We start with the conservation of mass, which we have already introduced in the steady-state case

where is the density and is the fluid velocity with radial, azimuthal and vertical components, respectively.

For a thin axisymmetric disc integrating (1) vertically and neglecting a vertical outflow from the disc we get

where is the surface density.

where is the half-thickness of the disc at any given radius, while is the volume density. The variation in the surface density at radius R conforms to the equation of continuity and gives

where represents the mass transfer rate at each radius given by

The time-dependent version of the angular momentum transferred per unit mass of a fluid can be derived from the Navier-Stoke’s equation and rewritten in general as

where pressure, kinetic viscosity, the gravitational potential the current density and the magnetic field. The viscosity in general is small, and we will only retain it where it plays a crucial role. The radial component of Navier-Stoke’s equation is

For a thin accretion disc as shown below and the dominant terms of the equation give us that

this shows that the disc rotates in a Keplerian fashion. That is the matter in the disc approximately flows along circular Keplerian orbits with this velocity.

The vertical component of the time dependent momentum transfer equation is

Neglecting vertical outflows since the motion in the disc along the z-direction are subsonic and assuming the magnetic field to be weak the equation reduces to the equation of hydrostatic equilibrium

Using the above equation the pressure is

but the hydrostatic equilibrium can also be expressed as

which shows that the Keplerian velocity is highly supersonic in a thin accretion disc assumed above, where

is the isothermal speed of sound. For a thin disc this can only be true if that is the radial derivative of pressure is small compared with gravitational and centrifugal forces.

The angular momentum transport can be found from the azimuthal component of Navier-Stoke’s equation (see Paper I & II) for detail derivations:

We neglect because the large-scale poloidal magnetic field is the stellar dipole field, which has only a vertical component in the stellar equatorial plane. Using continuity equation and assuming that

. Then the conservational angular momentum given by Equation (14) leads to the following height-integrated equation

where the right-hand side of Equation (15) accounts for the radial advection of angular momentum owing to the viscous and the magnetic torques. The magnetic term describes the exchange of angular momentum between the disc and the star via magnetosphere.

We assume that the vertical magnetic field is due to the dipolar field of the neutron star, so that its value in the equatorial plane is

where is the magnetic dipole moment of the star.

This term vanishes if is an even function of, but the shear between the disc and the stellar magnetosphere generates an odd whose value in the upper half of the disc is

where, is the angular velocity of the star, is a dimensionless parameter of the order of a few [12] [13] . We now add an extra large-scale toroidal field, which is generated by an internal dynamo in the accretion disc.

Such a dynamo is a natural consequence of the magnetohydrodynamic turbulence in the accretion disc [14] . In order to estimate the size of we will for the moment assume that the viscous stress in the accretion disc is due to the internal magnetic stress

where we use the [1] prescription for the viscosity in the last equality. Based on the results of numerical simulations of magnetohydrodynamic turbulence in the accretion discs (e.g. [15] [16] ) argued that

where. However this is the sum of the large-scale field and small-scale turbulent field, which is also contributing to stress through its correlation with a turbulent -field. Since the large-scale field might be a small fraction of the total field we multiply with a factor ϵ to get an estimate for:

where, and a negative value describes a magnetic field which is pointing in the negative -direction at the upper disc surface. We now replace the in Equation (15) with the sum of the toroidal fields that are generated by the shear, Equation (17), and the internal dynamo, Equation (20) gives

Solving Equation (15) for radial velocity yields

Inserting Equation (22) into continuity equation, then the equation for the evolution of the surface density of the accretion disc for the disc around the magnetised star can be found as

Equation (23) is an integral of the mass and angular momentum equations which is more general, and relies on the assumptions: conservation of mass, conservation of angular momentum, and that the potential is Keplerian. Since the time dependence enters only through Equation (23), it is essential to try to express the other quantities in terms of as far as possible, as they depend on t only through and not explicitly.

The Navier-Stoke’s equation for the conservation of internal energy can be written as:

Here the second term on the left describes the variations in the internal energy, while the first and third terms in the right-hand-side describes viscous heating and radiative cooling respectively and the second and fourth terms are respectively ohmic heating and heat conduction. Neglecting very small terms and retaining the dominant terms, then the energy equation gives

Integrating Equation (25) vertically over the disc we have

This is the conservation of energy.

The optical depth of the disc is given by :

where κ  is the general opacity given by

where is electron scattering opacity equal to

is free-free absorption given by Kramer’s law

with

.

The total pressure is the sum of gas pressure and radiation pressure and then given by the equation of state:

In Equation (32), , , , and are the mass of a proton, Boltzmann constant, the mean molecular weight for ionized gas, and familiar radiation constant, respectively. We can write Equation (31) as

In our model the viscous stress tensor is related according to

The conventional model assumes that

where is a viscous parameter which describes the strength of the viscous stress.

From Equation (35) the internal pressure becomes

Which we solve for the density of the gas

The time-dependent accretion discs with a-parameter viscosity were constructed by [2] and its flow is controlled by the size of the viscosity. Hence observations of time-dependent disc behaviour offer one of the few sources of quantitative information, it is important to consider the relative magnitudes of the various timescales over which accretion discs form and evolve and the effect of magnetic field is observed on timescales. We begin by identifying the typical timescales on which the disc structure may vary.

The dynamical time scale is simply related as:

This is the shortest time scale present in the disc, which is the Keplerian period, that ranges very small in the inner region and increases to outer disc and also the typical growth time of some important instabilities, such as the magneto-rotational instability. Using steady state equation we can write this as

s(50)

We see that the dynamical timescale depends on mass of the accretor, accretion rate, magnetic moment and radius and also measures the speed with which hydrostatic equilibrium in the vertical direction is established. From Equations (50) and (51) the dynamical time scale increases when the magnetic field is strong.

Thermal time scale which is responsible for readjustment to thermal equilibrium and mathematically given by

We thus see that

Viscous time scale (t_visc): a time in which angular momentum distribution changes due to torque caused by dissipative stresses. The viscous timescale sets the scale for the evolution of the surface density. From the analysis of time dependent models above, we have seen that this timescale is given by the viscous time scale

which gives the timescale on which matter diffuses through the disc under the effect of viscous torques. The viscosity has the effect of spreading the original ring in radius on a typical time scale, assuming that the typical length scale for surface density gradients in the disc is. This is an order of magnitude longer than other time scales. Using the -parameterization we can also write

The viscous and the dynamical time scales can be related as

Then from Equation (55) the thermal time scale is

The thermal time scale is large in the region of weak magnetic field or inner region of the disc (see Equations (56) and (57)) which leads to instability of the disc. Since α < 1 then the three time scales can be related by a well-defined hierarchy of timescales

As long as the disc is thin, we thus see that the various timescales are ordered in the following way:

which then shows that the centrifugal balance in the radial direction and hydrostatic balance in the vertical direction are very rapidly achieved, while the disc temperature generally evolves on a longer timescale, and finally, on an even longer timescale, one can see some evolution in the surface density profile.

Finally, note that all of the above timescales are a function of radius. In particular, if H/R are constant which is not our case (which is not generally true), then they all scale in the same way, and for a Keplerian disc, they increase with radius as. Thus the evolution of the inner disc is generally much more rapid than the evolution of the outer disc. Suppose now that a small perturbation is made to a generally considered equilibrium solution and that this perturbation continues to grow rather than being damped. Then the supposed steady solution is said to be unstable and cannot occur in reality. The sharp difference we have found in the various timescales means we can distinguish different types of instability. If for example the energy balance is distributed in the disc, any instability will grow on a timescale t_th, which is much less than t_visc. Since t_visc is the timescale for different changes in the surface density to occur, we can assume that is fixed during the growth time t_th. We refer to this as a thermal instability. For α < 1 we also have t_th > − t_z, so the vertical structure of the disc can respond rapidly, on a timescale t_z, to changes due to the thermal instability, and keep the vertical structure close to hydrostatic equilibrium.

In Paper I & II we calculated the solution of numerically by writing the relations in the form. To investigate the viscous stability of the solution we calculate the derivatives of using the results of the Paper I & II for a slightly perturbed value of. When this derivative is positive the solution is viscously stable, otherwise it is viscously unstable.

Equation (39) gives the viscosity integral for given, and other parameters as a function of r. In more general situation when the system is time-dependent, there is no fixed but in our work we use accretion rate in steady state, and time-independent equation has to be replaced by an explicitly time-dependent diffusion type equation for. In any case, in order to close the system of equations we need another equation that relates 𝚲 with. In the time-dependent case the sign of determines whether the solution is viscously stable (positive sign) or unstable (negative sign). In paper I Equations (39) and (40) we have the relations, R = rR_A, use of these relations in Equation (44) and dropping the subscript zero and the shear component of the magnetic stress with similar reason given in Section 5.4 we obtain

In the gas pressure dominated disc when the accretion rate increases the magnetic field due to dynamo increases (strong field is generated) as result sufficient angular momentum is transported which lead to stability of the disc properties. But in the radiation pressure dominated disc the inner edge of accretion disc close to the surface of the neutron star and the magnetic field due to dynamo decrease which cannot shut off the instability. From paper I Equation (47) 𝚲 and were related by Equation (60).

From the dispersion relation Equation (60) gives information for stability and instability of disc with positive and negative slope. Figure 2 shows that positive and negative slope indicate stability and instability of the disc. This is basically comes from the time-dependent equation of accretion disc around magnetized stars.

this is the case for steady-disc which shows the disc is stable.

For the time-dependent disc we have

where

and hence

which gives that

In this case steady disc flow is possible and which satisfies stability condition. Therefore, steady state model is stables against small perturbation in the gas pressure and free-free absorption dominated disc (that is positive slope). Since in the gas pressure dominated disc the magnetic field is strong than the radiation pressure dominated disc, the strong magnetic field generated by internal dynamo helps the disc to be stable. We can also show stability of the disc using the derivatives of temperature with respect to the surface density as shown in Figure 3

the disc is stable. When, and a viscous instability, this time known as the Lightman-Eardley instability. The time-dependent discs give us information to explain the torque reversal of X-ray pulsars by reversing the

magnetic field in the disc with similar magnitude of spin-up and spin down torques which depends on the inner and corotation radii. In linearly stable disc we can observe the stable torques, and it is non-linearly unstable because otherwise no torque reversal would occur. Since in the star disc interacting system presence of a disc dynamo produces a significant enhancement of the torque between the magnetic field of the neutron star and the accretion disc.

In this case steady disc flow is possible and which satisfies stability condition. Therefore, steady state model is stable against small perturbation in the gas pressure and free-free absorption dominated disc.

In the inner regions of the disc where the radiation pressure and an electron scattering opacity is dominant, then 𝚲 (dimensionless) and (in kg∙m⁻²) are related as (see Paper II Equation (52))

where,

.

Thus

which is the condition for viscous instability (negative slope).