ASTR 1102-002 2008 Fall Semester

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

17. Figure 19-1

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

19. Solar Oscillation Measurements Figure 16-5

20. Solar Neutrino Measurements Figure 16-6

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

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

23. L and T appear to be Correlated Nearest Stars

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

25. Hertzsprung-Russell (H-R) Diagram

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

27. 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!

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

29. Apply the “Age” Concept to Other Stars

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

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

32. 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.