98

STABILITY VALLEY FOR STRANGE DWARFS

Yu. L. Vartanyan, G. S. Hajyan, A. K. Grigoryan, and T. R. Sarkisyan

This is a study of the stability of strange dwarfs, superdense stars with a small self-confining core

(�

M.M core 020< ) containing strange quark matter and an extended crust consisting of atomic nuclei

and degenerate electron gas. The mass and radius of these stars are of the same orders as those of

ordinary white dwarfs. It is shown that any study of their stability must examine the dependence of the

mass on two variables, which can, for convenience, be taken to be the rest mass (total baryon mass) of the

quark core and the energy density trρ of the crust at the surface of the quark core. The range of variation

of these quantities over which strange dwarfs are stable is determined. This region is referred to as the

stability valley for strange dwarfs. The mass and radius from theoretical models of strange dworfs are

compared with observational data obtained through the HIPPARCOS program and the most probable

candidate strange dwarfs are identified.

Keywords: stars: strange dwarfs: superdense stars

1. Introduction

Strange quark matter may exist in a more bound state than the matter in atomic nuclei [1,2]. If the bag model

equation of state[3] is used for the quarks, then for certain values of the phenomenological parameters of the model,

a case may arise in which the average energy εb per baryon at some baryon concentration n = n

min has a negative

minimum ( ( ) 0<ε minb n ). In this case, the quark matter may be in a self-bound state, so that the existence of self-

confining cosmic objects, or so-called “strange stars” (ss), which can exist without gravitation, becomes possible [4].

This places a limit on the maximum mass of these configurations, which, as in the case of neutron stars, is on the

order of �M2 . Strange stars have been studied in Refs. 5-10. Research on strange quark matter and its relationship

Astrophysics, Vol. 55, No. 1, March , 2012

0571-7256/12/5501-0098 ©2012 Springer Science+Business Media, Inc.

Original article submitted September 16, 2011; accepted for publication November 23, 2011. Translated fromAstrofizika, Vol. 55, No. 1, pp. 113-126 (February 2012).

Yerevan State University, Armenia; e-mail: yuvartanyan@ysu.am

99

to compact stars have been reviewed by Weber [11].

At the surface of a strange star the electron density ne is several orders of magnitude below that of the quarks

and, since the electrons are confined only by an electrostatic field, some of them can move away from the quark

surface of the strange star by hundreds of fermis ( 31−en~l ) to form a thin charged layer where the electric field reaches

1017-1018 V/cm [5]. This field isolates the crust, which is made up of atomic nuclei and a degenerate electron gas

(“Ae” matter), and is not in thermodynamic equilibrium with the strange quark matter; it is coupled to the quark core

only by gravitation. A strange star can acquire a crust as it is formed or by accretion of matter. The probability

of tunnelling by atomic nuclei is so small that the crust and quark core can coexist forever. Since free neutrons, with

no electrical charge, can pass unimpeded through the electrostatic barrier and be absorbed by the strange quark matter,

the maximum density of the crust is limited by the rate of escape of neutrons from nuclei, dripρ . The numerical value

of trρ depends on the model equation of state for the matter in the crust. The formation and structure of the crust

in strange stars have been examined in Refs. 12 and 13. Models of strange stars with a crust have been examined

[14] over the entire range of variation of the central density of a star for two sets of parameters of the bag model,

on which the integral characteristics of the strange quark core depend, and for three values of the crust boundary

density. It was found that for strange stars with strange quark core masses 50.MM core >�, the thickness and mass

of the crust are negligibly small compared to the star’s radius and mass. The situation is different for strange stars

with low core masses ( 020.MM core <� ). In configurations of this sort, the crust swells greatly, and the mass and

radius are the same as for white dwarfs (wd), from which they differ in that they have a core in the form of a small-

radius, low-mass strange star and a crust in which the density can be two orders of magnitude greater than the limiting

density for white dwarfs. These models are referred to as strange dwarfs (sd) [15a, b].

We now discuss two circumstances which may be important for the formation of strange dwarfs. First,

following Ref. 16, we consider superconductivity in the strange quark matter. It has been shown [17,18] that at high

densities a strange quark plasma can be in a state of color superconductivity (CFL) with different amounts of u, d,

and s quarks. However, at relatively low densities, two color superconductivity (2SL), which has properties that are

extremely similar to the bag model [11], is more probable. Furthermore, even in the CFL model, the thermodynamic

potential of the quarks is given by efffree B−Ω=Ω , where freeΩ is the thermodynamic potential of paired quarks

and Beff

is expressed additively in terms of the bag constant B and 2μ , where μ is the average chemical potential

of the quarks [19]. Taking some sort of average value for this quantity, we can treat Beff

as a constant over the entire

range of variation in the density. With this simplification, accounting for the effect of superconductivity reduces

to replacing the bag constant B by Beff

, which, in turn, changes the structural parameters of the strange star by several

percent. If the CFL model, which is essentially electrically neutral, is taken for the superconducting quark plasma,

even for the entire volume of the ss, then, because of the surface effect [21], a narrow surface layer of strange quark

material acquires a positive charge [20], i.e., conditions are created for formation of a crust consisting of degenerate

electrons and atomic nuclei. Thus, the inclusion of quark superconductivity is not sufficient to exclude the possible

presence, in strange stars, of a crust with the composition of white dwarf matter.

The second circumstance that we wanted to consider separately, is related to the possible existence of a so-

called “mixed phase,” where droplet-shaped, rod-shaped, and slab-shaped quark configurations [22] can develop, as

100

the density varies continuously, at densities above the threshold for appearance of quarks. A mixed phase of quark

and nuclear matter can be energetically favorable as a function of the magnitude of the local surface and Coulomb

energy associated with the development of these configurations [23-25]. If the surface tension between the quarks

and nuclear matter is sufficiently strong, then formation of a mixed phase is energetically unfavorable; that is, a first

order phase transition takes place with a density jump (if ( ) 0>ε minb n ) or the formation of a strange star ss (when

( ) 0<ε minb n ). Strange dwarf sd models can be formed only when the development of a “mixed phase” is energetically

unfavorable, since otherwise the electric field at the surface of the strange star will be insignificant [26].

This paper is a study of the stability and observational manifestations of strange dwarfs.

2. The stability valley for strange dwarfs

Only stable equilibrium configurations of superdense stars are of physical interest. Glendenning, et al. [15],

have studied the stability of strange dwarfs sd by the method of small radial perturbations developed in the general

theory of relativity by Chandrasekhar [27,28]. They examined the dependence of the mass M on the central energy

density of the quark core, cρ , for a series of sd in which the energy density of the nuclear-electron crust at the surface

of the quark core, trρ , was set equal to its maximum value, i.e. on the curve ( )driptrcM ρ=ρρ , . It was found that,

as opposed to ordinary white dwarfs and neutron stars, sd configurations for which 0<ρcddM have 020 >ω ( 0ω

is the fundamental mode frequency for radial pulsations), i.e., they are stable, while configurations with 0>ρcddM

have 020 <ω , i.e., are unstable.

This, however, is nothing surprising, since complete information on the stability of sd can be obtained only

by examining the entire range of variation of cρ and trρ , i.e., we have to consider the functional surface, i.e., the

mass of the sd as a function of these variables, ( )trcM ρρ , . Here it is convenient to take u and trρ as the independent

variables rather than cρ and trρ , where u is the total number Ncore

of baryons in the quark core of the sd expressed

in solar masses, i.e., �MmNu core= , where ( ) 56Fe56Mm = and ∫ λπ=

0

0

224R

core drrneN , where n is the baryon

concentration, R0 is the radius of the quark core, and ( )λexp is the radial component of the metric tensor. In fact,

as opposed to neutron stars, the ( )cM ρ curve for bare (without a crust) strange dwarfs is not bounded below; it

approaches zero when the central density cρ approaches the density of self-bound quark matter neglecting gravitation.

The surface of an ss without a crust, as usual, is determined by the radius over which the pressure goes to zero:

( ) 0=ssRP . In the case of strange dwarfs, on the other hand, where there is an extended nuclear-electron crust over

the surface of a small quark core ( �M.u 020≤ ), the energy density trρ and pressure P

tr at the surface of a fixed quark

core will increase as the mass of the crust increases. In the bag model used for the quark core, the energy density

at the center of the quark core will increase extremely slowly. As noted above, the increase in trρ is limited by

maximum value driptr ρ≤ρ .

Each small quark core becomes the basis of a family of sd with different crust masses. Thus, in studying the

101

stability of strange dwarfs it is necessary to study the stability of individual families of this sort [29]. Here it becomes

possible to use a static criterion for stability developed by Zel’dovich [30] and generalized by Bisnovatyi-Kogan [31]

that is free of the cumbersome mathematical calculations characteristic of Chandrasekhar’s method. Since the

electrostatic field at the surface of the quark core inhibits the penetration of matter from the crust into the quark core,

for small radial oscillations the masses of both the crust and the core are unchanged. Thus, the static criterion is

applicable to series of this kind. At the maxima of the dependences of the sd mass on trρ for a fixed value of u

(the ( )truM ρ curves), the sd loses stability.

Therefore, as opposed to neutron stars and white dwarfs, models of the stability of strange dwarfs in M, u, trρ

space occupy the part of the ( )truM ρ , surface that is bounded above by the curve joining the maxima of the ( )truM ρ

curves (see Fig. 1). We refer to this region of the ( )truM ρ , surface as the stability valley for strange dwarfs.

3. Computational results

For the chosen equations of state of the quark core and crust, the integral parameters of an sd (mass, total

baryon number, radius) are uniquely determined by the energy density cρ at the center of the quark core and the

energy density trρ of the nuclear-electron matter at the boundary between the quark core on integrating the

Fig. 1. The mass M of strange dwarfs as a function

of the parameters u and trρ (explanations in text).

The leading part of the surface ),( truM ρ is

bounded above by the curve ecba – the stabilityvalley for strange dwarfs.

u

M/M

�

ρ tr (

g/sm

3 )

dripρ

102

relativistic equations of hydrostatic equilibrium, Tolman-Oppenheimer-Volkov (TOV) equations [32,33]. For the

equation of state of the quark core we have used a bag equation of state with the following model parameters: bag

constant B = 60 MeV/fm3, quark-gluon interaction constant 050.c =α , and strange quark mass ms

= 175MeV

(model 2 of Ref. 14), for which ( ) 628.nminb −=ε MeV and 2960.nn smin == fm-3, and ns is the baryon concentration

at the surface of the bare strange star. The Baym-Pethick-Sutherland equation of state [34] was used for the crust;

it was matched to the Feynman-Metropolis-Teller equation of state [35] for a density of 410=ρ g/cm3. For this

equation of state, 111034 ⋅=ρ .drip g/cm3.

As noted above, for sd models the central density cρ and pressure Pc are uniquely related to the energy density

trρ and pressure Ptr of the nuclear-electron matter above the surface of the quark core. Thus, it is convenient to take

trρ , rather than cρ , as the independent variable for examining sd models with a fixed parameter u. In fact, as trρ

ranges from 104 g/cm3 to 4.3·1011 g/cm3, the central density cρ will increase very little.

Many series of strange dwarfs with fixed values of u within 1010 4 .u ≤≤− were studied. Some of the results

of the calculations for typical series are shown in Fig. 1, where in M, u, trρ space (M is the sd mass), on the

( )truMM ρ= , surface we have plotted ( )truM ρ curves (the mass as a function of trρ ) for u = const, extended up

to the intersection with the coordinate plane driptr ρ=ρ . The mass increases with increasing trρ in the individual

series, and stability is lost at Mmax

. For the series with u = 0.013, the sd mass reaches Mmax

for driptr ρ=ρ (the point

c in Fig. 1). For series of strange dwarfs with u > 0.013, the curves intersect the driptr ρ=ρ plane when the mass

has no longer reached the maximum (segment bc in Fig. 1). For the series that intersect the driptr ρ=ρ plane in the

segment ab, where 020.u ≥ , the ( )truM ρ curves are horizontal, i.e., for hem the mass of the crust is negligible

compared to the core and uMM ≈� .

The segments of the individual ( )truM ρ curves for stable configurations of strange dwarfs in M, u, trρ space

on the saddle surface ( )truMM ρ= , , occupy the stability valley for the strange dwarfs. This region is bounded by

the curves ec (the maxima of the individual series) and cb (the front portion of the surface in Fig. 1). The part of

this saddle surface that applies to unstable configurations and is formed by the segments of the ( )truM ρ curves after

the maximum points, intersects the driptr ρ=ρ plane along the dc curve, where uM ∂∂ and, therefore, cM ρ∂∂ are

greater than zero. Unstable configurations of this sort naturally have 020 <ω [15]. But this does not mean that the

quark cores of the individual series in the segment dc cannot form stable sd configurations.

Strange dwarf configurations for which the masses of the individual series ( )truM ρ have not reached the

maximum values, so that 020 >ω for them, lie in the driptr ρ=ρ plane along the curve bc. And there was no

justification for the surprise of Glendenning, et al. [15], that this is so, for, on going from one series to another along

the bc curve, uM ∂∂ and, therefore, cM ρ∂∂ , are less than zero. Similar results will be obtained for the intersection

of the ( )truM ρ , surface with planes parallel to driptr ρ=ρ for smaller values of trρ , as was done by Glendenning,

et al. [15b]

Vartanyan, et al. [29] pointed out that, although 020 >ω for sd on the segment bc, the fact that the condition

103

driptr ρ=ρ holds for them makes them analogous to the configurations of series with u < 0.013, for which the mass

is maximal and 020 =ω (the configurations lying on the curve ec of Fig. 1). Thus, if the crust density is increased

even slightly for these configurations (e.g., as a result of radial pulsations), trρ will become greater than dripρ and

free neutrons will be born at the surface of the quark core and move into the quark core, increasing its mass (the total

number of baryons). Since 0<∂∂ uM on the segment cb in this case, the new quark core with its larger mass cannot

form and sd with the initial baryon number. Thus, this kind of configuration loses equilibrium. It undergoes a

transition to a branch of strange stars ss in a state with the same number of baryons, but with a thin nuclear-electron

crust (an extension of the ab branch in Fig. 1). The radius Rss of these configurations is on the order of 10 km, as

in typical neutron stars. An energy ssG RGM~W 2Δ is released during this transition; this is of the same order as

the energy released in supernova explosions. The error in the conclusion that strange dwarfs with 020 >ω and

driptr ρ=ρ can be stable arises from the fact that Chandrasekhar’s method is applicable when no irreversible processes

take place involving matter in the star during the pulsations. This condition is violated when neutrons pass into

the quark core. These configurations are at the edge of stability loss. In this case, for each fixed quark core with

0130.u ≥ , the distance of the sd from the critical state (the stability “margin”) is greater when the difference

0>ρ−ρ trdrip is larger.

While the dependence of the mass of stable configurations on the central density (the ( )cM ρ curve) for

ordinary white dwarfs is smooth over the entire range of variation of cρ , Fig. 1 shows that for strange dwarfs the

( )truM ρ curves have spikes. The mass of the crust over a large part of the variation of trρ is negligible, there is

Fig. 2. The radius R of strange dwarfs as a functions of

u and trρ . The points indicate configurations for which

stability is lost.

R (

km)

uρ tr

(g/cm

3 )

104

a steep rise in the mass to the maximum value, where stability is lost. Here, as u increases from 10-4 to 10-2, the limiting

sd mass decreases by two-three percent, while trρ , at which stability is lost, increases from 1.9·109 g/cm3 for

u = 10-4 to 2.3·1011 g/cm3 for u = 10-2.

Figure 2 shows plots of ( )truR ρ, , the sd radius as a function of trρ for a fixed quark core, in R, u, trρ space

(R is the strange dwarf radius) on the ( )truR ρ surface; here the dots indicate the configurations for which stability

is lost. These curves are analogous to graphs of the radii of ordinary white dwarfs as a function of central density

( )cR ρ . Thus, for low masses, in the individual series with increasing trρ , an increase in the sd mass is accompanied

by an increase in the radius, which reaches a maximum Rmax

at some value of trρ , where the configuration is stable,

and then the radii of the configurations in this series decrease with increasing mass until stability is lost. As u

increases, Rmax

decreases from 23000 km for u = 10-4 to 13058 km for u = 10-2. With regard to the definition of the

sd radius, we note the following. The equation of state of “Ae” matter is specified in tabular form [34]. Intermediate

(untabulated) values of these data are approximated when integrating the TOV equations. Unlike their mass, the

radius of strange dwarfs is more sensitive to the way this approximation is made. Here we have chosen an

approximation technique that yields the values of the radius given in Refs. 34 and 39 for configurations without a

quark core, i.e., for ordinary white dwarfs.

Figure 3 shows trρ as a function of u, ( )utrρ , for configurations with the maximum mass, at which stability

is lost (smooth curve) and for configurations with the maximum radius (dashed curve). The radii (km, top row) and

sd masses (in solar masses) for the characteristic configurations are indicated next to the curves.

Figure 4 shows the mass as a function of radius, Mu(R), in M, R, u space on the M(R, u) surface for different

series of strange dwarfs with fixed quark cores. The curves for the different series are extended to configurations for

Fig. 3. trρ as a function of the parameter u for

configurations with the maximum radius Rmax

(dashedcurve) and with the maximum mass M

max (smooth

curve). The radius (km, upper row) and mass (in solarmasses) are indicated for typical configurations.

u

ρ tr (

g/cm

3 )

����� ����� ���� ���� �����

�

%�

�%�

�%�

�%�

�%�

Mmax

Rmax

105

which driptr ρ=ρ (see Fig. 1). The configurations are stable up to the maximum mass. On the individual series, the

points indicate the configurations for which stability is lost. For the series with 0010.u ≤ the Mu(R) curves are

extremely close to one another, as well as to the analogous curves for ordinary white dwarfs. It is clear from this

figure that if these curves are close in for medium masses ( 8050 .MM. ≤≤�

), then for low masses ( 20.MM ≤�

)

the radii of sd with a relatively large quark core ( 0130.u ≥ ) are smaller by almost a factor of two than in the case

of white dwarfs with the same mass. Observations of objects of this kind would confirm the existence of strange

dwarfs. Unfortunately, these kinds of observations have been made only for medium masses (see the next section),

and no such data exist for low masses. In this regard, we note that in theoretical models of hot configurations with

low masses for which the radii are large, the thickness of the nondegenerate layers may approach 15% of the radius.

4. Comparison with observations

In the above discussion we have used the notation ss for strange stars, sd for strange dwarfs, and wd for

theoretical models of white dwarfs consisting of atomic nuclei and a degenerate electron gas (“Ae” matter) that were

first discussed by Chandrasekhar [36] and subsequently developed by many researchers. While retaining this

notation, we introduce the notation owd – for observed white dwarfs. The European Space Agency’s HIPPARCOS

Fig. 4. The mass M as a function of the strange dwarfradius R for different values of u. The individual series

are extended to configurations with driptr ρ=ρ . The

points indicate configurations for which stability is lost.

M/M

�

u R (

km)

106

satellite has played a major role in determining and refining the radii and masses of owd. HIPPARCOS data have

been analyzed [37,38] to obtain improved values for the mass and radius of 22 owd with masses of �

M.. 0140 ÷ .

The masses were determined with some accuracy for those owd that form part of visual binaries or for which the

gravitational red shift and parallax have been determined with sufficient accuracy. The accuracy in determining the

radii of owd depends on the accuracy with which their luminosity, parallax, and effective atmospheric temperature

have been measured [37,38].

In these papers, data on the radii and masses of the owd with the associated measurement errors were shown

in a plot of the theoretical dependence [39] of the radius on wd mass, obtained separately for four atomic nuclei,4He, 12C, 24Mg, and 56Fe; these results are shown in Fig. 5. While the R(M) curves are very similar for the first three

nuclei (for which Z/A = 1/2), in the case of 56Fe, Z/A = 0.46 and this curve is significantly lower and, for a given

mass, the corresponding wd radius is the smallest. Note that the R(M) curve for 56Fe in this range of masses coincides

with the analogous curve for white dwarfs derived from the equation of state [34] used here. This curve is indicated

by the dashed curve in Fig. 5.

For this equation of state, Vartanyan, et al. [14], have compared data for owd with the R(M) curve for strange

dwarfs that have driptr ρ=ρ . It was found that three owd, EG-50, G238-44, and Procyon B lie very close to this curve

and, therefore, may be candidate strange dwarfs. Note that Procyon B was included in this list by mistake, since the

refined estimate of its mass from Ref. 38 was not taken into account in that analysis.

The same problem has been examined in more detail in Ref. 16, where a series of sd with driptr ρ=ρ for a

crust containing 12C nuclei was examined for two different equations of state of the quark core. It was found that

Fig. 5. Radius R as a function of mass M for white (wd)and strange (sd) dwarfs. The curves 4He, 12C, 24Mg, and56Fe are for wd and use data from Ref. 39. The othernotation is explained in the text.

�MM

R (

km)

��� ��� ���

����

����

����

���

����

����

����

107

the latter have little effect on the position of the R(M) curve. As might be expected, a quark core reduces the star’s

radius for a fixed star mass. R(M) for sd is considerably lower than the analogous curve for wd containing 56Fe nuclei.

It was shown that, of the total number of 22 owd, 8 may be strange dwarfs with a 12C crust. However, this sort of

identification is not unique, since the R(M) curve for wd is lower in the case of 56Fe nuclei and it is not impossible

that some of the possible sd candidates with a 12C core might be 56Fe-containing wd, although an evolutionary path

for formation of configurations of this type is unlikely [16].

For completeness of the comparison, it is necessary to examine R(M), not just for one series of sd with

driptr ρ=ρ , but also the curves for characteristic fixed values of u from the stability valley calculated for the different

elements. This is of special interest for 56Fe nuclei, since only sd configurations can lie below the R(M) curve for

wd of this element.

Under the R(M) curve for 56Fe-containing wd in Fig. 5 we have plotted R(M) for series of stable sd with

u = 0.005, 0.01, 0.013, and 0.016 based on the present work. The curves are lower for larger u. The last series

corresponds to the maximum sized quark core, for which an extended crust can develop, i.e., an sd can be formed.

The R(M) curve for the sd series with driptr ρ=ρ is also shown here.

Thus, under R(M) curve for iron wd in Fig. 5, there is a limiting strip on which only sd can lie. If the owd

include candidates with masses and radii and associated measurement errors that end up below the R(M) curve for

wd consisting of 56Fe, then these owd may be identified with strange dwarfs. Thus, there is pressing need for

refinements in the boundaries of this region. Thus, when rotation is included, the radius of a star may be grater and

shift this region toward larger radii. As noted above [16], in this mass range, there can hardly be a significant change

in the equation of state for the quark core.

Of the owd noted here, only EG-50 is very close to satisfying this requirement. There is another candidate,

G238-44, which was also noted in Ref. 14, for which the mass has been found with small error to be close to this

zone. The data for these stars are given in Fig. 5 and here:

�MM

�RR

EG50 0.50 ± 0.020 0.01040 ± 0.0006

G238-44 0.420 ± 0.010 0.01200 ± 0.0010

5. Conclusion

In order to obtain complete information on the stability of strange dwarfs, we have examined the dependence

of the mass M of these configurations on the parameter u (�

MMu core= , where Mcore

is the rest mass (total number

of baryons) of the quark core which contains the strange quarks) and on the energy density trρ of the nuclear-electron

crust at the surface of the quark core. This makes it possible to apply static stability criteria that are free of

cumbersome mathematical calculations.

The range of variation of u and trρ in which strange dwarfs are stable, i.e., their valley of stability, is

determined on the ( )truM ρ , curves surface. To do this, we have examined the dependence of the mass on trρ ,

( )truM ρ , for individual series with u = const. With increasing trρ the mass increases and at Mmax

the loss of stability

108

occurs. For series with u > 0.013 the ( )truM ρ curves intersect the plane 111034 ⋅=ρ=ρ .driptr g/cm3, near which free

neutrons are born in the nuclear-electron plasma, when the mass has not yet reached its maximum value. Although

the fundamental frequency for radial pulsations for limiting configurations of this kind, 020 >ω , they are on the edge

of losing stability. Their transition to the stable branch for strange stars, with radii on the order of ten kilometers,

is accompanied by an energy release equivalent to that in a supernova explosion.

The limiting region for the existence of stable strange dwarfs has been determined on plots of the radius of

strange dwarfs as a function of their mass. A comparison with data from the HIPPARCOS satellite of the European

Space Agency shows that the most likely candidate strange dwarf among the observed white dwarfs is EG-50.

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