Models

Draft

A model is a deliberate simplification. Recording one means recording what was simplified away, because that is what determines when the model can be used and when it cannot.

Structure

Field	Content
Purpose	What question the model answers
Governing equations	The mathematics, stated fully
Assumptions	Every simplification, with its justification
Validity range	The parameter range where the assumptions hold
Inputs	Parameters, their units, sources and uncertainties
Calibration	Data used to fit any free parameters
Validation	Independent data the model was tested against
Known failures	Where it is known to be wrong, and by how much
Alternatives	Other models for the same problem, and when to prefer them

The known failures field is the one that distinguishes a useful model record from a description. Every model is wrong somewhere, and recording where is more valuable than recording where it works.

Assumptions and validity

The two are the same thing stated differently. An assumption defines a boundary, and outside it the model gives answers that are wrong without indicating that they are.

Model	Key assumption	Fails when
Two-body orbital motion	Point masses, no other forces	Over days in low orbit; near a third body; under thrust
Clohessy-Wiltshire relative motion	Circular reference orbit, small separation	Eccentric reference orbit; separations beyond tens of km
Exponential atmosphere	Constant scale height	Across large altitude ranges; during geomagnetic storms
Ideal rocket nozzle	Isentropic, frozen composition	Near flow separation; with significant boundary layer
Blackbody radiator	Grey surface, uniform temperature	Wavelength-dependent surfaces; large gradients
Linear control design	Small perturbations about an operating point	Large slews; actuator saturation

The pattern is that models fail gracefully in appearance and severely in substance. A two-body propagation of a low orbit produces a perfectly reasonable-looking trajectory that is kilometres wrong after a day.

Calibration and validation

These are different operations and conflating them is a common error.

Calibration fits free parameters to data. Validation tests the calibrated model against different data it has not seen.

A model validated on its calibration data has demonstrated only that the fitting worked. This distinction is the same one made in Thermal control and Structures and mechanisms, where models are correlated against test data and then used predictively.

The loop back from known failures is the part that makes a model record cumulative. Each time the model is found to be wrong somewhere, the validity range narrows and the record improves.

Fidelity

Higher fidelity is not automatically better. A model should be as simple as the question allows.

Question	Appropriate fidelity
Is this mission roughly feasible?	Two-body, impulsive burns, order-of-magnitude budgets
What is the delta-v to within a few per cent?	Patched conics, or a simple perturbed propagation
Where will the spacecraft be next week?	Full force model, numerically integrated
Will it collide with a tracked object?	High-fidelity propagation with covariance

A high-fidelity model applied to a question that needs a rough answer wastes effort and, worse, produces false confidence. Six significant figures from a model whose inputs are known to one is a presentation problem that becomes a reasoning problem.

Preferring analytical models

Where an analytical model exists, it is usually worth having alongside a numerical one.

An analytical expression shows how the result depends on each input, which a numerical result does not. Seeing that radiator area scales as $T^{-4}$ conveys something no table of computed values does, and it means the scaling can be applied without rerunning anything.

The numerical model then serves to check the analytical one and to handle the cases where its assumptions fail.

References

Models cite their own sources. This page is a convention.