Hierarchy of Timescales
There are two dynamical timescales for a rotating star. Gravity provides the free-fall timescale τFF
τFF ~ (R3/M)1/2,
where M is the gravitational mass of the star, and R is the radius (Here and throughout we set G=c=1). If the star is rotating, its rotation period Prot provides another important timescale:
Prot = 2π/Ω,
where Ω is the angular frequency. Dynamical instabilities will act on the above timescales.
The stars we study are dynamically stable initially, so their structure is altered on secular timescales. Secular effects will in general take many rotation periods to significantly affect the structure or velocity profile of a differentially rotating star.
Viscosity will redistribute angular momentum on a viscous timescale τvis
τvis ~ ρ R2 <η>-1 >> τFF,
where ρ is the density and <η> is an averaged value of η (the shear viscosity) across the star.
Thermal energy is radiated away, primarily by neutrinos in the case of a neutron star. Depending on the temperature and the nature of the viscosity, the cooling timescale may be greater than or less than the viscous timescale. If τvis<<τcool, then the heat generated by viscosity will build up inside the star ("no cooling"). If τcool<<τvis, the heat will be radiated away as quickly as it is generated ("rapid cooling"). We study both limits in our simulations.
The secular timescales are so much longer than the dynamical timescales that the star can be thought of as evolving quasi-statically. However, using this approximation is highly nontrivial in full general relativity. Instead, the strength of the viscosity is artificially amplified so that the viscous timescale is short enough to make numerical treatment with our relativistic hydrodynamic code tractable. We keep the viscous timescale sufficiently long that the hierarchy of timescales is maintained, and the secular evolution still proceeds in a quasi-stationary manner.