# Is 0–2 proto-O-stars within 2 kpc expected? (2026-07-02)

Observed: N_proto(L>1e5, embedded, <2 kpc) = 0–2 (W3 IRS5; RCW38 IRS2 if counted despite
being already ionizing; NGC6334 only at region level).

## Expected count: N = SFR(<2 kpc) × n(>m)_IMF × τ(L>1e5 phase)

**SFR within π(2 kpc)²:**
| method | SFR (M☉/yr) |
|---|---|
| (a) uniform share: 2 M☉/yr × π2²/π(13²−3²) | 0.050 |
| (b) local Σ_SFR = (1–5)e−9 M☉/yr/pc² (KE12/Lada+10/Mor+19) | 0.013–0.063 |
| (c) Σ_gas=10–14 M☉/pc² → K98 KS | 0.08–0.13 (KS overpredicts locally) |
| (d) inverted from ALS alive O2–O7 count (125 <2kpc, t_MS 4–5 Myr) | ~0.014–0.02 |

Adopt 0.015–0.05.

**IMF (Kroupa):** L_ZAMS>1e5 ⇔ m≳30 M☉ → n = 1 per ~600 M☉ formed.
(Choosing m>20, "will become O6+": 1 per ~330.)

**Result:** N_exp = (0.015–0.05)/600 × τ.
- τ = 3e4 yr → N_exp = 0.8–2.5 → **0–2 observed is exactly expected for fast formation.**
- τ = 1e5 yr → N_exp = 2.5–8.4 → marginal (P(≤2)≈0.02–0.5).
- τ = 1 Myr (P&G-like: time spent at m>30 while growing to 60 on t∝√m, τ0=4 Myr) →
  N_exp = 25–84 → **excluded at P(≤2) < 1e−7 even at the lowest SFR.**

## Limits set (95% Poisson, SFR=0.04)
- N_obs=0: τ(L>1e5 embedded phase) < 4.5e4 yr
- N_obs=1: < 7.1e4 yr (best est 1.5e4)
- N_obs=2: < 9.4e4 yr (best est 3e4)
At the most conservative SFR=0.015: limits ×2.7 → τ < 1.2–2.5e5 yr.

## Cross-check (independent clock, no SFR needed)
τ = (N_proto/N_alive) × t_MS = (1–2)/125 × 5 Myr = 4–8e4 yr. Agrees.

## Conclusions
1. The absence/scarcity of proto-O stars within 2 kpc is **expected** iff the luminous
   (L>1e5) embedded phase lasts ≲1e5 yr — consistent with our LF-ratio result
   (0.015–0.02 × t_MS) and the ALS-clock estimate.
2. It **independently excludes** multi-Myr growth through the luminous embedded phase
   (P&G-style m>30 growth time ~1 Myr predicts 25–84 such objects in the 2 kpc circle;
   ≤2 seen). Third convergent argument (reservoirs, LF fill-in, source counts).
3. Sensitivity: dominated by SFR(<2 kpc) (×3 spread) and the L↔m mapping (accretion
   luminosity adds to L, shortening the implied τ further — conservative direction).

## 2026-07-02 extension: the 2<r<3 kpc annulus as an independent sample
Area π(3²−2²)=5π kpc² (1.25× the inner disk); uniform-share SFR = 0.0625 M☉/yr.
(Secondary SFR estimates — star counts, gas+KS — are more uncertain here: the annulus
samples R_gal ≈ 5.5–8.5 kpc including the Sagittarius arm, so the solar-neighborhood
Σ_SFR and Σ_gas no longer strictly apply; uniform-share is the cleanest.)

Counts (embedded L≥1e5, distance-revised):
- **r<2 kpc: N=4** — M17/G015.03 (1.91±0.13), W3 IRS5 (1.95±0.14), W3(OH)/G133.7184
  (1.95±0.14), RCW38 (1.70). Strict pre-ionizing proto: 1–2 (W3 IRS5; RCW38 hybrid).
- **2<r<3 kpc: N=3** — W33/G012.81 (2.40±0.17), NGC7538/G111.54 (2.65±0.19),
  NGC3576 (2.40, L=1.0e5). Strict proto: 0–1.
- Boundary caution: three of the r<2 members sit at 1.91–1.95±0.14 — the 2 kpc line is
  soft; the combined count is robust.

| zone | SFR (M☉/yr) | N_emb | τ_embedded (best, 95% CI) | τ strict-proto |
|---|---|---|---|---|
| r<2 | 0.050 | 4 | 4.8e4 [1.3e4, 1.2e5] yr | ~1.8e4 (<8.6e4) |
| 2–3 annulus | 0.0625 | 3 | 2.9e4 [5.9e3, 8.4e4] yr | ~4.8e3 (<5.3e4) |
| combined <3 | 0.1125 | 7 | **3.7e4 [1.5e4, 7.7e4] yr** | ~1.1e4 (<4.7e4) |

The two zones are statistically consistent (ratio 1.7, well within Poisson) — the annulus
independently reproduces the clock. P&G-like growth (~1 Myr spent at m>30) predicts
**105 embedded L>1e5 objects in the annulus alone; 3 observed: P = 7e−41.** Even
τ = 1e5 yr is mildly disfavored by the annulus (P(N≤3)=0.007); τ = 3e4 yr fits (P=0.6).

Machinery in this file's companion commands; numbers reproducible via the snippet in
session log (Kroupa integration + Poisson).