Last Time... We discussed energy sources to a shell of stellar material:

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Cora O’Neal’
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1 Energy Sources We are straying a bit from HKT Ch 6 initially, but it is still a good read, and we'll use it for the summary of rates at the end. Clayton is a great source

2 Last Time... We discussed energy sources to a shell of stellar material: The gravitational term represents local expansion or contraction For the nuclear term, we focused (so far) on non-resonant reactions Neutrinos are generally an energy sink (except for neutron stars)

3 Mass Excesses We can instead deal with binding energies:

4 Binding Energies The binding energies are related to the mass excesses Models for the binding energy exist: e.g. liquid drop model (Wikipedia/Fastission)

5 Non-Resonant Rates We argued that the rate (# of reactions/time/volume) is In the CM frame, the relative velocity obeys a Maxwell-Boltzmann distribution: The cross section will have strong dependence due to tunneling and the de Broglie cross section, which we factored out, giving: This gives:

6 Cross Section (Clayton)

7 Cross Section Notice how smooth S is in our region of interest we can take it to be ~ constant (Clayton)

8 Gamow Peak Note how well the Gaussian approximation fits the Gamow peak:

9 Non-Resonant Rates We can assume that S is constant Integrate the Gaussian (extending lower limit to - ) Energy generation rate: Finally, we estimated the temperature sensitivity in the form:

10 Resonant Rates At a resonance, the cross section is strongly peaked Energy corresponds to an energy level in the compound nucleus resulting from X + a Dominates the cross section Integrate assuming the M-B term doesn't change much in this range Different T dependence results in the rate (Clayton)

11 12 13 C(p,γ) N Q = MeV (just difference the mass excesses) Two parts to the rate Low temperature (note the dominant terms): Resonance: if Total rate is the sum for 0.25 < T9 < 7 (T9 ~ 0.6 for this reaction)

12 Weak Rates That reaction leaves us with 13N We could do 13N(p,γ)14O (Q = MeV) next But 13N is unstable and will beta decay We need to consider timescales Consumption via 13N(p,γ)14O changes 13N number density as: Destruction timescale is From your text, for T6 = 20, X ~ 1, and ρ ~ 10, we have τ ~ 107 yr.

13 Weak Rates But what about beta decay? has a half-life of 10 min We can ignore the 13N(p,γ)14O reaction in this case Evolution of 13N consists of creation and destruction (via beta decay) as: Here, Note: there will be corresponding evolution equations for p, 12C and 13C. Even through 13N is unstable, an equilibrium amount will be present, given by:

14 number of protons (Z) Chart of Nuclides number of neutrons (N) (Taken from National Nuclear Data Center/BNL)

15 Weak Rates There are also electron captures Consider pp chain: Rate requires detailed modeling and understanding of the weak interaction Your text lists the following effective rate: Q value would be MeV but neutrinos carried almost all of this out of the star Electron captures also important in core-collapse supernovae

16 Integrating Reaction Networks Consider the reaction: A + A C + d Evolution is: Note the 2 in the destruction of A, since 2 A's are consumed for each reaction Note all have the 1/2 in the actual rate, to account for double counting This system needs to be integrated together Other rates that affect these species will appear as additional terms on the RHS

17 Integrating Reaction Networks Typically the evolution equations are written in terms of molar fractions : Recalling number density: Our system becomes: Molar fractions are nice because the number of each particle consumed still appears on the RHS, making comparison to the reaction easy

18 Integrating Reaction Networks The quantity: is what is normally tabulated in rate compliations. Finally, what about mass fractions:

19 Integrating Reaction Networks Since we have

20 Integrating Reaction Networks There are tradeoffs in which variables to use For reacting flows: the X's are the natural choice for hydrodynamics, since you can write a conservation law for them. Y's are more natural for the reaction network itself

21 Integrating Reaction Networks The system of reaction rates tends to be stiff Consider the ODE (example from Byrne & Hindmarsh 1986): This has the exact solution: Looking at this, we see that there are two characteristic timescales for change, τ1 = 1 and τ2 = 10-3 A problem with dramatically different timescales for change is called stiff Stiff ODEs can be hard for some ODE integration methods Generally this requires an implicit ODE integrator

22 Integrating Reaction Networks What about a system of equations? Consider the following: This is a linear system of ODEs This models reactive flow with forward and inverse reactions (system from F. Timmes summer school notes) In matrix form: Or:

23 Integrating Reaction Networks This problem has the solution: for Characteristic timescale for change here is t = 1/( + ) Long term behavior: YB/YA / Also, YA + YB = 1 and d(ya + YB)/dt = 0

24 Integrating Reaction Networks Backward-Euler discretization of this system: Solving for the new state: This is a matrix inverse Alternately, we can write this as a linear system: Analogous to Ax = b

25 Integrating Reaction Networks Notice the R-K 4 method does well when we step at < characteristic timescale of change

26 Integrating Reaction Networks At 5 the characteristic timescale, R-K 4 fails completely. Backward-Euler still has a large error (it's only first-order), but remains stable.

27 Integrating Reaction Networks For stiff systems, robust integration packages exist. VODE is a popular one (F77, but also in python/scipy) Uses a 5th-order implicit method, with error estimation (alternately, a 15th order explicit method) Can either use a user-supplied Jacobian or compute one numerically Two types of tolerances: absolute and relative these have a big influence on your solution Will do adaptive timesteps to reach your specified stop time

28 Integrating Reaction Networks Variable timestep methods like VODE should do well here, since the solution is flat for most of the time, allowing larger timesteps.

29 Reaction Rate Libraries JINA ReacLib (Cyburt et al. 2010) >4500 nuclei w/ > reactions

Approximate Networks Carrying lots of nuclei is computationally expensive For helium-rich environments, alpha-chains provide the bulk of the energetics, but.

30 Approximate Networks Carrying lots of nuclei is computationally expensive For helium-rich environments, alpha-chains provide the bulk of the energetics, but... (α,p)(p,γ) may be faster than (α,γ) Intermediate nuclei doesn't last long, so we can just carry the end result using the rates for the sequence (α,p)(p,γ) (Frank Timmes)

31 Approximate Networks Jacobian for a 19-isotope network (handles pp, CNO, and alpha) (Frank Timmes)

32 Screening Electron screening: cloud of electrons shield the nuclear charges easier to overcome the Coulomb potential Two important scales: Wigner-Seitz radius (inter-ion spacing): Debye radius (measure of the distance over which electrostatic effects act): Weak screening: or equivalently:

33 Screening Basic idea: In weak screening regime, the electric potential of an ion surrounded by e- cloud is We are interested in small radii (near the classical turning point), so Since electrostatic PE is Screening reduces Coulomb barrier by Alternately, KE in CM frame of particles boosted by this amount Approximate screening effect by: End result (see your text):

34 Screening Strong screening is harder to deal with Ex: carbon burning in pre-sne Ia WDs (see Woosley et al. 2004)

35 pp Chain There are 3 different pp chains all end with 4He PP-I PP-II PP-III with increasing T All start with p+p reaction (slowest of all) PP-I: Last reaction: 3He + 3He is the next slowest can get equilibrium abundance PP-II: T increase drops 3He equilibrium abundance He always present so 4He + 3He takes over 4 (HKT)

36 pp Chain PP-III kicks in when p capture on 7Be faster than e- capture Be highly unstable Overall rate can be taken as proportional to initial p+p rate Crossing point of two reactions gives T6 ~ 20 8 Q value depends on mix of reactions, because of neutrino losses (HKT)

37 pp Chain in the Sun All of the neutrino producing reactions in the Sun (Wikipedia)

38 CNO cycle (Wikipedia)

39 CNO cycle There are 3 (4?) CNO cycles (Frank Timmes)

40 CNO cycle Single nucleus decay faster than the first reaction in pp chain All end with 4He Very sensitive to abundance of C,N,O Overall rate set by 14N(p,γ)15O This will be the most abundant CNO nucleus in equilibrium (Zelik & Smith) Mix of proton captures and beta decays

41 Helium Burning He + 4He 8Be is endothermic 4 Lifetime only s Equilibrium abundance at higher T: use Saha eq. to find it Be(α,γ)12C captures to excited state 8 At high T, some will decay to ground state rather than doing the inverse reaction Screening for the 3-alpha rate is hard to work out Next up: 12C(α,γ)16O This rate has a large uncertainty very hard to measure experimentally Other alpha captures happen at higher T You book discusses other reactions, we'll see them as needed

42 Neutrino Emission Neutrinos can be created by means other than weak reactions Pair Annihilation: Electron-photon scattering that gives neutrino/antineutrino pair instead of photon Plasma neutrinos High temperatures (kt ~ mec2) generate Photoneutrinos and Bremsstrahlung neutrinos Except for NS, these are an energy loss Electromagnetic waves become plasmons that can decay into neutrino/antineutrino pairs Many references exist which give rates for these different mechanisms

Fundamental Stellar Parameters. Radiative Transfer. Stellar Atmospheres. Equations of Stellar Structure

Fundamental Stellar Parameters Radiative Transfer Stellar Atmospheres Equations of Stellar Structure Nuclear Reactions in Stellar Interiors Binding Energy Coulomb Barrier Penetration Hydrogen Burning Reactions