gin{aligned}
{\boldsymbol{F}}^{(0)}(t)=
-\frac{g_p}{\pi^2a^2}\int d{\boldsymbol{z}} e^{-\frac{z^2}{2 a^2}} \int d{\boldsymbol{z}}' K_0(2|{\boldsymbol{z}}-{\boldsymbol{z}}'|)
\left[\left(\partial_t-{\boldsymbol{V}}_p\cdot\nabla_{{\boldsymbol{z}}'}-\frac{\gamma}{2}\nabla^2_{{\boldsymbol{z}}'}\right)\delta{\boldsymbol{w}}^{(0)}({\boldsymbol{z}}',t) + {\boldsymbol{\dot V}}_p(t) \right] .
\label{eq:F0full}\end{aligned}$$

The above expression is a weighted average of contributions from properties of the fluid velocity in a neighborhood of the impurity center-of-mass position (${\boldsymbol{z}} = 0$ in the comoving frame). The size of this neighborhood is given by the combination of the range of the Bessel function kernel, which in dimensional units would be the correlation length $\xi$, and the range of the Gaussian potential, $a$, giving an effective particle size. In classical fluids, the analogous force on a spherical particle involves the average of properties of the undisturbed velocity field within the sphere size [@parmar2012equation], and there is no equivalent to the role of $\xi$.

As in the classical case [@maxey1983equation; @parmar2012equation], if fluid velocity variations are weak at scales below $a$ and $\xi$, we can approximate the condensate velocity by a Taylor expansion near the impurity, i.e.: $$\begin{aligned}
\delta w_i^{(0)}(\mathbf z',t)&\approx& \delta
w_i^{(0)}(t)+\sum_j
e_{ij}(t) z'_j \nonumber\\
&+&\frac{1}{2}\sum_{jk}e_{ijk}(t)  z'_j z'_k + \ldots,\end{aligned}$$ where the indices $i,j,k=x,y$ denote the coordinate components. $e_{ij}(t)=\partial_j \delta w_i^{(0)}(\mathbf z,t)|_{{\boldsymbol{z}}=0}$ and $e_{ijk}(t)=\partial_j
\partial_k \delta w_i^{(0)}(\mathbf z,t)|_{{\boldsymbol{z}}=0}$ are gradients of the unperturbed condensate relative velocity. Inserting this expansion into Eq. (\[eq:F0full\]), and performing the integrals of the Gaussian and of the Bessel function (using for example $\int K_0(2|{\boldsymbol{z}}|)d{\boldsymbol{z}}=\pi/2$ and $\int z_i z_j K_0(2|{\boldsymbol{z}}|)d{