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Constraining the Binarity of Massive Black Holes in the Galactic Center and Some Nearby Galaxies via Pulsar Timing Array Observations of Gravitational Waves

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Future pulsar timing array observations can independently reveal or exclude intermediate-mass black hole companions orbiting the massive black holes in the Galactic Center, M31, and M87, and, if pulsars are found near those nuclei, could…

desk verdict A careful, internally consistent sensitivity forecast for PTA detection of MBH companions in nearby nuclei; the headline mass-ratio thresholds are white-noise ideals that red spin noise will shift upward, but the qualitative rankings survive. read the letter →

arxiv 2411.14150 v1 pith:ZXW7Z3BM submitted 2024-11-21 astro-ph.HE astro-ph.GA

Xiao Guo , Qingjuan Yu , Youjun Lu This is my paper · ORCID
classification astro-ph.HEastro-ph.GA
keywords pulsartimingarraysgravitationalwavesmassiveblackholebinariesGalacticCenterintermediate-massholesSagittariusA*eccentricSKA-PTA
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks a concrete question: if the massive black hole at the Galactic Center has a lighter black hole companion on a months-to-years orbit, can pulsar timing arrays (PTAs) detect its gravitational waves, and what can a non-detection prove? It derives the expected signal-to-noise ratio for circular and eccentric binaries in the nanohertz band and maps the parameter space that a future Square Kilometer Array PTA (SKA-PTA) could probe in the Galactic Center and in the LMC, M31, M32, and M87. The central quantitative claims are that a 20-year SKA-PTA campaign with 1,000 pulsars at 10 ns timing precision could reveal or rule out a companion of roughly 500–5,000 solar masses (mass ratio $q\sim10^{-4}$–$10^{-3}$) around Sgr A* at separations of 20–3,000 AU, a companion with $q\gtrsim10^{-4}$ in M31, and one with $q\gtrsim10^{-5}$ in M87. The LMC and M32 would not be reachable with Earth-based pulsars for small mass ratios, but if a few millisecond pulsars are found within a parsec of the central black hole, a center-based PTA could push the detectable companion mass down to about 100 solar masses. The paper thus positions PTAs as an independent observational route to the binarity of nearby massive black holes, complementing stellar-orbital and dynamical constraints.

What carries the argument

The machinery that carries the argument is an effective characteristic strain $h_{c,\mathrm{eff}}$ defined at the peak harmonic frequency of an eccentric binary, $f_{\mathrm{pk}}=n_{\mathrm{pk}}f_{\mathrm{orb}}$ with $n_{\mathrm{pk}}=2(1+e)^{1.1954}/(1-e^2)^{1.5}$, together with two signal-to-noise estimators (matched-filtering and cross-correlation) that sum over all gravitational-wave harmonics inside the PTA frequency band. This definition converts any eccentric binary into a point on the same plot as a PTA sensitivity curve, letting the authors draw S/N contours in the mass-ratio versus semimajor-axis plane. A second key element is the near-field geometric factor $\chi$ that correctly handles the case where pulsars sit much closer to the gravitational-wave source than the Earth does; this factor is what gives center-based PTAs their large sensitivity boost over Earth-based arrays.

What would settle it

Observe Sgr A* with a 20-year, 1,000-pulsar SKA-PTA at 10 ns timing precision: a null search for a monochromatic or harmonic gravitational-wave signal would directly test the claimed exclusion region ($q\sim10^{-4}$–$10^{-3}$, $a\sim20$–$3{,}000$ AU), while a deep radio survey that finds far fewer than 5–10 pulsars within 1 pc of Sgr A* would falsify the GC-PTA sensitivity projections.

Watch

Extended reading notes

Core claim

The paper's central claim is that pulsar timing arrays, particularly the future SKA-PTA, can provide an independent way to reveal or exclude low-mass-ratio massive black hole binaries in the Galactic Center, M31, and M87. For the Galactic Center, a non-detection after 20 years of SKA-PTA observations would suggest the absence of an intermediate-mass black hole with mass $\gtrsim 500$–$5{,}000\,M_\odot$ (or $q\sim10^{-4}$–$10^{-3}$) on a semimajor axis of $\sim20$–$3{,}000$ AU; for M31 the reach is $q\gtrsim10^{-4}$ at $a\sim10^2$–$10^4$ AU, and for M87 it is $q\gtrsim10^{-5}$ at $a\sim10^3$–$2\times10^4$ AU. The LMC and M32 are expected to stay out of reach for the SKA-PTA when $q\ll1$, but hypothetical PTAs built from 5–10 millisecond pulsars within 0.1–1 pc of the central black hole would reveal companions with masses down to about $100\,M_\odot$, close to the stellar-mass regime. The paper also shows that high orbital eccentricity does not prevent detection, because the gravitational-wave power is distributed into high harmonics that can fall inside the PTA band.

Load-bearing premise

The most spectacular results assume future surveys find and stably time 5–10 millisecond pulsars within about 0.1–1 parsec of the central black hole in each galaxy, a population that has not yet been confirmed beyond a single magnetar near Sgr A*.

Editorial extensions

If this is right

  • A 20-year SKA-PTA non-detection toward the Galactic Center would independently rule out an intermediate-mass black hole of roughly 500–5,000 solar masses orbiting Sgr A* at 20–3,000 AU.
  • For M31 the same campaign would reveal or exclude companions with $q\gtrsim10^{-4}$ at $10^2$–$10^4$ AU, and for M87 companions with $q\gtrsim10^{-5}$ at $10^3$–$2\times10^4$ AU, within about 20 years.
  • If several millisecond pulsars are discovered within 0.1–1 pc of the central black hole in the GC, LMC, M31, or M32, a center-based PTA could detect companions down to roughly 100 solar masses, close to stellar masses.
  • Highly eccentric binaries remain detectable because their gravitational-wave power moves into high harmonics that can fall inside the PTA band, extending the reachable semimajor axis.
  • The effective characteristic strain formalism gives future searches a common yardstick for reporting detection or exclusion limits on the mass-ratio versus semimajor-axis plane.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If these sensitivity estimates hold, a null SKA-PTA result would become an independent, dynamics-free constraint on intermediate-mass black holes in the Galactic Center, complementing limits from stellar orbits and proper-motion measurements.
  • The same near-field enhancement that powers the GC-PTA idea implies that any future discovery of pulsars within a parsec of a galactic nucleus would make that nucleus a high-value gravitational-wave laboratory, even before a planet-scale array improves.
  • Because the paper deliberately omits red intrinsic spin noise, adding that noise to the model would raise the effective noise floor and shift the quoted mass-ratio boundaries upward; the 100-solar-mass reach should therefore be read as an optimistic ceiling.
  • The formalism is general enough to be turned on stellar-mass bodies: the same center-based method that reaches ~100 solar masses could eventually search for gravitational waves from stellar remnants or even S-stars orbiting Sgr A*, a direction the paper only notes in passing.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a signal-to-noise ratio (S/N) formalism for pulsar timing array (PTA) searches for gravitational waves from an eccentric massive black hole binary (BBH) with a low-mass secondary, and applies it to hypothetical binaries in the Galactic center (GC), LMC, M31, M32, and M87. Using standard Peters-type harmonic decomposition, matched-filter and cross-correlation estimators, and assumed PTA configurations (IPTA, CPTA, SKA-PTA, and hypothetical center-based PTAs), the authors derive sensitivity contours in the mass-ratio versus semimajor-axis plane. Their central quantitative claims are that a future SKA-PTA with 20-year observations could reveal or constrain a GC companion with q ~ 1e-4 to 1e-3 (M_BH,2 ~ 500 to 5000 solar masses), a M31 companion with q ~ 1e-4, and a M87 companion with q ~ 1e-5, while center-based PTAs of a few MSPs within ~0.1-1 pc of the central black holes could push detectable companion masses down to about 100 solar masses.

Significance. If the assumptions hold, the paper provides a useful forecasting framework for an independent probe of low-mass-ratio massive black hole binaries in nearby galactic nuclei. The formalism is internally consistent: the eccentric-orbit harmonic decomposition follows standard Peters/Maggiore expressions, the matched-filter and cross-correlation S/N estimates are explicit, and the comparison with existing dynamical limits in Figure 5 gives context for the claimed constraints. The paper also credits and uses prior work for the near-field geometric factor and the GWB model, and it clearly labels the center-based PTA scenarios as conditional on future discoveries. The main value is the identification of parameter space where future PTAs could complement stellar-dynamics and S-star constraints on a hypothetical IMBH around Sgr A*, and similar bounds for M31 and M87.

major comments (3)
  1. [Sections 4.1 and 5.1, Eqs. (20)-(26)] The noise model is white shot noise plus the GWB only; red intrinsic spin noise is explicitly omitted as 'quite uncertain' (Section 4.1, Eqs. (20)-(22)). At the 1-30 nHz frequencies that set the quoted q ~ 1e-4 to 1e-3 thresholds, measured red spin noise in current PTA datasets is often comparable to or larger than a 10 ns white-timing floor. Because h_c,i is proportional to q for q << 1 and h_n appears in the denominator of Eqs. (23)-(26), including red noise raises the minimum detectable q by approximately h_n,red/h_n,white for the matched-filter method and by the square root of that ratio for the cross-correlation method. The headline statement in Section 5.1 that a 20-year SKA-PTA non-detection 'would suggest independently that there is no IMBH with mass greater than about 500 to 5,000 Msun' is therefore an idealized white-noise sensitivity bound, not a realistic non-detection forecast. This is not an internal inconsistency, but the quoted mass boundaries should be presented as optimistic limiting sensitivities and preferably supplemented with a representative red-noise case.
  2. [Table 1, Sections 5.2, 6.1-6.3] The center-based PTA scenarios (GC-PTA, LMCC-PTA, M31C-PTA, M32C-PTA) assume the existence of 5-10 timing-stable MSPs within 0.1-1 pc of each central MBH (Table 1, column r_p-BBH). This assumption is the decisive factor behind the headline result that companions down to about 100 solar masses could be revealed. For the GC only a single magnetar at roughly 0.1 pc is currently known; for M31, M32, and the LMC no such pulsars are confirmed. The paper's abstract does hedge with 'if a number of milli-second stable pulsars ... can be detected in future', but Sections 5.2 and 6.2-6.3 present the 100-solar-mass reach as a main result without a quantitative assessment of the probability or feasibility of discovering and timing such pulsars in the dense nuclear environment. These claims should be explicitly separated as a highly speculative projection, or accompanied by a discussion of scattering, dispersion measure variations, and timing stability requirements.
  3. [Section 4.4 and Figures 3-9] The interpretation of the S/N contours as exclusion or discovery boundaries uses a single deterministic threshold (rho_th = 1) with no accounting for detection probability, false-alarm rate, or noise realization. For example, the text after Figure 3 states that 'the parameter space below the brown curves cannot be ruled out' and that a non-detection 'would suggest' no IMBH above the quoted mass. A sensitivity curve with rho_th = 1 is a reasonable projection tool, but converting it into a non-detection statement requires a statistical framework (e.g., detection probability as a function of the source parameters, or a Bayesian upper limit). As written, the exclusion language overstates what a single S/N threshold actually delivers. I recommend rephrasing these conclusions as sensitivity boundaries or adding a brief statistical treatment of the non-detection case.
minor comments (5)
  1. [Page 10, text after Figure 5] There is a typo: 'can also be imited by some dynamical arguments' should read 'can also be limited by some dynamical arguments'.
  2. [Section 6.1, footnote 6] The footnote says 'the black hole in the LMC is in the mass rage of IMBHs'; 'rage' should be 'range'.
  3. [Section 6.2, first paragraph] The sentence 'We hypothesize that a BBH exists in the M31 center, with total mass M_BBH = 1.4 x 10^8 Msun and an mass ratio q' contains a grammatical error: 'an mass ratio' should be 'a mass ratio'.
  4. [Figures 3-13] The brown curves (GWB-removed cases) are described in the captions but not labeled directly in the figure panels; adding a legend entry or a label to the brown curves would improve readability.
  5. [Section 5.2, Eq. (33)] The remark that H_i in Eqs. (24) and (26) is 'the GW strain at Earth with r = 8 kpc' while Eq. (9) already includes this distance is a potential source of confusion; a short clarifying sentence on how the near-field chi modifies the S/N would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quoted PTA thresholds follow from standard GW strain and PTA-sensitivity formulas with explicitly stated assumptions; the self-citations are to published derivations/inputs, not to the claimed result.

full rationale

The paper's derivation chain is self-contained: Eqs. (1)-(10) give the GW strain from standard quadrupole/Peters formulas; Eqs. (14)-(17) define the characteristic strain; Eqs. (18)-(22) define the noise from white timing residuals and the GWB; Eqs. (23)-(26) give the S/N for matched-filter and cross-correlation methods. The Section 5.1 thresholds (q ~ 1e-4 to 1e-3, M_BH,2 ~ 500-5000 Msun) are obtained by setting S/N=1 and inverting h_c proportional to q, not by fitting any parameter to the target claim. The PTA parameters in Table 1 are stated assumptions, and the paper explicitly flags the omission of red spin noise and the speculative nature of center pulsars; these are limitations on realism, not circularity. Citations to Guo et al. (2022) for the geometric factor chi and to Chen et al. (2020, 2023) for the GWB model are published derivations/measured inputs that do not incorporate the paper's conclusion, so they do not make the argument circular. There is no fitted input renamed as a prediction and no self-citation chain forcing the result.

Assumptions & free parameters 7 free parameters · 8 assumptions · 0 invented entities

The paper is a sensitivity forecast built from standard GW formulas and assumed instrument parameters. The free parameters are the chosen PTA timing precision, cadence, observation span, pulsar counts, pulsar distances, GWB noise shape, and detection threshold. No new physical entities are introduced: the hypothetical companions are existing astrophysical hypotheses, and the near-center pulsar PTAs are observational scenarios, not new physics.

free parameters (7)
  • PTA timing residual sigma_t = IPTA 100 ns; CPTA 20 ns; SKA-PTA 10 ns; hypothetical center PTAs 100 ns
    Chosen timing precision of pulsars (Table 1); directly sets the shot-noise floor and the sensitivity curves.
  • Number of MSPs N_pl = 49 (IPTA), 100 (CPTA), 1000 (SKA-PTA), 10 (GC-PTA), 5 (LMCC/M31C/M32C PTAs)
    Assumed pulsar counts; sensitivity scales roughly as sqrt(N_pl) in matched filtering and as N_pl^(1/4) to N_pl^(1/2) across methods.
  • Observation cadence Delta_t = 0.04 yr for IPTA/CPTA/SKA-PTA; 0.02 yr for hypothetical center PTAs
    Sets the high-frequency cutoff of the PTA band; chosen in Table 1.
  • Observation span T_obs = 20 yr for IPTA/CPTA/SKA-PTA/GC-PTA; 10 yr for LMCC/M31C/M32C PTAs
    Sets the low-frequency cutoff and the number of cycles; chosen in Table 1.
  • Pulsar distance to the source r_pl = 1 pc for GC/M31/M32; 0.1 pc for LMC
    Drives the near-field geometric factor chi, which scales as r/r_pl and produces the large S/N boost for center PTAs.
  • GWB amplitude and spectral shape parameters = A = 2.5e-15, f_bend = 1.15e-10 Hz, kappa_gw = 3.70, gamma_gw = 0.19
    Adopted from Chen et al. 2020 and 2023, from the same research group, to model GWB confusion noise; affects low-frequency sensitivity.
  • S/N threshold rho_th = 1, with contours also at 3, 10, and 100
    Detection threshold used to draw sensitivity curves; changing it shifts the quoted parameter-space boundaries.
assumptions (8)
  • standard math Standard gravitational-wave emission formulas for eccentric binaries, including Peters 1964 and Peters-Mathews harmonic decomposition with Bessel functions.
    Used throughout Sections 2 and 3 to compute strain amplitudes and harmonic power.
  • domain assumption Newtonian Keplerian orbits with slow GW-driven inspiral, so the orbital phase follows Keplerian motion and the binary is treated as stationary over the observation.
    Stated after Equation (4): the GW decay timescale is much longer than the orbital period for all systems considered.
  • domain assumption PTA noise consists only of white timing noise plus GWB confusion; red intrinsic spin noise is ignored.
    Section 4.1 states that red spin noise is uncertain and is not considered, which can make sensitivity forecasts optimistic.
  • domain assumption The geometric factor chi for the pulsar-Earth-source configuration follows Guo et al. 2022, including near-field enhancement when pulsars sit close to the GW source.
    Used in Sections 4.2 and 5.2, Equation (33); central to the large S/N boost for center-based PTAs.
  • domain assumption The GWB signal can be modeled and subtracted from PTA data in the optimistic scenarios.
    Section 4.1 and Figures 3 to 4: if the GWB can be well detected, it is treated as a known signal and removed; this has not yet been demonstrated at the required precision.
  • domain assumption Adopted central black hole masses and distances for the GC, LMC, M31, M32, and M87 from cited observations.
    Sections 5 and 6 use MBBH = 4.26e6, 2.4e4, 1.4e8, 2.4e6, and 6.5e9 solar masses respectively; these set the strain amplitudes.
  • ad hoc to paper Hypothetical future pulsar populations within about 0.1 to 1 pc of the central MBHs exist and can be discovered and timed.
    Defines GC-PTA, LMCC-PTA, M31C-PTA, and M32C-PTA in Table 1 and Sections 5.2 and 6.1 to 6.3; this is not yet established observationally.
  • domain assumption A fixed S/N threshold of 1 (or 3, 10, 100) marks a detection, without false-alarm statistics.
    Used to draw sensitivity curves and quoted constraints; practical searches typically require a higher threshold. Location: Section 4.4 and figures.

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Cite this review

Pith. "Pith review of Constraining the Binarity of Massive Black Holes in the Galactic Center and Some Nearby Galaxies via Pulsar Timing Array Observations of Gravitational Waves." pith.science (2026). https://pith.science/paper/ZXW7Z3BM

@misc{pith2026241114150,
  author       = {Pith},
  title        = {Pith review of: Constraining the Binarity of Massive Black Holes in the Galactic Center and Some Nearby Galaxies via Pulsar Timing Array Observations of Gravitational Waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZXW7Z3BM}},
  note         = {Machine review of arXiv:2411.14150}
}
abstract

Massive black holes (MBHs) exist in the Galactic center (GC) and other nearby galactic nuclei. As natural outcome of galaxy mergers, some MBHs may have a black hole (BH) companion. In this paper, assuming that the MBHs in the GC and some nearby galaxies are in binaries with orbital periods ranging from months to years (gravitational-wave frequency $\sim1-100$\,nHz), we investigate the detectability of gravitational-waves from these binary MBHs (BBHs) and constraints on the parameter space for the existence of BBHs in the GC, LMC, M31, M32, and M87, that may be obtained by current/future pulsar timing array (PTA) observations. We find that a BBH in the GC, if any, can be revealed by the Square Kilometer Array PTA (SKA-PTA) if its mass ratio $q\gtrsim10^{-4}-10^{-3}$ and semimajor axis $a\sim20-10^3$\,AU. The existence of a BH companion of the MBH can be revealed by SKA-PTA with $\sim20$-year observations in M31 if $q\gtrsim10^{-4}$ and $a\sim10^2-10^4$\,AU or in M87 if $q\gtrsim10^{-5}$ and $a\sim10^3-2\times10^4$\,AU, but not in LMC and M32 if $q\ll1$. If a number of milli-second stable pulsars with distances $\lesssim0.1-1$\,pc away from the central MBH in the GC, LMC, M32, or M31, can be detected in future and applied to PTAs, the BH companion with mass even down to $\sim100M_\odot$, close to stellar masses, can be revealed by such PTAs. Future PTAs are expected to provide an independent way to reveal BBHs and low-mass MBH companions in the GC and nearby galaxies, improving our understandings of the formation and evolution of MBHs and galaxies.

Figures

Figures reproduced from arXiv: 2411.14150 by the authors.

Figure 1
Figure 1. shows some examples for the GW waveform emitted from a hypothetical BBH in the GC, with q = 0.01, a = 100AU, e = 0 (top panel), 0.5 (middle panel), or 0.9 (bottom panel). As shown in this figure, with increasing eccentricity e, the amplitude of the GW strain emitted at the pericenter increases, but the time duration for the peak GW radiation amplitude becomes shorter. This suggests that the GW signal from a highly e… view at source ↗
Figure 2
Figure 2. Sensitivity curves (hsc) of different PTAs and the effective characteristic strains (hc,eff ) for GW signals radiated from a hypothetical GC BBH with different eccentricities. Left and right panels represent the results obtained by the matched-filtering (MF) and the cross-correlation (CC) method, respectively. The two vertical black dotted lines in each panel indicate the upper limits and the lower limits for the PT… view at source ↗
Figure 3
Figure 3. Contours of the expected S/N (ϱ) in the q/(1 + q) versus a parameter space of the hypothetical BBH in the GC, monitored by the assumed future SKA-PTA (see Tab. 1). The hypothetical BBH is assumed to have a mass ratio q, semimajor axis a, and eccentricity e. Top-left, top-right, bottom-left, and bottom-right panels show the cases with e = 0.2, 0.5, 0.7, and 0.9, respectively. For each panel, the left and the right ve… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Expected S/N of a hypothetical BBH in the GC with mass ratio q, semimajor axis a, and eccentricity e, monitored by an assumed PTA composed of MSPs in the GC (see GC-PTA in Tab. 1). Legends are similar to those for [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Possible constraints on the parameter space for the existence of a BBH in the GC expected from future PTA observations. The blue, red, and brown lines represent those BBHs (e = 0.2) with S/N ϱ = 1 estimated for the CPTA, SKA-PTA, and GC-PTA, respectively, with paramete…
Figure 6
Figure 6. Figure 6: Expected S/N of a hypothetical BBH in the LMC center with mass ratio q, semimajor axis a, and eccentricity e monitored by an assumed PTA composed of MSPs in the LMC center (see LMCC-PTA in Tab. 1). Legends are similar to those for [PITH_FULL_IMAGE:figures/full_fig_p01…
Figure 7
Figure 7. Figure 7: Expected S/N of a hypothetical BBH in the center of M31 with mass ratio q, semimajor axis a, and eccentricity e, monitored by the IPTA (see Tab. 1). Legends are similar to those for [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Expected S/N of a hypothetical BBH in the center of M31 with mass ratio q, semimajor axis a, and eccentricity e, monitored by the CPTA (see Tab. 1). Legends are similar to those for [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Expected S/N of a hypothetical BBH in the center of M31 with mass ratio q, semimajor axis a, and eccentricity e, monitored by the SKA-PTA (see Tab. 1). Legends are similar to those for [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Expected S/N of a hypothetical BBH in the M31 center with mass ratio q, semimajor axis a, and eccentricity e, monitored by an assumed PTA composed of MSPs in the M31 center (see M31C–PTA in Tab. 1). Legends are similar to those for [PITH_FULL_IMAGE:figures/full_fig_p…
Figure 11
Figure 11. Figure 11: Expected S/N of a hypothetical equal-mass BBH in the center of M32 as a function of the orbital semimajor axis a, monitored by the SKA-PTA (see Tab. 1). Red, green, and cyan solid (or dashed) lines show the results obtained for a BBH with eccentricity e = 0, 0.5, and …
Figure 12
Figure 12. Figure 12: Expected S/N of a hypothetical BBH with mass ratio q, and semimajor axis a in the center of M32, monitored by a PTA with MSPs in the center of M32 (see M32C-PTA in Tab. 1). The legend is similar to that for [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: Expected S/N of a hypothetical BBH in the center of M87 with mass ratio q, semimajor axis a, and eccentricity e, monitored by the SKA-PTA. The legend is similar to that for [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]

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