Landing footprint

Mathematical explanation

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{|\Sigma|}} \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\in0,2\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.

References

References

1. Lakdawalla, Emily. (13 May 2008). "Landing Ellipse". The Planetary Society.
2. Tooley, Jeff. (2006-08-21). "Stardust Entry: Landing and Population Hazards in Mission Planning and Operations". American Institute of Aeronautics and Astronautics.
3. Golombek, M.. (2017-10-01). "Selection of the InSight Landing Site". Space Science Reviews.