The Luminosity Functions and Timescales of MYSOs and Compact HII regions

We present a determination of the luminosity functions of massive young stellar objects (MYSOs) and compact (C)HII regions within the Milky Way Galaxy using the large, well-selected sample of these sources identified by the Red MSX Source (RMS) survey. The MYSO luminosity function decreases monotonically such that there are few with $L\gtrsim 10^{5}$Lsol, whilst the CHII regions are detected up to ~10$^{6}Lsol. The lifetimes of these phases are also calculated as a function of luminosity by comparison with the luminosity function for local main-sequence OB stars. These indicate that the MYSO phase has a duration ranging from 4x10$^{5}$ yrs for 10$^{4}$Lsol to ~7x10$^{4}$ yrs at 10$^{5}$Lsol, whilst the CHII region phase lasts of order 3x10$^{5}$ yrs or ~3-10% of the exciting star's main-sequence lifetime. MYSOs between 10$^{4} Lsol and ~10$^{5}$ Lsol are massive but do not display the radio continuum or near-IR \HI{} recombination line emission indicative of an HII region, consistent with being swollen due to high ongoing or recent accretion rates. Above ~10$^{5}$ Lsol the MYSO phase lifetime becomes comparable to the main-sequence Kelvin-Helmholtz timescale, at which point the central star can rapidly contract onto the main-sequence even if still accreting, and ionise a CHII region, thus explaining why few highly luminous MYSOs are observed.