where a gas bubble can rise (or fall) far and rapidly due to being hotter, less dense, and more buoyant (or cooler, denser, and less buoyant) than its surroundings. Helioseismology, the study of the Sun’s vibrations, has determined the thickness today of the convection zone, l in Fig. 4, as 0.287 ± 0.003 of the solar radius (Phillips 1999, p. 97). In the convection zone, the temperature gradient (the change of temperature T with radius r) is slightly steeper than the adiabatic (no heat exchange) temperature gradient: Here g is the local acceleration of gravity, g = Gm(r)/r2 , m(r) is the mass within radius r, G is Newton’s gravitational constant, γ is usually 5/3 (but nearer the surface it approaches 1), μ is the average atomic mass of the solar plasma (counting free electrons), 0.61 a. m. u., and k is Boltzmann’s constant. (All the temperature gradients are negative, so I am using absolute values for clarity.) With the gradient of the medium slightly steeper than adiabatic, a rising and adiabatically expanding bubble will aways find the surrounding gas slightly cooler than it, so it will continue to rise. For more details on these things, see Phillips (1999). In the steady-state conditions of the Sun today, the temperature gradients in the core and radiation zone are shallower than the adiabatic gradient. That means a rising bubble expanding adiabatically will immediately find itself cooler and denser than its surroundings, and it will stop rising. In that zone, radiation transfers heat upward by diffusion with a flux J proportional to the temperature gradient: where σ is the Stefan-Boltzmann constant, c is the speed of light, ρ is the mass density, and κ is the opacity. This process is much slower than convection; heat generated in the core today would take about ten million years to rise to the bottom of the convection zone (Noyes 1982). From that height, convection bubbles carry heat upwards at more than several km/s (Stein and Nordlund 1998), reaching the surface in a day or less and making the granulation in Fig. 5. By setting the radiative temperature gradient of eq. (4) equal to the adiabatic temperature gradient of eq. (3), Phillips (1999, pp. 97-98) shows that there is a critical value of the power generated per unit mass in a core of radius r: (5) Above this value, convection must occur. L(r) is the power flowing out of the core (today equaling the luminosity at the Sun’s surface), m(r) is the mass of the core, P is the hydrostatic pressure, and the Figure 3. Probability of fusion (in parts per billion) for a single deuteron-deuteron collision in the Sun’s core depends on the square-well deuteron radius R. The radius today is about 2 fm. Figure 4. Cross-section of the Sun, showing the energy-generating core, diffusive radiation zone, and convection zone. Figure 5. Solar granulation from rising convection cells. Note size scale. A granule lasts about five minutes before it sinks and is replaced by another granule. (5) (3) (4) HUMPHREYS Cause of large post-Flood jump in 14C 2023 ICC 282
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