I399 | Is the hot phase's escape set by the potential well, or by the wind?

01 · Question

In one sentence: is the hot-phase gas's "stay or go" written in the potential well, or in the wind?

Star formation heats gas to millions of degrees and drives it out of the disk. This hot-phase wind has two possible fates: breaking free of gravity into the intergalactic medium (escape), or falling back onto the disk to be recycled. The entire mass-loading literature writes "can it escape" as a property of the wind—launch speed, mass loading, driving mechanism—with the potential well appearing only as a threshold to be compared against.

This idea puts that assumption itself on stage. Define the dimensionless ratio B = (5 kT / μ m_p) / v_esc²: the specific enthalpy of the hot phase over the binding energy at its location. Measure one B for each of four galaxies—same instrument, same pipeline, same matching radius r_ref (in units of r/r_25, frozen before any unfreezing)—then fit a slope to log B = α · log(v_esc/200) + c. α is the estimator.

02 · Why it matters

Two named physical endpoints, and every point along the line between them is a result

This is not "fit a power law". Each endpoint of α is a named physical hypothesis, and whose published conclusions would move can be written down:

α = −2 (potential well decides): kT_mw is the same in every wind—a universal launch/thermalization temperature. B then scales as 1/v_esc² only. This means given a rotation curve, any galaxy's hot-phase fate can be predicted without taking an X-ray image. What needs revision is every mass-loading prescription that defends a steepening low-mass end with "shallower well, easier escape"—that defense requires α<0, and this measures it directly.
α = 0 (universal bound state): kT_mw ∝ μ m_p v_esc²/5, the hot phase is virialized with its host, and B is universal. This means Boettcher & Hodges-Kluck 2024 and XRISM 2026 (escape side) and Oppenheimer+2010 and Mitchell+2020 (recycling side) do not actually contradict each other on the hot phase—they report the same universal boundary state in galaxies of different depths. This is the strongest result obtainable here, because it says the debate has been measuring an unchanging quantity all along.
In between: Partial coupling. The interval itself is the number—how closely thermalization tracks the potential well—which is exactly the quantity a feedback prescription needs.

No overclaiming: this is four galaxies, one CCD, one thermal (not kinematic) estimator. It does not "validate" any cosmological prescription; it constrains α on these four galaxies, at this projected scale, at this precision. Its value: this is the first time this ratio is measured with a single method, in more than one galaxy, with independently constrained potential wells.

03 · The number to measure, and the lever arm

The two hypotheses differ by a factor of six in the observable; NGC 55 carries the lever arm

The sample spans v_esc = 222–541 km/s, a factor of 2.43. Under α=−2 this produces a factor-5.9 spread in B; under α=0, zero. Two hypotheses a factor of six apart can be separated with roughly 20% B precision per target. In the figure below, the two curves are the two endpoints and the four points are each galaxy's predicted position at its own v_esc.

Decision plot of B against v_esc. The solid curve is α=−2 (B falls as the inverse square of v_esc); the dashed line is α=0 (universal B). Each of the four galaxies marks both predictions at its own v_esc; the double-headed arrow at NGC 55 marks the factor-5.9 lever arm. — **Decision plot.** Vertical axis B (logarithmic), horizontal axis v_esc. Filled circles are the α=−2 prediction, open squares the α=0 prediction, anchored to NGC 253 with kT_mw=0.30 keV (B=0.842, v_esc=533 km/s).Key reading: NGC 891 and NGC 253 differ in v_esc by only 1.5%—they alone cannot separate the two endpoints; NGC 4945 adds only a factor of 1.2. **The lever arm is almost entirely in NGC 55**—which is why I398 is staged, and why without NGC 55's outer ring the whole synthesis degenerates to three points spanning a factor of 1.20 in v_esc, unfalsifiable. All numbers come from the actually-run `r2_sample_arithmetic.py` (see sources at the end).

B is a dimensionless energy ratio measured from emission under model conditions, not a direct readout; freezing r_ref (I399-G2) before any component unfreezes is precisely to prevent "choosing the radius after seeing the result". The curves in the figure are endpoint illustrations; the four points' B values are what will be measured.

04 · The sample is real, and archive-covered

Four galaxies, one instrument, one response family—keeping the cross-calibration term out of the slope

This sample exists because EPIC's 30′ field of view and 0.5–1 keV effective area let the same estimator be measured with one instrument on four galaxies spanning 1.99–9.50 Mpc. Mixing instruments would put a cross-calibration term into the slope—this is forbidden by design, so only XMM is used here; Chandra only contributes point-source masking within components, and Suzaku/Chandra never enter σ_tot.

Sample plot: the four galaxies in the v_esc–distance plane, with point area proportional to archival PN exposure. NGC 55 is nearest with the lowest v_esc; NGC 891 is farthest with the highest v_esc. — **Sample and archival coverage.** Horizontal axis v_esc, vertical axis distance, point area proportional to existing PN exposure (NGC 253: 110.5+97.6 ks; NGC 4945: 238.3 ks; NGC 891: 496.6 ks; NGC 55: 578.6 ks). Three components (I394/I396/I397) are pure archive; I398's inner ring comes from the existing 578.6 ks, and only its outer ring needs one offset pointing—this is the sample's only dependence on new time, explicitly marked. The sample's ceiling is also written here: a target needs an H I rotation curve undisturbed by mergers, and the 0.8 kpc annulus width must exceed the EPIC 16.6″ HEW, which limits the reachable sample to D≲11 Mpc—no amount of extra exposure adds a fifth galaxy.

05 · If the answer is the opposite, or unmeasurable

α≈0 is not "nothing"—it is the one result that dissolves the debate

If B is flat across the sample (α≈0), that is not a boring null. It says: every disk galaxy's hot phase sits at the same point relative to its own binding energy, no galaxy is special, and the escape-side and recycling-side camps have been measuring the same universal boundary state in galaxies of different depths and reporting it as a disagreement. This is a more disruptive answer than α=−2.

If the four galaxies do not lie on a single relation at all—different driving mechanisms (NGC 4945 hosts an AGN, NGC 891 has no starburst) put them on different tracks—then α is not a meaningful parameter, and the honest product is four points with faithfully reported scatter. That is a positive result about heterogeneity, to be reported as such, not folded into INCONCLUSIVE.

Unmeasurable does not mean absent. If one component's B interval is wide enough to cover both endpoints simultaneously (the I399-G1 failure mode), then that component yields a "below S"-style constraint at its aperture, band, and achieved sensitivity—not "this galaxy has no hot halo". If sensitivity was never qualified, the honest write-up is INCONCLUSIVE—an honest INCONCLUSIVE beats a dressed-up null. This statistical-power question (I399-G1) must be quantified before component execution, not after—this is exactly what this page asks the PI to adjudicate.

06 · Cost, and what is still unknown

This page's single request: decide whether this cross-target synthesis is worth entering component qualification

Known cost: Three components are pure archive (zero new observations); I399 itself performs no new data reduction—its product is a joint fit to the four B(r) profiles. The only dependence on new time is I398's outer ring (one offset pointing), explicitly marked.

Still unknown, and must be quantified before component execution:

I399-G1 (blocking): The statistical power of the slope. Four points, and the endpoints separate by a factor of 5.9 only if every B is good to about 20%; if components only deliver 50% errors, α's interval contains both endpoints and the synthesis says nothing.
I399-G2: Freezing r_ref (in units of r/r_25, fixed before any unfreezing).
I399-G3: The four H I rotation curves come from four different studies of varying quality; systematics in the v_esc derivation do not cancel and enter the slope directly.
I399-G4: Systematics shared across components (σ_CX handling, the same APEC/atomic data) make the slope error bar no longer the naive value.

Single point of failure: The lever arm is concentrated in NGC 55. Losing NGC 55 is a single point of failure; losing NGC 891 or NGC 4945 costs precision, not the test.

This is a "one page for one hour" decision: this page is the one product the pipeline currently asks a human to read. If α's interval is destined to contain both endpoints, the honest answer is to say so now and save the component computations.

Sources and reproducibility

The evidence boundary of this page

All B, v_esc, and lever-arm numbers: python3 cycles/cycle-obs3/materials/R2_P005_sample_geometry_20260906/r2_sample_arithmetic.py (actually re-run this shift, reproducing the card §C table byte for byte).
PN exposures: I394 (NGC 253, 110.5+97.6 ks), I396 (NGC 4945, 238.300 ks), I397 (NGC 891, 496.586 ks), I398 (NGC 55, 578.628 ks), from the respective component cards and this repository's shared/survey/per-target/.
Literature (original language preserved): Martin 2005 (2005ApJ...621..227M)、Heckman+2015 (2015ApJ...809..147H)、Strickland & Heckman 2009 (2009ApJ...697.2030S)、Boettcher & Hodges-Kluck 2024 (2024ApJ...975..128B)、XRISM 2026 (2026arXiv260324674X)、Oppenheimer+2010 (2010MNRAS.406.2325O)、Mitchell+2020 (2020MNRAS.497.4495M)、Muratov+2015 (2015MNRAS.454.2691M)。
This page is a cross-target synthesis with no single-sky WCS figure; both figures are scalar plots generated by the deterministic script make_figures.py, with geometry/numbers recorded in fig/figure_geometry.json.