Inaccurate estimates of the thermospheric density are a major source of error in low Earth orbit prediction. Therefore, real‐time density estimation is required to improve orbit prediction. In this work, we develop a dynamic reduced‐order model for the thermospheric density that enables real‐time density estimation using two‐line element (TLE) data. For this, the global thermospheric density is represented by the main spatial modes of the atmosphere and a time‐varying low‐dimensional state and a linear model is derived for the dynamics. Three different models are developed based on density data from the TIE‐GCM, NRLMSISE‐00, and JB2008 thermosphere models and are valid from 100 to maximum 800 km altitude. Using the models and TLE data, the global density is estimated by simultaneously estimating the density and the orbits and ballistic coefficients of several objects using a Kalman filter. The sequential estimation provides both estimates of the density and corresponding uncertainty. Accurate density estimation using the TLEs of 17 objects is demonstrated and validated against CHAMP and GRACE accelerometer‐derived densities. The estimated densities are shown to be significantly more accurate and less biased than NRLMSISE‐00 and JB2008 modeled densities. The uncertainty in the density estimates is quantified and shown to be dependent on the geographical location, solar activity, and objects used for estimation. In addition, the data assimilation capability of the model is highlighted by assimilating CHAMP accelerometer‐derived density data together with TLE data to obtain more accurate global density estimates. Finally, the dynamic thermosphere model is used to forecast the density.

A novel low-thrust trajectory design method is developed based on the velocity hodograph of a spacecraft. The trajectory design is done by shaping the velocity components during the transfer. For this purpose, velocity functions are used that consist of a sum of simple base functions. These base functions can be integrated analytically, such that the change in position can be obtained analytically. Doing so, the departure and rendezvous conditions on position and velocity can be solved very easily without the need of iterative computations. Extra parameters can be added to make the transfer design more flexible and optimize the trajectory. Two different methods have been developed: one that shapes the velocity as a function of time and another one that shapes as a function of the polar angle. To obtain minimum-ΔV trajectories, the free parameters in the velocity functions have been optimized and the search for the optimal departure date and time of flight is done by stepping through the flight window using a grid. Both hodographic-shaping methods have been tested for missions to Mars, the near-Earth asteroid 1989ML, comet Tempel 1, and Mercury and have been compared with results of other shape-based methods. Nomenclature c i = coefficient in velocity function, m∕s;m∕rad;s∕rad f = thrust acceleration, m∕s 2 K = intermediate vector L = intermediate vector N = number of complete revolutions n = number of base functions P = position, m R = radial-velocity function, m∕rad r = radial distance, m s = distance from the Sun, m T = time-evolution function, s∕rad t = time, s V = velocity, m∕s v i = velocity base function Z = axial-velocity function, m∕rad z = axial distance, m θ = polar angle, rad ψ = transfer angle, rad Subscripts 0 = initial value f = final value r = radial component z = axial component θ = transverse component Superscripts T = transpose of a vector _ □= derivative with respect to time t □ = second derivative with respect to time t □ 0 = derivative with respect to polar angle θ □ 0 0 = second derivative with respect to polar angle θ□ = integral with respect to time t □ = integral with respect to polar angle θ

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Thermospheric density is estimated using a reduced-order density model and radar 7 range and range-rate tracking data and GPS measurements. 8 • The estimated densities are validated against accurate SWARM density data. 9 • The obtained densities are significantly more accurate than NRLMSISE-00 and 10 JB2008 modelled densities and TLE-estimated densities.