6805: 1094 Activities 7

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

Approximating an integral by an asymptotic expansion

Goal: Try out a new way to approximation certain integrals.

1. In the Mathematica notebook Gamow_peak_explorations_part_1.nb, with the Maxwell-Boltzmann distribution, you were asked to find the probability at temperature 15 million degrees that the energy is greater than 150 keV. Most people found that the integral evaluated to zero. First, check that if you used NIntegrate instead of Integrate that you can get a numerical answer. What is the result?
   
   
   
   
2. How could we get this answer approximately by hand (so to speak)? Yes, Mathematica could do this as well, but let's pretend we are on a desert island with only paper and pencil and need to do this integral to be rescued. We will derive an asymptotic expansion using integration by parts. Define a simpler function (set constants to 1) for practice (in Mathematica notation):
      f[t_] := NIntegrate[Sqrt[x] Exp[-x],{x,t,Infinity},Assumptions->{t>0}]
   which we want to evaluate for t much greater than 1 (e.g., t=100).
3. Recall the formula to integrate by parts:
      ∫ u  dv = uv - ∫ v  du,
   where the integration limits have been suppressed. Let's take u = x^1/2 in f(t). Figure out dv and then apply the formula. What do you get?
   
   
   
   
   
   
   
   
4. The first term (from uv) is our leading approximation. Calculate the f[100] and compare to this approximation for t=100 using Mathematica. Write down the results.
   
   
   
   
   
   
   
   
5. Finally, let's find an upper bound to the second term. If you replaced the Sqrt[x] factor by Sqrt[t], you could take it out of the integral, which you now can do by hand. What is the result? Argue that the actual answer must be less than this. [Hint: compare the integrands; would the approximate integrand be larger or smaller at each x if we put back Sqrt[x]?] Test your upper bound with Mathematica. How close is your approximation when t=100?
   
   
   
   
   
   
6. Extra: How would you get another term in the series in 1/t?
   
   
   
   
   
   

Discussion questions from the "Breakthrough video"

Goal: Answer (as best you can!) some basic questions about relativistic heavy-ion collisions.

1. There is a collision from 0:47 to 1:08 of the video. The time scale for a fireball is said to be "1 billionth of 1 billionth of 1 millionth of a second". How far does a particle travel at the speed of light in this time (in fm)? Is this consistent with the size of the fireball in the video (use the size of the initial nucleus as a ruler)?
   
   
   
   
2. What do you think is flying off in the collision and what are the "remnants" that will be detected? E.g., where are the quarks and gluons and where are hadrons like pions, protons, and neutrons? Are there photons?
   
   
   
   
   
   
3. In the simulation, the colliding nuclei don't stop but go seem to go through each other. So where does the energy for creating the fireball come from?
   
   
   
   
4. What are the lines in the Star detector "measurement" at the 1:13 mark (see 1:58 to 2:06 for more)? How does this relate to detection by a Geiger counter?
   
   
   
   
5. From 1:22 to 1:45 there is a brief discussion of the nature of the matter produced. We'll come back to the "liquid vs. gas" issue and focus instead on the energy. The temperature is said to reach 4 trillion degrees. What is this in MeV (use Mathematica notebook from last time)? Remember this number for below!
   
   
   
   

How close to the speed of light?

Goal: Approximate the speed of colliding gold nuclei, recalling a basic Taylor series along the way.

1. In the videos, they say that the colliding nuclei at RHIC are moving at very close to the speed of light before the collision. Let's calculate how close. We'll use the relativistic formula E = γ m, where we have set c=1 and γ² = 1/1-v². If a gold nucleus is accelerated to 200 GeV/nucleon, what is γ for each nucleon (roughly, no calculator or computer!)?
   
   
   
   
   
   
2. Solve for v as a function of γ. [Hint: square 1/γ, solve for v², then take the square root.]
   
   
   
   
   
   
3. γ is large so 1/γ is much smaller than 1. Expand the result for v in a Taylor series for two terms, by hand. [Hint: recall that (1+t)^a = 1 + at + O(t²).]
   
   
   
   
   
   

Questions about the "RHIC the Perfect Liquid" video and Decadal survey QGP slides 1 and 2

Goal: Discuss some basic questions about the nature of matter produced in relativistic heavy-ion collisions.

1. Reminder: What are hadrons, leptons, baryons, and mesons? Under which category is a pion, proton, neutron, photon, graviton, electron?
   
   
   
   
   
   
2. Slide 1, "The Phases of QCD slide" has many interesting features; we'll touch on a few of them here. The x-axis is the baryon chemical potential, which is related to the excess of quarks over anti-quarks. Do baryons have such an excess? Do all hadrons?
   
   
   
   
   
   
3. If the baryon chemical potential is zero, does that mean there are no hadrons? Does it mean there are no baryons?
   
   
   
   
   
   
4. If the colliding nuclei mostly pass through each other, do you expect the baryon chemical potential in the fireball to be large or small? What if the nuclei were mostly stopped?
   
   
   
   
   
   
5. Based on your answer earlier to the temperature of the fireball, where is the simulated collision on the phase diagram? What happens as it expands?
   
   
   
   
   
   
6. In the video from 1:06 to 1:18, a side-by-side simulated view of a liquid and gas is shown. Characterize the difference in behavior.
   
   
   
   
   
   
7. The claim is that the quark-gluon plasma (QGP) might be the most perfect liquid there is. But the viscosity of the QGP is 10¹⁴ times that of water! Reconcile the claim and this fact. [Hint: Look at slide 2 and pages 81-82 in the Decadal survey.]