84 Chapter 3: v$_{2}$ vs. R$_{AA}$ as a Constraint on sQGP Dynamics I$_{2}$ = ∫ ∞ 0 d l l$_{0}$ l + l$_{0}$ ρ$_{part}$ ( x + ( l + l$_{0}$ ) n$_{x}$, y + ( l + l$_{0}$ ) n$_{y}$ ) . (3.8) the 1D expanding medium: Note that I$_{2}$ has one less power of l in the integrand; this allows a unique determination of the two free parameters ( κ, k ) when fitting the model to a particular ( R$_{AA}$, v$_{2}$ ) point. Using the same hard sphere geometry as before, we are able to fit the 20-30% centrality Phenix π 0 R$_{AA}$ ( φ ) data point with κ = . 5 and k = . 25. Keeping κ and k fixed, Fig. 3.2 shows the results when the impact parameter is varied (long-dashed, blue curve); one can see that this matches the Rhic R$_{AA}$ and v$_{2}$ trends quite well. Taking Eq. (3.7) seriously, we invert the relationships and solve for dN$_{g}$/dy . Recalling that µ = gT , λ − 1 ≈ 9 πα 2 $_{s}$( ρ/ 2) 1 / $^{3}$/ 4, and the density for an expand- ing medium is ρ = dN$_{g}$ dy / $^{(}$A$_{⊥}$ ( L $_{2}$) $^{)}$, we find that, as a function of parton energy E , path length L , formation time l$_{0}$ , and α$_{s}$ , where ˜ v$_{1}$ and ˜ v$_{2}$ correspond to the bracketed terms in the energy loss and energy gain approximations, Eqs. (3.2) and (3.6), respectively. For our fitted values of κ and k , the choice of E = 6 GeV, L = 5 fm, dN rad g dy ∼ κ 4 E 9 πC$_{R}$α 3 $_{s}$˜ v$_{1}$ l$_{0}$L l$_{0}$ + L N$_{part}$ (3.9) dN abs g dy ∼ k 4 E 2 3 πC$_{R}$α 2 $_{s}$˜ v$_{2}$ l$_{0}$L l$_{0}$ + L N$_{part}$, (3.10)