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[ kd ] hwk05 Ast 4001, 2015 October 13 Homework set 5 -- Stellar interiors Turn in your solution and explanation for problem 3 on Thursday October 22. __________________________________________________________________________________________ 1. -- A fallacy- puzzle, no calculations are needed. Here’s a fraudulent derivation of hydrostatic equilibrium in a star. Consider a thin shell inside it. If subscripts 1 and 2 denote the inner and outer boundaries of the thin shell, then the inner pressure force pushing the shell outward is area pressure = 4 r 12 P 1 , while the outer pressure force pushing it inward is 4 r 22 P 2 . The net outward force due to pressure is the difference between 4 r 12 P 1 and 4 r 22 P 2 . In the limit as r = r 2 r 1 becomes infinitesimally small, this difference is (net pressure force) = 4 { d ( r 2 P ) / d r } r . Meanwhile the shell is also pulled inward by gravity. The volume of the thin shell is 4 r 2 r , so its mass is 4 r 2 r . (Here is mass density as usual.) If g is the local gravitational acceleration, then gravity exerts a radial force g m = 4 r 2 g r on the shell. Summing the pressure and gravity forces, we see that static equilibrium requires 4 { d ( r 2 P ) / d r } r + 4 r 2 g r = 0 , which simplifies to d( r2 P ) / d r r 2 g . This looks credible but it’s wrong ! The correct formula is d P / d r g . Anyone who tries to use our erroneous result will get poor models for stellar interiors. So: Where’s the fallacy in the above derivation ?? If you don’t see it immediately, make a serious effort to figure it out before asking someone who already knows. Two hints: (1) The answer isn’t some little technicality in elementary calculus, taking a limit wrong or something like that. It’s a question of basic physics, not math. (2) Julius Caesar would have guessed the answer from his bridge-building experience. __________________________________________________________________________________________ 4001hwk05 - p2 __________________________________________________________________________________________ 2. (Practice only; useful for problem 3.) Suppose that the Sun consists of 72% hydrogen and 28% helium by mass. Ignore heavier elements, because they have nearly the same effect as helium for this problem. By "hydrogen" and "helium" we mean the familiar isotopes with mass 1 and 4. Assuming that the gas is a fully ionized plasma, calculate the gas constant in both c.g.s. and SI units. Aim for a precision of 1% or better. Don't forget the free electrons! __________________________________________________________________________________________ 3. (Turn in your results for this problem on Thursday October 22.) Let’s estimate the total binding energy of the Sun, based on very little information and a few simple assumptions. Advice: You can minimize the ugly algebra here by working with dimensionless variable x = r / R instead of r . ( R = the Sun’s radius. ) (a) Assume that the density distribution inside the Sun is ( r ) = { 1 r / R } 0 . (Why? Because it's the simplest formula that has ( R ) = 0, increases monotonically toward the center, and gives a finite value at r = 0.) Deduce a formula for m ( x ) where x = r / R . Evaluate the central density 0 that is consistent with the Sun’s known mass M and radius R . (b) Do another integral to find a formula for the total gravitational energy E G , a negative quantity, in terms of total mass M and outer radius R . Then use the Virial Theorem to convert this to a formula for the Sun’s total energy including both gravitational and thermal energy. Assume that it’s a perfect monatomic gas so U = 1.5 P . (c) Integrate the hydrostatic equation to estimate P ( 0 ), the pressure at the center of the Sun. Let’s interpret this in terms of temperature. Work out a reasonable gas constant for the ideal gas law, P = T . (Problem 2 above.) Use these results to estimate the temperature at the center of the Sun in this idealized model. __________________________________________________________________________________________