# How much sky does one IceCube track alert occupy?

For these 20 supplied 2026 maps, the recalculated 50% regions span approximately **0.27–37.5 full-Moon areas**, with a median of **1.24**. The 90% regions span **1.09–121.3 full-Moon areas**, with a median of **4.20**. These are separate probability levels. They describe uncertainty in the incoming direction, not the probability that any particular astronomical object produced the event.

The sample comprises the supplied maps dated 2026-01-15 through 2026-09-30. It is not a verified complete or latest catalog as of 2026-10-09.

![Areas of the 20 supplied alerts](icecube-2026-areas.png)

## Conversion and definitions

A Moon with angular diameter 0.5 degrees has apparent area π(0.25°)² = **0.19634954 square degrees**. Thus N_Moon = 5.09295818 × A_deg². The exact spherical-cap Moon area differs by only approximately 0.00016%, far below the map discretization effects. A hypothetical 1-degree-radius circle occupies about 16 Moons; a 1-degree-diameter circle occupies about 4.

The supplied copy of [GCN's IceCube documentation](https://gcn.nasa.gov/missions/icecube) describes several distinct quantities:

| Quantity | Meaning and treatment |
|---|---|
| Map PROBDENSITY | Directional probability density per steradian. Multiply by each pixel's solid angle to obtain probability mass. |
| 50% and 90% map regions | Every supplied FITS header explicitly calls these highest-posterior-density credible regions. The documentation also describes confidence contours and simulation-calibrated coverage. This report uses the files' HPD probability definition; it does not independently recalibrate frequentist coverage. |
| RA/Dec asymmetric errors | Error-rectangle boundaries at the stated level; not independent Gaussian standard deviations. |
| Revised notice ra_dec_error | Approximate circularized 90% radius from mean RA/Dec errors, corrected for cos(declination). It need not enclose 90% of the map probability. |
| Preliminary vs revised uncertainty | The documentation says revision 0 excludes systematics and normally has no map URL; revision 1 includes systematics and links a map. The archive lacks event notices, so exact record numbers cannot be verified. |
| p_astro / signalness | An astrophysical-origin metric defined using simulated signal/background populations and an assumed spectrum. It is separate from directional containment and is not a probability assigned to a specific counterpart. |

At a given probability level, the highest-density region is the smallest-area region under the piecewise-constant map model. A 90% region includes more of the directional probability than a 50% region. There is no universal area ratio: the recalculated A90/A50 ranges from **3.23 to 4.11** in this sample. The circular Gaussian value, ln(10)/ln(2) ≈ 3.32, is a model assumption, not a rule for real maps.

## Calculation and integrity audit

The ZIP SHA-256 is `6bff0e4fecdbed2e146249abc2cf9ecfae0aa3c123a9c9b0d8089acb3e40a7d0`. All 20 compressed-map byte counts and SHA-256 checksums match the supplied manifest; all 20 decompressed checksums also match. This verifies consistency with the supplied manifest, not independent authentication against the live server. The manifest supplies direct URLs under [the official alert directory](https://roc-2.icecube.wisc.edu/public/alerts/); those URLs and checksums are retained in the CSV.

For each NUNIQ pixel, decode its HEALPix order k and use Ω = 4π/(12 × 4^k) steradians. This uses each pixel's actual resolution rather than treating every row as having the header NSIDE. Check that densities are nonnegative where finite and that pixels have no duplicate or overlapping sky coverage. All maps pass those checks.

As directed by the headers, ignore non-finite density pixels. Let S = ΣρᵢΩᵢ over the remaining pixels, and normalize the retained density by S. Sort by descending **density**, not descending pixel probability mass. Sum ρᵢΩᵢ/S to the desired probability and sum the selected solid angles. Interpolate linearly within the crossing pixel to quote an exact target-probability area. This fractional-pixel convention yields a scalar area but does not determine where inside that pixel the partial region lies. The CSV also records the areas excluding and including the crossing pixel. The shape figure displays whole crossing pixels.

The original finite-pixel probability sums range from **0.978338 to 1.001875**, rather than all being exactly unity. Using unnormalized cumulative mass thresholds changes some 90% areas by as much as **9.95%** relative to normalized integration. Both versions are retained in the CSV. Normalizing defines a probability distribution on the supplied valid support; it cannot recover missing probability or repair an underlying failed reconstruction.

Four files contain non-finite pixels: **260610A (2), 260807A (20), 260827A (21), and 260919A (29)**. The files instruct readers to ignore such values, attributing them to minimizer convergence failures. Six maps' native pixel sets cover slightly less than the full sphere: 260610A, 260722A, 260805A, 260807A, 260821A, and 260919A. The minimum represented fraction is 0.999893. Unrepresented or non-finite locations are not scientifically established zero-probability directions; the normalization and reported regions are conditional on the supplied valid values.

Relative to CONTOUR_AREA_50 and CONTOUR_AREA_90, normalized interpolated areas differ by:

| Level | Minimum signed difference | Median signed difference | Maximum signed difference |
|---|---:|---:|---:|
| 50% | −4.08% | +2.22% | +5.89% |
| 90% | −2.63% | +1.67% | +4.03% |

The values are close, but not identical. Only 10 of 20 header 50% areas and 7 of 20 header 90% areas fall within the corresponding crossing-pixel bracket expanded by 0.005 square degrees for two-decimal rounding. Therefore, simple rounding plus this particular boundary-pixel convention does not explain every discrepancy. Differences in normalization, contour construction, and map sampling are possible contributors; the exact upstream convention cannot be established from these inputs. Neither header areas nor recalculated areas have been silently substituted for the other.

Pixel size limits precision. For 260807A, the crossing pixel covers 0.8393 square degrees (4.27 Moons). Its 50% area bracket is 6.9537–7.7930 square degrees, and its 90% bracket is 23.7658–24.6051. For 260919A, the corresponding brackets are 0.04918–0.05246 and 0.20982–0.22294 square degrees. These are discretization brackets, not additional statistical confidence intervals.

## Event results

“Map” values below use normalized probability and fractional crossing pixels. Header values are copied without modification. “Circle mass” is the actual normalized map probability inside the circularized nominal 90% radius, integrated with subpixel refinement; it is not the target 90% assigned by construction.

| IceCube event | Map 50%, deg² | Map 50%, Moons | Header 50%, deg² | Map 90%, deg² | Map 90%, Moons | Header 90%, deg² | Circle mass |
|---|---:|---:|---:|---:|---:|---:|---:|
| 260115A | 1.9547 | 9.96 | 1.92 | 6.5120 | 33.17 | 6.48 | 81.9% |
| 260125A | 0.3116 | 1.59 | 0.31 | 1.0466 | 5.33 | 1.03 | 89.1% |
| 260217A | 0.2180 | 1.11 | 0.21 | 0.7382 | 3.76 | 0.72 | 88.6% |
| 260315A | 0.2360 | 1.20 | 0.23 | 0.7961 | 4.05 | 0.77 | 88.7% |
| 260425A | 0.3150 | 1.60 | 0.31 | 1.0580 | 5.39 | 1.05 | 89.0% |
| 260504A | 0.2115 | 1.08 | 0.21 | 0.7120 | 3.63 | 0.70 | 88.6% |
| 260610A | 1.3254 | 6.75 | 1.30 | 4.5901 | 23.38 | 4.61 | 87.4% |
| 260622A | 0.2791 | 1.42 | 0.27 | 0.9439 | 4.81 | 0.92 | 88.9% |
| 260704A | 0.2010 | 1.02 | 0.19 | 0.6763 | 3.44 | 0.67 | 89.1% |
| 260708A | 0.2015 | 1.03 | 0.20 | 0.6970 | 3.55 | 0.67 | 88.1% |
| 260712A | 0.1947 | 0.99 | 0.19 | 0.6631 | 3.38 | 0.64 | 88.8% |
| 260722A | 0.5836 | 2.97 | 0.58 | 2.0293 | 10.34 | 2.05 | 87.1% |
| 260805A | 0.3933 | 2.00 | 0.41 | 1.3499 | 6.87 | 1.36 | 87.3% |
| 260807A | 7.3666 | 37.52 | 7.24 | 23.8162 | 121.30 | 24.46 | 64.1% |
| 260821A | 0.2012 | 1.02 | 0.19 | 0.6824 | 3.48 | 0.67 | 88.9% |
| 260825A | 0.2524 | 1.29 | 0.25 | 0.8520 | 4.34 | 0.84 | 89.1% |
| 260827A | 0.1896 | 0.97 | 0.18 | 0.6829 | 3.48 | 0.68 | 88.2% |
| 260916A | 0.2189 | 1.11 | 0.21 | 0.7426 | 3.78 | 0.73 | 89.1% |
| 260919A | 0.0521 | 0.27 | 0.05 | 0.2140 | 1.09 | 0.21 | 89.6% |
| 260930A | 0.2573 | 1.31 | 0.25 | 0.8689 | 4.43 | 0.85 | 88.9% |

All figures in this table are numerical summaries of the supplied maps; the displayed digits do not imply finer physical localization than their pixels or calibrated reconstruction allow. Sample medians include all 20 supplied maps under the documented non-finite-pixel handling, because none is verified as retracted in the supplied evidence. That is not an assertion that none has been retracted.

## What circularization misses

The GCN documentation gives the revised 90% radius as

r90 = sqrt(mean(|ΔRA90|) × cos(dec) × mean(|ΔDec90|)),

where each mean averages the plus and minus errors. Calculate the circle's spherical area as 2π[1−cos(r90)] steradians. Its small-angle form is πr90² square degrees, or about 16r90² Moons for radius in degrees. The 50% circles in the area plot use the same formula applied to the 50% bounds as an explicitly derived comparison, not an independently documented 50% notice field.

![Map shapes and circular approximations](icecube-2026-shapes.png)

For **260115A**, the circle has radius **1.4394°** and area **33.15 Moons**, almost identical to the 90% map area's **33.17 Moons**. Yet it encloses only **81.9%** of the normalized map probability. The circle spends area above and below the elongated region and misses probability at its sides.

For **260807A**, the circle has radius **2.8818°** and area **132.85 Moons**, **9.53% larger** than the actual 90% map area of **121.30 Moons**. It nevertheless contains only **64.1%** of the normalized map probability. The long, asymmetric extension extends far outside the circle. Coarse pixels and invalid or absent map samples also produce visible irregularities; those should not automatically be interpreted as separate astrophysical sources.

The two headline circle probabilities were checked using recursive HEALPix subdivision to orders 12 and 15. Both changed by less than 0.00006 in probability (0.006 percentage point). This validates numerical integration of the supplied piecewise-constant maps, not the underlying reconstruction. The CSV retains a simpler native-pixel-center estimate as a diagnostic, but the quoted circle probabilities use the refined integral. In particular, the native-center estimate of 62.4% for 260807A is too crude; the refined result is 64.1%.

The maps are plotted in local sky-offset coordinates, with equal horizontal and vertical angular scales within each panel. Panel scales differ; each includes a 0.5-degree Moon disk. The blue region is nested within the orange 90% region. Whole threshold pixels are shown, whereas numerical areas interpolate within the last pixel.

## Revisions, retractions, and missing maps

The archive contains exactly 20 distinct events and one map per event, plus one GCN HTML document and the manifest. No map listed in the manifest is missing or unreadable. There are no event-specific GCN notices, circulars, revision histories, retraction records, or a dated complete directory listing.

The FITS headers do not provide revision or retraction status. Map availability is consistent with the documented revised-map workflow, but does not prove which notice revision is represented or that a map remains current. No alert has been labeled “not retracted.” Later updates, retractions, or additional 2026 maps cannot be verified from this archive. Live access was denied earlier and was not retried.

A missing map for a preliminary notice can be expected: GCN documents healpix_url = null for revision 0. A map missing from this ZIP, however, cannot be distinguished from an uncollected event without a complete external inventory.

There are two additional metadata observations: 260115A lacks ALERT-STREAM, and 260217A repeats ALERT-STREAM 77 times with Bronze values. Neither changes the density calculation; neither is evidence of a revision or retraction.

The HTML document's SHA-256 is `db82beea310f608d4e55eaf018c37c452c1b4e104fc75358b567ff5186c8fbdb`. Its contents were read locally as the supplied copy of the linked GCN documentation, not independently refreshed online.

## Detecting a particle versus identifying a source

IceCube observes Cherenkov light from charged particles produced after a neutrino interacts in or near the detector. A track event allows the incoming direction to be reconstructed, but finite light measurements, propagation through the ice, detector and reconstruction uncertainties, and interaction kinematics limit its precision. The maps quantify that directional uncertainty under the reconstruction model.

An astronomical source is an additional inference. Several objects may lie in a localization region; the true source may be uncataloged or electromagnetically faint; and the event may have an atmospheric origin. A credible spatial and temporal counterpart, repeated neutrinos, and an assessment of chance coincidence can strengthen an association. A 90% directional region is not a 90% probability that a named galaxy or blazar is the source.

## Reproduction

The accompanying CSV retains normalized areas, unnormalized-threshold sensitivity, crossing-pixel brackets, untouched header areas, circular radii and areas, circle probability integrals, density normalizations, invalid-pixel counts, source URLs, and map checksums.

The accompanying Python script runs entirely offline with NumPy, pandas, Astropy, and astropy-healpix. Astropy 8.0.1 and astropy-healpix 2.0.1 were used here. Its calculations were executed against the supplied archive and reproduced all 20 normalized 50% and 90% areas and both circle integrals to numerical tolerance. It reproduces numerical results rather than the figure layout.

```sh
python icecube-localization-analysis.py icecube-2026-localization-maps.zip --output reproduced-results.csv
```
