Waste-Heat Constraint Plane

Paper E's Figure 2, interactive. An engineered swarm that processes power P_comp radiates a waste luminosity L_waste = (1 − f_sink)·P_comp at a blackbody temperature set by its radius. Which telescope bounds it depends on where that emission peaks, so the mid-infrared limit is radius-dependent. Drag a source through the (r, L_waste) plane and the (P_comp, 1 − f_sink) plane to read where it sits against the JWST/WISE/Spitzer wedges, the transport floor, and the fuel ceiling.

🔬 Closed-form
The mid-infrared exclusion is radius-dependent, not a single number. A warm inner swarm (r ≲ a few AU, T_eff ≳ 150 K) peaks inside JWST/MIRI and inherits a sub-L☉ point-source limit. A cool outer swarm (r ~ 10²–10³ AU) peaks past 20 μm where the operative limits are WISE and Spitzer/MIPS, confusion-limited in the ω Cen core, weakening to ~10²–10⁴ L☉. This tool reproduces the piecewise wedge of fig2_constraint_plane.py exactly; the load-bearing assumptions behind it are listed under "Caveats" below.
Temperature-resolved plane · (r, L_waste)
Red: excluded by the radius-appropriate mid-infrared limit L_lim_mir(T_eff). Dotted: T_eff contours. Blue dashed: fuel ceiling (L_waste if the ambient Bondi supply were fully radiated). White marker: the source set by the sliders below. Mirrors panel (b) of fig2_constraint_plane.py.
Source in the (r, L_waste) plane
1.0 AU
1.0 L☉
Readout
Effective temperature T_eff
—
Operative instrument
—
Its point-source limit L_lim
—
computing…
Warm-swarm plane · (P_comp, 1 − f_sink)
Red: excluded by the JWST warm-swarm limit, L_waste = P_comp·(1 − f_sink) > 1 L☉ (r ≲ 3 AU). Grey hatch below the dashed line: the adopted 10⁻⁴ entropy-transport floor (an adopted parameter, not a data boundary). Blue dashed: the 8×10⁵ L☉ ambient Bondi fuel ceiling. Dash-dot: L_Edd. Mirrors panel (a) of fig2_constraint_plane.py.
Source in the (P_comp, 1 − f_sink) plane
10² L☉
10⁻³
Readout
Radiated L_waste = P_comp·(1 − f_sink)
—
Below the 10⁻⁴ transport floor?
—
P_comp vs fuel ceiling / L_Edd
—
computing…

The two planes

A swarm processing power P_comp and delivering a fraction f_sink of its generated entropy to the horizon radiates the remainder, L_waste = (1 − f_sink)·P_comp, thermally at T_eff = (L_waste / 4π r² σ_SB)^(1/4). That is roughly 300 K for 1 L☉ at r ~ 1 AU but only ~50 K at 10³ AU. The (r, L_waste) plane resolves this temperature axis: which telescope limits a source depends on where its blackbody peaks. The (P_comp, 1 − f_sink) plane is the warm-swarm slice, where the JWST sub-L☉ limit maps to a simple hyperbola P_comp·(1 − f_sink) > 1 L☉.

The instrument wedge (radius-dependent limit)

The piecewise mid-infrared point-source limit L_lim_mir(T_eff), degraded for crowding at the 5.43 kpc distance of the ω Cen core, is taken verbatim from common.py:

  • T_eff ≥ 150 K (peak < ~20 μm): JWST/MIRI, sub-arcsec PSF → ~1 L☉
  • 50–150 K (peak 20–60 μm): WISE W3/W4, Spitzer/MIPS 24 μm, 6–18 arcsec PSF, confusion-limited → ~10² L☉
  • T_eff < 50 K (peak > 60 μm): Spitzer/MIPS 70 μm only, heavily confused in the core → ~2×10⁴ L☉

The red wedge is the locus L_waste > L_lim_mir(T_eff(L_waste, r)), computed on the same L-grid the figure uses. A cool outer swarm evades the bound entirely: past 60 μm the only photometry is MIPS 70 μm, and it is insensitive at the L☉ level in this field.

Caveats (load-bearing, not decoration)

Three assumptions carry the constraint. They are stated in Paper E §2.7 and are the reason the wedge sits where it does:

  • Unit covering fraction, single-sided radiators. T_eff(r) assumes the swarm fully intercepts and re-radiates. A sparse swarm of covering fraction f_cov runs hotter by f_cov^(−1/4), moving mass into the JWST wedge. The paper reports this shifts the composite ln K to −0.34 at f_cov = 10⁻², still inside the quoted band.
  • Peak-band limits. Each wedge limit is the point-source sensitivity in the band where a blackbody of that T_eff peaks. A real spectral energy distribution spreads flux across bands; the single-band peak limit is the conservative adjudication the paper adopts.
  • MIPS-70 insensitivity. The cold wedge (T < 50 K) rests on there being no L☉-level far-infrared photometry of the core past 60 μm. Spitzer/MIPS 70 μm is the only instrument and is heavily confused, so the ~2×10⁴ L☉ floor is what a cool outer swarm has to clear. Far-infrared sensitivity is where this bound would tighten.

Transport floor and fuel ceiling

The dashed line at 1 − f_sink = 10⁻⁴ is the Appendix A.3 entropy-transport bound, drawn (as in the figure) as an adopted parameter over hatching rather than a data-driven boundary: below it is disfavoured by the A.3 accounting, not excluded by data. A lower true floor weakens the mid-infrared charge and moves ln K toward zero; a higher floor tightens the P_comp ceiling. The blue dashed 8×10⁵ L☉ fuel ceiling is the ambient Bondi accretion supply (ṁ_Bondi·c², MAD efficiency ~1); imported mass moves it right at the cost of visibility. L_Edd for the 2×10⁴ M☉ hole is shown for reference.

A note on σ

The fuel ceiling is computed with the paper's central velocity dispersion σ = 21 km/s (from common.py), the value Paper E adopts. The site's other calculators use a fiducial σ = 18.2 km/s; the offset is documented and deliberate (A5 decision) and is not silently harmonized. The ceiling scales weakly with σ through the Bondi rate, so the ~8×10⁵ L☉ figure is robust to the difference.

OCS connection

This is the waste-heat leg of the engineered-IMBH feasibility argument (Paper E). The (P_comp, 1 − f_sink) and (r, L_waste) planes together define the surviving parameter volume the paper's Bayesian adjudication marginalizes over. See also the Paper E feasibility scenario and Compute-in-Space.

v1.0 — 2026-07-23 · Code MIT · Prose CC BY 4.0 · Swanson 2026, "Engineered IMBH Systems" (Paper E) §2.7/3.1; figs/fig2_constraint_plane.py + common.py. Instrument limits: Chen et al. 2025 (JWST); WISE; Spitzer/MIPS.