The relativistic three dimensional evolution of SN 1987A

The high velocities observed in supernovae require a relativistic treatment for the equation of motion in the presence of gradients in the density of the interstellar medium. The adopted theory is that of the thin layer approximation. The chosen medium is auto-gravitating with respect to an equatorial plane. The differential equation which governs the relativistic conservation of momentum is solved in numerically and by recursion. The asymmetric field of relativistic velocities as well the time dilation are plotted at the age of 1 yr for SN 1987A.


Introduction
The expansion velocities in supernovae (SN) are quite high and, as an example, a time series of eight spectra in SN 2009ig reports that the velocity at the CA II line, decreases from 32000 kms −1 to 21500 kms −1 , in 12 day, see Fig. 9 in [1]. Another example is given by SN 2009bb in which the velocity of expansion has been evaluated to be ≈ 255000 kms −1 , see [2].
We briefly recall that the corrections in special relativity (SR) for stable atomic clocks in satellites of the Global Positioning System (GPS) are applied to satellites which are moving at a velocity of ≈ 3.87 kms −1 . The problem of the aspherical SN, such as SN 1987A , is to find an acceptable model which can reproduce the observed complex morphology of the aspherical SN 1987A and this was done in a classical framework by [3]. In this paper we shall discuss a relativistic treatment of the thin layer approximation in the presence of an auto-gravitating medium.

Relativistic conservation of momentum
The chosen auto-gravitating profile is where n 0 is the density in the equatorial plane (θ = 0), R is the radius of the advancing shell, θ is the latitude angle (θ = 0 at the equator and θ = ±90 at the two poles) and h is a parameter which characterizes the gradient. The chosen symmetry imposes that the motion is independent of the azimuthal angle in spherical coordinates but depends only on the latitude angle and the time. The classical conservation of momentum in the presence of an auto-gravitating medium was treated in [3] and therefore we will not duplicate the results already obtained. The relativistic conservation of momentum, see [4,5,6], is formulated as with and c being the velocity of light, here M (R 0 ; b) is a first mass between 0 and R 0 and M (R; b) is a second mass between 0 and R. We know already that M (R; b) = (I m (R)) 1/p where the integral I m (R) has been defined in eq. (15) of [3] and p is a parameter to be found. The fundamental Eq. (2) can be first solved for β 2 where and P the polylog operator, which is defined by The value of β is or This first order differential equation can be solved with the Runge-Kutta method, see FORTRAN SUBROUTINE RK4 in [7]. Another approach separates the variables The previous integral does not have an analytical solution and we treat the previous result as a non-linear equation to be solved with the FORTRAN SUBROUTINE ZRIDDR in [7]. The presence of an analytical expression for β as given by Eq. (8) allows setting up the recursive solution where R n , V n , ∆t are the temporary radius, the relativistic velocity, and the interval of time, respectively. An interesting application of SR is the time delay: given an interval of time, ∆t, in the laboratory frame the interval of time, ∆t , in a frame that that is moving with velocity v in the x-direction is We can therefore introduce the following ratio which measures the time dilation, and lies between 0 and 1.

Astrophysical application
We numerically solved the non-linear equation, Eq. (9) even if the same results can be obtained by solving the differential equation (8) or implementing the recursive relationship as given by Eq. (10), see Table 1 for the adopted data. The complicated structure of SN 1987A is due to the great variety of shapes obtained when the point of view of the observer changes. One way to parametrize the point of view of the observer is the introduction of the Euler angles (Φ, Θ, Ψ), as an example, Fig. 1 shows the 3D advancing shell after 23 years. In order to avoid complicated changes of framework for the field of velocity we limit ourselves to the nonrotated image. This choice is already widely used by astronomers in order to reduce the data of η-Carinae , see Fig. 4 in [8]. The progressive increase of the asymmetry is clearly outlined in Fig. 2, in which sections of the expansion are drawn at time steps of 1 yr. The difference in velocity between the polar direction and equatorial direction are oulined in Fig. 3. The relativistic field of velocity in the various points of SN 1987A after 1 yr was shown in Fig. 4. The relativistic time dilation is mapped in Fig. 5 where the velocity of expansion perpendicular to the observer (x-direction) is considered.

Conclusions
We have covered the evolution of a SN in an auto-gravitating medium in a relativistic framework. The initial shape is represented by a sphere of radius R 0 = 0.011 pc. After 1 yr, the asymmetry between the radius in the equatorial plane and the radius   Table 1. This is a non-rotated image and the three Euler angles characterizing the orientation are Φ=180 • , Θ=90 • and Ψ=0 • .  in the polar direction is well defined and Fig. 4 summarizes both the asymmetrical shape and the anisotropic field of velocity. The time dilation at 1 yr as represented by the parameter D varies between a minimum of 0.9975 and a maximum of 1.