. : http://www.kiam1.rssi.ru/pubs/prep_2013_066.pdf
: Tue Oct 8 14:46:34 2013
: Thu Feb 27 19:56:13 2014
:
..

.. , .. , .. , .. , ..

-

-- 2013


.., .., .., .., .. - - , . . . , . . . : , ,

Tuchin A.G., Komovkin S.V., Lavrenov S.M., Tuchin D.A., Yaroshevsky V.S. Celestial mechanics interpretation of radio measurements of slant range and radial velocity Celestial mechanics interpretation is defined as calculation parameters of spacecraft motion from observed quantities obtained at the tracking stations. The range and range rate models are considered. Particular attention has been paid to the Doppler effect. One-way, two-way and three-way Doppler measurements are investigated. Results are given for media corrections for the computed values of the observables. The ionospheric and tropospheric errors of measurements have been studied. Key words: radio measurements, Doppler effect, celestial mechanics interpretation


3 1. () : - ; - ; - - ; - , . - , - . - . , , , . . - , . . . , , , , , . , . 1988-89 . - [1]. -1,2. . - [2]. - [3] () , . , -X 12- , [1]. - - [4]. X-


4 . , . - . . 2. : . : - , ; - ; - , . . [1] . f min , . t . tm , t : 1 t = tm + q f min q . , .. q. ta :

ta - tm ) ) round ( x) x. f min 150 . , 2000 . , . . , , . .
min

q = round ( f

(


5 . , , . . . ( ) . , . , () , 3109 (.. 1 ). 35 .. : t + t2 D + D2 (1) =c 1 D= 1 2 2
t1 () ; t2 ; . c ( tb , rb ) ( te , re ) rb re [5]:

c ( te - tb ) = re - rb +

2S re + rb + re - rb ln re + rb - re - rb c2

(2)

S . ; t1 ; t2 ; t3 rsnd (t1 ) t1 (snd sender );


6 (sc spacecraft ); rrcv (t3 ) (rcv receiver ).

rsc (t2 )

(2), : 1 t1 = rsc (t2 ) - rsnd (t1 ) + D1 , c 1 t2 = rsc (t2 ) - rrcv (t3 ) + D2 , c

(3)

t2 = t1 + t1, t3 = t2 + t2 = t1 + t1 + t2 ,

D1 = D2 = 2S rsc (t2 ) + rsnd (t1 ) + rsc (t2 ) - rsnd (t1 ) ln , rsc (t2 ) + rsnd (t1 ) - rsc (t2 ) - rsnd (t1 ) c2 2S rsc (t2 ) + rrcv (t3 ) + rsc (t2 ) - rrcv (t3 ) ln . rsc (t2 ) + rrcv (t3 ) - rsc (t2 ) - rrcv (t3 ) c2

(3) (1), : 1 D = ( rsc (t2 ) - rsnd (t1 ) + rsc (t2 ) - rrcv (t3 ) +D1 +D2 ) 2 ( D1 + D2 ) 2 . t3 . t2 t1 : r (t ) - rrcv (t3 ) D2 , t2 = t3 - sc 2 - 2c c r (t ) - rsnd (t1 ) D1 . t1 = t2 - sc 2 - 2c c : c t2 , t1 . , t2 :
(0 1) t2 ) = t3 .

(i 2) t2 ) (i 1) :

t

( i +1) 2

= t3 -

(i rsc t2 ) - rrcv ( t3

()

)

c

-

(i D2 rsc t2 ) , rrcv ( t3

(()
2c

)

)

(4)


7 :
( i +1 (i t2 ) - t2 ) ,

.
( i +1 t 2 = t2 ) . t1 : (0 3) t1 ) = t2 ( i +1 t1 ) = t2 - (i rsc ( t2 ) - rsnd t1 )

()

i +1 i ( i +1 t1( ) - t1( ) . t1 = t1 ) .

c

-

D1R rrc ( t2 ) , rs 2c

(

nd

(t( ) ))
i 1

(5)

, ~ 10-6 . (4) (5) . , (4) (5) [6]. rsc ( t ) . rrcv ( t3 ) rsnd ( t1 ) , , IAU2000 [7]. , , , [8].
3. 3.1. . fsnd . fsnd + f D1 . f D1 . , M 2 ( ). f rcv = M 2 ( fsnd +f D1 ) +f D2 . f D2 . f med = f rcv - f h . f h . , f med . f med

f

nom med

( 70 ). f

med

f

nom med


8
nom M 2f D1 + f D2 = f med - f med . , . : , .. 2, . , . [1] . . :

( tmb , t

me

)=

tme

f

med

(t ) dt

tmb

tmb , tme . . , . , , . , , , .
3.2. . , .. 2, . , . . . , A t . , A . (. 1) .


9 : ; * t (t ) , t ; * ( ) , ; , .
X, Y, Z 1 2

t (tm t)

t (tm)

tm t

tm

t

. 1. 1 ; 2 ; tm ; t
, m - m , : 1 A=
m m -



d * 1 d = d

m m -



* ( m ) - * ( m -) d =
*

, . [5]:


10
2 vrcv ( - S = 1 - 2 - c r ( - ) 2c 2 rcv * *

)

t ,

2 vrcv * ( m -) S ( ) - ( m - ) = 1 - 2 - c rrcv * ( m -) 2c 2

(

)

(

)

t * (tm ) - t * (tm - t ) , rrcv ( - ) , vrcv ( - ) ; rrcv * ( m -) , vrcv * ( m -)

(

)

(

)

(

)

. , A : t * ( tm ) - t * ( tm -t ) A= t .
3.3. , , t , [9]: t 1- A (6) D tm - = c , 2 1+ A t D tm - : 2 t D - D1H + D2 H - D2 K D t m - = 1K , 2 2 (r - r , v ) D1H = 2 1 1 | r2 - r1 | : ;


11 (r2 - r1 , v2 ) | r2 - r1 | : ; (r - r , v ) D2K = 3 2 2 | r3 - r2 | : ; (r - r , v ) D2H = 3 2 3 | r3 - r2 | : ; r1 ( x1 , y1 , z1 ) ; r2 ( x2 , y2 , z2 ) ; r3 ( x3 , y3 , z3 ) ; d v1 = r1 dt1 t1 ; d v2 = r2 dt 2 t2 ; d v3 = r3 dt3 t3 . . 2 () () . ( r1, v1 ), ( r2 , v2 ) ( r3 , v3 ), . : ( r2 - r1 ) ( r3 - r2 ) D1K =

. 2. ,


12 ,
dt : dt . : S c ( t3 - t2 ) = r3 ( t3 ) - r2 ( t2 ) + 2 ( t3 , t2 ) c r + r2 + r3 - r2 ( t3 , t2 ) = 2 ln 3 r3 + r2 - r3 - r2 t3 , :
*

dt dt c 1 - 2 = v3 - v2 2 , dt3 dt3 dt 2 r - r v2 , 3 2 - c = v3 , dt3 r3 - r2 : dt2 D2 H - c = + dt3 D2 K - c c 2 (

r3 - r2 S dt ( t3 , t2 ) + r3 - r2 c 2 dt3 r3 - r2 S dt ( t3 , t - c + r3 - r2 c 2 dt2

2

)

S dt ( t3 , t D2 K - c ) dt3

2

)

: S (7) c ( t2 - t1 ) = r2 ( t2 ) - r1 ( t1 ) + 2 ( t2 , t1 ) , c r + r1 + r2 - r1 ( t2 , t1 ) = 2 ln 2 . r2 + r1 - r2 - r1 t1 , : dt2 D1K - c S dt = +2 ( t2 , t1 ) . dt1 D1H - c c ( D1H - c ) dt1 : dt3 ( D1K - c ) ( D2 H - c ) . = dt1 ( D1H - c ) ( D2 K - c ) , : (8)

1 , (8) c2


13

dt * D - D1H + D2 H - D2 K = 1 - 1K + dt c ( D1H + D2K ) ( D1H + D2H - D1K - D2K ) + + c2 DD DD + 1K 2 2 H - 1H 2 2 K . c c , t t 1 m dt * 1 m D1K - D1H + D2 H - D2 K A= dt = 1 - dt + c t t t dt t t t - -
m m

1 + t +

tm tm -t



( D1H + D2 K )( D1H + D2 H - D1K - D2 K ) + c2

D1K D2 H D1H D2K - dt c2 c2

: 1 D= t
tm tm -t



D dt .

(6) : 1- A 1+ A , 1- A D 1 =+ 1 + A c t 1 + t , t D tm - 2 1- A c 1+ A
tm tm -t

1 , : c2



2 2 2 2 D1K - D1H - D2 K + D2 H dt + 4c 2 2 tm t m - t

tm tm -t



1 D dt - c t



D2 dt. c2 (9)


14

1 t

tm tm -t



2 2 D12K - D12H - D2K + D2H dt + 4c

2 t t 1m 1m 2 D dt - D dt t t t t m - tm -t . : t t t D(t ) = D tm - + D tm - t - tm - + 2 2 2 (10) 2 t t 1 + D tm - t - tm - + ... 2 2 2 tm ( ); t . tm -t tm (10) :

1 + c

tm tm -t



t D(t ) dt = D tm - 2

t t +D tm - 2

3 t + ... 24

, 1 D= t , t D D tm - 2 t t 2 D tm - . 2 24 (9) (11), , (6) - : t 1- A D tm - = c 2 1+ A D , :
tm tm -t



t D(t ) dt = D tm - 2

t + D tm - 2

2 t + ... 24

(11)


15 1 D = t
tm tm -t



2 2 2 2 D1K - D1H - D2 K + D2 H dt + 4c
tm tm -t

1 + c

1 t



1 Ddt - t

2

tm tm -t



D dt +
2

t t 2 . + D tm - 2 24 ( 10 ) . D 0.01 /. . . .
3.4. , o : D ( tm ) - D ( tm - t ) = c (1 - A ) (12) t , , tm , D ( tm ) : ; D ( tm -t ) , , tm -t , : . t * ( t m ) - t * ( t m - t ) c (1 - A ) = c 1 - = t (13) t m - t * ( t m ) - ( t m - t - t * ( t m - t ) ) = c . t (12) (13), ..


16

D (t

m

)

= c tm - t * (tm ) ,

(

D ( tm -t ) = c tm - t - t * (tm -t ) .
3.5. , . , . . D ( t0 )

(

)

)

t t , t0 , D t0 - D t0 + 2 2 t t t0 + . , (7), t0 - 2 2 t t t0 - , t0 + 2 2 t t D t0 + - D t0 - 2 2 A (t ) = 1 -
0

c t

, (9). , : 1 - A ( t0 ) N ( t0 ) = D ( t0 ) - 1 + A ( t0 )
. , N ( t0 ) , . , , N ( t0 ) , : N ( t0 ) N ( t0 ) N ( t0 ) N ( t0 ) N ( t0 ) N ( t0 ) , , , , , x y z vx v y vz N ( t
0

)

,


17 : 0.5 10-9 c -1 , 0.5 10-4 ( ). , , , , , , 1 /, 1000 .
3.6. . , , . , -. , , . , (- ), , , , . , , :

1 A2 =


m m -



d* sc d d

(14)

* ( sc





m

; ) , , ; ; . (14), :


18

A2 =

* ( sc

m

) - *c ( s


m -

)


(15)
(16)

. (15) : * ( m ) - * ( m - ) * ( m ) - * ( m - ) sc A2 = sc * ( m ) - * ( m - )
* (
m

)

,

, ( m , m - ). (16) , . , , [5]: 2 2 vsc - v rv sc - rv * ( m ) - *c ( m - ) sc s (17) =1- + * ( m ) - * ( m - ) 2c 2 c2 vsc , v rv ; sc , rv , . (16). , , : t *(tm ) - t *(tm -t ) t (18) = *( m ) - *( m -) tm ; * t (t ) , t , ( tm , tm - t ); t . (17) (18), (16) :


19
v2 - v2 - t * ( t m ) - t * ( t m - t ) A2 = 1 - sc 2 rcv + sc 2 rcv = t 2c c (19) 2 2 vsc - v rcv sc - rcv 1 tm dt * = 1 - + dt. t dt c2 2c 2 tm -t . . 8.4 (X) [1] . C-.
3.7. , dt * dt : dt * c - D2H = (20) dt c - D2K (r - r , v ) D2K = 3 2 2 r3 - r2 : ; (r - r , v ) D2H = 3 2 3 r3 - r2 : ; r2 ( x2 , y2 , z2 ) ;

r3 ( x3 , y3 , z3 ) ; d v2 = r2 ; dt d v3 = r3 . dt 1 , 2 , (20) c :


20
D2H dt c = 1 - D2H + D2 K - D2H D2 K = 1 = D dt c c c 1 - 2K c D2 = D2 H - D2 K . (21) (19) : v2 - v2 - A2 = 1 - sc 2 rcv + sc 2 rcv c 2c
*

1-

-

D2 D2H D2K , - c c2

(21)




K

D 1 1 - 2 - c t

tm tm -t



1 D2H ( t ) D2 c2

(

t ) dt .

(22)

, : 1 D A2 = 1 - 2 - c t 1 D2 = t
tm tm tm -t tm tm -t

1 , (22) c2
2 rcv



D2H ( t ) D2 c2

K

(t )

v2 - v dt - sc 2 2c D2 ( t ) dt

+

sc - c2

rcv

,

(23)



(23) D2 , :
2 vsc - v r2cv sc - rcv + dt - (24) . 2c c tm -t (24) , .

1 D2 = c (1 - A2 ) - t

D2

H

(t )

D2 c

K

(t )

3.8. . (, ) . ( ) , . , , . , , , , .


21 ( , ).
3.9. [10] . -: , , one-way Doppler ( ). [11] , , , . , : 1,2 D = D1 ( t ) - D1 ( t - t ) - D2 ( t ) + D2 ( t - t ) ; t D1 ( t ) , D2 ( t ) 1 2;

. [10] , Mars Pathfinder, . X. (1) 0.05 /c. [11] . [11], D2 ( t ) - D1 ( t ) () ():

t

D1 - D2 L1,2 cos cos(1,2 - ) + z1,2 sin

L1,2 , 1 2, ; z1,2 , 1 2, , ; 1,2 , 1 2. , , , .


22
4. , S-, X- Qu- . : . , . . , , . , . [13] , L , H , : H n ( h ) ( R + h ) L = dh, (25) 2 0 ( R + h ) - R cos

; ; . , .
4.1. , , .
ht

n ( h R h

)

1;

Rt = n(h N ht

(1 - n ( h ) )
0

dh , 1 - n ( h ) = 10-6 N ( h ) ,

) (h)

, , .


23 Rt , . N ( h ) . [13], : 10 - , , ~ 93; - , , : 77.6 4810 Bs Ns = Ps + , Ts Ts Ps , ; Ts , ; Bs , , , . N ( h ) = N s e -h , (26) h ; 1 93 10 , = - ln . 10 N s Deep Space Network (DSN) NASA [14-16], . , : 77.6 4810 B 77.6 77.6 4810 B N (h) = P ( h) + . P ( h) + = T (h) T ( h) T (h) T 2 (h) (dry refractivity) ND ( h ) , (wet NW ( h ) Rtw . , refractivity) NW ( h ) . ND ( h ) Rtd ,


24

Rtd = ND(h) dh, Rtw = NW (h) dh,
0 0

ht

ht

Rt = Rtd + Rtw .

, 0.90

Rtd R 0.01 tw 0.1. 0.99; Rt Rt Rtd :

R Rtd = 77.6 Ps , g Ps , ; R , ( ) ; g , 2 ; rt , . (26) (25). R , 1 , : R + h dh = 1 - cos 2 0 , Rt rt =
H



* N 0 10-6 e

-h

* N 0 10-6 1 - e

(

-H

)

sin

* N 0 10-6 Rt = . sin sin

(27)

, Rt , rt : rt = Rti . sin -2 2 22 . , , (27) : cos Rt rt = . sin 2 , Rt , Rt , : cos Rti rt = . sin 2 1 2 . ,


25 td tw . 2 + 2w td t cov rt ( 1 ) rt ( 2 ) = sin sin ; 1 2 cov rt ( 1 ) rt ( 2 ) = 1 2 cos 1 cos sin 1 sin
2 2 2 2 2

(

2d + t

2 tw

)

;

2 cos 2 cov rt ( 1 ) rt ( 2 ) = sin 2 sin 2 1

(

2 + td

2 tw

)

.

4.2. , . , , , , . . , . , GPS-, , . GPS- . , TEC ( Total Electron Content) [17]. , 1 2 . TECU. TECU 1016 /2. GPS- [12]. n : N ( h) n ( h ) = -40.4 e 2 , f

N e ( h ) , 1 3 ; f , .


26 ngr : dn ngr = n + f . df : N (h) (28) ngr ( h ) = 40.4 e 2 . f , , . ri rri /c, . , , . (28) (25) : h N e ( h ) ( R + h ) 40.4 i ri = 2 dh, f 0 ( R + h )2 - R 2 cos 2

hi . hm , [13]:
hi 0



N e ( h ) ( R + h

)
2

(

R + h

)

2

-

2 R

cos

dh

R + hm

hi

( )

R + hm

)

2

-

2 R hi

cos

2

Ne ( h )
0

dh .

: 40.4 ri = 2 f
R + hm

(

R + hm

2

-

2 R

cos

2

Ne ( h )
0

dh.

(29)

, , . (29) : h 2 40.4 ( R + hm ) R cos sin i N h dh. ri = 2 3 e( ) f 2 ( R + hm ) - R2 cos 2 2 0

(

)


27 ( = 0 )

40.4 i Ri = - 2 N ( h ) dh. f0 Ri , : 2 R sin 2 Ri , ri = ri = Ri . 3 2 2 ( R + hm ) R cos 2 2 1- R cos 2 2 2 2 1 - ( R + hm ) ( R + h )2 m Ri , Ri , ri ri : 2 Ri R sin 2 ri = , ri = . 3 Ri 2 2 2 ( R + hm ) R cos 2 1- R cos 2 2 2 2 1 - ( R + hm ) ( R + h )2 m 1 2 . : 1 cov ri ( 1 ) ri ( 2 ) = 2 R 1- cos 2 1 2 ( R + hm )

h

1- cov ri ( 1 ) ri ( 2 ) =
4 RE

1 ( R + hm )
2 R 2

cos 2
2

2 i

4 ( R + hm

)

4

sin 2 1 1
2 R cos 2 1 1 - ( R + h )2 m

sin 2 2

2



3 2

2 R cos 2 2 1 - ( R + h )2 m



3 2



2 i


28
cov ri ( 1 ) ri ( 2 ) =
2 RE

2 ( RE + hm 1

)

2

sin 2 2
2 3 2

2 2 R cos 2 1 R cos 2 2 1 - 1 - ( R + h )2 ( R + h )2 m m , , 3 1017 / 2 hm = 300 1 2.

1 2



2 i

, . ri , , . rri , /

1 20 1.1 40 0.7 60 0.6 80 0.5 2 80 0.01

10 0.15

20 0.12

40 0.05

60 0.02

1. 2.

3.

4. 5. 6.

. . / . .. . .: . 2012. 216 . .., .., .., .., .. - -. // . .. . 2002. 2. 20 . URL: http://library.keldysh.ru/preprint.asp?id=2002-2 .., .., .., .., .., .. -: , // . . 44, 1, 2010, . 29-40. .. . . . // . .. . 2012, 3, . 4-21. URL: http://vestnik.laspace.ru/archives/03-2012/ .. . .: , 1972. 382 . .., .. // . . 1, . 1., 1963, .5-50.


29 7. 8. URL: http://www.kiam1.rssi.ru/~den/lib_akim.html .. . : 2 2006. 480 . Moyer T.D. Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation, John Wiley & Sons, Inc., Hoboken, New Jersey. 2003. URL: http://descanso.jpl.nasa.gov/Monograph/series2/Descanso2_all.pdf .. . // , . 23, . 1, 1985, . 4962. Bhaskaran S. The application of noncoherent Doppler data types for deep space navigation. // TDA Progress Report 42-121. JPL, 1995. Thurman S.W. Deep space navigation with differenced data types. Part 1: Differenced range information content // The telecommunications and data acquisition progress report 42-103, JPL, 15, 1990. P. 47-60. .., .., .., .. . // . .. . . . 3. . 2012. . 20-26. .., .., .. . .: , 1969. 155 . Berman A.L. A New Approach to the Evaluation and Prediction of Wet Tropospheric Zenith Wet Refraction. JPL DEEP SPACE Progress Report 42-25. Berman A.L. The Prediction of Zenith Range Refraction From Surface Measurements of Meteorological Parameters. JPL Technical Report 32-1602, 1976. 40 p. Estefan J.A., Folkner W.M. Sensitivity of Planetary Cruise Navigation to Earth Orientation Calibration Errors. TDA Progress Report 42-123. Opperman B.D.L. Reconstructing ionospheric TEC over South Africa using signals from a Regional GPS network. // A thesis submitted in fulfillment of the requirements for the degree Doctor of philosophy of Rhodes University. 28 Nov. 2007.

9. 10. 11. 12.

13. 14. 15. 16. 17.


30


1 2 3 ............................................................................................................ 3 ................................................................... 4 ..................................... 7 3.1 ................................................. 7 3.2 ......................................... 8 3.3 .............................................................................. 10 3.4 ................ 15 3.5 .................................................... 16 3.6 .................................. 17 3.7 .............................................................................. 19 3.8 ........................................ 20 3.9 ........................ 21 4 ......................................................................................... 22 4.1 ................................................. 22 4.2 ................................................... 25 ............................................................................................................. 28