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Stellar Structure Section 4: Structure of Stars Lecture 8 – Mixing length theory The three temperature gradients Estimate of energy carried by convection Approximation near centre of stars … … which doesnt work near the surface Surfaces are hard to treat! Definition of surface via optical depth

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Mixing length theory – basic ideas (see blackboard for detail) Assumptions: can define a mean temperature profile rising and falling elements have temperature excess (or deficit) ΔT elements lose identity after a mixing length Blob moves adiabatically with mean temperature excess ΔT and mean speed v Blob rises through mixing length, stopped by collisions, gives up heat Can calculate ΔT and v as a function of Assume what proportion of material is in rising and falling blobs Can then calculate the convective flux, as a function of mixing length and local quantities (for details see Schwarzschilds book, pp. 47-9)

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What is the mixing length? (see blackboard for detail) Mixing length theory has a gap – it does not provide an estimate for the mixing length itself, so is a free parameter. The only scale in the system is the pressure scale height, defined by: It is the distance over which P changes by a significant factor (~e) It is common to write: = H P (4.42) where = O(1). Typical values are in range 0.5 to 2 – for Sun, fix by choosing value that reproduces correct solar radius. Can we nonetheless make progress?

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The three temperature gradients (4.43) (4.44) (4.45) When convection is occurring, expect: ad < < rad (4.46) (see blackboard, including sketch). How big is - ad ?

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Estimate of energy carried by convection Approximately: L conv 4 r 2 c p ΔT v (4.47) Here c p ΔT is the average excess thermal energy per unit volume, being transported with mean speed v through a spherical shell of radius r. Substituting for c p and using values for, r near the centre of the Sun (see blackboard), we find: L conv 2.5 10 26 ΔT v W. This allows the solar luminosity to be carried with (e.g., from ML theory)v 40 m s -1 ( 10 -4 v s ) and ΔT 0.04 K. Hence we find (see blackboard): - ad 10 -8 ad 0.

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Structure equations for efficient convection ( - ad 0) In regions of efficient convection, replace energy transport equation by = ad ; structure then determined by 4 equations: (4.3) (4.4) (4.50) (3.25) If radiation pressure negligible, and = 5/3, this is a polytrope of index 3/2. Note absence of any equations involving luminosity.

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Energy carried by (efficient) convection Energy equations can now be used to find L conv : Re-write (4.40) as: L conv = L – L rad (4.51) Terms on RHS can be found from the radiative energy equations (not yet used): (3.32) (4.52) This works only for efficient convection, usually confined to convection in stellar cores. What about convective envelopes?

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Convection in stellar envelopes What is L conv near surface of Sun? Now find (see blackboard): L conv 2.5 10 20 ΔT v W. This needs much larger ΔT and v to carry the solar luminosity: ΔT 100 K (about T/50), even for v v 1.5 10 4 m s -1 For v smaller than this, ΔT would need to be even larger. Convection may not be able to carry all the energy. Here we need: ad < rad full mixing length theory.

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Interiors versus surfaces Mixing length theory much messier than simple - ad = 0 Generally: deep interiors of stars easier to treat than surface regions For convective cores (but not for envelopes), homologous solutions can be found – see Problem Sheet 3 Surface also provides difficulty with boundary conditions – how do we improve simple zero ones? See next lecture…… ……but heres a start:

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Surface boundary conditions How can we improve the simple zero boundary conditions? One obvious better condition is: T = T eff at the surface.(4.53) But what is the surface? visible surface = surface from which radiation just escapes. What is mean free path of a photon from this photosphere? Suggestions?

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Radiative transfer and optical depth (see blackboard for detail) In terms of intensity of radiation, mean free path of photon corresponds to e-folding distance of the intensity. Writing as the monochromatic absorption coefficient, we can write down a formal expression for the monochromatic intensity that shows why the e-folding distance is a useful concept, and define a monochromatic mean free path by an integral expression. Integrating over frequency, and taking the frequency-integrated mean free path to be infinite, we call this integral the optical depth,. Then: photosphere, or visible surface, is equivalent to = 1.

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1 The structure and evolution of stars Lecture 4: The equations of stellar structure.

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