1. The Arithmetic

This section contains one formula, five inputs, and one output. You can verify the output in five minutes. Everything that follows — whether the physics is right, whether the result means anything — is built on this arithmetic.

1.1 The formula

Bondi-Hoyle accretion rate for a mass M moving through a medium of density ρ at velocity v:

dM/dt \= 4πG²M²ρ / (v² + cₛ²)³˲

1.2 The five inputs

Note: v ≫ cₛ, so the sound speed is negligible. The result is dominated by orbital velocity.

1.3 The output

dM/dt \= 4.93 × 10¹⁸ kg/s \= 7.82 × 10⁻⁵ M☉/yr

t_accrete \= M / (dM/dt) \= 1,278 Gyr \= 93 × t_Hubble

This number was independently verified by a stress test script (comp_023_stress_test_3.py). The arithmetic is correct.

1.4 Sensitivity

That is the end of the arithmetic. Everything below is physics.

2. The Bug We Found

The original simulation made two claims: (1) TDGs can’t accrete DM fast enough (1,280 Gyr), and (2) even if they did, TDGs can’t thermalize it into a condensate (10⁹ Gyr). Both were supposed to support the same conclusion.

Claim 2 is wrong. The thermalization calculation had a unit conversion bug. The original script used sigma_over_m \= 0.01e-4 \= 10⁻⁶ m²/kg. The correct conversion of 0.01 cm²/g to SI is 10⁻³ m²/kg. The cross-section was 1,000× too small.

The stress test recalculated with the actual Berezhiani-Khoury parameters:

With the correct BK cross-section (σ \~ 2.8 × 10⁻⁸ m², which is 10¹² times larger than atomic cross-sections), thermalization is instantaneous. Any DM a TDG manages to capture thermalizes into a condensate in minutes, not gigayears.

This leaves claim 1 standing but reframes the problem. The bottleneck is purely accretion rate. The question becomes: can a TDG capture enough dark matter?

3. Why the Physics May Be Wrong

The 1,278 Gyr number is arithmetically correct for Bondi-Hoyle with those inputs. But there are two reasons the physical setup itself may not apply.

3.1 Bondi-Hoyle assumes a classical gas

Bondi-Hoyle describes a gravitating point mass moving through a weakly-interacting medium that streams past it. The formula was derived for classical gas accretion (Bondi 1952).

Superfluid dark matter is not a classical gas. It has:

• A finite sound speed and quantum pressure

• Long-range phonon-mediated forces

• Quantum coherence over macroscopic scales (healing length)

• A self-interaction cross-section (σ \~ 2.8 × 10⁻⁸ m²) that is enormous — 10¹²× atomic

Whether the standard Bondi-Hoyle formula captures the relevant accretion physics for a quantum superfluid is an open theoretical question. A stress test on a subsequent run (run 023) explicitly flagged this: “the Bondi-Hoyle analogy to quantum condensate accretion is explicitly unvalidated.”

The accretion rate could be higher (if coherent infall or phonon-mediated attraction enhances capture) or lower (if quantum pressure creates an effective barrier). We do not know which.

3.2 TDGs may already be inside the condensate

This may be the more important objection. In Berezhiani & Khoury’s model, the superfluid condensate of a Milky Way-mass halo extends to a condensate radius Rₜ ≈ 157 kpc. Observed TDGs orbit at 12–85 kpc from their parent galaxy.

Most TDGs sit inside the host’s condensate.

If a TDG is already immersed in superfluid dark matter, it does not need to accrete anything. The phonon-mediated MOND-like force arises from the superfluid medium itself. The TDG experiences modified gravity because it is embedded in a condensate, not because it has built its own. The entire accretion calculation becomes moot.

Under this reading, SFDM actually predicts that TDGs should follow the RAR (at least while inside the host condensate), which is what observations show. The tension with SFDM dissolves.

4. What Remains

Here is an honest inventory of what we have and what we don’t.

4.1 What is solid

• The Bondi-Hoyle arithmetic: 1,278 Gyr for fiducial parameters. Verified independently. You can check it in five minutes.

• The thermalization bug: the original 10⁹ Gyr claim was wrong by 23 orders of magnitude. With correct BK parameters, thermalization is \~260 seconds.

• The sensitivity analysis: no combination of fiducial-range parameter variations brings the Bondi-Hoyle timescale comfortably below Hubble time.

4.2 What is open

• Whether Bondi-Hoyle applies to superfluid accretion at all. This is a physics question, not a numbers question. It requires theoretical work on superfluid accretion dynamics.

• Whether TDGs need to accrete in the first place, or inherit modified gravity from the host’s condensate. This depends on the condensate radius relative to TDG orbital radii — a calculable quantity that should be checked against the specific SFDM parameters assumed.

• The observational sample: reliable TDG rotation curves exist for a handful of objects (NGC 5291 system, VCC 2062). More data are needed.

• Whether model extensions (different boson mass, cooling during formation, seeded condensation from parent halo) change the picture.

4.3 The three-way discriminant

Despite the caveats, the TDG test remains useful because three frameworks make distinct predictions:

The critical observation would be a TDG far enough from its parent galaxy to be outside the condensate radius. If such a TDG still follows the RAR, that would constrain SFDM more tightly. If it doesn’t, that would distinguish SFDM from MOND.

5. For Your Own Calculation

Everything you need to reproduce or challenge this result:

Bondi-Hoyle accretion (verified): Plug G, M \= 10⁸ M☉, ρ \= 0.01 GeV/cm³, v \= 200 km/s, cₛ \= 5 km/s into dM/dt \= 4πG²M²ρ/(v²+cₛ²)³˲. You should get 7.82 × 10⁻⁵ M☉/yr, giving t \= 1,278 Gyr.

Thermalization (corrected): τ \= 1/(nσv) with n \= 3.8 × 10¹⁴ m⁻³, σ \= 2.8 × 10⁻⁸ m², v \= 10⁵ m/s. You should get \~260 seconds. The original script’s 10⁹ Gyr used σ that was 1,000× too small (unit bug: 0.01 cm²/g was converted as 10⁻⁶ instead of 10⁻³ m²/kg).

Condensate radius: For a MW-mass halo with BK fiducial parameters, Rₜ ≈ 157 kpc. Compare this to TDG orbital radii (typically 12–85 kpc) to determine whether the accretion question is even relevant.

The physics question: Does Bondi-Hoyle apply to a quantum superfluid with σ \~ 10⁻⁸ m²? This is the interesting open problem.

Source scripts: runs/run_022b/simulations/sim_tdg_accretion.py (original), comp_023_stress_test_3.py (correction).

References

Berezhiani, L., & Khoury, J. (2015). Theory of dark matter superfluidity. Phys. Rev. D, 92, 103510.

Berezhiani, L., & Khoury, J. (2016). Dark matter superfluidity and galactic dynamics. Phys. Lett. B, 753, 639.

Bondi, H. (1952). On spherically symmetrical accretion. MNRAS, 112, 195.

Lelli, F., et al. (2015). Gas dynamics in tidal dwarf galaxies. A\&A, 584, A113.

McGaugh, S.S., et al. (2016). Radial Acceleration Relation. Phys. Rev. Lett., 117, 201101.