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Testing the star formation scaling relations in the clumps of the North American and Pelican cloud complexes

The processes which regulate the star-formation within molecular clouds are still not well understood. Various star-formation scaling relations have been proposed to explain this issue by formulating a relation between star-formation rate surface density ($\rm Σ_{SFR}$) and the underlying gas surface density ($\rm Σ_{gas}$). In this work, we test various star formation scaling relations, such as Kennicutt-Schmidt relation, volumetric star-formation relation, orbital time model, crossing time model, and multi free-fall time scale model towards the North American and Pelican Nebulae complexes and in cold clumps associated with them. Measuring stellar-mass from young stellar objects and gaseous mass from CO measurements, we estimated mean $\rm Σ_{SFR}$, star formation rate per free-fall time, and star formation efficiency (SFE) for clumps to be 1.5 $\rm M{_\odot}~yr^{-1}~kpc^{-2}$, 0.009, 2.0$\%$, respectively, while for the entire NAN complex the values are 0.6 $\rm M{_\odot}~yr^{-1}~kpc^{-2}$, 0.0003, and 1.6$\%$, respectively. For clumps, we notice that the observed properties are in line with the correlation obtained between $\rm Σ_{SFR}$ and $\rm Σ_{gas}$, and between $\rm Σ_{SFR}$ and $\rm Σ_{gas}$ per free-fall time and orbital time for Galactic clouds. At the same time, we do not observe any correlation with $\rm Σ_{gas}$ per crossing time and multi free-fall time. Even though we see correlations in former cases, however, all models agree with each other within a factor of 0.5 dex, and discriminating between these models is not possible due to the current uncertainties in the input observables. We also test the variation of $\rm Σ_{SFR}$ versus the dense gas, but due to low statistics, a weak correlation is seen in our analysis.