Three-Body Gravitational Slingshot Dynamics

Duration: 1 year 5 monthsStarted: Sep 2024Completed: PresentCategory: astrophysics

Project Narrative

This project explores gravitational slingshot dynamics beyond the classical two-body approximation by modeling hyperbolic flybys in full three-body systems. I developed a unified analytical and numerical framework to quantify energy ceilings in close-in gravitational slingshots, combining closed-form hyperbolic scattering theory with large-scale three-body Monte Carlo simulation in a hot Jupiter system. Starting from first principles, together with my PI, Dr. David Kipping, we derived strict two-body kinematic limits and geometric vector plateaus, then I built a high-fidelity integrator to test how real trajectories behave in a deep stellar potential.

The results show that while no trajectory exceeds the stellar two-body energy ceiling, three-body dynamics can amplify energy gains far beyond classical planet-only bounds through structured angular momentum exchange consistent with conservation of the Jacobi constant. This project demonstrates rigorous theoretical derivation, invariant-based diagnostics, and large-scale numerical validation to uncover efficiency frontiers in multi-body orbital systems.

Analytical Foundations

Hyperbolic Scattering and Energy Ceilings

At the core of the project is a closed-form solution to two-body hyperbolic scattering. In the rest frame of a gravitating body, the asymptotic velocity magnitude is conserved:

|\mathbf{v}_{\infty}^{(+)}| = |\mathbf{v}_{\infty}^{(-)}| = v_{\infty}

The encounter only rotates the velocity vector by a deflection angle

\theta = 2 \arctan\!\left( \frac{\mu}{b v_{\infty}^2} \right)

where $\mu$ is the standard gravitational parameter and $b$ is the impact parameter.

Transforming back to the inertial frame yields a strict kinematic ceiling for an unpowered gravity assist:

\Delta V_{\max} = 2 v_{\infty}

This is the true kinematic ceiling and establishes the fundamental single-encounter speed gain limit, independent of numerical simulation.

Numerical Engine

Large-Scale Three-Body Monte Carlo Simulation

To explore behavior beyond idealized two-body theory, I built a high-fidelity three-body integrator modeling a starâ€“planetâ€“particle system. Over 24,000 trajectories were simulated using adaptive integration, randomized initial conditions, and strict physical filtering (unbound states, stellar clearance constraints, planetary encounter thresholds).

For each trajectory, I tracked:

Star-centric specific orbital energy
Planet-relative hyperbolic excess velocity
Angular momentum evolution
Velocity vector change magnitude

Energy Efficiency Frontiers

Scalar Speed vs. Vector Rotation Limits

Two distinct ceilings emerge:

Scalar speed gain ceiling

\eta_{\text{speed}} = \frac{\Delta{V}}{{v}_{\infty,\text{p}}} \leq 1

Vector change plateau

\eta_{\text{vector}} = \frac{|\Delta{V}|}{2\sqrt2{v}_{\infty,\text{p}}}

The first represents a physically meaningful energy frontier. The second reflects a purely geometric upper bound in Euclidean velocity space.

In close-in hot Jupiter systems, trajectories frequently approach the scalar ceiling and cluster near vector-efficiency plateaus, revealing near-optimal angular deflection geometries.

Breaking the Planet-Only Ceiling

Three-Body Amplification in a Deep Stellar Potential

Classical gravity assist analysis assumes the planet dominates locally. In a close-in system, this assumption fails: the stellar potential remains strong throughout the encounter.

Simulations demonstrate that three-body trajectories exceed the planet-only two-body energy ceiling by over two orders of magnitude, yet remain strictly below the stellar two-body bound.

The planet does not supply energy directly; rather, it acts as a geometric lever that redirects trajectories inside a deep stellar potential well, unlocking kinetic energy otherwise inaccessible in a purely two-body framework.

Invariant Diagnostics

Energyâ€“Angular Momentum Coupling via Jacobi Constant

To confirm physical consistency, results were analyzed in the framework of the Circular Restricted Three-Body Problem (CR3BP), where the conserved quantity is the Jacobi constant:

C = -2\Omega - |\dot{{r}}|^2

In dimensional form, inertial energy changes are coupled to angular momentum exchange:

\Delta\mathcal{E} \propto {\Omega} \cdot \Delta{L}

Monte Carlo results exhibit this correlation explicitly: large energy boosts correspond to large changes in angular momentum delivered by planetary torque.

This confirms that observed amplification arises from structured three-body dynamics rather than numerical artifacts.

Related Writings

Gravitational Slingshot Dynamics in Three-Body Systems

Writing Tab

Technologies & Tools

PythonSymplectic N-body IntegratorNumPySciPyMatplotlibMCMC (emcee)Monte Carlo SamplingOrbital Mechanics