Although the absorption efficiency at $\lambda >100~\mu$m is modified, we call the grain species “silicate” and “graphite” according to the adopted absorption efficiency at $\lambda <100~\mu$m.

The number density of grains with sizes between $a$ and $a+\mathrm{d}a$ is denoted as $n_i(a)\,\mathrm{d}a$, where the subscript $i$ denotes a grain species (silicate or graphite). Since there is little knowledge on the grain size distribution in BCDs, we simply assume a power-law form for $n_i(a)$: $$\begin{aligned}
n_i(a)=\mathcal{C}_ia^{-K}~~~(a_\mathrm{min}\leq a\leq
a_\mathrm{max})\, ,
\label{eq:size_dist}\end{aligned}$$ where $a_\mathrm{min}$ and $a_\mathrm{max}$ are the upper and lower cutoffs of grain size, respectively, and $\mathcal{C}_i$ is the normalizing constant. We assume $K=3.5$, $a_\mathrm{min}=3.5$ Å and $a_\mathrm{max}=0.25~\mu$m for both graphite and silicate [@mathis77; @li01]. As long as we treat the FIR colours of a single species (Section \[subsec:colour\]), the constant $\mathcal{C}_i$ cancels out. Because the dust composition in BCDs is still unclear, we consider silicate and graphite separately as two possible dust species.

Following HHS07, we adopt the multi-dimensional Debye models for the heat capacities of grains [@draine01]. The physical parameters necessary to characterize the heat capacities can be found in HHS07.

Calculation of FIR colours {#subsec:colour}
--------------------------

The FIR intensity $\mathcal{J}_\nu^i(\lambda )$ (per frequency per hydrogen nucleus per solid angle) of 