– 1 –1. Star Formation At Low Metallicity1.1. The Baryons Collapse Within the HalosThe dark matter is collisionless and dominates t he total mass (hence gravity), so itscollapse proceeds without pressure effects. But pressure must be included when consideringthe collapse of baryons within a dark matter halo. We assume the dark matter has alreadycollapsed and virialized at redshift zvir, with a radial density profile of the dark matter ofthe form calculated by Navarro , Frenk and White. This produces a gravitational potentialφ(r) inside the dark matter object. We need to calculate the resulting gas distribution,ignoring cooling.At z < 100 the gas temperature is decoupled from the CMB, and its pressure evolvesadiabatically. Assuming the gas obeys the condition of hydrostatic equilibrium within thehalo, we findρb< ρb>= (1 −25µmpφk < T >)3/2where the mean temperature < T > refers to the background gas temperature a nd isµmp/k. Setting Tvir= −13mpφ/k as the virial temperature for the gravitational potentialdepth −φ, the overdensity of baryons at the virialization redshift isδb=ρb< ρb>− 1 = [1 +65Tvir< T >]3/2− 1.If we require δb> 100 (see §1.3 of the notes on cosmology and dark matterhalos for a justification of this choice), then Tvir> 17.2 < T >. For z∼< 160,< T > ≈ 170[(1 + z )/100]2K. So the required overdensity of baryons is achieved onlywithin halos where Tvir> 3 × 103[(1 + z)/1002] K , which defines a minimum mass for– 2 –baryonic objects ofMmin= 5 × 103[Ωmh20.15]−1/2[Ωbh20.022]−3/5[1 + z10]3/2M⊙≈ 5 × 103[1 + z10]3/2M⊙This minimum mass is quite close to the naive Jeans mass calculation from lineartheory given above.1.2. Cooling of 0Z GasFor solar metallicity gas, although the “metals” ar e only 2% of the total, it is themetals that generate dust, molecules, and cooling via atomic lines at temperatures below10,000 K, below which H is mostly neutral. The metals control the gas thermodynamics.Normally the timescale for thermal equilibrium is much shorter than typical dynamicaltimescales, and hence the equilibrium temperature is set at that of cold, dense clouds, whichis about 1 0 K. In effect, the gas self-regulates and is isothermal.However, primordial gas has no “metals”, and therefore cannot cool effectively once Hrecombines. Neutral H a nd He have few energy levels, all those of H a r e many eV abovethe ground state, a nd hence offer few possibilities for line radiation. They a re very poorradiators for T < 104K, and unless some other cooling mechanism enters, the cloud wouldcontract adiabatically. If this happens, the adibatic index of the gas is big enough so that apressure gradient develops which halts the gr avitational collapse; t he cloud is subesquentlyvirialized, and then evolves slowly within the long cooling time scale.Cooling at 0Z at any temperature involves the following processes: radiativerecombinations, collisional io nizations where the thermal energy of the electron is convertedinto ionizing the atom, bound-bound transitions which become rare below 10−4K as 9 eV– 3 –is required to excite the n = 2 level of H, and bremstrahlung emission (radiation due toacceleration of a charge in the Coulomb field of anot her charge).The cooling rate is parameterized by Λ(T )/n2H. In Ay 121 the rates of these variousprocesses were examined. At T > 105.5K, bremstrahlung dominates. For 103< T < 105.5K,collisional excitation of H and He dominate. Below 104K, the cooling rate for 0Z gas dropsvery rapidly.We consider the cooling time to be tc= 3π/[2nΛ(T )], where n is the tot al numberdensity for the gas. The free-fall time is tff=q32Gρ/(3π), and the Hubble time is a lsorelevant, tH= H(z)−1.Thus the sharp dro p in the coo ling rate for T < 104K f or 0Z gas causes the collapsetime to increase substantially, in effect preventing the collapse of primordial gas clouds.Below 104K, we must turn to cooling by H2, which can make a major contribution tothe cooling ra te even though the ratio of H2/H is very low, and even though H2moleculeshave no dipole moment. Another consequence of relying on H2for co oling is that the clouddoes not become optically thick until densities much higher than would occur in a solarmetallicity gas cloud of the same mass, so at least initially the gas is optically thin. Thedetailed cooling rates depend on the population of the low vibrational and rotational levelsof the molecule by collisional excitation; the deexcitation rates contain terms from bothcollisional and radiative interactions.The critical density is that at which the collisional excitation ra te equals that of theradiative decay rate, for H2it is about 104/cm3for the lowest r otational transition. Thenumber density of H2molecules ∝ n2(H) at low density with respect to the critical densityand to n(H) at high density, while the emissivity is proportio na l to the number density ofmolecules and to the radiative deexcitation rate. Detailed calculations are required to take– 4 –into account additional terms fr om dissociation of the molecules etc.Molecules normally f orm on the surface of dust grains. Since there is no dust at 0Z, themolecules must fo r m in the gas, which is much harder to do. A detailed consideration ofthe various reactions involved is required to eva luate the ratio of n(H)/n(H2). The relevantreactions require free electrons, which can only come from relics of ionization after theepoch of recombination. Eventually after the recombination time exceeds the Hubble time,the electron fraction remains fixed (“freezes out”); this occurs at z ∼ 70. After that, in theabsence of metals, this fraction is about 10−6, which is not enough to produce much cooling,and the virial temperature of objects which can cool is not decreased much below 10,000 K.To get enough cooling, we need this fraction to be about 10−3. This occurs as a resultof the collapse of dark matter structures, with the baryons following along later. Thefraction of H2is then boosted to the necessary level. Detailed calculations must include e,H, H+, H−, H2, and H+2, and are rather complex.The results of such calculations delineate the regime of T and redshift at which objectscan cool (tc<< tff). Calculations with a full set of the above species and reaction ratessuggest that H2cooling allows 3σ fluctuations at z ∼ 30 to collapse. These correspond tohalos with virial temperature of about 3000 K. Adding HD to the mix, even though it hasa