We present a model of star formation in self-gravitating turbulent gas. We treat the turbulent velocity $v_T$ as a dynamical variable, and assume that it is adiabatically heated by the collapse. The theory predicts the run of density, infall velocity
, and turbulent velocity, and the rate of star formation in compact massive gas clouds. The turbulent pressure is dynamically important at all radii, a result of the adiabatic heating. The system evolves toward a coherent spatial structure with a fixed run of density, $rho(r,t)torho(r)$; mass flows through this structure onto the central star or star cluster. We define the sphere of influence of the accreted matter by $m_*=M_g(r_*)$, where $m_*$ is the stellar plus disk mass in the nascent star cluster and $M_g(r)$ is the gas mass inside radius $r$. The density is given by a broken power law with a slope $-1.5$ inside $r_*$ and $sim -1.6$ to $-1.8$ outside $r_*$. Both $v_T$ and the infall velocity $|u_r|$ decrease with decreasing $r$ for $r>r_*$; $v_T(r)sim r^p$, the size-linewidth relation, with $papprox0.2-0.3$, explaining the observation that Larsons Law is altered in massive star forming regions. The infall velocity is generally smaller than the turbulent velocity at $r>r_*$. For $r<r_*$, the infall and turbulent velocities are again similar, and both increase with decreasing $r$ as $r^{-1/2}$, with a magnitude about half of the free-fall velocity. The accreted (stellar) mass grows super-linearly with time, $dot M_*=phi M_{rm cl}(t/tau_{ff})^2$, with $phi$ a dimensionless number somewhat less than unity, $M_{rm cl}$ the clump mass and $tau_{ff}$ the free-fall time of the clump. We suggest that small values of p can be used as a tracer of convergent collapsing flows.
We use the WMAP maximum entropy method foreground emission map combined with previously determined distances to giant HII regions to measure the free-free flux at Earth and the free-free luminosity of the galaxy. We find a total flux f_ u=54211 Jy an
d a flux from 88 sources of f_ u=36043 Jy. The bulk of the sources are at least marginally resolved, with mean radii ~60 pc, electron density n_e ~ 9 cm^{-3}, and filling factor phi_{HII}=0.005 (over the Galactic gas disk). The total dust-corrected ionizing photon luminosity is Q=3.2x10^{53} photons/s, in good agreement with previous estimates. We use GLIMPSE and MSX 8 micron images to show that the bulk of the free-free luminosity is associated with bubbles having radii r~5-100 pc, with a mean ~20 pc. These bubbles are leaky, so that ionizing photons from inside the bubble excite free-free emission beyond the bubble walls, producing WMAP sources that are larger than the 8 micron bubbles. We suggest that the WMAP sources are the counterparts of the extended low density HII regions described by Mezger (1978). Half the ionizing luminosity from the sources is emitted by the nine most luminous objects, while the seventeen most luminous emit half the total Galactic ionizing flux. These 17 sources have 4x10^{51} < Q <1.6x10^{52}, corresponding to 6x10^4M_odot < M_*< 2x10^5M_odot; half to two thirds of this will be in the central massive star cluster. We convert the measurement of Q to a Galactic star formation rate dM/dt=1.3M_odot/yr, but point out that this is highly dependent on the exponent Gamma~1.35 of the high mass end of the stellar initial mass function.
The masses of star clusters range over seven decades, from ten up to one hundred million solar masses. Remarkably, clusters with masses in the range 10^4 to 10^6 solar mases show no systematic variation of radius with mass. However, recent observatio
ns have shown that clusters with masses greater than 3x10^6 solar masses do show an increase in size with increasing mass. We point out that clusters with m>10^6 solar masses were optically thick to far infrared radiation when they formed, and explore the hypothesis that the size of clusters with m> 3x10^6 solar masses is set by a balance between accretion powered radiation pressure and gravity when the clusters formed, yielding a mass-radius relation r~0.3(m/10^6M_odot)^{3/5} pc. We show that the Jeans mass in optically thick objects increases systematically with cluster mass. We argue, by assuming that the break in the stellar initial mass function is set by the Jeans mass, that optically thick clusters are born with top heavy initial mass functions; it follows that they are over-luminous compared to optically thin clusters when young, and have a higher mass to light ratio Upsilon_V=m/L_V when older than ~1 Gyr. Old, optically thick clusters have Upsilon_V~ mcl^{0.1-0.3}. It follows that L_V~sigma^{beta}, where sigma is the cluster velocity dispersion, and beta~4. It appears that Upsilon_V is an increasing function of cluster mass for compact clusters and ultra-compact dwarf galaxies. We show that this is unlikely to be due to the presence of non-baryonic dark matter, by comparing clusters to Milky Way satellite galaxies, which are dark matter dominated. The satellite galaxies appear to have a fixed mass inside a fiducial radius, M(r=r_0)=const.
