The paper I found postulates that the rarefaction wave will travel to the left with a velocity of $C_h$ and be replaced by an exponential decay function of $e^{-\frac{x}{C_ht}}$. Obviously, this is only valid up until $d/2 = C_ht$, at which point the wave will have converted the entire initial distribution into an exponential decay function; let us call the value for time that makes this equation true $\tau$. If we assume that the wave travels up to $x_f = 5.517C_h\tau$ (as shown in prior derivations), we can find the amount of mass that can be accounted for within the distribution:

In my last post, I used a couple of simple assumptions about the modified density distribution of the plasma to motivate an expression for $n_{max}$, the maximum density of the new target. Using this result, we can derive the new effective thickness of the target, $d_{eff}$, to pass into the Fuchs function.

The Fuchs function, while providing an incredibly useful basis with which a proton energy distribution resulting from TNSA can be estimated, does not account for the target deformation from the prepulse shot. While the prepulse shot is several orders of magnitude lower than the main pulse shot in intensity, the temporal separation between the prepulse and main pulse shots allows for a significant change to the shape of the target. Consider the following initial density distribution:
(image of rectangular target of height $n_0$ and width d)
Due to the conservation of mass, we can assume that there exists a conserved quantity, $N$, which is found by integrating the linear density with respect to position: