text: /* ///////////////////////////////////////////////////////////////////// */
  /*!
    \file
    \brief Templeate file for the \c PVTE_LAW EoS.
    
    Collect the basic set of functions required by the 
    \c PVTE_LAW equation of state:
  
    - InternalEnergyFunc() defines the gas internal energy as a function
      temperature and ionization fractions (for non-equilibrium chemistry) 
      or temperature and density (in Local Thermodynamic Equilibrium - LTE -
      or Collisional Ionization Equilibrium - CIE).
    - GetMu() computes the mean molecular weight (in LTE or CIE).
    
    \author A. Mignone (mignone@ph.unito.it)\n
            B. Vaidya
    \date   20 June, 2014
  
  */
  /* /////////////////////////////////////////////////////////////////// */
  #include "pluto.h" 
  
  /* ***************************************************************** */
  double InternalEnergyFunc(double *v, double T)
  /*!
   *  Compute the gas internal energy as a function of temperature
   *  and fractions (or density):
   *  - <tt> rhoe = rhoe(T,rho) </tt> in LTE or CIE;
   *  - <tt> rhoe = rhoe(T,X) </tt> in non-equilibrium chemistry.
   *
   *  \param [in]   v   1D Array of primitive variables containing
   *                    density and species. Other variables are
   *                    ignored.
   *  \param [in]   T   Gas temperature
   *
   *  \return The gas internal energy (\c rhoe) in code units.
   ******************************************************************* */
  {
  }
  
  /* ********************************************************************* */
  void GetMu(double T, double rho, double *mu)
  /*!
   *  Calculate the mean molecular weight for the case in which 
   *  hydrogen fractions are estimated using Saha Equations.
   *
   *  \param [in]   T    Gas temperature in Kelvin.
   *  \param [in]   rho  Gas density (code units)
   *  \param [out]  mu   Mean molecular weight
   *
   *********************************************************************** */
  {
  }
  
  /* ********************************************************************* */
  double Gamma1(double *v)
  /*!
   *  Calculate the value of the first adiabatic index:
   *  \f[
   *     \Gamma_1 = \frac{1}{c_V}\frac{p}{\rho T} \chi_T^2 + \chi_\rho^2
   *     \qquad{\rm where}\quad
   *     \chi_T = \left(\pd{\log p}{\log T}\right)_{\rho} = 
   *              1 - \pd{\log{\mu}}{\log T} \,;\quad
   *     \chi_\rho = \left(\pd{\log p}{\log\rho}\right)_{T} = 
   *                  1 - \pd{\log{\mu}}{\log\rho} \,;\quad
   *  \f]
   *  where \c p and \c rho are in c.g.s units.
   *  Note that if species are evolved explicitly (non-equilibrium chemistry),
   *  we set \c chi=1.
   *
   *  The heat capacity at constant volume, \c cV, is defined as the 
   *  derivative of specific internal energy with respect to temperature:
   *  \f[
   *      c_V = \left.\pd{e}{T}\right|_V
   *  \f]
   *  and it is computed numerically using a centered derivative.
   *
   *  This function is needed (at present) only when computing the 
   *  sound speed in the Riemann solver.
   *  Since this is only needed for an approximated value, 5/3 (upper
   *  bound) should be ok.
   *
   *  \b Reference 
   *     - D'Angelo et. al  ApJ 778, 2013 (Eq. [26-27])
   *
   * \param [in]    v       1D array of primitive quantities
   * 
   * \return Value of first adiabatic index Gamma1.
   *********************************************************************** */
  {
    return 1.6667; 
  }
  