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Nechvile-Transformed Spacecraft Dynamics and Propellant Computation in the 3-Body Problem

Michael J. Dixon, Isaac M. Ross

arXiv:2608.29271v1math.OCeess.SYmath.NA

TL;DR

The paper addresses whether standard force and propellant formulations remain valid after Nechvile transformation in the elliptic restricted three-body problem. It derives the transformed force and mass-flow structure, develops corresponding cost and rocket-equation modifications, and reports substantial differences for cislunar mission analysis. The paper also notes that impulse-location effects complicate interpreting attenuation as a universally optimal placement rule.

  • Problem

    Standard rocket-equation and propellant calculations appear applicable to the ER3BP, but the paper identifies a need to revisit them after Nechvile transformation.

  • Method

    The paper combines Nechvile-transformed dynamics with a modified mass-flow model and an engine-agnostic L1-based propellant functional.

  • Results

    The transformed rocket equation depends on true anomaly, and impulse-location effects can attenuate or amplify effective Delta-V relative to the circular case.

  • Takeaways & Limitations

    Nonzero eccentricity should not be ignored in early cislunar mission design because it can produce dramatically different optimized trajectories and may provide pathways to lower propellant consumption.

  • Takeaways & Limitations

    Impulse-location attenuation alone does not determine the propellant-optimal maneuver location once complete mission boundary conditions are imposed.

Abstract

from arXiv · show

The uncontrolled equations of motion in the Nechvile frame for the restricted three-body problem have been well-known since at least the 1960s. It would seem that adding an external force to these equations is quite trivial: simply add an external force per mass term to the acceleration equations. Here we show that the last statement is not true. In fact, we show that the additive generic external force must be multiplied by the inverse of $(1 + e \cosθ)^3$ where $e$ is the relative eccentricity of the primaries and $θ$ is the true anomaly of the rotating frame located at the barycenter. Furthermore, when this result is combined with the mass flow rate equation, it generates several surprising results due to the mismatch between the resulting quadratic term and the cubic term in the equations of motion. This leads to a corresponding modification of the rocket equation itself. A Birkhoff-theoretic solution to an illustrative cislunar space mission problem shows propellent savings of 80% with the use of the correct cost functional. The popular quadratic cost utilizes more than $2X$ the minimum propellant consumption.

I. Introduction

The paper shows that standard propellant and motion formulations require revision in the elliptic restricted three-body problem because Nechvile’s transformation introduces eccentricity- and anomaly-dependent scaling. These modifications affect thrust, mass flow, Delta-V, control-space modeling, and mission design.

  • Motivation: Standard rocket mass-flow integration and Tsiolkovsky calculations do not directly carry over to the ER3BP equations examined here.The paper revisits these foundations because the widely used ER3BP equations require a different treatment.
  • Transformation effects: Nechvile’s transformation changes time to true anomaly and makes the rotating frame pulsating while keeping the Lagrange points relatively stationary.The associated thrust term is scaled by the inverse of a cubic factor involving eccentricity and true anomaly.
  • Propellant model: The rocket mass-flow equation uses an inverse quadratic factor, creating a mismatch with the cubic factor in the transformed equations of motion.This discrepancy requires modified propellant-consumption calculations and changes the rocket equation.
  • Optimization: The modified L1 cost functional and Delta-V calculation account for the discrepancy between mass flow and transformed dynamics, whereas popular quadratic costs are described as erroneous proxies for propellant consumption.The classic Delta-V computation also acquires a linear (1 + e cos θ) term.
  • Mission implications: Ignoring the transformed-force and mass-flow nuances can increase propellant consumption or cause mission failure in a numerically illustrated cislunar mission.Artificially boxing the control space can limit a practical rocket to less than 75% of its maximum capability when e ≃ 0.05.
  • Derivation scope: The paper derives the spacecraft model from first principles because adding external forces to uncontrolled ER3BP equations is more nuanced than simply appending control terms.The derivation follows Szebehely’s framework while using standard inertial-to-rotating-frame coordinate transformations.

B. Basic Equations of Motion

The basic ER3BP model describes a spacecraft relative to two primaries rotating about their barycenter, with gravity, rotating-frame kinematics, and a spacecraft-only external force.

  • Coordinate setup: The spacecraft position is expressed in a local rotating frame centered at the barycenter, while the primaries lie on the local x axis.The position vector is represented by the local-frame coordinates (x̄, ȳ, z̄).
  • Forces: The spacecraft experiences Newtonian inverse-square gravitational forces from the primaries and a generic external force acting only on the spacecraft.The external-force components in the local frame are Fx, Fy, and Fz.
  • Equations of motion: The rotating-frame equations include angular-acceleration, Coriolis, and centrifugal terms in the spacecraft acceleration components.The displayed component equations contain terms involving ω̇, ω, and ω^2.

C. Nechvile’s Transformation

Nechvile’s transformation rescales spacecraft position and replaces time with true anomaly, producing a rotating and pulsating frame in which the primaries remain fixed.

  • Transformation: Nechvile’s transformation scales the spacecraft position components and changes the independent variable from time to true anomaly.This transformation is applied to the local rotating-frame formulation.
  • Frame properties: The transformed frame both rotates and pulsates, while the primaries occupy fixed normalized positions (−µ, 0, 0) and (1−µ, 0, 0).In the non-pulsating rotating frame, the primaries instead pulsate nonuniformly with orbital frequency ω.
  • Derivation: The transformation uses the primaries’ relative equation of motion and angular-momentum conservation to eliminate first- and second-time derivatives of their separation.This simplifies the transformed left-hand side of the spacecraft equations.
  • Derivation: The semilatus rectum p and angular momentum h provide an additional relation used in simplifying the transformed dynamics.This relation supports the reduction of the transformed equations after the derivative-elimination step.

D. The Transformed Equations of Motion

The transformed equations are organized through the ER3BP pseudo-potential and reveal that external forces acquire a cubic eccentricity-dependent scaling. This scaling becomes consequential whenever the primaries’ orbit is noncircular.

  • Equation construction: The transformed equations are obtained by substituting the simplified transformed acceleration into the spacecraft equations and rewriting the result using the primary separation relation.The derivation also rewrites the centrifugal contribution before combining the component equations.
  • Pseudo-potential: The pseudo-potential function provides a concise representation of the gravitational and centrifugal forces in the transformed model.The transformed component equations are written more succinctly using this function and the constant AC.
  • External forcing: Adding an external force produces a cubic multiplicative term in the transformed equations, which becomes unity when e = 0.For nonzero eccentricity, the term has far-reaching consequences for the dynamics.

E. Constant and Nechvile Units

The paper defines constant and Nechvile units for distance, time, frequency, and acceleration, while noting that Nechvile coordinates describe pulsating rather than physical motion. These units preserve relative stationarity of Lagrange points but make the Nechvile distant unit and frequency time-varying.

  • The constant distant unit DC is defined consistently with the semilatus rectum.
  • The Nechvile distant unit DN is time-varying.
  • The Nechvile frequency ωN is time-varying and defines the derived Nechvile unit of time τN.
  • Nechvile coordinates describe pulsating spacecraft motion that differs from physical motion in nonpulsating rotating coordinates.The L2 point is fixed in the pulsating representation but not in the nonpulsating rotating frame.
  • The paper derives constant and Nechvile units of acceleration from the distance and time transformations.

III. ER3BP Models For a Thrusting Spacecraft

The paper formulates thrusting-spacecraft dynamics in Nechvile variables by transforming the rocket mass-flow relation and augmenting the transformed equations of motion. The resulting Model A exposes a mismatch between quadratic mass-flow scaling and cubic dynamical scaling.

  • The rocket mass-flow equation uses the ℓp-norm of thrust together with specific impulse Isp and sea-level gravitational acceleration g0.
  • The mass-flow equation is transformed from time to true anomaly before being combined with the Nechvile equations.
  • Model A is obtained by introducing a dimensionless spacecraft mass variable and augmenting the transformed mass equation with the equations of motion.
  • Model A contains a discrepancy between the quadratic term in the rocket mass-flow equation and cubic terms in the remaining equations of motion.The paper identifies this mismatch as having far-reaching consequences.

A. Models A Versus B

Model B removes the cubic thrust factor from the displayed dynamics but transfers its effects into a cosine-dependent mass-flow relation and a time-varying control space. Model eB restores a time-invariant control space by imposing an artificial thrust constraint, substantially reducing usable thrust capacity.

  • A. Models A Versus B: Model B scales thrust by the cubic Nechvile factor, while its mass-flow equation retains the linear term (1 + e cos θ).
  • A. Models A Versus B: Models A and B are mathematically and physically equivalent, but their bounded-thrust implementations have different practical consequences.The finite engine limit is Tmax.
  • A. Models A Versus B: Model B has a time-varying control space, whereas Model A has a time-invariant control space.The independent variable is true anomaly, used interchangeably with time in this discussion.
  • A. Models A Versus B: Model B’s dynamical simplicity does not provide a computational advantage because it shifts the difficulty into time variability of the control space.
  • B. Model eB as an Alternative to Model B: Model eB retains Model B dynamics but imposes Model A’s time-invariant control space.
  • B. Model eB as an Alternative to Model B: A constant transformed thrust limit can exceed the feasible actual thrust Tmax, including at θ = 0 and over half an orbit.
  • B. Model eB as an Alternative to Model B: Model eB artificially restricts actual thrust below Tmax for most true-anomaly values and most severely at θ = π.
  • B. Model eB as an Alternative to Model B: In cislunar space, Model eB utilizes less than 75% of the engine’s thrust capability.

C. Introducing Model C: An Engine-Agnostic Dynamical Model

Model C represents propellant consumption with an engine-agnostic control formulation based on an ℓp variant of the L1 norm. The resulting proxy accounts for finite-burn effects while avoiding dependence on a particular engine.

  • Model C formulation: Model C uses a control vector u as a thrust proxy, with bounds defined by the dynamical model.The components ux, uy, and uz serve as control variables, and u is bounded according to the model constraints.
  • Propellant model: The proposed propellant model is the L1 norm of the time-varying ℓp norm of u(t), with p ∈ {1, 2, ∞}.The choices p = 1 and p = 2 correspond to common thruster configurations, while p = ∞ is also considered mathematically.
  • Propellant proxy: ∆prox is an engine-agnostic proxy for propellant consumption rather than the actual propellant amount.Actual consumption ∆prop is computed by integrating the rocket mass flow rate equation, while ∆prox measures Isp-agnostic consumption.
  • Propellant proxy: ∆prox incorporates the transformed ER3BP weighting and has nondimensional velocity units that account for finite-burn losses.For p = 1 it corresponds to six orthogonally arranged engines, whereas p = 2 corresponds to a single engine.
  • Cost functional choice: Quadratic cost functionals can produce non-fuel-optimal solutions with propellant consumption up to 50% above the optimum.The paper states that elementary control-variable transformations can avoid this issue while providing continuously differentiable formulations.

IV. A Modified Formula for Impulsive ∆V Computation

The paper modifies impulsive ∆V computation for the ER3BP by incorporating true-anomaly-dependent effects into the rocket equation and mission cost. These effects disappear in the circular problem but can alter impulse sizing, placement, and mission-level optimization in the elliptic case.

  • A. An ER3BP-Modification to the Rocket Equation: The modified rocket equation is derived by retaining only thrust and second-derivative terms from the ER3BP dynamics.The derivation follows a textbook rocket-equation approach and then integrates under the infinitesimally short-burn assumption.
  • A. An ER3BP-Modification to the Rocket Equation: The resulting ∆V uses nondimensional velocity units and depends on the circular speed vC and the impulse application anomaly θI.The mass change is ∆m = m(θ0)−mf(θf), while θI denotes the impulse location.
  • B. Discussion of Eq. (56): When e = 0, the true-anomaly dependence vanishes and the modified expression reduces to Tsiolkovsky’s formula after multiplication by vC.Thus, the CR3BP does not expose the anomaly-dependent modification present in the ER3BP.
  • B. Discussion of Eq. (56): At nonzero eccentricity, apoapse produces the largest attenuation and periapse the largest amplification of the effective ∆V.No attenuation or amplification occurs when the impulse is applied at the semilatus rectum, θI = π/2.
  • B. Discussion of Eq. (56): The mission ∆V must weight each impulse by its application anomaly rather than treating all impulse locations equivalently.The individual impulse locations θi and magnitudes ∆Vi enter the mission-level cost.
  • B. Discussion of Eq. (56): Using the circular-problem cost with e = 0 in an ER3BP can overestimate or underestimate total ∆V and produce incorrect impulse numbers, locations, or magnitudes.The error affects the mission solution, not merely the reported total cost.
  • B. Discussion of Eq. (56): The anomaly-dependent impulse factor cannot be optimized in isolation because mission boundary conditions may require impulses at otherwise unfavorable locations.Apoapse attenuation does not guarantee global mission optimality when the complete set of boundary conditions is enforced.

C. Connections Between ∆prop, ∆prox and ∆V

The paper relates exact propellant consumption to engine-agnostic cost measures in the ER3BP, where the quadratic cosine correction changes both optimized trajectories and their control profiles. A cislunar example shows that omitting this correction or using a quadratic cost substantially increases propellant consumption.

  • Connections Between ∆prop, ∆prox and ∆V: ∆prop is the exact propellant measure, but its Isp dependence motivates engine-agnostic mission design using ∆prox.The L1 formulation can account for finite-burn losses without selecting a propulsion system.
  • Connections Between ∆prop, ∆prox and ∆V: The three cost measures compared for identical mission boundary conditions are a quadratic functional, an L1 functional without the cosine term, and an L1 functional with it.The optimization uses six bounded thrusters and a Birkhoff-theoretic trajectory-optimization implementation.
  • Connections Between ∆prop, ∆prox and ∆V: The quadratic cosine term substantially changes the optimized trajectory even though it is only a correction in the cost functional.The J1 and ∆prox trajectories differ substantially, while J1 and JQ appear close despite using incorrect propellant measures.
  • Connections Between ∆prop, ∆prox and ∆V: 81% excess propellant results from omitting the quadratic cosine term, while the quadratic cost incurs 114% excess.These percentages apply to the cislunar mission trajectories reported in Table 1.
  • Connections Between ∆prop, ∆prox and ∆V: The reported propellant values apply only to the illustrated mission, although J1- and JQ-optimized trajectories are theoretically no better than ∆prox.Different mission scenarios can produce different absolute values.
  • Connections Between ∆prop, ∆prox and ∆V: The control profiles differ across the three trajectories, including an extra impulse in the #3 direction for J1 relative to ∆prox.JQ produces continuously modulated control, whereas J1 and ∆prox have substantially different control solutions.

VI. Conclusions

The conclusions identify the cubic-quadratic mismatch created by Nechvile’s transformation as central to ER3BP trajectory-optimization behavior. They emphasize that even small nonzero eccentricities can produce large changes and may offer routes to lower propellant consumption.

  • VI. Conclusions: The cubic-quadratic mismatch in ER3BP thrust terms is identified as the main mathematical source of disrupted trajectory-optimization behavior.The effect is linked to adding the mass-flow equation to Nechvile’s equations and to pulsation.
  • VI. Conclusions: Small eccentricities can have big impacts in trajectory optimization, so cislunar mission design should not ignore them.The conclusion applies to cislunar trajectories because Earth-Moon eccentricity is nonzero.
  • VI. Conclusions: Earth-Moon eccentricity may be harnessed to lower propellant consumption through new pathways.This conclusion is stated within the scope of cislunar space trajectories.
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