🛰️ Cislunar Custody

Custody Horizons for Ground-Based Angles-Only Tracking of Cislunar Periodic Orbits

How long can an object near the Moon go unobserved before Earth's telescopes lose it for good?
A computational study of NRHO, halo, Lyapunov and DRO orbits in the Earth–Moon three-body problem.

Bishwaswarup Nayak · Indian Institute of Science, Bangalore · independent project

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The Result

A cislunar object's custody is decided by how long it was tracked before the gap, not by the size of the search.

Seen from Earth, cislunar targets sit only 4–14° from the Moon. The Moon's glare and the lunar phase therefore impose a monthly 8.5–15.7 day observation blackout that no ground network or bigger telescope can close. We define the custody horizon as the longest gap after which the target is still inside the telescope search pattern with 99 % probability:

$$T_c(N) = \max\left\{ \Delta t \;:\; \Pr\left[ \angle(\hat{\mathbf u}_{\mathrm{target}}, \hat{\mathbf c}) \le r_N \right] \ge 0.99 \right\}, \qquad r_N = \phi \sqrt{N/\pi}$$

Here the search pattern is $N$ fields of view of side $\phi = 1^\circ$. The post-fit covariance $P_0$ at the gap start is the Cramér–Rao bound of the preceding tracking arc, which the filters reach:

$$\mathcal{I}(t_0) = \sum_k \Phi(t_k,t_0)^{\top} H_k^{\top} R^{-1} H_k \, \Phi(t_k,t_0), \qquad P_0 = \Phi \, \mathcal{I}^{-1} \, \Phi^{\top}$$

Propagating $P_0$ through gaps of 0.1–30 days (linear, unscented and Monte Carlo) gives:

Orbit	Stability	$T_c$, 10 fields, after 1 / 3 / 7-day arcs	Blackouts survived (1 / 3 / 7-day arcs)
9:2 NRHO (Gateway)	near-stable	> 30 d	78 % / 100 % / 100 %
DRO (~70 000 km)	stable	> 30 d	85 % / 100 % / 100 %
L1 halo ($A_z$ ≈ 30 000 km)	unstable	11.0 / 13.2 / 16.5 d	89 % / 96 % / 100 %
L2 Lyapunov	unstable	12.5 / 14.9 / 17.4 d	74 % / 97 % / 100 %

The values are medians over arcs through the year (Tables 3 and 4 of the paper). Further findings:

• The horizon is predictable without Monte Carlo. The unscented transform of the post-fit covariance (13 sigma points) predicts $T_c$ with log-$R^2$ = 0.96 / 1.00 / 0.93 for 1 / 10 / 100 fields. The orbit's finite-time Lyapunov exponent alone does not predict it (log-$R^2$ ≤ 0.04), because stretching only matters along the directions the post-fit covariance actually excites.
• The operator's horizon equals the ideal one. Centring the search on the UT prediction instead of the (unknowable) true mean gives the same $T_c$ (median ratio 1.00). Across 25 members of five orbit families, $T_c$ falls steadily with the stability index ν (Spearman ρ = −0.92 for ν > 10).
• Linear prediction gives false custody. After 14 days on the L1 halo, an EKF-style 99 % ellipse contains only 5.4 % of the true probability (12.6 % in the ephemeris model), against 98 % for the UT ellipse. The UT claims too long a horizon in only 2 of 254 arcs.
• Watch the NRHO perilune. The NRHO's sky-plane uncertainty swells at every perilune passage (every 6.56 d) and shrinks again, so a single-field search can lose the target there briefly.
• Filter choice decides reacquisition. After a 3.2-day gap on the L1 halo, the EKF ends consistent in 25 % of 100 runs, the UKF in 55 %, a particle filter in 91 % and a Gaussian-mixture UKF in 99 %.

All of this holds in a JPL DE440 ephemeris model with solar radiation pressure, where each orbit is transitioned by multiple shooting (horizon ratio 0.93–1.01).

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Figures

Targets hug the Moon. Angle from the Moon as seen from Earth; dashed lines are the exclusion angles.	The lunar-phase blackout. First blocking constraint per site for the 9:2 NRHO over 60 days.
Uncertainty growth. True 99 % sky radius after 7-day arcs, against the search radii for 1, 10 and 100 fields.	Does custody survive? Custody horizon against blackout length; points above the line survive.
Predicting Tc. Orbit-only FTLE (left) fails; UT-propagated covariance (right) matches Monte Carlo.	False custody. Probability mass inside linear (dotted) and UT (solid) 99 % ellipses.
Custody vs stability. Operator horizon (UT-centred circle, 10 fields, 7-day arcs) across five families against stability index ν (paper Fig. 14).	Operator vs ideal. Horizon with the search centred on the UT prediction (circle, strip) against the true-mean horizon.
After a 3.2-day gap. Particles vs EKF/UKF ellipses vs GM-UKF components (L1 halo).	Ephemeris check. Custody horizon in the CR3BP vs DE440 + SRP.

_{Earth–Moon CR3BP families: L1 halo, L2 halo → NRHO, L1/L2 Lyapunov and DRO; the 9:2 NRHO is in black.}

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How it works

flowchart LR
    A[CR3BP orbit catalogue<br/>Richardson + continuation] --> B[Sky geometry<br/>real Sun/Moon, GMST]
    B --> C[Ground sensors<br/>4 sites, moonlight, photometry]
    C --> D[Tracking arc<br/>CRLB, EKF/UKF/GM-UKF/PF]
    D --> E[Gap propagation<br/>linear / UT / Monte Carlo]
    E --> F[Custody horizon T_c<br/>and blackout survival]
    F --> G[DE440 + SRP check<br/>multiple shooting]

Loading

Stage	What is done
Dynamics	Earth–Moon CR3BP ($\mu$ = 0.0121505842), STM verified to $3\times10^{-10}$. Families built from Richardson seeds by differential correction and pseudo-arclength continuation. 9:2 NRHO: $T$ = 6.5624 d, perilune 3249 km, apolune 71 222 km.
Sensors	Hanle · La Palma · Haleakala · Siding Spring. Twilight, 20° elevation, Moon exclusion and occultation, shadows, Lambertian photometry, Krisciunas–Schaefer moonlight.
Estimation	Fisher information from the real visible schedule. EKF, UKF, GM-UKF (entropy-triggered splits along the max-stretch direction), regularised PF with progressive correction and a UKF handover.
Custody	Post-fit covariance through 0.1–30 d gaps (121-point grid, converged to 0.15 d). Ideal, actual and claimed horizons; UT-centred circle and ellipse-aligned strip searches; Gaussian containment; FTLE vs UT predictors; 525 cases (394 distinct arcs).
Ephemeris	DE440 Sun and Moon + cannonball SRP, exact gravity-gradient STM, Levenberg–Marquardt + Newton multiple shooting.

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Operational rules

1. Shrink the Moon-exclusion angle before growing the aperture. At 10° exclusion, L1/L2 targets are never visible; at 5°, 21–33 % of the year.
2. Track at least 3 (preferably 7) days before every predicted blackout. Extra sites do not remove it.
3. Plan searches with unscented or Gaussian-mixture prediction. On the L1 halo, linear prediction is safe only for gaps shorter than about 10 days.
4. Reacquire with a Gaussian-mixture filter. It costs about 2× a UKF and was the only filter to reacquire at least 99 % of runs.
5. On the NRHO, time narrow searches away from perilune, where the sky-plane uncertainty briefly swells.

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Quick start

git clone https://github.com/Bishwaswarup/cislunar-custody.git && cd cislunar-custody
python -m venv .venv && source .venv/bin/activate
pip install -e ".[dev,ephem]"
curl -o data/de440s.bsp https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/de440s.bsp
pytest                                    # 52 tests, ~15 s

Reproduce every result (click to expand)

#	Command	Time	Produces
1	python scripts/build_catalogue.py then python scripts/plot_catalogue.py	~20 s	orbit catalogue, fig01–02
2	python scripts/visibility_study.py	~10 s	one-year visibility, fig03–05
3	python scripts/observability_study.py	~10 s	CRLB of tracking arcs, fig06–07
4	python scripts/filter_study.py	~1 min	EKF/UKF Monte Carlo, fig08–09
5	python scripts/nonlinear_filter_study.py	~10 min	GM-UKF / PF reacquisition (100 runs), fig10–11
6	python scripts/custody_study.py	~12 min	custody horizons, fig12–15
7	python scripts/ephemeris_check.py	~6 min	DE440 + SRP cross-check, fig16–17
8	python scripts/sensitivity_check.py	~5 min	Monte Carlo sample-size check (300 vs 2000) and gap-grid check (37 vs 121 points vs uniform 0.02 d)
9	python scripts/family_sweep.py	~25 min	operator horizons across whole families vs stability index, fig18–19 (paper Fig. 14–15)
10	python scripts/consider_study.py	~20 min	site biases, clock offsets and SRP errors as consider parameters, fig20 (paper Fig. 16)
11	python scripts/verify_claims.py	~30 s	checks every number in paper/main.tex against data/ (needs family_sweep.csv, consider_cases.csv)

Or run everything with bash scripts/run_all.sh (logs in logs/); bash scripts/run_all.sh --check only re-checks the paper.

Each script prints its summary tables. The steps depend on the earlier ones, so run them in order. Figures are written twice: figures/*.png (previews) and figures/jas/FigN (PDF + 600 dpi TIFF, 84/174 mm wide, for the journal). custody_study.py, ephemeris_check.py and nonlinear_filter_study.py accept --plots-only to redraw figures from saved data.

Repository layout

src/cislunar_custody/
├── dynamics/        CR3BP EOM + STM, periodic orbits, continuation, Richardson seeds
├── catalogue.py     orbit families (L1/L2 halo, NRHO, Lyapunov, DRO)
├── frames/          analytic Sun/Moon, GMST, sites, CR3BP → inertial
├── sensors/         telescopes, photometry, moonlight, visibility, RA/Dec
├── observability/   Fisher information and CRLB
├── filters/         EKF · UKF · GM-UKF · PF→UKF
├── scenario.py      measurement generation and a common filter runner
├── custody/         gap propagation and custody horizons
├── plotstyle.py     journal figure style (fonts, widths, FigN export)
└── ephem/           DE440, ephemeris + SRP dynamics, multiple shooting
scripts/             one script per milestone
tests/               52 pytest tests
.github/workflows/   CI: pytest on Python 3.10–3.14 with the DE440 kernel
paper/               LaTeX manuscript for JAS (Springer sn-jnl template: main.tex, references.bib)

Assumptions and limitations

• Truth and filters share the dynamics. The CR3BP study places orbits in the real Earth–Moon direction; the DE440 check confirms the results.
• Limiting magnitudes of 19.5 / 21.5 / 23.0 (0.36 / 1 / 2 m), extinction and dark-sky brightness are nominal; the first is anchored to LANL's 36 cm CAPSTONE detection.
• The post-fit covariance is the CRLB of the preceding arc. The filters reach it in tracking, but not always immediately after a gap.
• Targets are passive, non-manoeuvring 2 m spheres; data association is not modelled.
• The NRHO ephemeris counterpart keeps a 50–124 km continuity residual at perilune.
• $T_c$ is the first gap at which the 99 % radius exceeds the search radius. On the NRHO a brief perilune spike can end it, so the N = 1 NRHO horizon is sensitive to the Monte Carlo quantile (one arc moves from 5.3 to 17.6 d with 2000 samples).
• The strip search gains over the circle mainly where the cloud is sub-arcsecond thin across track, which relies on the idealised CRLB covariance (no biases, process noise or SRP uncertainty).

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Verification

Every number in the manuscript is re-computed from data/ by scripts/verify_claims.py, which reports each claim as PASS, FAIL (the paper is wrong), WEAK (right but statistically thin: small n, wide confidence interval or a non-significant difference) or TEXT (the paper no longer states it). Current report: data/claims_report.txt.

Check	What it tests
Claim verifier	~217 numbers in text and tables against the saved study outputs (0 FAIL, 6 WEAK)
Confidence intervals	95 % Wilson intervals on filter success rates and blackout survival; Fisher exact tests between filters
Bootstrap	95 % intervals on every log-R² of the custody-horizon predictors
Distinct arcs	pooled statistics count each tracking arc once (131 of the 525 cases repeat an arc)
Monte Carlo convergence	sensitivity_check.py: 300 vs 2000 samples change T_c by a median 0.09–0.14 d, at most 0.9 d for N = 10 and 100
Grid convergence	121-point gap grid vs a uniform 0.02-d grid: T_c changes by at most 0.15 d (N = 1, 10) and 0.6 d (N = 100)
Continuous integration	GitHub Actions runs the 52 tests on Python 3.10–3.14 on every push
Run count	reacquisition re-run with 100 Monte Carlo runs per filter (was 10)
Operator horizon	T_c with the search centred on the UT prediction (circle and ellipse-aligned strip), not on the true mean
Family sweep	T_c for members of all five families, with random orbital phase, against stability index
Reproducibility	every script re-runs deterministically from fixed seeds; run_all.sh regenerates everything

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Roadmap

• CR3BP catalogue · sensor model · observability · filters · custody horizons · ephemeris check
• Manuscript in the Springer template, figures to JAS specifications (paper/, figures/jas/)
• Claim verifier, sample-size check, confidence intervals
• Operator horizon and family sweep: code, fig18–19, grid refinement (milestone 8)
• Operator horizon as the headline metric and family sweep in the paper
• Consider covariance (measurement bias, timing, SRP uncertainty) and filter-derived post-fit covariance
• Submission to The Journal of the Astronautical Sciences
• Real-data validation with Indian observatories (IIA Hanle, ARIES Devasthal, GROWTH-India)
• Custody-aware sensor tasking using the UT-predicted $T_c$

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Bishwaswarup Nayak · Department of Physics, Indian Institute of Science, Bangalore
📧 bishwaswarup@iisc.ac.in · 🆔 ORCID 0009-0001-9926-5329

_{Manuscript prepared for The Journal of the Astronautical Sciences. To cite, use Cite this repository in the sidebar (from CITATION.cff). Code: MIT License.}