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2. Convection in Stars

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2. Convection in Stars 2. Convection in Stars 2.1 The Schwarzschild criterion for the onset of convection Consider a bubble of gas that is accidentally displaced upwards by some disturbance. • If the bubble is heavier than the gas in its new surroundings it will sink; if lighter, it will rise. • In the first case, where the bubble returns to its original position, the gas is said to be convectively stable; in the latter case, when the bubble continues to rise, the environment is convectively unstable . Consider Figure 2.01. Let r describe the outward, radial direction, and let the bubble be displaced outward from point a to point b by the distance δr. At its initial location, the bubble was in local thermodynamic equilibrium with its surroundings. • We assume that the displacement happens sufficiently fast that the bubble does not have a chance to exchange heat with the surrounding gas: that is, it behaves adiabatically . • However, the rate of rise is slow enough to allow the internal pressure of the bubble to adjust to that of the ambient surroundings during the rise. T rising dT a dr ad dT b dr gas ∆T ∆T d dT c dr ad falling r Figure 2.01: Temperature stratification of the atmospheric model discussed in the text So, under what conditions will the bubble continue to rise? • Clearly, it will do so if it is lighter than its surroundings, i.e ., if the internal density remains smaller than the external density. • If it is to remain in pressure equilibrium with the surrounding gas, this requires the temperature of the bubble to be higher than ambient. This follows from the perfect gas law: ρ ℜT P = µ , (R2.01) where P is the pressure of the gas, T its temperature, µ is the mean molecular weight per particle, and ℜ the gas constant. From Figure 2.01, at point b the bubble is ∆T hotter than the surrounding gas; its temperature will be given by: dT T T r bubble = a + δ , dr ad where ( dT / dr )ad is the temperature gradient under adiabatic conditions and Ta is the temperature at the undisplaced location a. The temperature of the surrounding gas at b will be given by: dT T T r gas = a + δ . dr gas The temperature excess of the bubble will therefore correspond to: dT dT ∆T = T − T = δ r ⋅ − bubble gas . (2.02) dr ad dr gas For the bubble to be buoyant and rise, ∆T > 0: this gives us the condition for convective instability, i.e ., dT dT < , (R2.03) dr dr ad where I have now dropped the ‘gas’ subscript in the ‘surrounding’ gradient. If we choose to think about the absolute values of the gradients we then have: dT dT > . (R2.04) dr dr ad Conditions 2.03 and 2.04 tell us that for convection to occur , the ambient temperature gradient must be steeper (i.e ., more negative ) than the adiabatic temperature gradient. With reference to the perfect gas law, our treatment here has assumed that the mean molecular weight, µ, remains constant ( i.e ., changes to µ can arise from alterations to the ionisation state of the gas). From Figure 2.01, we could have also considered a falling or sinking bubble of gas. • It must follow the path c to d if it is to maintain a higher density than its surroundings and continue to fall. • The rising, hotter elements and the falling, cooler elements therefore maintain a net upward transfer of heat flux in a convectively unstable layer. In reality, the rising and falling bubbles do interact a little with their surroundings. In Figure 2.02, we show the resulting temperature stratification for a rising element. The temperature gradient for the bubble is steeper than the adiabatic value because it interacts with the environment and loses some of its energy. T dT dT dr ad dr bubble dT T dr ∆ gas r Figure 2.02: Temperature stratification of a rising convective element, allowing for energy exchange 2.2 Convectively unstable layers in stars Stars can be convectively unstable in their outer layers, and in their cores. The Schwarzschild condition clearly tells us that for convection to take place either: 1. The radiative gradient must become steep enough; or 2. The adiabatic gradient must become shallow enough. We therefore have potentially two processes that can give rise to conditions favourable to the onset of convection. 2.2.1 Convection in Outer Envelopes Figure 2.03 shows the variation of temperature within the outer 30 per cent or so by radius of the Sun. This is the extent of the solar sub-surface convection zone. Just beneath the visible surface of the Sun (or photosphere, at fractional radius 1.0) the temperatures and pressures are just right for hydrogen atoms to ionise. At progressively greater depths, helium ionises, first once then twice. Figure 2.03: Temperature variation in a model convection zone (data from Stix, 1989) Since energy is used up in ionising these species, the opacity of the layers rises. The solar luminosity must be maintained, i.e., the layers must readjust to compensate for the increased opacity. The radiative heat flux can be shown to depend upon the temperature gradient, i.e. dT ℑ ∝ R dr . (R2.05) The absolute value of the temperature gradient therefore increases sufficiently over the range of fractional radii from 0.71 to 1.00 to give the required flux and the Schwarzschild condition to be met. The outer third by radius of the Sun is therefore convectively unstable . The adiabatic gradient is also affected in the outer layers. To help see this, we re-express the Schwarzschild condition in terms of double logarithmic gradients . First, the condition can be given in terms of gradients made with respect to pressure: dT dT > . dP dP ad If we introduce the logarithmic gradient: d log T ∇ = d log P , (R2.06) we can then recast the condition for the onset of convection as: ∇ > ∇ ad . (R2.07) It can be shown that the adiabatic gradient is related to the adiabatic exponent, γ, by: γ −1 ∇ = ad γ . (2.08) This exponent helps to describe the motion of a bubble that moves adiabatically (i.e., so that it does not exchange heat with its surroundings). The bubble will always obey the relation: −γ Pρ = constant (R2.09) The exponent is the ratio of the specific heat capacities at constant pressure and constant volume respectively, i.e. γ = cP / cV . (R2.10) • The heat capacity at constant volume, cV, is the amount of heat energy needed to heat one mol of gas by one degree Kelvin at constant volume. • The heat capacity at constant pressure, cP, corresponds to the amount of energy that must be transferred to one mol of the gas at constant pressure in order to raise its temperature by one degree Kelvin. − When heated at constant pressure the gas will expand (cf. Equation 2.01). It must therefore ‘do work’ against the pressure of the ambient surroundings, and this requires the input of more heat energy − As a result, cP > cV. Under conditions where ionisation is taking place, a large part of the energy available will go into bringing about the change of state; there is less available to simply heat the gas. So we require a greater amount of total energy to increase the temperature of the gas by a specified amount: the specific heats increase. Since it can also be shown that: cP = cV + ℜ , (2.11) where ℜ is the gas constant, the adiabatic exponent will approach unity, and ∇ad becomes very small. 2.2.2 Convection in Cores Convection can also start in the cores of stars. • As M increases, so does T and the CNO nuclear reaction cycle starts to dominate energy production. • Because of the high temperature dependence of the cycle, the energy generation gets highly concentrated in the central part of the core. • This increases d T / dr , and eventually convective instability sets in. • This happens for stars with masses greater than M about 1.1 . 2.2.3 Trends in Convection Zone Properties First, in stars hotter than Teff =9000K, we have core zones but no discernible surface zones. For the surface zones: hydrogen ionizes at temperatures around Teff =6000K, while helium starts to ionize at temperatures of 20,000K and above. Provided stars have temperatures lower than about Teff =7500K, we will have thin surface convection zones. These zones get thicker as temperatures decrease. In very cool, the zones may extend down into the cores. We saw in Section 1.5.6 that in the atmospheres of stars, temperature ca n be described as a function of optical depth, i.e ., 1/ 4 3τ 1 T(τ ) ≈ T + eff 4 2 (1.32) This means that: 4 4 T (τ ) ∝ τ T eff . So: • For a star with a lower Teff , the temperature will increase more slowly as one moves deeper into the atmosphere. So to reach a given optical depth (and temperature) one must go deeper where densities and pressures are higher (for higher pressures, one needs higher temperatures to get the same ionization). • The sub-surface ionisation zones are deeper: since these are what give the conditions for the onset of convection, the convection zones get deeper. In contrast, for higher Teff : • A given optical depth is encountered higher in the atmosphere, where pressures are lower. So ionisation starts relatively higher in the atmosphere and the zones are thinner and the convection less efficient (lower density). • When Teff reaches ≈ 7500 K the zones become so thin that the stratification is indistinguishable from radiative: the sub-surface zones begin to disappear.
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