text: The governing equations are
  where  is the density,  is the pressure,  is the entropy and  is the radial velocity. In place of the pressure , we can use the sound speed  using the relation .
  To obtain the self-similar equations, we introduce
  Note that since both  and  are negative, . Formally the solution has to be found for the range . The boundary conditions at  are given by
  The boundary conditions at  can be derived from the observation at the time of collapse , wherein  becomes infinite. At the moment of collapse, the flow variables at any distance from the origin must be finite, that is to say,  and  must be finite for . This is possible only if 
  Substituting the self-similar variables into the governing equations lead to 
  From here, we can easily solve for  and  (or, ) to find two equations. As a third equation, we could two of the equations by eliminating the variable . The resultant equations are
  where  and . It can be easily seen once the third equation is solved for , the first two equations can be integrated using simple quadratures. 
  The third equation is first-order differential equation for the function  with the boundary condition  pertaining to the condition behind the shock front. But there is another boundary condition that needs to be satisfied, i.e.,  pertaining to the condition found at . This additional condition can be satisfied not for any arbitrary value of , but there exists only one value of  for which the second condition can be satisfied. Thus  is obtained as an eigenvalue. This eigenvalue can be obtained numerically. 
  The condition that determines  can be explained by plotting the integral curve  as shown in the figure as a solid curve. The point  is the initial condition for the differential equation, i.e., . The integral curve must end at the point . In the same figure, the parabola  corresponding to the condition  is also plotted as a dotted curve. It can be easily shown than the point  always lies above this parabola. This means that the integral curve  must intersect the parabola to reach the point . In all the three differential equation, the ratio  appears implying that this ratio vanishes at point  where the integral curve intersects the parabola. The physical requirement for the functions   and  is that they must be single-valued functions of  to get a unique solution. This means that the functions  and  cannot have extrema anywhere inside the domain. But at the point ,  can vanish, indicating that the aforementioned functions have extrema. The only way to avoid this situation is to make the ratio  at  finite. That is to say, as  becomes zero, we require  also to be zero in such a manner to obtain . At ,
  Numerical integrations of the third equation provide  for  and  for . These values for  may be compared with an approximate formula , derived by Landau and Stanyukovich. It can be established that as , . In general, the similarity index  is an irrational number.