text: For spherical vessel, there is no known explicit solution, so Frank-Kamenetskii used numerical methods to find the critical value. Here , then
  If the transformations  and , where  is the maximum temperature which occurs at  due to symmetry, are introduced
  The above equation is nothing but Emden–Chandrasekhar equation, which appears in astrophysics describing isothermal gas sphere. Unlike planar and cylindrical case, the spherical vessel has infinitely many solutions for  oscillating about the point , instead of just two solutions, which was shown by Israel Gelfand. The lowest branch will be chosen to explain explosive behavior.
  From numerical solution, it is found that the critical Frank-Kamenetskii parameter is . The system has no steady state( or explodes) for  and for , the system goes to a steady state with very slow reaction. The maximum temperature  occurs at  and maximum critical temperature is .