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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 doesn’t work near the surface Surfaces are hard to treat!
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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 doesn’t work near the surface Surfaces are hard to treat! Definition of “surface” via optical depth
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 Schwarzschild’s book, pp. 47-9)
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: = HP (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?
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 ?
Estimate of energy carried by convection • Approximately: Lconv≈ 4r 2cpΔTv (4.47) Here cpΔT is the average excess thermal energy per unit volume, being transported with mean speed v through a spherical shell of radius r. • Substituting for cp and using values for , r near the centre of the Sun (see blackboard), we find: Lconv 2.51026ΔTv W. • This allows the solar luminosity to be carried with (e.g., from ML theory) v≈ 40 m s-1 (≈ 10-4vs) and ΔT≈ 0.04 K. • Hence we find (see blackboard): - ad 10-8ad 0.
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.
Energy carried by (efficient) convection • Energy equations can now be used to find Lconv: • Re-write (4.40) as: Lconv = L – Lrad (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?
Convection in stellar envelopes • What is Lconv near surface of Sun? • Now find (see blackboard): Lconv 2.51020ΔTv W. • This needs much larger ΔT and v to carry the solar luminosity: ΔT≈ 100 K (about T/50), even for v ≈ v ≈ 1.5104 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.
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 here’s a start:
Surface boundary conditions • How can we improve the simple zero boundary conditions? • One obvious better condition is: T = Teff 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?
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.
