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.
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 [
We assume a one-dimensional motion with a resistive force of Stokes type [
v ( t ) = v 0 e − A t e − t 0 A , (1)
which has an analytical solution in an explicit form
v ( t ; A , v 0 , t 0 ) = v ( t ) = v 0 e − A t e − t 0 A , (2)
where v 0 is the velocity at t = t 0 . The equation of motion in the explicit form is
r ( t ; A , v 0 , t 0 , r 0 ) = − v 0 ( e − A t − e − t 0 A ) e t 0 A A + t 0 (3)
where r 0 is the distance at t = t 0 . The numerical value of the constant A is
A = ln ( v 1 v 0 ) t 0 − t 1 , (4)
where v 1 is the velocity at t = t 1 .
We assume a one-dimensional motion with a resistive force of Stokes type, F r e s = − A m 0 v ( t ) , where A is a constant, m 0 is the considered rest mass and v ( t ) is the velocity. Newton’s second law in special relativity is:
F = d d t ( m 0 v ( t ) 1 − v ( t ) 2 c 2 ) , (5)
where F is the force, m 0 is the rest mass, c is the velocity of light and v ( t ) is the velocity; see Equation (7.16) in [
d d t v ( t ) ( 1 − ( v ( t ) ) 2 c 2 ) 3 2 = − A ( v ( t ) ) . (6)
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
t = N D , (7)
N = ( − 2 c 3 + 2 c v 0 2 ) c 2 − v 2 − ( ( − 2 t 0 A + ln ( c 2 − v 2 − c ) − ln ( c + c 2 − v 2 ) − ln ( c 2 − v 0 2 − c ) + ln ( c + c 2 − v 0 2 ) ) c 2 − 2 c c 2 − v 0 2 − v 0 2 ( − 2 t 0 A + ln ( c 2 − v 2 − c ) − ln ( c + c 2 − v 2 ) − ln ( c 2 − v 0 2 − c ) + ln ( c + c 2 v 0 2 ) ) ) ( c − v ) ( c + v ) . (8)
and
D = 2 A ( c 2 − v 2 ) ( c 2 − v 0 2 ) , (9)
where v 0 is the velocity at t = t 0 . The constant A can be derived from the following formula
A = N N D D , (10)
where
N N = ( − 2 c 3 + 2 c v 1 2 ) c 2 − v 0 2 + ( c + v 0 ) ( c − v 0 ) ( ( ln ( c 2 − v 1 2 − c ) − ln ( c + c 2 − v 1 2 ) − ln ( c 2 − v 0 2 − c ) + ln ( c + c 2 − v 0 2 ) ) c 2 + 2 c 2 − v 1 2 c − v 1 2 ( ln ( c 2 − v 1 2 − c ) − ln ( c + c 2 − v 1 2 ) − ln ( c 2 − v 0 2 − c ) + ln ( c + c 2 − v 0 2 ) ) ) , (11)
and
D D = 2 ( t 0 − t 1 ) ( c 2 − v 1 2 ) ( c 2 − v 0 2 ) , (12)
where v 1 is the velocity at t = t 1 .
The first order differential equation for the mildly-relativistic motion is
d d t v ( t ) + 3 v ( t ) 2 ( d d t v ( t ) ) 2 c 2 = − A v ( t ) , (13)
which has solution
v ( t ; t 0 , v 0 ) = e − A t − W ( 3 2 c 2 ( e B ) 2 ( e A t ) 2 ) 2 − B , (14)
where W is the Lambert W function [
B = − 4 t 0 A c 2 + 4 ln ( v 0 ) c 2 + 3 v 0 2 4 c 2 , (15)
with v 0 being the velocity at t = t 0 . The trajectory in the mildly relativistic case is
r ( t ; t 0 , r 0 , v 0 ) = − e t 0 A c 2 v 0 ( W ( D ) + 3 ) ( e v 0 2 c 2 ) 3 4 − 3 e A t e W ( D ) ( ( A r 0 + v 0 ) c 2 + v 0 3 2 ) 3 e W ( D ) e A t c 2 A , (16)
where
D = 3 v 0 2 e − 4 ( t − t 0 ) c 2 A + 3 v 0 2 2 c 2 2 c 2 , (17)
with r 0 being r at t = t 0 . The constant A can be derived in the mildly relativistic case by the following formula
A ( t 0 , t 1 , v 0 ) = − v 0 2 ( 4 ln ( v 0 v 1 ) c 2 + 3 v 0 2 − 3 v 1 2 ) 4 v 0 2 c 2 ( t 0 − t 1 ) , (18)
where v 1 is the velocity at t = t 1 .
The mechanical relativistic luminosity is
L m , r = 4 π r ( t ) 2 1 1 − β ( t ) 2 ρ 0 ( r 0 r ) d c 3 β ( t ) , (19)
where r ( t ) is the temporary radius of the expansion, r 0 is the radius at t = t 0 , ρ 0 is the density at t = t 0 , d is a shape parameter and β ( t ) = v ( t ) c . The observed luminosity, L o b s , is assumed to scale as
L o b s = C o b s L m , r ( 1 − e − τ ν ) , (20)
where C o b s is a constant that allows the match between theory and observations, and − τ ν is the optical thickness.
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 χ 2
χ 2 = ∑ j ( r t h − r o b s ) 2 σ o b s 2 ,
where r t h , r o b s and σ o b s are the theoretical radius, the observed radius and the observed uncertainty, respectively.
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
| model | values | χ 2 |
|---|---|---|
| Stokes’s | r 0 = 3.0 × 10 − 3 pc ; v 0 = 13800 km / s ; A = 0.07 1 years | 85.7 |
A second example is applied to the LC in X-ray of GRB 120521C 2, see
A third example is given by the LC in X-ray of GRB 130606A, see
| GRB name | theoretical parameters |
|---|---|
| GRB 130427A | r 0 = 9.9 × 10 − 5 pc ; t 0 = 1.0 × 10 − 3 year ; β 0 = 0.9 ; A = 1 1 pc ; d = 3.1 |
| GRB 120521C | r 0 = 1.0 × 10 − 4 pc ; t 0 = 1.0 × 10 − 6 year ; β 0 = 0.9 ; A = 10000 1 pc ; d = 3 |
| GRB 130606A | r 0 = 1.0 × 10 − 4 pc ; t 0 = 1.0 × 10 − 6 year ; β 0 = 0.9 ; A = 1000 1 pc ; d = 2 |
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.
The author declares no conflicts of interest regarding the publication of this paper.
Zaninetti, L. (2021) Relativistic Motion with Viscosity: II Stokes’s Law of Resistance. International Journal of Astronomy and Astrophysics, 11, 481-488. https://doi.org/10.4236/ijaa.2021.114025