Derivative analysis and algorithm modification of the transverse- eccentricity-based Lambert's problem

The classical Lambert's problem can be parameterized and solved through the transverse eccentricity component. A further study is conducted to calculate the derivative of the transverse-eccentricity-based Lambert's problem and to modify its algorithm. Results show that the derivative of a direct Lambert's problem is positive and continuous, which verifies that the transfer-time monotonically increases with the transverse eccentricity; however, the derivative of a multi-revolution Lambert's problem increases from negative to positive, indicating that the transfer-time decreases to the minimum firstly, and then increases to infinity. The original solution algorithm is promoted by introducing the analytic derivative. Numerical simulations for different cases show that, compared with the two existing transverse-eccentricity-based methods, the average computational time cost decreases by 65.5% and 39.8%, respectively.