V.V. Skobelev

Moscow State Industrial University, Russia

ON POSSIBLE NEUTRON STAR EQUILIBRIUM RADIUS DECREASEDUE TO INFLUENCE OF SUPERSTRONG MAGNETIC FIELD

 

           ABSTRACT. There was formulated the necessary condition of neutron -decay suppression in degenerate magnetized electron gas. On this basis it is shown, that in a superstrong magnetic field of  the equilibrium radius of the neutron star appears to be approximately few times less, than in absence of the field. Therefore, one may assume, that short period pulsars are under suspicion in existence of such fields.    

 

               It is well known that the main contribution in the total pressure in neutron stars comes from degenerate neutron gas [1], and equilibrium radius of approximately ten  is formed namely under this condition. On the other hand, there possibly exists a superstrong magnetic field up to  [2, 3]. Therefore, it is interesting to investigate the influence of such a field on the equilibrium radius by means of its interaction with the matter of the neutron star, in particular, with degenerate electron gas; the last one may be connected with star dynamics in indirect way by means of its influence on neutron -decay and inverse reaction (the so called direct URCA processes, that dominate at a sufficiently high concentration of electrons and protons[4]), i.e. on the process of “neutronisation” and equilibrium radius (in the present work we do not discuss the role of “modified” URCA processes [5] with one extra nucleon in this reactions).

   We shall start with observation, that electron wave function in the homogeneous magnetic field (directed along the third axis) on the ground Landau level is equal to [6] (see also ref. [7]):

                               ,                        (1)

where  is a two-component spinor in a subspace (0,3), is the momentum along the field, is normalizing length, is the quasimomentum, defining the position of the wave package center on x-axis :

                              .                                                                              (2)

As a consequence, the integration over gives :

                               .                                                             (3)

  Further we formulate the condition under which the neutron -decay is forbidden in degenerate magnetized electron gas.

 For this purpose we shall note that the total number of electrons may be presented as a sum over all quantum states [8](the spin projection on the ground Landau level is fixed):

                                   ,                                                  (4)

where

                                                                                                   (5)

is the distribution function, and

                                                                                                              (6)

                                  

is the electron relativistic energy on the ground Landau level.

Using the relation (3) we obtain for the concentration:

                              .                                                                        (7)

For the totally degenerate () electron gas we find

                                ,                                                                           (8)

     where                                                                                             

                                                                                                                           (8b)

is the Fermi momentum.

    One can see from the general expression for the electron energy in a magnetic field

                                         ,                                             (9)

that the states are not exited, if  , where the “critical” field may be found  from the equation

                                         ,                                                                        (10)

or

                                          .                                                                (10a) 

 From this equation we find:

                               ,,                      (11)

  ( is electron Compton length)   and for a typical electron concentration , which is approximately [9] of neutron concentration in the center of the star at the nuclear density we obtain. Therefore, for our assessments at we may believe that majority of electrons are on the ground Landau level.   

Then one must add  the above “field” condition by the “temperature” condition

                             ,                                                                        (12)

which seems to be well carried out for the “old” neutron stars with temperature  [10].                          

     The neutron -decay is forbidden, when >, where the minimal concentration can be found from the equation

                                 ,                                                                         (13)

and expression (8b). Here stands for the energy output in neutron - decay. Using equations (8b), (13), and the maximum field value , we have

                                  .                                                                (14)

On the other hand, in the absence of the magnetic field one can obtain from equation (13) and usual relation [8] the following estimate

                                        .                                                       (15) 

     Thus, in the process of the star gravitational compression the “neutronisation” of the matter in the absence of magnetic field will take place at the largest star radius , then of  their existence . Particularly, we have the following result for star radius at the start of “neutronisation” :

                                           .                                   (16)

      We shall also note that the interaction of the neutron anomalous magnetic moment with the magnetic field is characterized by a dimensionless parameter  (see Appendix [11]), which is equal to unity in the absence of magnetic field and has the same order up to the field  of .Therefore, one may neglect the influence of the magnetic field on the neutron pressure, and, in this sense, on the equilibrium radiusas a final result of gravitational compression. If we assume also, that “equilibrium radius”  is proportional to “neutronisation radius” and superstrong magnetic field  formed before “neutronisation”, this and (16) leads us to definitive result :

                                                 .                                                    (17)     

  Most likely, this value is highly overestimated, because the gravitational compression continues after “neutronisation” too, but it seems that the essential difference in sizes  really exists and may be experimentally observed. Namely, the pulsars with a short period and, hence, with a small radius are under suspicion of presence of the superstrong magnetic field. On the contrary, the pulsars with a long period must have a big radius and no superstrong magnetic field. 

       As a result of additional calculations at the field value  we generalized equations (14), (16) as follows :

                                         ,                         (18)

                                             .                                                        (19)

The results follow from equations. (8b), (9)(13) and  condition of exited states  suppression for a final electron: .

       At a particular case       we have  the maximal results (14),(16) again.

        Note also, that in our approximation of totally degenerate electron gas on the ground

Landau level there must be the extra restriction (besides of (12)) on the temperature value:

                                .                                                                    (20)

The minimal value of is equal to  , and the condition (20) reduce

to the more strong restriction

                                 ,                                                                         (21)

which is rudely valid for the values   but not for the value . Even in the last case relations (16),(17)are approximately correct, because increase and decrease  at in the same degree  and our main conclusion does not change.

      As to neutron decay suppression  because of nonrelativistic proton gas it occurs (if really occurs) at and therefore electron component of the star in any case dominates in this sense  with again our conclusion without change. Thus, at  there is a double interdiction on neutron decay because of  both electron and proton gas and it is possible to assume on the base of our calculations, that neutron stars with  superstrong magnetic field can reach such concentration in the process of gravitational compression.  

    The star objects under consideration may be called as minimagnetars and in according with eq. (9.16) their radius(of the order 1km, instead of 10 km in the absence of superstrong magnetic field) tends to the gravitational one; therefore this objects may be identified because of gravitational red shift in radiation.

 

     

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