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Uncertainty#

Two studies repeat a calculation many times with inputs drawn at random: the engine uncertainty on the Analysis page and the flight dispersion on the Flight page. Both are only as meaningful as the distributions you give them.

Distributions#

Quantities that must stay positive (thrust, Isp, burn time, drag coefficient, chamber pressure, throat area) are multiplied by a lognormal factor with mean one and the stated coefficient of variation \(c\):

\[ \ln X\sim\mathcal N\!\left(-\tfrac12\sigma^2,\ \sigma^2\right),\qquad \sigma^2=\ln(1+c^2). \]

For small \(c\) this is indistinguishable from a normal distribution; for large \(c\) it never produces a negative thrust.

Other inputs:

Input Distribution
Gamma (engine) Normal, truncated to \(1<\gamma<2\)
Rail elevation Normal about the nominal value
Fin misalignment Uniform in \([-\delta,\delta]\)
Wind speed Normal, truncated at zero
Wind direction Normal about the mean, or uniform over the compass

The inputs are independent. If yours are correlated, thrust and burn time for a given total impulse for instance, the spread will be overstated.

Engine uncertainty#

Chamber pressure, throat area and gamma are drawn around the last engine result and pushed through the constant-\(\gamma\) nozzle relations with the chamber temperature and molar mass held fixed. The output is the distribution of thrust and Isp. With a minimum thrust set, the share of samples above it is reported as reliability.

Samples where the nozzle cannot be evaluated are counted as failed and left out of the statistics; they are never replaced with the nominal value. The result states attempted, valid and failed counts.

Flight dispersion#

Each case flies the full six-degree-of-freedom model with its own thrust, Isp, burn time, drag coefficient, rail elevation, fin misalignment and wind. From the cases that land before the time limit:

  • mean impact point and the sample standard deviation (\(N-1\)) in east and north;
  • CEP, the median distance of the impacts from the mean impact point;
  • confidence ellipse from the eigenvectors of the impact covariance matrix, with semi-axes \(\sqrt{\lambda_i\,\chi^2_2(P)}\) at confidence \(P\), 99.7 % by default.

Apogee statistics use every case, landed or not.

A fixed seed reproduces the study exactly, in one process or many.

How many runs#

The mean converges quickly; the tails do not. As a rule of thumb, fifty cases give a usable mean and standard deviation, a few hundred are needed before the 99.7 % ellipse settles. Run the study twice with different seeds: if the ellipse changes noticeably, run more.

What this is not#

These studies propagate the scatter you put in. They do not account for the model being wrong (constant drag coefficient, frozen nozzle flow, the atmosphere on the day) and they do not infer distributions from data. A study that will be relied on should record where each distribution came from.

Sources#

  • NASA-STD-7009B, Standard for Models and Simulations, 2024.
  • NASA-HDBK-7009, Handbook for Models and Simulations, 2013.