Fragmentation goes on as long as the energy from the cloud contraction is radiated away. When this is no longer the case, the process stops. Let us estimate the size of the smallest fragment in the opacity-limited fragmentation. The gravitational energy
|ß| » GM2/R of a cloud of mass M, radius R and average density q is liberated in a time of the order of 1/-/Gg . Thus, the gravitational power produced is
. GM2 i / 3 \ 2 G3 M5 £grav » GRr (G Q)1 = (¿j ^. (18.47)
The radiated power is at most that of the black body
where a is the Stefan constant and f is a numerical factor < 1. The collapse is isothermal as long as Érad > ¿¡,grav. Indeed, we notice from (18.47) that £grav does not depend on the mass of the collapsing configuration, since for a succession of fragments always at the limit of Jeans mass, the fragment masses vary linearly with radius. On the contrary, the radiated energy Èrad grows with R2, i.e., like M2. Thus, large configurations are always able to radiate their energy. The transition toward adiabatic collapse occurs for a mass small enough, i.e., when Èrad « Eigrav. This gives a limiting mass
64a;3 a2f2T8R9 1 G3
By eliminating the radius with R = (3/4k)1/3(Mj/q)1/3, we get c 64a3 a2f2T8 / 3 \3 /Miimx3
Fragmentation stops when the Jeans mass is equal to the mass limit, below which the contracting configuration is unable to radiate the gravitational power. The density at the Jeans limit (18.5) is
With this expression, we get for the lowest mass which can radiate the energy liberated by the collapse,
This is an approximate estimate of the lowest mass which can form by cloud fragmentation. This important result leads to several remarks:
1. For f « 1 and T « 10 K, the smallest fragment is ~ 0.03 M0. Thus the minimum stellar mass resulting from contraction and fragmentation of interstellar clouds is of the order of 10-2 M0.
2. This minimum stellar mass is lower than the mass MH = 0.08 M0, above which nuclear fusion of hydrogen is active. Stars with masses between Mlim and MH are brown dwarfs (Sect. 26.4.1).
3. For regions of lower metallicities Z, dust grains are less abundant, thus contracting clouds radiate less energy, i.e., f is likely smaller and Mljm larger. Thus, we might wonder whether very low-mass stars, such as brown dwarfs, also form at lower Z (as shown in Sect. 23.1.3, although the fragmentation process does not occur at Z = 0, very small hydrostatic cores nevertheless initially form, these small initial cores then lead to stars of different final masses depending on the amount of matter further accreted).
4. Objects with M < 0.01 M0, such as giant planets, do not result from fragmentation, but from the process of accumulation. In a protostellar disk, small condensations of rocks (in the core) and ice grow by collisions and accumulation of materials from the surrounding disk.
5. For objects near the limit M ~ 0.01 M0 in multiple systems, the reality is likely more complex than the rather schematic distinction between the processes of fragmentation and accumulation.
The initial mass spectrum or initial mass function, the so-called IMF, can be approximated by which is often called &(M) or |(M). The original Salpeter's slope obtained from stellar counts in open clusters is x = 1.35. A classical study  gives x = 0.4,1.5,1.7-2.0 in the ranges M=0.1-1 M0, M=1-10M0 and above 10 M0, respectively. Recent works [293, 333] confirm a nearly flat slope from about 0.2 to 0.6 M0 which turns into a power law above 1 M0 up to high masses with a slope equal or steeper than the Salpeter's law. There is no convincing differences in the IMF due to metallicity Z, e.g., between the Galaxy and the Magellanic Clouds. Indications exist that some regions of intense star formation, like starbursts, may be relatively richer in massive stars; in that case, one often speaks about a "top heavy IMF".
The thermodynamic state of the gas, which determines the density distribution in the clouds, appears to critically influence the mass spectrum and the range of stellar masses produced . Studies of cloud collapse  suggest that there are two different regimes for mass accretion onto protostars.
1. If the gas dominates the gravitational potential, as is the case initially, the motions of the stars and of the gas are the same. There is no large systematic motion differences between the two components and the accretion on a protostar is determined by the local gravitational potential of the protostar.
2. Later when stars dominate the gravitational potential, they virialize and have different motions from the gas, i.e., with larger relative velocities. The accretion in this case results mainly from the sweeping of the interstellar gas by the fast moving stars (the so-called Bondi-Hoyle accretion). The accretion is less dependent on the local gravitational potential. These two regimes of accretion may lead to different IMF slopes: (1 + x) = -1 -5 in the first case, which applies mainly to the low-mass stars and -2-5 in the second, which concern mostly the massive stars.
Among other results from numerical simulations, we note the following:
- The IMF is also shaped by chaotic interactions between protostars . This allows bursts of star formation to occur locally.
- Brown dwarfs and other stars could be formed from dynamical ejection of small fragments from unstable multiple systems.
- Star-disk encounters may form binaries, while stellar encounters rather destroy multiple systems.
- Close binaries may result from hardening of initially wider systems through successive encounters.
At present, the relative importance of fragmentation, collisions and accretion in shaping the different parts of the IMF is still uncertain.
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