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Constraining the Mass of a Hypothetical Secondary Black Hole in M87 with the NANOGrav 15-Year Data Set

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read If either periodic jet wobble in M87 is a binary's orbit, the companion's mass ratio is pinned between 0.69% and 4.2% (11-year period) or between 3.7% and 100% (0.9-year period), and VLBI astrometric tracking of the jet base is the…

desk verdict A transparent, conditional constraint on a hypothetical M87 secondary; the headline ranges hinge on an ad hoc 1 rg reflex-motion threshold, so treat them as scenario-dependent, not robust measurements. read the letter →

arxiv 2506.12313 v1 pith:QBPB4ARI submitted 2025-06-14 astro-ph.HE astro-ph.GAgr-qc

Motoki Kino , Masahiro Nagashima , Hyunwook Ro , Yuzhu Cui , Kazuhiro Hada , Jongho Park This is my paper
classification astro-ph.HEastro-ph.GAgr-qc
keywords ActivegalacticnucleiGravitationalwaveastronomyRadiointerferometrySupermassiveblackholesVerylongbaselineSMBHbinaryM87NANOGrav
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

This paper asks how heavy a hypothetical companion black hole to M87's supermassive black hole could be. Recent monitoring with the East Asian VLBI Network (EAVN) has caught two periodic behaviors in the M87 jet — an 11-year precession and a 0.9-year transverse oscillation — and the paper treats either as a candidate for the orbital period of a supermassive-black-hole binary. Combining the gravitational-wave merger limit, the NANOGrav 15-year background-strain limit, and the requirement that the primary's reflex motion be large enough to imprint the jet, it finds the mass ratio $q = M_2/M_1$ must satisfy $6.9 \times 10^{-3} \le q \le 4.2 \times 10^{-2}$ for the 11-year period and $3.7 \times 10^{-2} \le q \le 1$ for the 0.9-year period. The result matters because M87 is the nearest and most massive black hole for which such a companion could be directly confirmed by very-long-baseline interferometry (VLBI) astrometry, turning a generic prediction of galaxy-merger models into a testable nearby case.

What carries the argument

The load-bearing device is a two-dimensional exclusion diagram in the $(q,a)$ plane built from three curves. The merger floor $a_{\rm merge}$ comes from the Peters (1964) gravitational-wave timescale, with the binary merging once $t_{\rm GW}$ drops below the orbital period. The NANOGrav background floor $a_{\rm GWB} = 1.4 \times 10^{17} (M_1/10^9 M_\odot)^2 (D/10\,{\rm Mpc})^{-1} (h_{\rm GWB}/10^{-14})^{-1} q$ cm converts the measured background strain into a mass-ratio-dependent lower bound on $a$ through the chirp-mass strain formula $h_c = 2.76 \times 10^{-14} (M_{\rm ch}/10^9 M_\odot)^{5/3} (D/10\,{\rm Mpc})^{-1} (f_{\rm GW}/10^{-8}\,{\rm Hz})^{2/3}$ with $M_{\rm ch} = M_1 q^{3/5}/(1+q)^{1/5}$. The reflex-motion floor $a_1 = q a/(1+q) \ge 1\,r_g$, where $r_g = GM_1/c^2$, is the requirement that the primary's wobble be large enough to imprint the jet. The candidate periods enter through the Keplerian relation $a_T = (GM_1 T^2/4\pi^2)^{1/3}(1+q)^{1/3}$, and the quoted $q$ ranges are the intersections of these period lines with the surviving region of the plane.

What would settle it

Phase-referenced VLBI astrometric monitoring of the M87 jet base at 22–43 GHz (and eventually 86 GHz), with per-epoch precision near $4\,\mu$as or better over at least one full cycle of each candidate period: if no periodic reflex displacement of the core appears at the predicted amplitude ($\sim 4$–$40\,\mu$as for the allowed $q$ ranges) on either the 11-year or the 0.9-year timescale, the orbital interpretation of the jet periodicity is falsified. In the opposite direction, a matched-filter search of future pulsar-timing data for a continuous gravitational-wave source at $f = 2/T \approx 5.7$ nHz with strain at or above the value predicted by $h_c$ in Eq. (10) would confirm the 11-year binary.

Watch

Extended reading notes

Core claim

On the assumption that M87 hosts a circular supermassive-black-hole binary whose primary ($M_1 \approx 6.5 \times 10^9 \, M_\odot$, from the EHT shadow measurement) launches the observed jet, the paper derives the allowed region in the plane of mass ratio $q = M_2/M_1$ and semimajor axis $a$ by imposing three exclusion conditions. First, the binary must not already be merging: requiring the gravitational-radiation timescale to exceed the orbital period sets the floor $a > a_{\rm merge}$, with $t_{\rm GW}$ from Peters (1964). Second, the binary's characteristic strain $h_c$, set by the chirp mass $M_{\rm ch} = M_1 q^{3/5}/(1+q)^{1/5}$, must not exceed the NANOGrav 15-year background amplitude $h_{\rm GWB} \approx 1 \times 10^{-14}$ in the 2–30 nHz band, giving the floor $a \ge a_{\rm GWB}$. Third, the reflex motion of the primary must be able to drive the jet's periodicity, which the paper encodes as $a_1 \ge 1 \, r_g$. Intersecting these excluded regions with the Keplerian period lines $a_T(T_{\rm prec})$ and $a_T(T_{\rm trans})$ yields $6.9 \times 10^{-3} \le q \le 4.2 \times 10^{-2}$ for $T = 11.2$ yr and $3.7 \times 10^{-2} \le q \le 1$ for $T = 0.94$ yr, i.e., a secondary of roughly $4.5 \times 10^7$–$2.7 \times 10^8 \, M_\odot$ in the long-period case. The paper further identifies a 'gap-window' between the NANOGrav and merger limits where $q \sim 1$ remains viable, and concludes that direct VLBI astrometric monitoring of the jet base is the essential next test.

Load-bearing premise

The central numerical ranges rest on the paper's chosen threshold that the primary's reflex motion must be at least one gravitational radius ($a_1 \ge 1\,r_g$) to imprint a detectable periodicity on the jet; the paper states this threshold without deriving it, and a smaller true threshold would let the companion mass be arbitrarily small.

Editorial extensions

If this is right

  • If the 11-year precession is the orbital period, the secondary is a $4.5 \times 10^7$–$2.7 \times 10^8\,M_\odot$ black hole, and the upper end of that range is set directly by the NANOGrav 15-year background strain.
  • If the 0.9-year oscillation is the orbital period, the allowed range extends up to an equal-mass binary ($q=1$), in which case the primary's reflex motion is of order 10 gravitational radii, a few tens of microarcseconds on the sky, and is directly testable with current VLBI astrometry within a few years.
  • The 'gap-window' between the NANOGrav and merger limits, corresponding to orbital periods of roughly two years and $q \sim 1$, is the most promising place to hunt for the binary, and the paper proposes a 1–2 year VLBI astrometric pilot study of the M87 jet base as the first step.
  • Long-term astrometric monitoring, rather than the short-duration epochs available so far, is required to separate a genuine reflex motion from jet flares, annual parallax, and structural changes in the jet base.

Reading between the lines

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

  • My reading: the lower ends of both $q$ ranges are governed entirely by the assumed reflex-motion floor $a_1 \ge 1\,r_g$, which the paper adopts rather than derives; if jets can show periodicity for smaller reflex amplitudes, the lower bounds shrink toward zero and only the NANOGrav-based upper bound ($q \le 4.2 \times 10^{-2}$ for 11 yr) remains a robust exclusion.
  • My reading: the same exclusion-diagram machinery transfers to other low-redshift massive supermassive black holes with periodic jets, but the location of the gap-window depends on the pulsar-timing frequency band, so for M87 the window is as much a statement about NANOGrav's 2–30 nHz band as about the source itself.
  • My reading: the paper's logic yields a concrete prediction — astrometric monitoring that reaches a few $\mu$as precision and covers at least one full cycle of either candidate period should reveal a periodic core wobble if the binary interpretation is right, and its absence would effectively close both period cases.
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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 considers a hypothetical supermassive-black-hole binary in M87, with the primary mass fixed to the EHT measurement M1 = (6.5 +/- 0.7) x 10^9 Msun. Using three constraints - (i) a lower bound on the semimajor axis from the merger criterion t_GW < T, (ii) a lower bound on the semimajor axis from the NANOGrav 15-year GWB amplitude to avoid overproduction, and (iii) a lower bound a1 >= 1 rg on the primary's reflex-motion semimajor axis - the authors derive allowed ranges for the mass ratio q and semimajor axis a. Assuming one of the two EAVN jet periodicities (T_prec = 11.24 +/- 0.47 yr or T_trans = 0.94 +/- 0.12 yr) equals the binary orbital period, they obtain the central quantitative results 6.9e-3 <= q <= 4.2e-2 for T = T_prec and 3.7e-2 <= q <= 1 for T = T_trans, stated in Eqs. (14) and (15). The paper also discusses future VLBI astrometric strategies to test the binary hypothesis.

Significance. If the quoted constraints were robust, the paper would provide a useful first step connecting EHT, EAVN, and PTA observations to bound the mass of a hypothetical secondary black hole in one of the nearest giant ellipticals. The derivation is simple, transparent, and reproducible from the stated inputs, and the falsifiable prediction that VLBI astrometry of the M87 jet base can test the allowed parameter space is valuable. The main weakness is that the lower bounds in Eqs. (14) and (15) rest entirely on the ad hoc threshold a1 = 1 rg, which has no physical or observational justification, and the upper bound for T = T_prec depends on a fixed, un-uncertainty-quantified choice of the NANOGrav amplitude. As presented, the central quantitative claim is therefore not as robust as the abstract implies.

major comments (3)
  1. [Section 4.2.1, item 3] The lower bounds in Eqs. (14) and (15), q_min = 6.9e-3 and 3.7e-2, are directly determined by the assumed threshold a1 = 1 rg. The statement “below this threshold, the jet is unlikely to exhibit a distinct periodic signature” is unsupported by any physical calculation, simulation, or observational calibration. The VLBI astrometric limits in Section 3.2 are upper limits of about 8–12 rg over short epochs, so they cannot justify a lower limit. For fixed T, the relation a1 = q a_T/(1+q) implies that q_min scales linearly with the chosen threshold; for a1_min = 10 rg the lower bounds would become roughly 7e-2 and 0.37, substantially changing the scientific interpretation, especially for T = T_trans where the allowed range would shrink toward q ~ 1. The authors should derive a1_min from the observed amplitude of the jet periodicities, use the VLBI upper limits in a more careful way, or present q constraints as explicit functions of a1_min.
  2. [Section 3.3 and Eq. (12)] The GWB amplitude is fixed to h_GWB = 1e-14 without justification and without uncertainty propagation. The published NANOGrav 15-yr amplitude at 1 yr^-1 is approximately A = 2.4e-15, and for the SMBHB spectral index of -2/3 the characteristic strain at f ~ 5.6 nHz (the GW frequency corresponding to T_prec) is about 5e-15, not 1e-14. Because the upper bound on q from the condition a_T = a_GWB is proportional to h_GWB, this choice overestimates the upper limit in Eq. (14) by roughly a factor of two. Additionally, the quoted ranges (6.9e-3 <= q <= 4.2e-2 and 3.7e-2 <= q <= 1) are presented as exact inequalities with no propagation of uncertainties in h_GWB, M1, D, or T. The authors should use the NANOGrav posterior or at least a quoted 1-sigma range, and provide error bars on the derived q intervals.
  3. [Section 4.2.1] The central results assume that either T_prec or T_trans equals the binary orbital period. The paper does not discuss whether the physical mechanism for the observed jet precession and transverse oscillation (e.g., disk precession, Lense-Thirring precession, or bending waves) would produce a period equal to the orbital period; many such mechanisms yield timescales that differ from the orbital period. Because the entire interpretation of the q ranges in Eqs. (14) and (15) depends on this equality, the authors should either provide a physical argument for why T = T_orb or explicitly state that the constraints are derived under a simplifying assumption and discuss the resulting caveat.
minor comments (5)
  1. [Section 3.3] The text before Eq. (12) says “one can obtain the lower limit a_GWB” and then “the NANOGrav 15-year dataset can place an upper limit through the GWB strain amplitude.” These statements are contradictory; a_GWB is a lower limit on the semimajor axis needed to avoid overproduction, so please clarify the wording.
  2. [Section 3.4] The definition of t_merge is incomplete. The text should state explicitly that t_merge is the common timescale at which t_GW(a) = T(a), and then report the resulting value for M87 (approximately 0.3 yr for q = 1).
  3. [Throughout] There are several typographical issues: “EA VN” appears with a space; the title contains “NANOGrav 15-Y ear”; Section 4.2.1 contains “and and aT”; Section 5.2 contains “gravitational Browninan motion”; and Eq. (19) uses f_GW without defining it in that section (it is defined in Eq. (11)).
  4. [Figure 3] The legend entries “a (for a1 = 1rg)” and “a (for a1 = 6rg)” denote curves of a versus q, not single values; consider renaming them to something like “a1 = 1 rg” and “a1 = 6 rg” for clarity.
  5. [Section 5.2] The discussion of fuzzy dark matter Brownian motion is interesting but tangential to the paper's main claim; consider shortening it or moving it to an appendix to keep the manuscript focused.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the q ranges combine external NANOGrav and EAVN inputs with an explicitly labeled a1 threshold assumption.

full rationale

I find no circular step in the derivation chain. The upper bounds on q come from the external NANOGrav 15-year GWB amplitude via the strain inequality h_GWB >= h_c and Eq. (12), while the periods T_prec and T_trans are taken from the independent EAVN monitoring papers (Ro et al. 2023; Cui et al. 2023) and fix the orbital semimajor axis through Eq. (9). The lower bounds in Eqs. (14) and (15) are obtained by solving a1(q, T) = 1 rg, where 1 rg is explicitly introduced in Section 4.2.1 item 3: 'we assume a lower limit of a1 = 1 rg... Below this threshold, the jet is unlikely to exhibit a distinct periodic signature.' This means the lower limits are algebraically determined by that assumption, not by a derived first-principles result. However, the paper is transparent that this value is assumed rather than measured, so this is a robustness limitation, not a circular reduction. The VLBI astrometric measurements cited in Section 3.2 are upper limits on core-position stability and cannot independently justify the 1 rg threshold, but the paper does not claim they do. The self-citations in the paper (Hada, Cui, Ro, Kino) provide observational data and are not load-bearing in the sense of a uniqueness theorem or an unverified premise that is needed to make the central constraints true. The NANOGrav, EHT mass, and merger-limit inputs are all external to the present work. No prediction or allowed range is equivalent to its input by construction, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central constraints rest on the NANOGrav amplitude (an external input, not fitted), the assumed a1 = 1 rg threshold (ad hoc), and the conditional association of jet periods with the orbital period. The main free parameter is the reflex-motion threshold; hGWB is a rounded observational input.

free parameters (2)
  • a1_min (reflex-motion threshold) = 1 rg ≈ 9.6e14 cm
    Hand-chosen in Section 4.2.1; sets the lower bound on q for both period cases. No physical model is given for this threshold.
  • hGWB (assumed GWB amplitude) = 1e-14
    Rounded input from NANOGrav 15-yr, used in Eq. (12) and Figure 3; not fitted here, but the choice directly affects the upper bound on q for T = Tprec.
assumptions (7)
  • domain assumption The SMBH binary is on a circular orbit.
    Invoked in Section 2.1 and used in Eqs. (2), (6), and (10); a standard simplification but not justified for M87.
  • domain assumption The primary mass is the EHT-measured mass, M1 = 6.5e9 Msun, and the EHT ring corresponds to the primary.
    Stated in Section 3; if M87 is a binary, the EHT shadow likely traces the primary but the total mass could differ.
  • domain assumption The EAVN jet periodicities (11.24 yr precession, 0.94 yr transverse oscillation) are caused by the reflex motion of the primary in a binary.
    The central hypothesis, explicitly made conditional in the abstract and Section 4.2; no independent evidence that these periods are orbital.
  • domain assumption The GWB strain amplitude from M87 must not exceed the NANOGrav15 total background, hGWB ≈ 1e-14.
    Valid as an upper bound if M87 contributes to the background, but hGWB is an order-of-magnitude value with no uncertainty propagated into the final q ranges.
  • domain assumption The binary separation must exceed the merger limit, defined by tGW < T at a = amerge.
    Uses the Peters (1964) timescale; standard, but ignores possible eccentricity and gas effects, as discussed in Section 4.1.
  • ad hoc to paper a1 >= 1 rg is required for the jet to show a distinct periodic signature.
    Introduced in Section 4.2.1 item 3; unsupported by a physical model and directly sets the lower bound on q.
  • domain assumption The secondary black hole produces no jet; only M1 anchors the observed jet.
    Assumed in Section 4 and Figure 1; if M2 also launched a jet, the interpretation of the periodicity would change.

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Cite this review

Pith. "Pith review of Constraining the Mass of a Hypothetical Secondary Black Hole in M87 with the NANOGrav 15-Year Data Set." pith.science (2026). https://pith.science/paper/QBPB4ARI

@misc{pith2026250612313,
  author       = {Pith},
  title        = {Pith review of: Constraining the Mass of a Hypothetical Secondary Black Hole in M87 with the NANOGrav 15-Year Data Set},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QBPB4ARI}},
  note         = {Machine review of arXiv:2506.12313}
}
abstract

Galaxy mergers, each hosting a supermassive black hole (SMBH), are thought to form SMBH binaries. Motivated by recent observations from the East Asian VLBI Network (EAVN) showing periodic behavior in the M87 jet, a precession of about 11 years and a transverse oscillation of about 0.9 years, we constrain the mass of a hypothetical secondary black hole orbiting the primary SMBH in M87. To constrain the mass ratio between the primary SMBH ($M_{1}$) and the secondary black hole ($M_{2}$) defined as $q \equiv M_{2}/M_{1} \leq 1$, and the length of the semimajor axis of the binary system ($a$), we impose the following three constraints: (i) the lower limit of $a$, below which the SMBH binary is expected to merge. (ii) the strain amplitude of the gravitational wave background (GWB) at nanohertz frequencies shown in the NANOGrav 15-year dataset. (iii) a finite length of the semimajor axis of $M_{1}$, that can induce periodic behavior in the jet. By combining these constraints, we obtain the allowed parameter space for $q$ and $a$. If either of the EAVN-detected periods ($T$) corresponds to the binary's orbital period, the allowed range of $q$ is $6.9 \times 10^{-3} \le q \le 4.2 \times 10^{-2}$ for $T \approx 11$ years, and $3.7\times 10^{-2} \le q \le 1$ for $T \approx 0.9$ years. VLBI astrometric monitoring of the jet base of M87 is essential to explore the allowed parameter space for $q$ and $a$.

Figures

Figures reproduced from arXiv: 2506.12313 by the authors.

Figure 1
Figure 1. An illustration of the basic geometry of the hypothetical SMBH binary system considered in this study, with M87 used as a prime example. The primary black hole (M1) generates the prominent radio jet observed at low frequencies, with the jet base anchored to M1, while the secondary black hole (M2) does not produce a jet. The reflex motion of M1 likely induces periodic behaviors in the jet, such as precessing motion, … view at source ↗
Figure 2
Figure 2. Comparison of characteristic timescales in M87. The binary orbital period (T ∝ a 3/2 ), represented by the thick line, is the most important timescale, as given by Eq. (2). The timescale for GW radiation, tGW ∝ q −1 a 4 , is represented by a dark-gray thick line for 0.01 ≤ q ≤ 1. We also plot the two blue lines representing Tprec and Ttrans. When T = Tprec, the corresponding semimajor axis is approximately a ≈ 1 × 1… view at source ↗
Figure 3
Figure 3. Allowed parameter space for q and a in M87. All the gray-shaded regions represent the excluded areas for q and a, while the remaining white region is the allowed parameter space for q and a. In particular, the allowed ranges for the Cases of T = Tprec are T = Ttrans are shown in blue lines. The upper limit of a is constrained by the condition of a < ahard. The lower limit of a is constrained by the condition of a > … view at source ↗

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