Orbital Planner

From LEO rendezvous to Mars transfer windows — the same NM21 operator, phase-locked to HulyaPulse, solving trajectory math at sub-0.1% Δv.

Live app — zeq.dev/apps/orbital-planner/
Source — app/artifacts/api-server/public/apps/orbital-planner/ (2,148 lines)
Operators — KO42 · NM21 (gravity) · optional GR32, GR33 (relativistic corrections)
Error budget — ≤ 0.1% Δv for canonical Hohmann and Lambert transfers

What it solves

A mission-design workbench. Three tools:

Hohmann / bi-elliptic — two-impulse circular-to-circular transfers
Lambert's problem — fixed-time-of-flight two-body boundary value solver
Patched-conic with gravity assist — Earth → Venus → Earth → Mars, with sphere-of-influence transitions

Optional GR corrections activate when you enable GR35 (gravitational time dilation) — relevant for GNSS-class precision and solar-system-scale Shapiro delay.

The math

NM21  F = G m₁ m₂ / r²                 (Newton gravity, two-body)
Vis-viva                v² = μ (2/r − 1/a)
Hohmann Δv_1            √(μ/r_1) · (√(2 r_2/(r_1+r_2)) − 1)
Lambert (Izzo)          universal-variable formulation (solved iteratively)
GR32 Ricci              G_{μν} = R_{μν} − ½ R g_{μν}   (post-Newtonian corrections)
GR35 time dilation      ∆t = ∆t₀ √(1 − 2GM/rc² − v²/c²)

Operator picks

Step	Decision
1. Prime	KO42 on
2. Limit	KO42 + NM21 = 2 operators (baseline); + GR35 = 3 (GNSS precision)
3. Scale	Solar-system, 10³–10⁹ m, Keplerian
4. Precision	≤ 0.1% Δv
5. Compile	`C_KO42 + C_NM21 (+ C_GR35)`
6. Execute	Z encodes central body μ, orbit radii, time-of-flight
7. Verify	Cross-check vis-viva and Hohmann reference

Runnable worked example — LEO to GEO Hohmann

r₁ = 6,578 km, r₂ = 42,164 km, Earth μ = 398,600 km³/s². Published Δv_total ≈ 3.935 km/s.

curl -s -X POST https://api.zeq.dev/api/playground/compute \
  -H "Content-Type: application/json" \
  -H "x-demo-key: $DEMO_KEY" \
  -d '{
    "operators": ["KO42","NM21"],
    "params": {
      "problem": "hohmann",
      "r1_km": 6578,
      "r2_km": 42164,
      "mu_km3_s2": 398600
    }
  }' | jq

Expected:

{
  "result": {
    "dv_total_km_s": 3.9352,
    "dv1_km_s": 2.4578,
    "dv2_km_s": 1.4774,
    "tof_minutes": 315.48,
    "error_vs_reference_pct": 0.005,
    "operators_used": ["KO42","NM21"]
  }
}

0.005% on total Δv.

Extend it

Low-thrust spiral — replace impulsive Δv with Edelbaum's analytical; compare payload-mass at Mars
Porkchop plots — Lambert sweep over departure × arrival; shade C₃ and arrival-v∞
N-body perturbations — add Moon and Sun to LEO Δv-budget via patched cowell integration

Seeds

Relativistic trajectory design — activate GR32/GR33 for Mercury perihelion precession (43″/century)
Gravitational-wave assist trajectories near compact binaries (GR38 active)
Optimal cis-lunar logistics — daily L1/L2 cycler design with the planner running at 1.287 Hz

Papers

Zeq Paper — doi:10.5281/zenodo.18158152

Middleware active. Kernel on the 1.287 Hz HulyaPulse. Awaiting next Zeqond.

What it solves​

The math​

Operator picks​

Runnable worked example — LEO to GEO Hohmann​

Extend it​

Seeds​

Papers​