6805: 1094 Activities 6

Write your name and answers on this sheet and hand it in at the end.

Is 15 million degrees Celsius hot?

Goal: Make some estimates to put this temperature in perspective.

1. In the short nucleosynthesis video, the temperature 15 million degrees Celsius came up. Sounds big. But you always need to ask: "big on what scale?" First: What is this temperature in Kelvin?
   
   
   
   
2. Second: let's make a general conversion from K to energy in MeV in the form E(MeV) = α T(K). That is, you give me temperature in Kelvin and I'll give you energy in MeV. What are the units of α? Find α given that the average energy associated with temperature is T is roughly E = k_BT. [Shortcut: try Googling "Boltzmann constant in MeV/K".]
   
   
   
   
   
   
   
   
3. What is a typical energy for a nucleus? (E.g., recall what we know about nuclear binding energies.) What is T in K for this energy? So at 15 million degrees, will nuclei be dissociated?
   
   
   
   
   
   
   
   
4. What is a typical energy for an atom? (E.g., typical ionization energy for an atom.) What is T in K for this energy? So at 15 million degrees, will atoms be ionized?
   
   
   
   
   
   
   
   
5. Connect your last result to the claim in the video that it will take several hundred thousand years from the Big Bang before nuclei become neutral atoms.
   
   
   
   
   
   
   
   

Maxwell-Boltzmann distribution for the energy

Goal: Explore the energy available for reactions in the sun.

1. Open the Mathematica notebook Gamow_peak_explorations.nb and look at the first section on "Maxwell-Boltzmann distribution for the energy". Does the conversion factor used in finding kBTsun agree with what you used in the last section?
   
   
   
   
2. The function phiMB defines the Maxwell-Boltzmann distribution for the energy instead of the speed. Compare to slide 6 of Tunneling_and_Gamow_peak_slides.pdf. Write down this function below. Make sure you can see how the speed (not velocity!) distribution gets transformed into the energy (Ecom, where "com" means "center-of-mass") distribution. There is a v² in the speed distribution; why doesn't this become Ecom instead of the square root of Ecom?
   
   
   
   
   
   
   
   
3. Verify that the Mathematica function is normalized. What integral did you do?
   
   
   
   
   
   
   
   
4. We want to verify that the behavior for the energy distribution at 10 and 20 million K corresponds to that shown for the speed in slide 7 of Tunneling_and_Gamow_peak_slides.pdf. Plot the function for these temperatures (remember to convert to MeV!); you will need to appropriately adjust the x-axis range (which is in MeV) to be able to see a meaningful graph. What range did you use? Where are the peaks (roughly, in MeV or keV) and how far out are the tails non-negligible?
   
   
   
   
   
   
5. What is the probability at 15 million degrees that the energy is greater than 150 keV?
   
   
   
   
   
   
6. Show with Mathematica that the average value of the energy is 3 k_BT/2 (symbolically, not with numbers substituted). Did you succeed?