Constellation design

The constellation-design kind builds one or more satellite constellations around the Earth, the Moon, Mars or any other body in the solar-system catalogue (src/body.rs: the eight planets, Pluto, the Moon, Phobos, Deimos, the four Galilean moons and Titan, with the same constants the solar-system and body-pnt kinds use; see SOLAR-SYSTEM.md) and maps their coverage and dilution of precision (DOP) over a latitude/longitude grid. Source: src/constellation.rs.

kshana scenarios/constellation-multi-gnss-coverage.toml
# constellation-design: 102 satellites (GPS, Galileo, BeiDou, GLONASS) around the Earth;
#   availability 100.00% (PDOP <= 6 at 10 deg mask), worst site 100.00%, median PDOP 0.95,
#   mean 31.0 in view; 648 grid points x 95 epochs (MODELLED)

The body is chosen with body (default earth). The run spans duration_s (default one sidereal day, 86 164 s) in steps of step_s (default 600 s), above mask_deg (default 5 deg), on a grid of grid_step_deg (default 10 deg) that lat_min_deg, lat_max_deg, lon_min_deg and lon_max_deg may restrict; j2 = true adds the secular drift from J2, the second zonal harmonic of the body's gravity field.

What goes in#

• Walker shells in the T/P/F convention: T satellites in P equally spaced planes with phasing F. A delta pattern spreads the ascending nodes over 360 deg, a star pattern over 180 deg. Plane k sits at node raan0 + k·ΔΩ and satellite j of it at mean anomaly m0 + j·360·P/T + k·360·F/T.
• Explicit element sets: semi-major axis or altitude, eccentricity, inclination, right ascension of the ascending node (RAAN), argument of periapsis and mean anomaly.
• Presets from published nominal elements (Earth only; a preset around another body is an error):

The GPS table's RAAN is inertial at its epoch; the preset converts it to an Earth-fixed node longitude with the Greenwich hour angle the table states (100.765 deg). GLONASS states absolute (Greenwich) node longitudes directly. The Galileo document states no hour angle, so its RAAN is used as the node longitude. The BeiDou standard does not state the MEO phase or the IGSO nodes; the preset starts the MEO pattern at node 0 and mean anomaly 0 and puts the three IGSOs on one ground track crossing the equator at 118 deg E. Those two choices are modelled.

Several constellations can share one run. With clock = "per-constellation" (the default) the receiver estimates one clock per system in view, as a multi-GNSS (Global Navigation Satellite System) receiver does; clock = "common" shares one clock.

What comes out#

Per grid cell: mean, fewest and most satellites above the mask; the share of epochs with a fix and with position DOP (PDOP) at or below pdop_threshold; mean geometric, position, horizontal and vertical DOP (GDOP, PDOP, HDOP, VDOP); largest PDOP. Globally, weighted by the cosine of latitude so each cell counts by its area: availability, worst-site availability, and the mean, median, 90th, 95th and 99th percentile and maximum of each DOP. Per satellite: a downsampled ground track (track_points samples, default 24, for at most max_tracks satellites, default 120). The chart is two maps, availability and mean PDOP.

How it scales#

A satellite at radius r is above an elevation mask ε exactly when its central angle from the user is at most λ = arccos((R/r)·cos ε) − ε, so the visibility test is one dot product against cos λ. Each epoch the satellites are also sorted by sub-satellite latitude, and a grid row at latitude φ only tests the satellites within λ_max of φ. Both are exact on a spherical body, and a unit test holds every map to a brute-force scan with the direct elevation test. The work block reports the pair tests a brute-force scan would make and the ones made. The unit test's 5 000-satellite design on a 10 deg grid (648 points, 6 epochs) keeps 9.9 % of the 19 440 000 brute-force pair tests; the bundled leo-pnt-mega-shell scenario (5 000 satellites, 648 points, 12 epochs) runs in about 0.14 s with the release command-line binary on an Apple-silicon laptop. Wall-clock times vary with the machine and its load. The engine uses no threads, clock or filesystem, so it runs unchanged as WebAssembly.

What is checked, and against what#

VALIDATED, against published documents:

• the Walker generator reproduces all 24 rows of Galileo OS SDD Table 23, and the GLONASS ICD slot formula is the same set of 24 slots as a Walker 24/3/1;
• the GPS preset reproduces the groundtrack equatorial crossing column of SPS PS Tables 3.2-1 and 3.2-2: 35 of 36 locations within 0.0108 deg against a 0.015 deg rounding bar. E3F is held to a looser 0.06 deg bar, because the table's own E3F row is inconsistent with its RAAN and argument of latitude: its argument of latitude is 14.60 deg ahead of E3's, which moves the crossing by 7.30 deg, while the table moves it by 7.36 deg;
• the GPS baseline under the SPS PS Appendix B conditions (one sidereal day, 287 steps of five minutes, a 4 deg grid, all in view, 5 deg mask) gives a global HDOP median of 0.940, 90 % 1.165, 95 % 1.255, 98 % 1.370 and mean 0.965, against the published 0.94, 1.16, 1.25, 1.37 and 0.96 (bar 0.03, fixed before the first run). PDOP median 1.795 is below the 1.815 of Table B.3-1, which describes a degraded 20-24 satellite mix. PDOP of 6 or less holds 100 % of the time, against 98 % global and 88 % worst site in Table 3.8-1.

Checked against a hand derivation, not an external document: a single-epoch DOP matches the closed form for a zenith satellite and three at 30 deg elevation (PDOP 8/3, GDOP 3.0732).

MODELLED: two-body orbits with an optional secular J2 drift; a spherical body; geometric visibility only (no signal power, satellite health or terrain); no third-body perturbation, which a real lunar frozen orbit depends on. The worst single time-space point depends on the grid: HDOP 2.40 and VDOP 5.22 here against 2.49 and 5.43 published; these are reported, not pinned. The four systems' presets come from different reference epochs, so their relative phase in one run is not a snapshot of any date.

Bundled scenarios#

Each line gives what the run prints with kshana 0.29.1.

• scenarios/constellation-multi-gnss-coverage.toml: GPS, Galileo, BeiDou and GLONASS (102 satellites) over one day above a 10 deg mask, one clock per system. Availability (PDOP 6 or less) 100 % globally and at the worst site, median PDOP 0.95, 31.0 satellites in view on average.
• scenarios/leo-pnt-mega-shell.toml: 5 000 satellites in four low Earth orbit (LEO) shells for positioning, navigation and timing (PNT), with J2, above a 20 deg mask. Availability (PDOP 3 or less) 99.22 % globally and 0 % at the worst site, median PDOP 1.11, 39.4 satellites in view on average.
• scenarios/lunar-relay-constellation.toml: 8 satellites in elliptical lunar frozen orbits and a 6-satellite circular shell around the Moon. Availability (PDOP 6 or less) 21.33 % globally and 0 % at the worst site, median PDOP 5.22, 3.2 satellites in view on average.

LEO-PNT coverage for polar and Arctic users, against medium Earth orbit GNSS, is the leo-pvt kind's polar mode (scenarios/polar-arctic-leo-coverage.toml; see LEO-PNT.md).