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Download Homework #2 1. There are two ways to estimate the energy carried
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Homework #2 1. There are two ways to estimate the energy carried by convection. The first is that the energy flux is Fc ≈ 1/2ρvc3 ≡ Fc,1 where vc is the characteristic velocity of the convective motions. This is the KE flux carried by moving blobs. The other estimate is that Fc ≈ ρ∆Evc ≡ Fc,2 where ∆E is the difference in the thermal energy of a rising hot blob (or sinking cool blob) relative to the background star (where E is per unit mass). Show that these two expressions are equivalent, to order of magnitude (which is the accuracy of mixing length theory). 2. The solar convection zone contains very little mass (only ≈ 2% of the mass of the sun). Consider a model in which we neglect the mass of the convection zone in comparison to the rest of the sun. Model the convection zone as a polytrope with P = Kργ . The radius of the base of the convection zone is Rc . a) Solve for the density, temperature, and pressure as a function of radius in the convection zone. Do not assume that the convection zone is thin (i.e., even though Mr = constant = M , because r changes significantly in the convection zone, do not assume that the gravitational acceleration is constant). b) In detailed solar models, the pressure at the base of the convection zone is ≈ 5.2 × 1013 dyne/cm2 and the density is ρ ≈ 0.175 g cm−3 . Estimate the radius of the base of the convection zone Rc . Compare this to the correct answer of Rc ≈ 0.71R . c) In your model, what is the temperature of the sun at 0.99R , 0.9R , and at the base of the solar convection zone. This gives you a good sense of how quickly the temperature rises from its surface value of ≈ 5800 K as one enters the interior of the sun. d) Using a simple publically available stellar structure code, I find that the zero-age main sequence (ZAMS) solar model has the following properties (ZAMS is when the star first starts to undergo fusion, so no H has yet been converted into He; this is t = 0 in stellar models, vs. t ≈ 4.5 billion years for the sun today). log(Tef f ) log(L/L ) log(Tc ) log(ρc ) log(R/R ) 3.7256 -0.3287 7.1287 1.8444 -0.0886 For the same code, if I turn off convection, so that energy is always transported by radiation even when the solution is in reality convectively unstable, I find the following solar model,1 log(Tef f ) log(L/L ) log(Tc ) log(ρc ) log(R/R ) 3.6978 -0.3292 7.1286 1.8443 -0.033 The luminosity, central temperature, and central density do not change significantly (not surprising since the convection zone has so little mass). However, the radius of the model sun increases by about 14% and the effective temperature decreases as a result by about 7%. To understand this, consider the structure of a radiative atmosphere in which gas pressure dominates and the enclosed mass, enclosed luminosity and opacity are all constant (constant Mr and Lr are quite reasonable at large radii, while constant κ is only approximately valid and is made for simplicity). Use hydrostatic equilibrium and the radiative diffusion equation to derive a relationship betweeen pressure and density in such an atmosphere. Use this and your model atmosphere from a) to explain the numerical results above, namely that the radius of the sun would be larger if the outer convection zone were absent. Note: You showed in HW 1 that an n = 3 polytrope is a pretty good model of the radiative parts of the sun. This problem should help you see why this is a reasonable first approximation. 1 This was not done with MESA. We had trouble doing this in MESA. Extra credit if you can do so. 1 3. In class, we discussed Kelvin-Helmholz (KH) contraction of fully convective stars to the main sequence. a) Argue that, for KH contraction to occur, the timescale for KH p contraction tKH must be longer than the gravitational free-fall time of the cloud, tf f ≈ 1/ Ghρi, where hρi is the mean density of the cloud. What happens if tKH < tf f ? b) Estimate the critical radius Rc (in R ) at which tKH ≈ tf f , i.e, at which KH contraction begins, for a given cloud of mass M (in M ). Assume, as we did in class, that the cloud is fully convective at early times. Show that for R < Rc , the cloud undergoes KH contraction according to your criterion from a). Recall that the luminosity of a fully convective star is L ≈ 0.2L (M/M )4/7 (R/R )2 . In lecture we showed that moderately massive stars become radiative well before they reach the main sequence. c) Does the transition from convective to radiative first happen at the center of the star or the outside (i.e., is the transition inside-out or outside-in)? To determine this, consider where the star is likely to first become stable by the criterion d ln T /d ln P |rad < 2/5. During KH contraction the energy source for the star (gravity) depends only weakly on density and temperature so assume that Lr /Mr ∼ constant, independent of position in the star. In the rest of this problem, we will derive the KH contraction of a radiative star. Assume for simplicity that electron scattering dominates the opacity of the star. d) Derive R(t,M) for radiative stars undergoing KH contraction. e) If stars reach the main sequence when R/R ≈ M/M , what is the time to reach the main sequence, tM S , as a function of stellar mass? 4. The treatment of convection, even within the simplified framework of mixing length theory, has numerous complications. In this problem, we will use MESA to explore a few of them. We will consider the main sequence evolution (core hydrogen burning) of a 3 M star. We will adjust some of the convective parameters used by MESA and assess their impact on the mass of the convective core (as a function of time) and the main sequence lifetime of the star. So that you don’t have to run a dozen MESA models yourself, each student will be assigned a model to run. Complete your assigned model by Wed Feb 20 and send it to Josiah, who will collect them and distribute a full set to everyone for the interpretation parts of the problem. For each part below, use qualitative physical arguments to explain the variation found in the MESA calculations in the size of the convective core and main sequence lifetime with the relevant convective parameter. a) When using mixing length theory, one must choose the mixing length parameter α. Explore the effects of varying α; explain physically what you find. b) A rising (or falling) blob will not instantaneously halt when it reaches the edge of the region of convective instability. Its inertia will cause it to penetrate some distance into the stable region. This convective overshooting introduces additional mixing.2 Explore the effects of varying the overshooting parameter fov ; explain physically what you find. c) The Schwarzchild criterion for convection (which is what we derived in lecture) omits the possible effects of composition gradients; the Ledoux criterion, which we will develop in this problem, takes them into account. 2 In MESA, the overshooting parameter f ov is the length scale (in units of one-half the pressure scale height) for the exponential decay of the convective diffusion coefficient. 2 i) Consider an ideal gas. Relate the logarithmic density change δ ln ρ = δρ/ρ to the logarithmic changes in P, T, µ. The condition for whether a blob will be convectively unstable can be expressed (as in class) in terms of the Brunt-Väisälä frequency via the density contrast N2 = g (ρblob − ρbackground ) <0 ρbackground δr In addition to the assumptions of adiabaticity and pressure equilibrium from class, assume that the blob does not mix with the background medium to simplify your expression for N 2 and write it in terms of background gradients. ii) What is the sign of (d ln µ/dr)background that you expect to develop during core H burning? Given your result from (i), will the composition gradient have a stabilizing or destabilizing effect on convection? iii) Explore the effect of using the Ledoux convective criterion instead of the Schwarzchild criterion; explain physically what you find. d) Regions that are stable by the Ledoux criterion, but unstable by the Schwarzchild criterion are referred to as semiconvective. While dynamically stable, instabilities can grow on longer timescales in these regions and cause some mixing to occur.3 The exact efficiency of mixing by semiconvection is uncertain and needs to be calculated using multi-dimensional simulations. It is typically modeled with a parameterized efficiency αsc analogous to the mixing length α in part a).4 Explore the effect of varying αsc ; explain physically what you find. 3 Semiconvective regions are overstable on the thermal diffusion timescale. A blob which is perturbed will oscillate around its equilibrium position. But as the blob rises, it is not perfectly adiabatic and so loses some heat. When it returns to its old equilibrium position, it is now cooler than the surroundings and so proceeds to sink deeper, where it is heated and when it returns to the old equilibrium is now hotter than the surroundings and so rises to an even higher position, etc. 4 In MESA, the diffusion coefficient associated with mixing of the composition by semiconvection scales linearly with αsc . 3