Computing Differentially Rotating Neutron Stars Obeying Realistic Equations of State by Using Hartle ’ s Perturbation Method

In this paper, we use the well-known Hartle’s perturbation method in order to compute models of differentially rotating neutron stars obeying realistic equations of state. In our numerical treatment, we keep terms up to third order in the angular velocity. We present indicative numerical results for models satisfying a particular differential rotation law. We emphasize on computing the change in mass owing to this differential rotation law.


Introduction
In [1] we have implemented the well-known Hartle's perturbation method ( [2][3][4]), by keeping terms up to third order in the angular velocity, to the computation of differentially rotating neutron stars simulated by generalrelativistic polytropic models (angular momentum, moment of inertia, rotational kinetic energy, and gravitational potential energy are quantities drastically corrected by the third-order approach).The present study is continuation of [1] regarding models of differentially rotating neutron stars obeying realistic "equations of state" (EOS, EOSs).
For clarity and convenience, we adopt here the definitions and symbols used in [1].
The motivation for computing models of differentially rotating neutron stars obeying realistic equations of state by implementing Hartle's pertrurbation method arises from the fact that, to the extent of the knowledge of the authors, respective results seem to be "missing" from the bibliography; and this seems to be true even for the alternative (iterative) methods, except for certain computations regarding maximum mass models as in [5].Thus, interested readers may have the opportunity to take into account our results or to compare with respective results of their own numerical methods.

Preliminaries
Theory and computations are extensively presented in [1] (Sections 2-5).The Schwarchild metric of a nonrotating spherical object, expressed in spherical coordinates

Results and Discussion
As said above, the computations of the present study proceed as in [1] (Section 5).We shall not repeat here such details; interested readers can find the full numerical treatment there.We present numerical results for the EOSs AU, C, F, FPS, and G.All models computed here have central density .When resolved in the state of uniform rotation ( [1], Section 6, Equation ( 209)), all models are assumed to be rotated with their Keplerian angular velocity K  .Several methods have been developed for the computation of K  (for a general discussion on such methods, see e.g.[14], Section 3.7).In this study, in order to compute K  for our models, we use the well-known RNS package [15].
When resolved in the state of differential rotation ([1], Section 6, Equation ( 210)), all models have angular velocity at the equatorial surface, e  , equal to their Keplerian angular velocity, e K .   The differential rotation function on the equatorial plane ) increases from zero at the equatorial surface up to a maximum at the center, which (maximum) depends on the dimensionless model parameter describes a uniform rotation, while its gradually increasing values 0.3, 0.5, and 0.7 (studied in this work) yield differential rotations with respectively increasing maxima and the ratio c e   is written as Tables 2-6 give numerical results for certain significant quantities of our models.In each table, there are two groups of quantities: The first group, labeled "corrections due to differential rotation", contains the changes occured in the three kinds of mass due to differential rotation.The second group, labeled "total quantities and significant ratios", presents total values of certain significant quantities and, also, values of some interesting ratios.Emphasizing on the increase in the baryonic mass Table 1.Realistic EOSs used in the computations.

EOS Description Reference
C Modified Reid soft core potential [6] F Modified Reid potential [7] G Modified Reid soft core potential [8] AU AV14  UVII; Wiringa, Fiks and Fabrocini  owing to differential rotation, we find that the ratio    takes its larger value for the EOS AU which is the most stiff among the EOSs examined.It is worth reminding here that, considering a group of EOSs, most stiff is the EOS yielding the larger pressure for a particular density, and most soft is the EOS yielding the smaller pressure for that density among the EOSs of the group.Note, in addition, that the EOSs AU, C, and FPS are compatible in their stiffness for the density 15 3 1 10 g cm    , next is the EOS F and most soft in the group is the EOS G; for larger densities, AU clearly becomes the most stiff, C and FPS continue to be compatible, F is the next one, and G remains the most soft in the group.Sorting our EOSs this way, we find that the change in mass decreases with the stiffness of the EOSs.
The "equatorial differential rotation profiles" (EDRP)           "meridional cross sections" of our models when the parameter 1 ˆ  takes its larger value studied here, i.e.

Figure 2 .
Figure 2. Meridional cross section for the model obeying the EOS AU and for the value ˆ1 = 0.7   (bold line; soft line: the unperturbed model).

Figure 4 .
Figure 4. Meridional cross section for the model obeying the EOS C and for the value ˆ1 = 0.7   (bold line; soft line: the unperturbed model).

Figure 6 .
Figure 6.Meridional cross section for the model obeying the EOS F and for the value ˆ1 = 0.7   (bold line; soft line: the unperturbed model).

Figure 8 .
Figure 8. Meridional cross section for the model obeying the EOS FPS and for the value ˆ1 = 0.7   (bold line; soft line: the unperturbed model).

Figure 10 .
Figure 10.Meridional cross section for the model obeying the EOS G and for the value ˆ1 = 0.7   (bold line; soft line: the unperturbed model).

Table 2 . Numerical results given in cgs for the EOS AU; differential rotation for three values of the model parameter
ˆ1   .

Table 3 . Numerical results given in cgs for the EOS C; differential rotation for three values of the model parameter
ˆ1   .

Table 4 . Numerical results given in cgs for the EOS F; differential rotation for three values of the model parameter
ˆ1   .

Table 5 . Numerical results given in cgs for the EOS FPS; differential rotation for three values of the model parameter
ˆ1   .

Table 6 . Numerical results given in cgs for the EOS G; differential rotation for three values of the model parameter
ˆ1   .