Fr.: critère de Schwarzschild
The condition in stellar interior under which → convection occurs. It is expressed as: |dT/dr|ad < |dT/dr|rad, where the indices ad and rad stand for adiabatic and radiative respectively. This condition can also be expressed as: ∇ad<∇rad, where ∇ = d lnT / d lnP = P dT / T dP with T and P denoting temperature and pressure respectively. More explicitly, in order for convection to occur the adiabatic temperature gradient should be smaller than the actual temperature gradient of the surrounding gas, which is given by the radiative temperature gradient if convection does not occur. Suppose a hotter → convective cell or gas bubble rises accidentally by a small distance in height. It gets into a layer with a lower gas pressure and therefore expands. Without any heat exchange with the surrounding medium it expands and cools adiabatically. If during this rise and → adiabatic expansion the change in temperature is smaller than in the medium the gas bubble remains hotter than the medium. The expansion of the gas bubble, adjusting to the pressure of the medium, happens very fast, with the speed of sound. It is therefore assumed that the pressure in the gas bubble and in the surroundings is the same and therefore the higher temperature gas bubble will have a lower density than the surrounding gas. The → buoyancy force will therefore accelerate it upward. This always occurs if the adiabatic change of temperature during expansion is smaller than the change of temperature with gas pressure in the surroundings. It is assumed that the mean molecular weight is the same in the rising bubble and the medium. See also → Ledoux's criterion; → mixing length.
Named after Karl Schwarzschild (1873-1916), German mathematical physicist (1906 Göttinger Nachrichten No 1, 41); → criterion.