  Impact craters are among the most prominent topographic features on planetary
bodies. Crater scaling laws allow us to extract information about the impact
histories on the host bodies. The pi-group scaling laws have been constructed
based on the point-source approximation, dimensional analysis, and the results
from impact experiments. Recent impact experiments, however, demonstrated that
the scaling parameters themselves exhibits complex behavior against the change
in the impact conditions and target properties. Here, we propose an
alternative, fully analytical method to predict impact outcomes based on impact
cratering mechanics. This approach is based on the Maxwell Z-model and the
residual velocity. We present analytical expressions of (1) the proportionality
relation between the ejection velocity and the ejection position, (2) the
radius of a growing crater as a function of time, and (3) the transient crater
radii. Since we focused on obtaining analytical solutions, a number of
simplifications are employed. Due to the simplifications in the strength model,
the accuracy of the prediction in the strength-dominated regime is relatively
low. Our model reproduces the power-law behavior of the ejecta velocity
distribution and the approximate time variation of a growing crater. The
predicted radii under typical impact conditions mostly converge to a region
between the two typical scaling lines by the pi-group scaling laws, strongly
supporting the notion that the new method is one of the simplest ways to
predict impact outcomes, as it provides analytical solutions. Our model could
serve as a quick-look tool to estimate the impact outcome under a given set of
conditions, and it might provide new insights into the nature of impact
excavation processes.
