Astrophysical Systems Analysis Projects

ÓBob Field 2005

Background on Solar Structure and Evolution

Thermonuclear fusion makes stars like our Sun shine (luminosity) and produces (nucleosynthesis) chemical elements beyond hydrogen and helium, which compose rocky planets like Earth. The rate at which energy is transferred from the interior of the Sun to the surface controls the rate of fusion, the stability, the longevity, and the luminosity of the Sun. Interactions of energy and matter determine the energy transfer rate and ultimately control the formation, composition, structure, functioning, and evolution of the Sun.

Thermonuclear fusion cannot occur at ordinary temperatures. How did the Sun become hot enough to initiate fusion? Why don't all nuclei fuse in a brief time period, or in other words, why is the Sun stable for billions of years? Why is the average density of the Sun so low when its mass is so great and its gravitational field is so intense? Does any nucleus in the Sun have enough kinetic energy to overcome the intense electrostatic repulsion of its neighboring nuclei in order to approach close enough for fusion to occur? How does the Sun lose the energy generated by fusion when the vacuum of space is an ideal thermal insulator? How does the Sun trap heat deep in its interior when photons travel at the speed of light? Is electron scattering a minor contributor to solar opacity? Is free electron absorption of photons the dominant source of opacity in the core of the Sun? Is bound electron absorption of photons by the so-called metallic ions the dominant source of opacity in the radiative zone of the Sun? How important is electron scattering of other electrons in minimizing thermal conduction? These studies investigate processes that control the rate at which energy is transferred from the interior of the Sun to the surface because this rate is fundamental to the long-term stable thermonuclear fusion process.

For a “fun” discussion of these questions, read Father Sun’s Fusion Factory.

Potential Student Projects related to the Sun

These projects investigate the transport of electromagnetic and thermal energy in the interior of stars and/or planets. Students may receive college-based fee support or course credit for special problems, advanced topics, senior projects, or possibly undergraduate seminar. The results could include a Mathcad computer model, a written report, a presentation, an exhibit, and a website. A possible theme of the work is that interactions of energy and matter control the formation, composition, structure, and evolution of stars and planets. These processes influence the origin, evolution, diversity, abundance, and distribution of life in the universe. Current efforts focus on the Sun.

1. I have obtained a copy of a Los Alamos National Lab solar astrophysics code for building a pre-main sequence star and following its evolution. This project involves compiling and running a powerful Fortran code, defining cases to study and problems to solve, and interpreting the results. This is a great tool and a fantastic opportunity!

2. Develop interpolation method for LLNL and LANL solar opacity tables. Generate functions of composition, density, and temperature from the tables. Compare to textbook opacity graphs and formulas for electron scattering and free and bound electron absorption. Estimate the importance of each effect throughout the Sun. Calculate photon mean free paths. Determine the influence of metallic ions on solar luminosity and lifetime.

3. Use the numerical outputs of existing standard solar models of Guzik and/or Bahcall to develop analytical expressions for density, temperature, and other functions. Develop clear and accurate explanations of major features associated with solar structure and evolution including energy storage and transport. Click Graphs to view my plots of numerical data provided by Guzik using her LANL solar evolution codes.

4. Explain solar core contraction, envelope expansion, and changes in luminosity and surface temperature.

5. Analyze gravitational energy conversion and energy storage and transport during the Sun’s formation.

6. Investigate convective processes and instability criteria and apply to the Sun or to a purely convective star.

7. Examine atomic processes that make thermal conduction insignificant and study local thermal equilibrium. Study electron mean free paths and transition rates that influence absorption and emission of photons.

8. Investigate processes that influence fusion rates in the solar core including collision cross-sections, kinetic energy distributions, and electromagnetic repulsion effects. Relate processes to the known formula for energy production rate as a function of composition, density, and temperature. Apply to solar structure and evolution.

9. Examine existing standard solar models to understand the mathematical methods, accuracy, and complexity. Use or develop numerical and/or analytical models to solve the equations of stellar structure and evolution. Relate the zero age Sun to the current Sun and examine stars with similar masses and compositions.

10. Analyze energy transport in the solid Earth, oceans, atmosphere, and other common physical systems, including, for example, an isothermal sphere radiating energy into space.

Project History on Solar Structure and Evolution

In the fall of 2003 I started a project to investigate the transport of radiant energy within the Sun. In the winter of 2004 I prepared an animated slide show to explain how the Sun creates, stores, and radiates energy. In the summer of 2004 I started a feasibility study to determine if the non-linear differential equations of stellar structure and evolution can be solved using MathCad or other simple mathematical techniques. My goal is to improve my understanding of the inner workings of the Sun and to define a reasonable senior project. A most interesting problem to pursue is to find a solution to the zero age solar structure and to relate composition change to spatially varying energy production rates. By stepping this model through time, the current composition can be estimated. The model can use existing time-dependent models of surface temperature and radius. We made great progress in understanding the physical processes within the Sun and the mathematical complexities of modeling the processes, but developing a complete math model was beyond the scope of the project.

The MathCad analysis developed in 2003 was based on Guzik’s 1994 temperature and density graphs in Carroll and Ostlie’s book. Analytical functions based mostly on Gaussian line shapes were fit to the data and were used to estimate the radiant and thermal energy trapped in the Sun and the time for radiant energy to escape from the solar interior. In addition, the gas pressure was estimated from the ideal gas law and from the equation of hydrostatic equilibrium. Opacity equations for electron scattering and absorption of photons were used to estimate photon mean free paths and to investigate the diffusion of radiant and thermal energy within the radiative zone. Efforts to reconcile the mean free paths with the radiative transport equation are incomplete.

In the summer of 2004, progress was made on numerous tasks.

1. We studied many books (Carroll, Bowers, Kippenhahn, Clayton, Collins, Harwit), papers, and websites (Bahcall, Dhillon, OPAL tables) to explore particle interactions (opacity and energy production) in the Sun.

2. We studied analytical models of opacity and energy production and numerical methods to solve differential equations including the use of polytropes to relate density and pressure to temperature.

3. I reviewed and revised my 2003 model to eliminate the previously unnoticed unphysical non-zero derivative at the center of the Sun and to improve the analytical fits by using Lorentz line shapes.

4. Several polytrope models were studied including one with a polytropic index that varies as the radius cubed.

5. Models were improved to include opacity and energy production as functions of composition, density, and temperature. Opacity is difficult to model accurately, especially in the convective zone where heavy ions may have high absorption bound electrons due to relatively low temperatures.

6. Nucleosynthesis reduces the number of particles in the core, raising the temperature. Core contraction increases density to maintain the pressure gradient. Higher core density and temperature result in higher fusion rates despite the reduction in hydrogen composition.

7. Previous analyses erroneously indicated that photons scatter many times before being absorbed. In fact electron scattering is uncommon and opacity is dominated by absorption by free electrons and by electrons bound in so-called metals. The full implications of this will require further study.

8. The Sun is a self gravitating ideal gas of known (or assumed) composition in hydrostatic and local thermal equilibrium. Density and temperature distributions can be determined by solving the non-linear stellar structure differential equations. The solution is complicated by the fact that the equation of radiative transport involves opacity and energy production, which depend on composition, density, and temperature. Furthermore the opacity function is very complicated and hard to determine.

9. Indefinite integrals and derivatives of analytical functions that approximate the temperature and density data were investigated with partial success.

10. Additional models were developed to try to solve the differential equations of stellar structure. MathCad has many features and functions that may be applicable, but none have proven successful so far because of the extreme complexity and non-linearity of this problem. Feasibility has been studied for a simplified system.

11. A flow chart was constructed to suggest an approach to solving stellar equations using trial functions for density or temperature. So far, the solve blocks have been too computationally intensive for MathCad.

Mathematical Considerations on Solar Structure and Evolution

Consider the four static stellar structure equations. One equation relates mass to density and another one relates luminosity to energy production (less any non-luminous energy loss like neutrinos). These equations can be used to eliminate explicit M and L dependence in the other two equations. In addition, the ideal gas law (which relates pressure to composition, density, and temperature) can be used to eliminate explicit pressure dependence from the equation of hydrostatic equilibrium. Approximate analytical expressions for opacity and energy production rates as functions of composition, density, and temperature can eliminate them from the equation of radiative transport.

In the end, the two integro-differential equations of local mechanical and thermal equilibrium can be expressed in terms of a known composition distribution function and two unknown functions for density and temperature. Imposing some initial values and/or boundary conditions can lead to solutions in principle. These two equations can be reduced to one equation by solving the ideal gas law for temperature and using that to eliminate explicit temperature dependence from the radiative transport equation making it a function of density only. To do this the explicit temperature dependence is also eliminated from the analytical expressions for opacity and energy production rate. These operations apply to a self-gravitating ideal gas in hydrostatic and local thermal equilibrium.

Physical Processes affecting Solar Structure and Evolution

Another perspective on the problem comes from separating the physical concepts from the mathematical complexities. It is often stated that the structure and evolution of the Sun results from the initial composition and mass of a dilute cool gas under the influence of all four fundamental interactive forces of physics. Gravity is the only long range force in an electrically neutral gas. As the gas contracts, its particles accelerate and radiate some of the energy away allowing a stable bound state to form eventually. Particles in close proximity are subject to electromagnetic interactions that result in elevated temperatures and pressures, which stabilize the structure from additional gravitational collapse.

As the hot gases radiate energy into space, electromagnetic effects of scattering and absorption produce an optical opacity which traps energy locally. Unlike some stars, electron scattering is uncommon everywhere within the Sun despite many popular descriptions to the contrary; opacity is dominated by absorption by free electrons and by electrons bound in so-called metals. High particle scattering cross-sections keep thermal conductivity low and minimize energy transport by conduction. In the Sun, unlike many stars, convective heat transport is insignificant except in the outer one percent of the mass (two-thirds of the volume due to low mass density) of the Sun where optical opacity is extremely high due to interactions of photons with bound electrons in so-called metals (any element other than hydrogen and helium).

Coulomb repulsion of positive nuclei prevents rapid fusion to occur at ordinary stellar temperatures resulting in stable long lives for ordinary stars. The densities and temperatures are sufficiently high to permit quantum tunneling to occur so that nuclear interactions can release energy as a by-product of thermonuclear fusion. This energy replaces energy that slowly diffuses to the surface where it is radiated into space.

I think I finally cracked the core contraction mystery and can explain why the Sun grows hotter and brighter even though the core hydrogen abundance decreases over time. My explanation is not simple, but it might be true. Hydrostatic support depends on pressure gradients, not pressure! So the issue is not whether pressure would drop if something else didn’t happen – the issue is maintaining a pressure gradient to provide hydrostatic support. Nucleosynthesis reduces the core hydrogen abundance. The decrease in core particles does not decrease the local energy density or pressure because the temperature rises as the average energy per particle rises. Radiant energy diffusion increases due to the increase in temperature gradient and perhaps due to reduced opacity as the number of electrons decreases. Energy diffusion increases the temperature just beyond the core. The higher temperature increases the pressure which compresses the core and expands the envelope. The core density, energy density, and pressure rise. Core density increases as gases contract gravitationally to maintain the pressure gradient required for hydrostatic support. So there are three factors that affect the fusion rate: one reduces it and two increase it, presumably the latter more than compensating for the former, resulting in the overall increase in fusion rate over time predicted by the codes. Specifically, decreases in core hydrogen abundance reduce the fusion rate because fewer protons are available and they encounter fewer protons. However, increases in core density and temperature increase the fusion rate. And naturally the increased thermal diffusion increases the Sun’s luminosity as its surface radius and temperature grow over billions of years.

In the summer of 2004, progress was made on numerous tasks including the following section.

Solar Formation and Evolution

How can the Sun grow brighter over time while the core hydrogen abundance decreases?

Core contraction is the dominant cause of the increase in solar luminosity with time, but core contraction itself is a response to an increased outward flow of energy caused by the fusion-induced reduction in core hydrogen abundance.

My detailed explanation of core contraction is not simple, but it does explain solar evolution. Fusion reduces the abundance of hydrogen and electrons in the core. The internal energy is redistributed among fewer particles, resulting in more energy per particle or higher core temperatures. Despite the lower hydrogen abundance, higher temperatures increase the fusion rate and allow more energy to flow outward. Rising temperatures increase the pressure, forcing the core to contract and the envelope to expand. Contraction maintains the pressure gradient necessary for hydrostatic support and raises the density and temperature of the core, further increasing fusion.

1. Energy generated by fusion replaces energy diffusing from the core to the surface.

2. Nucleosynthesis reduces the core hydrogen abundance and particle density.

3. Some core electrons are annihilated by positrons produced during nucleosynthesis.

4. Core opacity decreases as temperature rises and density of core electrons decrease.

5. The decrease in core particles does not decrease the local energy density or pressure.

6. The core temperature rises as the average energy per particle rises.

7. Decreases in core hydrogen abundance reduce protons available for fusion, but fusion rate increases slightly due to the increased core temperature.

8. Luminosity increases as the temperature and temperature gradient increase and opacity decreases.

9. Increased luminosity increases energy density and pressure at larger radii.

10. Pressure increase expands envelope and forces more particles into core.

11. Core contraction maintains the pressure gradient required for hydrostatic support.

12. Gravitational contraction increases core density, pressure, temperature, and energy density.

13. Fusion rate increases with core density and temperature – enough to sustain higher luminosity.

14. Solar envelope expands as its temperature rises, increasing the surface radius and temperature.

15. The Sun’s luminosity increases as its surface radius and temperature grow over billions of years.

Backtracking from solar evolution to solar formation, how did a cold dilute gas contract under gravitational attraction and produce a core hot enough and dense enough to sustain thermonuclear fusion? My simplified but detailed explanation of solar formation is more complete than most non-mathematical discussions.

1. Enormous molecular clouds resist gravitational contraction for billions of years with the help of kinetic energy, rotational energy, and magnetic fields until an external perturbation alters the properties of a portion of the cloud enough to trigger free fall contraction as gravitational attraction dominate other influences.

2. My simple explanation of solar formation will ignore rotation and magnetic effects and will assume the cloud is a cold dilute self-gravitating gas with uniform composition, density, and temperature and the mass of the Sun.

3. Gas particles in the cloud accelerate as they fall toward the center of mass because there is no hydrostatic support.

4. Gas density remains uniform as it increases because all particles have the same free fall time since velocity and acceleration increase linearly with radius since a = GM/r² = G(4πρr/3).

5. Collisions in the center raise the temperature, internal energy, and pressure producing temperature and pressure gradients as the opacity increases.

6. The developing pressure gradient provides some hydrostatic support for the increasingly dense core gases.

7. Falling particles continue to compress the core, increasing its density, pressure, and temperature.

8. The differential pressure reduces the contraction near the center producing a density gradient.

9. Gas opacity initially increases with density and temperature, trapping radiant energy in the interior.

10. Surface cooling by radiative transport also increases the interior temperature gradient.

11. The high opacity of the interior maintains the increased temperature gradient.

12. A convection instability forms and convection transports trapped interior heat from the core to the surface.

13. At very high temperature, opacity decreases as bound electrons are freed.

14. The core density increases enough to fuse hydrogen nuclei.

15. Radiative energy transport replaces convective energy transport except for the outer gases.

Web Resources

astrophysics book by Collins (who is now at CWRU: http://astrwww.cwru.edu/personal/collins/astrobook/

Bahcall’s Princeton home page: http://www.sns.ias.edu/~jnb/

LANL opacity models and codes and websites: http://www.t4.lanl.gov/

Lawrence Livermore OPAL website: http://www-phys.llnl.gov/Research/OPAL/opal.html

specific opal tables ftp://www-phys.llnl.gov/pub/opal/type1data/GN93/ascii/GN93hz

a solar model with a large table of parameters: http://www.ap.stmarys.ca/~guenther/solar/a_solar_model.html#anchor326506

OPAL table interpretation program xztrin21.f: http://www-phys.llnl.gov/Research/OPAL/existing.html

upgraded version of the Livermore code: http://www.krl.caltech.edu/~aib/kappa.html

Books on Astronomy and Astrophysics – introductory and advanced

1. Meir H. Degani, Astronomy Made Simple, QB44.2.D43 Doubleday 1976 edition.

2. 520 Robert Baker & Laurence Frederick, An Intro. to Astronomy, van Nostrand 1968. NHA

3. 523.2 The Solar System, Scientific American, 1975.

4. Gary Mechler, The Sun and the Moon, National Audubon Society Pocket Guide, Knopf 1995.

5. 523.2 Fred Whipple, Orbiting the Sun, QB601.W6 Harvard 1941-1981.

6. 522.2078 Fred Schaaf, Seeing the Solar System Telescopic Projects..., QB64.S4273 Wiley 1991.

7. 523 Fred Schaaf, The Starry Room, QB64.S43 Wiley 1988.

8. 523.3 Ernest Cherrington, Exploring the Moon through Binoculars, McGraw-Hill 1969.

9. Carl Sagan, Jonathan Leonard, Planets, Life Science Library Time Inc. 1966.

10. 520.93 Anthony Aveni, Ancient Astronomers, Smithsonian 1993. GN799.A8A84

11. 520.76 James Pickering, 1001 Questions about Astronomy, reference, Blanchard Community Library.

12. Guy Ottewell, The Astronomical Companion, 1979 -1992.

13. 522.076 Peter Duffett-Smith, Practical Astronomy with your Calculator, 3rd ed., Cambridge 1988. QB62.5.D83

14. Dale Ostlie & Bradley Carroll, Intro. to Modern Stellar Astrophysics, Addison-Wesley 1996 QB801.C25

15. Bradley Carroll & Dale Ostlie, Modern Astrophysics, Addison-Wesley 1996 QB461.C35

16. Richard Bowers & Terry Deeming, Astrophysics I Stars, Jones & Bartlett 1984 QB461.B64

17. R. Kippenhahn & A Weigert, Stellar Structure & Evolution, Springer-Verlag 1991 QB808.K57

18. Martin Harwit, Astrophysical Concepts, Springer 1998 QB461.H37

19. Donald Clayton, Principles of Stellar Evolution & Nucleosynthesis, McGraw-Hill 1968 QB80??

20. Martin Schwarzchild, Structure & Evolution of the Stars, Dover 1962

21. A.J. Meadows, Stellar Evolution 2nd Ed., Pergamon 1978 QB806.M4

22. V.C. Reddish, The Physics of Stellar Interiors, Edinburgh Univ. Press 1974 QB808.R42

23. V.Kourganoff, Intro. to The Physics of Stellar Interiors, D. Reidel 1973 QB808.K6813

24. John Cox & R.Thomas Giuli, Principles of Stellar Structure Vol.1 Physical Principles, Gordon & Breach 1968 QB801.C65 v.1