Recovering the Kill Sequence of the Doomsday Battle from Oort Debris Alone: A Single-Probe Trajectory Fit to Fragment Velocities and Hull-Breach Geometry

1. Introduction

The 38 cabin chronometers recovered from the Oort debris field stopped at readings that differ from the reconstructed rupture times by 0.95 s RMS, and the same reconstruction places the whole battle at 211 s. Of the 1,712 parent hulls identified in the survey catalogue, 1,336 have a kill rank that is placed to within five places (1σ) by fragment positions and velocities alone, and that share is 78.0%.

This paper is written in the Deterrence Era, Year 57 (DE 57), and every figure in it rests on survey data that were released before that date. The Doomsday Battle is history. Its record is thin. The telemetry of the Combined Fleet was lost with the fleet, and no complete record of the order of loss survives in the Academy archives. The debris field is the only surviving witness, and it has been worked for six seasons by the autonomous survey vehicles of the Oort Debris Field Survey Office, an organ of the Fleet International Academies (Oort Debris Field Survey Office, 56 DE).

Survey effort so far has been spent on cataloguing. For example, each fragment is given a position, a velocity vector and a residual charge, and then it is left in the catalogue until the next season. The catalogue has been used to count hulls (Marchetti, 55 DE) and to estimate losses from the survey volume (Taneja and Ferreira-Nunes, 54 DE). It has not been used to ask what the droplet did and in what order.

The answer matters for strategy. A single-probe path through a fleet is a sequence of positions, and the sequence fixes how long the remaining ships had to respond after the first loss. For example, a pass of 211 s leaves no useful interval for any ship that is still intact, while a pass of ten minutes would leave a great deal.

A method for recovering that sequence from debris is tested here against stopped chronometers, and the places where it fails are stated with their causes.

2. System Description

The survey vehicles are small and energy-limited, and their missions are replanned while running by the heuristic of an earlier survey paper (Vesnin-Wu, 56 DE). A fragment is ranged by each vehicle with a laser and a Doppler pair, its orbit is fitted over several passes, and a six-component state vector is stored. The catalogue is restricted to fragments above 2 kg, for which velocity precision after three passes is 0.1 mm/s (Halvorsen and Ketterman, 52 DE). Over some six decades since the battle, that precision limits the back-propagated position error of a single fragment to roughly 180 km. The mean spacing of kills along the path fitted in Section 3 is 127 km.

Position error is brought down by grouping. The catalogue holds 10,203 fragments above 2 kg that have been ranged on three survey passes so far. They are a sample of the debris and not a count of it. Of these, 316 (3.1%) were removed by a collision screen before grouping, and 247 more, held in groups of fewer than three, were set aside. Fragments that share a centre-of-mass velocity within the ejection scatter are taken to belong to one parent hull (Marchetti, 55 DE). The remaining 9,640 fall in 1,712 groups, a mean of 5.6 per group. Independent fragment errors average as the square root of the group size, so the centroid error is 76 km.

Breach geometry comes from a subset of the catalogue. Of the 9,640 grouped fragments, 2,118 carry a readable entry hole and exit hole, and these fall in 1,509 of the 1,712 groups, or 88.1%. The local direction of the probe through the hull is fixed by the axis from entry hole to exit hole. The petalling of the exit hole gives the speed through a calibration on laboratory hypervelocity shots into hull coupons (Taneja, 53 DE). Those shots reached about 12 km/s, so the calibration is extrapolated by a factor of 86 to the speed fitted here, and Section 3 carries that extrapolation into the speed error. A pre-encounter formation plan, compiled from archived staffing orders (Dlamini, 31 DE), is used as a nominal frame. It supplies a prior on centroid positions and no ordering, and a fit without it is reported in Section 3.

Independent timing is provided by recovered cabin chronometer modules. A module stops when the hull around it ruptures. The last synchronisation tick from the fleet time signal before the encounter is stored in each module, and the offset between that tick and the module oscillator was read out and subtracted, which removes the epoch offset between ships. Oscillator drift was corrected by the procedure of Ketterman (50 DE). The readings were withheld from every fitting step.

3. Analysis / Model

A single probe is assumed. Its path is modelled as a chain of K straight segments joined at turn points. Each hull group is assigned an arc-length coordinate along the chain, and the kill time is that coordinate divided by the local speed, while group centroids and breach axes are combined in one weighted least-squares objective and the nominal formation frame is applied only as a prior on centroid positions.

The number of segments was chosen by the Bayesian information criterion, with n = 2,118 breach observations. Five parameters, a start point and two direction angles, are carried by the first segment, since its length is set by the groups that it holds. Three more are added by each further segment, namely two direction angles and a length. The lowest score was given by four segments. Relative to that score, one, two, three, five and six segments scored 503.0, 235.3, 74.3, 19.8 and 40.0 higher, so a fifth segment is not supported by the breach observations.

A second probe was tested in the same way, as a competing explanation of the breach axes that fit the path badly. A two-probe model with eleven additional parameters reduced the residual sum of squares by 0.57%, and its criterion score was 72.2 higher than the single-probe score. The second probe is excluded by a wide margin. The fit alternates between the path and the assignment of groups to arc length.

Without the formation-plan prior, the same fit assigns a rank to 1,298 hulls, or 75.8%, which is 38 fewer than the 1,336 with the prior. Its Kendall correlation with the chronometers is 0.977 against 0.980. Over the hulls that both fits rank, the orders agree at a Kendall correlation of 0.991 with a mean shift of 1.9 places. The ordering step was compared with an exact branch-and-bound solution of the same objective on 25 subsets of 40 groups, the largest size for which the exact method finished. The heuristic returned the identical order in all 25 subsets. The shortest exact run took 258 s and the longest heuristic run took 1.21 s, so the minimum speed ratio is 213 times. Exact solution of the full field was not possible.

The rank interval of a group is its one-standard-deviation interval along the path, from the centroid error of the group and the local path-model residual, converted to places at the local hull spacing. A hull counts as ranked when the 1σ half-width of that interval is no more than five places, or 635 km. The centroid term is 0.074 s, which is 76 km at a speed of 1,030 km/s, and the path-model residual averages 0.545 s over the full catalogue, so the typical half-width is 566 km, or 4.5 places.

Total span follows from path length and speed. The fitted path is 2.17 × 10⁵ km long and the breach-derived speed averages 1,030 km/s, so the span is 211 s. 3.5% of the length and 3.7% of the speed are contributed by the fit. A further 8.0% is added to the speed by the extrapolated calibration. That allowance holds only if the petalling law keeps its form across 86 times its tested range. It is conditional and not a measured bound. The three terms combine to 9.5%, and the 95% interval is 172 to 250 s. The speed is in effect anchored by the chronometer scale check of Section 4, 0.987, and the interval given here is the debris-only statement made before that check. Path length and speed are both estimates of this paper from the debris, and neither is offered as an established property of the droplet.

A path of this length with 127 km between consecutive kills implies a formation extended over a scale of the order of 10⁵ km along the path. That is larger than the dense array of the nominal plan. No dispersal within a 211 s pass could produce it, and the fit does not resolve the discrepancy. The manoeuvring caveat of Section 6 applies.

4. Validation Against Field Data

Of 61 recovered chronometer modules, 49 were readable and 44 could be tied to a catalogued hull group through fragment association. Of those 44, 38 lie in ranked groups and 6 in unranked ones.

The 38 modules in ranked groups define the validation set, and they validate only the ranked subset. An RMS residual of 2.9 s is found for the six modules in unranked groups, and the pooled residual over all 44 is 1.39 s. The choice of rankable modules therefore flatters the headline figure, and the pooled value is the more cautious summary of the whole catalogue.

Reconstructed times were compared with stopped readings after one epoch offset and one linear scale factor were fitted over the 38 modules. The fitted scale is 0.987, which corresponds to a span of 208 s and lies inside the interval of Section 3. The chronometer intervals are measured without the petalling law, so the scale anchors the speed. The span reported throughout is still the uncorrected 211 s from the debris alone.

Two modules were carried by each of eight hulls, so the 38 modules sit on 30 hulls and 8 of the 703 module pairs share a reconstructed rank. The 8 tied pairs lie inside the validation set of 38 modules. The Kendall rank correlation (τ-b) between reconstructed and chronometer order is 0.980 from 5 discordant pairs among the 695 untied pairs. A bootstrap over the 38 modules, with 2,000 resamples, gives a 95% interval of 0.94 to 0.995. At a mean separation of about 45 places between validation hulls, the rank correlation tests gross order. The five-place resolution rests on the time residual decomposed below.

The overall RMS time residual is 0.95 s. In the worst case the residual was 2.6 s, and it occurred in the opening segment. The other three segments stayed at or below 1.1 s RMS, as Table 1 shows. A residual of 0.95 s (0.946 s unrounded) converts to 974 km of path, or 7.7 hull places, which is larger than the five-place criterion. The difference is a clock effect. The disagreement within the eight module pairs is 1.10 s RMS, so the clock term of one module is 0.78 s. Added in quadrature to the centroid term of 0.07 s and the path-model term of 0.545 s, it gives 0.95 s. The debris is credited with 0.55 s, or 566 km and 4.5 places, and about two thirds of the residual variance is attributed to the chronometers.

For rank accuracy within the ranked subset, the debris-only figure of 0.55 s is the relevant one.

5. Failure Modes

Three failures account for all 376 unranked hulls, and Table 2 divides them by cause and by path segment. A hull lost at a join between segments is assigned to the later segment.

The first failure is at the opening of the pass. In the first segment, 41 hulls are mixed with fragments of Mantis, the vessel that had captured the droplet, and the contact there was slower and more complex than a straight probe path allows. The zone is small. Its breach axes disagree with each other by more than the fit tolerance, and these 41 hulls, 2.4% of the catalogue, are removed from the rank assignment.

Loss of fragments is the second. Groups whose fragments drifted sunward out of the survey volume (Taneja and Ferreira-Nunes, 54 DE) survive in the catalogue with too few members, a mean of 3.2 fragments, of which 31% carry a readable breach. That is about one readable breach per group. The local path axis is poorly constrained. The mean local path residual of these groups rises to 0.70 s, against 0.52 s for the rest of the catalogue. With a centroid error of 101 km, their 1σ half-width is 728 km, or 5.7 places. Hulls that lost every fragment are not among the 1,712 at all, so the catalogue is a lower bound on the fleet. In all, 273 groups, or 15.9% of the catalogue, are affected. By segment the share is 1.9% in the second, 36.8% in the third and 27.4% in the fourth, because the third and fourth segments lie on the side of the field that drifts out. A fragment that has left the survey volume cannot be recovered by any routing of vehicles.

Turn points are the third. A group that lies within the centroid error of two adjacent segments can be projected onto either one. Near each of the three joins the path-model residual rises from 0.545 s overall to 0.75 s, which is 6.1 hull places at a spacing of 127 km, and a projection onto the wrong segment moves a group further still. Rank was lost for this reason by 62 hulls. That is 3.6% of the catalogue. The turns are real, and they are the places where order is least certain, because a four-segment model cannot settle from centroids alone which segment receives a hull near a join.

Secondary collisions among fragments are a smaller risk, since they would corrupt the common-velocity grouping. The screen of Section 2 removed 3.1% of catalogued fragments before grouping. The effect on rank assignment was below the resolution of the chronometer test. The 38 validation modules lie in ranked groups, so the validation covers the 78.0% that the method claims and makes no statement about the remainder.

6. Conclusion

This paper reconstructed the kill sequence of the Doomsday Battle from the survey catalogue of the Oort debris field. A single-probe path of four straight segments was fitted to fragment velocities and hull-breach geometry from 9,640 grouped fragments, and 1,712 parent hulls were placed along it. The kill rank of 1,336 hulls is placed to within five places (1σ) by the method, and the full pass is put at 211 s. Against 38 independent chronometers, gross rank order agrees at 0.980 and time agrees to 0.95 s RMS, of which 0.55 s in quadrature belongs to the debris and about two thirds of the variance to the chronometer clocks. The remaining hulls are lost in three gaps that have been named and bounded.

The result is firm for the ranked hulls. The model was selected over one to six segments and over one or two probes, and the ordering heuristic matched an exact solver in every case that the solver could finish. A fit without the formation-plan prior ranks 38 fewer hulls (1,298 against 1,336) and agrees with the prior fit at a Kendall correlation of 0.991, with a mean shift of 1.9 places. For the 78.0% of hulls whose rank is placed, the debris is a sufficient record of the order of loss.

Wrecks struck in sequence by a single fast object and left to drift hold the order of their loss in the geometry of their breaches, so the method carries beyond this field. The open question is how few fragments with readable breaches are enough to recover it, and no survey has yet tried to lower the bound below the 2,118 used here.

Subsequent work will add forecasts of fragment charge to the routing so that vehicles reach the sunward-drifting fragments first. One simplification remains to be removed from the model. The formation is treated as static during the pass, although the ships were manoeuvring, and a moving-frame prior will replace the nominal plan in the next survey season.