6805: 1094 Activities 6
Write your name and answers on this sheet and hand it in at the
end.
Work with others at your table on these activities. Argue about
the answers but work efficiently!
Is 15 million degrees Celsius hot?
Goal: Make some estimates to put this temperature in perspective.
 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?

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_{B}T.
[Shortcut: try Googling "Boltzmann constant in MeV/K".]
 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?
 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?
 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.
MaxwellBoltzmann distribution for the energy
Goal: Explore the energy available for reactions in the sun.
 Open the Mathematica notebook Gamow_peak_explorations.nb
and look at the first section on "MaxwellBoltzmann distribution
for the energy". Does the conversion factor used in finding
kBTsun agree with what you used in the last section?

The function phiMB defines the MaxwellBoltzmann 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
"centerofmass") distribution. There is a v^{2}
in the speed distribution; why doesn't this become Ecom
instead of the square root of Ecom?
 Verify that the Mathematica function is normalized. What integral did
you do?
 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 xaxis 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 nonnegligible?
 What is the probability at 15 million degrees
that the energy is greater than 150 keV?
 Show with Mathematica that the average value of the energy
is 3 k_{B}T/2 (symbolically, not with numbers substituted).
Did you succeed?
6805: 1094 Activities 6.
Last modified: .
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