 Mass-spectrum transport under quadratic accretion

If stars are neither created nor lost and all follow quadratic accretion in a central Keplerian potential, conservation of number gives $F(M,t)dM=F_0(M_0)d{M}_0$. Inverting the trajectory gives $M_0=M/(1+\lambda tM)$ and its Jacobian determinant is $(1+\lambda tM{)}^{-2}$. This proves the displayed mass-density-in-mass-space transformation. In a broad high-mass range with $\lambda tM\gg 1$, whose inverse images remain in a narrow interval where $F_0$ varies little, the power law is $F\propto M^{-2}$. A strictly monochromatic initial population remains monochromatic. Finite initial support imposes a finite upper cutoff before the earliest formal blow-up, so the power law is an intermediate asymptotic subject to the accretion model's limits.