Douglas Boyd & A.P.
Whitworth
Ouro Preto , Brazil, April 2003

We present a new analysis of the minimum mass for star formation, based on opacity limited fragmentation.
Our analysis differs from the standard one (e.g. Rees 1976), which considers hierarchical fragmentation of a 3-D medium, and yields M min
≈
10-15M
Jupiter
for
Population I star formation.
Instead we consider the more realistic scenario in which there is one-shot fragmentation of a shock compressed layer, of the sort which arises in starforming turbulent molecular clouds. In this situation, M min
is significantly different, and can be much smaller.

Opacity-limited 3D hierarchical fragmentation yields a minimum stellar mass of order
10-15M
Jupiter
† . However, it is now generally accepted (Elmegreen, 2002, ApJ,577,206;
Padoan & Nordlund, 2002, ApJ,576,870) that star-forming clouds are transient turbulent entities, and that star-formation occurs almost as soon as the cloud forms, in a crossing time. In this picture, prestellar cores are created where turbulent elements collide with sufficient ram-pressure for the resulting shock-compressed layer to be gravitationally unstable. The geometry of fragmentation is therefore quite different from that involved in 3D hierarchical fragmentation. We have modelled opacity-limited fragmentation of a shock-compressed layer, and find that the minimum mass for fragmentation is reduced below 3M
Jupiter
. Since layer fragmentation is more likely to be permanent ‡ , this suggests a way of forming planetary-mass objects by gravitational instability.
†
This result was first obtained to order of magnitude by Rees (1976, MN,176,483), but the value above is the result of a more accurate treatment of the dynamics of fragmentation.
‡ rather than 3D fragmentation, because fragments of a layer condense out faster than the layer as a whole.

•The fragmentation of a 2D layer is fundamentally different from the fragmentation of a three-dimensional medium. For a static layer (e.g. Larson 1985), the Jeans radius and mass are given in terms of the surface density of the layer. The time-scale for a
Jeans-unstable fragment to condense out has a minimum for R =2 R
Jeans
and M =4 M
Jeans
, i.e. fragments on this size scale condense faster than smaller fragments and faster than larger fragments.
•Given these differences between hierarchical fragmentation of a three-dimensional cloud and one-step fragmentation of a two-dimensional layer, we ask how, and at what mass scale, opacity limits the fragmentation of a layer?
•We consider two identical streams of gas of density ρ
0 a
0
, which collide at relative speed 2 v
0
and isothermal sound speed
(see Figure 1). We assume that a plane contact-discontinuity forms where the two streams meet. The two streams approach the contact discontinuity with velocities ± v
0
, and a self-gravitating plane-parallel layer forms symmetrically about the contact discontinuity, bounded by two accretion shocks and supported by hydrostatic forces.
•We assume that radiative cooling in the shocked gas is so efficient that the gas cools to its pre-shock temperature very quickly. Ram pressure due to the inflowing gas confines the layer in the direction perpendicular to the contact discontinuity.

In our model, we wish to take into account the material which continues to flow into the shocked layer after it has started to fragment. We also want to follow the development of a fragment as it condenses out of the layer. We do this by treating the condensing fragments as uniform-density spheroids of radius R and half-height Z .
Figure 1

Whilst the layer continues to accrete matter from the converging flow, the mass of the fragment will increase. There are two perpendicular pressures acting on the fragment: ram pressure along Z due to the inflowing gas and radial pressure along R from gas already inside the layer.
To determine the opacity limit of a condensing fragment, we compare
• the compressional heating rate due to the gravitational contraction;
• the heating arising from the dissipation of kinetic energy of the inflowing gas;
• the maximum (i.e. blackbody) cooling rate .
The opacity limit is reached when the total heating rate is equal to the cooling rate .
The evolution of a fragment, and hence its opacity limit, depend on five parameters:
• the sound speed in the gas, a
0
(fixed at 0.2kms
-1 , molecular Hydrogen at 10K);
• the density of the gas, ρ
0
;
• the inflow velocity of the accreting gas, v
0
;
• the initial radius of the fragment, R initial
;
• the initial time at which the fragment condenses out, t initial
.

In order to determine the minimum mass for opacity-limited fragmentation, we take the fragment mass at the point at which it becomes opaque.
We determine this by varying v density ρ
0
0
, ρ
0
, R initial
, t initial
to find the maximum possible gas before the fragment can no longer cool efficiently.
To be certain that a fragment will continue to condense and form a distinct object, we set a number of fragmentation conditions that must be satisfied at the opacity limit:
Fragment velocities in R and Z are both negative , i.e. the fragment is collapsing along both axes;
The final size (both radius R and half-height Z) of the fragment must be less than half its initial radial length R initial
.
Compressional heating is the dominant source of heating until v
0 above this, heating due to the inflow takes over.
≈ 6a
0
. At velocities
At the opacity limit, the condensing fragment may be either an oblate disc or a prolate spindle.
An increase in the local gas density leads to a lower final fragment mass.

This 3-dimensional plot shows the mass (in
Jupiter masses) of fragments that can be formed over a range of initial conditions.
Each point represents a fragment that satisfies our fragmentation conditions.
Here we have fixed v
0
and ρ
0
such that v
0
=7a
0
(1.4kms
-1 ), ρ
0
=2 × 10 -16 gcm -3 and varied t initial
and R initial
.
The lowest mass fragments have an initial radius at which they are just Jeansunstable (R initial
≥ R
Jeans
). Once a protofragment has accreted sufficient mass, it starts to condense out of the layer and collapse gravitationally.
The gaps in the plot indicate that under certain conditions, a distinct fragment may not form because it has not collapsed to half its initial radius or its height in Z is still growing.

The upper figure shows the variation of maximum gas density ρ
0 velocity, v
0 with inflow
. This curve provides an upper limit for the density at which a fragment may form for a particular inflow velocity.
The lower figure shows the variation in fragment mass as a function of inflow velocity at the maximum density indicated in the figure above. This curve gives a lower limit for fragment mass .
We find a minimum mass of ≈ 2.6M
for an inflow velocity ≈ 6.4a
and a density ≈ 6 × 10 -16 gcm -3 .
0
Jupiter
(1.3kms
-1 )

In 3D hierarchical fragmentation, the minimum mass is thought to be of the order of 10-15M
Jupiter
.
We have modelled opacity-limited fragmentation in a 2D shockcompressed layer. We find that the minimum mass is much smaller than is possible in 3D hierarchical fragmentation and, under certain conditions, can be lower than 3M
Jupiter
.
This new analysis demonstrates that low-mass objects can be formed quite easily through gravitational fragmentation, the same process that is widely believed to form stars.
References
Larson R.B., 1985, Mon.Not.R.Astron.Soc
, 214 , 379
Rees M.J., 1976, Mon.Not.R.Astron.Soc
, 176 , 483