from the collapse of a cloud core which is identical to that in the previous section, but which is initially in solid-body rotation with angular frequency $\Omega=7.6 \times 10^{-14}$ rad s$^{-1}$. Thus, the ratio of rotational energy to the magnitude of the gravitational potential energy is $\beta=0.005$ (i.e. the cloud is rotating quite slowly).

The evolution of the calculation is as follows (Figure \[bate.rotmassdens\]). The initial collapse is isothermal. When the density surpasses $10^{-13}$ g cm$^{-3}$, the gas in the center is assumed to become optically thick to infrared radiation and begins to heat ($t=1.009~t_{\rm ff}$). The heating stops the collapse at the center and the first hydrostatic core is formed ($t=1.015~t_{\rm ff}$) with maximum density $\approx  2 \times 10^{-11}$ g cm$^{-3}$, mass $\approx 0.01 {\rm \ M}_\odot$ ($\approx 3 \times 10^3$ particles), and radius $\approx 7$ AU. As the first core accretes, its maximum density only slowly increases at first. However, the first core is rapidly rotating, oblate, and has $\beta \approx 0.34 > 0.274$, making it dynamically unstable to the growth of non-axisymmetric perturbations [@Durisenetal86; @ImaDurPic99]. At $t \approx 1.023~t_{\rm ff}$, after about 3 rotations, the first core becomes violently bar-unstable and forms trailing spiral arms (Figure \[bate.bar\]). 