Scaling laws for multi-object spectrographs: empirical relationships and the photonic advantage of CAWSMOS

Conventional multi-object spectrographs (MOS) incur masses and costs that scale unfavourably with channel count~$N_{\mathrm{c}\mathrm{h}}$ and resolving power~$\mathcal{R}$. A dataset of eight fibre-fed MOS-VIS instruments (1995–2024) is compiled and analysed. Ordinary least-squares fitting yields a mass function $M\propto N_{\mathrm{c}\mathrm{h}}^{0.78\pm 0.28}$; the resolving-power exponent is empirically consistent with zero ($\gamma =0.20\pm 0.37$) but is predicted analytically to lie in ${\mathcal{R}}^{0.8\text{–}2.0}$. Leave-one-out cross-validation confirms sub-linear channel-count scaling ($\beta \in [0.59,1.13]$, mean 0.79), with HERMES and AAOmega – sharing the same telescope and $N_{\mathrm{c}\mathrm{h}}=392$ across a factor of~7 in $\mathcal{R}$ at essentially equal estimated mass – providing a direct empirical confirmation that $\gamma \approx 0$. Cost per channel decreases as $N_{\mathrm{c}\mathrm{h}}^{-0.22\text{ to }-0.53}$ for all assumed cost–mass exponents $k\in [0.6,1.0]$, demonstrating a genuine economy of scale. The photonic integrated-circuit (PIC) approach embodied by the arrayed waveguide grating (AWG)-based PAWS demonstrator and the proposed CAWSMOS instrument decouples dispersive element size from $N_{\mathrm{c}\mathrm{h}}$ and $\mathcal{R}$, with chip-area scaling as ${\mathcal{R}}^{1\text{–}1.5}$ instead of ${\mathcal{R}}^3$. Applied to the Wide-field Spectroscopic Telescope (WST), the model predicts $\sim$140 tonnes for a conventional NIR spectrograph system against $\sim$200 kg for a CAWSMOS equivalent, with projected costs of EUR 300–700M conventional versus EUR 40–120M photonic ($6$–$15\times$ reduction). AWG-based photonics are identified as a strategially important enabling technology for WST.