Abstract
The force acting upon a moving mass element is resolved mainly into the pressure force and the buoyancy force of Archimedes. The resolution is meaningful relative to both the duration of the motion, and the creation of vorticity. The dynamic time-scale of a star is introduced as the characteristic time needed for a mass element to travel from the star’s surface over a fraction of the total radius due to a deviation of the dominant pressure force from its equilibrium value. The energy exchange between moving mass elements is shown to be generally small in a star, except near the surface. Therefore, the isentropic approximation can be adopted for stellar perturbations. Finally, a local criterion for stability with respect to convection is derived by consideration of the slow isentropic motion of a bubble under the influence of the buoyancy force of Archimedes.
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Notes
- 1.
Now we perform a calculation with simple assumptions, which gives the oscillation period of the air as a function of the gradients of temperature. We assume that a quantum of air is removed from its equilibrium layer by a rise or a descent, whereby its temperature changes adiabatically. The force, which drives the quantum of air back in the equilibrium layer, be proportional to the difference of the concerned density of the quantum of air, and of the surrounding layer. We use the following denominations:
z: height of the quantum of air above its equilibrium level,
θ0: the absolute temperature of the air at the equilibrium level,
ρ′, θ′: density and temperature of the quantum of air at the height z,
ρ, θ: density and temperature of the surrounding air at the height z,
γ: gradient of temperature of the air, assumed as constant,
γ0: the adiabatic gradient of temperature,
g: the gravitational acceleration.
It is
$$\theta = {\theta }_{0} - \gamma \,z\;,{\theta }^{{\prime}} = {\theta }_{ 0} - {\gamma }_{0}\,z.$$Since the pressure in the particular quantum of air is assumed to be equal to the pressure of the surrounding air at the same height, we have ρθ=ρ′θ′, from which
$${\rho }^{{\prime}}- \rho = {\rho }^{{\prime}}\,{ \left ({\gamma }_{0} - \gamma \right )z \over {\theta }_{0}} ,$$where the higher powers of \({ \gamma \,z \over {\theta }_{0}}\) are not taken into consideration. The oscillating force acting on the quantum of air is also for the mass unit
$$-{ g\left ({\gamma }_{0} - \gamma \right ) \over {\theta }_{0}} \,z ={ {d}^{2}z \over {\mathit{dt}}^{2}} .$$The equation yields after integration
$$z ={ A \over a} \,\sin \,a\,t,$$where
$${a}^{2} ={ g\left ({\gamma }_{0} - \gamma \right ) \over {\theta }_{0}} .$$Tacitly we have assumed that γ<γ0.
- 2.
When we displace ... an arbitrary mass of air without any external supply of heat in a higher layer of the atmosphere, it expands because of the decrease of the external pressure, and its temperature sinks at the same time. If the decrease of the temperature resulting from Poisson’s tension theorem is larger than the atmospheric one, which is related to the height reached, and if our quantum of air is cooled down below the temperature of its new surroundings, then that same quantum of air, when it is left to itself, must sink again downwards to its earlier layer. The equilibrium of the air is then a stable or lasting equilibrium. On the contrary, the mass of air will rise still higher, when its decrease of temperature is smaller than that of the atmosphere, and when it therefore remains warmer than the surrounding layer of air; the equilibrium is in this case an unstable or vacillating equilibrium.
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Smeyers, P. (2010). Deviations from the Hydrostatic and Thermal Equilibrium in a Quasi-Static Star. In: Linear Isentropic Oscillations of Stars. Astrophysics and Space Science Library, vol 371. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13030-4_4
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