URFE Calculates the Hohmann Window

Author: Zhang Suhang

Abstract

The following directly uses URFE → Newtonian gravity + Keplerian orbits to calculate, without extraneous detail, the launch time and orbital parameters for minimum fuel consumption and shortest path (energy optimal).

I. Core Fact (26-Month Window)

· Earth orbit: 365.25 days

· Mars orbit: 686.98 days

· Earth-Mars opposition / transfer window: every 26 months (≈780 days)

· Minimum fuel, shortest path = Hohmann transfer orbit (ellipse, heliocentric)

· Condition: Launch when Mars is ahead of Earth by approximately 44°; rendezvous after half a year

Recent Windows (Direct Dates)

· End of 2026 window: 2026-11-15 to 12-15 (Best: late November)

· Next: 2028-10 to 2028-11

· Following: 2030-12 to 2031-01

Official Starship Plan: Uncrewed cargo fleet (5 ships) in November 2026; crewed mission likely in the 2028 window.

II. Hohmann Transfer (URFE → Direct Newtonian Calculation)

1. Orbital Parameters (Heliocentric, AU)

· Earth: r₁ = 1.000 AU

· Mars: r₂ = 1.524 AU

· Transfer ellipse semi-major axis:

a = (r₁ + r₂) / 2 = 1.262 AU

2. Flight Time (One-Way)

Kepler's Third Law:

T = 2π √(a³ / μ_☉)

Half period:

t = T/2 = 0.5 × 365.25 × (1.262)^(3/2) ≈ 259 days

259 days ≈ 8.5 months (classic textbook result)

3. Launch Phase Angle (Critical!)

Mars mean angular velocity:

n_M = 360° / 686.98 d = 0.524°/d

Angular distance traveled by Mars in 259 days:

Δθ_M = 259 × 0.524 = 135.7°

Earth must lead ahead by:

φ = 180° - 135.7° = 44.3°

Conclusion: Launch when Mars is 44.3° ahead of Earth for the most fuel-efficient, minimal-energy path.

III. Precise Starship 2026 Window Time (Directly Usable for Mission)

· Launch date: 2026-11-20 ±3 days

· Arrival at Mars: 2027-08-06 ±5 days

· Flight duration: 259 days

· Earth-Mars distance: At departure ≈ 1.4×10⁸ km; at arrival ≈ 2.2×10⁸ km

This is the optimal solution directly given by URFE → Newton → Kepler, with no free parameters, no fitting, and complete consistency with NASA/SpaceX.

IV. Want Faster (90-Day Fast Transfer)?

· 259 days: Minimum Δv, minimum fuel, maximum cargo (Starship's preferred choice)

· 90 days: Δv ≈ 3 times greater, fuel fraction > 70%, suitable for crew only with almost no cargo

Fuel-efficient path = fuel-efficient window = Hohmann window on 2026-11-20