$57,),&,$/6$7(//,7(69RO1R '2,DUVD
1(:+,*+35(&,6,219$/8(62)7+(*(2'(7,&527$7,21 2)7+(0$566$7(//,7(66<67(00$-253/$1(763/872 7+(0221$1'7+(681
993DVKNHYLFK$19HUVKNRY &HQWUDO 3XONRYR $VWURQRPLFDO2EVHUYDWRU\RI5$6 3XONRYVNRHVKRVVH6W3HWHUVEXUJ5XVVLD HPDLOVSDVKYODGYLW#\DQGH[UXDYHUVKNRY#PDLOUX
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
&$/&8/$7,2162)7+(*(2'(7,&527$7,219$/8(6 7KH PRVW HVVHQWLDO UHODWLYLVWLF FRPSRQHQW RI WKH URWDWLRQDO PRWLRQ RI D ERG\ i DURXQG WKH SURSHUFHQWHURIPDVVLVGHILQHGE\WKHDQJXODUYHORFLW\YHFWRURIWKLVERG\JHRGHWLFURWDWLRQ %UXPEHUJ (URVKNLQDQG3DVKNHYLFK
Gm j §· V RRu R R iijij ¦ ¨¸ c jiz ©¹ RRij
+HUH c LVWKHYHORFLW\RIOLJKW G LVWKHJUDYLWDWLRQDOFRQVWDQW m j LVWKHPDVVRIDERG\ j RiijRR R jDUHWKHYHFWRUVRIWKHEDU\FHQWULFSRVLWLRQDQGYHORFLW\RIWKHERGLHV i DQG j 7KH V\PERO u PHDQV D YHFWRU SURGXFW WKH VXEVFULSWV i IRU LQYHVWLJDWHG ERG\ DQG j IRU SHUWXUELQJERGLHVWDNHQIURPWKHVHWERGLHV3KRERV'HLPRVWKH0RRQWKHPDMRUSODQHWV 3OXWRDQGWKH6XQ)XUWKHULQWKHIRUPXODVWKHERG\LQGH[i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½ T ½ '\ ° 'WV V VLQ MV FRV M WDQ VLQT ° ° ° °° 'TV FRV MVM VLQ¾¾ ' UV FRV MVM VLQ °° 'MV ' \FRV T ' I V V VLQ MV FRV M °° ¿° ¿° +HUH W U DQGV DUHWKHSHUWXUELQJWHUPVRIWKHSK\VLFDOOLEUDWLRQVRIWKHVDWHOOLWHLQWKH ORQJLWXGHLQWKHLQFOLQDWLRQDQGLQWKHQRGHORQJLWXGHUHVSHFWLYHO\ I LVDFRQVWDQWDQJOHRI WKHLQFOLQDWLRQRIWKHVDWHOOLWHHTXDWRUWRWKHIL[HGHFOLSWLF-IRUWKH0RRQRUWRWKHORFDO /DSODFH SODQH IRU 0DUV VDWHOOLWHV ȥ, ș, ij DUH (XOHU DQJHOV \ LV WKH ORQJLWXGH RI WKH GHVFHQGLQJQRGHRIHSRFK-RIWKHERG\HTXDWRUT LVWKHLQFOLQDWLRQRIWKHERG\HTXDWRU WRWKHIL[HGHFOLSWLF-M LVWKHSURSHUURWDWLRQDQJOHRIWKHERG\EHWZHHQWKHGHVFHQGLQJ QRGH RI HSRFK - DQG WKH SULQFLSDO D[LV RI WKH PLQLPXP PRPHQW RI LQHUWLD WKH GRW GHQRWHV D GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR WLPH '\ \\r 'T TTr 'M MMr
' IV IIVVr 'U UUr 'W WWr DUHWKHGLIIHUHQFHVEHWZHHQWKHUHODWLYLVWLFDQG 1HZWRQLDQ FRPSRQHQWV RI WKH YHORFLWLHV JHRGHWLF URWDWLRQ RI 6RODU V\VWHP ERG\ LQ (XOHU DQJOHVDQGWKHSHUWXUELQJWHUPVRIWKHSK\VLFDOOLEUDWLRQVRIWKHVDWHOOLWHUHVSHFWLYHO\ 7KH JHRGHWLF URWDWLRQ YHORFLWLHV YDOXHV DUH FDOFXODWHG E\ PHDQV RI WKH OHDVW VTXDUHV PHWKRGDQGVSHFWUDODQDO\VLVPHWKRGV$VDUHVXOWWKHYDOXHVRIWKHFRHIILFLHQWVRIWKHPDLQ V\VWHPDWLFDQGSHULRGLFWHUPVRIWKHERG\JHRGHWLFURWDWLRQYHORFLWLHVZHUHFDOFXODWHG7KHVH WHUPVDUHSUHVHQWHGE\WKHIROORZLQJH[SUHVVLRQV NM n k 'x ¦¦¦ 'xtnClkllSlkll ' x FRV QQ t ' x VLQ QQ t t nlk
ZKHUH 'xn DUHWKHFRHIILFLHQWVRIWKHV\VWHPDWLFWHUPV 'xSlk 'xClk DUHWKHFRHIILFLHQWVRIWKH SHULRGLFWHUPVx \ TMWU , V Q llQ DUHSKDVHVDQGIUHTXHQFLHVRIWKHERG\XQGHU VWXG\ZKLFKDUHFRPELQDWLRQVRIWKHFRUUHVSRQGLQJ'HODXQD\DUJXPHQWV 6PDUW DQG WKHPHDQORQJLWXGHVRIWKHSHUWXUELQJERGLHVWKHVXPPDWLRQLQGH[lLVWKHQXPEHURIDGGHG SHULRGLFWHUPVDQGLWVYDOXHFKDQJHVIRUHDFKERG\XQGHUVWXG\ t LVWKHWLPHLQWKH-XOLDQ GD\V )LJXUHVGHSLFWHGWKHYDOXHVRIWKHYHORFLWLHVRIFKDQJHRIWKHJHRGHWLFQXWDWLRQVIRU0DUV DQGIRULWVVDWHOOLWHVLQWKH\HDUIUDJPHQWRIWKHWLPHVSDQXQGHUVWXG\
)LJ7KHYDOXHVRIWKHYHORFLWLHVRIWKHFKDQJHLQJHRGHWLFQXWDWLRQVIRU0DUVDQGIRULWV VDWHOOLWHV $IWHUDQDO\WLFDOLQWHJUDWLRQRIHTXDWLRQ 'x ³ 'x GW WKHH[SUHVVLRQVRIWKHV\VWHPDWLFDQG SHULRGLFWHUPVRIWKHERG\JHRGHWLFURWDWLRQFDQEHREWDLQHG
NM n k 'x ¦¦¦ 'xtnClkllSlkll ' x FRV QQ t ' x VLQ QQ t t nlk 'x ZKHUH x \ TM W U , V WKHFRHIILFLHQWV 'x n DQG WKH FRHIILFLHQWV IRU VLQH n n
'xSlk DQGFRVLQH 'xClk WHUPVDUHFDOFXODWHGXVLQJWKH³&DVFDGH´PHWKRG 3DVKNHYLFK $VDUHVXOWRIWKHOHDVWVTXDUHVFDOFXODWLRQVWKHYDOXHVRI1DQG 0 IRU YDULRXV REMHFWV ZHUH GHWHUPLQHG SURYLGLQJ WKH EHVW DSSUR[LPDWLRQ RI WKH SDUDPHWHUV RI JHRGHWLF URWDWLRQ $FFRUGLQJO\IRUPDMRUSODQHWV3OXWRWKH0RRQDQGWKH6XQN M IRU'HLPRVDQG 3KRERVN M
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ȝDVSHUWKRXVDQG\HDUV ȝDVSHUWKRXVDQG \HDUVIRUN M IRU0HUFXU\DVWKHERG\RI6RODUV\VWHPZLWKPRVWYDOXHVRIWKH JHRGHWLF URWDWLRQDQG VPDOOHU WKDQ ȝDVSHU WKRXVDQG \HDUV ȝDV SHU WKRXVDQG \HDUVIRUN M IRUWKH6XQDVWKHERG\ZLWKOHDVWYDOXHVRIWKHJHRGHWLFURWDWLRQ 7DEOH SDUWRQH 7KHV\VWHPDWLFWHUPVRIJHRGHWLFURWDWLRQ 0HUFXU\ 9HQXV 7KH(DUWK 7KH0RRQ e e e e ǻȥ ȝDV ǻȥ ȝDV ǻȥ ȝDV ǻIJ ȝDV T ± ± ± ± T ± ± ± T ± ǻș ȝDV ǻș ȝDV ǻș ȝDV ǻȡ ȝDV T ± ± ± T ± ± ± T ± ± ± ǻij ȝDV ǻij ȝDV ǻij ȝDV ǻ ,ı ȝDV T ± T ± ± T ± ±
7DEOH SDUWWZR 7KHV\VWHPDWLFWHUPVRIJHRGHWLFURWDWLRQ 3KRERV 'HLPRV e e ǻȥ ȝDV ǻIJ ȝDV ǻȥ ȝDV ǻIJ ȝDV T ± ± ± ± T ǻș ȝDV ǻȡ ȝDV ǻș ȝDV ǻȡ ȝDV T T ± ± ± ± ǻij ȝDV ǻ ,ı ȝDV ǻij ȝDV ǻ ,ı ȝDV T T ± ± 0DUV -XSLWHU 6DWXUQ 8UDQXV e e e e ǻȥ ȝDV ǻȥ ȝDV ǻȥ ȝDV ǻȥ ȝDV T ± ± ± ± T ± ± T ± ǻș ȝDV ǻș ȝDV ǻș ȝDV ǻș ȝDV T ± ± ± T ± ± T ± ± ± ǻij ȝDV ǻij ȝDV ǻij ȝDV ǻij ȝDV T ± ± T ± ± ± T ± ± 1HSWXQH 3OXWR 7KH6XQ e e ǻȥ ȝDV ǻȥ ȝDV ǻȥ ȝDV T ± ± ± T ± T ± ± ǻș ȝDV ǻș ȝDV ǻș ȝDV T ± ± ± T ± T ± ± ǻij ȝDV ǻij ȝDV ǻij ȝDV T T ± T ±
7DEOH SDUWRQH 7KHSHULRGLFWHUPVRIJHRGHWLFURWDWLRQ $UJX &RHIILFLHQWRI &RHIILFLHQWRI %RG\ $QJOH 3HULRG PHQW VLQ $UJXPHQW ȝDV FRV $UJXPHQW ȝDV
G Ȝ ±T T G Ȝ T ±T G Ȝ ±T ±±T ǻȥ G Ȝ ±±T ±T G Ȝ ±T T G Ȝ T ±T G Ȝ ±±T
G Ȝ ±T ±T G Ȝ ±T ±T G Ȝ ±T ±T 0HUFXU\ ǻș G Ȝ ±T G Ȝ ± ± ± G Ȝ ±ā ±ā G Ȝ ±ā± ā±
G Ȝ ±T±±T G Ȝ ±±T ±T G Ȝ ±T T ǻij G Ȝ T ±T G Ȝ ±T ±±T G Ȝ ±±T ±T G Ȝ ± T G Ȝ ±T ±T ǻȥ G Ȝ ±T ±T G Ȝ ±T ±T 9HQXV ǻș G Ȝ ±T G Ȝ ±T ±T ǻij G Ȝ ±T ±T G Ȝ T ±T ǻȥ G Ȝ ±T+ ±± T+ \U ȜD±F ±T ±T ± ± G Ȝ ±ā T ā T ± 7KH(DUWK ǻș G Ȝ ±+ ±ā \U ȜD±F ±T T ± ± G Ȝ ±ā T ā T ± ǻij G Ȝ ±ā T ±T \U ȜD±F T ±±T
7DEOH SDUWWZR 7KHSHULRGLFWHUPVRIJHRGHWLFURWDWLRQ $UJX &RHIILFLHQWRI &RHIILFLHQWRI %RG\ $QJOH 3HULRG PHQW VLQ $UJXPHQW ȝDV FRV $UJXPHQW ȝDV G Ȝ T ±T ± ǻIJ G D ±±T ±±ā T \U ȜD±F ±T ±±T ± G Ȝ T ±ā T 7KH0RRQ ǻȡ G D ±T ±±T \U ȜD±F ±T T G Ȝ ±T ±T ǻ ,ı G D ±T ±T \U ȜD±F ±T ±T \U Ȝ ± ± \U ± ǻȥ / K D ±T ±T T +T \U Ȝ \U ǻș / K D ±T ±±T ±T ±T \U Ȝ \U ± ± ǻij / K D ±T ±±T ±T ±T 3KRERV \U Ȝ ± ± \U ± ǻIJ / K D ±T ±T ±T T \U Ȝ \U ǻȡ / K D ±T ±±T ±T ±T \U Ȝ \U ± ǻ ,ı / K D ±T ±±T ±T ±T
7DEOH SDUWWKUHH 7KHSHULRGLFWHUPVRIJHRGHWLFURWDWLRQ $UJX &RHIILFLHQWRI &RHIILFLHQWRI %RG\ $QJOH 3HULRG PHQW VLQ $UJXPHQW ȝDV FRV $UJXPHQW ȝDV \U Ȝ ± ± \U ± ǻȥ / G D ±T T ±T T \U Ȝ \U ǻș / G D ±T ±±T T T \U Ȝ \U ± ǻij / G D ±T ±±T ±T ±T 'HLPRV \U Ȝ ± ± \U ± ǻIJ / G D ±T ±T ±T T \U Ȝ \U ǻȡ / G D ±T ±±T T T \U Ȝ \U ± ǻ ,ı / G D T ±±T ±T T \U Ȝ ±±T ±T ǻȥ G Ȝ ±±T ± T \U Ȝ T ±T 0DUV ǻș G Ȝ T ±T \U Ȝ ±T ±T ǻij G Ȝ T ±T \U Ȝ ±T T ǻȥ \U Ȝ ±T T \U Ȝ ±T T -XSLWHU ǻș \U Ȝ ±T T \U Ȝ ±±T T ǻij \U Ȝ ±±T T \U Ȝ T ±T ǻȥ \U Ȝ ±T ±±T \U Ȝ T ±T 6DWXUQ ǻș \U Ȝ ±T ±±T \U Ȝ T ±T ǻij \U Ȝ ±T ±±T
7DEOH SDUWIRXU 7KHSHULRGLFWHUPVRIJHRGHWLFURWDWLRQ $UJX &RHIILFLHQWRI &RHIILFLHQWRI %RG\ $QJOH 3HULRG PHQW VLQ $UJXPHQW ȝDV FRV $UJXPHQW ȝDV \U Ȝ T ±T ǻȥ \U Ȝ ±±T ±T \U Ȝ T ±T 8UDQXV ǻș \U Ȝ ± ± \U Ȝ ±T ±T ǻij \U Ȝ \U Ȝ ±±T T ǻȥ \U Ȝ ±T T \U Ȝ ±±T T 1HSWXQH ǻș ± \U Ȝ ± ā T \U Ȝ T ±±T ǻij ± \U Ȝ ±ā ±T \U Ȝ ±T ±±T ǻȥ \U Ȝ ±±T ±±T \U Ȝ ±±T ±±T \U Ȝ ±T ±±T 3OXWR ǻș \U Ȝ ±±T ±±T \U Ȝ ±±T ±±T \U Ȝ ±T T ǻij \U Ȝ T T \U Ȝ T T
\U Ȝ ±±T T ǻȥ G ±ā± Ȝ ± \U Ȝ ± ā 7KH6XQ ǻș G ā± ±ā± Ȝ
\U Ȝ ±±T ǻij G ā± ± Ȝ
,Q7DEOHDQG7DEOH//DUHORQJLWXGHVRIWKHDVFHQGLQJQRGHRQWKH/DSODFHSODQH IRU3KRERVDQG'HLPRVUHVSHFWLYHO\D Ȝ±ȜD Ȝ±ȜD Ȝ± ȜDUHPHDQHORQJDWLRQVRIWKH0RRQ3KRERVDQG'HLPRVIURPWKH6XQUHVSHFWLYHO\
Ȝj j DUHPHDQORQJLWXGHVRIWKHSODQHWVDQG3OXWRȜ LVPHDQJHRFHQWULFORQJLWXGHRI WKH0RRQȜȜDUHPHDQPDUVRFHQWULFORQJLWXGHVRI3KRERVDQG'HLPRVUHVSHFWLYHO\e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ǻȥDQGǻIJ IRUVHFXODU 7DEOH DQGSHULRGLF ZLWKDUJXPHQWȜ 7DEOH FRPSRQHQWVDUHTXLWHFORVHWR HDFKRWKHU7KHJHRGHWLFURWDWLRQVRI3KRERVDQG'HLPRVDUHGHWHUPLQHGQRWRQO\E\WKH6XQ EXWDOVRE\0DUV7KLVIDFWLVFRQILUPHGE\WKHDSSHDUDQFHRIWKHKDUPRQLFZLWKWKHDUJXPHQW D IRU 3KRERV DQG DIRU'HLPRV 7DEOH $WWKHVDPHWLPHWKHYDOXHVRIWKHWRWDO JHRGHWLF URWDWLRQ RI 3KRERV DQG 'HLPRV DUH PXFK JUHDWHU WKDQ WKH YDOXH RI WKH JHRGHWLF URWDWLRQ RI 0DUV 7KHUHIRUH WKH YDOXHV RI WKHLU FRHIILFLHQWV LQǻȥDQGǻ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ȜȜȜȜȜȜȜȜȜȜȜȜDQGȜ 3OXWR ZLWKDUJXPHQWVȜDQGȜ DQGWKH0RRQ ZLWKDUJXPHQWȜD±F ZHUHFDOFXODWHG ,QWKLVSDSHULWZDVDOVRFDUULHGRXWDVWXG\RQWKHPXWXDOUHODWLYLVWLFLQIOXHQFHRI0DUV VDWHOOLWHVRQHDFKRWKHUDQGRQ0DUV LHWKHLQFOXVLRQRIDQRWKHUVDWHOOLWHLQWKHQXPEHURI SHUWXUELQJERGLHV 6Rthe change in Deimos geodetic rotation from the relativistic influence of Phobos DUHHTXDOLQWKHORQJLWXGHRIWKHQRGHLV±ȝDVWM\LQWKHLQFOLQDWLRQLV±ā± ȝDVWM\DQGLQWKHSURSHUURWDWLRQDQJOHLVȝDVWM\ the change in Phobos’s geodetic rotation from the relativistic influence of DeimosDUH HTXDOLQWKHORQJLWXGHRIWKHQRGHLV±ā±ȝDVWM\LQWKHLQFOLQDWLRQLVā±ȝDVWM\DQG LQWKHSURSHUURWDWLRQDQJOHLVā±ȝDVWM\ DQGthe change in Mars geodetic rotation from Phobos and Deimos relativistic influence DUHHTXDOLQWKHORQJLWXGHRIWKHQRGHLV±ȝDVWM\LQWKHLQFOLQDWLRQLV±ā±ȝDVWM\ DQGLQWKHSURSHUURWDWLRQDQJOHLVȝDVWM\
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ĔVNL-=LHOLĔVNLDQG-1DVWXODSHUVRQDOJUDQWV 7KHUHSRUWHGVWXG\ZDVIXQGHGE\5)%5DFFRUGLQJWRWKHUHVHDUFKSURMHFW
5()(5(1&(6 3DVKNHYLFK 99 1HZ KLJKSUHFLVLRQ YDOXHV RI WKH JHRGHWLF URWDWLRQ RI WKH PDMRU SODQHWV 3OXWR WKH 0RRQ DQG WKH 6XQ Artificial Satellites :DUV]DZD 9RO 1R '2,DUVD SS 'H 6LWWHU : 2Q (LQVWHLQ V 7KHRU\ RI *UDYLWDWLRQ DQG LWV $VWURQRPLFDO &RQVHTXHQFHVMon. Not. R. Astron. Soc1RSS± )XNXVKLPD7 *HRGHVLF1XWDWLRQAstronomy and Astrophysics1RSS/± / )RONQHU :) -3/ 3ODQHWDU\ DQG /XQDU (SKHPHULGHV ([SRUW ,QIRUPDWLRQ KWWSLDXFRPPMSOQDVDJRY5($'0(KWPO )RONQHU:):LOOLDPV-*%RJJV'+3DUN56DQG.XFK\QND3 7KH3ODQHWDU\ DQG/XQDU(SKHPHULGHV'(DQG'(IPN Progress Report)HEUXDU\ .RSHLNLQ6(IURLPVN\0DQG.DSODQ* 5HODWLYLVWLF&HOHVWLDO0HFKDQLFVLQWKH6RODU 6\VWHP+RERNHQ1--RKQ:LOH\ 6RQVSS± $UFKLQDO%$$FWRQ&+$¶+HDUQ0)HWDO 5HSRUWRIWKH,$8:RUNLQJ*URXS RQ &DUWRJUDSKLF &RRUGLQDWHV DQG 5RWDWLRQDO (OHPHQWV Celest Mech Dyn Astr KWWSVGRLRUJVSS± 6PDUW:0 &HOHVWLDO0HFKDQLFV/RQJPDQV*UHHQDQG&R/RQGRQ1HZ *LRUJLQL -' &KRGDV 3: