Gravitational Instability of Thin Disk

Disks are common in the Universe.
Spiral and barred spiral galaxies have disks where stars are formed.
In more small scale, gas and dust disks are often found around protostars.
Moreover, such a disk may become a proto-planetary disk.
It is valuable to study how the self-gravity works in such thin structures.
Here, we assume a thin disk extending in - and -directions
whose surface density is equal to
, in other word
the density is written using the Dirac's delta function as

(2.55) |

where we assumed , , and took the first order terms (those contain only one ).

Outside the disk, the rhs of equation (2.58) is equal to zero.
It reduces to the Laplace equation

(2.60) |

(2.61) |

On the other hand, integrating equation (2.58) from to or in other word, applying the Gauss' theorem to the region containing the surface, it is shown that the gravity has a jump crossing the surface as

Equations (2.62) and (2.63) lead a final form of the potential as

Putting this to equations (2.57), and using equations (2.56) and (2.57), we obtain the dispersion relation for the gravitational instability in a thin disk as

This reduces to the dispersion relation of the sound wave for the short wave . While for a longer wave than , an exponential growth of is expected. The dispersion relation is shown in Fig.2.2.