The Arp 240 Galaxy Merger: A Detailed Look at the Molecular Kennicutt-Schmidt Star Formation Law on Sub-kpc Scales

The molecular Kennicutt-Schmidt (mK-S) Law has been key for understanding star formation (SF) in galaxies across all redshifts. However, recent sub-kpc observations of nearby galaxies reveal deviations from the nearly unity slope (N) obtained with disk-averaged measurements. We study SF and molecular gas (MG) distribution in the early-stage luminous infrared galaxy merger Arp240 (NGC5257-8). Using VLA radio continuum (RC) and ALMA CO(2-1) observations with a uniform grid analysis, we estimate SF rates and MG surface densities ($Σ_{\mathrm{SFR}}$ and $Σ_{\mathrm{H_2}}$, respectively). In Arp 240, N is sub-linear at 0.52 $\pm$ 0.17. For NGC 5257 and NGC 5258, N is 0.52 $\pm$ 0.16 and 0.75 $\pm$ 0.15, respectively. We identify two SF regimes: high surface brightness (HSB) regions in RC with N $\sim$1, and low surface brightness (LSB) regions with shallow N (ranging 0.15 $\pm$ 0.09 to 0.48 $\pm$ 0.04). Median CO(2-1) linewidth and MG turbulent pressure (P$_{\mathrm{turb}}$) are 25 km s$^{-1}$ and 9 $\times$10$^{5}$ K cm$^{-3}$. No significant correlation was found between $Σ_{\mathrm{SFR}}$ and CO(2-1) linewidth. However, $Σ_{\mathrm{SFR}}$ correlates with P$_{\mathrm{turb}}$, particularly in HSB regions ($ρ>$0.60). In contrast, SF efficiency moderately anti-correlates with P$_{\mathrm{turb}}$ in LSB regions but shows no correlation in HSB regions. Additionally, we identify regions where peaks in SF and MG are decoupled, yielding a shallow N ($\leq$ 0.28 $\pm$ 0.18). Overall, the range of N reflects distinct physical properties and distribution of both the SF and MG, which can be masked by disk-averaged measurements.