Chameleon stars

Chameleon stars

Vladimir Dzhunushaliev, 111 Email: vdzhunus@krsu.edu.kg Vladimir Folomeev, 222Email: vfolomeev@mail.ru Douglas Singleton 333Email: dougs@csufresno.edu Institute for Basic Research, Eurasian National University, Astana, 010008, Kazakhstan
Institute of Physicotechnical Problems and Material Science of the NAS of the Kyrgyz Republic, 265 a, Chui Street, Bishkek, 720071, Kyrgyz Republic
Physics Department, CSU Fresno, Fresno, CA 93740-8031
Abstract

We consider a gravitating spherically symmetric configuration consisting of a scalar field non-minimally coupled to ordinary matter in the form of a perfect fluid. For this system we find static, regular, asymptotically flat solutions for both relativistic and non-relativistic cases. It is shown that the presence of the non-minimal interaction leads to substantial changes both in the radial matter distribution of the star and in the star’s total mass. A simple stability test indicates that, for the choice of parameters used in the paper, the solutions are unstable.

I Introduction

In cosmology scalar fields play a key role in models of both the inflation era of the early Universe Linde () and the current accelerated expansion Copeland:2006wr (). Scalar fields are also widely used to study smaller scale objects such as the effect of a dark matter, scalar field on the structure of galaxies Bertone:2004pz (). Moving to still smaller scales there have been studies of boson stars Schunck:2003kk () – compact (usually spherically symmetric) configurations formed from gravitating scalar field(s). The sizes of such configurations varies from the microscopic (“gravitational atom”) up to sizes corresponding to massive black holes such as those presumed to exist at the center of many galaxies including our own. In this connection, it is quite natural to investigate the role that such scalar fields may have in the processes of formation and evolution of galaxies and more compact astrophysical structure – stars and their clusters. As applied to the stars, it is possible to imagine a situation where scalar fields can exist inside usual stars consisting of ordinary matter. The existence of such scalar fields would undoubtedly have an impact on the star’s inner structure. One such example is the model of a star consisting of both ordinary polytropic matter and a ghost scalar field Dzhunushaliev:2011xx (). The presence of the field leads to the appearance of a tunnel at the center of such stellar configurations which provides new geometric and physical properties to such objects. Another example of such stellar objects involves fermion fields interacting with a real scalar field not only gravitationally but also through the Yukawa coupling of the type , where is a Yukawa type coupling constant Lee:1986tr (). Further examples of mixed configurations consisting of both boson and fermion fields can be found in references Henriques:1989ar (); Henriques:1989ez (); Jetzer:1990xa ().

The investigation in the present work is along the direction of the above works – we consider a scalar field interacting non-minimally with ordinary matter in the form of a perfect polytropic fluid. An interaction similar to this was used to describe the evolution of dark energy within the framework of chameleon cosmologies Farajollahi:2010pk (); Cannata:2010qd (). In chameleon cosmologies the properties of the scalar (chameleon) field depend strongly on the environment Khoury:2003rn () into which it is embedded. For example, the mass of the scalar field can change depending on the background environment Khoury:2003rn (). As will be shown below, the presence of such an interaction between the polytropic fluid and the scalar field results in a substantial change of the inner structure of the polytropic star. This change is related to the fact that the behavior of the scalar field depends the “environment” (i.e. the fluid) into which it is embedded as is the case with chameleon cosmology models. Thus we call these configurations chameleon stars.

In building our model of a chameleon star we will use a real scalar field. In the bulk of the literature complex (charged) scalar fields are more often used in constructing models of boson stars. The charge of the complex scalar field generically leads to repulsion which counteracts the attraction of gravity and thus gives a physical basis for the existence of these boson star solutions. Real scalar fields have received less attention because they do not carry a charge, and constructing regular stable solutions is more difficult task in this case. The known static solutions for real scalar fields are either singular Wyman:1981bd (); jetzer (), or use phantom scalar fields Kodama:1978dw (); Kodama:1979 (); Dzhunushaliev:2008bq (). Singular solutions with trivial topology and non-phantom fields have been found for both massless fields Wyman:1981bd () and for self-interacting fields jetzer (). The latter ordinary scalar field solutions are asymptotically flat but the issue of their stability is not quite clear (for further discussion see reference clayton ()). In Torii:1999uv () regular, static solutions in the presence of the cosmological constant were found. In this case the space-time asymptotically approaches de Sitter space-time. The linear stability analysis of these solutions indicate that they are unstable. In contrast the paper Dzhunushaliev:2010bv () found regular, static solutions which asymptotically approached anti-de-Sitter space-time, which were stable under a linear stability analysis. In the present paper the real scalar field non-minimally interacts with the polytropic fluid. As will be shown below, this leads to the existence of regular, asymptotically flat solutions.

In the absence of gravity, Derrick’s theorem derrick (); rajaraman () forbids the existence of non-trivial, stable, regular static dimensional solutions from only scalar fields. This theorem assumes that the potential energy of the system is some non-negative function, , vanishing only at its absolute minima. A partial extension of Derrick’s theorem to the case of general relativity has been studied in Bronnikov:2002kw (); Bronnikov:2005gm () where it was shown that the existence of asymptotically flat particle-like solutions with a regular center and normal (non-phantom) scalar fields is also impossible if . The distinction between the non-gravitational form of Derrick’s theorem derrick (); rajaraman () and the gravitational variant Bronnikov:2002kw (); Bronnikov:2005gm () is that the latter does not touch on the stability of the solutions.

If one allows other fields besides just scalar fields (e.g. vector or fermion fields) then there are static solutions when the spatial dimensionality is three or more. In the absence of gravity, one has the ’t Hooft-Polyakov monopole which is a finite energy, stable solution for Yang-Mills vector fields plus scalar fields tHooft (); Polyakov (). In the presence of gravity plus a real scalar field there are boson star solutions (for a review see Schunck:2003kk ()) which however have one or more of the bad features listed above (e.g. non-trivial topology, not asymptotically flat, not stable)

In this paper we study the gravitating system of a real scalar field plus a perfect fluid and show that for this system there are regular, three-dimensional, static solutions. The most important feature of this system, aside from the gravitational interaction, is the presence of the non-minimal coupling between the scalar field and the perfect fluid. It is this interaction which allows us to avoid the restrictions of the gravitational version of Derrick’s theorem Bronnikov:2002kw (); Bronnikov:2005gm (), and find regular static solutions even if the potential energy of the scalar field satisfies . We take a polytropic equation of state for the perfect fluid. Stellar models using polytropic fluids were investigated in detail both within the framework of Newtonian gravitational theory Chandra:1939 () and for strong gravitational fields Tooper:1964 (). Both of these studies hinted at the existence of finite size, regular solutions. These solutions were successfully used to give a description of both non-relativistic stars and stars where relativistic effects were important e.g. neutron stars. As will be shown below, the inclusion of the non-minimal interaction between the polytropic fluid and the scalar field leads to regular solutions which, however, are generally different from usual polytropic stars.

The paper is organized as follows: In section II the general equations describing a static configuration consisting of a real scalar field coupled to a perfect polytropic fluid are derived. In section III these equations are written for a particular case when the potential energy of the scalar field is chosen to have a quadratic mass term and a quartic self-interaction term. For this potential energy, in section IV we give the results of the numerical calculations for this potential for both relativistic and non-relativistic cases, and discuss the issue of the stability of the solutions obtained. Next, in section V we present a simple analytical solution for the non-relativistic case with the scalar field taken to be massless and for a special choice of the coupling function. Finally, in section VI we summarize the main results and give some speculations about the physical applications of these chameleon star configurations to neutron stars and living stars (i.e. stars still on the main sequence).

Ii Derivation of equations for a static configuration

As discussed in the introduction we consider a gravitating system of a real scalar field coupled to a perfect fluid. The Lagrangian for this system is

(1)

Here is the real scalar field with the potential ; is the Lagrangian of the perfect isotropic fluid i.e. a fluid with only one radial pressure; is some function describing the non-minimal interaction between the fluid and the scalar field. The case corresponds to the absence of the non-minimal coupling, but even in this case the two sources are still coupled via gravity.

We choose the Lagrangian for the isentropic perfect fluid to have the form Stanuk1964 (); Stanuk (). Using this Lagrangian, the corresponding energy-momentum tensor is (details are given in Appendix A)

(2)

where and are the density and the pressure of the fluid, is the four-velocity (here and throughout the paper we set ). We take the static metric of the form

(3)

where is the metric on the unit 2-sphere. The and components of the Einstein equations for the metric (3) and the energy-momentum tensor (2) are

(4)
(5)

The equation for the scalar field coming from the Lagrangian (1) is

Using this field equation above with the perfect fluid , and the metric (3), gives the following scalar field equation

(6)

where the prime denotes differentiation with respect to . The Einstein field equations are not all independent because of the relation . The component of this equation has the form

(7)

Taking into account the expressions

and , from (4), (5) and using (6), allows us to write (7) as

(8)

The matter Lagrangian used here, , is not the only possibility. Other variants can be found in Bertolami:2008ab (). However, the choice has the simplifying feature (8) does not contain additional terms involving the coupling function .

For a polytropic equation of state one has

(9)

where are constants. Now one can introduce the new variable defined as Zeld ()

(10)

Here is the central density, and the constant , the polytropic index, is related to via . Putting these definitions together gives equation (9) in the form

(11)

Using (11) in equation (8) leads to

(12)

with and is the pressure of the fluid at the center of the configuration. This equation may be integrated to give in terms of :

(13)

where is the value of at the center of the configuration where . The integration constant , corresponds to the value of at the center of the configuration. It is determined by requiring at infinity that i.e. that the space-time is asymptotically flat.

The gravitating system of a real scalar field interacting with a perfect fluid is characterized by three unknown functions – and . These three functions are determined by the three equations (4), (5) and (6), and also by the relation (13). We now rewrite these equations by the introduction of a new function Tooper:1964 ()

(14)

Here is the mass of the configuration within the range , where is the boundary of the fluid where . Using this function, equation (4) becomes

(15)

From (14) one can define which can be interpreted as the total mass of the configuration in the range . This mass has contributions from both the fluid and the scalar field, within a sphere of coordinate radius . To avoid a singularity in at the origin, one has to put Tooper:1964 (). This corresponds to the fact that the mass at the origin is equal to zero i.e. .

In anticipation of analyzing the system of equations – (4), (5) and (6)) – numerically, we introduce the following dimensionless variables

(16)

has the dimensions of an inverse length. With this one can rewrite equations (4) and (5) in the form

(17)
(18)

where is the dimensionless potential energy of the field.

Next, using (12), one can rewrite equation (6) as follows:

(19)

Thus the static configuration under consideration is described by the three equations (17)-(19).

Iii Configuration with a mass and a quartic self-interaction term

In this section we will show that there are non-singular, finite-mass solutions of equations (17)-(19). First we specify the boundary conditions. Using the above dimensionless variables we “normalize” to unity at the center of the configuration

(20)

From equation (17) one can show that like as . Combining this with equation (18) one in turn finds as . Bearing in mind that we are looking for regular solutions, we define the boundary conditions in the vicinity of as

(21)

where corresponds to the initial value of the scalar field , the parameters are arbitrary, and the value of the coefficient is defined from equation (19) as

The index denotes that the values of the functions are taken at .

Using the boundary conditions (21), we proceed to solve the system (17)-(19) numerically. The behavior of the solution will depend both on the parameters of the polytropic fluid and the form of the potential energy of the scalar field , and the coupling function . One of the simplest and most commonly used choices for the potential energy is that of a scalar field with mass and a quartic self-interaction term ()

It was pointed out in jetzer () that the system with such potential has only singular static solutions. Below we show that an inclusion of the non-minimal interaction between the scalar field and the polytropic fluid allows one to obtain regular static solutions. As an example, let us choose the coupling function in dimensionless form as follows

(22)

Then, using the dimensionless variables introduced in the previous section, the potential can be rewritten as

(23)

where the new dimensionless constants are given as

Using the above potential and the function , equations (17)-(19) take the form

(25)
(26)

The parameter can be absorbed by introducing the rescalings , , , . Bearing this in mind, we assume in further calculations. One can see from these equations that the presence of the interaction between the scalar field and the fluid is defined by the parameter . In the absence of the fluid, this system only has singular solutions jetzer (). The inclusion of the fluid changes the situation since the presence of the term on the right hand side of (26) for the scalar field corresponds to an effective mass term whose sign depends both on the behavior of the fluid density and the values of the parameters and . We note that the present system of gravitating real scalar field plus perfect fluid turns out to be similar to the gravitating complex scalar scalar field system considered in Colpi (). The gravitating complex scalar field studied in Colpi () had a potential of the type (23) and it gave regular, stationary solutions. Thus this is already a hint that we can expect regular, stationary solutions for the present system.

Iv Numerical results

iv.1 Relativistic case

The results of the numerical calculations for the system (25)-(26) with the boundary conditions (21) are presented in tables 1 and 2. The solutions were started near the origin (i.e. near ) and solved out to a point where the function becomes zero. From equations (9) and (10) this is where the fluid vanishes and it is this point, , that we define to be the surface of the star. Beyond the point we continued the numerical solutions with only the gravitational and scalar fields while the fluid was set to zero. The interior and exterior solutions were then connected to one another. Details of this are given below. Previous numerical studies with a fluid having a polytropic equation of state Tooper:1964 () found regular, relativistic star-like configurations. These star-like solutions of Tooper:1964 () where found for the values of the parameters and . Since our system has two additional parameters () we restrict ourselves to examine just two sets of the parameters (i.e. and ) at different values of and when looking for regular solutions. Below we show that these two sets of solutions differ considerably in the behavior of their physical characteristics.

The parameters of the system under consideration for the chosen values of and are given in tables 1 and 2. The procedure for finding these parameters is the following: given the values of and , we seek eigenvalues (i.e. values of the parameters and ) for which the function goes to zero at some finite value of which as mentioned above we take to correspond to the surface of the star. Then since (where is the total mass of the star between and its surface at ) it is required that . Using this we can evaluate the function from (16) at

(27)

This quantity defines the total mass of the configuration through the parameters of the fluid , and which determine the parameter .

The value of the coordinate , corresponding to the boundary of the configuration, does not represent the radius of the star as measured by a distant observer. To define this radius it is necessary to make a coordinate transformation to a new dimensionless variable which is defined as follows

or taking into account (14) and (16)

(28)

Then the observable radius of the configuration in dimensional variables is defined as in accordance with the data presented in tables 1 and 2.

1500 10200 0.37 0.4017 0.0269 0.6273 0.1344
1250 8500 0.41 0.4450 0.0295 0.7786 0.1344
1000 6900 0.47 0.5092 0.0326 1.0609 0.1341
750 5200 0.54 0.5851 0.0376 1.3972 0.1341
500 3500 0.66 0.7150 0.0458 2.0939 0.1340
250 1800 0.95 1.0277 0.0637 4.4834 0.1337
150 1130 1.25 1.3494 0.0801 8.1296 0.1331
100 790 1.58 1.7012 0.0955 13.7681 0.1326
75 620 1.83 1.9678 0.1075 19.0029 0.1322
60 525 2.13 2.2830 0.1160 27.7771 0.1316
41 395 2.70 2.8820 0.1330 49.3308 0.1308
30 325 3.35 3.5573 0.1449 86.4977 0.1299
20 260.5 4.70 4.9472 0.1594 217.0848 0.1287
10 195.24 9.00 9.3259 0.1798 1351.2011 0.1269
5 162.1683 21.00 21.4257 0.1983 15568.1073 0.1229
3 148.8345216 30.00 30.4723 0.2102 42819.2173 0.1192
1500.0 8700 0.210 0.2315 0.0313 0.0985 0.1229
1250.0 7250 0.230 0.2535 0.0343 0.1181 0.1229
1000.0 5800 0.257 0.2833 0.0384 0.1473 0.1230
500.0 2910 0.362 0.3992 0.0543 0.2909 0.1231
350.0 2050 0.430 0.4742 0.0648 0.4090 0.1231
200.0 1200 0.567 0.6250 0.0846 0.7181 0.1229
150.0 920 0.655 0.7216 0.0965 0.9712 0.1227
100.0 640 0.800 0.8805 0.1152 1.4810 0.1222
50.0 350 1.110 1.2204 0.1559 2.9250 0.1217
30.0 238 1.400 1.5362 0.1880 4.8644 0.1209
20.0 180 1.650 1.8084 0.2160 6.9318 0.1203
10.0 126 2.160 2.3567 0.2551 13.1668 0.1185
6.8 109 2.442 2.6577 0.2724 17.8189 0.1175
5.0 100 2.670 2.8992 0.2825 22.4626 0.1167
3.0 90 3.000 3.2474 0.2950 30.5069 0.1157
1.0 80 3.500 3.7719 0.3094 46.1887 0.1145
0.3 76 3.670 3.9511 0.3169 52.0002 0.1141
0.1 75 3.750 4.0345 0.3186 55.1796 0.1140
Table 1: The parameters of the relativistic configurations with the constant central value of the scalar field
,
Relativistic star without a scalar field
2.699 0.9604 6.8270 0.2207
Relativistic star with the scalar field
0.30 2101.0 1642.5 3.055 0.1022 92.9763 0.1325
0.35 617.0 868.0 3.040 0.1186 78.9467 0.1323
0.40 184.0 531.0 3.030 0.1306 70.9796 0.1316
0.44 54.5 385.0 3.025 0.1376 67.0397 0.1308
0.45 34.9 355.0 3.025 0.1396 66.0795 0.1304
0.47 6.6 317.0 3.025 0.1402 65.8043 0.1297
,
Relativistic star without a scalar field
2.277 1.1430 3.4430 0.2108
Relativistic star with the scalar field
0.27 510.15 275.10 2.424 0.2986 15.8983 0.1262
0.30 300.40 225.90 2.425 0.2947 16.1309 0.1250
0.33 178.90 190.30 2.427 0.2903 16.4159 0.1237
0.35 124.50 169.95 2.430 0.2889 16.5532 0.1228
0.39 57.70 142.00 2.436 0.2794 17.2429 0.1207
0.41 36.00 130.00 2.442 0.2769 17.5278 0.1197
0.43 19.30 119.00 2.442 0.2739 17.7255 0.1185
0.45 6.80 109.00 2.442 0.2724 17.8189 0.1175
0.46 1.90 105.00 2.442 0.2714 17.8864 0.1171
Table 2: The parameters of the relativistic configurations as a function of the central value of the scalar field for fixed observable radius of the star and for given values of and .
Figure 1: The dependence of the function (which is proportional to the total mass ) on the value of the coupling parameter from (23) at the fixed central value and . The data are taken from table 1.
Figure 2: The dependence of the function on the central value of at the fixed for and for , and for both graphs. The data are taken from table 2.
Figure 3: The density and the pressure of the fluid, the scalar field , the metric functions , and the current mass as functions of the relative invariant radius for and respectively. To provide the asymptotic flatness of the solutions, i.e.  as , the value of the constant needed to be chosen as for and for .
Figure 4: The distributions of the total energy density in units of from (29) for different the initial values of , from top to bottom.
Figure 5: The distributions of the total energy density in units of from (29) for the different initial values of , from top to bottom.

Also the ratio of the central density (of the fluid only) to the average density (of both the fluid and the scalar field) is presented in the tables. We define the average density as Tooper:1964 ()

Using the expression for from (27), one can express the quantity in terms of the central density and the boundary value of the mass function. Then we obtain the relation

Finally, in the last column of both tables the gravitational potential energy expressed in units of is shown (see Tooper:1964 () for more details):

where the expression for the energy density is defined from (2). Using equations (14), (16), (22) and (23) one can write the energy density in the following form (in units of )

(29)

The absolute value of the potential energy represents the work that would have to be done on the system to diffuse its mass to infinity.

Next, using the data from the tables, in figure 2 we plot , which is proportional to the total mass of the configuration , on the value of the self-coupling parameter at some constant central . In figure 2 we plot as a function of the value of central for different values and . From figure 2 one can see that, while increases, the masses of the configurations decrease both for and . Initially, for small , the masses are very different for and ; asymptotically, for large , they approach comparable values. This is because for large the total mass is defined by the scalar field but not by the contribution from the fluid. The sizes of the configurations are different for the two cases (which has a size characterized by ) and (which has a size characterized by ).

On the other hand if one fixes the size of the star to equal the size of a corresponding relativistic configuration without a scalar field, and changes the central value of so as to get regular solutions, it is necessary to choose appropriate eigenvalues of and . In this case as a function of is given in figure 2. From this figure one sees that as increases the mass of the configuration can either increase (for the case) or decreases (for the case).

It follows from tables 1, 2 and figures 2, 2 that the masses of the configurations, computed for the values of the system parameters used here, are considerably smaller than the masses of the stars without a scalar field. The maximal dimensionless masses of the configuration with the scalar field for and at are and , respectively; without a scalar field the masses of the same configurations are and , respectively. Our attempts to increase the masses of the system by varying were not successful. From figure 2 the implication is that the masses change only slightly with .

Next, in figure 3 the metric functions , and the mass distribution (as a function of the dimensionless radius ) are given. In plotting these functions we used the following procedure:

Inside the Star: The function was plotted using (13); the function was plotted from equation (14), in terms of the dimensionless variables from (16). Explicitly

(30)

From the interior solution parts of figure 3 (i.e. the region ) we see the reason for the terminology “chameleon star” for the present solutions – the scalar field mimics the behavior of the fluid in the interior region. Even in the exterior region where the fluid goes to zero the scalar field is asymptotically going to zero.

Outside the star: The solution goes to the Schwarzschild solution

(31)

The mass function is defined as

Finally, in figures 5 and 5 we plot the energy density, , from the expression (29). The definition of the mass function given just above takes into account the energy of the fluid and the scalar field from to . Although the fluid vanishes at the scalar field does not vanish. Thus in principle one should include the contribution to the mass function of the scalar field from to . However from the figures 3-5 one can see that and rapidly go to zero as and thus the mass function is given essentially just by the fluid and scalar field energy density between and .

We now turn the the question of connecting the interior region (i.e. where one has both fluid plus scalar field as a source) with the exterior region (i.e. where one has only a rapidly vanishing scalar field). For this purpose, we write down the Einstein equations of (4) and (5), and the scalar field equation of (6), without the fluid source i.e.  so that from (9) and (10) we have zero pressure and density. This leads to the following system of equations:

(32)
(33)
(34)

which, by analogy with the transformations made above, can be rewritten in terms of the dimensionless variables and as follows

(35)
(36)
(37)

where is given by (30), and is taken from (23). Note that this system of equations (32)-(34) (or (35)-(37)) is essentially the same as those obtained in jetzer () which considered a real scalar field with quartic self-interaction. The above system contains the parameter as a trace of the influence of the fluid on the external solution. The solution sought beginning from the surface of the star at using, as the boundary conditions, the values of and obtained from the solution of the equations (25)-(26) for the internal part of the configuration. This allows one to determine the value of the integration constant from (13) by requiring to be equal to unity at infinity, providing asymptotical flatness of the space-time. (The values of for the examples shown in figures 3 are given in the caption.) Thus the complete solution for the configuration under consideration is derived by matching of the internal fluid solutions given by equations (25)-(26) with the external solutions obtained from the system (35)-(37).

The system (35)-(37) has obvious asymptotically flat solutions in the form: with taken from (31) that corresponds to the fact that, by choosing the eigenvalues of the parameters and presented in tables 1 and 2, the external scalar field makes negligible contribution to the total mass of the configuration. It allows using the solution for the metric functions in the form of (31) as a good approximation; , where the constant is made equal to zero by the corresponding choice of (see above); . The numerical calculations of the system (35)-(37) confirm this asymptotic behavior.

From figures 3, 5 and 5 we can draw the following conclusions about the metric functions, and the mass distribution :

(1) Asymptotically, as , the space-time becomes flat, i.e. . To get this asymptotic behavior of the metric function , it is necessary to choose a specific value for the central value, (see the caption of figures 3). The asymptotic behavior of follows from equation (30) and the fact that as .

(2) As shown in Tooper:1964 (), the relativistic configurations without a scalar field are characterized by a greater concentration of matter toward the center than in the non-relativistic case. By including a scalar field, we obtain even greater concentration of matter toward the center. This can be seen by comparing figures 3 with figures 1 and 2 from the paper Tooper:1964 (). The main reason for the increased concentration of mass is the non-minimal coupling between the scalar field and the fluid with the choice of the coupling function given by (22). The functional form of this coupling gives the required regular solutions, lying in the range , only for the eigenvalues of the parameter (see tables 1, 2). This leads to the fact that near the origin, when , a greater concentration of mass occurs due to the presence of the term in the expression for the energy density (29). At the same time both and give relatively small contributions to the energy density compared to the term coming from the non-minimal scalar-fluid coupling. This fact will become important, below, when we consider the stability of the solutions.

(3) The total energy density presented in figures 5 and 5 strongly depends on the polytropic index . At comparable central values of the scalar field the energy density for the case is several times greater than for the case . In turn, both values of yield a greater energy density at the center of the configurations than in the case of the relativistic stars without a scalar field Tooper:1964 () when the energy density was found to be 1.

(4) Despite the higher concentration of matter at the center of the configurations with the addition of the scalar field, their masses are considerably smaller than the masses of the relativistic stars of the same size but without a scalar field (see table 2). This occurs as a consequences of the fact that the external region of the stars, with only the scalar field as a source, is strongly rarefied due to the rapid vanishing of the scalar field in the region .

iv.2 Non-relativistic case

In this section we consider the non-relativistic limit of the system (25)-(26). The non-relativistic limit corresponds to , and . First, we recall some results about this system in the absence of the scalar field Zeld (). With no scalar field the system of equations (25)-(26) reduces to the well known Lane-Emden equation

(38)

This equation has solutions which describe finite size configurations for different values of the parameter Zeld (). The non-relativistic limit for the case when there a scalar field is obtained by omitting all terms with in equations (25)-(26) except for the term in the scalar field equation (26). This term must be kept since the product is generally a non-zero quantity even for . Taking all the above into account we rewrite the system (25)-(26) in the following form:

(39)
(40)
(41)

Here, as in the relativistic case the presence of the term is important – it leads to a change in sign of the effective mass term for certain values of . This feature is important for the existence of regular solutions.

Proceeding as in the previous section, we obtain numerical results for the non-relativistic limit which we present in table 3. We chose the parameter to be . Such a small value of requires that be large enough so that the first term on the right hand side of equation (41) is negative, which is the necessary condition to have regular solutions. In the non-relativistic case, as in the relativistic case, we need to find eigenvalues of two parameters and in order to have regular solutions for a given values of the polytropic index, , and the central value of the scalar field, . From table 3 we do indeed find that there are values of large enough to make the first term on the right hand side of (41) negative thus yielding regular solutions.

,
Non-relativistic star without a scalar field
3.6538 2.7141 5.9907 0.0000
Non-relativistic star with the scalar field
0.039 1040 20600 3.6538 0.4384 37.0928 -0.0009
0.038 1850 21800 3.6538 0.4373 37.1840 -0.0009
0.036 3980 24500 3.6538 0.4354 37.3411 -0.0009
0.034 7010 27600 3.6538 0.4346 37.4105 -0.0009
0.030 17575 35500 3.6538 0.4349 37.3895 -0.0009
0.028 25895 39500 3.6538 0.4462 36.4388 -0.0009
0.027 31270 41700 3.6538 0.4556 35.6855 -0.0009

,
Non-relativistic star without a scalar field
3.1416 3.1416 3.2899 0.0000
Non-relativistic star with the scalar field
0.039 190 10700 3.1416 0.6568 15.7364 -0.0008
0.038 595 11000 3.1416 0.6687 15.4556 -0.0008
0.036 1645 12000 3.1416 0.6761 15.2872 -0.0008
0.034 3070 13100 3.1416 0.6868 15.0493 -0.0008
0.030 7645 15700 3.1416 0.7166 14.4227 -0.0008
0.025 20685 21000 3.1416 0.7457 13.8609 -0.0008
Table 3: The parameters of the non-relativistic configurations with a variable central value of the scalar field . The observable radius of a star is fixed. The choice of the parameters and is indicated in the table.

Using the data from table 3, the dependence of the on the central value of the scalar field is given in figure 7. Figure 7 uses the date from table 3 to plot the ratio of the energy density to the central density, as a function of the dimensionless, normalized radius. Also from table 3 one can see that we have fixed the radius of a star to be equal to the size of a non-relativistic star without a scalar field. From the figure we can draw the following observations:

(1) The masses of the stars with a scalar field are considerably smaller than the masses of the stars without a scalar field. In the case the masses are 4-5 times smaller, and in the case about 6 times smaller.

(2) The mass is a fairly slowly varying function of , especially when .

(3) There is a greater concentration of the mass density towards the center of the stellar configurations. This can be seen explicitly by comparing with the distribution of the mass density for a star without a scalar field which is also presented in figure 7.

All these results are very similar to those of the relativistic case.

Figure 6: The dependence of the function on the central value of at the fixed for and for , and for both graphs. The data are taken from table 3.
Figure 7: The plot of the total energy density, , in units of in the non-relativistic case () for different central values of from table 3. The solid lines correspond to the case of , the dashed lines to the case of . The top two sets of curves correspond to the configurations with the scalar field. For comparison the energy densities of the non-relativistic configurations without a scalar field are also shown by the two bottom, less peaked curves.

iv.3 Stability of the solutions

In this section we discuss the issue of the stability of the regular solutions obtained above. There are two basic approaches to studying the stability: (i) The energy approach to the theory of equilibrium for a star Zeld (); (ii) A more rigorous dynamical stability approach based on studying the stability of linear and nonlinear time-dependent perturbations. In this paper we will use the first approach. Proceeding along the lines of reference Tooper:1964 (), we define the total energy of the system, including the internal and gravitational energies, as

(42)

where is defined by (2) and corresponds to the total energy density of the system (recall that ). Next, we consider a system consisting of a gas of particles having a rest-mass density . Its relativistic energy density is the sum of the rest energy plus the density of internal energy. For the special case of an adiabatic process which assumes the absence of heat flow terms in the energy-momentum tensor (2), it is possible to obtain a relation between the gas density and the total mass density or alternatively between and , .

To do this we use the first law of thermodynamics which, in our case, takes the form

(43)

where is the total pressure, and is the specific volume (this symbol should not be confused with the potential energy used earlier). Since , equation (43) gives the following relation

(44)

In general, when and are functions of , this equation cannot be integrated. However, for our configuration, the numerical calculations performed in the previous sections show that the main part of the energy is provided by the term containing the non-minimal coupling but not by and . This allows us to neglect the terms containing the scalar field kinetic energy and potential as compared with and in the and components of the energy-momentum tensor (2), respectively. In this approximation we have the following expressions for the total mass density and the pressure:

Substituting these expressions into (44) and taking into account that and , we have from (44):

This equation differs from the case without a scalar field by the presence of the second term on the right-hand side containing the coupling function . In the absence of the non-minimal coupling, i.e. when , this term vanishes. Integrating, we find

(45)

where is the value of the gas density at the center of the configuration. This constant may be evaluated as follows Tooper:1964 (): near the boundary of the configuration, where , equation (45) becomes approximately

where is the point starting from which the approximation becomes valid. This equation can be rewritten in differential form as

(46)

Since is a function of , the term on the right-hand side can be evaluated as follows

Taking this expression into account, equation (46) takes the form

Near the boundary the internal energy density is small compared to the rest-mass energy density, so that . Comparing the above equation with the expression , we obtain

and equation (45) becomes

(47)

Using expression (47), the corresponding proper energy of the gas may be defined as the integral of with respect to proper volume for the metric (3) as follows

(48)

The quantity is proportional to the total number of particles in the configuration, , where is Avogadro’s number. Using the dimensionless variables (16) and expressions (30), (47), we obtain the proper energy of the gas (48) in units of the total energy in the following form

(49)

This expression differs from the corresponding expression in Tooper:1964 () through the presence of an extra factor under the integral which comes from the non-minimal coupling in the system. Stability of the configuration can tested via the sign of the expression Tooper:1964 (); Zeld ()

(50)

The necessary (but not a sufficient) condition for stability of the system that i.e. that the total system energy is less that the energy of the non-gravitationally interacting gas, , making the energetically preferred state. The condition in (50) amounts to requiring that the Binding Energy defined in this equation be positive. In the paper Tooper2 (), the question of stability of adiabatic polytropic configurations was considered. It was shown that such configurations may have both positive and negative binding energies depending on the value of the polytropic index , and the parameter (configurations with and were considered).

When a non-minimally coupled scalar field is included, we considered a very narrow range of these parameters, and , and also only one choice of the coupling function given by the form (22). In this case the numerical calculations indicate that , i.e. we have the negative binding energy, and correspondingly the configurations under consideration are unstable. The same values of and without a scalar field give configurations with the positive binding energy Tooper2 (). Obviously, the difference between the present results and those of Tooper2 () is connected with the presence of the extra factor under the integral in the expression (49) whose value, as numerical calculations indicate, is much less than unity for the parameters , used in the paper, and for the coupling function of the form (22).

The behavior of in (49) can be approximately estimated as follows: Since we are looking only for solutions with , the value of may be approximated as giving a finite contribution to the value of integral in the form of some constant factor , i.e. . Then one can see that