Mon. Not. R. Astron. Soc. 37, 1549 1574 (006) doi:10.1111/j.1365-966.006.10870.x TEMPO, nw pulsr timing pckg II. Th timing modl nd prcision stimts R. T. Edwrds, G. B. Hobbs nd R. N. Mnchstr Austrli Tlscop Ntionl Fcility, CSIRO, PO Box 76, Epping, NSW 1710, Austrli Accptd 006 July 5. Rcivd 006 July 4 ABSTRACT TEMPO is nw softwr pckg for th nlysis of pulsr puls tims of rrivl. In this ppr, w dscrib in dtil th timing modl usd by TEMPO, nd discuss limittions on th ttinbl prcision. In ddition to th intrinsic slow-down bhviour of th pulsr, TEMPO ccounts for th ffcts of binry orbitl motion, th sculr motion of th pulsr or binry systm, intrstllr, Solr systm nd ionosphric disprsion, obsrvtory motion (including Erth rottion, prcssion, nuttion, polr motion nd orbitl motion), troposphric propgtion dly, nd grvittionl tim-diltion du to binry compnions nd Solr systm bodis. W bliv tht th timing modl is ccurt in its dscription of prdictbl systmtic timing ffcts to bttr thn 1 ns, xcpt in th cs of rltivistic binry systms whr furthr thorticl dvlopmnt is ndd. Th lrgst rmining sourcs of potntil rror r msurmnt rror, intrstllr scttring, Solr systm phmris rrors, tomic clock instbility nd grvittionl wvs. Ky words: mthods: dt nlysis stromtry clstil mchnics tim pulsrs: gnrl. 1 INTRODUCTION Th cor of ny pulsr timing softwr pckg is th mthmticl modl for th tims of rrivl (TOAs) of pulss t th obsrvtory. As th pulsr popultion grows mor divrs nd obsrving systms incrs in snsitivity, th ccurcy dmndd of timing modls grows ccordingly. For mny yrs, th stndrd choic for pulsr timing ws th TEMPO 1 softwr pckg. Accurt t th 100-ns lvl, th timing modl of TEMPO ws dqut for numrous importnt discovris (.g. Tylor, Fowlr & McCulloch 1979; Tylor & Wisbrg 1989; Wolszczn & Fril 199; Kspi, Tylor & Ryb 1994; vn Strtn t l. 001). Howvr, motivtd primrily by th possibility of dirctly dtcting grvittionl rdition through its ffct on puls TOAs, currnt obsrving cmpigns r stting highly mbitious gols in trms of dt prcision nd volum. Ths projcts im to obtin of th ordr of 10 4 indpndnt TOA msurmnts with typicl root-mn-squr (rms) uncrtinty of 100 ns (.g. Jnt t l. 005), dictting mximum llowbl systmtic rror of 1 ns. Motivtd by th nd for improvd ccurcy s wll s grtr flxibility in fitting nd nlysis procdurs, w hv dvlopd nw pulsr timing pckg clld TEMPO. Th rchitctur, cpbilitis nd usg of TEMPO r dscribd by Hobbs, Edwrds & Mnchstr (006) (hrftr Ppr I). Th purpos of this work is to dscrib th timing modl in dtil, nd provid stimts of th ccurcy of ch componnt of th modl. Sction bgins with n ovrviw of th modl nd dfinitions of th vrious rltivistic coordint frms nd propr tims (Sction.). This is followd by n xpnsion of th complt gomtric propgtion dly ccurt to bttr thn 1 ns (Sction.3). Th rmindr of Sction discusss ll owthr dly trms in th timing formul, nd provids dtild dscription of how ch contribution is vlutd ccording to th prmtrs of th modl nd othr informtion. Sction 3 givs dtild ccount of th ccurcy with which ch trm is computd, nd discusss possibl vnus for furthr improvmnt of th timing modl. Finlly, for th convninc of comprison with th commonly usd TEMPO softwr pckg, Sction 4 dtils th diffrncs btwn TEMPO nd TEMPO. Tbulr summris of timing modl prmtrs nd vribls r providd in Appndix A. E-mil: Russll.Edwrds@csiro.u 1 http://www.tnf.csiro.u/rsrch/pulsr/tmpo/. C 006 Th Authors. Journl compiltion C 006 RAS
1550 R. T. Edwrds, G. B. Hobbs nd R. N. Mnchstr TIMING MODEL.1 Ovrviw Pulsr timing consists of th msurmnt of th puls TOAs t th obsrvtory nd th fitting of ths TOAs to modl. Th timing modl rlts th msurd TOA to th tim of mission t th pulsr, from which puls phs of mission is computd vi modl of th intrinsic vritions in th puls priod. If th timing modl is corrct, in th bsnc of msurmnt nois ll clcultd puls phss will b n intgr (whn xprssd in units of cycls or turns). In prctic, th prmtrs of th modl r not known priori, nd msurmnt nois is prsnt, so tht th dvitions of ths puls phss from th nrst intgr (th rsiduls ) r finit. Onc th timing modl prmtrs r rfind to optimiz som sttistic of th rsiduls in TEMPO, this is th rms or χ (s Ppr I) nd ssuming thr r no shortcomings in th structur of th timing modl, th rsidul dvitions r du only to msurmnt nois. Th focus of this ppr is th timing modl, which rlts msurd TOA to tim of mission, nd from thr computs corrsponding puls phs of mission. Th lttr stp is dscribd in Sction.8. Th formr prt compriss th bulk of th complxity of th modl, nd it too cn b hlpfully brokn into two min prts. First, bcus tht prt of th modl rlts propr TOA t th obsrvtory to propr tim of mission t th pulsr, gnrl rltivistic frm trnsformtion is ncssry. This is prformd in squnc of mngbl stps, outlind in Sction.. Scondly, th light trvltim must b ccountd for. This is drivd in Sction.3 undr th ssumption of Euclidn spc tim nd zro rfrction or disprsion. Th rmining prts of Sction driv in full mthmticl dtil th vrious prts of th frm trnsformtion nd light trvltim, including disprsion, rfrction nd gnrl rltivistic ffcts long th propgtion pth.. Coordint frms nd tim vribls Th rltivistic frm trnsformtion btwn obsrvtory propr tim nd pulsr propr tim is computd in svrl stps. This sction outlins th stps, givs th nms of th vrious rfrnc frms nd introducs th nottions usd to rfr to tim vribls in ch frm. Th tim vribl for ch frm my ppr subscriptd with th lttr or, whr dnots TOA nd tim of mission, t th origin of th frm dnotd by th suprscript. A tim vribl my lso ppr without subscript, mning tht it is n rbitrry coordint tim vribl. As such, tim vribls without subscripts r rltd by coordint trnsformtions, whrs subscriptd tim vribls dditionlly includ propgtion dlys. Sptil thr-dimnsionl vctors r dfind in trms of right-hndd Crtsin systm. Th pulsr timing procdur bgins with msurd TOA, which ftr corrcting for imprfctions in th obsrvtory clock yilds TOA t obs. Th clock-corrction procss in TEMPO oprts through n intrpoltion of tbls of msurd offsts btwn pirs of clocks nd thir vrition with tim (s Ppr I), nd includs offsts rlting to th fct tht obsrvtory clocks usully pproximt Coordintd Univrsl Tim (UTC ) rthr thn Trrstril Tim (TT). Th corrctd TOA is msurd in th TT tim-scl, which is dfind to diffr from coordint tim t th gocntr by constnt rt. It should b notd thn tht tims rfrrd to TT r only pproximtly propr tims of th obsrvtory. This trnsformtion to Gocntric Coordint tim-scl (TCG) is don in th sm stp s trnsformtion to Brycntric Coordint Tim (TCB), so no vribl is ssignd to this tim scl. Th sptil prt of th trrstril frm is th Intrntionl Trrstril Rfrnc Systm (ITRS), whil th gocntric countrprt is th Gocntric Clstil Rfrnc Systm (GCRS). Coordint tims t SSB in th frm of th Solr systm brycntr (SSB) r obtind by ppliction of th grvittionl rdshift nd spcil rltivistic tim-diltion intgrl ( Einstin dly ) dscribd in Sction.5.1. Coordint tim (TCB) in this frm rprsnts th propr tim tht would b msurd by n obsrvr loctd t th SSB wr th grvittionl fild of th Sun nd plnts not prsnt. Th sptil prt of th brycntric frm is th Brycntric Clstil Rfrnc Systm (BCRS), whil dirctions in this frm dfin th Intrntionl Clstil Rfrnc Systm (ICRS). Th origin of th BCRS is th SSB, hnc t SSB rfrs to th TOA of th puls t th brycntr. For furthr informtion on th dfinitions of TT, TCG, TCB, th ITRS, GCRS, BCRS nd ICRS, s th Rsolutions of th XXIIIrd nd XXIVth Gnrl Assmbly of th Intrntionl Astronomicl Union (IAU; Andrson 1999; Rickmn 001) nd th 003 Convntions of th Intrntionl Erth Rottion nd Rfrnc Systms Srvic (IERS; McCrthy & Ptit 004). Trcing th propgtion bckwrds towrds th sourc, th nxt rfrnc frm is tht which is comoving with th pulsr, or if it is mmbr of binry systm, th binry brycntr (BB). Coordints in this frm r dfind such tht th coordint tim t BB would b qul to th propr tim of n obsrvr t th origin of th frm, wr th grvittionl fild of th pulsr nd its compnion not prsnt. Th trnsformtion of th TOA to this frm is discussd in Sction.6.3. Th orinttion of this frm is chosn such tht th Lorntz trnsformtion rlting it to th SSB frm is rottion-fr, so tht to zroth ordr th corrsponding bsis vctors r prlll. Th sptil origin of this frm is th BB. Th tim origin is chosn such tht t BB hs zro offst from t SSB t t BB = t SSB = t pos, whr th lttr is n poch of position st by th usr. Th tim-scl for msurmnts is th propr tim t psr msurd t th pulsr, or mor prcisly t its cntr, wr its grvittionl fild not prsnt. For isoltd pulsrs t psr = t BB. For binry pulsrs, t psr t BB is th binry Einstin dly (Sction.7.4). As with th SSB BB cs, th BB pulsr trnsformtion is rottion-fr. Th sptil origin of this frm is th cntr of grvittionl fild of th pulsr, so tht t psr is th quivlnt tim of mission from th cntr of th str. This diffrs from th ctul tim of mission by (to first ordr) th light trvltim to th tru point of mission, which is ssumd to b stbl on th tim-scl of intrst. Du to copyditing rrors, in Ppr I th cronyms UTC, TCG, TCB nd TDB wr indvrtnty rfrrd to s UCT, CGT, BCT nd BDT, rspctivly.
TEMPO: timing modl nd prcision 1551.3 Gomtric propgtion dly Although it is convnint to brk th propgtion dly into sprt contributions nd trt thm in th ordr in which thy occur long th photon pth, som trms in th modl dpnd jointly on th motion of th obsrvtory nd th motion of th pulsr (rltiv to th SSB). For this rson, in this sction, w driv th complt gomtric propgtion dly, tht is th tim of propgtion nglcting disprsion, rfrction nd rltivistic dlys. Th rsultnt trms r thn ssignd to diffrnt prts of th propgtion pth nd trtd in th rlvnt sctions blow. Th gomtric propgtion dly is simply th Euclidn distnc from th obsrvtory to th pulsr, dividd by th vcuum spd of light, c. Th displcmnt vctor from th obsrvtory to th pulsr is th sum of th brycntric position of th obsrvtory (r), th brycntric position of th pulsr (or BB) (R 0 ) t givn poch (t pos ), th position of th pulsr with rspct to th BB (b, zro for isoltd pulsrs), nd th displcmnt (k) of th BB (or th pulsr itslf, if isoltd) in th tim lpsd sinc th poch t pos, owing to th initil vlocity of th systm (rltiv to th SSB) nd cclrtion in th Glctic or clustr grvittionl fild: R = R 0 + b + k r. (1) Squring qution (1) nd xpnding th squr root to third ordr, R = [ R 0 + b + k + r + (R 0 b + R 0 k R 0 r + b k b r k r) ] 1/ () = R 0 +A A R 0 + A3 R 0 +..., (3) whr A = 1 [ b + k + r + (R 0 b + R 0 k R 0 r + b k b r k r) ]. (4) R 0 Nglcting trms of th ordr of R 0 3 nd dnoting th initilly rdil nd trnsvrs componnts of ch vctor using subscripts, tht is, = R 0, = R 0 / R 0 nd hnc, b = ( R 0 ) (b R 0 ), th following rltion is found: R = R 0 +k r + b + 1 R 0 ( k k r + k b + r r b + b )( 1 k R 0 + r R 0 b R 0 ) + O( R 0 3 ). (5) Th first four trms in qution (5) r th initil SSB BB distnc (Sction.6.), th sculr displcmnt in th initilly rdil dirction (Sction.6.), th projction of th obsrvtory SSB vctor on th initil lin of sight (Sction.5.3), nd th projction of th binry motion on th initil lin of sight (Sction.7.1). Th trms in th first pir of prnthss corrspond to th Shklovskii ffct (Sction.6.), nnul propr motion (Sction.5.3), sculr chngs in th pprnt orbitl viwing gomtry (Sction.7.1), nnul prllx (Sction.5.3), nnul-orbitl prllx (Sction.7.1) nd orbitl prllx (Sction.7.1). Th scond, third nd fourth trms in th scond pir of prnthss rprsnt vritions in th ffcts just discussd du to rdil motion of th pulsr or binry systm, th motion of th Erth round th SSB, nd th motion of th pulsr bout th BB. Although ths trms giv ris to n dditionl 18 contributions to th timing formul, to rch th sttd ccurcy gol of 1 ns, for 0-yr obsrving cmpign on ny of th prsntly known milliscond pulsrs only thr of thm nd to b considrd: th sculr chng (Sction.6.) nd nnul modultion (Sction.5.3) of th Shklovskii ffct nd th sculr chng of th nnul propr motion (Sction.5.3). Th displcmnt du to sculr motion, k my b brokn into first nd scond drivtivs: k = μ R 0 ( ) t BB ( ) t pos + t BB t pos, (6) whr μ is th vlocity dividd by th distnc, or in ssnc thr-dimnsionl propr motion, nd is n cclrtion vctor ccounting for Glctic diffrntil rottion nd grvittionl cclrtion. Th cclrtion is of th ordr of 10 11 ms (Bll & Bils 1996), contributing only 500 km to k t ithr nd of 0-yr obsrvtion cmpign cntrd on t pos. Eqution (5) cn b xpndd in trms of this prmtriztion of k, giving ris to n dditionl 7 trms. W hv stimtd th mgnitud of ths trms for ll pulsrs in th Austrli Tlscop Ntionl Fcility (ATNF) ctlogu 3 (Mnchstr t l. 005), including n llownc for grtr cclrtion for pulsrs in globulr clustrs ( 10 8 ms ; Frir t l. 001). Only two of th xtr trms involving xcd 1 ns for 0-yr obsrving cmpign, th first contributing to k nd th scond contributing to th Shklovskii trm of qution (5). For ll othr trms of qution (5) involving k, th cclrtion is nglctd..4 Top-lvl timing formul Hving dscribd th coordint frms usd nd drivd th bsic Euclidn propgtion lngth, it rmins to spcify th dtils of th clcultion of th vrious dly trms ssocitd with frm trnsformtions, th vcuum propgtion dly, nd corrctions du to non-unity rfrctiv indics nd spc tim curvtur long th propgtion pth. For convninc, th timing formul is dividd into thr min contributions, which will b furthr brokn down in th sctions tht follow: t psr = t obs IS B. (7) 3 http://www.tnf.csiro.u/rsrch/pulsr/psrct/.
155 R. T. Edwrds, G. B. Hobbs nd R. N. Mnchstr (Sction.5) includs th coordint trnsformtion to th SSB frm, vcuum propgtion dlys ssocitd with th Erth s orbitl motion nd th spin, prcssion nd nuttion of th Erth (on which th obsrvtory is loctd) nd xcss dlys owing to th pssg of th signl through th Erth s tmosphr nd th Solr systm. This trm rlts th msurd TOA to th TCB TOA t th SSB: t SSB t obs. IS (Sction.6) includs th trnsformtion to th BB frm, vcuum propgtion dlys du to th sculr motion of th systm, nd xcss propgtion dlys du to pssg of th signl through th intrstllr mdium (ISM). This trm rlts th TCB TOA t th SSB to th BB coordint TOA t th BB: t BB = t SSB IS. B includs th trnsformtion to th pulsr frm, vcuum dlys du to th binry orbitl motion, nd xcss dlys du to pssg of th signl through th grvittionl fild of th compnion. This trm rlts th BB coordint TOA t th BB to th pulsr propr tim of mission: t psr = t BB B. Not tht sinc som of th gomtric dly trms involv two of th thr componnts (Solr systm, sculr or binry) of th rltiv motion of th obsrvtory nd th pulsr, th choic of which of ths trms r ssignd to ch of th thr top-lvl trms is somwht rbitrry. S Sction.3 for dtild listing of whr ch trm is discussd. Onc th tim of mission is obtind, it rmins to comput th phs of th puls trin t tht tim. This rltionship, φ(t psr ), is discussd in Sction.8. For th corrct st of modl prmtrs nd in th bsnc of msurmnt nois, th phs for vry computd tim of mission is n intgr numbr of cycls (turns). Th frctionl prt of φ(t) is th timing rsidul, which is mor oftn xprssd in tim units through division by th puls frquncy. =.5 Forming brycntric TOAs Th trm dnotd by in qution (7) includs ll ffcts prtinnt to th trnsformtion of th obsrvd TOA (t obs ) to n quivlnt TOA for th sm puls wvfront t th SSB. This, in turn, cn b brokn into svrl stps. Ths r: tmosphric dlys, vcuum rtrdtion du to obsrvtory motion ( Romr dly nd prllx), xcss propgtion dly du to disprsion, th ffcts of rltivistic frm trnsformtion ( Einstin dly ), nd th xcss dly xprincd by rys s thy pss through th grvittionl potntil of Solr systm bodis ( Shpiro dly ). Ths trms r dnotd s follows: = A + R + p + D + E + S (8).5.1 Einstin dly Th spc tim coordints of puls rrivl vnts r spcifid in th coordint frm of th obsrvtory, in which puls timing nomlis my mnifst du to spin nd orbitl cclrtion nd vritions in grvittionl potntil. To void ths ffcts, puls rrivl vnts r trnsformd to th qusi-inrtil frm of th SSB. Th two frms r rltd by rltivistic four-dimnsionl spc tim trnsformtion. Th sptil prt of th vnt is th displcmnt of th obsrvtory from th gocntr t th instnt of rcption, which is ltrd by ngligibl mount du to spcil nd gnrl rltivistic lngth contrction. Th ffcts of rltivistic tim-diltion, on th othr hnd, cnnot b nglctd. TEMPO uss th numricl tim-diltion rsults of Irwin & Fukushim (1999), who usd th DE405 Solr systm phmris (Stndish 1998) to comput th tim-diltion intgrl: E = 1 t [ ] U c + v t 0 + L(PN) C + L (A) C dt, (9) whr U is th grvittionl potntil t th gocntr du to ll Solr systm bodis xcpt th Erth, nd v is th vlocity of th gocntr rltiv to th SSB. Th first two trms dscrib, to ordr 1/c, th grvittionl rdshift nd spcil rltivistic tim-diltion, rspctivly. Ths contribut in roughly :1 rtio ( 1 U v 10 8 c ), for mn drift of 1.5 10 8,or 0.5 s yr 1. Th min sourc of timvrition in th intgrnd is du to th orbitl cclrtion of th Erth, which vi consrvtion of nrgy cuss pproximtly qul chngs (of th sm sign) in th first two trms, mounting to 3 10 10 in rt mplitud, or ms in intgrtd dly mplitud. Th rmining trms pply corrction for highr-ordr rltivistic trms [ L (PN) C = 1.097 10 16 ; Fukushim 1995] nd stroids [ L (A) C = 5 10 18 ; Fukushim 1995], md in mn rt only. Th tim-diltion intgrl of qution (9) rlts th coordint tims of clocks t th gocntr nd th SSB. For obsrvrs loctd on th surfc of th Erth, corrctions must b md for th diffrntil tim-diltion nd grvittionl rdshift btwn th obsrvtory nd th gocntr, yilding E = E + s ṙ + W 0 t obs, (10) c whr s is vctor from th gocntr to th obsrvtory, ṙ is th vlocity of th gocntr with rspct to th brycntr, nd W 0 = 6.969 90 134 10 10 c pproximts th grvittionl plus spin potntil of th Erth t th goid. (S Sction.5.3 for th clcultion of s nd ṙ.) Not tht, whil th vlu of th potntil vris gogrphiclly nd tmporlly, th most rcnt dfinition of TT (Rickmn 001), th trrstril tim-scl to which TEMPO rfrs puls TOA msurmnts, tks th xct vlu of W 0 /c quotd bov s dfinitiv of th rt diffrnc btwn TT nd SI coordint tim t th gocntr. Givn common poch of t 0 = 43 144.000 375 (Modifid Julin Dt), qutions (9) nd (10) rlt msurd TT to th IAU rcommndd coordint tim-scls, TCG (gocntric) nd TCB (brycntric).
TEMPO: timing modl nd prcision 1553 It should b notd tht th tim phmris of Irwin & Fukushim (1999) in fct tks its tim rgumnt, T ph, in rfrnc frm tht is linr trnsformtion of TCB, mking th computtion of th Einstin dly strictly n invrs problm. In prctic, sinc th rgumnt tim-scl hs zro mn rt diffrnc to TT, th lttr cn b substitutd with ngligibl rror in th rsult (Irwin & Fukushim 1999). In ddition, th vlution of qution (10) rquirs consulting th Solr systm phmris (Sction.5.3) which lso tks T ph tims s its rgumnt. Sinc T ph is computd from TCB (qution 10), circulr dpndncy problm riss. This is solvd to sufficint ccurcy by two psss through n itrtiv rfinmnt procss..5. Atmosphric dlys Th group vlocity of rdio wvs in th tmosphr diffrs from th vcuum spd of light. Rfrctivity is inducd both by th ionizd frction of th tmosphr (minly in th ionosphr) nd th nutrl frction (minly in th troposphr). Th troposphric propgtion dly b sprtd into th so-clld hydrosttic nd wt componnts. In Vry Long Bslin Intrfromtry (VLBI) nd Globl Positioning Systm (GPS) pplictions, ths componnts r typiclly modlld s th product of th dly inducd t th znith, nd so-clld mpping function, which spcifis th rtio btwn th znith dly nd th dly in givn dirction. For plnr tmosphr, th mpping function is simply givn by csc whr is th sourc lvtion ngl. Mor dvncd mpping functions tk into ccount th curvtur of th tmosphr, ssuming zimuthl symmtry. Th ffct of zimuthl symmtry is typiclly lss thn nnoscond vn t vry low lvtion ngls (.g. McMilln 1995), nd nd not b considrd for th prsnt purpos. Th hydrosttic componnt contributs pproximtly 90 pr cnt of th totl dly, nd my b computd priori, ssuming givn mixtur of gss in hydrosttic quilibrium. TEMPO uss th formul of Dvis t l. (1985), r-writtn s tim-dly P/43.91 kp hz = c(1 0.00 66 cos ϕ 3.6 10 6 m/h), (11) whr hz is th hydrosttic znith dly, P is th surfc tmosphric prssur, ϕ is th godtic ltitud of th sit, nd H is its hight bov th goid. In combintion with th mpping function, this lds to timing trm of mplitud 7.7 ns csc, mostly on diurnl tim-scl. If tmosphric prssur dt r unvilbl, TEMPO uss cnonicl vlu of on stndrd tmosphr (101.35 kp). TEMPO uss th Nill mpping function (NMF; Nill 1996), which is of comprbl ccurcy to othr publishd mpping functions but dos not rquir mtorologicl dt: it dpnds only on th sourc lvtion. Th wt componnt of th troposphric propgtion dly is highly vribl nd cnnot b prdictd ccurtly. Fortuntly, it is smll. Th znith wt dly (ZWD; wz ) my b msurd using pproprit nlyss of rdiosond, wtr vpour rdiomtr, GPS or VLBI obsrvtions (.g. Nill t l. 001), of which only GPS is likly to b obtinbl on routin bsis t most rdio obsrvtoris. If such msurmnts r vilbl, TEMPO cn mk us of thm in conjunction with th NMF to ccount for th ffcts of troposphric wtr vpour on puls TOAs. Th ZWD could concivbly b includd s fr prmtr in th pulsr timing modl, but bcus th ffct is smll nd vris from dy to dy, this would not b possibl with th snsitivity of prsntly ttinbl obsrving systms. For this rson, if no tbultd ZWD informtion is vilbl, th ffct is nglctd. Nglcting disprsion (blow), th totl tmosphric dly is writtn s A = m h ( ) hz (P) + m w ( ) wz (P), (1) whr m h ( ) nd m w ( ) r th NMFs for th hydrosttic nd wt componnts. Vlus for P nd wz r to b providd s input dt to TEMPO, or dfult vlus of P = 101.35 kp nd wz = 0 r usd. Th pssg of th signl through th ionosphr inducs disprsiv dly tht vris strongly with th solr ctivity cycl nd lso on ssonl, diurnl nd shortr tim-scls. Th intgrtd column dnsity of lctrons ( totl lctron contnt, TEC) in th ionosphr typiclly lis in th rng 5 100 TECU (1 TECU =10 16 m 3. 10 7 cm 3 pc), corrsponding to vribl propgtion dlys of th ordr of 7 130 ns (f/1 GHz), whr f is th obsrving frquncy. Modls of th ionosphr r vilbl (.g. Schr 1999; Bilitz 001); howvr, th prdictd TEC is ccurt to only 3 0 TECU (Mmoru, Kondo & Kwi 003). In ny cs, ionosphric disprsion is insprbl from nd smllr thn uncrtintis in intrplntry (Sction.5.4) nd intrstllr (Sction.6.1) disprsion, ncssitting th inclusion of fittbl, fully tim-vribl disprsion prmtr (Sction.6.1)..5.3 Romr dly nd prllx Th Romr dly is th simpl vcuum dly btwn th rrivl of th puls t th obsrvtory nd th SSB, not including ffcts rltd to th binry motion of, or finit distnc to, th pulsr: R = r ˆR BB, (13) c whr ˆR BB R BB / R BB is unit vctor in th dirction of th BB t th tim of obsrvtion. TEMPO prforms this clcultion in th BCRS. Th vctor ˆR BB is constructd from th sphricl coordint ngls of th ICRS sourc dirction, right scnsion (α) nd dclintion (δ), t tim t pos, nd th Crtsin componnts of th propr motion of th sourc in th pln of th sky (μ α nd μ δ ) nd long th lin of sight
1554 R. T. Edwrds, G. B. Hobbs nd R. N. Mnchstr (μ ), ll of which r fittbl prmtrs in TEMPO: ( ) ( ) ˆR BB = ˆR 0 + μ t BB 1 (t t pos μ ) BB ˆR 0 + μ μ t pos, (14) whr cos α cos δ ˆR 0 = sin α cos δ sin δ (15) μ = μ α ˆα + μ δ ˆδ, ˆα = sin α cos α, 0 nd cos α sin δ ˆδ = sin α sin δ. cos δ Altrntivly, cliptic coordints (λ, β, μ λ, μ β ) my b usd, in which cs: cos λ cos β ˆR 0 = E sin λ cos β, sin β nd μ = μ λ ˆλ + μ β ˆβ, whr (0) E = 1 0 0 0 cos ɛ 0 sin ɛ 0, (1) 0 sin ɛ 0 cos ɛ 0 sin λ ˆλ = E cos λ 0, () nd cos λ sin β ˆβ = E sin λ sin β. (3) cos β Hr ɛ 0 = 84 381.405 78 rcsc (Hrd & Fukushim 004) is th mn obliquity of th cliptic t J000.0. Th four trms tht rsult from th xpnsion of qution (14) driv from corrsponding trms in th xpnsion of qution (5), such tht r ˆR BB = r + k r R 0 r k R 0 k k r R 0. Hr w hv ssumd tht k =μ R 0 (t BB (Sction.3). In principl, t BB t pos ), sfly nglcting ny cclrtion or highr-ordr sculr motion of th pulsr or BB cnnot b computd without first knowing R ; this is rsolvd by initilly stting t BB to th TCB TOA t th obsrvtory, nd itrtivly rfining nd IS until convrgnc. Eqution (4) consists of four trms corrsponding to th xprssion in qution (14) of th instntnous sourc dirction s th sum of four contributions. Ths r th initil dirction (yilding th first trm of qution 4), th propr motion (scond trm), corrction to mintin unit lngth ftr ddition of th propr motion (third trm) nd th cclrtion or dclrtion of th propr motion owing to th fct tht ovr tim, th propr motion movs th sourc vctor, cusing prt of th initilly rdil vlocity to cquir trnsvrs componnt (fourth trm). Th third trm cn lso b intrprtd s th propr-motion-inducd scond-ordr rduction in th projction of th sourc dirction on th rdil componnt of r tht ccompnis th first-ordr incrs in its projction on th trnsvrs componnt of r. Anothr ltrntiv intrprttion of th third trm is tht it corrcts th Shklovskii trm (Sction.6.) for th distnc vritions inducd by th Erth s orbitl motion. Th fourth trm, first introducd by N. Wx (1997, unpublishd contribution to TEMPO), provids on of two possibl wys of ccssing th lusiv rdil vlocity, th intgrtd first-ordr contribution of which (k ) is othrwis insprbl from Dopplr shift of th puls nd orbitl frquncis nd thir drivtivs. Th scond mthod is dscribd in Sction.6.. Th first trm of qution (13) dscribs th diffrnc in TOA of puls t th SSB comprd to tht t th point obtind by projcting th SSB obsrvtory vctor on th SSB BB vctor. Th TOA t tht point diffrs gin from th TOA t th obsrvtory du to th curvtur (16) (17) (18) (19) (4)
TEMPO: timing modl nd prcision 1555 of sphricl wvfronts conncting photons mittd simultnously from th pulsr. This trm is rfrrd to s th prllx trm, lthough it diffrs somwht from stllr prllx: positionl stromtry msurs th thr-dimnsionl orinttion of th wvfront norml, whrs pulsr timing msurs th position of th wvfront in its intrsction with th cliptic. To first ordr, th curvtur is proportionl to th squr of th ltrl displcmnt of th obsrvtory from th SSB BB vctor, yilding n xcss dly corrsponding to on of th trms of qution (5): p = r, (5) cd p Hr th prllx distnc, d p is substitutd for R 0 to llow this ffct to b sprtd from othrs involving th sourc distnc. Th convntionl prllx 1u/d p, bing th ngl subtndd from th sourc by lin of 1 u in lngth t th SSB, is fittbl prmtr in TEMPO. Its vlu is spcifid in units of millircsconds, corrsponding numriclly to th rciprocl of th distnc in kiloprscs. In computing th Romr nd prllx dlys, th vctor r is constructd in two stps, from th SSB to th gocntr nd from th gocntr to th obsrvtory, tht is, r = r + s. Th first prt (r ) is ccomplishd with th id of numricl Solr systm phmris. Th currnt dfult phmris is DE405 (Stndish 1998), which is lignd within known uncrtinty to th ICRS, but uss tim-scl nd lngth-scl slightly diffrnt from th SI scond nd mtr (Stndish 1998). TEMPO uss th qutions nd constnts of Irwin & Fukushim (1999) to ppropritly trnsform th TCB sit TOA (Sction.5.1) for input to th phmris (using n offst nd scl), nd ppropritly scls th output vctors to SI units. It should b notd tht th us of ltrntiv phmrids tht rquir diffrnt trnsformtion procdur will introduc rrors (s Sction 3.5). Th scond prt of th obsrvtory position vctor, s, is obtind by trnsformtion of th trrstril obsrvtory coordints into th GCRS (which to sufficint ccurcy is quivlnt to th BCRS). This is ccomplishd using n ITRS sit position vctor (s ITRS ) supplid to TEMPO, trnsformd to th GCRS using lgorithms consistnt with th IAU 000 Rsolutions. Th trnsformtion procds s follows: s = Q ( ) ( ) ( ) t obs R t obs W t obs sitrs, (6) whr th mtrics W, R nd Q ccount for polr motion, Erth rottion, nd th motion of th Erth spin xis in th ICRS. Th lttr trm, in turn, consists of product of frm bis nd th IAU000B prcssion-nuttion mtrix (McCrthy & Luzum 003). Dtils of th construction of ths mtrics r vilbl in th IERS Convntions (McCrthy & Ptit 004) nd in th documnttion of th IAU Stndrds of Fundmntl Astronomy (SOFA) softwr librry 4 usd by TEMPO to comput thm. Polr motion nd Erth rottion both xhibit unprdictbl vritions tht must b corrctd post fcto using msurd prmtrs. TEMPO uss th C04 sris of Erth orinttion prmtrs, providd by th IERS, s input to th SOFA routins. For th purposs of trnsforming th obsrving frquncy to th brycntric frm (Sction.6.1), th tim-drivtiv of th Romr dly is ndd: d R (ṙ + ṡ ) ˆR 0 =, (7) dt c whr dots dnot th tim-drivtiv nd th ffcts of propr motion nd prllx r ngligibl. Th brycntric vlocity of th gocntr, ṙ, is providd by th Solr systm phmris. Th brycntric vlocity of th sit is domintd by th vlocity imprtd by Erth rottion. This is clcultd using ṡ ITRS = ( ω W 1 ẑ ) s ITRS. (8) Hr unit North vctor (ẑ) is trnsformd using th invrs polr motion mtrix nd multiplid by th instntnous Erth ngulr rottion rt, ω, to giv th ngulr vlocity of th Erth in th ITRS. Th cross-product of this with th sit rdius vctor givs th tngntil vlocity, which is trnsformd to th GCRS in th sm mnnr s s. Of th nglctd trms in this pproximtion, th lrgst is th prcssionl vlocity, t ngligibl 10 13 c..5.4 Solr systm disprsion Rdio signls ncountr significnt disprsion in th intrplntry mdium, du to th lctron contnt of th solr wind. Th lctron distribution follows roughly r form consistnt with sphricl xpnsion (Issutir t l. 1998). Intgrting long th lin of sight yilds disprsion msur (DM): DM = 0 n 0 [ 1u r(s) ] ds = n 0 (1 u) ρ r sin ρ, (30) whr r is th hliocntric rdius of th obsrvtory, ρ is th pulsr-sun-obsrvtory ngl, nd n 0 is n ovrll scl prmtr, which cn b spcifid in TEMPO prmtr fils s NE1AU. [It should b notd tht qution (30) is n pproximtion tht ignors th ffcts on th (9) 4 http://www.iu-sof.rl.c.uk/.
1556 R. T. Edwrds, G. B. Hobbs nd R. N. Mnchstr disprsion rltion du to th bulk plsm vlocity nd its vrition long th lin of sight. Howvr, thr r othr much mor significnt sourcs of rror s discussd blow.] Th dfult vlu for n 0 is 4 cm 3, consistnt with rcnt msurmnts (Issutir t l. 1998, 001; Splvr t l. 005) nd significntly lowr thn th vlu of 9.961 cm 3 usd by TEMPO. Th computd DM is usd to clcult th dly du to intrplntry disprsion: D = DM.410 10 16 cm 3 pc ( f SSB ), (31) whr f SSB is th obsrving frquncy trnsformd to th brycntric frm (s Sction.6.1). This corrction is of limitd ccurcy. Rcnt studis with th Ulysss spccrft hv rvld xtrordinry complxity in th solr wind. At solr minimum, th lctron dnsity (scld by r ) ws found to show rms tmporl vritions of 10 50 pr cnt, dpnding on hliocntric ltitud (Issutir t l. 1998), whil t solr mximum th mn lctron dnsity incrsd but ws subjct to vry strong modultion (Issutir t l. 001). Th dfult corrction of qution (31) is thrfor uncrtin within t lst fctor of, corrsponding to minimum of 130 ns of rror t f = 1 GHz for sourc loctd t n cliptic pol, incrsing to mny microsconds for sourcs within fw dgrs of th cliptic. Errors in th prdictd Solr systm disprsion dly r of concrn not only bcus thy dd rndom nois to th timing rsiduls nd fittd prmtrs, but lso bcus portion of th rror trm will xhibit nnul priodicitis tht could corrupt th fittd pulsr position, propr motion nd prllx (Splvr t l. 005). It is lso possibl tht othr priodicitis in this trm could b misintrprtd s rising du to plntry compnion to th pulsr (Schrr t l. 1997, though s lso Wolszczn t l. 000). Prvious ttmpts to improv th modl hv focusd on dding fittbl dgrs of frdom (.g. Cognrd t l. 1996; Splvr t l. 005); howvr th complx bhviour sn in th Ulysss obsrvtions indicts tht th only wy to dqutly rmov th ffct is to msur it dirctly using multifrquncy obsrvtions (s Sction.6.1)..5.5 Shpiro dly Th Shpiro dly ccounts for th tim-dly cusd by th pssg of th puls through curvd spc tim. Th totl dly obtind by summing ovr ll th bodis in th Solr systm (Bckr & Hllings 1986): Gm j S = ln( ˆR r c 3 j + r j ) + S, (3) j whr ˆR is unit vctor in th dirction of th pulsr, m j is th mss of body j, r j is vctor from body j to th tlscop, nd S is scond-ordr corrction discussd blow. Th first trm my b r-writtn in th following mor convnint form: Gm j S = ln r c 3 j ( ) 1 cos ψ j + S, (33) j whr ψ j is th pulsr tlscop objct ngl for th jth objct. TEMPO includs th first-ordr Shpiro dly for th Sun, Vnus, Jupitr, Sturn, Urnus nd Nptun. For obsrvtions of sourcs vry clos to Solr systm bodis, highr-ordr ffcts my bcom importnt. Th lrgst of ths is th gomtricl xcss pth lngth du to grvittionl light bnding (Richtr & Mtznr 1983), which mounts to 9.1 ns for ry with trjctory grzing th solr limb. 5 From th drivtion of Hllings (1986), ssuming gnrl rltivity nd r-prmtrizing in trms of ψ for vry distnt sourc, this trm is givn by S = 4G m c 5 r tn ψ sin ψ (34) 4G m c 5 r ψ. For pulsr timing purposs, only th gomtric dly du to th Sun nds to b considrd. (35).6 Propgtion through intrstllr spc Th propgtion tim of th signl from th pulsr (or binry systm brycntr) to th SSB is convnintly brokn into thr contributions: th vcuum propgtion dly, tht is, th pth lngth dividd by th vcuum spd of light, nd th xcss dly du to disprsion nd othr frquncy-dpndnt dlys: IS = VP + ISD + FDD + ES. (36) Th fourth trm ccounts for th spcil rltivistic tim-diltion btwn th SSB rfrnc frm nd th binry rfrnc frm. 5 Hllings (1986) quotd figur of 36 ns, rfrring to th clcultion of Richtr & Mtznr (1983). Howvr, tht clcultion incurrd n xtr pir of fctors of two, on du to th fct tht th clcultion rfrrd to round trip, nd th othr du th fct tht th rflctor ws loctd 1 u pst th Sun, rthr thn bing vry distnt sourc.
TEMPO: timing modl nd prcision 1557.6.1 Intrstllr disprsion nd othr frquncy-dpndnt dlys Th intrstllr disprsion dly ntrs th timing modl undr th stndrd rltion: D ISD = ( f SSB ), (37) whr D is th so-clld disprsion constnt, nd f SSB is th frquncy of th rdition t th SSB. Th brycntric frquncy diffrs from th frquncy t th obsrvtory, du to simpl Dopplr shift ( 10 4 in mgnitud) nd scond-ordr rltivistic corrction ( 10 8 ), plus th ffcts of grvittionl rdshift ( 10 8 ). Th first trm is simply th tim-drivtiv of th Romr dly (th Romr rt ), whil th sum of th lttr two is drivtiv of th Einstin dly (th Einstin rt ): ( f SSB = 1 + d R + d ) E f. (38) dt dt Although D is th dirctly msurbl quntity of qution (37), it is customry to spk in trms of th infrrd column dnsity of lctrons long th lin of sight, tht is, th DM, DM = k D D. Th vlu of k D usd by TEMPO (nd TEMPO)isk D.410 10 16 cm 3 pc; s Ppr I for furthr discussion. Mny pulsrs xhibit tmporl vrition in th intrstllr DM. TEMPO follows othr softwr pckgs in modlling ths vritions by mns of Tylor sris: DM (n) ( ) DM = t SSB n t D, (39) n! n 0 whr DM (n) is th (fittbl) nth drivtiv of th DM, nd t D is usr-spcifid poch. This pproch is limitd to modlling vritions of low to modrt complxity. At th lvl of ccurcy dmndd by currnt highprcision timing cmpigns, significnt unmodlld vritions will ppr on short tim-scls, du to ionosphric (Sction.5.), intrplntry (Sction.5.4) nd intrstllr lctrons (Fostr & Cords 1990). An ltrntiv nd rcommndd cours of ction is to us th strid fit ftur of TEMPO (Ppr I), to fit diffrnt DM to vry group of simultnous (or contmpornous) multifrquncy TOA msurmnts. TEMPO lso llows for th fitting of n dditionl frquncy-dpndnt dly trm: FDD = k f ( f SSB ) ζ, (40) whr k f nd ζ dfin th scl nd spctrl indx of th trm. This trm my b usd in conjunction with th strid fit ftur to modl, for xmpl, dlys du to rfrction in th ISM (Fostr & Cords 1990) or dvitions from th cold plsm disprsion lw (Phillips & Wolszczn 199)..6. Vcuum propgtion dly Th vcuum propgtion dly is ffctd by th distnc to th pulsr nd th vrition of this with tim. Th full gomtric distnc ws drivd in Sction.3. Th first trm from qution (5) to b ssignd to IS, R 0 /c, is in fct not msurbl to vn rmotly sufficint prcision, but is constnt nd so cn b droppd from th timing formul. A sid ffct of this is tht pochs of msurd puls frquncis, tims of pristrons, glitchs nd so on r, in fct, rtrdd by th initil light propgtion tim, c R 0. Th rmining prts ssignd to th intrstllr dly r s follows: s IS = k + k R 0 k k R 0. (41) Th first trm on th right-hnd sid of qution (41) simply rprsnts th displcmnt of th systm in th initilly rdil dirction. This rcivs potntilly significnt contributions from th rdil vlocity of th pulsr nd th rdil cclrtion in th grvittionl potntil of th globulr clustr or Glctic nvironmnt (Dmour & Tylor 1991). Th scond trm dscribs th so-clld Shklovskii ffct (Shklovskii 1970), whrby initilly trnsvrs motion ttins rdil componnt du to th chng in th dirction of th lin of sight cusd by th trnsvrs motion of th pulsr. Tht is, th pulsr movs tngntilly, whil pth of constnt distnc dscribs circl cntrd on th SSB. Th distnc btwn th two is th xcss propgtion lngth. Th third trm corrsponds to sculr chng in th siz of th Shklovskii ffct s th rdil motion incrss or dcrss th distnc, nd hnc chngs th curvtur of th lin of constnt distnc just mntiond. An ltrntiv intrprttion of this trm is tht it is th cumultiv ffct on th Shklovskii trm cusd by th incrs or dcrs in th pprnt propr motion, owing to th incrsing trnsvrs componnt of th initilly rdil motion (Sction.5.3). To our knowldg, this ffct ws first notd by vn Strtn (003), in th contxt of its mnifsttion s n pprnt scond drivtiv of th spin nd orbitl priods. Ths ffcts r oftn nglctd in timing formul, bcus s simpl constnt or sculrly incrsing Dopplr shifts, thy r insprbl from th puls nd orbitl priods nd thir drivtivs. A common procdur is to corrct th pprnt priod drivtivs by stimting th contributions of trnsvrs motion nd grvittionl cclrtion nd subtrcting thm to obtin th intrinsic vlu (.g. Dmour & Tylor 1991; Cmilo, Thorstt & Kulkrni 1994). Altrntivly, in crtin binry systms whr th tru orbitl priod drivtiv is xpctd to b ngligibl, th msurd drivtiv cn b ttributd ntirly to ths ffcts to obtin, in combintion with msurmnt of th trnsvrs propr motion, n stimt of R 0 (Bll & Bils 1996). Likwis, msurd scond spin frquncy drivtiv could, in principl, yild msurmnt of th rdil vlocity (vn Strtn 003) vi th third trm of qution (41).
1558 R. T. Edwrds, G. B. Hobbs nd R. N. Mnchstr As n ltrntiv to modlling ths ffcts vi rtificil contributions to spin nd binry prmtrs, TEMPO cn optionlly includ thm dirctly in th following form: VP = v ( ) t BB + d Shk μ ( ) t pos + t BB 1 ( t pos + μ v Shkdot μ )( ) t BB 3 t pos, (4) c c c whr v Shkdot,, d Shk nd μ μ r fittbl prmtrs tht spcify th rdil vlocity, rdil cclrtion, Shklovskii distnc, nd componnt of cclrtion in th dirction of th trnsvrs propr motion. Th trms involving cclrtion ris du to th xpnsion of k in trms of propr motion, distnc nd cclrtion (qution 6). Th circulr dpndnc of VP nd t BB is rsolvd by itrtion s dscribd in Sction.5.3. Th prmtrs nd d Shk r insprbl, nd highly covrint with simultnous chng to th spin frquncy drivtiv nd orbitl priod drivtiv. It is thrfor imprtiv tht t lst on of th first two prmtrs plus ny on of th rmining thr prmtrs r hld fixd t som indpndntly dtrmind vlu whn fitting. Ths prmtrs r includd sprtly bcus ch of thm is potntilly subjct to xtrnl constrints. Th trnsvrs propr motion, μ, is typiclly immun from this covrinc du to its pprnc in th nnul propr motion trm (Sction.5.3). Likwis, v Shkdot nd μ r insprbl, nd highly covrint with th spin frquncy scond drivtiv, so only on of ths prmtrs should b llowd to vry in fit. Thy r includd sprtly bcus v my b dtrmind from μ (Sction.5.3) nd on or mor of th distnc prmtrs, whil my b stimtd from modls of th Glctic rottion nd grvittionl potntil. Th first-ordr rdil vlocity trm is discussd in th following sction..6.3 Trnsformtion from SSB to BB coordint tim Th finl trm contributing to IS ccounts for th spcil rltivistic tim-diltion owing to th rltiv vlocitis of th Solr systm nd BBs: t BB t pos = t SSB t pos 1 v /c, (43) whr v is th rltiv vlocity of th frms. This trnsformtion is pplid to th TOA of th puls t th BB, so tht t BB = t SSB ISD FDD VP ES, (44) whr ES = v ( ) t SSB c t pos ISD FDD VP, (45) nglcting trms of th ordr of v 4 /c 4 nd highr. Hr, th nottion ES rfrs to th fct tht th trm is nlogous to th Solr systm nd binry Einstin dlys (Sctions.5.1 nd.7.4) nd rlts to th rltiv sculr motion of th SSB nd BB. This trnsformtion combins with th much lrgr v /c trm of qution (4) to ffct Dopplr shift btwn th two frms: SSB dt ( ) Ɣ = t SSB dt BB = t pos (46) = 1 v /c 1 v /c. An unknown Dopplr shift of this kind cnnot b distinguishd from r-scling of vrious prmtrs intndd to rfr to th BB frm, mning tht nithr v nor v cn b msurd vi thir contribution to Ɣ. In fct, bcus th numricl vlus usd for c nd G r th sm in ll frms, th ffct of nglcting ES is th sm s chng of unit systm involving r-scling of th lngth mss nd tim units, nd physicl tsts involving only msurd (wrong) prmtrs nd ths constnts will rmin vlid (Dmour & Drull 1986). In th timing modl prsntd hr (nd in contrst to tht of Dmour & Drull 1986), v cn, in principl, b dducd from prmtrs msurd vi othr trms of qution (5). For this rson, it my b includd by spcifying non-fittbl vlus for th initil rdil vlocity, v, nd trnsvrs spd, v, in th prmtr fil, which corrct for th Dopplr ffct vi qution (4) nd th following formultion of th rltivistic scond-ordr Dopplr shift: ES = v + v ( ) t SSB c t pos ISD FDD VP. (48) Howvr, it must b rmmbrd tht, in prctic, th uncrtinty of mny prmtrs msurd in th frm of th BB or pulsr will b domintd by th uncrtintis in v nd v. (47).7 Th ffcts of binry compnion Th ffcts of binry compnion r sprtd into svrl componnts: th gomtric Romr dly, which ccounts for xcss th vcuum light trvltim du to th Euclidn displcmnt of th pulsr, psudo-dly tht ccounts for th brrtion of th rdio bm by binry motion, th Einstin dly (combind grvittionl rdshift nd spcil rltivistic tim-diltion in th pulsr frm) nd Shpiro dly (grvittionl tim-diltion in th vicinity of th compnion): B = RB + AB + EB + SB. (49)
TEMPO: timing modl nd prcision 1559 Th binry modl usd by TEMPO is bsd on thos of Blndford & Tukolsky (1976), Dmour & Drull (1986; hrftr DD), Tylor & Wisbrg (1989), Wx (1998 unpublishd contribution to TEMPO; s lso Lng t l. 001) nd Wx (1998), with xtr trms s dscribd by Kopikin (1995, 1996)..7.1 Romr dly nd Kopikin trms Th vrition in th distnc of th pulsr du to binry motion contributs four significnt trms to th full gomtric pth lngth of qution (5): s B = b + 1 ( k b r b + b ). (50) R 0 Th first trm of qution (50) is th wll-known first-ordr gomtric dly du to th initilly rdil componnt of th displcmnt du to binry motion. Th rmining trms r much smllr in mgnitud nd hv only provn msurbl in fw pulsrs to dt. Hrftr, w will rfr to ths collctivly s th Kopikin trms. Th scond trm of qution (50) dscribs chngs to th pprnt viwing gomtry of th orbitl motion, du to th propr motion of th systm (Arzoumnin t l. 1996; Kopikin 1996). An pproch tkn in th pst to ccount for this ffct is to llow it to b bsorbd in drivtivs of th projctd smimjor xis, x, nd longitud of pristron, ω, with th msurd vlus bing convrtd into constrints on th position ngl of scnding nod,, nd th inclintion, i (.g. Sndhu t l. 1997; Nic, Splvr & Stirs 001). Altrntivly, th binry modl cn b modifid to xplicitly includ linr chngs to x nd ω s prmtrizd by th fittbl prmtrs nd i (.g. vn Strtn 004). Hr, w tk third pproch which w bliv is clnr: rthr thn dfining th orbitl prmtrs in rotting rfrnc frm, w stt xplicitly tht th orbitl prmtrs rfr to th rfrnc frm dfind by th initil gomtry t tim t BB, nd hnc ccount for th ffct dirctly vi th scond trm of qution (50). Th third trm of qution (50) dscribs th so-clld nnul-orbitl prllx (Kopikin 1995). This my b intrprtd ithr s th modultion of th Solr systm Romr dly (Sction.5.3) du to th trnsvrs orbitl motion of th pulsr, or s th modultion of th binry Romr dly (first trm of qution 50) du to th trnsvrs orbitl motion of th Erth. As with th sculr chngs to th pprnt viwing gomtry, lthough th nnul-orbitl prllx hs bn modlld in th pst by prturbing th orbitl prmtrs with trms involving r (vn Strtn t l. 001; Splvr t l. 005), w choos to mintin consistnt rfrnc frm for th orbitl prmtrs nd instd modl th ffct dirctly vi th third trm of qution (50). Th fourth trm of qution (50) is th orbitl prllx (Kopikin 1995), which ccounts for th fct tht trnsvrs binry motion ltrs th lin of sight nd thrby cquirs n pprntly rdil componnt: this is th orbitl quivlnt of th Shklovskii ffct (Sction.6.). To our knowldg, this is yt to b msurd in ny pulsr, nd nturlly cnnot b bsorbd in tim-vrying orbitl prmtrs sinc th chngs thmslvs occur on th orbitl tim-scl. Th contributions of th four trms of qution (50) r includd in th timing modl s follows: RB = RB + KB, whr RB = b (5) c nd KB = 1 ( ) t BB t pos μ B b r b + b, (53) c cd AOP cd OP whr d AOP nd d OP r stimts of th initil distnc which cn b ithr hld fixd t som indpndntly dtrmind vlu (or omittd from th prmtr fil, to nglct th ffct), tid to othr fittbl distnc prmtrs (.g. d p nd/or d Shk ), or, givn sufficint constrints on th rlvnt orbitl prmtrs, fittd to obtin indpndnt distnc stimts. Likwis, μ B cn b ithr tid to μ, or omittd from th prmtr fil to nglct th ffct. Not tht ths prmtrs, in fct, rfr to th quntitis s obsrvd in frm tht is comoving with th BB, in contrst to corrsponding distnc nd propr motion prmtrs in prcding sctions. Howvr, th trnsformtion cn b nglctd for frctionl rror of O(v /c ) < 10 5 in thos prmtrs, corrsponding to chngs in th tim of mission clcultion wll blow our 1-ns gol. (51).7. Post-Nwtonin orbitl kinmtics Clcultion of th vrious orbitl dlys dpnds on knowldg of th displcmnt b of th pulsr from th binry systm brycntr, or t lst its rdil componnt b, dpnding on th dmndd ccurcy. TEMPO follows th gnrlizd post-nwtonin trtmnt of DD for th clcultion of b, with th ddition of sculr drivtivs for th orbitl priod, ccntricity nd projctd smimjor xis ftr Tylor &
1560 R. T. Edwrds, G. B. Hobbs nd R. N. Mnchstr Wisbrg (1989). Spcificlly, from qutions (16) nd (17) of DD sin cos 0 b = ( ˆ 1 ˆ ˆR 0 ) cos sin 0 1 0 0 b cos θ 0 cos i sin i b sin θ, (54) 0 0 1 0 sin i cos i 0 whr b =(1 r cos u), (55) n ( t psr T 0 ) = u sin u, θ = ω + A θ (u), [ ( ) ] 1 + A (u) = tn 1 tn u, 1 (56) (57) (58) r = (1 + δ r ) (59) θ = (1 + δ θ ) (60) n = π + πṗ ( ) b t psr T 0, (61) P b0 P b0 ω = ω 0 + ka (u), k = ω n, (6) (63) = 0 + ė ( ) t psr T 0, (64) whr P b0 nd Ṗ b r th initil vlu of th orbitl priod nd its drivtiv, ω 0 nd ω r initil longitud of pristron nd th mn of its drivtiv, T 0 is th propr tim of pristron, 0 nd ė r th initil propr tim ccntricity nd its drivtiv, nd δ r nd δ θ prmtriz rltivistic dformtions of th orbit. Th mtrics of qution (54) in th right- to lft-hnd ordr ccount for th inclintion of th orbit to th lin of sight (i), rottion bout th lin of sight ( ), nd projction from rdil trnsvrs bsis to frm rottionlly lignd to th ICRS (Sction.). Th mtrics r chosn such tht for i = 0, th ngulr momntum vctor of th orbit is ntiprlll to ˆR 0, nd msurs th position ngl of th scnding nod with rspct to ˆ, in th sns of rottion into ˆ 1. Th dfinitions of i nd thrfor corrspond to stronomicl convntion if ˆ 1 nd ˆ r st nd north vctors. This is th cs by dfult in TEMPO: ˆ 1 = ˆα, ˆ = ˆδ. Altrntivly, if cliptic coordints r in us (Sction.5.3), ˆ 1 = ˆλ nd ˆ = ˆβ, so tht position ngls r msurd countrclockwis from n cliptic mridin. Not tht th solution to Kplr s qution (56) involvs th propr tim of mission, t psr. This dpnds on th quivlnt coordint TOA t th BB, which, in turn, is rltd to th propr tim of rcption t th obsrvtory by ll of th trms in th timing modl, including thos involving t psr. This circulr dpndnc is rsolvd by strting with tpsr = t BB, computing th orbitl dlys, mking n updtd stimt of t psr, nd itrting this procss until th chng in th orbitl dly is lss thn 100 ps. Using th bov xprssion for b nd writing it in trms of its trnsvrs nd rdil componnts, th gomtric dly of qution (51) cn b found xplicitly. Bginning by xpnding S b / sin θ nd C b / cos θ, S = sin ω (cos u r ) + cos ω ( 1 θ) 1/ sin u, (65) nd C = cos ω (cos u r ) sin ω ( 1 θ) 1/ sin u, (66) whr w hv followd th ppndix of DD in mking minor modifiction to th dfinition of to simplify th xprssions. Writing qution (54) in trms of th projctions of b on ˆ 1, ˆ nd ˆR 0 s function of S nd C, b 1 = (C sin + S cos cos i), (67) b = (C cos S sin cos i), nd b = Ssin i. Th bsic Romr dly my thn b writtn s RB = xs, whr x = x 0 + ẋ ( ) t psr T 0, (68) (69) (70) (71)
TEMPO: timing modl nd prcision 1561 nd x /c sin i is th projctd smimjor xis s vcuum light trvltim, dfind in trms of its initil vlu, x 0 (DD) nd its drivtiv, ẋ (Tylor & Wisbrg 1989). For th Kopikin trms, following qution (53), KB = SR + AOP + OP, (7) whr SR = x ( ) t BB t pos [(μ1 sin + μ cos )C csc i + (μ 1 cos μ sin )S cot i], (73) AOP = x [(r ˆ 1 sin + r ˆ cos )C csc i + (r ˆ 1 cos r ˆ sin )S cot i], (74) d AOP nd OP = cx (C csc i + S cot i). (75) d OP Th propr motion componnts r dfind in trms of fittbl prmtrs tht cn optionlly flot indpndntly to th nnul propr motion prmtrs (Sction.5.3), in ordr to llow th lttr to b sprtd from th sculr chng of binry viwing gomtry. For qutoril coordints, μ 1 = μ αb nd μ = μ αb, whil for cliptic coordints μ 1 = μ λb nd μ = μ βb. In grmnt with stronomicl convntion, th prmtr corrsponds to th position ngl of pristron in trms of stwrd rottion from north, whil th prmtr i is dfind such tht i = 0 corrsponds to binry orbitl ngulr momntum vctor dirctd towrds th obsrvr. As pointd out by Splvr t l. (005), ths diffr from th dfinitions of Kopikin (1995, 1996). W not tht som cution is rquird in this r owing to vritions in dfinitions in th litrtur: for othr xmpls of th us of th unconvntionl dfinition of i, s, for xmpl, DD, Dmour & Tylor (199), Wisbrg & Tylor (00) nd Stirs, Thorstt & Arzoumnin (004)..7.3 Abrrtion Owing to th rltiv trnsvrs vlocity of th pulsr nd th Erth, th dirction of th obsrvr s sn from th pulsr diffrs from tht which would b msurd in frm comoving with th obsrvr. Th dirction is brrtd ccording to th Lorntz trnsformtion tht rlts th two frms. Undr th ssumption tht th sourc of pulss is th priodic rottion of n mission bm, ny such chng in th dirction of th obsrvr ltrs th rottionl phs corrsponding to rdition rcivd t givn tim. Following DD (qution 7), in TEMPO this is convrtd to n quivlnt tim-dly: AB = A {sin[ω + A (u)] + sin ω} + B {cos[ω + A (u)] + cos ω}, (76) whr A nd B r prmtrs rltd to th orinttion of th pulsr spin xis nd th siz of th orbit (s qutions 38 nd 39, nd sction 3., of DD). As discussd by DD, ths prmtrs r highly covrint with th othr binry prmtrs, nd my only b sprtd on th tim-scl of godtic prcssion. Not, hr th rltiv motion of th obsrvtory nd th BB is nglctd..7.4 Post-Nwtonin dlys In ddition to th modifid Romr dly of qution (51) nd th brrtion psudo-dly, undr th post-kplrin formlism of DD thr r two dditionl dly trms: th Einstin dly nd th Shpiro dly. Th Einstin dly is th diffrnc btwn th propr tim of mission nd th coordint tim in th qusi-inrtil frm of th BB: t BB = t psr + EB. (77) This is du to th combind ffct of grvittionl rdshift nd tim-diltion. Following DD nd in contrst to th TEMPO trtmnt of th Solr systm Einstin dly, th binry Einstin dly spcificlly xcluds linr trm by scling t psr in such wy tht th orbitl priod is numriclly th sm in ithr tim-scl. It is xprssd in thory-indpndnt mnnr in trms of timing modl prmtr, γ ; from qution (19) of DD: EB = γ sin u. (78) Th Shpiro dly is cusd by th curvtur of spc tim in th grvittionl fild of th binry compnion. Aftr qution (6) of DD: SB = r log { 1 cos u s [ sin ω (cos u ) + (1 ) 1/ cos ω sin u ]}. (79) Hr th fittbl prmtrs r th thory-indpndnt rng, r, nd shp, s. Undr gnrl rltivity, s = sin i nd r = Gm /c 3, whr m is th mss of th binry compnion. In TEMPO, r is xprssd in units of T = GM /c 3, so tht its vlu is numriclly qul to th gnrl rltivistic (GR) compnion mss in units of solr msss. (T is hlf th light trvltim cross th solr Schwrzschild rdius.) An ltrntiv prmtriztion is vilbl, whr th prmtr s is rplcd by th SHAPMAX prmtr, z s ln (1 s), This drivs from modifiction to TEMPO dsignd to void difficultis in fitting highly inclind orbits nr sin i =1 (Krmr t l. 006). In this cs, th bov rltion pplis ftr substituting s = 1 xp( z s ).