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ASTR 1102-002 2008 Fall Semester

ASTR 1102-002 2008 Fall Semester. Joel E. Tohline, Alumni Professor Office: 247 Nicholson Hall [Slides from Lecture10]. Modeling the Sun. Building a mathematical model ( part 1 )

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ASTR 1102-002 2008 Fall Semester

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  1. ASTR 1102-0022008 Fall Semester Joel E. Tohline, Alumni Professor Office: 247 Nicholson Hall [Slides from Lecture10]

  2. Modeling the Sun • Building a mathematical model (part 1) • We know mass (M), radius (R), luminosity (L), surface temperature (Tsurf), and surface composition (74% H; 25% He; 1% other) • Assume… • Uniform density (r), given by M and volume (4pR3/3) • Uniform composition (same as surface) • Specify internal temperature throughout, T(r), so that hydrostatic equilibrium is achieved throughout [NOTE: At each location “r”, pressure (P) can be calculated from your model specification of density, temperature, and composition at “r”.] • Does this T(r) and r(r) produce a proper thermal equilibrium throughout as well as correct surface L? • If not, readjust T(r) and r(r), while holding M and Tsurf fixed, until both hydrostatic & thermal equilibrium are simultaneously achieved along with correct L.

  3. Modeling the Sun • Building a mathematical model (part 1) • We know mass (M), radius (R), luminosity (L), surface temperature (Tsurf), and surface composition (74% H; 25% He; 1% other) • Assume… • Uniform density (r), given by M and volume (4pR3/3) • Uniform composition (same as surface) • Specify internal temperature throughout, T(r), so that hydrostatic equilibrium is achieved throughout [NOTE: At each location “r”, pressure (P) can be calculated from your model specification of density, temperature, and composition at “r”.] • Does this T(r) and r(r) produce a proper thermal equilibrium throughout as well as correct surface L? • If not, readjust T(r) and r(r), while holding M and Tsurf fixed, until both hydrostatic & thermal equilibrium are simultaneously achieved along with correct L.

  4. Modeling the Sun • Building a mathematical model (part 1) • We know mass (M), radius (R), luminosity (L), surface temperature (Tsurf), and surface composition (74% H; 25% He; 1% other) • Assume… • Uniform density (r), given by M and volume (4pR3/3) • Uniform composition (same as surface) • Specify internal temperature throughout, T(r), so that hydrostatic equilibrium is achieved throughout [NOTE: At each location “r”, pressure (P) can be calculated from your model specification of density, temperature, and composition at “r”.] • Does this T(r) and r(r) produce a proper thermal equilibrium throughout as well as correct surface L? • If not, readjust T(r) and r(r), while holding M and Tsurf fixed, until both hydrostatic & thermal equilibrium are simultaneously achieved along with correct L.

  5. Modeling the Sun • Building a mathematical model (part 1) • We know mass (M), radius (R), luminosity (L), surface temperature (Tsurf), and surface composition (74% H; 25% He; 1% other) • Assume… • Uniform density (r), given by M and volume (4pR3/3) • Uniform composition (same as surface) • Specify internal temperature throughout, T(r), so that hydrostatic equilibrium is achieved throughout [NOTE: At each location “r”, pressure (P) can be calculated from your model specification of density, temperature, and composition at “r”.] • Does this T(r) and r(r) produce a proper thermal equilibrium throughout as well as correct surface L? • If not, readjust T(r) and r(r), while holding M and Tsurf fixed, until both hydrostatic & thermal equilibrium are simultaneously achieved along with correct L.

  6. Modeling the Sun • Building a mathematical model (part 1) • We know mass (M), radius (R), luminosity (L), surface temperature (Tsurf), and surface composition (74% H; 25% He; 1% other) • Assume… • Uniform density (r), given by M and volume (4pR3/3) • Uniform composition (same as surface) • Specify internal temperature throughout, T(r), so that hydrostatic equilibrium is achieved throughout [NOTE: At each location “r”, pressure (P) can be calculated from your model specification of density, temperature, and composition at “r”.] • Does this T(r) and r(r) produce a proper thermal equilibrium throughout? • If not, readjust T(r) and r(r), while holding M and Tsurf fixed, until both hydrostatic & thermal equilibrium are simultaneously achieved.

  7. Modeling the Sun • Building a mathematical model (part 1) • We know mass (M), radius (R), luminosity (L), surface temperature (Tsurf), and surface composition (74% H; 25% He; 1% other) • Assume… • Uniform density (r), given by M and volume (4pR3/3) • Uniform composition (same as surface) • Specify internal temperature throughout, T(r), so that hydrostatic equilibrium is achieved throughout [NOTE: At each location “r”, pressure (P) can be calculated from your model specification of density, temperature, and composition at “r”.] • Does this T(r) and r(r) produce a proper thermal equilibrium throughout? • If not, readjust T(r) and r(r), while holding M and Tsurf fixed, until both hydrostatic & thermal equilibrium are simultaneously achieved.

  8. Modeling the Sun • Building a mathematical model (part 2) • Usually, after completing “part 1”, you discover that, • the total radius, R, of your model is too large • the central density & temperature of your model are not sufficient to ignite nuclear reactions • Hence, you have constructed a viable “equilibrium” model, but it will be subject to Kelvin-Helmholtz contraction, that is, it is not a proper model of our Sun • Repeat the steps in “part 1”, assuming different initial values for the density, until the model’s central “core” that is • hot and dense enough to “burn” hydrogen via nuclear fusion • large enough in size such that energy (E = mc2) is being generated at a rate sufficient to replace the heat being lost at the surface (L) • A model that satisfactorily meets all of these stringent conditions will have a radius, R, that is slightly smaller than the Sun’s present radius; it provides the correct detailed description of a “zero-age” Sun!

  9. Modeling the Sun • Building a mathematical model (part 2) • Usually, after completing “part 1”, you discover that, • the total radius, R, of your model is too large; L too large also! • the central density & temperature of your model are not sufficient to ignite nuclear reactions • Hence, you have constructed a viable “equilibrium” model, but it will be subject to Kelvin-Helmholtz contraction, that is, it is not a proper model of our Sun • Repeat the steps in “part 1”, assuming different initial values for the density, until the model’s central “core” that is • hot and dense enough to “burn” hydrogen via nuclear fusion • large enough in size such that energy (E = mc2) is being generated at a rate sufficient to replace the heat being lost at the surface (L) • A model that satisfactorily meets all of these stringent conditions will have a radius, R, that is slightly smaller than the Sun’s present radius; it provides the correct detailed description of a “zero-age” Sun!

  10. Modeling the Sun • Building a mathematical model (part 2) • Usually, after completing “part 1”, you discover that, • the total radius, R, of your model is too large; L too large also! • the central density & temperature of your model are not sufficient to ignite nuclear reactions • Hence, you have constructed a viable “equilibrium” model, but it will be subject to Kelvin-Helmholtz contraction, that is, it is not a proper model of our Sun • Repeat the steps in “part 1”, assuming different initial values for the density, until the model’s central “core” that is • hot and dense enough to “burn” hydrogen via nuclear fusion • large enough in size such that energy (E = mc2) is being generated at a rate sufficient to replace the heat being lost at the surface (L) • A model that satisfactorily meets all of these stringent conditions will have a radius, R, that is slightly smaller than the Sun’s present radius; it provides the correct detailed description of a “zero-age” Sun!

  11. Modeling the Sun • Building a mathematical model (part 2) • Usually, after completing “part 1”, you discover that, • the total radius, R, of your model is too large; L too large also! • the central density & temperature of your model are not sufficient to ignite nuclear reactions • Hence, you have constructed a viable “equilibrium” model, but it will be subject to Kelvin-Helmholtz contraction, that is, it is not a proper model of our Sun • Repeat the steps in “part 1”, assuming different initial values for the density, until the model’s central “core” is … • hot and dense enough to “burn” hydrogen via nuclear fusion • large enough in size such that energy (E = mc2) is being generated at a rate sufficient to replace the heat being lost at the surface (L) • A model that satisfactorily meets all of these stringent conditions will have a radius, R, that is slightly smaller than the Sun’s present radius; it provides the correct detailed description of a “zero-age” Sun!

  12. Modeling the Sun • Building a mathematical model (part 2) • Usually, after completing “part 1”, you discover that, • the total radius, R, of your model is too large; L too large also! • the central density & temperature of your model are not sufficient to ignite nuclear reactions • Hence, you have constructed a viable “equilibrium” model, but it will be subject to Kelvin-Helmholtz contraction, that is, it is not a proper model of our Sun • Repeat the steps in “part 1”, assuming different initial values for the density, until the model’s central “core” is … • hot and dense enough to “burn” hydrogen via nuclear fusion • large enough in size such that energy (E = mc2) is being generated at a rate sufficient to replace the heat being lost at the surface (L) • A model that satisfactorily meets all of these stringent conditions will have a radius, R, that is slightly smaller than the Sun’s present radius; it provides the correct detailed description of a “zero-age” Sun!

  13. Figure 19-2

  14. Sun’s Internal Structure Figure 16-4

  15. Figure 19-1

  16. Checking Accuracy of Solar Model • Solar Oscillation measurements • Solar Neutrino measurements

  17. Solar Oscillation Measurements Figure 16-5

  18. Solar Neutrino Measurements Figure 16-6

  19. Chapter 19: Stellar Evolution:On & after the Main Sequence

  20. Plot “L vs. T” for 27 Nearest Stars Data drawn from Appendix 4 of the textbook.

  21. L and T appear to be Correlated Nearest Stars

  22. L and T appear to be Correlated A few of the brightest stars in the night sky

  23. Hertzsprung-Russell (H-R) Diagram

  24. Hertzsprung-Russell (H-R) Diagram “main sequence”

  25. Checking Accuracy of Solar Model • Solar Oscillation measurements • Solar Neutrino measurements • Specify a different mass, M, and construct a new mathematical model  resulting model has an L and Tsurf that also falls on the main sequence! And in accordance with observed masses of stars along the main sequence!

  26. Apply the “Age” Concept to Other Stars • How long can other stars live? • tage = fMc2/L • (tage /1010 years) = (M/Msun)/(L/Lsun)

  27. Apply the “Age” Concept to Other Stars

  28. How does a Star’s StructureChange as it Ages (Evolves)?

  29. More complete mapping of stars onto the H-R Diagram

  30. Determining the Sizes of Stars from an H-R Diagram • Main sequence stars are found in a band from the upper left to the lower right. • Giant and supergiant stars are found in the upper right corner. • Tiny white dwarf stars are found in the lower left corner of the HR diagram.

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