MHD in protostellar discs is modified by the Hall current when the ambipolar diffusion approximation breaks down .
Here I examine the Balbus-Hawley ( magnetorotational ) instability of a weak , vertical magnetic field within a weakly-ionized disc .
Vertical stratification is neglected , and a linear analysis is undertaken for the case that the wave vector of the perturbation is parallel to the magnetic field .

The growth rate depends on whether the initial magnetic field is parallel or antiparallel to the angular momentum of the disc .
The parallel case is less ( more ) unstable than the antiparallel case if the Hall current is dominated by negative ( positive ) species .
The less-unstable orientation is stable for \chi \la 0.5 , where \chi is the ratio of a generalised neutral-ion collision frequency to the Keplerian frequency .
The other orientation has a formal growth rate of order the Keplerian angular frequency even in the limit \chi \rightarrow 0 !
In this limit the wavelength of the fastest growing mode tends to infinity , so the minimum level of ionization for instability is determined by the requirement that a wavelength fit within a disc scale height .
In the ambipolar diffusion case , this requires \chi > v _ { A } / c _ { s } ; in the Hall case this imposes a potentially much weaker limit , \chi > v _ { A } ^ { 2 } / c _ { s } ^ { 2 } .