$v_0$ and angles $\theta$ is excellent for the largest fragment, however, there are large fluctuations of the value of the second largest mass at impact velocities just below the critical point. Above the critical point all curves merge nicely together.

![\[avgfra\] The average fragment mass $\overline M$ as a function of the impact velocity $v_0$. The inset shows the same curves plotted as a function of the normal component of $v_0$. For oblique fragmentation when the velocity approaches the critical point always fluctuations arise, however beyond the critical velocity all curves merge nicely. ](figures/velnor_avgfra_all.eps){width="50.00000%"}

More information about the evolution of fragment sizes with varying impact angle $\theta$ and velocity $v_0$ can be obtained by studying the moments of fragment masses [@instable; @transition; @univer; @proj]. The $k$th moment $M_k$ of fragment masses is defined as $$\label{eq:m_k}
M_k=\sum_i^{N_f} m_i^k-M_\mathrm{max}^k,$$ where $N$ denotes the total number of fragments, $m_i$ is the mass of fragment $i$ and $M_\mathrm{max}$ is the largest fragment mass. The definition Eq. (\[eq:m\_k\]) means that the $k$th power of the largest mass is extracted from the sum of the $k$th power of the fragment masses. The average mass of fragments $\overline M$ can be defined as the ratio of the 