# 11/19/08. Gravitational equilibrium: The outward push of pressure balances the inward pull of gravity. Weight of upper layers compresses lower layers

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1 Gravitational equilibrium: The outward push of pressure balances the inward pull of gravity Weight of upper layers compresses lower layers Gravitational equilibrium: Energy provided by fusion maintains the pressure Gravitational contraction: Provided energy that heated core as Sun was forming Contraction stopped when fusion began 1

2 Stellar Structure Hydrostatic Equilibrium Assume stars are in equilibrium Stars are spherical, so physics depends only on radius where Need ρ(r) to determine M(r) and then P(r) The Sun 5 The Sun 6 Check Calculate the average density of the sun. Pressure Gas Thermal Pressure Particles/m 3 The central density in the sun is 110 times its average density. What is the central density? The Sun 7 The Sun 8 2

3 Mass Fractions We can express mean molecular weight in terms of mass ratios, or mass fractions: Mean Molecular Weight For a neutral gas: For an ionized gas: where A = # of protons + # of neutrons The Sun 9 The Sun 10 Pressure Radiation Pressure photons have momentum Central pressure in Sun The pressure in the center of the sun equals the weight per unit area of the material on top. We can approximate central pressure: A more accurate calculation yields: The Sun 11 The Sun 12 3

4 Central Temperature in Sun Group problem: Using our estimates for central pressure and density, what is central temperature of the sun? Assume the sun consists of 90% H, 10% He (ignore the 1% heavier elements), and that everything is ionized in the core. What is peak wavelength of blackbody spectrum for this temperature? Radiative Transfer You should find the radiation field in core contains mostly x-rays on average, photons travel 0.5 cm before scattering random walk out to surface of sun average energy drops from x-ray to visible at photosphere, photons go from walking to flying if luminosity transfer is too high for radiative transfer, convection results The Sun 13 The Sun 14 Think-Pair-Share The total photon flight time is Compute the flight time assuming l=0.5 cm and compare this with a free-flight time, R/c, if photons could fly straight from the center of the sun to the surface. Source of Sun s Energy In the sun, the thermal energy of a plasma is 10 3 times greater than instantaneous energy content in radiation field. Photons leak out in 3 x 10 4 yr. Current thermal energy would leak away in 10 3 x3x10 4 yr = 3 x 10 7 yr. Must be another energy source! The Sun 15 The Sun 16 4

5 Gravitational Energy? Kelvin-Helmholtz, nineteenth century Virial theorem: thermal E + grav. pot. E = E = 1/2 grav. pot. E If sun contracted from much larger to its present size, it would yield Energy Requirements We know the Earth is ~5 billion years old. We can estimate total energy output of sun at: The number of atoms in the sun is: Assuming current energy output, how long would sun shine? The Sun 17 The Sun 18 Energy Requirements Energy required per atom: In comparison, rest-mass energy is: You need m c 2! Thermonuclear Fusion Process by which heavy elements are built up from light ones via nuclear reactions at high temperatures. At center of sun, T~15 million degrees only H can fuse implies core must be most H clue that predominant form of matter in universe is H Limited energy source implies finite lifetime The Sun 19 The Sun 20 5

6 Controlled Fusion How does nuclear fusion occur in the Sun? Balance between pressure and gravity ideal gas not true for degenerate matter Radius of sun maintains T c -> nuclear reaction rate to balance energy loss due to random walk leakage Fission Fusion The Sun 21 Big nucleus splits into smaller pieces (Nuclear power plants) Small nuclei stick together to make a bigger one (Sun, stars) High temperature enables nuclear fusion to happen in the core Sun releases energy by fusing four hydrogen nuclei into one helium nucleus 6

7 Proton-Proton Chain Solar Thermostat (10 10 yr) Temperature Decreases Fusion Rate Decreases Temperature Restored Core compresses The Sun 25 Solar Thermostat CNO Cycle Temperature Increases Fusion Rate Increases Temperature Restored Core expands Requires star to have some C still fuses H -> He but uses C as a catalyst Proposed by Hans Bethe in 1938 The Sun 28 7

8 Triple Alpha Reaction More Fusion be He capture Energy generation rate 3-body interaction b/c Be rapidly decays if not struck with another He nuclei very strong T dependence T > 10 8 K Carbon burning: T > 6 x 10 8 K Oxygen burning: T > 10 9 K Only H and He made in big bang all other elements made in stellar interiors The Sun 29 The Sun 30 Nuclear Fusion H fusion at 15 million K difficult to get from He to C very sensitive to T Heavier nuclei require higher T The Sun 31 8

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