To create a landing footprint for a spacecraft, the standard approach is to use the
Monte Carlo method to generate
distributions of
initial entry conditions and atmospheric parameters, solve the reentry
equations of motion, and catalog the final
longitude/
latitude pair (\lambda,\phi) at
touchdown. It is commonly assumed that the resulting distribution of landing sites follows a
bivariate Gaussian distribution: :f(x) = \frac{1}{2\pi\sqrt} \exp\left[ -\frac{1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu) \right] where: • x = (\lambda,\phi) is the vector containing the longitude/latitude pair • \mu is the
expected value vector • \Sigma is the
covariance matrix • |\Sigma| denotes the
determinant of the covariance matrix Once the parameters (\mu,\Sigma) are
estimated from the numerical simulations, an ellipse can be calculated for a
percentile p . It is known that for a real-valued vector x\in\mathbb{R}^{n} with a multivariate Gaussian joint distribution, the square of the
Mahalanobis distance has a
chi-squared distribution with n degrees of freedom: :(x-\mu)^{T}\Sigma^{-1}(x-\mu) \sim \chi_{n}^{2} This can be seen by defining the vector z = \Sigma^{-1/2}(x-\mu) , which leads to Q = z_{1}^{2}+\cdots+z_{n}^{2} and is the definition of the chi-squared statistic used to construct the resulting distribution. So for the bivariate Gaussian distribution, the boundary of the ellipse at a given percentile is z^{T}z = \chi_{2}^{2}(p) . This is the
equation of a circle centered at the origin with radius \sqrt{\chi_{2}^{2}(p)} , leading to the equations: :z = \sqrt{\chi_{2}^{2}(p)} \begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix} \Rightarrow x(\theta) = \mu + \sqrt{\chi_{2}^{2}(p)} \Sigma^{1/2} \begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix} where \theta\in[0,2\pi) is the angle. The
matrix square root \Sigma^{1/2} can be found from the
eigenvalue decomposition of the covariance matrix, from which \Sigma can be written as: :\Sigma = V \Lambda V^{T} \implies \Sigma^{1/2} = V \Lambda^{1/2} V^{T} where the
eigenvalues lie on the diagonal of \Lambda . The values of x then define the landing footprint for a given level of confidence, which is expressed through the choice of percentile. == See also==