Integrators¶

Integrators advance spacecraft state through time under configured force models. Both integrators share a common interface and produce dense-output records.

IIntegrator Interface¶

Method	Description
`Initialize(StateVector)`	Set initial conditions and force model context
`IntegrateSegment(pos, vel, baseEpoch, duration, eventDetectors)`	Advance state, returning `IntegrationResult`

IntegrationResult contains the completed PropagationSegment and optional event information if a detector triggered during the segment.

AcceptedStep¶

Both integrators record AcceptedStep entries containing position, velocity, and acceleration at the start and end of each step. These records enable cubic Hermite interpolation for dense output between steps.

VVIntegrator¶

VVIntegrator is the default fixed-step symplectic integrator using the Velocity-Verlet scheme. CentralBodyPropagator creates one with the propagator's deltaT as step when no integrator is given.

Constructor	Description
`VVIntegrator(TimeSpan deltaT)`	Fixed step; forces are added by the propagator
`VVIntegrator(IEnumerable<ForceBase> forces, TimeSpan deltaT, StateVector initialState)`	Fixed step, forces and initial state given directly

Property	Description
`DeltaT`	The configured step (`TimeSpan`)
`DeltaTs`	The step in seconds

Symplectic: conserves energy over long durations for conservative systems.
Second order: the global error grows with the square of the step (see below).
Event detection occurs at step boundaries only (no sub-step refinement).

var integrator = new VVIntegrator(TimeSpan.FromSeconds(1.0));

RK78Integrator¶

Choosing an integrator

RK78Integrator is the default choice for high-fidelity work. Prefer VVIntegrator when a fixed step and symplectic energy behaviour matter more than accuracy per step.

RK78Integrator implements the Prince-Dormand 7(8) method: a 13-stage, 8th-order integrator with an embedded 7th-order error estimate for adaptive step control.

Adaptive Constructor¶

Parameter	Type	Default	Description
`absoluteTolerance`	`double`	`1e-9`	Absolute error tolerance
`relativeTolerance`	`double`	`1e-9`	Relative error tolerance
`initialStepSize`	`double`	`60.0`	Initial step size (s)
`minStepSize`	`double`	`1e-6`	Minimum allowed step size (s)
`maxStepSize`	`double`	`86400.0`	Maximum allowed step size (s)
`safetyFactor`	`double`	`0.9`	PI controller safety factor
`minFactor`	`double`	`0.2`	Minimum step-size reduction factor
`maxFactor`	`double`	`5.0`	Maximum step-size growth factor

Fixed-Step Constructor¶

var fixedStep = new RK78Integrator(fixedStepSize: 30.0);

When created with a fixed step size, AdaptiveMode is false and the integrator uses constant steps.

Properties¶

Property	Description
`AbsoluteTolerance`	Configured absolute tolerance
`RelativeTolerance`	Configured relative tolerance
`AdaptiveMode`	`true` for adaptive PI control, `false` for fixed step

Key Features¶

Adaptive PI step control: Adjusts step size to maintain error within tolerances.
Sub-step event refinement: Uses BisectionEventFinder to locate event times to ~1e-10 s precision within a step via Hermite dense output.
Accuracy: on the 24-hour conformance cases, 3.6 m (SSO) to 13.1 m (LEO) from the GMAT references with tolerances 1e-11, the rest being model differences between the two tools; see Validation.

// Adaptive (default)
var integrator = new RK78Integrator(
    absoluteTolerance: 1e-10,
    relativeTolerance: 1e-10,
    initialStepSize: 30.0);

// Fixed-step
var fixedIntegrator = new RK78Integrator(fixedStepSize: 10.0);

Accuracy And Cost¶

Velocity-Verlet is second order: halving the step divides the global error by four. Measured on 2026-10-04 on the LEO orbit of conformance case propagator_24h_leo_grav10_001:

Integrator	Two-body position error after 24 h	Time per orbit, EGM2008 10x10 + Moon + Sun
Velocity-Verlet, 0.5 s	69.9 m	21.0 ms
Velocity-Verlet, 1 s	279.5 m	10.5 ms
Velocity-Verlet, 2 s	1,118 m	about 5 ms or about 24 ms, bimodal (#346)
RK7(8), tolerances 1e-11	not measured in two-body; 13.1 m from the full-model reference	5.5 ms

Sources: errors from VVIntegratorTests.GlobalErrorIsSecondOrderInTheStep, against the analytic Keplerian solution; times from VVBenchmarks and RK78Benchmarks.LeoOneOrbit_EGM10_SunMoon (BenchmarkDotNet 0.14.0, medium run, AMD Ryzen 7 5800X, Fedora Linux 44, .NET 10). The step of VVIntegrator is also the output cadence of the propagator; RK7(8) outputs every 10 s here.

With the full force model, the 1 s Velocity-Verlet error on this case is 294 m, of which 280 m is truncation. For results below about ten meters, use RK78Integrator: on this case it is both more accurate and faster than Velocity-Verlet at 1 s.