While standard Kalman-based filters, Gaussian assumptions, and covariance-weighted metrics are very effective in data-rich tracking environments, their use in the data-sparse environment of space surveillance is more limited. To properly characterize non-Gaussian density functions arising in the problem of long-term propagation of state uncertainties, a Gaussian sum filter adapted to the two-body problem in space surveillance is proposed and demonstrated to achieve uncertainty consistency. The proposed filter is made efficient by using only a onedimensional Gaussian sum in equinoctial orbital elements, thereby avoiding the expensive representation of a full six-dimensional mixture and hence the "curse of dimensionality." Additionally, an alternate set of equinoctial elements is proposed and is shown to provide enhanced uncertainty consistently over the traditional element set. Simulation studies illustrate the improvements in the Gaussian sum approach over the traditional unscented Kalman filter and the impact of correct uncertainty representation in the problems of data association (correlation) and anomaly (maneuver) detection. Nomenclature A= lower-triangular Cholesky factor of a covariance matrix a = semimajor axis a pert = perturbing acceleration a; h; k; p; q; ' = equinoctial orbital elements c = parameter controlling the accuracy of the Gaussian sum filter f = system dynamics vector G = process noise shape matrix h = measurement function vector k = (subscript) time index '= mean longitude N = number of mixture components N = Gaussian probability density function n = mean motion n; h; k; p; q; '= alternate equinoctial orbital elements P = covariance of a Gaussian distribution PE = prediction error p = probability density function Qt = process noise covariance matrix R k = measurement noise covariance matrix r = Cartesian Earth-Centered Inertial (ECI) position coordinates _ r = Cartesian ECI velocity coordinates r = Cartesian ECI acceleration coordinates t 0 , t = initial and current times u = orbital element coordinates (sixdimensional) fv 1 ; . . . ; v k g = Gaussian white noise sequence wt = Gaussian white noise process w ; w ; . . . = mixture weights x = dynamic state vector Z k fz 1 ; . . . ; z k g = measurement sequence ; ; . . . = (subscripts) indices of mixture components kj = Kronecker delta symbol = mean of a Gaussian distribution = Earth gravitational constant (398600:4418 km 3 =s 2 ) = standard deviation of a univariate Gaussian distribution = inverse solution flow r x = gradient operator with respect to x (column oriented)

While orbital propagators have been investigated extensively over the last fifty years, the consistent propagation of state covariances and more general (non-Gaussian) probability densities has received relatively little attention. The representation of state uncertainty by a Gaussian mixture is well suited for problems in space situational awareness. Advantages of this approach, which are demonstrated in this paper, include the potential for long-term propagation in data-starved environments, the capturing of higher-order statistics and more accurate representation of nonlinear dynamical models, the ability to make the filter adaptive using real-time metrics, and parallelizability. Case studies are presented establishing uncertainty consistency and the effectiveness of the proposed adaptive Gaussian sum filter.

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While standard Kalman-based filters, Gaussian assumptions, and covariance-weighted metrics work remarkably well in data-rich tracking environments such as air and ground, their use in the data-sparse environment of space surveillance is more limited. In order to properly characterize non-Gaussian density functions arising in the problem of long term propagation of state uncertainties in the two-body problem, a framework for a Gaussian sum filter is described which achieves uncertainty (covariance) consistency and an accurate approximation to the Fokker-Planck equation up to a prescribed accuracy. The filter is made efficient and practical by (i) using coordinate systems adapted to the physics (i.e., orbital elements), (ii) only requiring a Gaussian sum to be defined along one of the six state space dimensions, and (iii) the ability to initially select the component means, covariances, and weights by way of a lookup table generated by solving an offline nonlinear optimization problem. The efficacy of the Gaussian sum filter and the improvements over the traditional unscented Kalman filter are demonstrated within the problems of data association and maneuver detection.

We reexamine the accuracy of the Landau-Zener approximation, whose analytic simplicity has made it often cited and applied over the past 70 years in describing nonadiabatic transitions. We find that this approximation may be in substantial disagreement with exact numerical results for a model physical system and a wide range of coupling strengths and relative velocities. We conclude that the Landau-Zener approximation should be used with caution for quantitative estimates of nonadiabatic transition probabilities or cross sections.