In the Milky Way there are thousands of stellar clusters each harboring from a hundred to a million stars. Although clusters are common, the initial conditions of cluster formation are still not well understood. To determine the processes involved in
the formation and evolution of clusters it is key to determine the global properties of cluster-forming clumps in their earliest stages of evolution. Here, we present the physical properties of 1,244 clumps identified from the MALT90 survey. Using the dust temperature of the clumps as a proxy for evolution we determined how the clump properties change at different evolutionary stages. We find that less-evolved clumps exhibiting dust temperatures lower than 20 K have higher densities and are more gravitationally bound than more-evolved clumps with higher dust temperatures. We also identified a sample of clumps in a very early stage of evolution, thus potential candidates for high-mass star-forming clumps. Only one clump in our sample has physical properties consistent with a young massive cluster progenitor, reinforcing the fact that massive proto-clusters are very rare in the Galaxy.
While not generally a conservation law, any symmetry of the equations of motion implies a useful reduction of any second-order equationto a first-order equation between invariants, whose solutions (first integrals) can then be integrated by quadratur
e (Lies Theorem on the solvability of differential equations). We illustrate this theorem by applying scale invariance to the equations for the hydrostatic equilibrium of stars in local thermodynamic equilibrium: Scaling symmetry reduces the Lane-Emden equation to a first-order equation between scale invariants un; vn, whose phase diagram encapsulates all the properties of index-n polytropes. From this reduced equation, we obtain the regular (Emden) solutions and demonstrate graphically how they transform under scale transformations.
