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q_0^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{1 - (\rho/\rho_0)^2 \cdot \exp \left(  -L/ L^* \right)}{2 \ln (\rho_0/\rho) + fL/d \cdot (1- \exp \left(  - L/ L^* \right))/(L/L^*)}

c^* \, \rho^* = c _0 \, \rho_0

L^* = c^* \, \rho^* \, G \, L

\rho/\rho_0 = \frac{1+ c^* p}{1+ c^* p_0}

\dot m^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{\rho_0^2 - \rho^2 \cdot \exp \left(  -L/ L^* \right)}{2 \ln (\rho_0/\rho) + fL/d \cdot (1- \exp \left(  - L/ L^* \right))/(L/L^*)}

c^* \, \rho^* = c _0 \, \rho_0

L^* = c^* \, \rho^* \, G \, L

\rho/\rho_0 = \frac{1+ c^* p}{1+ c^* p_0}

\rho = \rho_0 \, \exp (c^* \rho^* \, G \, L) \cdot \sqrt{ 1 - \frac{f}{2d} \cdot \frac{j_m^2}{G \, \rho_0^2} \cdot ( 1 - \exp(-2 \, c^* \rho^* \, G \, L))}

p(L) = \frac{1}{c^*} \cdot \left[ 
-1 + (1+c^* p_0) \cdot \exp (c^* \rho^* \, G \, L) \cdot \sqrt{  1 - \frac{f}{2d}  \cdot \frac{j_m^2}{G \rho_0^2} \cdot \big(1 - \exp (-2 \, c^* \rho^* \, G \, L) \big) }
\right]

p(L) = \frac{-1 + (1+c^* \, p_0) \cdot \exp(c^*  \rho^*  G \, L)}{c^*}