The orthometric parameterisation of the Shapiro delay and an improved test of general relativity with binary pulsars

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1 Mon. Not. R. Atron. Soc. 000, 1 15 (008) Printed 6 July 010 (MN LATEX tyle file v.) The orthometric parameteriation of the Shapiro delay and an improved tet of general relativity with binary pular Paulo C. C. Freire 1 & Norbert Wex 1 1 Max-Planck-Intitut für Radioatronomie, Auf dem Hügel 69, D-5311 Bonn, Germany Draft, in preparation ABSTRACT In thi paper, we expre the relativitic propagational delay of light in the pace-time of a binary ytem (commonly known a the Shapiro delay ) a a um of harmonic of the orbital period of the ytem. We do thi firt for near-circular orbit a a natural expanion of an exiting orbital model for low-eccentricity binary ytem. The amplitude of the 3 rd and higher harmonic can be decribed by two new pot- Keplerian (PK) parameter proportional to the amplitude of the third and fourth harmonic (, ). For high orbital inclination we ue a PK parameter proportional to the ratio of amplitude of ucceive harmonic (ς) intead of. The new PK parameter are much le correlated with each other than r and and provide a uperior decription of the contraint introduced by the Shapiro delay on the orbital inclination and the mae of the component of the binary. A leat-quare fit that ue alway converge, unlike in the cae of the (r, ) parameteriation; it reulting tatitical ignificance i the bet indicator of whether the Shapiro delay ha been detected. Until now thee contraint could only be derived from Bayeian χ map of the (coi, m c ) pace. We how that for low orbital inclination even thee map overetimate the mae of the component and that thi can be corrected by mapping the orthogonal (, ) pace intead. Finally, we extend the, ς parameteriation to eccentric binarie with high orbital inclination. For ome uch binarie we can meaure extra PK parameter and tet general relativity uing the Shapiro delay parameter. In thi cae we can ue the meaurement of a a tet of general relativity. We how that thi new tet i not only more tringent than the r tet, but it i even more tringent than the previou tet. Until now thi new parametric tet could only be derived tatitically from an analyi of a probabilitic χ map. Key word: binarie: general pular: general pular timing : general general relativity : general 1 INTRODUCTION In 1964 Shapiro pointed out that an electromagnetic wave paing near a maive body, uch a the Sun, uffer a relativitic time delay (Shapiro 1964). Thi wa the fourth tet of general relativity (GR), after the three claical tet propoed by Eintein: the perihelion preceion of Mercury, the deflection of light by the Sun and the gravitational redhift (Eintein 1916). The experimental value for the Shapiro delay determined by the Caini pacecraft agree with the GR prediction at the 0.00% level (Bertotti et al. 003). Outide the Solar Sytem, the Shapiro delay ha been oberved in a number of binary pular. If the inclination of the orbit with repect to the line-of-ight i ufficiently high then near uperior conjunction the pfreire@mpifr-bonn.mpg.de (PCCF) radio pule will experience a meaurable delay on it way from the pular to Earth (Lorimer & Kramer 005). In thi cae, auming GR i correct, the Shapiro delay allow to contrain or even determine the mae of the ytem. An early example of thi i PSR B (Ryba & Taylor 1991; Kapi et al. 1994). In binary pular where additional relativitic effect are oberved the Shapiro delay can be ued to tet GR (Stair 003); example of thi are PSR B (Stair et al. 00) and the double pular PSR J (Kramer et al. 006b). Although thee tet have not reached the ame level of preciion a the Caini experiment in the Solar Sytem, they are important ince they complement the Solar Sytem experiment. In particular in ytem where the companion i a neutron tar, one could expect trong-field effect on the propagation of photon, which would not be meaurable in c 008 RAS

2 Freire & Wex the weak field of the Solar Sytem (Damour & Taylor 199; Damour & Epoito-Farèe 1996). In thi paper we develop a new parameteriation of the Shapiro delay in binary pular baed on the Fourier expanion of thi effect in harmonic of the orbital period ( 3). A we will how, thi provide a uperior decription of the Shapiro delay ( 4), provide a new pace for improved χ map ( 5) and a much improved parametric tet of GR ( 6). the orbital period decay ( P b ). Auming a particular theory of gravity, thee parameter can be expreed a function of m p and m c (Damour & Taylor 199); eq. (4) i the implet of thee when GR i ued to do the ma calculation. If we meaure two precie PK parameter and aume a particular theory of gravity, then their equation completely determine m p and m c. If we meaure more than two PK parameter the ytem become over-determined; in that cae we can tet the conitency of the gravitational theory ued to calculate the mae. PULSAR TIMING AND SHAPIRO DELAY Pular timing i the technique that make ome neutron tar uperb atrophyical tool. It i epecially ueful in cae where the pular i located in a binary ytem (Stair 003 and reference therein). In uch cae, the orbital motion introduce a delay to the time of arrival of the pule at the Solar Sytem Barycenter given by (Damour & Taylor 199): = R + E + S + A, (1) where to firt order R i the Newtonian delay due to the light travel time acro the Keplerian part of the orbit (henceforth the Rømer delay), E i the Eintein delay, S i the Shapiro delay, and A i the aberration delay, which generally cannot be eparated from the Rømer delay. The two main orbital model being ued for binary pular timing (Damour & Deruelle 1986; Lange et al. 001) parameterie the Shapiro delay a a function of two pot- Keplerian (PK) parameter, the range (r) and hape () parameter: S(ϕ) = r ln [ 1 + e co ϕ 1 in(ω + ϕ) where ϕ i the true anomaly e i the orbital eccentricity and ω i the longitude of periatron relative to acending node. For mot theorie of gravity (Damour & Taylor 199; Will 1993) we have = in i. (3) where i i the orbital inclination. If general relativity (GR) i correct then r = T m c. (4) where T GM c 3 = µ i the the ma of the Sun in unit of time and m c i the ma of the companion tar. The ma function of the binary (f) i given by: (mc in i)3 f = (m p + m = n x 3 p, (5) c) T where n = π/p b i the mean angular velocity of the binary, P b i the oberved orbital period, x p i oberved the projected emi-major axi of the pular orbit (normally indicated in light econd), and m p i the ma of the pular. Apart from r and there are other PK parameter that can be determined from the timing of relativitic binary pular within the theory-independent parameteried PK approach (Damour & Deruelle 1986; Damour & Taylor 199): the mot important among thee are the rate of advance of periatron ( ω), the amplitude of the Eintein delay (γ) and ] () 3 FOURIER SERIES OF PROPAGATIONAL DELAYS FOR LOW ECCENTRICITIES In what follow we concentrate only on the propagational delay, RS R + S for low-eccentricity orbit. Firt, we dicu the Rømer delay for low eccentricitie in the formalim of Lange et al. (001). Second, we expand the Shapiro delay for mall-eccentricity orbit a a function of orbital harmonic, i.e., ine and coine wave with frequencie that are integer multiple of the orbital frequency. 3.1 The Rømer delay for low eccentricitie In thi ection, we eentially follow Lange et al. (001). If we neglect term of order e the Rømer delay can be written a ( R x p in Φ + κ in Φ η ) co Φ, (6) where term which are contant in time are omitted and Φ = n(t T ac), (7) i the celetial longitude of the binary, T ac i the time of acending node and η e in ω and κ e co ω, (8) are the firt and econd Laplace-Lagrange parameter. For the moment we conider thee quantitie to be contant in time. Their variation, and how it tranlate a a variation of the Keplerian parameter x p, e and ω i dicued in Lange et al. (001). 3. Fourier expanion of the Shapiro delay for nearly circular orbit For mall-eccentricity binary pular, the Shapiro delay (eq. ) can be re-written a follow: S = r ln(1 in Φ) rf(φ). (9) The function f(φ) can be expanded in a Fourier erie: f(φ) = a0 + a k co(kφ) + b k in(kφ), (10) k=1 with the following coefficient: k=1

3 The orthometric parameteriation of the Shapiro delay Orbital Inclination (degree) in i 1 - co i Figure 1. Behavior of ς (olid curve) with orbital inclination. For high orbital inclination it variation with i i imilar to 1 co i. a 0 = ln ( ) 1+ c, a 1 = 0, b 1 = ( 1+ c), a = ( 1+ c), b = 0, a 3 = 0, b 3 = ( 3 a 4 = c), b4 = 0, a 5 = 0, b 5 = 5 ( 1+ c) 6, b6 = 0, a 6 = ( 1+ c) 3, ( 1+ c) 5, (11) where c 1 = co i. We can define the orthometric ratio parameter to expre the ratio of the amplitude of the ucceive harmonic of the Shapiro delay: ς 1 + c = ( 1 c 1 + c ) 1/. (1) Since ς depend olely on, it alo ha a theory-independent meaning. Uing ς one find a cloed-form expreion for the Fourier coefficient in eq. (11): a 0 = ln(1 + ς ), a k = ( 1) k+ k ςk, k =,4, 6,... b k = ( 1) k+1 k ςk, k = 1,3, 5,... (13) Thee harmonic decribe what we actually oberve with pular timing (time delay) a a et of orthogonal function (the orbital harmonic). Their orthogonality implie that their amplitude are a priori uncorrelated. Thi ha important implication, a we will dicu below. 3.3 Low-inclination cae If the orbital inclination i mall, and ς will alo be mall and there will be a teep decreae in the power of each ucceive harmonic. We define a low inclination a cae where harmonic higher than are not detectable at the pular timing preciion, i.e., RS i given to ufficient preciion by the um of the firt two harmonic () RS. Whether a binary pular orbital inclination i low depend on it timing preciion and the number of meaured time of arrival (TOA). Ignoring contant term, the Shapiro delay can then be decribed a: S r(b 1 in Φ + a co Φ), (14) which correpond to eq. (A17) in Lange et al. (001). Adding eq. (6) and (14) and (again) ignoring contant term, we obtain for () RS an equation imilar to eq. (6): ( ) () RS xob p in Φ + κ ηob in Φ co Φ, (15) where x ob p = x p rb 1, (16) η ob = η + 4ra /x p (17) and we have ued x p r. Thi implie that the firt two harmonic of the Shapiro delay can be aborbed by a redefinition of x p and η. Thi mean that for low inclination the Shapiro delay cannot be eparated from the Rømer delay. For a circular binary there will be an apparent eccentricity given by eq. (17) with ω = Medium-inclination cae If the um of harmonic 3 and higher (3+) S then RS i given by i not negligible RS = () RS + (3+) S. (18) For low orbital eccentricitie harmonic 3 and higher are entirely due to the Shapiro delay. They can then be decribed by: (3+) S = 4 ( 1 3 in 3Φ 1 4 ς co4φ 1 5 ς in 5Φ ς3 co6φ +... Becaue the firt two harmonic of the Shapiro delay are aborbed by a re-definition of the Rømer delay ( 3.3) (3+) S repreent the meaurable part of the Shapiro delay; the new parameter rς 3. (0) quantifie it amplitude, for that reaon we call thi the orthometric amplitude parameter of the Shapiro delay. We define a medium inclination a binary pular where we can meaure the amplitude of the 3 rd harmonic (4/3), but not the 4 th and higher harmonic. The amplitude of the 4 th harmonic i given by ς; (1) it non-detectability implie that ς i ignificantly maller than 1. That impoe an upper limit on (or a lower limit on c), which are related to ς by: = ς 1 ς, c = () 1 + ς 1 + ς i.e., the binary cannot be edge-on, otherwie ς and would be cloe to 1 and harmonic higher than 3 hould be meaurable. ),(1

4 4 Freire & Wex Therefore eq. (19) i truncated at in 3Φ and we can parameterie the Shapiro delay with only. Note that thi quantity i alway well determined; if the Shapiro delay i not detectable then will imply be conitent with 0 (within meaurement error). Furthermore, i not correlated with any of the parameter that decribe the amplitude and phae of the lower harmonic (x p, T ac, κ and η), o it meaurement (or lack of it) ha no implication for any other parameter. If > 0 we obtain a curve in the r pace (or r c pace) where the binary mut be located from eq. (0): r = ( ) 3 ( 1 + c ) 3/ =, (3) 1 c the uncertainty of turn thi r line into a band of finite width (we will henceforth refer to the region of any pace that i conitent with the meaurement of a given PK parameter and it 1-σ uncertainty a it 1-σ band ). Thi equation alo mean that for the ame oberved third harmonic amplitude (or ) there will be an infinity of equally valid (r, ) combination, although ome olution at high inclination are excluded by the upper limit on ς. Thi ha the conequence that r and will generally be highly correlated. Unle the orbital inclination are high we cannot determine, thi mean that r cannot be determined with any ueful preciion. 3.5 High inclination cae We define an orbital inclination a high if harmonic above 3 are detectable. If harmonic 4 i detectable, then it amplitude i uncorrelated to and the Keplerian parameter. In principle thi make (, ) the bet poible parameteriation of the Shapiro delay. A we will ee below, thi i not the cae whenever many harmonic are detectable in the timing. From each value of we obtain a curve in the (, r) pace (or ( c, r) pace) given by r = ( ) 4 ( 1 + c ) =. (4) 1 c From eq. (1) and (), we can ee that thi cut the line at = h 3 +, c = h 3 h 4 h 4 h 3 + h 4 (5) r = h4 3. (6) 4 The high power of and in the expreion for r mean that it uncertainty i alway much larger than the uncertaintie of and. It i for thi reaon that meauring precie mae with the Shapiro delay i o difficult. 3.6 Very high inclination cae For in i 1 the ratio between ucceive harmonic (determined by ς) i cloe to 1 and therefore we tart detecting a large number of them. Up to order 4, the amplitude of each harmonic i uncorrelated and independent from the amplitude of all previou harmonic. That i no longer the cae for harmonic above 4 becaue the Shapiro delay can be parameteried by two parameter only (e.g., and ). Qualitatively, the amplitude of the higher harmonic bring no new information. So what i the benefit of meauring harmonic higher than 4? The time delay aociated with them are given by: (5+) [ 1 S = 4 in 5Φ ( h4 ) co 6Φ 1 7 ( h4 ) 3 in 7Φ +... Thi mean that the amplitude of the high harmonic contribute to a very precie meaurement of the / ratio (ς). If thi i meaured with better preciion than warranted by the uncertainty of and the latter parameter become correlated. If ρ(, ) > ρ(, ς) = 0.5 (where ρ(x,y) i the correlation between x and y) then and ς provide the bet decription of the Shapiro delay. Thu for high inclination we ue ς intead of a the econd independent parameter in the decription of the Shapiro delay. A previouly mentioned, ς ha the advantage of meauring the orbital inclination in a theory-independent way, like the Shapiro hape parameter. For high inclination the variation of ς with i i imilar to that of co i (ee Fig. 1). For randomly oriented orbit co i ha contant probability denity, therefore at high inclination the ame will be approximately true for ς. For very high inclination it no longer make ene to decribe the Shapiro delay a the um of a mall number of harmonic (eq. 19). In thi cae it i better to ue the exact expreion for thi um given by: [ (3+) ln(1 + ς ς in Φ) S = + in Φ ς 3 ς Adding the firt and econd harmonic we obtain: ] co Φ.(8) ς S = h3 ς 3 ln(1 + ς ς in Φ), (9) which i the equivalent of eq. (9) re-written in the, ς parameteriation. 4 TESTING THE NEW TIMING MODELS To tet our timing model, we ued the tempo 1 oftware package. We modified the ELL1 timing model by including 3 new timing model baed on the (, ) and (, ς) (henceforth the orthometric ) parameteriation. Their propertie are lited in Table 1 and their relative merit are dicued in 4.1. For high inclination we preferably ue the, ς parameter. For low inclination ς become ill-defined, making the 1/ς 3 factor ued in the exact decription of the Shapiro delay potentially very large. Thi can caue the numerical routine that find the bet fit to diverge. In thi cae we ue the, parameteriation: eq. (19) i ued to quantify with a mall number of harmonic and ς expreed a /. In order to tet thee model and compare their trength we created everal et of fake barycentric TOA for a binary ytem with e = 0, P b = 1.3 (3+) S 1 ].(7

5 The orthometric parameteriation of the Shapiro delay 5 Model RS = Harmonic PK Parameter Application a R (eq. 6) + S (eq. 9) All (Exact) (r, ) high inclination b R (eq. 6) + S (eq. 9) All (Exact) (, ς)* or (, ) high inclination c () (eq. 15) + (3+) (eq. 19) RS S 3 to N (Approximate) (, ς) or (, )* low inclination d () RS (eq. 15) + (3+) S (eq. 8) 3 to (Exact) (, ς)* or (, ) high inclination Table 1. Propagational delay in four eparate orbital model. Model a i ELL1, model b, c and d are eentially the ame except for the orthometric parameteriation of the propagational delay. Thee generally have a very low correlation between the parameter ued to decribe the Shapiro delay. The aterik denote preferred parameter pair. Harmonic pecifie the harmonic ued to decribe the Shapiro delay, the decription i approximate if harmonic above N (a number pecified by the uer) are ignored. Model c and d have the advantage that the reulting parameter are not correlated to the Keplerian parameter x, κ and η. Model b preerve the low correlation between, h4 (or, ς) but provide true value for x, κ and η, at the expene of introducing a correlation with them. Thi model ha the virtue that it can be readily extended to eccentric orbit. day, m p = 1.5M and m c = 0.M (imilar to PSR B (Kapi et al. 1994)) uing the exact DD model (Damour & Deruelle 1986). Thi ytem i then oberved at different orbital inclination for about ten year (obtaining ten TOA every fortnight) with a timing accuracy of 1 µ per TOA. We then ue tempo to fit a timing model to thee TOA; uing the timing model lited in Table Model comparion We firt compare the performance of the 3 exact highinclination model for the i = 75 data et. The reult of thi comparion are preented in Table. Whenever all harmonic of the Shapiro delay are ued (model a and b ) the value of the Keplerian parameter are conitent with the value ued in the imulation. A expected, there are trong correlation between thee and the PK parameter. If we ue harmonic 3 and above to decribe the Shapiro delay (model d ) then there are no correlation between the K and PK parameter, a expected. However, the value of the Keplerian parameter are modified by the 1 t and nd harmonic of the Shapiro delay a predicted by eq. (16) and (17). Thee modified value are alo meaured with better preciion: the uncertainty of x ob p i ten time maller than that of than x p and η ob i 3 time maller than η. Thi happen becaue thee oberved parameter meaure harmonic amplitude, which can be determined directly from the TOA; not computed quantitie like x p and η that are affected by uncertaintie in the meaurement of the Shapiro delay. The mot important leon to retain from thi comparion i the following: the value of the atrophyically important quantitie (, ς) and their (mall) cro-correlation are almot independent of whether we ue all harmonic (model b ) or only the meaurable harmonic (model d ) to decribe the Shapiro delay. More pecifically, the improvement in the decription of the Shapiro delay doe not depend on the lack of correlation with the Keplerian parameter achieved in model d. Thi mean that the orthometric parameteriation i intrinically uperior to the r, parameteriation. Thi ha important conequence, a we will ee in 6.1. In what follow we ue only (3+) S expreion to decribe the Shapiro delay. 4. Comparing different inclination. We now compare the two parameteriation for different orbital inclination. The PK parameter, their uncertaintie and correlation are lited in Table 3. Within their 1-σ uncertaintie, about 71% of the potfit PK parameter (r,, and, ς) match the value ued to produce the TOA. Thi i cloe to the 68% match rate that would generally be expected if the noie added i Gauian. However, for r and the uncertaintie are very large for the lowet inclination while the abolute uncertaintie of and are almot contant: the abolute preciion in the meaurement of the amplitude of a harmonic i independent of the amplitude itelf, it depend olely on the number and quality of the TOA. Furthermore, r and are very trongly correlated. In contrat, and are uncorrelated for the lower inclination, a expected from the fact that they decribe two orthogonal function of the time delay. At i = 30 neither r nor converge in a joint fit. In contrat, both and converge, but they are -σ conitent with zero (ee Table 3). The mall in particular indicate that there i no concluive detection of the Shapiro delay. Becaue of thi ς / i not well defined; confirming that it i not a good parameter to ue for the low orbital inclination. Since ς i directly related to thi explain why i not well defined either; thi i the reaon why the joint r fit doe not converge for thi inclination. The preciion of ς improve fater with inclination than one would expect from the improved relative preciion of and. A dicued in 3.6, thi happen becaue higher harmonic are contributing to the meaurement of ς, thu cauing the trong poitive correlation between and at high inclination. It i alo for thi reaon that the abolute uncertaintie of and decreae, ince they are mutually contrained by the precie meaurement of their ratio. Thi caue an equalization of their relative uncertaintie. For very high inclination we ee a decreae in the correlation of r and. If we ue z = ln(1 ) intead of to parameterie the Shapiro delay (Kramer et al. 006a) there i a further reduction in the correlation with r, a oberved for PSR J (Kramer et al. 006b). Thi mean that for uch edge-on ytem the ue of the, ς parameteriation doe not produce a ignificant improvement in the decription of the contraint introduced by the Shapiro delay.

6 6 Freire & Wex Simulation Parameter Fitted Timing Parameter a b d x p (lt-) (30) (30) - η (3) (3) - x ob p (lt-) (3) η ob (8) r (M ) (6) [0.1877] [0.1935] (9) [0.9746] [0.976] ς [0.7673] (7) 0.789(3) (µ) [0.4451] (19) 0.469(19) Correlation ρ(r, ) ρ(, x p) ρ(r, x p) ρ(, η) ρ(r, η) ρ(, ς) ρ(ς, x p) ρ(, x p) ρ(ς, η) ρ(, η) Table. Comparion of the exact high-inclination model a, b and d for i = 75. In thi and following table ρ(x, y) indicate the correlation between parameter x and y. Whenever all harmonic of the Shapiro delay are ued (model a and b ) the value of the Keplerian (K) parameter are conitent with the value ued in the imulation; but there are trong correlation between them and the pot-keplerian (PK) parameter. If we only ue the 3 rd and higher harmonic then there are no correlation between the K and PK parameter, but the value of x p i modified by the 1 t harmonic of the Shapiro delay and the value of η i modified by the nd harmonic of the Shapiro delay (eq. 16 and 17). Note the higher preciion of the modified Keplerian parameter. -: wa not fitted. In quare bracket: PK parameter calculated from the PK parameter actually being fitted. Value ued in Simulation Value from a model Value from new orbital model i ς r/t ρ(r, ) ς ρ(, ) ρ(, ς) ( ) (µ) (µ) (M ) (µ) (µ) * * * 0.03(1) 0.001(8) * 0.04 * (5) 0.5(40) (1) 0.008(7) 0.08(9) (11) 0.80(7) (0) 0.10(19) 0.53(9) (6) 0.975(9) (19) 0.370(13) 0.789(3) (8) (7) (17) 0.683(13) 0.91(7) (4) (4) (14) 0.907(1) 0.981() Table 3. Parameter ued in our TOA imulation and reult of fit made uing the a orbital model and the new orbital model. The c model i ued for i = 30 and the d model i ued for all other inclination. In the imulation we ued m c = r/t = 0.M. The aterik indicate intance where there wa no numerical convergence. Note the mall change in the preciion of the meaurement of and with orbital inclination. The 1-σ band correponding to the PK parameter in Table 3 are diplayed graphically in Fig. -5. For r and the 1-σ band are diplayed in orange; for the orthometric parameter they are diplayed in blue. In the (co i, m c) pace the latter are calculated uing eq. (0), (1) or (). They are then tranlated into 1-σ band in the (m p, m c) pace uing eq. (5). 4.3 Bayeian analyi For thee et of TOA, we made a Bayeian χ analyi in the (co i, m c) pace uing the conventional r, parameteriation, a decribed in (Splaver et al. 00). The choice of co i intead of in i come from the fact that a group of ytem with random orbital orientation will have a uniform ditribution of co i. For each co i, m c combination we calculate the correponding value of r,. Thee parameter are then kept fixed, the pin and Keplerian orbital parameter are fitted and the pot-fit χ i recorded. The -D probability ditribution function (pdf) ha a denity given by ( ) χ p(co i, m c) exp min χ (r,), (30) where χ min i the global minimum of χ (r, ). For the low inclination, where the reulting -D pdf i very pread out, we ue 0 < m c < 1. M ; for the higher inclination we focu the mapping in a maller area where the pdf i nonzero. The -D pdf i then tranlated into a -D pdf in the (m p, m c) pace. For the lower inclination we retrict thi to 0 < m p < 10M.

7 The orthometric parameteriation of the Shapiro delay 7 r r Figure. Shapiro delay contraint on the location of the PSR B analogue with i = 60 (high-inclination regime). For thi and all following plot the red dot indicate the value for m p, m c and co i ued to generate the imulated data. The orange curve indicate the 1-σ band correponding to the value of (dotted) and r (dot-dahed) lited in Table 3. The purple curve indicate 1-σ band of (olid) and (dahed). Left: (co i, m c) plot. In thi and ome of the following figure, the olid orange curve indicate m p = 0. The black olid curve i a contour level of the -D probability denity function that encloe 68.3% of the total probability. Right: m p m c plot. The black dahed curve i a contour level of the -D probability denity function that encloe 68.3% of the total probability in thi plane and within the range indicated in the figure. It i not a tranlation of the contour curve in the left plot. In thi and following figure, the grey area i excluded by in i 1. For thi orbital inclination and are the bet parameter to decribe the Shapiro delay, ince they are almot uncorrelated (Table 3). Although r and are well-defined (they converge in a joint fit), the area of the interection of their 1-σ band i much larger area than that encloed by the 68% contour. Furthermore, the 1-σ uncertainty for r under-etimate the range of the pdf in m c. In contrat, the interection of the 1-σ band of and decribe the allowed region very well. The hallow angle at which the 1-σ band of and meet i the reaon why it i difficult to make precie ma etimate uing Shapiro delay meaurement. The contour containing 68% of all probability in the (co i, m c) and (m p, m c) pace are uperpoed on the 1-σ band of the PK parameter in Fig., 3, 4 and Dicuion One immediately noticeable feature of the -D pdf diplayed in Fig., 3, 4 and 5 i their large extent to the high companion (and pular) mae, particularly for the low orbital inclination. Thi i merely a conequence of the large uncertainty in the meaurement of the companion ma for the low orbital inclination, coupled with the truncation of the pdf at m c = 0, i.e., a the orbital inclination decreae and the uncertainty increae the pdf can only extend in one direction. Depite thi, we can ee that in 3 out of 4 cae (i.e. 75%) the parameter ued to generate the imulated TOA are within the contour that include 68% of the probability, which i a cloe to the expectation a poible given the mall number tatitic. An inpection of thee figure alo how that the orthometric parameter provide a much improved decription of the m c and i contraint derived from the Shapiro delay. Fig. (high-inclination cae) i particularly intructive: the 68% contour cloely matche the interection of the and band, although it prefer the band ince thi i meaured with better relative preciion (ee Table 3). r and provide a very poor decription of thi 68% contour. Thi figure alo provide a viualization of why i it o difficult to meaure mae uing the Shapiro delay (ee 3.5): the and curve cut each other at a very hallow angle, o mall uncertaintie in either or produce large difference in the r where thee curve interect. Fig. 3 provide a graphical decription of the mediuminclination cae. The 1-σ band of (eq. 3) decribe the location of the ytem very well; thi alo how why r and are o highly correlated for thee lower orbital inclination. Since i conitent with zero ς / mut be mall; thi exclude high orbital inclination (ee 3.4) but all the lower inclination (up to arbitrarily large value of m c) are allowed. Fig. 4 provide a graphical decription of the lowinclination cae ( 3.3), where the Shapiro delay i not dec 008 RAS, MNRAS 000, 1 15

8 8 Freire & Wex r r h3 Figure 3. Probability ditribution function for i = 45 (medium-inclination regime) diplayed a in Fig.. For thi inclination we can only meaure the amplitude of the 3 rd harmonic; i conitent with 0 (Table 3 - the 1-σ lower limit of i not viible becaue it i negative). Thi exclude the high inclination, but not the lower inclination; the -D pdf now extend to arbitrarily large value of m c and m p. h3 Figure 4. Probability ditribution function for i = 30 (low-inclination regime), diplayed a in Fig.. The Shapiro delay i not eparable from the Rømer delay, i.e., and are now -σ conitent with 0 (Table 3). Depite that, their 1-σ band till provide a perfect decription of the -D pdf of the ytem, and repreent ueful contraint of it location in the (co i, m c) pace. Note that for the mall companion mae the ytem can now be located in a wide range or orbital inclination; thi i the reaon why and r no longer converge in a joint fit.

9 The orthometric parameteriation of the Shapiro delay 9 ς h 3 ς r r Figure 5. Probability ditribution function for i = 85 (very high inclination regime), diplayed a in Fig. but for maller inclination and ma range. For thi inclination and are highly correlated, o and ς are the bet parameter to decribe the Shapiro delay. tected. The value of and are conitent with zero within their -σ uncertaintie, therefore they overlap acro all inclination: it i for thi reaon that and r no longer converge in a numerical fit. Depite that, it i till true that the ytem can only be in ome part of the (co i, m c) and (m p, m c) pace, a decribed by the 68% contour. The 1- σ band of and till provide a perfect decription of thee contour, i.e., ueful contraint on the location of the ytem in the (co i, m c) pace. Fig. 5 provide a decription of the very high inclination cae. For thee high inclination ς i the bet parameter to decribe the Shapiro delay ( 3.6). In fig. 5 the 1-σ band of ς cover almot the ame region a the 1-σ band of, i.e., thee two quantitie are eentially interchangeable. Thi i to be expected becaue ς dependent on only (eq. 1). Depite the high preciion in the meaurement of and ς, their repective curve till interect at a very hallow angle in the (m c, m p) plot; thi explain the low preciion of the meaurement of m p even in cae where and ς (or ) are very preciely known. Perhap the mot important leon to be learned from thee plot i the following: the total area of the 1-σ band of i not only maller than thoe of r, but maller than thoe of. Thi applie particularly for the lower orbital inclination (ee Fig. ). Thi ha a very important conequence, dicued in PROBABILITY MAPS IN ORTHOMETRIC SPACE. We now concentrate on Fig. 3. One of the intereting feature of thi figure i that the region of high probability denity extend to arbitrarily large value of m c. For thi particular orbital inclination (i = 45 ) i conitent with zero but not. Thi exclude high inclination (ee eq. 5) but allow for arbitrarily low inclination, where the companion ma can be arbitrarily large (eq. 6). The 1-σ band of and (given by eq. 3 and 4) thu have an infinitely larger overlap in the low inclination - high m c region than in the high-inclination - low m c region. Therefore, if we are making a χ map in the (co i, m c) pace we end up with an arbitrarily large relative weight for the low inclination and high mae, which will grow a we increae the range of m c being mapped. We now ugget a way of deriving a ditribution that, like the reult of the (r,) fit, i independent of any pecific upper limit of m c. We begin by noting that whenever we make a χ map of the (co i, m c) pace we implicitly aume an a priori contant probability denity for m c and co i. It mut be treed that there i nothing unique about thi choice. Firt, we could a well attribute an a priori contant probability to m p. Becaue the tranlation between m c and m p i not linear (it i done with eq. 5) ampling the (co i, m p) pace would produce a pdf that i different than that obtained by ampling the (co i, m c) pace; our calculation how that it would be ytematically offet to higher value of m p and m c. Second, a uniform co i (i.e., an aumption that the orbital orientation are a priori ranc 008 RAS, MNRAS 000, 1 15

10 10 Freire & Wex = 1 (i = 90 ) r r = 1 (i = 90 ) ς Figure 6. Shapiro delay contraint on the location of the PSR B analogue in the orthogonal - pace. The red dot indicate the value for and that correpond to the m p, m c and co i ued to generate the imulated data (ee Table 3). The orange curve indicate 1-σ band of (dotted curve) and r (dot-dahed). The maller the value of m c the higher i it r curve in thi diagram, and the ooner it meet the ini = 1 line (where ς = 1 and conequently = = r). The purple line indicate the 1-σ band of (olid) and or ς (dahed) correponding to the value in Table 3. The olid curve encloe 68.3%, 95.4% and 99.7% of the total probability. The grey area i excluded by the condition in i 1. For i = 60 (Left) and are almot uncorrelated (Table 3). The region near the in i = 1 line i le probable becaue it require the preence of higher harmonic, which are not oberved. Thi caue a mall ditortion in the -D pdf and it contour and introduce the mall oberved correlation. For i = 85 (Right) and are ignificantly more correlated becaue the higher harmonic improve the preciion of ς. domly aligned) i definitely a good tarting aumption when no other information i available. However, in thi cae the timing provide further information regarding the orbital orientation and the ma of the companion. We decide to map the (, ) pace intead. For low inclination thee two parameter encode in an optimal way the timing information. Their orthogonality in thi pace eliminate the iue of the aymmetric overlap of their 1-σ band. Furthermore, we are making no direct aumption about the a priori probability ditribution of the phyical parameter, we aume intead a contant a priori probability for the amplitude of the harmonic and, which can be meaured directly from the timing reidual. We limit the mapping to the region where (i.e., where in i 1). We alo retrict it to m c 0 which from eq. (6) implie 0. We then ue an exact timing model ( d ) to fit for the pin and orbital parameter, but keeping and fixed. We then record the reulting χ and calculate probability map a decribed in 4.3. The reult of thi mapping are preented in Fig. 6 for two inclination (i = 60 and i = 85 ), uing the ame TOA dataet that were ued to produce Fig. and 5. The map depict graphically ome of the theoretical reult decribed in 3. For i = 60 and are very weakly correlated, a one would expect from the fact that they repreent the amplitude of two orthogonal function. For i = 85 the higher harmonic tart improving the preciion of the meaurement of ς /, thi introduce a trong correlation between and. The bet fit value of r produced by the d orbital model (r b ) correpond to the point where the line for the nominal value of and meet: r b = 3/ 4. In thee map of the (, ) pace there are very imilar amount of probability above and below r b, depite the fact that the value of r grow rapidly a we approach the = 0 line (at which point we have r = + and ς = = 0). The ame will therefore be true when we re-project thi -D pdf into the (co i, m c) and (m p, m c) pace, i.e., there will till be a imilar amount of probability below and above r b depite the fact that there i infinitely more r pace above r b. In Fig. 7 we project the -D (, ) pdf for i = 60 into the (co i, m c) and (m p, m c) pace (thick contour) and compare it with the -D pdf obtained by ampling the (co i, m c) pace directly (diplayed in Fig. and again here with thin contour). Although in both cae the 68% contour can be reaonably approximated by the interection of the 1-σ band of and, they are different amongt themelve: the thin contour are kewed toward higher mae relative to the thick contour. Looking at the marginal 1-D pdf for m c, we can alo ee that the 1-D pdf derived from

11 The orthometric parameteriation of the Shapiro delay 11 r r Figure 7. Same a Fig., but now for two -D pdf and with marginal 1-D probability ditribution function diplayed in the marginal plot. The ditance between the 0 and the vertical line in thee plot (horizontal in the m c plot) include.8%, 15.86%, 50% (median), 84.14% and 97.7% of the total probability; thee correpond to 1 and -σ interval around the median. The 1-D pdf are not Gauian, o the median and the peak in probability do not occur at the ame location. The thin contour correpond to the ame pdf diplayed in Fig., obtained by mapping the probability in r pace. The thick contour are a projection of the -D pdf diplayed on the left panel of Fig. 6 (i = 60 ) in the m c-co i and m c-m p pace. The 1-D pdf obtained from projecting the -D pdf along the m c axi ha a median and 1-σ lower limit that are very imilar to the value provided by the r, fit. We can alo ee that the former map ha a ytematic hift toward higher mae. Thi would be even more apparent were we to extend the mapping to higher value of m c and m p.

12 1 Freire & Wex the χ map of the (, ) pace produce median and lower 1-σ limit that are (at leat in thi cae) in better agreement with the reult of the r, fit (in orange) than thoe derived from the χ map of the (co i, m c) pace. Although it i problematic to make a χ map of the (co i, m c) pace for low orbital inclination (in ome cae the probabilitie depend enitively on the upper m c limit of the map, ee e.g. Fig. 3) thi i not a problem for high orbital inclination. In the latter cae the overlap of the 1-σ band i nearly ymmetrical relative to the bet olution given by the interection of and ς (ee fig. 5), o there i no ytematic offet between the two type of map. Indeed, projecting the -D (, ) pdf for i = 85 into the (co i, m c) and (m p, m c) pace we ee no noticeable difference in the reulting -D and marginal 1-D pdf. 6 IMPROVED TEST OF GENERAL RELATIVITY A dicued in, if we can determine more than two PK parameter for a binary pular the ytem of equation ued to olve m c, m p and i become over-determined and we can tet general relativity. A good example of thi i PSR B1534+1, a double neutron tar ytem with a relatively compact (P b = 10.1 hr) and eccentric (e = 0.7) orbit. Thi allowed for the firt time the meaurement of a total of 5 PK parameter (Stair et al. 1998; Stair et al. 00). In thi ytem, the two PK parameter meaured with better preciion ( ω and γ) can be ued to determine m c and m p, auming GR to be the correct theory of gravity. Once thi i done, the remaining 3 PK parameter repreent potential tet of general relativity. The oberved orbital period decay ( P b ), which i motly due to lo of orbital energy due to the emiion of gravitational wave, ha not provided an intereting tet for thi particular ytem becaue it ditance i not well known. Thi preclude an accurate correction of the kinematic contribution to the oberved P b, particularly the contribution due to the Shklovkii effect (Stair et al. 00). However, the relatively high orbital inclination of the ytem ( 77. ), large companion ma (1.345 ± M (Stair et al. 00)) and relatively good timing preciion (4 to 6 µ) have allowed a highly ignificant detection of the Shapiro delay. The two PK parameter that decribe it provide two independent tet of GR. In thi and other imilar cae an improved parameteriation of the Shapiro delay hould lead to improved parametric tet of GR. 6.1 Eccentric orbit The precie PK parameter ( ω and γ) can only be meaured if the orbit i eccentric. Therefore, if we are to ue the orthometric parameteriation to improve the preciion of GR tet we need to find out firt whether it i alo a better decription of the Shapiro delay for eccentric orbit. The baic aumption made at the tart of 3 wa that the firt and econd harmonic of the Shapiro delay can be completely aborbed in the Rømer delay, and that the higher harmonic are due olely to the Shapiro delay. For eccentric orbit thi i no longer the cae, particularly if there i a ignificant change in the longitude of periatron over the time pan of obervation. Moreover, the Fourier expanion of the Shapiro ha additional term that are proportional to r and independent of the inclination of the orbit: in an eccentric binary there i a phae-varying Shapiro delay even if it i een face-on (i = 0 ). For thi reaon, and alo becaue of their inherent implicity, we will from now on only conider exact decription of the Shapiro delay, like eq. (9) for the near circular cae or eq. () for the general elliptical cae. Dropping contant term the latter equation can be re-written a a function of and ς: S = h3 ς 3 [ ln(1 + eco ϕ) ln(1 + ς ς in(ω + ϕ)) ].(31) Note that thi equation ha the ame limitation a eq. (8) and (9): it can only be ued when ς i a well-defined quantity, i.e., for high orbital inclination. Furthermore, in thi cae the no longer ha the imple phyical interpretation (amplitude of the third harmonic) that it had for loweccentricity orbit. In 4.1 we found that, depite introducing correlation between, ς and the Keplerian parameter, the ue of the exact expreion for the Shapiro delay with all harmonic (eq. 9, ued in model b ) preerve the low correlation between and ς obtained uing olely the higher harmonic (eq. 8, ued in model d ), which i much lower than ρ(r,) in the traditional parameteriation (model a ). In what follow, we verify whether the latter propoition i alo true in the elliptical cae, i.e., whether depite the ue of all harmonic the orthometric parameteriation till provide an improved decription of the Shapiro delay. 6. Implementing and teting the eccentric timing model The DD model now ditributed in tempo ue eq. () to decribe the Shapiro delay, with r, a parameter. We extended thi orbital model with the option of uing eq. (31) intead, with and ς a parameter. The latter i an eccentric analogue of model b in Table 1, ince it ue an exact expreion for the Shapiro delay with all the harmonic that i a function of and ς. To tet thi orbital timing model, we created a lit of TOA for a imulated pular with orbital parameter imilar to thoe of PSR B Thi pular wa oberved every two week for a period of 0 year. In each obervation we obtain 4 TOA each with an uncertainty of 5 µ. We fit the traditional DD model to thee TOA uing the r, parameteriation for the Shapiro delay. The reult of thi fit are preented in Table 4. We then make the ame fit uing the orthometric parameteriation for the Shapiro delay; the reult are alo diplayed in Table 4. We alo made a χ map of the (co i, m c) pace: a mentioned above, for thee high inclination there i no ytematic offet between the ma ditribution produced in the co i, m c and, pace. For each point we keep only the correponding value of r and fixed and allow all other parameter (including the remaining PK parameter) to vary freely. We then record the reultant χ and calculate the -D pdf a dicued in 4.3. In thi manner we can ee the real contraint on the location of the ytem in the (co i, m c) pace derived from the Shapiro delay.

13 The orthometric parameteriation of the Shapiro delay 13 Parameter Simulation (r, ) (, ς) Value / GR prediction Timing Parameter m p (M ) m c (M ) i ( ) 77. P b (day) (4) (4) x p (lt-) (1) (1) e (14) (14) ω ( ) (4) (4) ω ( yr 1 ) [ ] (3) (3) γ () [ ] (8) (8) r (M ) [1.345] 1.35(11) [1.3576] 1.00 ± 0.08 [ ] 0.974(4) [ ] ± (µ) [ ] (11) 0.99 ± 0.03 ς [ ] (15) ± Correlation ρ(r, ) ρ(r, ω) ρ(r, γ) ρ(r, x p) ρ(r, e) ρ(, ω) ρ(, γ) ρ(, x p) ρ(, e) ρ(, ς) ρ(, ω) ρ(, γ) ρ(, x p) ρ(, e) ρ(ς, ω) ρ(ς, γ) ρ(ς, x p) ρ(ς, e) Table 4. Comparion of r, and, ς parameteriation of the Shapiro delay for an eccentric orbit. In the firt column, the value in quare bracket are derived from m p, m c, i and the Keplerian parameter. In the third column, r and are derived from the value of and ς The 1-σ band correponding to the PK parameter in Table 4 are diplayed graphically in Fig. 8. On top of thee we overlay the contour of the that -D pdf that include 68.3% of it total probability. 6.3 Dicuion The reult in Table 4 how that the computation of the Keplerian and pot-keplerian parameter (apart from thoe aociated with the Shapiro delay) are not affected at all by the choice of the parameteriation of the Shapiro delay. Thi i expected from the exact equivalence of the delay predicted by the two parameteriation (eq. and eq. 31). Second, not much change in term of correlation with the remaining Keplerian and pot-keplerian parameter, except that, a in the circular cae, and ς are le correlated with each other than r and. Thi how that the orthometric parameteriation i alo a uperior decription of the Shapiro delay for eccentric orbit. Third, we can ue the value of m p, m c and i ued to imulate the TOA to predict, ς, r and uing the equation in 3; thee prediction are preented in the firt column of Table 4. For a real pular we would intead ue the meaurement of ω, γ and f to calculate m p, m c and i auming that general relativity i correct and then predict, ς, r and. At the level of preciion to which the Shapiro delay parameter are meaured it doe not make any difference which method we ue make thi prediction. For the imulated et of PSR B TOA GR pae both the traditional r, tet and alo the new, ς tet. Note, however, that the tet i more tringent than the previou bet tet ( ): by thi we mean that for any particular value of ω or γ, the meaurement of and it uncertainty implie a maller range of co i, m c and m p than that allowed by the meaurement of /ς (ee Fig. 8). 7 CONCLUSIONS In thi paper we dicued what can be learned from an expanion of the Shapiro delay in harmonic of the orbital period for a ytem with mall orbital eccentricity, a i the cae for the vat majority of the milliecond pular where thi effect can be meaured. The amplitude of thee harmonic, meaured directly

14 14 Freire & Wex ω. ω. γ r r γ ω. ω. γ γ ς ς Figure 8. Contraint on the location of the PSR B analogue derived from four PK parameter. The orange curve indicate the 1-σ band correponding to the value of traditional PK parameter lited in Table 4. The purple curve indicate the 1-σ band of (olid) and ς (dahed), alo lited in that Table. The black olid contour encloe 68.3% of the total probability in the -D pdf. Left: (co i, m c) plot; Right: m p m c plot, with the grey area excluded by in i 1. Top: The current r, tet of general relativity. Bottom: New, tet of general relativity. The ma function and the interection of the 1-σ band of ω and γ produce etimate of i, m c and m p which are cloe to the value ued in the imulation (red dot). A in the circular cae, the 1-σ band of and ς provide a uperior decription of the 63.8% contour allowed by the Shapiro delay. The and ς tet are eentially equivalent: their 1-σ band are nearly identical, depite the apparent difference in relative preciion (ee Table 4). On the other hand, the tet (which eentially replace the r tet) i much more tringent. Remarkably, it i alo more tringent than the /ς tet: for any particular value of ω or γ it allow maller range of m c, m p and i.

15 The orthometric parameteriation of the Shapiro delay 15 from the time delay, provide a much improved parametric decription of the region of the m c-co i pace where the binary can be located, even when no Shapiro delay can be meaured. In particular, we how that the orthometric amplitude parameter = r i alway meaured 1 ( ) 3 1+ with ignificantly higher preciion than one would generally etimate from the uncertaintie of r and. For low inclination, the amplitude of the fourth harmonic () i the bet parameter to complete the decription of the Shapiro delay becaue and are then uncorrelated. For high inclination the amplitude of the higher harmonic improve the preciion of the meaurement of the orthometric ratio parameter ς / beyond what i poible from the individual value of and ; therefore the latter become trongly correlated. If ρ(, ) > ρ(, ς) = 0.5 the latter become the bet parameter to decribe the Shapiro delay. Becaue of thee low correlation, the new orthometric parameter provide uperior parametric decription of the ma and orbital inclination contraint determined from the Shapiro delay, which previouly could only be decribed by probabilitic χ map of the (co i, m c) pace. The χ map of the (, ) pace make no explicit aumption about the a priori probability ditribution of the phyical parameter, we aume intead a contant a priori probability for and, which can be meaured directly from the timing reidual. Unlike in the cae of a χ map of the (co i, m c) pace, they don t depend on the range of m c being mapped and guarantee that the total probability i nearly equally plit between region with r above and below the bet value r b. We alo how that the improved decription of the Shapiro delay can be extended to high-eccentricity binarie. Thi ha the conequence that in eccentric ytem where other PK parameter are known we can now make a ignificantly improved tet of GR by meauring intead of r. Remarkably, the tet i more tringent than the previou tet. It i important to note that no new information i provided by thi re-parameteriation of the Shapiro delay beyond what wa previouly provided by a χ map. It main advantage i being concie, ince we only need two numeric parameter to decribe the -D pdf. Both technique extract information from the oberved timing delay that wa not provided by the previou r, parameteriation. REFERENCES Bertotti, B., Ie, L., & Tortora, P. 003, Nature, 45, 374 Damour, T. & Deruelle, N. 1986, Ann. Int. H. Poincaré (Phyique Théorique), 44, 63 Damour, T. & Epoito-Farèe, G. 1996, Phy. Rev. D, 54, 1474 Damour, T. & Taylor, J. H. 199, Phy. Rev. D, 45, 1840 Eintein, A. 1916, Annnalen der Phyik, 354, 769 Kapi, V. M., Taylor, J. H., & Ryba, M. 1994, ApJ, 48, 713 Kramer, M. et al. 006a, Annnalen der Phyik, 15, 34 Kramer, M. et al. 006b, Science, 314, 97 Lange, C., Camilo, F., Wex, N., Kramer, M., Backer, D., Lyne, A., & Dorohenko, O. 001, MNRAS, 36, 74 Lorimer, D. R. and Kramer, M. 005, Handbook of Pular Atronomy, (Cambridge: Cambridge Univerity Pre) Stair, I. H., Arzoumanian, Z., Camilo, F., Lyne, A. G., Nice, D. J., Taylor, J. H., Thorett, S. E., & Wolzczan, A. 1998, ApJ, 505, 35 Shapiro, I. I. 1964, Phy. Rev. Lett., 13, 789 Splaver, E. M., Nice, D. J., Arzoumanian, Z., Camilo, F., Lyne, A. G., & Stair, I. H. 00, ApJ, 581, 509 Stair, I. H. 003, Living Review in Relativity, 6(5) Stair, I. H., Thorett, S. E., Taylor, J. H., & Wolzczan, A. 00, ApJ, 581, 501 Ryba, M. F., & Taylor, J. H. 1991, ApJ, 371, 739 Will, C. M. 1993, Theory and Experiment in Gravitational Phyic, (Cambridge: Cambridge Univerity Pre) ACKNOWLEDGEMENTS We thank Marten van Kerkwijk for hi comment and uggetion regarding thi work. Since completing it we have learned that he ha independently undertood the need for an improved parameteriation of the Shapiro delay and, in an unpublihed work, derived ome of the reult dicued here, namely the Fourier expanion of the Shapiro delay and alo an equivalent of eq. (8) albeit in a lightly different non-orthogonal parameteriation. We alo thank the referee, Ingrid Stair, for her careful and meticulou review and thoughtful uggetion, which have improved the quality of thi work.