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

Background

The TRAPUM survey (Transferring Astronomical Precision to Radio Astrometry 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 eight millisecond pulsars in the cluster.

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_max(R) = G M / R² This maximum occurs when the pulsar lies in the plane of the sky (no projection correction needed for the upper bound). 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.

Reported TRAPUM 2026 pulsar sample (illustrative; see paper for full values):

PulsarR (arcmin)R (pc)|ȧ_los| (m/s²)
OC-MSP-A0.150.24< 8.5 × 10⁻⁶
OC-MSP-B0.310.49< 4.1 × 10⁻⁶
OC-MSP-C0.550.87< 2.3 × 10⁻⁶
OC-MSP-D0.801.27< 1.5 × 10⁻⁶

Note: Projected radii computed at d = 5.43 kpc. Values are illustrative; use the tool for precise model-dependent limits.

Problems

1. Point-mass acceleration limit

Using the point-mass formula above and OC-MSP-A at R = 0.24 pc:

  1. What is the maximum central mass M consistent with amax(0.24 pc) ≤ 8.5 × 10−6 m/s²? Express in M☉. [6 marks]
  2. Repeat for OC-MSP-C at R = 0.87 pc and amax ≤ 2.3 × 10−6 m/s². Which pulsar places the stronger constraint and why? [5 marks]
  3. The Häberle (2024) detection mass is 8,200 M☉. Is this mass ruled out by either pulsar above? Show your calculation. [4 marks]
↗ Pulsar Acceleration Mapper — pre-filled M = 50,000 M☉, point model
Show solution
Part (a): M_max = a_max × R² / G R = 0.24 pc = 0.24 × 3.086×10¹⁶ = 7.41×10¹⁵ m M_max = 8.5×10⁻⁶ × (7.41×10¹⁵)² / 6.674×10⁻¹¹ = 8.5×10⁻⁶ × 5.49×10³¹ / 6.674×10⁻¹¹ = 7.00×10²¹ kg = 7.00×10²¹ / 1.989×10³⁰ M☉ ≈ 3,520 M☉
Part (b): R = 0.87 pc = 0.87 × 3.086×10¹⁶ = 2.685×10¹⁶ m M_max = 2.3×10⁻⁶ × (2.685×10¹⁶)² / 6.674×10⁻¹¹ = 2.3×10⁻⁶ × 7.21×10³² / 6.674×10⁻¹¹ = 2.48×10²² kg ≈ 12,500 M☉ OC-MSP-A (closer to center) gives a stronger constraint (3,520 M☉) because the acceleration falls as R⁻², so inner pulsars constrain much tighter even for the same measured limit.

Part (c): OC-MSP-A gives M_max ≈ 3,520 M☉ — below the Häberle value of 8,200 M☉. Under the point-mass model, OC-MSP-A does nominally rule out 8,200 M☉. However, this illustrative table uses round numbers; the actual TRAPUM limit at 90% CL is M < 10⁵ M☉, suggesting the real pulsars are at somewhat larger R and/or the constraint on a_los is less tight. The key point: inner pulsars can, in principle, reach the Häberle mass scale.
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 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): For a Plummer model, the enclosed mass at R is M(
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 not a point source but is distributed — e.g. 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 — 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☉) is two orders of magnitude above the Häberle detection of 8,200 M☉. Reaching the Häberle mass scale with pulsar timing requires either (a) discovering inner pulsars at R < 0.1 pc or (b) achieving a much tighter alos limit for known pulsars.

  1. At what projected radius R would a pulsar need to lie for a point-mass 8,200 M☉ IMBH to produce amax = 1 × 10−7 m/s² (a plausible SKA timing precision target)? Express in pc and arcsec at d = 5.43 kpc. [5 marks]
  2. OC's core radius is ~1.4 pc. Is the radius you computed in (a) inside the core? What does this imply for pulsar discovery rates with MeerKAT vs. SKA? [5 marks]
Show solution
Part (a): R = sqrt(G M / a_max) M = 8200 × 1.989×10³⁰ = 1.631×10³⁴ kg R = sqrt(6.674×10⁻¹¹ × 1.631×10³⁴ / 1×10⁻⁷) = sqrt(1.088×10³⁷) = 1.043×10¹⁸·⁵ ... let's compute: = sqrt(1.088e37) = 3.298×10¹⁸ m Convert: R = 3.298×10¹⁸ / 3.086×10¹⁶ = 106.9 pc Wait — this seems too large. Let me recheck with a_max = 1e-7 m/s²: R = sqrt(6.674e-11 × 1.631e34 / 1e-7) = sqrt(1.088e38) ... = 1.043e19 m Hmm, let me redo: G M = 6.674e-11 × 1.631e34 = 1.089e24 m³/s² R = sqrt(1.089e24 / 1e-7) = sqrt(1.089e31) = 1.044e15·⁵ Actually: sqrt(1.089e31) = sqrt(10.89e30) = 3.30e15·⁵ Let me be precise: 1.089×10³¹; sqrt = 3.300×10¹⁵·⁵ = 10^(31/2) × sqrt(1.089) = 10^15.5 × 1.044 = 3.162×10¹⁵ × 1.044 = 3.30×10¹⁵ m Convert to pc: 3.30×10¹⁵ / 3.086×10¹⁶ = 0.107 pc ≈ 0.11 pc Converting to arcsec: 0.107 pc / 5.43 kpc × (180/π × 3600) arcsec/rad = 0.107 / 5430 × 206,265 = 4.1 arcsec.

Part (b): The required R ≈ 0.11 pc is well inside the core radius of 1.4 pc — deep in the crowded central region. MeerKAT has found pulsars at R ~ 0.2–1 pc in OC; the innermost ≲0.1 pc is heavily confused by stellar crowding. SKA's higher sensitivity and resolution (~3× improvement in sensitivity per beam) could reveal pulsars at R ~ 0.1 pc that MeerKAT misses due to scintillation and crowding. However, the discovery rate scales roughly with survey depth; the probability of finding a pulsar at exactly the right radius depends on the cluster's pulsar density profile, which peaks in the core but is not well constrained inside 0.05 pc.

References