and In section IV, we showed that c$_{1}$ and c$_{2}$ in (25), which are expansions in the small parameter ϵ$_{LP}$x , are excellent approximations to the speeds of first and second sound at all temperatures (see figure 1). We thus expect (B6) and (B7) to be similarly good approximations for the velocity fields at all temperatures. Using (B4) to remove the dependence of these expressions on v$_{¯}$ $_{s}$, they reduce to and When ϵ$_{LP}$ is set to zero, we recover the results in (B2), describing pure in-phase and out- of-phase oscillations of the superfluid and normal fluid components. However, we note that even though ϵ$_{LP}$ vanishes as T → 0, so does the normal fluid density ρ$_{n}$$_{0}$ , and it is not obvious that √ ρ$_{s}$$_{0}$ϵ$_{LP}$x/ρ$_{n}$$_{0}$ is small at low temperatures. Conversely, at high temperatures, T → T$_{c}$ , the factor of √ ρ$_{n}$$_{0}$/ρ$_{s}$$_{0}$ multiplying √ ϵ$_{LP}$x in (B8) and (B9) will enhance the small parameter √ ϵ$_{LP}$x . It is thus crucial to know the temperature dependencies of ρ$_{s}$$_{0}$/ρ$_{n}$$_{0}$ and ϵ$_{LP}$x in both limits. At low temperatures, the thermodynamics is dominated by Goldstone phonons. Using standard results in the phonon regime, one can show (see A) that in the T → 0 limit, for a unitary Fermi gas. This non-zero limiting value as T → 0 arises from the fact that both ϵ$_{LP}$x and ρ$_{n}$$_{0}$ vanish as T $^{4}$. For a Fermi gas at unitarity in the T → 0 limit, using (B10) and the fact that x = 1 / 3 (see A), (B8) and (B9) give even though ϵ$_{LP}$ vanishes as T → 0 (as shown in figure 3). If we had simply set ϵ$_{LP}$ = 0, as often done in the superfluid literature, we would instead obtain the results given by (B2). In figure 10, we plot the velocity ratios in (B8) and (B9) for a superfluid unitary Fermi gas. It is clear that the superfluid and normal fluid velocities which are involved in first 28 v (1) s v (1) n ≃ 3 5 , ρ$_{s}$$_{0}$v (2) s ρ$_{n}$$_{0}$v (2) n ≃ − 2 , (B11) √ ρ$_{s}$$_{0}$ ρ$_{n}$$_{0}$ ϵ$_{LP}$x ≃ 2 3 (B10) v (1) s v (1) n ≃ 1 − √ ρ n 0 ρ s 0 ϵ$_{LP}$x 1 + ρ s 0 ρ n $_{0}$ϵ$_{LP}$ x (B8) ρ$_{s}$$_{0}$v (2) s ρ$_{n}$$_{0}$v (2) n ≃ −$^{(1}$ − x ) + √ ρ s 0 ρ n $_{0}$ϵ$_{LP}$ x (1 − x ) − ρ n 0 ρ s 0 ϵ$_{LP}$x . (B9) ρ$_{s}$$_{0}$v (2) s ρ$_{n}$$_{0}$v (2) n ≃ − v 2 ¯ $_{s}$(1 − x ) + ρ s 0 ρ n $_{0}$b 2 v 2 ¯ $_{s}$(1 x ) − b 2 . (B7)