1. ASTR 1102-002 2008 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 (T surf ), and surface composition (74% H; 25% He; 1% other) –Assume… Uniform density (  ), given by M and volume (4  R 3 /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) produce a proper thermal equilibrium throughout as well as correct surface L? –If not, readjust T(r) and  (r), while holding M and T surf 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 (T surf ), and surface composition (74% H; 25% He; 1% other) –Assume… Uniform density (  ), given by M and volume (4  R 3 /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) produce a proper thermal equilibrium throughout as well as correct surface L? –If not, readjust T(r) and  (r), while holding M and T surf 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 (T surf ), and surface composition (74% H; 25% He; 1% other) –Assume… Uniform density (  ), given by M and volume (4  R 3 /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) produce a proper thermal equilibrium throughout as well as correct surface L? –If not, readjust T(r) and  (r), while holding M and T surf 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 (T surf ), and surface composition (74% H; 25% He; 1% other) –Assume… Uniform density (  ), given by M and volume (4  R 3 /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) produce a proper thermal equilibrium throughout as well as correct surface L? –If not, readjust T(r) and  (r), while holding M and T surf 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 (T surf ), and surface composition (74% H; 25% He; 1% other) –Assume… Uniform density (  ), given by M and volume (4  R 3 /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) produce a proper thermal equilibrium throughout? –If not, readjust T(r) and  (r), while holding M and T surf 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 (T surf ), and surface composition (74% H; 25% He; 1% other) –Assume… Uniform density (  ), given by M and volume (4  R 3 /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) produce a proper thermal equilibrium throughout? –If not, readjust T(r) and  (r), while holding M and T surf 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 = mc 2 ) 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 = mc 2 ) 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 = mc 2 ) 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 = mc 2 ) 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 = mc 2 ) 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. “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 T surf 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? –t age = fMc 2 /L –(t age /10 10 years) = (M/M sun )/(L/L sun )

29. Apply the “Age” Concept to Other Stars

30. How does a Star’s Structure Change 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.