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 ofordinary white dwarfs. It is shown that any study of their stability must examine the dependence of themass on two variables, which can, for convenience, be taken to be the rest mass (total baryon mass) of thequark core and the energy density tr of the crust at the surface of the quark core. The range of variationof these quantities over which strange dwarfs are stable is determined. This region is referred to as thestability valley for strange dwarfs. The mass and radius from theoretical models of strange dworfs arecompared with observational data obtained through the HIPPARCOS program and the most probablecandidate strange dwarfs are identified.Keywords: stars: strange dwarfs: superdense stars
Strange quark matter may exist in a more bound state than the matter in atomic nuclei [1,2]. If the bag modelequation of state 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 b per baryon at some baryon concentration n = nmin has a negativeminimum ( ( ) 0
to compact stars have been reviewed by Weber .At the surface of a strange star the electron density n
e 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 ( 31en~l ) to form a thin charged layer where the electric field reaches1017-1018 V/cm . 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 coreonly by gravitation. A strange star can acquire a crust as it is formed or by accretion of matter. The probabilityof tunnelling by atomic nuclei is so small that the crust and quark core can coexist forever. Since free neutrons, withno 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, drip . The numerical value
of tr depends on the model equation of state for the matter in the crust. The formation and structure of the crustin strange stars have been examined in Refs. 12 and 13. Models of strange stars with a crust have been examined 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 stars radius and mass. The situation is different for strange stars
with low core masses ( 020.MM core
the density varies continuously, at densities above the threshold for appearance of quarks. A mixed phase of quarkand nuclear matter can be energetically favorable as a function of the magnitude of the local surface and Coulombenergy associated with the development of these configurations [23-25]. If the surface tension between the quarksand nuclear matter is sufficiently strong, then formation of a mixed phase is energetically unfavorable; that is, a firstorder phase transition takes place with a density jump (if ( ) 0> minb n ) or the formation of a strange star ss (when
( ) 0cddMhave 020
stability of strange dwarfs it is necessary to study the stability of individual families of this sort . Here it becomespossible to use a static criterion for stability developed by Zeldovich  and generalized by Bisnovatyi-Kogan that is free of the cumbersome mathematical calculations characteristic of Chandrasekhars method. Since theelectrostatic 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 isapplicable to series of this kind. At the maxima of the dependences of the sd mass on tr for a fixed value of u(the ( )truM curves), the sd loses stability.
Therefore, as opposed to neutron stars and white dwarfs, models of the stability of strange dwarfs in M, u, trspace occupy the part of the ( )truM , surface that is bounded above by the curve joining the maxima of the ( )truM curves (see Fig. 1). We refer to this region of the ( )truM , 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, totalbaryon number, radius) are uniquely determined by the energy density c at the center of the quark core and theenergy density tr 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 functionof the parameters u and tr (explanations in text).The leading part of the surface ),( truM isbounded above by the curve ecba the stabilityvalley for strange dwarfs.
relativistic equations of hydrostatic equilibrium, Tolman-Oppenheimer-Volkov (TOV) equations [32,33]. For theequation of state of the quark core we have used a bag equation of state with the following model parameters: bagconstant B = 60 MeV/fm3, quark-gluon interaction constant 050.c = , and strange quark mass ms = 175MeV
(model 2 of Ref. 14), for which ( ) 628.nminb = MeV and 2960.nn smin == fm-3, and ns is the baryon concentrationat the surface of the bare strange star. The Baym-Pethick-Sutherland equation of state  was used for the crust;it was matched to the Feynman-Metropolis-Teller equation of state  for a density of 410= g/cm3. For this
equation of state, 111034 = .drip g/cm3.
As noted above, for sd models the central density c and pressure Pc are uniquely related to the energy density
tr and pressure Ptr of the nuclear-electron matter above the surface of the quark core. Thus, it is convenient to take
tr , rather than c , as the independent variable for examining sd models with a fixed parameter u. In fact, as trranges from 104 g/cm3 to 4.31011 g/cm3, the central density c will increase very little.
Many series of strange dwarfs with fixed values of u within 1010 4 .u were studied. Some of the resultsof the calculations for typical series are shown in Fig. 1, where in M, u, tr space (M is the sd mass), on the
( )truMM = , surface we have plotted ( )truM curves (the mass as a function of tr ) for u = const, extended upto the intersection with the coordinate plane driptr = . The mass increases with increasing tr in the individual
series, and stability is lost at Mmax
. For the series with u = 0.013, the sd mass reaches Mmax
for driptr = (the pointc in Fig. 1). For series of strange dwarfs with u > 0.013, the curves intersect the driptr = plane when the masshas no longer reached the maximum (segment bc in Fig. 1). For the series that intersect the driptr = plane in thesegment ab, where 020.u , the ( )truM curves are horizontal, i.e., for hem the mass of the crust is negligiblecompared to the core and uMM .
The segments of the individual ( )truM curves for stable configurations of strange dwarfs in M, u, tr spaceon the saddle surface ( )truMM = , , occupy the stability valley for the strange dwarfs. This region is bounded bythe curves ec (the maxima of the individual series) and cb (the front portion of the surface in Fig. 1). The part ofthis saddle surface that applies to unstable configurations and is formed by the segments of the ( )truM curves afterthe maximum points, intersects the driptr = plane along the dc curve, where uM and, therefore, cM are
greater than zero. Unstable configurations of this sort naturally have 020 for them, lie in the driptr = plane along the curve bc. And there was no
justification for the surprise of Glendenning, et al. , that this is so, for, on going from one series to another alongthe bc curve, uM and, therefore, cM , are less than zero. Similar results will be obtained for the intersection
of the ( )truM , surface with planes parallel to driptr = for smaller values of tr , as was done by Glendenning,et al. [15b]
Vartanyan, et al.  pointed out that, although 020 > for sd on the segment bc, the fact that the condition
driptr = holds for them makes them analogous to the configurations of series with u < 0.013, for which the mass
is maximal and 020 = (the configurations lying on the curve ec of Fig. 1). Thus, if the crust density is increasedeven slightly for these configurations (e.g., as a result of radial pulsations), tr will become greater than drip andfree neutrons will be born at the surface of the quark core and move into the quark core, increasing its mass (the totalnumber of baryons). Since 0 and
driptr = can be stable arises from the fact that Chandrasekhars method is applicable when no irreversible processes
take place involving matter in the star during the pulsations. This condition is violated when neutrons pass intothe 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 difference0> trdrip is larger.
While the dependence of the mass of stable configurations on the central density (the ( )cM curve) forordinary white dwarfs is smooth over the entire range of variation of c , Fig. 1 shows that for strange dwarfs the
( )truM curves have spikes. The mass of the crust over a large part of the variation of tr is negligible, there is
Fig. 2. The radius R of strange dwarfs as a functions ofu and tr . The points indicate configurations for whichstability is lost.
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 limitingsd mass decreases by two-three percent, while tr , at which stability is lost, increases from 1.9109 g/cm3 foru = 10-4 to 2.31011 g/cm3 for u = 10-2.
Figure 2 shows plots of ( )truR , , the sd radius as a function of tr for a fixed quark core, in R, u, tr space(R is the strange dwarf radius) on the ( )truR surface; here the dots indicate the configurations for which stabilityis lost. These curves are analogous to graphs of the radii of ordinary white dwarfs as a function of central density
( )cR . Thus, for low masses, in the individual series with increasing tr , an increase in the sd mass is accompaniedby an increase in the radius, which reaches a maximum R
max at some value of tr , where the configuration is stable,
and then the radii of the configurations in this series decrease with increasing mass until stability is lost. As uincreases, R
max 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 . Intermediate(untabulated) values of these data are approximated when integrating the TOV equations. Unlike their mass, theradius of strange dwarfs is more sensitive to the way this approximation is made. Here we have chosen anapproximation technique that yields the values of the radius given in Refs. 34 and 39 for configurations without aquark core, i.e., for ordinary white dwarfs.
Figure 3 shows tr as a function of u, ( )utr , for configurations with the maximum mass, at which stabilityis lost (smooth curve) and for configurations with the maximum radius (dashed curve). The radii (km, top row) andsd 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. tr as a function of the parameter u forconfigurations with the maximum radius R
curve) and with the maximum mass Mmax
(smoothcurve). The radius (km, upper row) and mass (in solarmasses) are indicated for typical configurations.
which driptr = (see Fig. 1). The configurations are stable up to the maximum mass. On the individual series, thepoints indicate the configurations for which stability is lost. For the series with 0010.u the Mu(R) curves areextremely close to one another, as well as to the analogous curves for ordinary white dwarfs. It is clear from thisfigure 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 caseof white dwarfs with the same mass. Observations of objects of this kind would confirm the existence of strangedwarfs. 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 withlow 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 fortheoretical models of white dwarfs consisting of atomic nuclei and a degenerate electron gas (Ae matter) that werefirst discussed by Chandrasekhar  and subsequently developed by many researchers. While retaining thisnotation, we introduce the notation owd for observed white dwarfs. The European Space Agencys HIPPARCOS
Fig. 4. The mass M as a function of the strange dwarfradius R for different values of u. The individual seriesare extended to configurations with driptr = . Thepoints indicate configurations for which stability is lost.
u R (k
satellite has played a major role in determining and refining the radii and masses of owd. HIPPARCOS data havebeen 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 thegravitational red shift and parallax have been determined with sufficient accuracy. The accuracy in determining theradii of owd depends on the accuracy with which their luminosity, parallax, and effective atmospheric temperaturehave been measured [37,38].
In these papers, data on the radii and masses of the owd with the associated measurement errors were shownin a plot of the theoretical dependence  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 threenuclei (for which Z/A = 1/2), in the case of 56Fe, Z/A = 0.46 and this curve is significantly lower and, for a givenmass, the corresponding wd radius is the smallest. Note that the R(M) curve for 56Fe in this range of masses coincideswith the analogous curve for white dwarfs derived from the equation of state  used here. This curve is indicatedby the dashed curve in Fig. 5.
For this equation of state, Vartanyan, et al. , have compared data for owd with the R(M) curve for strangedwarfs that have driptr = . 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 therefined 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 = 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.
the latter have little effect on the position of the R(M) curve. As might be expected, a quark core reduces the starsradius 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 ofidentification is not unique, since the R(M) curve for wd is lower in the case of 56Fe nuclei and it is not impossiblethat some of the possible sd candidates with a 12C core might be 56Fe-containing wd, although an evolutionary pathfor formation of configurations of this type is unlikely .
For completeness of the comparison, it is necessary to examine R(M), not just for one series of sd withdriptr = , 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 forwd 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 withu = 0.005, 0.01, 0.013, and 0.016 based on the present work. The curves are lower for larger u. The last seriescorresponds 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 = 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 forwd consisting of 56Fe, then these owd may be identified with strange dwarfs. Thus, there is pressing need forrefinements in the boundaries of this region. Thus, when rotation is included, the radius of a star may be grater andshift this region toward larger radii. As noted above , in this mass range, there can hardly be a significant changein 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 thiszone. The data for these stars are given in Fig. 5 and here:
EG50 0.50 0.020 0.01040 0.0006G238-44 0.420 0.010 0.01200 0.0010
In order to obtain complete information on the stability of strange dwarfs, we have examined the dependenceof 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 tr of the nuclear-electroncrust at the surface of the quark core. This makes it possible to apply static stability criteria that are free ofcumbersome mathematical calculations.
The range of variation of u and tr in which strange dwarfs are stable, i.e., their valley of stability, isdetermined on the ( )truM , curves surface. To do this, we have examined the dependence of the mass on tr ,
( )truM , for individual series with u = const. With increasing tr the mass increases and at Mmax the loss of stability
occurs. For series with u > 0.013 the ( )truM curves intersect the plane 111034 == .driptr g/cm3, near which freeneutrons 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 > , 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 ofstrange dwarfs as a function of their mass. A comparison with data from the HIPPARCOS satellite of the EuropeanSpace Agency shows that the most likely candidate strange dwarf among the observed white dwarfs is EG-50.
1. A. R. Bodmer, Phys. Rev. D 4, 160 (1971).2. E. Witten, Phys. Rev. D 30, 272 (1984).3. A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorne, and V. F. Weisskopf, Phys. Rev. D 9, 3471 (1974).4. E. Farhi and R. L. Jaffe, Phys. Rev. D 30, 2379 (1984).5. C. Alcock, E. Farhi, and A. Olinto, Astrophys. J. 310, 261 (1986).6. P. Hansel, J. L. Zdunik, and R. Shaeffer, Astron. Astrophys. 160, 121 (1986).7. L. A. Kondratyuk, M. I. Krivoruchenko, and V. V. Martemyanov, Pisma v Astron. zh. 16, 954 (1990).8. F. Weber, N. K. Glendenning, and M. K. Weigel, Astrophys. J. 373, 579 (1991).9. F. Weber and N. K. Glendenning, LBL-33066 (1992).
10. Yu. L. Vartanyan, A. R. Arutyunyan, and A. K. Grigoryan, Astrofizika 37, 499 (1994); Pisma v Astron. zh. 21,136 (1995).
11. F. Weber, Prog. Part Nucl. Phys. 54,193 (2005).12. J. Miralda-Escude, P. Haensel, and B. Paczynski, Astrophys. J. 362, 572 (1990).13. N. K. Glendenning and F. Weber, Astrophys J. 400, 647 (1992).14. Yu. L. Vartanyan, A. K. Grigoryan, and T. R. Sarkisyan, Astrofizika 47, 223 (2004).15. N. K. Glendenning, Ch. Kettner, and F. Weber, Phys. Rev. Lett. 74, 3519 (1995a); Astrophys, J. 450, 253 (1995b).16. G. J. Mathews, L. S. Suh, B. O. Gorman, et al., J. Phys. G 32, 747 (2006).17. M. Alford, K. Rajagopal, and F. Wilczek, Nuc. Phys B 537, 433 (1999).18. K. Rajagopal and F. Wilczek, Phys. Rev. Lett. 86, 3492 (2001).19. G. Lugones and J. E. Horvath, Astron. Astrophys. 149, 239 (2003).20. V. V. Usov, Phys. Rev. D 70, 067301 (2004).21. J. Madsen, Phys. Rev. Lett. 87, 172003 (2001).22. N. K. Glendenning, Phys. Rev. D 46, 1274 (1992).23. H. Heiselberg, C. J. Pethick, and E. F. Staubo, Phys. Rev. Lett. 70, 1355 (1993).24. C. P. Lorenz, D. G. Revenhall, and C. J. Pethick, Phys. Rev. Lett. 70, 379 (1993).25. H. Heiselberg and M. Hjorth-Jensen, nucl.-th/9902033 (1999).26. P. Jaikumar, S. Reddy, and A. W. Steiner, Phys. Rev. Lett. 96, 041101 (2006).
27. S. Chandrasekhar, Phys. Rev. Lett. 12, 114 (1964a); Astrophys. J. 140, 417 (1964b).28. S. Chandrasekhar, Phys. Rev. Lett. 14, 241 (1966).29. Yu. L. Vartanyan, G. S. Hajyan, A. K. Grigoryan, and T. R. Sarkisyan, Astrofizika 52, 325 (2009); 52, 481 (2009).30. Ya. B. Zeldovich, Voprosy kosmologii 9, 36, Izd. AN SSSR (1963).31. G. S. Bisnovatyi-Kogan, Astron. Astrophys. 31, 3910 (1974).32. R. C. Tolman, Phys. Rev. 55, 354 (1939).33. J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55, 374 (1939).34. G. Baym, C. Pethick, and P. Sutherland, Astrophys. J. 170, 299 (1971).35. R. P. Feynman, N. Metropolis, and E. Teller, Phys. Rev. 75, 1561 (1949).36. S. Chandrasekhar, Stellar structure, Chicago (1939).37. J. L. Provencal, H. L. Shipman, E. Hog, and P. Thejll, Astrophys. J. 494, 759 (1998).38. J. L. Provencal, H. L. Shipman, D. Koester, F. Wesemael, and P. Berg, Astrophys. J. 568, 324 (2002).39. T. Hamada and F. E. Salpeter, Astrophys. J. 134, 683 (1961).
STABILITY VALLEY FOR STRANGE DWARFS1. Introduction2. The stability valley for strange dwarfs3. Computational results4. Comparison with observations5. ConclusionREFERENCES