Time of Flight (ToF) and Transfer-Time Equations

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

Time of flight (ToF) is the time a spacecraft takes to travel between two specified states — typically two position vectors, an initial parking orbit and a target orbit, or two points on a manifold. In the Lambert problem, ToF is one of the three boundary conditions (along with the two position vectors) that select a unique conic; in the CR3BP, it is a property of the invariant-manifold trajectory chosen. Transfer-time equation and flight-time equation refer to the explicit relations that give ToF as a function of orbit parameters in Lambert-style solutions. Time-to-go ( $t_\text{go}$ ) is the remaining ToF from the current state to the terminal condition, used as a parameter in explicit guidance laws. Time of permanence is the duration a ballistic-capture trajectory stays captured before escaping the lunar sphere of influence.

Lambert's problem: ToF as a boundary condition

Given two position vectors $\mathbf r_0$ , $\mathbf r$ and the transfer angle $\Delta\nu$ between them, Lambert's theorem states that the ToF depends only on the semi-major axis $a$ , the sum $r_0 + r$ , and the chord length $c = \|\mathbf r - \mathbf r_0\|$ — not on the individual orbit (Vallado 2022, §7.6). With the semiperimeter $s = (r_0 + r + c)/2$ and the variables

\sin\tfrac{\alpha_e}{2} = \sqrt{\tfrac{s}{2a}}, \qquad \sin\tfrac{\beta_e}{2} = \sqrt{\tfrac{s-c}{2a}},

the elliptical flight-time equation (Lagrange's form) is

\Delta t = \sqrt{\tfrac{a^3}{\mu}} \big[\, 2\pi n_\text{rev} + \alpha_e - \sin\alpha_e \mp (\beta_e - \sin\beta_e)\,\big],

with the sign selecting short vs. long way and $n_\text{rev}$ counting complete revolutions. This transfer-time equation $\Delta t = T(a)$ is what Lambert solvers iterate on: given $\Delta t$ , find $a$ .

The universal-variable form (Bate, Mueller & White; Battin 1999) replaces $\alpha_e, \beta_e$ with a universal variable $\chi$ and works uniformly across ellipse, parabola, and hyperbola:

\sqrt{\mu}\,\Delta t = \tfrac{r_0 r}{\sqrt{a}} \sin\!\tfrac{\Delta\nu}{2}\, \chi + (1 - \lambda\cos\!\tfrac{\Delta\nu}{2})\,\chi^3 C(z) + \mu D(z),

where $z = \chi^2/a$ and $C, D$ are Stumpff functions. Robust initial-guess schemes for $\chi$ (e.g., Thorne's transfer-angle-based estimate) keep the iteration accurate even for multi-revolution cases.

ToF in the CR3BP: stable-manifold transfer times

In the restricted three-body problem, transfer duration is set by the geometry of the stable/unstable manifolds rather than by a Lambert conic. For transfers from the lunar surface to a halo orbit around $L_1$ or $L_2$ along the stable manifold (Alessi et al. 2009):

Direct transfers (no lunar loop): ToF ≈ 10 d, of which ≈ 5 d is spent asymptotically approaching the nominal orbit; when within 1000 km of the halo orbit the relative speed is below 0.25 km/s.
Each additional lunar loop adds ≈ 10 d.
Integration is started when the spacecraft is 70–90 km from the reference halo orbit; without that cut-off, the manifold definition would give infinite time.

Low-energy cislunar transfers along Sun–Earth manifolds take much longer — typically 70–120 d (Parker & Anderson 2013) — but cost less $\Delta v$ .

The minimum-ToF stepped law (cislunar)

For trajectories from a minimum parking orbit (MPO, radius ≈ 0.20 in normalized units) to the lunar surface, Liang et al. (2016) computed the minimum ToF at fixed Jacobi constant $C$ :

$C$	2.08	2.28	2.48	2.68	2.78	2.88	2.98	3.08	3.18
min ToF (d)	3.13	3.37	4.25	4.31	4.80	30.8	45.9	41.8	67.4

The ToF does not grow smoothly with $C$ but in steps: ≈ 3–5 d in the direct-transfer regime ( $C \le 2.78$ ), then a jump to 30–46 d, then to > 60 d. The jumps arise because the stable manifold of the $L_1$ Lyapunov orbit intersects the MPO only at certain periapses — when it does not intersect at the first periapsis, the spacecraft must wait through extra Keplerian twists (~11 d each, half the Keplerian period at half the Earth–Moon distance) before finding an opening (Liang et al. 2016, §4).

Time-to-go in guidance

In explicit and optimal-guidance laws, $t_\text{go}$ is the remaining ToF from the current state to the terminal condition; it appears as a parameter in the guidance command. For time-optimal problems with a Hamiltonian structure, $t_\text{go}$ is recovered analytically from the terminal condition on the Hamiltonian (often via a quartic equation); its accuracy directly determines the precision of the issued command (Zhao et al. 2021).

Time of permanence in ballistic capture

A ballistic-capture trajectory enters a temporary capture around the Moon without a insertion burn. The time of permanence — from the first perilune to escape from the lunar sphere of influence — quantifies the capture's quality. In the patched three-body model of Sousa-Silva, Terra & Ceriotti (2018), integrations run up to $t_\text{max} = 180$ d; capture lasting more than 90–120 d is achievable for selected Jacobi constants and perilune altitudes in the 90–400 km band. Longer permanence means more robust capture and more flexible downstream options (descent, orbit insertion, sample return).

References

Vallado, 2022, Fundamentals of Astrodynamics and Applications, §7.6 — Lambert problem, transfer-time/flight-time equations, multi-revolution solutions.
Battin, 1999, An Introduction to the Mathematics and Methods of Astrodynamics (rev. ed.) — universal-variable formulation and Battin's Lambert algorithm.
Alessi, Cascioli, Colombo & Lizia, 2009, "Leaving the Moon by means of invariant manifolds of libration point orbits" — manifold transfer-time statistics (direct ≈ 10 d, +10 d per loop, 5 d asymptotic phase, 0.25 km/s at 1000 km).
Liang, Xu & Xu, 2016, "The classification of cislunar trajectories and its applications in the Earth–Moon system," Astrophysics and Space Science 361:230 — Table 1 of minimum ToF vs. Jacobi energy; stepped-increase mechanism via stable-manifold/MPO intersections.
Parker & Anderson, 2013, Low-Energy Lunar Trajectory Design — typical ToF ranges for direct and low-energy transfers.
Sousa Silva, Terra & Ceriotti, 2018, "Fast Earth–Moon transfers with ballistic capture," Astrophys. Space Sci. 363:210 — time of permanence up to 90–120 d.
Zhao Hongqian et al., 2021, 基于动态规划的月面定点着陆快速制导方法 — time-to-go via the terminal Hamiltonian condition.