Relativistic Motion with Viscosity: II Stokes’s Law of Resistance ()
1. 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.
2. The Equation of Motion
2.1. 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
is the velocity. The differential equation which governs the motion is
(1)
which has an analytical solution in an explicit form
(2)
where
is the velocity at
. The equation of motion in the explicit form is
(3)
where
is the distance at
. The numerical value of the constant A is
(4)
where
is the velocity at
.
2.2. The Relativistic Case
We assume a one-dimensional motion with a resistive force of Stokes type,
, where A is a constant,
is the considered rest mass and
is the velocity. Newton’s second law in special relativity is:
(5)
where F is the force,
is the rest mass, c is the velocity of light and
is the velocity; see Equation (7.16) in [6]. The first order differential equation in the velocity which governs the relativistic motion is
(6)
An analytical solution to the above first order differential does not exist; however, a solution exists for
in an implicit form for the time
(7)
(8)
and
(9)
where
is the velocity at
. The constant A can be derived from the following formula
(10)
where
(11)
and
(12)
where
is the velocity at
.
2.3. The Mildly-Relativistic Case
The first order differential equation for the mildly-relativistic motion is
(13)
which has solution
(14)
where W is the Lambert W function [7] and
(15)
with
being the velocity at
. The trajectory in the mildly relativistic case is
(16)
where
(17)
with
being r at
. The constant A can be derived in the mildly relativistic case by the following formula
(18)
where
is the velocity at
.
2.4. Astrophysical Luminosity
The mechanical relativistic luminosity is
(19)
where
is the temporary radius of the expansion,
is the radius at
,
is the density at
, d is a shape parameter and
. The observed luminosity,
, is assumed to scale as
(20)
where
is a constant that allows the match between theory and observations, and
is the optical thickness.
3. 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
. A test for the quality of the fits is represented by the merit function
where
,
and
are the theoretical radius, the observed radius and the observed uncertainty, respectively.
3.1. Application to SN 1993J
Figure 1 reports the numerical trajectory, of SN 1993J for which observational parameters are available [8] [9] with data as in Table 1.
3.2. 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.
Figure 1. Numerical radius (full line) and astronomical data of SN 1993J with vertical error bars.
Figure 2. Flux in the X-ray as a function of time in seconds for GRB 130427A (empty stars) and theoretical curve as given by Equation (20) (full line) when
with data are as in Table 2.
Table 1. Numerical values for the parameters of Stokes’s theoretical model applied to SN 1993J.
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.
Figure 3. Flux in the X-ray as function of time in seconds for GRB 120521C (empty stars) and theoretical curve as given by Equation (20) (full line), with
as in Figure 4 and with data as in Table 2.
Figure 4. The time dependence of
(empty stars) for GRB 120521C and a logarithmic polynomial approximation of degree 5 (full line). Parameters as in Table 2.
Table 2. Numerical values of the parameters for the theoretical model.
Figure 5. Flux in the X-ray as a function of time in seconds for GRB 130606A (empty stars) and theoretical curve as given by Equation (20) (full line), with
as in Figure 6 and with data as in Table 2.
Figure 6. The time dependence of
(empty stars) for GRB 130606A and a logarithmic polynomial approximation of degree 5 (full line). Parameters as in Table 2.
4. 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.