Post-Keplerian perturbations of the orbital time shift in binary pulsars: an analytical formulation with applications to the Galactic Center

We develop a general approach to analytically calculate the perturbations $\Delta\delta\tau_\textrm{p}$ of the orbital component of the change $\delta\tau_\textrm{p}$ of the times of arrival of the pulses emitted by a binary pulsar p induced by the post-Keplerian accelerations due to the mass quadrupole $Q_2$, and the post-Newtonian gravitoelectric (GE) and Lense-Thirring (LT) fields. We apply our results to the so-far still hypothetical scenario involving a pulsar orbiting the Supermassive Black Hole in in the Galactic Center at Sgr A$^\ast$. We also evaluate the gravitomagnetic and quadrupolar Shapiro-like propagation delays $\delta\tau_\textrm{prop}$. By assuming the orbit of the existing S2 main sequence star and a time span as long as its orbital period $P_\textrm{b}$, we obtain $\left|\Delta\delta\tau_\textrm{p}^\textrm{GE}\right|\lesssim 10^3~\textrm{s},~\left|\Delta\delta\tau_\textrm{p}^\textrm{LT}\right|\lesssim 0.6~\textrm{s},\left|\Delta\delta\tau_\textrm{p}^{Q_2}\right|\lesssim 0.04~\textrm{s}$. Faster $\left(P_\textrm{b} = 5~\textrm{yr}\right)$ and more eccentric $\left(e=0.97\right)$ orbits would imply net shifts per revolution as large as $\left|\left\langle\Delta\delta\tau_\textrm{p}^\textrm{GE}\right\rangle\right|\lesssim 10~\textrm{Ms},~\left|\left\langle\Delta\delta\tau_\textrm{p}^\textrm{LT}\right\rangle\right|\lesssim 400~\textrm{s},\left|\left\langle\Delta\delta\tau_\textrm{p}^{Q_2}\right\rangle\right|\lesssim 10^3~\textrm{s}$, depending on the other orbital parameters and the initial epoch. For the propagation delays, we have $\left|\delta\tau_\textrm{prop}^\textrm{LT}\right|\lesssim 0.02~\textrm{s},~\left|\delta\tau_\textrm{prop}^{Q_2}\right|\lesssim 1~\mu\textrm{s}$. The expected precision in pulsar timing in Sgr A$^\ast$ is of the order of $100~\mu\textrm{s}$, or, perhaps, even $1-10~\mu\textrm{s}$.

The results for the mass quadrupole and the Lense-Thirring field depend, among other things, on the spatial orientation of the spin axis of the Black Hole. The expected precision in pulsar timing in Sgr A * is of the order of 100 µs, or, perhaps, even 1 − 10 µs. Our method is, in principle, neither limited just to some particular orbital configuration nor to the dynamical effects considered in the present study.

Introduction
In a binary hosting at least one emitting pulsar 1 p, the time of arrivals τ p of the emitted radio pulses changes primarily because of the orbital motion about the common center of mass caused by the gravitational tug of the unseen companion c which can be, in principle, either a main sequence star or an astrophysical compact object like, e.g., another neutron star which does not emit or whose pulses are, for some reasons, not detectable, a white dwarf or, perhaps, even a black hole (Wex & Kopeikin 1999). Such a periodic variation δτ p ( f ) can be modeled as the ratio of the projection of the barycentric orbit r p of the pulsar p onto the line of sight to the speed of light c (Damour & Schaefer 1991;Konacki, Maciejewski & Wolszczan 2000). By assuming a coordinate system centered in the binary's center of mass whose reference z-axis points toward the observer along the line of sight in such a way that the reference {x, y} plane coincides with the plane of the sky, we have δτ p ( f ) = r p z c = r p sin I sin u c = a p 1 − e 2 sin I sin u c (1 + e cos f ) = m c p sin I sin (ω + f ) m tot c (1 + e cos f ) . (1) In obtaining Equation (1), which is somewhat the analogous of the range in Earth-Moon or Earth-planets studies (Damour & Schaefer 1991), we used the fact that, to the Keplerian level, the barycentric semimajor axis of the pulsar A is In a purely Keplerian scenario, there is no net variation ∆δτ p over a full orbital cycle.
In this paper, we illustrate a relatively simple and straightforward approach to analytically calculate the impact that several post-Keplerian (pK) features of motion, both Newtonian (quadrupole) and post-Newtonian (1pN static and stationary fields), have on such a key observable. As such, we will analytically calculate the corresponding net time delays per revolution ∆δτ p ; the instantaneous shifts ∆δτ p ( f ) will be considered as well in order to cope with systems exhibiting very long orbital periods with respect to the time spans usually adopted for data collection. Our strategy has a general validity since, in principle, it can be extended to a wide range of dynamical effects, irrespectively of their physical origin, which may include, e.g., modified models of gravity as well. Furthermore, it is applicable to systems whose constituents may have arbitrary masses and orientations of their spin axes, and orbital configurations. Thus, more realistic sensitivity analyses, aimed to both re-interpreting already performed studies and designing future targeted ones, could be conducted in view of a closer correspondence with which is actually measured. We we will also take into account the Shapiro-like time delays due to the propagation of the electromagnetic waves emitted by the visible pulsar(s) throughout the spacetime deformed by axisymmetric departures from spherical symmetry of the deflecting bodies (Damour & Deruelle 1986;Klioner 1991;Damour & Taylor 1992;Doroshenko & Kopeikin 1995;Kopeikin 1997;Wex & Kopeikin 1999;Klioner 2003;Zschocke & Klioner 2011).
Our results, which are not intended to replace dedicated, covariance-based real data analyses, being, instead, possible complementary companions, will be applied to the so far putative scenario involving emitting radiopulsars, not yet detected, orbiting the Supermassive Black Hole (SMBH) in the Galactic Center (GC) at Sgr A * (Angélil, Saha & Merritt 2010;Eatough et al. 2013;Zhang, Lu & Yu 2014;Kramer 2016;Johannsen 2016;Psaltis, Wex & Kramer 2016;Goddi et al. 2017). Moreover, we will perform also quantitative sensitivity analyses on the measurability of frame-dragging and quadrupolar-induced time delays in such a hypothesized system. In principle, our results may be applicable even to anthropogenic binaries like, e.g., those contrived in past concept studies to perform tests of fundamental physics in space (Nobili, Milani & Farinella 1988;Sahni & Shtanov 2008), or continuously emitting transponders placed on the surface of some moons of larger astronomical bodies.
The paper is organized as follows. Section 2 details the calculational approach. The 1pN Schwarzschild-type gravitoelectric effects are calculated in Section 3, while Section 4 deals with the 1pN gravitomagnetic ones. The impact of the quadrupole mass moment of the SMBH is treated in Section 5. Section 6 summarizes our findings.

Outline of the proposed method
If the motion of a binary is affected by some relatively small post-Keplerian (pK) acceleration A, either Newtonian or post-Newtonian (pN) in nature, its impact on the projection of the orbit onto the line of sight can be calculated perturbatively as follows. Casotto (1993) analytically worked out the instantaneous changes of the radial, transverse and out-of-plane components r ρ , r σ , r ν of the position vector r, respectively, for the relative motion of a test particle about its primary: they are In Equations (3) to (5), the instantaneous changes ∆a where the time derivatives dκ/dt of the Keplerian orbital elements κ are to be taken from the right-hand-sides of the Gauss equations evaluated onto the Keplerian ellipse given by and assumed as unperturbed reference trajectory; the same holds also for entering Equation (6). The case of the mean anomaly M is subtler, and requires more care. Indeed, in the most general case encompassing the possibility that the mean motion n b is time-dependent because of some physical phenomena, it can be written as 2 (Milani, Nobili & Farinella 1987;Brumberg 1991;Bertotti, Farinella & Vokrouhlický 2003) the Gauss equation for the variation of the mean anomaly at epoch is 3 (Milani, Nobili & Farinella 1987;Brumberg 1991;Bertotti, Farinella & Vokrouhlický 2003) If n b is constant, as in the Keplerian case, Equation (14) reduces to the usual form In general, when a disturbing acceleration is present, the semimajor axis a does vary according to Equation (7); thus, also the mean motion n b experiences a change 4 which can be calculated in terms of the true anomaly f as by means of Equation (7) and Equation (13). Depending on the specific perturbation at hand, Equation (18) does not generally vanish. Thus, the total change experienced by the mean anomaly M due to the disturbing acceleration A can be obtained as where In the literature, the contribution due to Equation (21) has been often neglected. An alternative way to compute the perturbation of the mean anomaly with respect to Equation (19) implies the use of the mean longitude λ and the longitude of pericenter ̟. It turns out that 5 (Soffel 1989) where the Gauss equations for the variation of ̟, ǫ are (Milani, Nobili & Farinella 1987;Soffel 1989;Brumberg 1991;Bertotti, Farinella & Vokrouhlický 2003) d̟ dt = 2 sin 2 I 2 It must be remarked that, depending on the specific perturbing acceleration A at hand, the calculation of Equation (21) may turn out to be rather uncomfortable.
The instantaneous change experienced by the projection of the binary's relative motion onto the line of sight can be extracted from Equations (3) to (5) by taking the z component ∆r z of the vector ∆r = ∆r ρρ + ∆r σσ + ∆r νν (25) expressing the perturbation experienced by the binary's relative position vector r. It is It is possible to express the true anomaly as a function of time through the mean anomaly according to Brouwer & Clemence (1961, p. 77) where J k (se) is the Bessel function of the first kind of order k and s max , j max are some values of the summation indexes s, j adequate for the desired accuracy level. Having at disposal such analytical time series yielding the time-dependent pattern of Equation (26) allows one to easily study some key features of it such as, e.g., its extrema along with the corresponding epochs and the values of some unknown parameters which may enter the disturbing acceleration. The net change per orbit ∆r z can be obtained by calculating Equation (26) with f = f 0 + 2π, and using Equation (6) and Equations (19) to (21) integrated from f 0 to f 0 + 2π.
In order to have the change of the times of arrival of the pulses from the binary's pulsar p, Equation (26) and its orbit averaged expression have to be scaled by m c m −1 tot c −1 . In the following, we will look at three pK dynamical effects: the Newtonian deviation from spherical symmetry of the binary's bodies due to their quadrupole mass moments, and the velocity-dependent 1pN static (gravitoelectric) and stationary (gravitomagnetic) accelerations responsible of the time-honored anomalous Mercury's perihelion precession and the Lense-Thirring frame-dragging, respectively.

The 1pN gravitoelectric effect
Let us start with the static component of the 1pN field which, in the case of our Solar System, yields the formerly anomalous perihelion precession of Mercury of̟ = 42.98 arcsec cty −1 (Nobili & Will 1986).
The 1pN gravitoelectric, Schwarzschild-type, acceleration of the relative motion is, in General Relativity, (Soffel 1989) By projecting Equation (28) onto the radial, transverse, out-of-plane unit vectorsρ,σ,ν, its corresponding components are Here, we use the true anomaly f since it turns out computationally more convenient.
If, on the one hand, Equation (33) is the well known relativistic pericenter advance per orbit, on the other hand, Equation (34) represents a novel result which amends several incorrect expressions existing in the literature (Rubincam 1977;Iorio 2005Iorio , 2007, mainly because based only on Equation (15). Indeed, it turns out that Equation (21), integrated over an orbital revolution, does not vanish. By numerically calculating Equation (34) with the physical and orbital parameters of some binary, it can be shown that it agrees with the expression obtainable for ∆M from Equations (A2.78e) to (A2.78f) by Soffel (1989, p. 178) in which all the three anomalies f, E, M appear. It should be remarked that Equation (34) is an exact result in the sense that no a-priori assumptions on e were assumed. It can be shown that, to the zero order in e, Equation (34) is independent of f 0 .
We will not explicitly display here the analytical expressions for the instantaneous changes ∆κ GE ( f ) , κ = a, e, I, Ω, ω, ∆M GE ( f ) because of their cumbersomeness, especially as far as the mean anomaly is concerned. However, ∆κ GE ( f ) , κ = a, e, I, Ω, ω can be found in Equations (A2.78b) to (A2.78d) of Soffel (1989, p. 178). Equations (A2.78e) to (A2.78f) of Soffel (1989, p. 178)  4 1 − e 2 5/2 8 (−9 + 2ξ) + 4e 4 (−6 + 7ξ) + e 2 (−84 + 76ξ) + It should be noted that Equation (35) is independent of the semimajor axis a, depending only on the shape of the orbit through e and its orientation in space through I, ω. Furthermore, Equation (35) does depend on the initial epoch t 0 through f 0 . In the limit e → 0, Equation (35) does not vanish, reducing to In view of its cumbersomeness, we will not display here the explicit expression of ∆δτ GE p ( f ) whose validity was successfully checked by numerically integrating the equations of motion for a fictitious binary system, as shown by Figure 1; see also Section 3.1.
We will not deal here with the Shapiro-like propagation delay since it was accurately calculated in the literature; see, e.g., Damour & Deruelle (1986); Damour & Taylor (1992) and references therein.
3.1. The pulsar in Sgr A * and the gravitoelectric orbital time delay An interesting, although still observationally unsupported, scenario involves the possibility that radio pulsars orbit the SMBH at the GC in Sgr A * ; in this case, the unseen companion would be the SMBH itself. Thus, in view of its huge mass, the expected time shift per orbit ∆δτ GE p would be quite large.
By considering a hypothetical pulsar with standard mass m p = 1.4 M ⊙ and, say, the same orbital parameters of the main sequence star S2 actually orbiting the Galactic SMBH (Gillessen et al. 2017), Equation (35) Eq. (35) allows to find the maximum and minimum values of the net orbital change per revolution of the putative pulsar in Sgr A * by suitably varying e, I, ω, f 0 within given ranges. By limiting ourselves to 0 ≤ e ≤ 0.97 for convergence reasons of the optimization algorithm adopted, we have Such huge orbital time delays would be accurately detectable, even by assuming a pessimistic pulsar timing precision of just 100 µs (Psaltis, Wex & Kramer 2016;Goddi et al. 2017); more optimistic views point towards precisions of the order of even 1 − 10 µs (Psaltis, Wex & Kramer 2016;Goddi et al. 2017).

The 1pN gravitomagnetic Lense-Thirring effect
The stationary component of the 1pN field, due to mass-energy currents, is responsible of several aspects of the so-called spin-orbit coupling, or frame-dragging (Dymnikova 1986;Thorne 1988;Schäfer 2004Schäfer , 2009).
The 1pN gravitomagnetic, Lense-Thirring-type, acceleration affecting the relative orbital motion of a generic binary made of two rotating bodies A, B is (Barker & O'Connell 1975;Soffel 1989) In general, it isŜ i.e. the angular momenta of the two bodies are usually not aligned. Furthermore, they are neither aligned with the orbital angular momentum L, whose unit vector is given byν. Finally, also the magnitudes S A , S B are, in general, different.
The radial, transverse and out-of-plane components of the gravitomagnetic acceleration, obtained by projecting Equation (42) onto the unit vectorsρ,σ,ν, turn out to be By using Equations (44) to (46) in Equation (6) and Equations (19) to (21) and integrating them from f 0 to f 0 + 2π, it is possible to straightforwardly calculate the 1pN gravitomagnetic net orbital changes for a generic binary arbitrarily oriented in space: they are It is interesting to remark that, in the case of Equation (42), both Equation (15) and Equation (21) yield vanishing contributions to ∆M LT . For previous calculations based on different approaches and formalisms, see, e.g., Barker & O'Connell (1975); Damour & Schafer (1988); Iorio (2011), and references therein.
Eq. (26), calculated with Equations (47) to (50), allows to obtain the net orbit-type time change per revolution of the pulsar p as Note that, contrary to Equation (35), Equation (51) does depend on the semimajor axis as a −1/2 . As Equation (35), also Equation (51) depends on f 0 . The instantaneous orbital time shift ∆δτ LT p ( f ) turns out to be too unwieldy to be explicitly displayed here. Its validity was successfully checked by numerically integrating the equations of motion for a fictitious binary system, as shown by Figure 2; see also Section 4.1.
The gravitomagnetic propagation time delay is treated in Section 4.2.

The pulsar in Sgr A * and the Lense-Thirring orbital time delay
Let us, now, consider the so-far hypothetical scenario of an emitting radio pulsar orbiting the SMBH in Sgr A * (Angélil, Saha & Merritt 2010;Goddi et al. 2017).
It turns out that, in some relevant astronomical and astrophysical binary systems of interest like the one at hand, the (scaled) angular momentum S A/B of one of the bodies is usually much smaller than the other one. Let us assume that the pulsar under consideration has the same characteristics of PSR J0737-3039A. By assuming (Morrison et al. 2004;Bejger, Bulik & Haensel 2005) I NS ≃ 10 38 kg m 2 (52) for the moment of inertia of a neutron star (NS), the spin of PSR J0737-3039A is The angular momentum of a NS of mass m NS can also be expressed in terms of the dimensionless parameter χ g > 0 as (Laarakkers & Poisson 1999) Thus, Equation (53) implies for PSR J0737-3039A. Since for the Galactic SMBH it is (Hansen 1974) we have S p S • ≃ 6 × 10 −9 .
Thus, in this case, the dominant contribution to Equation (42) is due to the pulsar's companion c. As far as the orientation of the SMBH's spin is concerned, we model it aŝ The angles i • , ε • are still poorly constrained (Broderick et al. 2009(Broderick et al. , 2011Yu, Zhang & Lu 2016), so that we prefer to treat them as free parameters by considering their full ranges of variation Also in this case, we assume for our putative pulsar the same orbital parameters of, say, the S2 star.
By using our analytical expression for ∆δτ LT p (t), calculated with S p → 0 in view of Equation (58) within the assumed ranges of variation for the angles i • , ε • provided by Equations (60) to (61).
To this aim, see Figure 2 which shows both the analytical time series, calculated with Equations (26) to (27) where we considered 5 yr ≤ P b ≤ 16 yr; the ranges of variation assumed for the other parameters I, ω, Ω, f 0 , i • , ε • are the standard full ones. The values of Equations (62) to (63) and Equations (64) to (65) should be compared with the expected pulsar timing precision of about 100 µs, or, perhaps, even 1 − 10 µs (Psaltis, Wex & Kramer 2016;Goddi et al. 2017).

The Lense-Thirring propagation time shift
The gravitomagnetic propagation delay for a binary with relative separation r and angular momentum S of the primary is (Kopeikin 1997;Wex & Kopeikin 1999) According to Figure 2 of Wex & Kopeikin (1999), their unit vector K 0 agrees with ourê z since it is directed towards the Earth. About the spin axis of the primary, identified with a BH by Wex & Kopeikin (1999), our i • coincides with their λ • . Instead, their angle η • is reckoned from our unit vectorl, i.e. it is as if Wex & Kopeikin (1999) set Ω = 0. On the contrary, our angle ε • is counted from the reference x direction in the plane of the sky whose unit vectorê x , in general, does not coincide withl. Furthermore, Wex & Kopeikin (1999) use the symbol i for the orbital inclination angle, i.e., our I. It is important to notice that, contrary to the orbital time delay of Equation (51), Equation (66) is a short-term effect in the sense that there is no net shift over one orbital revolution. It is also worth noticing that Equation (66) is of order O c −4 , while Equation (51) is of order O c −3 .
As far as the putative scenario of the pulsar in the GC is concerned, the emitting neutron star is considered as the source s of the electromagnetic beam delayed by the angular momentum of the SMBH. Thus, by calculating Equation (66) for a S2-type orbit and with S = S • , it is possible to obtain Such values are one order of magnitude smaller than Equations (62) to (63) for the orbital time delay calculated with the same orbital configuration of the pulsar. It turns out that values similar to those of Equations (67) to (68)

The quadrupole-induced effect
If both the bodies of an arbitrary binary system are axisymmetric about their spin axesŜ A/B , a further non-central relative acceleration arises; it is (Barker & O'Connell 1975) in which the first even zonal parameter J A/B 2 is dimensionless. In the notation of Barker & O'Connell (1975), their J A/B 2 parameter is not dimensionless as ours, being dimensionally an area because it corresponds to our J A/B 2 R 2 A/B . Furthermore, Barker & O'Connell (1975) introduce an associated dimensional quadrupolar parameter ∆I A/B , having the dimensions of a moment of inertia, which is connected to our Thus, ∆I A/B corresponds to the dimensional quadrupolar parameter Q A/B 2 customarily adopted when astrophysical compact objects like neutron stars and black holes are considered (Laarakkers & Poisson 1999;), up to a minus sign, i.e.
Thus, Equation (71) can be written as Projecting Equation (71) onto the radial, tranvserse and out-of-plane unit vectorsρ,σ,ν provides us with A straightforward consequence of Equations (75) to (77) is the calculation of the net quadrupole-induced shifts per revolution of the Keplerian orbital elements by means of Equation (6) and Equations (19) to (21), which turn out to be Also Equation (82), as Equation (34) (71), Equation (21) does not vanish when integrated over a full orbital revolution. Furthermore, contrary to almost all of the other derivations existing in the literature, Equation (82) is quite general since it holds for a two-body system with generic quadrupole mass moments arbitrarily oriented in space, and characterized by a general orbital configuration. The same remark holds also for Equations (78) to (81); cfr. with the corresponding (correct) results by Iorio (2011) in the case of a test particle orbiting an oblate primary.
According to Equation (26) and Equations (78) to (82), the net orbit-like time change of the pulsar p after one orbital revolution is 2m tot ca 1 − e 2 (1 + e cos f 0 ) It turns out that Equation (83) does not vanish in the limit e → 0. If, on the one hand, Equation (83) depends of f 0 as Equation (35) and Equation (51), on the other hand, it depends on the orbital semimajor axis through a −1 . As far as ∆δτ J 2 p ( f ) is concerned, it will not be displayed explicitly because it is far too ponderous. Also in this case, a numerical integration of the equations of motion for a fictitious binary system, displayed in Figure 3, confirmed our analytical result for the temporal pattern of ∆δτ J 2 p ( f ); see also Section 5.1. The propagation time delay is dealt with in Section 5.2.

The pulsar in Sgr A * and the quadrupole-induced orbital time delay
A rotating NS acquires a non-zero quadrupole moment given by (Laarakkers & Poisson 1999) the absolute values of the dimensionless parameter q < 0 ranges from 0.074 to 3.507 for a variety of Equations of State (EOSs) and m NS = 1.4 M ⊙ ; cfr. Table 4 of Laarakkers & Poisson (1999). It is interesting to note that Laarakkers & Poisson (1999) find the relation where the parameter α of the fit performed by Laarakkers & Poisson (1999) depends on both the mass of the neutron star and the EOS used. According to Table 7 of Laarakkers & Poisson (1999), it is for some of the EOSs adopted by Laarakkers & Poisson (1999). In the case of PSR J0737-3039A, Equation (84) yields Q A 2 = q A 1.04 × 10 37 kg m 2 .
According to Equation (55) and Equation (85), it is As a consequence of the "no-hair" or uniqueness theorems (Hawking 1972;Israel 1967;Robinson 1975), the quadrupole moment of a BH is uniquely determined by its mass and spin according to (Geroch 1970;Hansen 1974) in the case of the SMBH in Sgr A * , it is (χ g = 0.6) Eq. (88) and Equation (91) so that the quadrupole of a hypothetical emitting neutron star p orbiting the SMBH in Sgr A * can be completely neglected with respect to the quadrupole of the latter one in any practical calculation.
According to our analytical expression for ∆δτ Q 2 p (t) applied to a pulsar moving along a S2-type orbit, in view of the ranges of variation assumed in Equations (60) to (61) for i • , ε • , it is The bounds of Equations (94) (95) and Equations (96) to (97) seem to be potentially measurable in view of the expected pulsar timing precision of about 100 µs, or, perhaps, even 1 − 10 µs (Psaltis, Wex & Kramer 2016;Goddi et al. 2017).

The quadrupole-induced propagation time shift
The propagation delay δτ J 2 prop due to the quadrupole mass moment is rather complicated to be analytically calculated; see, e.g., Klioner (1991); Kopeikin (1997); Klioner (2003); Zschocke & Klioner (2011). No explicit expressions analogous to the simple one of Equation (66) for frame-dragging exist in the literature. Here, we will obtain an analytical formula for δτ J 2 prop which will be applied to the double pulsar and the pulsar-Sgr A * systems. The approach by Zschocke & Klioner (2011) will be adopted by adapting it to the present scenario. In the following, the subscripts d, s, o will denote the deflector, the source, and the observer, respectively. In the case of, say, the double pulsar, d is the pulsar B, while s is the currently visible pulsar A; in the pulsar-Sgr A * scenario, d is the SMBH and s is the hypothetical pulsar p orbiting it. See Figure  4 for the following vectors connecting d, s, o. The origin O is at the binary's center of mass, so that r s emi r s (t emi ) is the barycentric position vector of the source s at the time of emission t emi , is the barycentric position vector of the deflector d at t emi , is the relative position vector of the source s with respect to the deflector d at t emi . Thus, to the Newtonian order, it is where m s , m d are the masses of source and deflector, respectively. Furthermore, is the barycentric position vector of the observer o at the time of reception t rec , is the position vector of the observer o at t rec with respect to the deflector d at t emi , and is the position vector of the observer o at t rec with respect to the source s at t emi . With our conventions for the coordinate axes, it is where D is the distance of the binary at t emi from us at t rec , which is usually much larger than the size r emi of the binary's orbit. Thus, the following simplifications can be safely made To order, O c −2 , the impact parameter vector can be calculated as (Zschocke & Klioner 2011) ℓ d ≃ŝ × (r emi ×ŝ) = r emi −ŝ (r emi ·ŝ) .
In view of Equation (108), it turns out that ℓ d , evaluated onto the unperturbed Keplerian ellipse, lies in the plane of the sky, being made of the x, y components ofρ scaled by Equation (12).
It is interesting to note that Equation (124) does not vanish for circular orbits. Furthermore, from Equation (124) it turns out that there is no net quadrupolar propagation delay per cycle.
If the pulsar-SMBH in the GC is considered, the quadrupole Shapiro-type time delay is much smaller than the orbital time shift. Indeed, by using Equations (124) to (125) for a S2-type orbital configuration, it turns out that The bounds of Equations (126) to (127) should be compared with those of Equations (94) to (95), which are about four orders of magnitude larger. Values as little as those of Equations (126) to (127) should be hard to be detectable in view of the expected pulsar timing precision, even in the optimistic case of 1 − 10 µs (Psaltis, Wex & Kramer 2016;Goddi et al. 2017). If the orbital configuration of S2 is abandoned letting I, Ω, ω, f, i • , ε • freely vary within their full natural ranges, we get values which can reach the 100 µs level for 0 ≤ e ≤ 0.97, 5 yr ≤ P b ≤ 16 yr.

Summary and conclusions
In order to perform sensitivity studies, designing suitable tests and reinterpreting existing data analyses in a way closer to the actual experimental practice in pulsar timing, we devised a method to analytically calculate the shifts ∆δτ A p experienced by the orbital component of the time changes δτ p of a binary pulsar p due to some perturbing post-Keplerian accelerations A: Schwarzschild, Lense-Thirring and mass quadrupole. We applied it to the still hypothetical scenario encompassing an emitting neutron star which orbits the Supermassive Black Hole in Sgr A * ; its timing precision could reach 100 µs, or, perhaps, even 1 − 10 µs (Psaltis, Wex & Kramer 2016;Goddi et al. 2017). The main results of the present study are resumed in Table 2. By assuming a S2-like orbital configuration and a time span as long as its orbital period, the magnitude of the post-Newtonian Schwarzschild-type gravitoelectric signature can reach ∆δτ GE Among other things, we also explicitly calculated an analytical formula for the Shapiro-like time delay δτ prop due to the propagation of electromagnetic waves in the field of a spinning oblate body, which we applied to the aforementioned binary system. As far as the Lense-Thirring and the quadrupolar effects are concerned, the Shapiro-like time shifts δτ prop are, in general, much smaller than the orbital ones ∆δτ p which, contrary to δτ prop , are cumulative. In the case of the pulsar-Sgr A * scenario, we have, for a S2-type orbit, that the Lense-Thirring propagation delay is as little as δτ LT prop 0.02 s, while the quadrupolar one is of the order of δτ Q 2 prop 1 µs, both depending on the spin orientation of the Black Hole. Removing the limitation to the S2 orbital configuration yields essentially similar values for δτ LT prop , δτ Q 2 prop , even for highly eccentric and faster orbits. Finally, we remark that our approach is general enough to be extended to arbitrary orbital geometries and symmetry axis orientations of the binary's bodies, and to whatsoever disturbing accelerations. As such, it can be applied to other binary systems as probes for, say, modified models of gravity. In principle, also man-made binaries could be considered.   Table 3 of Gillessen et al. (2017); they are referred to the epoch 2000.0. D 0 is the distance to Sgr A * . The Schwarzschild radius of the SMBH is r g 2GM • /c 2 = 0.088 au, while the linear size of the semimajor axis of S2 is a = 1, 044 au = 11, 863.6 r g . We quote also the derived values of the SMBH's angular momentum and quadrupole mass moment calculated as (Geroch 1970;Hansen 1974 • due to the the "no-hair" theorems (Hawking 1972;Israel 1967;Robinson 1975). The dimensionless parameter χ g ≤ 1 is of the order of about 0.6 for the SMBH in Sgr A * (Psaltis, Wex & Kramer 2016). We display also the value f 0 inferred from Equation (27) Table 2: Maximum and minimum values for the orbital and propagation time shifts ∆δτ p , δτ prop over a full orbital revolution due to the gravitoelectric (GE), gravitomagnetic (LT) and quadrupole (Q 2 ) effects for a hypothetical pulsar-Sgr A * scenario. While the orbital time delay ∆δτ p is cumulative over the revolutions, the propagation shift δτ prop vanishes over one orbital period. For the putative pulsar orbiting the SMBH in the GC, the orbital configuration of the main sequence S2 star was adopted. The values of the system's physical and orbital parameters corresponding to the quoted maxima and minima are not reported here: see the text for details. Only the angular momentum S • and the quadrupole Q • 2 of the SMBH, playing the role of deflector d, were taken into account. The expected timing precision for a pulsar orbiting the Galactic SMBH is about 100 µs, or, perhaps, even 1 − 10 µs (Psaltis, Wex & Kramer 2016;Goddi et al. 2017).  , in s, of a hypothetical pulsar in Sgr A * as the outcome of the difference between two numerical integrations of the equations of motion in Cartesian coordinates over a time span ranging from, say, t 0 = 2003.271 to t 0 + P b . Both the integrations share the same (Keplerian) initial conditions for f 0 = 139.72 deg, corresponding to t 0 = 2003.271, and differ by the 1pN Schwarzschild-like acceleration, which was purposely switched off in one of the two runs. Lower row, red curve: ∆δτ GE p (t), in s, of a hypothetical pulsar in Sgr A * obtained analytically from Equations (26) to (27) and the instantaneous changes of the Keplerian orbital elements, not displayed in the text, induced by Equation (28). It turns out that the net shift after a full revolution starting at t 0 = 2003.271 amounts to ∆δτ GE p = 1, 722.6948 s. The orbital configuration of the S2 star, quoted in Table 1, was adopted for the putative pulsar in Sgr A * . Fig. 2.-Upper row, blue curve: ∆δτ LT p (t), in s, of a hypothetical pulsar in Sgr A * as the outcome of the difference between two numerical integrations of the equations of motion in Cartesian coordinates over a time span ranging from, say, t 0 = 2003.271 to t 0 + P b . Both the integrations share the same (Keplerian) initial conditions for f 0 = 139.72 deg, corresponding to t 0 = 2003.271, and differ by the 1pN gravitomagnetic acceleration, which was purposely switched off in one of the two runs. Lower row, red curve: ∆δτ LT p (t), in s, of a hypothetical pulsar in Sgr A * obtained analytically from Equations (26) to (27) and the instantaneous changes of the Keplerian orbital elements, not displayed in the text, induced by Equation (42). It turns out that the net shift after a full revolution starting at t 0 = 2003.271 amounts to ∆δτ LT p = −9.6439 s. The orbital configuration of the S2 star, quoted in Table 1, was adopted for the putative pulsar in Sgr A * along with i • = 20.9 deg, ε • = 317.9 deg. Fig. 3.-Upper row, blue curve: ∆δτ Q 2 p (t), in s, of a hypothetical pulsar in Sgr A * as the outcome of the difference between two numerical integrations of the equations of motion in Cartesian coordinates over a time span ranging from, say, t 0 = 2003.271 to t 0 + P b . Both the integrations share the same (Keplerian) initial conditions for f 0 = 139.72 deg, corresponding to t 0 = 2003.271, and differ by the quadrupole-induced acceleration, which was purposely switched off in one of the two runs. Lower row, red curve: ∆δτ Q 2 p (t), in s, of a hypothetical pulsar in Sgr A * obtained analytically from Equations (26) to (27) and the instantaneous changes of the Keplerian orbital elements, not displayed in the text, induced by Equation (71). It turns out that the net shift after a full revolution starting at t 0 = 2003.271 amounts to ∆δτ Q 2 p = −0.0391 s. The orbital configuration of the S2 star, quoted in Table 1, was adopted for the putative pulsar in Sgr A * along with i • = 146.7 deg, ε • = 148.8 deg. The symbols adopted differ from those used in, e.g., Zschocke & Klioner (2011). The origin O coincides with the binary's center of mass (C. M.).