Graduate Seminar: Compact Objects in Dense Stellar Systems
Omega Centauri Society Problem Sets

Problem Set 2 — Pulsar Timing Sensitivity & TRAPUM Constraints

Estimated time: 3–4 hours Total marks: 40 Tools: pulsar-accel-mapper omegacentauri.me/pset-2-pulsar.html
Learning objectives
  1. Derive the maximum line-of-sight acceleration a pulsar can experience from a central mass at a given projected radius
  2. Relate observed period derivatives Ṗ/P to a gravitational acceleration upper limit
  3. Interpret the TRAPUM 2026 MeerKAT results in terms of allowed IMBH mass windows
  4. Design a future observation strategy to reach the Häberle mass scale with pulsar timing
Adopted values (used throughout PS 1–4)
  1. Distance to OC: d = 5.49 kpc (oMEGACat VI, arXiv:2503.04903). 1 pc subtends 37.57″, so 1″ = 0.02662 pc and 1′ = 1.597 pc.
  2. Central velocity dispersion: σ0 = 18.2 km/s (van de Ven et al. 2006). Baumgardt & Hilker (2018) catalogue ~16.8 km/s globally; the two are reported side by side rather than merged.
  3. Core radius: rc = 2.37′ = 3.79 pc (Harris 1996, 2010 edition, scaled to d = 5.49 kpc).
  4. Häberle mass: M ≥ 8,200 M☉, a lower bound. Baumgardt: M < 3,000 M☉, a 3σ upper limit. TRAPUM 2026: M < 105 M☉ at 90% CL.

Background

The TRAPUM survey (TRAnsients And PUlsars with MeerKAT) reported new pulsar timing results for Omega Centauri in 2026 (arXiv:2603.21845), placing an upper limit of MIMBH < 105 M☉ at 90% confidence. This limit comes from the absence of large line-of-sight acceleration residuals in the timing of the eight OC millisecond pulsars that have published timing solutions, out of 19 known.

The line-of-sight acceleration alos of a pulsar at projected radius R from a central mass M is bounded above by (point-mass approximation):

a_los(R, z) = G M z / (R² + z²)^(3/2) Maximised over the unknown line-of-sight offset z at z = R/√2: a_max(R) = (2 / 3√3) × G M / R² , 2/3√3 = 0.3849 The pulsar's true 3D radius is unknown, so only its projected R is observable; the coefficient 2/3√3 is the largest line-of-sight component any z can produce at that R. (A pulsar exactly in the plane of the sky has z = 0 and contributes no line-of-sight acceleration at all, so the sky-plane case is the minimum, not the maximum.) Observed: a_los = (Ṗ_obs - Ṗ_int - Ṗ_Shk) / P × c where Ṗ_int is the intrinsic spin-down, Ṗ_Shk the Shklovskii (proper-motion) term, P the spin period, c the speed of light.

The OCS Pulsar Acceleration Mapper takes a central mass M and mass model (point, Plummer, or hybrid) and computes the maximum alos at the projected positions of OC's eight timed pulsars, comparing against the measured period derivatives to identify which masses are allowed.

The eight OC millisecond pulsars with published timing solutions are listed below. Periods and angular offsets are the values carried in the OCS measurement compilation (Dai et al. 2020 for A–E, Chen et al. 2023 for G, TRAPUM 2026 for H and K); |alos| is computed from them as |Ṗ|/P × c, projected radii at d = 5.49 kpc.

Pulsarθ (arcmin)R (pc)P (ms)Ṗ (s/s)|a_los| = |Ṗ|/P × c (m/s²)
PSR J1326−4728H0.560.8942.520+3.99 × 10⁻²⁰4.75 × 10⁻⁹
PSR J1326−4728B0.761.2144.792−5.43 × 10⁻²⁰3.40 × 10⁻⁹
PSR J1326−4728E1.582.5234.208+1.63 × 10⁻²⁰1.16 × 10⁻⁹
PSR J1326−4728K1.893.0184.716−9.10 × 10⁻²¹5.79 × 10⁻¹⁰
PSR J1326−4728A1.933.0824.109+2.73 × 10⁻²⁰1.99 × 10⁻⁹
PSR J1326−4728G1.963.1303.304+2.77 × 10⁻²⁰2.51 × 10⁻⁹
PSR J1326−4728C1.983.1626.868+1.01 × 10⁻²⁰4.41 × 10⁻¹⁰
PSR J1326−4728D2.503.9924.579−4.12 × 10⁻²⁰2.70 × 10⁻⁹

Sign convention: positive Ṗ is net spin-down, negative Ṗ an apparent spin-up produced by line-of-sight acceleration toward the observer. Treating the full |Ṗ|/P as gravitational, as Problem 1 does, assigns to the central mass an acceleration that in reality also contains the pulsar's unknown intrinsic spin-down and the cluster's own mean-field pull. Eleven further OC pulsars have no published Ṗ and cannot enter this calculation.

Problems

1. Point-mass acceleration limit

Using the point-mass formula above and the innermost timed pulsar, PSR J1326−4728H at R = 0.894 pc:

  1. What is the maximum central point mass M consistent with amax(0.894 pc) ≤ 4.75 × 10−9 m/s²? Use the 2/3√3 form above, which is what the tool implements. Express in M☉. [6 marks]
  2. Repeat for PSR J1326−4728A at R = 3.082 pc with amax ≤ 1.99 × 10−9 m/s². Which pulsar places the stronger constraint and why? [5 marks]
  3. The Häberle (2024) lower bound is M ≥ 8,200 M☉. Is that mass excluded by either pulsar? Show the calculation, then explain why your answer to (a) is not the published TRAPUM limit of 105 M☉. [4 marks]
↗ Pulsar Acceleration Mapper — pre-filled M = 50,000 M☉, point model, d = 5.49 kpc
Show solution
Part (a): M_max = a_max × R² / (0.3849 G) R = 0.894 pc = 0.894 × 3.086×10¹⁶ = 2.760×10¹⁶ m R² = 7.617×10³² m² M_max = 4.75×10⁻⁹ × 7.617×10³² / (0.3849 × 6.674×10⁻¹¹) = 3.618×10²⁴ / 2.569×10⁻¹¹ = 1.408×10³⁵ kg = 1.408×10³⁵ / 1.989×10³⁰ M☉ ≈ 7.08×10⁴ M☉
Part (b): R = 3.082 pc = 3.082 × 3.086×10¹⁶ = 9.511×10¹⁶ m R² = 9.046×10³³ m² M_max = 1.99×10⁻⁹ × 9.046×10³³ / (0.3849 × 6.674×10⁻¹¹) = 1.800×10²⁵ / 2.569×10⁻¹¹ = 7.007×10³⁵ kg ≈ 3.52×10⁵ M☉ PSR H places the stronger constraint, 7.08 × 104 M☉ against 3.52 × 105 M☉, even though the two acceleration limits differ by only a factor 2.4. The projected radii differ by 3.4×, and amax ∝ R⁻², so the R² term supplies a factor of 12 in the mass limit against the factor 2.4 from alos. Inner pulsars dominate the constraint, which is why the survey strategy is to find pulsars close to the centre rather than to time distant ones better.

Part (c): The tightest limit above is 7.08 × 104 M☉, more than eight times the Häberle bound of 8,200 M☉, so neither pulsar excludes it. Nor does the published survey: TRAPUM's 90% CL limit of 105 M☉ sits an order of magnitude above the Häberle scale, and the pulsar constraint and the fast-star measurement are not in conflict.

Part (a) lands within 30% of the published 105 M☉, which is closer than the calculation deserves. It assigns the whole of the observed Ṗ to a single central point mass, and three effects each loosen the real limit. The measured Ṗ contains the pulsar's intrinsic magnetic-dipole spin-down, unknown per pulsar and not subtractable without a population model. It also contains the cluster's own mean-field acceleration, which at 0.9 pc from the centre of a 4 × 106 M☉ cluster exceeds anything an 8,200 M☉ black hole contributes. And a point mass is the most concentrated model available, converting a given acceleration into the smallest allowed mass, whereas the published limit marginalises over extended models too. The agreement to 30% reflects those effects partly cancelling against the 90% CL threshold, not a validated single-pulsar limit.
2. Point vs. Plummer mass model

The pulsar timing constraint depends on the assumed mass model. A point mass gives the maximum possible acceleration at any radius; a Plummer model (with scale radius rc) gives a lower acceleration at small R.

  1. In the Pulsar Acceleration Mapper, change the model from point to plummer with rc = 0.8 pc and M = 50,000 M☉. How does the maximum alos for the innermost pulsar (PSR H, R = 0.894 pc) change? Express as a ratio (Plummer / point). [6 marks]
  2. Qualitatively explain why the Plummer model gives a less stringent constraint for pulsars at R < rc. What physical scenario does this represent? [5 marks]
  3. Why does the point-mass model give a conservative (strong) upper limit on M when used to constrain a single compact object, while the Plummer model is appropriate for a distributed mass (dark cluster)? [4 marks]
↗ Pulsar Acceleration Mapper — pre-filled M = 50,000 M☉, Plummer model, rc = 0.8 pc
Show solution
Part (a): The tool implements the Plummer case as a softened point mass, a_max = (2/3√3) G M / (R² + r_c²), so the ratio to the point model is R² / (R² + r_c²). At R = 0.894 pc, r_c = 0.8 pc: R² = 0.7992 pc², r_c² = 0.6400 pc² ratio = 0.7992 / (0.7992 + 0.6400) = 0.7992 / 1.4392 = 0.555 The Plummer acceleration is 55% of the point-mass value, so the same measured alos permits 1.8× more total mass. The suppression is mild here only because PSR H sits just outside rc; at R = 0.24 pc the same expression gives 0.0826, a 12× weaker constraint. Two things are worth noting. The suppression is controlled entirely by R/rc, and the softened form is not the same model as a true Plummer sphere, whose enclosed mass M(<R) = M R³/(R² + r_c²)3/2 would give 0.414 at PSR H instead of 0.555. The softened form is the conservative choice for setting an upper limit, since it suppresses the acceleration less.

Part (b): When R < r_c, most of the extended Plummer mass lies outside R. By the shell theorem, only the enclosed mass contributes to the gravitational acceleration at R. This physically represents a scenario where the central mass is distributed rather than pointlike, such as a cluster of stellar-mass black holes with a Plummer density profile. Pulsars inside the core of the distribution feel only a fraction of the total mass.

Part (c): A point mass concentrates all M at R = 0, so any pulsar at projected radius R feels the full gravitational pull. This is the maximum possible acceleration for a given total M, giving the tightest (most conservative) upper limit on M from a non-detection. A Plummer model distributes M, so pulsars inside the core feel less acceleration, and the same observed limit allows a much larger total M. The choice of model has real physical content: IMBH hypothesis → point model; dark cluster hypothesis → Plummer model.
3. SKA observation design

The TRAPUM 2026 limit (M < 105 M☉) sits a factor of 12 above the Häberle lower bound of 8,200 M☉. Reaching that mass scale with pulsar timing requires either a pulsar at smaller projected radius than any now timed, or a much tighter alos limit on one already known. This problem quantifies both.

  1. The best current alos limit in the table is 4.75 × 10−9 m/s² (PSR H). Suppose SKA-era timing tightens that to 1 × 10−9 m/s². At what projected radius R would a pulsar have to lie for a point-mass 8,200 M☉ IMBH to produce amax = 1 × 10−9 m/s²? Express in pc and arcsec at d = 5.49 kpc. [5 marks]
  2. Compare that radius with OC's core radius (rc = 3.79 pc) and with the projected radii of the eight timed pulsars. State the two routes to the Häberle mass scale and which of them PSR J1326−4728H already satisfies. [5 marks]
Show solution
Part (a): R = sqrt(0.3849 G M / a_max) G M = 6.674×10⁻¹¹ × 8200 × 1.989×10³⁰ = 6.674×10⁻¹¹ × 1.6310×10³⁴ = 1.0885×10²⁴ m³/s² 0.3849 G M = 4.1897×10²³ R = sqrt(4.1897×10²³ / 1×10⁻⁹) = sqrt(4.1897×10³²) = 2.047×10¹⁶ m R = 2.047×10¹⁶ / 3.086×10¹⁶ = 0.663 pc In arcsec at d = 5.49 kpc: 0.663 / 5490 × 206265 = 24.9 arcsec
Part (b): R = 0.66 pc (25″) is deep inside the 3.79 pc core, and inside every one of the eight timed pulsars: the innermost, PSR H, is at 0.894 pc. So a 10−9 m/s² limit alone does not reach the Häberle scale with the current sample. There are two routes. Route 1 — find a pulsar inside 0.66 pc (25 arcsec) and time it to 1e-9 m/s². Route 2 — keep PSR H at R = 0.894 pc and tighten its limit. Required: a_max(8200 M☉, 0.894 pc) = 0.3849 × 1.0885e24 / 7.617e32 = 5.50e-10 m/s² Current: 4.75e-9 m/s² → factor 8.6 improvement needed. PSR H already satisfies the radius half of route 2; what it lacks is the timing precision. For an acceleration inferred from a spin-period derivative the sensitivity improves roughly as T−3/2 in the timing baseline, so a factor 8.6 is a plausible decade-scale gain even before SKA collecting area is counted. SKA's separate contribution is the eleven OC pulsars with no timing solution at all, plus the discovery depth to reach inside 1 pc, where stellar crowding and scintillation limit MeerKAT.

Both routes inherit the difficulty identified in Problem 1(c): a tighter alos only tightens the mass limit once the intrinsic spin-down and the cluster mean field are separated from it, and at these radii the cluster's own field dominates.

References