCRS can be expressed as follows:
orientation of BCRS. Te metric of GCRS is required to
 �
take the same form as the barycentric one:  t ≡ T + (γ E − 1)dT + e X /c
j
0

 j



2W 2W −6  j i i j 1 i k k k i k 2
� 2 �  � ��
 G 00 = − 1 − + + O(c ) x ≡ x (T) + e X + a X X − a X X c


j
E
E
E
 c 2 c 4 2




 
 i  �
4W � −5 � 
G 0i =− + O c (7)  t ≡ T + (γ E − 1)dT + e

 c 3


� � (10)

 2W � �
 G ij = δ ij 1 + + O c
 −4

c 2 where
� 1

�
 1 1 � j j � 2
 0 2 2 2
 γ E ≡ e = 1 − (2w E + v ) + 2w + 8w v − 2w E v
 0 2 E 4 E E E E
 c c


 � � � �

 1 1 1 1 5 3 � �
 2 2 2 4 j j −6
 =1 + w E + v + w + w v + v − 4w v + O c

c 2 c 2 2 8
 2 E 4 E E E E E E

 (11)
j
i
e = γ E v /c
 0 E


 j 1 � j j � � �
 0 −5
e = γ E v /c + 2w v − 4w + O c

 j
 E 3 E E E
 c


� �
 � �
 1 1
 i i j −4
 e = δ ij 1 − w E + v v + O c
 j 2 2 E E
c 2c
i
and x , v , a are respectively the position, velocity, and
i
i
E
E
E
0
W = W is the scalar potential, which is the sum of the acceleration of the center of the Earth in BCRS, w , w
i
E
E
earth’s gravitational potential and the tide forces of the the scalar potential and the vector potential at the center
Sun and other external celestial bodies. Where the poten- of the Earth. It can be seen from the coordinate relation-
tial functions: ships Eqs. (4) and (10) that if the coordinates in BCRS
are considered as Euclidean linear coordinates, the coor-
µ
µ
k
µ
k
k
W (T, X ) =W (T, X )+W (T, X ) (8)
E ext dinates of GCRS are curvilinear coordinates. Moreover,
µ
µ
W , W represent the geocentric potentials and the the scope of application of GCRS is limited to the local
ext
E
external potentials. In the outer space of the Earth, the space near the Earth, which is much smaller than the
geocentric potentials can be expressed as: space range of the Earth-Moon system. Since the tidal
∞ l
 � �
 k GM E � � � a E � l
 W E (T, X ) = 1 + P lm (cosθ)[C lm cos(mφ) + S lm sin(mφ)]


R R (9)
l=2 m=0

 G j
W (T, X ) = ε ijk S X
 i k k

E
E
2R 3