Orbital dynamic admittance and earth shadow

The effects of orbital dynamics on a velocity distribution from a point source --- for example, a fragmentation --- may be studied by the Lambert -- transformation of variables technique introduced in the paper that this video accompanies. This technique permits the computation of a spatial density due to the velocity distribution as the sum over all routes between the source point and any point in space of the velocity density divided by the absolute value of the Jacobian determinant of the propagation map. The determination of all routes constitute finding all Lambert (orbital two point boundary value problem) solutions for the two points over a given elapsed time. In order to understand the observed density structures better, the dynamic admittance, the sum of the reciprocal absolute Jacobian determinants and therefore independent of the initial velocity distribution, is introduced. The bands, pinch point and other features seen in the dynamic admittance plots are analyzed. The effects of the earth in reducing the number of routes and thereby casting a dynamic shadow are demonstrated.