Title: On the Flux-Intensity Relation of Molecular Clouds

Abstract: In this work, we report a study on the relationship between flux and intensity for molecular clouds. Our analysis is established on high-quality CO images from the Milky Way Imaging Scroll Painting (MWISP) project. The new flux-intensity relation characterizes the flux variation of molecular clouds above specific intensity levels. We found that the flux-intensity relation exhibits two prominent features. First, the flux-intensity relation generally follows exponential shapes; secondly, hierarchical structures of molecular clouds are imprinted on flux-intensity relations. Specifically, 12CO flux-intensity relations are composed of one or more exponential segments, and for molecular clouds with segmented flux-intensity relations, the edge and the flux of the high-temperature component are strikingly consistent with 13CO emission. Further analysis shows that a similar relationship also exists between 13CO flux-intensity relations and C18O emission. The mean brightness temperature of molecular clouds is tightly associated with the decay rate of flux, the break temperature of exponential segments, and, to a certain extent, the flux fraction of the high-temperature component. Broadly, the flux-intensity relation of a molecular tracer, either in optically thick or in optically thin cases, has the capability to outline the silhouette of internal structures of molecular clouds, proving to be a potent tool for probing structures of molecular clouds.