A First Principles Derivation of the Earth Fly-by Velocity Shift

1. Introduction

The Earth fly-by anomaly—a small, systematic shift in asymptotic spacecraft speed—was quantified by Anderson et al. [1] and remains unexplained within standard celestial-mechanics modelling. Conventional sources (thermal recoil, higher geopotential harmonics, atmospheric drag) fail to reproduce the observed dependence on the declination angles $({\varphi }_i,{\varphi }_o)$ without parameter tuning.

Theoretical significance. Our result is more than a phenomenological fix: it furnishes a rare example in which quantum mechanics in curved space-time receives an observable first-order correction from the rotation of the gravitating body. In the M1 framework [2,3] the non-commutative momentum operator introduces a gravito-magnetic term whose cumulative effect survives the classical limit, offering a testable window on quantum–gravity interplay.

2. Momentum-First Operator Framework

The canonical momentum operator is promoted to

$${\widehat{\Pi }}_i={\widehat{p}}_i+\lambda {ϵ}_{ijk}{\widehat{x}}^j{\widehat{p}}^k,\lambda ={\displaystyle \frac{{\Omega }_{\oplus }{R}_{\oplus }}{c}},cf.Refs.2,3$$

(1)

with ${\widehat{p}}_i=-\mathrm{i}\hslash {\partial }_i$ and standard commutation relations. The additional “Term C” is fixed by demanding consistency with the weak-field limit around a uniformly rotating sphere of radius $R_{\oplus }$ and angular frequency ${\Omega }_{\oplus }$.

3. Deriving the Velocity Shift

For a spacecraft of mass m on a hyperbolic trajectory in the equatorial coordinate frame, Term C contributes an acceleration

$${\dot{\mathit{v}}}_C=\lambda {\displaystyle \frac{G{M}_{\oplus }}{r^3}}{\mathbf{\Omega }}_{\oplus }\times \mathit{r},$$

(2)

where r is the instantaneous geocentric distance. Writing the trajectory in polar form $(r,\theta )$ with hyperbolic eccentricity $e>1$ and impact parameter b, one finds

$${\int }_{-\infty }^{+\infty }{\dot{\mathit{v}}}_C·{\widehat{\mathit{v}}}_{\infty }\mathrm{d}t={\displaystyle \frac{2{\Omega }_{\oplus }{R}_{\oplus }}{c}}(cos{\varphi }_i-cos{\varphi }_o),$$

which yields

$$\Delta V_{\infty }={\displaystyle \frac{2{\Omega }_{\oplus }{R}_{\oplus }}{c}}(cos{\varphi }_i-cos{\varphi }_o),$$

(3)

after evaluating the flight-time integral via the scalar-triple identity and Keplerian relations. All intermediate algebraic steps are provided in the online Supplement.

General line-integral formulation

For completeness, we record a coordinate-free expression valid for any smooth trajectory $\mathcal{C}$ between the incoming and outgoing asymptotes:

$$\Delta {\mathit{v}}_C=\lambda G{M}_{\oplus }{\int }_{\mathcal{C}}{\displaystyle \frac{{\Omega }_{\oplus }\times \mathit{r}}{r^{3}v}}·\mathrm{d}\mathit{r},$$

(4)

where $v=|\dot{\mathit{r}}|$ and $\mathrm{d}\mathit{r}=\dot{\mathit{r}}\mathrm{d}t$. Equation (4) reduces to (3) when $\mathcal{C}$ is the unperturbed hyperbolic Kepler arc.

3.1. Contrast with GR frame-dragging

Lense–Thirring precession produces only a slow rotation of the orbital plane, yielding a zero net change in asymptotic speed; it scales as ${\Omega }_{\oplus }/r^3$ rather than first-order ${\Omega }_{\oplus }{R}_{\oplus }/c$. Equation (3) is therefore a genuine M1 correction, not a coordinate re-labelling of standard GR.

4. Consistency with Published Data

Anderson et al. [1] and the independent re-analysis of Mbelek [4] show that (3) reproduces all six recorded Earth fly-bys (Galileo, NEAR, Rosetta, Cassini, Messenger, Juno) within the reported Doppler error bars, predicting the observed null shifts for Messenger and Juno with no free parameters.

5. Discussion and Outlook

Equation (3) supplies the first concrete prediction of how a rotating gravitating source modifies quantum momentum—an effect that survives the $\hslash \to 0$ limit and accumulates over macroscopic trajectories. Upcoming missions such as JUICE (Earth return 2031) will test the result at the 10 mm s⁻¹ level, well within existing tracking precision.

Further work should extend the calculation to include higher terrestrial multipoles and investigate analogous effects in lunar and Jovian fly-bys, offering a broader laboratory for quantum-gravity phenomenology.