A QUANTITATIVE TEST OF THE NO-HAIR THEOREM WITH Sgr A* USING STARS, PULSARS, AND THE EVENT HORIZON TELESCOPE

Optical/IR observations of the orbits of stars in the vicinity of Sgr A* have led to a measurement of its mass, ${M}_{\bullet }$, and distance from the Earth, D. The uncertainties in the two measurements are significant and highly correlated (Ghez et al. 2008; Gillessen et al. 2009b). Because of the directions of these correlations, however, the uncertainty in the apparent size of the black hole shadow, which is the most relevant quantity for the EHT observations, is significantly smaller. In the following discussion, we set the mass of Sgr A* to ${M}_{\bullet }\quad =\quad 4.3\times {10}^{6}\;{M}_{\odot }$ and its distance to D = 8.3 kpc (Reid et al. 2014), such that the apparent opening angle of one gravitational radius (${{GM}}_{\bullet }/{c}^{2}$) at the distance of Sgr A* is equal to $5.1\;\mu$as and consistent with the most likely value derived from current observations (Psaltis et al. 2015b).

Because of the large orbital distances of the currently known optical/IR stars around Sgr A*, there have been no dynamical measurements of its spin magnitude, χ, or orientation. Comparisons of accretion flow models with spectroscopic and EHT imaging observations indicate low spins, when semi-analytic models are used (e.g., Broderick et al. 2011), or relatively high spins when GRMHD models are used (e.g., Dexter et al. 2010; Chan et al. 2015). Moreover, the small inferred size of the 1.3 mm image of Sgr A* supports the assumption that the black hole spin is inclined by $\simeq 50^\circ \mbox{--}60^\circ$ with respect to the line of sight and is aligned with the angular momentum vector of the stellar disk at ∼3 arcsec away from the black hole (Psaltis et al. 2015a). For the purposes of the present paper, we set the spin of Sgr A* to $\chi \quad =\quad 0.6$, which corresponds to a Kerr quadrupole moment of $q\quad =-0.36$. We picked these values such that the effects of both the spin and of the quadrupole moment are potentially observable, without being maximal. Clearly, we can perform tests of the no-hair theorem only if the black hole in the center of the Milky Way is spinning.

Advances in adaptive optics have revealed a large number of stars in orbit around Sgr A* (see Genzel et al. 2010; Ghez et al. 2012). One of these stars has been followed for at least one fully closed orbit (Ghez et al. 2008; Gillessen et al. 2009a) and the orbital parameters of several others (S0-16, S0-102, and S0-104) will eventually place them within a few thousand gravitational radii from the black hole (e.g., Meyer et al. 2012). Even though monitoring these orbits in the near future will most likely lead to the detection of periapsis precession, additional relativistic effects that will allow for a test of the no hair theorem will either be too small to be detected or masked by other astrophysical complexities.

It is expected that observations with future instruments, such as the adaptive-optics assisted interferometer GRAVITY on the Very Large Telescope (Eisenhauer et al. 2011) and new generation adaptive optics instruments on a 30-m class telescope (Weinberg et al. 2005), will lead to the discovery of stars with closer orbits. Monitoring the precession of their orbits and of their orbital planes will offer the possibility of measuring the spin and the quadrupole moment of the black hole and therefore of testing the no-hair theorem (Will 2008).

The distribution of stellar-mass objects within a few thousand gravitational radii from Sgr A* is very difficult to infer observationally at this point (see the detailed discussion in Merritt 2010). For the purposes of the current study, we will follow Merritt et al. (2010) and set the distribution of the semimajor orbital axes of stellar objects around the black hole such that

$$\begin{eqnarray}&&n(a)\quad =\quad \displaystyle \frac{{N}_{0}}{{a}_{0}^{3}}{(\displaystyle \frac{a}{{a}_{0}})}^{-\gamma }.\end{eqnarray} \tag{ 3 }$$

We will write our expressions in the general case of $\gamma \lt 3$, but evaluate them in the corresponding figures for $\gamma \quad =\quad 2$ (Merritt et al. 2010) and $\gamma \quad =\quad 7/4$ (Bahcall & Wolf 1976), to quantify the effect of this assumed parameter. Requiring that the total mass of stars inside the characteristic orbital separation a₀ is equal to M_*, i.e.,

$$\begin{eqnarray}&&{m}_{*}{\int }_{0}^{{a}_{0}}4\pi {a}^{2}n(a){da}\quad =\quad {M}_{*},\end{eqnarray} \tag{ 4 }$$

we obtain for the normalization constant

$$\begin{eqnarray}&&{N}_{0}\quad =\quad \displaystyle \frac{3-\gamma }{4\pi }\displaystyle \frac{{M}_{*}}{{m}_{*}}\end{eqnarray} \tag{ 5 }$$

and for the total number of stars inside an orbit with semimajor axis a

$$\begin{eqnarray}&&N(a)\quad =\quad {(\displaystyle \frac{a}{{a}_{0}})}^{3-\gamma }\displaystyle \frac{{M}_{*}}{{m}_{*}}.\end{eqnarray} \tag{ 6 }$$

The characteristic values for the mass m_* of each object and the total mass M_* enclosed inside an orbital separation a₀ are also poorly constrained from current observations. We will adopt here a conservative set of values (Merritt et al. 2010) for which ${m}_{*}\quad =\quad 1\;{M}_{\odot }$, ${a}_{0}\quad =\quad 1$ pc, and ${M}_{*}\quad =\quad {10}^{6}\;{M}_{\odot }$.

We can use this distribution to calculate the mass, angular momentum, and quadrupole moment due to the stellar cluster that is enclosed inside an orbit of a given semimajor axis. The ratio of these quantities to the black hole mass, angular momentum, and quadrupole moment will represent the limiting accuracies to which these black hole properties can be inferred using observations of orbits of stars and pulsars.

The mass of stars inside a circular orbit with semimajor axis a,

$$\begin{eqnarray}&&{M}_{*}(a)\quad =\quad {m}_{*}{\int }_{0}^{a}4\pi {a}^{2}n(a){da},\end{eqnarray} \tag{ 7 }$$

and the relative contribution to the mass of the black hole is

$$\begin{eqnarray}\displaystyle \frac{{M}_{*}(a)}{{M}_{\bullet }} & = & (\displaystyle \frac{{M}_{*}}{{M}_{\bullet }}){(\displaystyle \frac{a}{{a}_{0}})}^{3-\gamma }\\ & = & 4.8\times {10}^{-8}(\displaystyle \frac{{M}_{*}}{{10}^{6}\;{M}_{\odot }})\\ & & \times {(\displaystyle \frac{{a}_{0}}{1{\rm{pc}}})}^{-1}(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }}),\end{eqnarray} \tag{ 8 }$$

where in the last expression we set $\gamma \quad =\quad 2$.

The enclosed angular momentum due to the stellar cluster depends on the relative orientation of the orbits and the distribution of their eccentricities. We can obtain an upper limit to the enclosed angular momentum by assuming that all orbits are circular and aligned. In this case, the enclosed angular momentum is

$$\begin{eqnarray}&&{J}_{*}(a)\quad =\quad {m}_{*}{\int }_{0}^{a}4\pi {a}^{2}n(a){({{GM}}_{\bullet }a)}^{1/2}{da}.\end{eqnarray} \tag{ 9 }$$

The dimensional spin angular momentum of the black hole is ${S}_{\bullet }\equiv \chi \;{{GM}}_{\bullet }^{2}/c$ (cf. Equation (1)) and hence the magnitude of the relative contribution to the angular momentum due to the stellar cluster is

$$\begin{eqnarray}\chi \displaystyle \frac{{J}_{*}(a)}{{S}_{\bullet }} & = & \displaystyle \frac{2(3-\gamma )}{7-2\gamma }{(\displaystyle \frac{a}{{a}_{0}})}^{3-\gamma }(\displaystyle \frac{{M}_{*}}{{M}_{\bullet }}){(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{1/2}\\ & = & 3\times {10}^{-8}(\displaystyle \frac{{M}_{*}}{{10}^{6}\;{M}_{\odot }}){(\displaystyle \frac{{a}_{0}}{1{\rm{pc}}})}^{-1}{(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{3/2},\end{eqnarray} \tag{ 10 }$$

where in the last expression we set $\gamma \quad =\quad 2$.

The enclosed quadrupole moment due to the stellar cluster also depends on the orientation of the orbits. If we add an axisymmetric angular dependence to the distribution of orbits, i.e., denote this distribution by $n(a,\theta ,\phi )\quad =\quad n(a){n}_{\theta }(\theta )$, then the quadrupole mass moment of the stellar cluster becomes

$$\begin{eqnarray}{Q}_{*}(a) & = & \displaystyle \frac{{m}_{*}}{2}{\displaystyle \int }_{0}^{a}{a}^{2}n(a,\theta ,\phi )(3{\mathrm{cos}}^{2}\theta -1)\\ & & \times {a}^{2}{da}\;d\phi \;d\mathrm{cos}\theta \\ & = & \displaystyle \frac{{\tilde{n}}_{\theta }}{12}(\displaystyle \frac{a}{{a}_{0}}){M}_{*}{a}^{2},\end{eqnarray} \tag{ 11 }$$

where we have defined

$$\begin{eqnarray}&&{\tilde{n}}_{\theta }\equiv {\int }_{-1}^{1}(3{\mathrm{cos}}^{2}\theta -1){n}_{\theta }(\theta )d\mathrm{cos}\theta .\end{eqnarray} \tag{ 12 }$$

The dimensional quadrupole angular momentum of the black hole is ${Q}_{\bullet }\quad =\quad q\;{G}^{2}{M}_{\bullet }^{3}/{c}^{4}$ (cf. Equation (2)) and hence the magnitude of the relative contribution to the quadrupole moment due to the stellar cluster is

$$\begin{eqnarray}q\displaystyle \frac{{Q}_{*}(a)}{{Q}_{\bullet }} & = & \displaystyle \frac{(3-\gamma ){\tilde{n}}_{\theta }}{4(5-\gamma )}(\displaystyle \frac{{M}_{*}}{{M}_{\bullet }}){(\displaystyle \frac{a}{{a}_{0}})}^{3-\gamma }{(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{2}\\ & = & 4.0\times {10}^{-10}(\displaystyle \frac{{\tilde{n}}_{\theta }}{0.1})(\displaystyle \frac{{M}_{*}}{{10}^{6}\;{M}_{\odot }}){(\displaystyle \frac{{a}_{0}}{1{\rm{pc}}})}^{-1}\\ & & \qquad \qquad \qquad \qquad \qquad \qquad {(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{3},\end{eqnarray} \tag{ 13 }$$

where in the last expression we set $\gamma \quad =\quad 2$.

The fractional contributions to the mass, angular momentum, and quadrupole moment enclosed inside an orbit of semimajor axis a are shown in Figure 1. Our goal is to use orbits of stars and pulsars to measure the quadrupole moment of the black hole and test the no-hair theorem. Just imposing the requirement that the stellar cluster does not dominate the quadrupole moment of the gravitational field forces us to use circular orbits with orbital separations (or equivalently elliptical orbits with periapsis distances) that are inside a few times $\simeq 1000\;{{GM}}_{\bullet }/{c}^{2}$ (see also Merritt et al. 2010). For pulsars in highly eccentric orbits ($e\gtrsim 0.8$), as we will demonstrate in Section 4, we have, besides the secular precession of the orbit, an additional probe of the relativistic effects via the near-periapsis periodic contributions, which are less affected by external perturbations.

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For a number of observational and theoretical considerations, we expect a large number of neutron stars in the central part of the Galaxy. For a comprehensive review of the observational evidence and related theoretical considerations, we refer to Wharton et al. (2012) and references therein. Based on evidence for, e.g., the past star formation rate, the expected initial stellar mass function in the Galactic Center environment, and the observations of massive stars and stellar remnants, overall up to 100 normal pulsars and 1000 millisecond pulsars should be expected in the inner parsec. Earlier, Faucher-Giguère & Loeb (2011) pointed out that the high stellar density in the region also allows the effective creation of exotic binaries, like millisecond pulsar-stellar black hole binaries, which would be exciting laboratories in their own right (Wex & Kopeikin 1999; Liu et al. 2014).

Millisecond pulsars are old, recycled pulsars, which show typical periods between 1.4 and 30 ms, while normal pulsars have average periods of 0.5–1 s. Millisecond pulsars also have spin-down rates and estimated magnetic field strengths that are typically three orders of magnitude smaller than those of normal (unrecycled) pulsars. These properties make Millisecond pulsars superior—and hence preferred—clocks in pulsar timing experiments. For a normal pulsar, a typical timing precision is around 100 μs, while for the best Millisecond pulsars one can achieve a timing precision as good as 100 ns or better. In both cases, the final timing precision depends on the pulsar itself (e.g., the sharpness of its pulse shape, the intrinsic rotational stability) and the strength of the pulsar, as the error on an individual TOA measurement scales with the signal-to-noise ratio (S/N) of the observation (see Lorimer & Kramer 2004 for further details on pulsar properties and timing methods).

Despite concentrated efforts and dedicated searches in the Galactic Center region, the yield has been disappointingly low given the estimates. Until 2013, only five pulsars had been found within $15^{\prime}$ of Sgr A*, with the closest of these $11^{\prime}$ away, i.e., at a projected distance of about 25 pc (Johnston et al. 2006; Deneva et al. 2009; Bates et al. 2011). All of these were slow pulsars with dispersion measures up to 1500 pc cm⁻³. Given their distances to Sgr A*, none of these are suitable for the experiments described below.

The resulting perceived paucity of Galactic Center pulsars had been explained as a consequence of hyper-strong scattering of the radio waves at the turbulent inhomogeneous interstellar plasma in the region. The scattering leads to temporal broadening of the pulses with expected timescales of at least 2000 (ν/1 GHz)⁻⁴ s (Cordes & Lazio 2002), rendering their detection impossible at typical search frequencies, around 1–2 GHz. For this reason, a number of high-frequency searches were conducted in the past (Kramer et al. 2000; Klein et al. 2004; Johnston et al. 2006; Deneva et al 2009; Macquart et al. 2010; Bates et al. 2011; Eatough et al. 2013; Siemion et al. 2013) at frequencies as high as 26 GHz. However, even in these searches, no pulsar in the central parsec was found. The currently best limit (${S}_{\mathrm{min}}\lt 10\;\mu$Jy for a ${\rm{S}}/{\rm{N}}\sim 10$) is provided by observations with the 100-m Effelsberg telescope at 19 GHz (R. P. Eatough et al. 2015, in preparation).

The recent discovery of radio emission from the magnetar SGR J1745–29 by Eatough et al. (2013; see also Shannon & Johnston 2013), which had been first identified at X-rays (Kennea et al. 2013; Mori et al. 2013), provides an unexpected probe of the Galactic Center medium and the local pulsar population. The source that, with improved positional precision, is now named PSR J1745–2900, is located within 24 (or 0.1 pc projected) of Sgr A* (Bower et al. 2015) and is strong enough that even single pulses can be detected from a frequency of a few GHz (Spitler et al. 2014) up to an unprecedented 154 GHz (Torne et al. 2015). Below 1.1 GHz, the temporal broadening prevents a detection of the source (Spitler et al. 2014), while pulsed radio emission is detected up to 225 GHz, which is the highest frequency at which radio emission from a neutron star has been detected so far (Torne et al. 2015). The dispersion measure and the rotation measure of PSR J1745–2900 are the largest in the Galaxy (only the rotation measure of Sgr A* itself is larger; Eatough et al. 2013; Shannon & Johnston 2013), while the angular broadening of the source is consistent with that of Sgr A* (Bower et al. 2014, 2015), providing evidence for the proximity of the magnetar to the Galactic Center. While its rotational stability is unfortunately not sufficiently good to conduct precision timing experiments, it allows us to revisit the question of the hidden pulsar population.

Radio-emitting magnetars are a very rare type of neutron star and previously only three of them were known to exist in the Galaxy, i.e., less than 0.2% of all radio-loud neutron stars (Olausen & Kaspi 2014). The discovery of such a rare object adjacent to Sgr A* thereby supports the notion that many more ordinary radio pulsars should be present (Eatough et al. 2013; Chennamangalam & Lorimer 2014). A surprising aspect of the magnetar discovery is the relatively small scatter broadening that is observed (Spitler et al. 2014). With a pulse period of 3.75 s, its radio emission should not be detectable at frequencies as low as 1.1 GHz, if hyper-strong scattering were indeed present.

Imaging observations (Bower et al. 2015) resulted in the measurement of a proper motion that does not allow us yet to conclude as to whether the pulsar is bound to Sgr A*. It is possible that PSR J1745–2900 and the other five nearby pulsars originated from a stellar disk (see also Johnston et al. 2006) and that a central population of pulsars is still hidden. Indeed, Chennamangalam & Lorimer (2014) argue that, even if the lower-than-expected scattering in the direction of PSR J1745–2900 is representative of the entire inner parsec, the potentially observable population of pulsars in the inner parsec still has a conservative upper limit of $\sim 200$ members. They conclude that it is premature to assume that the number of pulsars in this region is small.

In contrast, Dexter & O'Leary (2014) come to a different conclusion. They also revisited the question about the central pulsar population given the new constraints provided by the magnetar and the non-detection of previous high-frequency surveys. Considering various effects like depletion of the pulsar population due to kick velocities exceeding the central escape velocity, pulsar spectra, and the apparent reduced scattering indicated by the magnetar observations (Spitler et al. 2014), they argue in favor of a "missing pulsar problem." They also concluded that the magnetar discovery in the center may imply, in turn, an efficient birth process for magnetars in the central region. Similarly, others suggested that normal pulsars are not formed since they may collapse into black holes on comparably short timescales by accreting of dark matter (Bramante & Linden 2014).

At the core of deciding between these possibilities is our ability to properly model and account for all selection effects in the previous surveys. There are in fact indications that this is not the case. First, continued monitoring of the scattering timescales for the magnetar indicates that the scattering time is highly variable. While it remains well below the prediction of hyper-strong scattering, it varies by a factor of 2–4 on timescales of months at frequencies between 1.4 and 8 GHz (L. G. Spitler et al. 2015, in preparation). This suggests that local "interstellar weather" certainly plays a role and that nearby scattering screens also affect the observed emission, making the resulting ability to observe sources overall line of sight dependent, especially at lower frequencies. This is not unexpected given the properties of the turbulent interstellar medium in the Galactic Center. Rather than dealing with a uniform single screen, it is likely that we see the effects of multiple finite screens. In this case, second, one expects a much shallower frequency dependence of the scattering time than the canonical $-4$ values (Cordes & Lazio 2001). This is indeed seen for high-DM pulsars (Löhmer et al. 2001, 2004), where the scattering index is typically around $\sim -3.5$ for large dispersion measures. L. G. Spitler et al. (2015, in preparation) find similar values for the magnetar. If this is indeed representative for a possible central pulsar or millisecond pulsar population, then the remaining scattering at 5, 14, or even 19 GHz would be underestimated in the analysis by Macquart et al. (2010) or Dexter & O'Leary (2014) by factors of 2.2, 3.7, or 4.3 respectively, when extrapolating from 1 GHz. Löhmer et al. (2001) measured even flatter frequency dependencies, which would make the discrepancy between real and estimated scattering times even larger. Unless more scatter broadening times in the Galactic Center are measured, this issue is difficult to settle. However, there is yet another, third effect that has usually been neglected in sensitivity calculations of pulsar surveys. As shown very recently by Lazarus et al. (2015) for the P-ALFA survey, red noise present in pulsar search data due to radio interference (RFI), receiver gain fluctuations, and opacity variations of the atmosphere cause a significant decrease in sensitivity for pulsars with periods above 100 ms or so, when compared to the standard radiometer-based equation (see their Figure 11). This would affect in particular a search for young pulsars, but also, of course, magnetars, which are nevertheless still easier to detect at high frequencies due to their much flatter flux density spectrum (Torne et al. 2015). This selection effect in particular would favor the detection of magnetars over that of normal, young pulsars and may explain in some respects the peculiarities of the current observational situation pointed out by Dexter & O'Leary. The work by Lazarus et al. demonstrates that the various selection effects are highly dependent on the individual surveys and that much more work is needed to understand the impact on the resulting search sensitivities.

Finally, none of the previous high-frequencies surveys has, to our knowledge, applied a fully coherent acceleration search. Such an acceleration search may be needed to account both for the movement of the pulsar around the central black hole, as well as for the presence of a binary companion. Indeed, due to the high stellar density, even exotic systems (e.g., millisecond pulsar-stellar mass BH binary) may be expected (Faucher-Giguère & Loeb 2011). An acceleration search is usually very computationally expensive, especially for long integration times as employed in the high frequency searches (e.g., by Macquart et al. 2010 or R. P. Eatough et al. 2015, in preparation), since the parameter range to be searched scales as $\propto {T}_{\mathrm{obs}}^{3}$. The lack of such an acceleration search contributes as a selection effect to the present non-detection of fast-spinning pulsars.

In order to model the selection effects (red noise, acceleration, scattering etc.) a more detailed study, taking the orientation of the possible orbits and the change in acceleration into account, is needed. This is beyond the scope of this paper and will be presented elsewhere. It is clear, however, that selection effects are not adequately modeled so far and that more work is required.

We conclude that three scenarios are still possible: (a) the scattering seen for the Galactic Center magnetar is representative of the inner parsec. In this case, the pulsar population may be dominated by Millisecond pulsars, for which this moderate scattering would still have prevented their detection at previous search frequencies. Higher frequency searches may therefore even allow the discovery and hence the exploitation of Millisecond pulsars orbiting Sgr A* (see also Macquart & Kanekar 2015). We note in passing that the discovery of a Millisecond pulsar population may settle an ongoing debate about a possible excess of GeV gamma ray photons from the Galactic Center. It is being discussed whether such an excess could arise from the presence of dark matter or a central population of unresolved young or Millisecond pulsars (see e.g., O'Leary et al. 2015 and references therein). Any pulsar discovery in the Galactic Center would make a dark matter discovery less likely. (b) There is a reduced number of pulsars in the Galactic Center region that is consistent with selection effects. For example, the lack of dispersion makes the discovery of unknown pulsars actually more difficult at high frequencies, as the signals are more difficult to distinguish from radio interference (see R. P. Eatough et al. 2015, in preparation), or (c) PSR J1745–2900 is indeed in front of a much more severe scattering screen but the scattering properties for particular lines of sight are changing with time due to "local weather" effects, signs of which have been already detected (L. G. Spitler et al. 2015, in preparation). In this case, search observations at even higher frequencies are required and still promising.

Given that we cannot distinguish between these scenarios based on the available data, high frequency searches will continue. The use of more sensitive instruments than available in the past, e.g., ALMA or the Square Kilometre Array (SKA), may therefore lead to the discovery of normal pulsars and even Millisecond pulsars. In considering how they can be used to measure the properties of Sgr A*, we will therefore assume a variety of obtainable timing precisions. For details, we refer the reader to Liu et al. (2012), who demonstrated possible precision levels as a function of observing frequency for the SKA and 100-m class telescopes. In their arguments, Liu et al. (2012) only considered normal pulsars and also assumed a hyper-strong scattering. If the latter is not present as we have discussed above, Millisecond pulsars may be detectable (although this may require proper acceleration searching). Hence, for the discussion of the measurable effects, we will also allow for this possibility that Millisecond pulsars will be detected.

There are a number of Millisecond pulsars in globular clusters at distances that are signifantly larger than that of the Galactic Center. It is not uncommon to achieve a timing precision of about 10 μs for these distant sources. The exact precision mainly depends on the strength of the pulsar signal and the sensitivity of the telescope, as well as the sharpness of some of the detetable profile features. If we need to go to high radio frequencies in order to beat interstellar scattering to see pulsars in the center of the Milky Way, the flux density decreases and timing precision decreases accordingly. This can be compensated by larger bandwidth or bigger telescopes. As shown in Eatough et al. (2015), a timing precision of 1 μs should be routinely possible with the SKA, even at distances of the Galactic Center at higher frequencies. Such a precision is certainly more challenging with existing instruments. Overall, in order to cover all three plausible scenarios discussed above, we will assume, in the following, that a Galactic Center pulsar can be timed with a precision of 1, 10, and 100 μs. As, in principle, only one pulsar is needed to extract the black hole parameters, we consider this to be a useful range to demonstrate the effects that we can expect to measure.

In describing the orbit of a stellar-mass object around Sgr A*, we will use the coordinate system and notation shown in Figure 2. In particular, we will denote by m_* the mass of the orbiting object and by a and e the semimajor axis and eccentricity of its orbit. We will use the vector ${{\boldsymbol{S}}}_{\bullet }$ to define the black hole spin and the vector ${{\boldsymbol{K}}}_{0}$ to denote the line of sight unit vector pointing from the Earth to the black hole. We will also denote the longitude of the periapsis of the orbit with respect to the equatorial plane of the black hole by ω, the location of the ascending node by ϒ, and the inclination of the orbit with respect to the black hole spin axis by Θ.

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With these definitions, the Newtonian period of the orbit is

$$\begin{eqnarray}P & = & 2\pi {(\displaystyle \frac{{a}^{3}}{{{GM}}_{\bullet }})}^{1/2}\\ & = & 133.1(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}\;{M}_{\odot }}){(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{3/2}{\rm{s}}.\end{eqnarray} \tag{ 14 }$$

Eccentric orbits of stars and pulsars precess on the orbital plane (relativistic periapsis precession). The leading term comes from the mass-monopole ${M}_{\bullet }$ and corresponds to the relativistic precession of the Mercury orbit (Einstein 1915). The advance of periapsis per orbit is ${\rm{\Delta }}\omega \quad =\quad 2\pi k$, where

$$\begin{eqnarray}&&k\quad =\quad \displaystyle \frac{3}{1-{e}^{2}}{(\displaystyle \frac{2\pi {{GM}}_{\bullet }}{{{Pc}}^{3}})}^{2/3}\quad =\quad \displaystyle \frac{3}{1-{e}^{2}}(\displaystyle \frac{{{GM}}_{\bullet }}{{c}^{2}a}).\end{eqnarray} \tag{ 15 }$$

This corresponds to a characteristic timescale for this precession of (see Merritt et al. 2010 for the definition, who denote this by ${t}_{{\rm{S}}}$)

$$\begin{eqnarray}{t}_{{\rm{M}}} & \equiv & \displaystyle \frac{\pi P}{{\rm{\Delta }}\omega }\quad =\quad \displaystyle \frac{P}{6}\;\displaystyle \frac{{c}^{2}a}{{{GM}}_{\bullet }}(1-{e}^{2})\\ & = & 22.18(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}\;{M}_{\odot }}){(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{5/2}(1-{e}^{2})\;{\rm{s}}.\end{eqnarray} \tag{ 16 }$$

In this expression, we have neglected the small contributions of the spin and of the quadrupole of the black hole.

Orbits with angular momenta that are not parallel to the spin ${{\boldsymbol{S}}}_{\bullet }$ of the black hole show a precession of the orbital angular momentum around the ${{\boldsymbol{S}}}_{\bullet }$ direction due to frame dragging (Lense–Thirring precession of the nodes). The location of the ascending node of the orbit, ϒ, advances per orbit by

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{\Upsilon }}\quad =\quad {{\rm{\Omega }}}_{\mathrm{LT}}P,\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mathrm{LT}}\equiv \displaystyle \frac{8{\pi }^{2}}{{(1-{e}^{2})}^{3/2}}\displaystyle \frac{{{GM}}_{\bullet }}{{c}^{3}{P}^{2}}\chi \end{eqnarray} \tag{ 18 }$$

is the Lense–Thirring frequency. The characteristic timescale for this process is (Merritt et al. 2010)

$$\begin{eqnarray}{t}_{{\rm{J}}} & \equiv & \displaystyle \frac{\pi P}{{\rm{\Delta }}{\rm{\Upsilon }}}\quad =\quad \displaystyle \frac{P}{4\chi }{[\displaystyle \frac{{c}^{2}a(1-{e}^{2})}{{{GM}}_{\bullet }}]}^{3/2}\\ & = & 33.27\;{\chi }^{-1}(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}\;{M}_{\odot }}){(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{3}{(1-{e}^{2})}^{3/2}\;{\rm{s}}.\end{eqnarray} \tag{ 19 }$$

Finally, tilted orbits also precess because of the quadrupole moment of the spacetime with a characteristic timescale (Merritt et al. 2010)

$$\begin{eqnarray}{t}_{{\rm{Q}}} & = & \displaystyle \frac{P}{3| q| }{[\displaystyle \frac{{c}^{2}a(1-{e}^{2})}{{{GM}}_{\bullet }}]}^{2}\\ & = & 44.35| q{| }^{-1}(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}\;{M}_{\odot }}){(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{7/2}{(1-{e}^{2})}^{2}\;{\rm{s}}.\end{eqnarray} \tag{ 20 }$$

Figure 3 shows the characteristic timescales of these relativistic orbital effects as a function of the semimajor axes and orbital periods of the orbits. A number of additional relativistic effects related to time dilation and photon propagation (Shapiro delay) can also be detected during timing observations of pulsars. We will discuss these effects and their dependence on the pulsar orbital parameters in Section 4.

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Performing tests of the no-hair theorem with orbits is hampered by a number of astrophysical complexities caused by non-relativistic phenomena that affect, in principle, the orbits. These included the self-interaction between the stars in the stellar cluster (Merritt et al. 2010; Sadeghian & Will 2011), the hydrodynamic drag between the stars and the accretion flow (Psaltis 2012), as well as stellar winds and tidal deformations (Psaltis et al. 2013). In order to identify the orbital parameters of stars that are optimal for performing the test of the no-hair theorem, we will first summarize and combine the results of these studies.

Interactions with Other Stars. Merritt et al. (2010) and Sadeghian & Will (2011) explored the decoherence of the orbit of a star (or pulsar) due to Newtonian gravitational interactions within the inner stellar cluster. They obtained an approximate expression for the decoherence timescale given by

$$\begin{eqnarray}&&{t}_{{\rm{N}}}\quad =\quad \displaystyle \frac{P}{{q}_{*}\sqrt{N(a)}},\end{eqnarray} \tag{ 21 }$$

where ${q}_{*}\equiv {m}_{*}/{M}_{\bullet }$ is the average ratio of the mass of a star (or compact object) in the cluster to that of Sgr A*, and N(a) is the number of stars inside the orbit of the object under consideration.

Using Equations (6), (14), and (21), we obtain for the decoherence timescale of orbits due to the self interaction between objects in the stellar cluster

$$\begin{eqnarray}{t}_{{\rm{N}}} & = & 2\pi {(\displaystyle \frac{{M}_{\bullet }^{2}}{{M}_{*}{m}_{*}})}^{1/2}{(\displaystyle \frac{a}{{a}_{0}})}^{-1/2}{(\displaystyle \frac{{a}^{3}}{{{GM}}_{\bullet }})}^{1/2}\\ & = & 12.6\times {10}^{8}{(\displaystyle \frac{{M}_{\bullet }}{4.3\times {10}^{6}{M}_{\odot }})}^{3/2}{(\displaystyle \frac{{M}_{*}}{{10}^{6}{M}_{\odot }})}^{-1/2}\\ & & \times {(\displaystyle \frac{{m}_{*}}{{M}_{\odot }})}^{-1/2}{(\displaystyle \frac{{a}_{0}}{1{\rm{pc}}})}^{-1/2}(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})\;{\rm{s}}.\end{eqnarray} \tag{ 22 }$$

Figure 3 compares the Newtonian decoherence timescale with those of the three relativistic effects discussed in Section 2. For the parameters of Sgr A* and of the stellar cluster around it, stars with orbital periods less than $\sim 1$ year are required in order for the Newtonian interactions not to mask the orbital plane precession due to frame dragging (see a more detailed discussion and simulations in Merritt et al. 2010).

Hydrodynamic Interactions with the Accretion Flow. In Psaltis (2012) we investigated the changes in the orbits of stars and pulsars caused by the hydrodynamic and gravitational interactions between them and the accretion flow around Sgr A*. For all cases of interest, we found that the hydrodynamic drag is the dominant effect. However, as we will show below, even the hydrodynamic drag is negligible for the orbital separations considered here.

When a star of mass m_* and radius R_* plows through the accretion flow of density ρ with a relative velocity ${u}_{\mathrm{rel}}$, it feels an effective acceleration equal to

$$\begin{eqnarray}&&{a}_{{\rm{d}}}\quad =\quad \displaystyle \frac{\pi {R}_{*}^{2}\rho {u}_{\mathrm{rel}}^{2}}{{m}_{*}}.\end{eqnarray} \tag{ 23 }$$

We can use this acceleration to define a characteristic timescale for the change of the orbital parameters as

$$\begin{eqnarray}&&{t}_{{\rm{d}}}\equiv \displaystyle \frac{{u}_{\mathrm{rel}}}{{a}_{{\rm{d}}}}\quad =\quad \displaystyle \frac{{m}_{*}}{\pi {R}_{*}^{2}\rho {u}_{\mathrm{rel}}}.\end{eqnarray} \tag{ 24 }$$

Setting the relative velocity equal to the orbital velocity of a circular orbit, and the density of the accretion flow to

$$\begin{eqnarray}&&\rho \quad =\quad {m}_{{\rm{p}}}{n}_{e}^{0}{(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{-1.1},\end{eqnarray} \tag{ 25 }$$

which has been inferred observationally (see discussion in Psaltis 2012), we obtain

$$\begin{eqnarray}{t}_{{\rm{d}}} & = & \displaystyle \frac{{m}_{*}}{\pi {R}_{*}^{2}{m}_{{\rm{p}}}{n}_{0}c}{(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{1.6}\\ & = & 7.5\times {10}^{15}(\displaystyle \frac{{m}_{*}}{10\;{M}_{\odot }}){(\displaystyle \frac{{R}_{*}}{10{R}_{\odot }})}^{-2}\\ & & \times {(\displaystyle \frac{{n}_{e}^{0}}{3.5\times {10}^{7}{{\rm{cm}}}^{-3}})}^{-1}{(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{1.6}\;{\rm{s}}.\end{eqnarray} \tag{ 26 }$$

Here, ${m}_{{\rm{p}}}$ is the mass of the proton and we assumed for simplicity that the orbit is circular. This timescale is significantly larger than all other timescales shown in Figure 3.

Stellar Winds. In Psaltis (2012) we explored the change in the orbital parameters of stars due to the launching of a wind that carries a fraction of the orbital energy and angular momentum. The semimajor axis and the eccentricity of the orbit change at a timescale comparable to ${t}_{{\rm{w}}}\equiv {m}_{*}/{\dot{M}}_{{\rm{W}}}$, where ${\dot{M}}_{{\rm{w}}}$ is the rate of wind mass loss, i.e.,

$$\begin{eqnarray}&&{t}_{{\rm{w}},-7}\quad =\quad 3.2\times {10}^{15}(\displaystyle \frac{{m}_{*}}{10\;{M}_{\odot }}){(\displaystyle \frac{| {\dot{M}}_{{\rm{w}}}| }{{10}^{-7}{M}_{\odot }{{\rm{yr}}}^{-1}})}^{-1}\;{\rm{s}},\end{eqnarray} \tag{ 27 }$$

where we have used the subscript "$-7$" to denote the exponent in the wind mass loss rate. As shown in Figure 3, the evolution of the stellar orbit due to the launching of a stellar wind is always negligible compared to the effects of the Newtonian interactions with the other stars in the cluster.

Tidal Evolution. Finally, in Psaltis (2013) we also explored the evolution of a stellar orbit due to the tidal dissipation of orbital energy during the periapsis passages. Even though tidal dissipation does not cause a significant precession in the orbit (see Sadeghian & Will 2011), it leads to an evolution of the semimajor axis that may be misinterpreted (due to the expected low signal-to-noise in the observations) as a change in the projected orbital separation caused by orbital precession.

The characteristic timescale for orbital evolution due to tidal dissipation is

$$\begin{eqnarray}{t}_{{\rm{d}}} & \equiv & \displaystyle \frac{E}{{\rm{\Delta }}E/{\rm{\Delta }}t}\\ & = & \displaystyle \frac{\pi {R}_{*}}{c}{(\displaystyle \frac{{{GM}}_{\bullet }}{{c}^{2}{R}_{*}})}^{6}(\displaystyle \frac{{m}_{*}}{{M}_{\bullet }}){(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{13/2}{(1-e)}^{6}{T}_{2}^{-1}\\ & = & 0.98\times {10}^{-4}{(\displaystyle \frac{{M}_{\bullet }}{4.3\times {10}^{6}{M}_{\odot }})}^{5}{(\displaystyle \frac{{R}_{*}}{10{R}_{\odot }})}^{-5}\\ & & \times (\displaystyle \frac{{m}_{*}}{10\;{M}_{\odot }}){(\displaystyle \frac{{{ac}}^{2}}{{{GM}}_{\bullet }})}^{13/2}{(1-e)}^{6}{T}_{2}^{-1}(\eta )\;{\rm{s}},\end{eqnarray} \tag{ 28 }$$

where the quantity ${T}_{2}(\eta )$ is defined and calculated in Psaltis et al. (2013). Also, if the star at periastron reaches inside the tidal radius

$$\begin{eqnarray}&&{R}_{{\rm{t}}}\quad =\quad {R}_{*}{(\displaystyle \frac{{M}_{\bullet }}{{m}_{*}})}^{1/3},\end{eqnarray} \tag{ 29 }$$

it gets disrupted. Both these effects, for two different orbital eccentricities, are shown in Figure 3.

Optimal parameters. Comparing the various constraints shown in Figure 3 to the characteristic timescales of the three relativistic effects allows us to identify the optimal orbital parameters of stars and pulsars for measuring the black hole spin and for testing the no-hair theorem.

Using the orbital precession of stellar orbits to measure the spin of Sgr A* simply requires sub-year orbital periods for the effects of the stellar perturbations to become negligible (as previously discussed in Merritt et al. 2010). On the other hand, measuring the black hole quadrupole requires stars in much tighter orbits ($\lesssim 0.1$ yr), for the stellar perturbations to be negligible, but with moderate eccentricities ($e\lesssim 0.8$), for tidal effects to not interfere with the measurements of the relativistic precessions (see also Will 2008).

For the case of pulsar timing, tidal effects do not alter the orbits and therefore only stellar perturbations can limit our ability to observe relativistic precessions. If we were to use pulsar timing to measure the black hole quadrupole by observing the pulsar orbital plane precess, we would still be limited to using only rather tight orbits ($\lesssim 0.1$ yr). However, in defining the characteristic timescale for quadrupole effects on the pulsar orbits (Equation (20)), we have only considered the secular precession of the orbit. The most promising way to extract the quadrupole moment from timing observations is through the periodic effects in the orbital motion of the pulsar caused by the quadrupolar structure of the gravitational field of Sgr A* (Wex & Kopeikin 1999; Liu et al. 2012). This is not only the case for the quadrupole but also for the relativistic precession of the periapsis due to the mass monopole (Damour & Deruelle 1985) and the precession of the orbit due to the frame dragging (Wex 1995). (See also the discussion in Angélil & Saha 2014). As argued by Liu et al. (2012), such unique periodic features in the timing of a pulsar around Sgr A* provide a powerful handle to correct for external perturbations. As we will demonstrate with mock data simulations in Section 4, timing a pulsar only during a small number of successive periapses passages is sufficient to measure both the spin and the quadrupole moment of Sgr A*.

It is well known that the measurement of post-Keplerian (PK) parameters in binary pulsars can provide highly accurate measurements of the masses of the system (Lorimer & Kramer 2004). The same can be expected for a pulsar in orbit around Sgr A*, where the situation is insofar different as the pulsar is like a test particle, whose mass is negligible in comparison to the companion's mass, i.e., the 4.3 million solar masses of Sgr A*. In such a situation the measurement of a single PK parameter allows the determination of the mass of Sgr A*, once a theory of gravity is assumed. The measurement of a second PK parameter already allows for a consistency check, since the inferred mass should agree with the one from the first PK parameter (Liu et al. 2012).

In the following, we quickly summarize the most important relativistic effects and their leading order expression within GR (see Lorimer & Kramer 2004, and references therein for details).

• 1.  
  The advance of periapsis per orbit is ${\rm{\Delta }}\omega \quad =\quad 2\pi k$, where k was defined in Equation (15). For a fast spinning black hole, k can have a significant contribution from frame-dragging effects, as we will discuss in more detail below.
• 2.  
  The time dilation (Einstein delay) has an amplitude of

  $$\begin{eqnarray}&&{\gamma }_{{\rm{E}}}\quad =\quad 2\displaystyle \frac{e}{n}{(\displaystyle \frac{{{GM}}_{\bullet }}{{c}^{3}}n)}^{2/3}.\end{eqnarray} \tag{ 49 }$$
• 3.  
  The signal propagation delay (Shapiro delay), which is proportional to the black hole mass, reads as a function of the true anomaly f as

  $$\begin{eqnarray}&&{{\rm{\Delta }}}_{{\rm{S}}}\quad =\quad \displaystyle \frac{2{{GM}}_{\bullet }}{{c}^{3}}\;\mathrm{ln}[\displaystyle \frac{1+e\mathrm{cos}f}{1-\mathrm{sin}i\mathrm{sin}(\tilde{\omega }+f)}].\end{eqnarray} \tag{ 50 }$$

A measurement of the Shapiro delay simultaneously gives ${M}_{\bullet }$ and $\mathrm{sin}i$ (see Figure 2 for the definition of the inclination angle i). The latter is connected to ${M}_{\bullet }$ via the so-called mass function

$$\begin{eqnarray}&&\mathrm{sin}i\quad =\quad {nx}{(\displaystyle \frac{{{GM}}_{\bullet }}{{c}^{3}}n)}^{-1/3}.\end{eqnarray} \tag{ 51 }$$

The Keplerian parameter $x\equiv a\mathrm{sin}i/c$ is the projected semimajor axis of the pulsar orbit and can be measured with high precision from pulsar timing. Consequently, there are two ways to extract the mass of Sgr A* from a measurement of the Shapiro delay.

Based on a consistent covariance analysis using mock data simulations, Liu et al. (2012) demonstrated that a pulsar in a close orbit (${P}_{b}\sim 1$ year) around Sgr A* allows for a very precise determination of its mass, ${M}_{\bullet }$. Even a moderate timing precision can lead to a ∼10⁻⁵ precision for ${M}_{\bullet }$, since there are various relativistic effects that can be utilized for mass determination. The simulations in Liu et al. (2012), however, were based on the assumption that the pulsar can be timed continuously over several years, covering at least a few full orbits. Later in this section, we relax this assumption and allow for a situation where, due to external perturbation, only the timing data near the black hole can be used in a phase-connected solution. As it turns out, in particular the Einstein and the Shapiro delay provide a robust determination of ${M}_{\bullet }$ that is only weakly affected by external perturbations or uncertainties in our knowledge of the spin of Sgr A*.

The dragging of inertial frames by the rotation of the black hole affects the precession of the periapsis of the orbit. Indeed, beyond leading order, Equation (15) would include a Lense–Thirring contribution ${k}_{\mathrm{LT}}$, where

$$\begin{eqnarray}&&{k}_{\mathrm{LT}}\quad =\quad -3\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{LT}}}{n}\mathrm{cos}{\rm{\Theta }}\end{eqnarray} \tag{ 52 }$$

(see also Equation (30)). An independent measurement of the mass ${M}_{\bullet }$, for instance, through the Shapiro delay, could then be used to compute $\chi \mathrm{cos}{\rm{\Theta }}$, within GR, where $\chi \mathrm{cos}{\rm{\Theta }}\leqslant 1$ is required by the cosmic censorship conjecture.

The most prominent effect of frame dragging in the pulsar motion is the Lense–Thirring precession of the orbital plane. In GR, the nodes of the orbit precess at an averaged rate of

$$\begin{eqnarray}&&\langle \dot{{\rm{\Upsilon }}}\rangle \quad =\quad {wn}\quad =\quad {{\rm{\Omega }}}_{\mathrm{LT}},\end{eqnarray} \tag{ 53 }$$

(see also Equation (31)). Although w is a small quantity ($\sim {10}^{-4}\;\chi$ for a 0.1 year orbit), given the large size of a pulsar orbit around Sgr A*, it has a tremendous impact on the timing residuals. In fact, for orbits $\lesssim 1$ year, it leads to a large change in the projected semimajor axis, x, giving rise to a significant time evolution of x, that can be measured as a first derivative $\dot{x}$ and even a second derivative $\ddot{x}$. Furthermore, it also leads to an observable second derivative in the advance of the periapsis, $\ddot{\tilde{\omega }}$ (see Liu et al. 2012 for details).

As can be seen from Equation (45), the location of the ascending node, ϒ, advances nonlinearly in f and t, which we will exploit for the first time in this paper. Instead of using only the secular contributions $\dot{x}$, $\ddot{x}$, and $\ddot{\tilde{\omega }}$ to model the changes in the orbit due to Lense–Thirring precession, we implement the full model Equations (41)–(45), therefore also accounting for the periodic contributions. The latter is of particular importance if the pulsar is in a highly eccentric orbit. Morover, this will turn out to be extremely valuable in the presence of external perturbations. In fact, Liu et al. (2012) have already argued that these distinctive near-periapsis contributions can be used to differentiate between frame dragging by the black hole and external contributions, which are more likely to affect the pulsar's motion near the apoapsis (see also Angélil & Saha 2014). Figure 5 illustrates such Lense–Thirring contributions to the pulsar timing residuals for a highly eccentric (e = 0.9), wide (P_b = 3 years) pulsar orbit. From Figure 5 it is already clear that, from a single periapsis passage that is covered by a dense observing campaign, we can already infer relevant constraints on the Sgr A* spin. We will present more detailed conclusions in the subsection on mock data simulations below.

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4.2.1. Mock Data Simulations

While from the above discussion it is already obvious that the Lense–Thirring drag of the rotating black hole leads to a characteristic signal in the timing data, which ultimately allows the determination of the spin, we still need to address the question of spin measurement in a more quantitative way. For this reason, we have conducted extensive mock data simulations, based on the potential timing capabilities discussed in Section 2.3. The simulated TOAs were fitted with a timing model based on the equations of motion, (41)–(45), that also include the relativistic effects discussed in Section 4.1. By this our simulations are based on a timing model that accounts for all the relevant effects to leading order. This model has been implemented in a timing software package, which is based on TEMPO and has been optimized for the timing analysis of Galactic Center pulsars. It allows for a fully phase-connected timing solution, providing a consistent parameter estimation. This is similar to the analysis presented in Liu et al. (2012); however, in addition, it properly accounts for the prominent near-periapsis features in the residuals, caused by the Lense–Thirring effect. As discussed above, this is of particular importance if there are external perturbations to the pulsar orbit; moreover it also helps in determining the black hole spin on a shorter observing time-span, during which the second derivatives $\ddot{x}$ and $\ddot{\tilde{\omega }}$ are not well measured. The latter is important in case the pulsar is only visible for a limited period of time, which will be the case for a pulsar at the Galactic Center, as discussed in Section 2.3.

Figure 6 presents the measurement precision of the dimensionless spin parameter χ as a function of the observed number of periapsis passages for a pulsar in a 0.5 yr eccentric (e = 0.8) orbit. We have simulated a dense observing campaign where one obtains three precision TOAs per day. Simulations have been conducted for three different TOA uncertainties: 1, 10, and 100 μs, corresponding to our discussion in Section 2.3. As is evident from Figure 6, for all these TOA uncertainties we should be able to measure the spin of Sgr A* with very high precision, even if the observing time-span covers only a few periapsis passages. We have to keep in mind though, that the secular precession of the orbit, which is used in Figure 6 to determine the spin, also has a contribution from the quadrupole moment of the black hole. For a Kerr black hole and the orbits considered here, this contribution is considerably smaller than the spin contribution (cf. Figure 3). Nevertheless, for a 10⁻³ precision (or better) in the spin measurement within a no-hair-theorem test, we need to account for the quadrupole contribution without a priori assuming a fixed relation between spin and quadrupole moment.

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As already argued in Wex & Kopeikin (1999) and demonstrated in detail in Liu et al. (2012), we have a good handle on the quadrupole from the characteristic periodic features in the timing residuals. In a consistent analysis we can, therefore, measure the quadrupole moment and simultaneously account for its contribution to the spin determination based on the secular precession of the pulsar orbit. Moreover, as we will discuss below in the subsection on the quadrupole measurement, the periodic residuals caused by the quadrupole moment have a signature that is quite different from the signature of the periodic spin contributions, which further helps to separate spin and quadrupole effects in the orbital motion.

In an ideal situation, the pulsar's motion around Sgr A* will only be affected by the gravitational field of the black hole, as we have assumed in the previous simulations. However, if the orbital motion of the pulsar is exposed to external perturbations, for instance by a nearby mass distribution due to stars or dark matter, then the orbit might show an additional precession, which a priori cannot be quantified (see Merritt et al. 2010 and discussion in Section 2). This is expected to be of particular importance around apoapsis, where the pulsar spends most of its time, and where the gravitational effects from the black hole are weaker. In such a case, all the information on the black hole spin has to come from the expectedly dominating spin effects near the periapsis. We have modeled such a situation in our simulations by taking only TOAs during the periapsis passages (time interval of only $\pm 0.05\;P$ around periapsis), The estimated measurement precision for the spin of Sgr A* is plotted in Figure 7. While the measurement precision is weaker than in Figure 6, it is obvious that it will still be possible to measure the spin of Sgr A* with high precision just based on the characteristic Lense–Thirring signal near periapsis (cf. Figure 5).

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Finally, if we are only able to observe a single periapsis passage, of a wide but highly eccentric orbit, like the one in Figure 5, a complete spin measurement might be out of range, due to the strong correlations with other timing parameters. Nevertheless, as our simulations show, we should still be able to get precise constraints on different spin-projections, like $\chi \mathrm{cos}{\rm{\Theta }}$ and $\chi \mathrm{sin}\lambda$, similar to the situation in Liu et al. (2012), if, for instance, none of the higher derivatives in $\tilde{\omega }$ and x can be measured.

4.2.2. Determining the Spatial Orientation of the Sgr A* Spin

Further constraints on the spin orientation, in particular on the direction of the projection of the spin into the plane of the sky, do come from the proper motion of Sgr A* with respect to the solar system barycenter (SSB). The transverse motion of Sgr A* with respect to the SSB modifies the observed Roemer delay by

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{{\rm{R}}}^{\mathrm{pm}}\quad =\quad \displaystyle \frac{1}{c}({{\boldsymbol{\mu }}}^{*}\cdot {\boldsymbol{r}})(t-{t}_{0}),\end{eqnarray} \tag{ 54 }$$

where ${{\boldsymbol{\mu }}}^{*}$ is the angular proper motion vector of Sgr A* in the sky (Kopeikin 1996). The contribution ${{\rm{\Delta }}}_{{\rm{R}}}^{\mathrm{pm}}$ has an impact on the arrival times of the pulsar signals that is distinctly different from Lense–Thirring contributions (see Figure 8), and depends on the orientation of the pulsar orbit with respect to the well known proper motion of Sgr A* (Reid & Brunthaler 2004). This can be easily demonstrated by looking at the orbital averaged changes to the semimajor axis (x) and the longitude of periapsis ($\tilde{\omega }$), which are given by

$$\begin{eqnarray}&&\dot{x}/x\quad =\quad {\mu }^{*}\mathrm{cot}i\mathrm{sin}{\rm{\Omega }},\end{eqnarray} \tag{ 55 }$$

$$\begin{eqnarray}&&\dot{\tilde{\omega }}\quad =\quad {-\mu }^{*}\mathrm{csc}i\mathrm{cos}{\rm{\Omega }},\end{eqnarray} \tag{ 56 }$$

where Ω denotes the longitude of the ascending node (measured clockwise in the sky, with respect to the direction of proper ${{\boldsymbol{\mu }}}^{*}$). Hence, the proper motion contribution to the Roemer delay gives access to the sixth Keplerian parameter, i.e., Ω, and therefore completely determines the 3D orientation of the orbit. Consequently, since we know the orientation of the black hole spin with respect to the pulsar orbit from timing its orbital motion, the 3D orientation of the black hole spin can be determined. This is valuable input for combining pulsar observations with the measurements of the Sgr A* shadow with the EHT, as we will discuss in Section 6.

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Once the mass and spin are measured, a Kerr spacetime is fully determined. Consequently, as discussed above, the measurement of any higher multipole moment is a test of the Kerr hypothesis. For a pulsar in orbit around Sgr A* one can hope for the measurement of the quadrupole moment, as the leading multipole moment, after ${M}_{\bullet }$ and ${S}_{\bullet }$. The quadrupole moment of Sgr A* leads to a distinct signal in the arrival times of the pulses, as it modifies the orbital motion of the pulsar in a characteristic way (Wex & Kopeikin 1999). Based on self-consistent mock data simulations, Liu et al. (2012) showed that, for a pulsar with an orbital period of a few months, it should be possible to extract the quadrupole of the Sgr A* spacetime from the timing residuals with high precision. Depending on the rotation of Sgr A* and the eccentricity of the orbit, this could be easily achieved with a precision of 1% or even better.

The simulations for the no-hair-theorem test in Liu et al. (2012) are based on the optimistic (cf. discussion in Section 2) assumption that the pulsar orbit does not experience any relevant external perturbations and therefore the secular precession of the orbit can be used to determine the spin of Sgr A*.⁶ In this section we relax this assumption, like we have done above for the spin measurement. For our simulations we added the implementation for the quadrupole moment of Liu et al. (2012) to our aforementioned extension of TEMPO. Figure 9 is the result of a simulation, where we use timing data only near periapsis, and allow for an undetermined overall precession of the orbit due to some unknown external perturbations. Figure 9 clearly shows that, after fitting for the pulsar spin parameters, orbital parameters, and frame dragging, there is still a distinctive signal in the residuals as a result of the quadrupole moment of the black hole. As a general result, depending on the timing-precision and the periapsis distance, we will still be able to extract the quadrupole moment of Sgr A*. Of course, this also depends on the actual value of the spin of Sgr A*, which is poorly constrained to date. In fact, the strength of the quadrupole effect scales with χ², and is therefore clearly less prominent for a slowly rotating black hole (see Figure 10). Depending on the timing precision, however, the quadrupole moment can still be determined with high precision.

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We have conducted extensive mock data simulations to study the joint measurability of spin and quadrupole moment. Like in the simulations for the spin measurement, we have assumed three TOAs per day. Figure 11 shows the timing coverage of the spin and quadrupole signature during one periapsis passage. Some of the results are illustrated in the contour plots of Figure 12. We conclude that, even for the conservative situation of a comparably low timing precision (${\sigma }_{\mathrm{TOA}}\quad =\quad 100\;\mu$s) and the presence of external perturbations, a quantitative test of the Kerr hypothesis is possible after only a few periapsis passages. If we have a better timing precision or can make use of timing measurements along the whole orbit, the spin and quadrupole moment can be determined with high precision after a few orbits. The latter agrees with the findings in Liu et al. (2012).

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Given the large size of the pulsar orbit ($\sim {10}^{2}\;\mathrm{AU}$), the orbital parallax (Kopeikin 1995), which is of the order of $\sim {a}^{2}/2{cD}$, will lead to a significant contribution to the timing observations, even for a moderate timing precision. This timing effect depends only on well determined orbital parameters and the distance to Sgr A*, D, and consequently can give independent access to D (cf. discussion in Section 4.1). The orbital parallax is a periodic signal in the timing residuals, and therefore, if we have N equally distributed TOAs with uncertainty ${\sigma }_{\mathrm{TOA}}$, its measurement scales proportional to ${\sigma }_{\mathrm{TOA}}$ and $\sqrt{N}$. Consequently we find

$$\begin{eqnarray}\delta D & \sim & 2\;\displaystyle \frac{c{\sigma }_{\mathrm{TOA}}}{\sqrt{N}}{(\displaystyle \frac{D}{a})}^{2}\\ & \sim & 20\;\mathrm{pc}(\displaystyle \frac{{\sigma }_{\mathrm{TOA}}}{{10}^{2}\;\mu {\rm{s}}}){(\displaystyle \frac{N}{{10}^{3}})}^{-1/2}{(\displaystyle \frac{a}{{10}^{2}\mathrm{AU}})}^{-2},\end{eqnarray} \tag{ 57 }$$

where we have used D = 8.3 kpc.

External perturbations can also lead to changes of the orbit, which could in principle partly mimic the above effects. This, however, depends highly on the specifics of the perturbation, and we will not discuss this in further detail in this paper. On the other hand, as argued by Liu et al. (2012), a precise measurement of the Sgr A* mass from pulsar timing can be converted into a precise determination of the distance to Sgr A*, when combined with high-precision astrometric observations in the infrared. For instance, a high precision measurement of ${M}_{\bullet }$ in combination with the (angular) size of the S2-star orbit in Gillessen et al. (2009a) can be converted into a direct measurement of the Galactic Center distance with an error of ∼100 pc. Future 10 μas astrometry promises a precision of the order of one parsec or even better.