Relativistic Motion with Viscosity: II Stokes’s Law of Resistance

The deduction of a relativistic and mildly relativistic equation of motion in the presence of a drag force proportional to the velocity is presented. The obtained results are used to model the trajectory of the supernova SN1993J and the light curves of gamma-ray bursts.


Introduction
A relativistic treatment of the equation of motion in the presence of a resistive force proportional to the velocity has been investigated in the following models: a model for the Newtonian scattering of photons [1], a motion through a uniform adiabatic medium on the steady-state accretion of matter onto a Schwarzschild black hole [2], an extreme mass-ratio inspirals around strongly accreting supermassive black holes [3], and ultra-relativistic detonations in the framework of the cosmological first-order phase transitions [4]. In Section 2, this paper explores the relativistic law of motion in the presence of viscosity proportional to the velocity. Section 3 is devoted to the astrophysical applications.

The Classic Case
We assume a one-dimensional motion with a resistive force of Stokes type [5], , where A is a constant, m is the considered mass and ( ) v t is the velocity. The differential equation which governs the motion is which has an analytical solution in an explicit form where 0 v is the velocity at The numerical value of the constant A is

The Relativistic Case
We assume a one-dimensional motion with a resistive force of Stokes type, , where A is a constant, 0 m is the considered rest mass and ( ) v t is the velocity. Newton's second law in special relativity is: where F is the force, 0 m is the rest mass, c is the velocity of light and ( ) v t is the velocity; see Equation (7.16) in [6]. The first order differential equation in the velocity which governs the relativistic motion is An analytical solution to the above first order differential does not exist; however, a solution exists for ( ) v t in an implicit form for the time The constant A can be derived from the fol- and

The Mildly-Relativistic Case
The first order differential equation for the mildly-relativistic motion is where W is the Lambert W function [7] and ( ) where International Journal of Astronomy and Astrophysics

Astrophysical Luminosity
The mechanical relativistic luminosity is where ( ) r t is the temporary radius of the expansion, 0 r is the radius at where obs C is a constant that allows the match between theory and observations, and ν τ − is the optical thickness.

Astrophysical Applications
The astrophysical units are chosen to be pc for the length and years for the time: the constant A is therefore expressed in 1 yr . A test for the quality of the fits is represented by the merit function 2 χ ( )  Figure 1 reports the numerical trajectory, of SN 1993J for which observational parameters are available [8] [9] with data as in Table 1.

Application to GRBs
A first example is applied to the light curve (LC) of GRB 130427A , which was the most luminous gamma-ray burst in the last 30 years; see Figure 1 in [10]. Figure 2 reports the X-flux as a function of the time and the relative theoretical data, with data as in Table 2.
L. Zaninetti International Journal of Astronomy and Astrophysics   Table 2.  A second example is applied to the LC in X-ray of GRB 120521C 2, see Figure   2 in [11], which is reported in Figure 3, with temporal behavior of the optical depth as in Figure 4.
A third example is given by the LC in X-ray of GRB 130606A, see Figure 2 in [11], which is reported in Figure 5, with the temporal behavior of the optical depth as in Figure 6. International Journal of Astronomy and Astrophysics  Figure 4 and with data as in Table 2.  Table 2.   Figure 6 and with data as in Table 2.  Table 2.

Conclusions
We analyzed the one-dimensional relativistic motion in the presence of a resistive force proportional to the velocity. An analytical solution for the velocity was derived in an implicit form, see Equation (7). In the mildly relativistic case, we derived an analytical solution for both the velocity, see Equation (14), and the distance, see Equation (16), in terms of the Lambert W function.
A first test to evaluate the constant A in an astrophysical environment is on SN 1993J. A full relativistic treatment of the LC for GRBs was done for GRB 130427A, GRB 120521C and GRB 130606A in the framework of the optical thickness with a time dependence.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.