Searching for Strange Quark Matter Objects Among White Dwarfs

Prepared for submission to JCAP

Abudushataer Kuerban,a,b Yong-Feng Huang,a,b,1 Jin-Jun Genga,b and Hong-Shi Zongc,d,e,f

aSchool of Astronomy and Space Science, Nanjing University, Nanjing 210023, People’s Republic of China bKey Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023, People’s Republic of China cDepartment of Physics, Nanjing University, Nanjing 210093, People’s Republic of China dJoint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, People’s Republic of China eNanjing Institute of Proton Source Technology, Nanjing 210046, People’s Republic of China f Department of Physics, Anhui Normal University, Wuhu, Anhui 241000, People’s Republic of China E-mail: [email protected]

Abstract. It has long been argued that the ground state of matter may be strange quark matter (SQM), not hadronic matter. A whole sequence of SQM objects, ranging from strange quark stars and strange quark dwarfs to strange quark planets, can stably exist according to this SQM hypothesis. A strange dwarf has a mass similar to that of a normal white dwarf, but could harbor an extremely dense SQM core (with the density as large as ∼ 4 × 1014 g cm−3) at the center so that its radius can be correspondingly smaller. In this study, we try to search for strange dwarfs among the observed “white dwarfs” by considering their diﬀerence in the mass-radius relation. Eight strange dwarf candidates are identiﬁed in this way, whose masses are in the range of ∼ 0.02M – 0.12M , with the radii narrowly distributed in ∼ 9,000 km – 15,000 km. The eight objects are SDSS J165143.45+364647.6, LSPM J0815+1633, LP 240-30, BD+20 5125B, LP 462-12, WD J1257+5428, 2MASS J13453297+4200437, and SDSS J085557.46+053524.5. Comparing with white dwarfs of similar mass, these candidates are arXiv:2012.05748v1 [astro-ph.SR] 10 Dec 2020 obviously smaller in radius. Further observations with large radio/IR/optical telescopes on these interesting candidates are solicited.

Keywords: stars, neutron stars, white dwarfs

1Corresponding author. Contents

1 Introduction1

2 Structure of strange dwarf2

3 Data Collection3

4 Identifying strange dwarf candidates4 4.1 Mass-Radius Relation4 4.2 Comparison with Observations5

5 Comparison with previous studies7

6 Discussion and Conclusions7

1 Introduction

It has long been argued that the ground state of matter may be quark matter [1,2] rather than hadronic form, because its energy per baryon could be less than that of the most stable atomic nucleus (56Fe). The existence of more exotic states such as strange quark matter (SQM – an approximately equal mixture of u, d, s quarks) in the core of compact stars was also speculated [1–4]. According to this SQM hypothesis, there could exist a whole sequence of SQM objects, ranging from strange stars (SSs, with masses similar to those of neutron stars) [3–5], to strange dwarfs (SDs) – stellar objects composed of a small SQM core and a thick normal matter crust which could be regarded as the counterpart of normal white dwarfs [6–8], and even to strange quark planets [6,7,9–12]. According to previous studies [5–7], SQM stars may be bare SQM objects, but they also may be covered by a crust of normal hadronic matter. From this point of view, crusted SQM objects share many common features with ordinary compact objects. The similarity between SQM stars and normal neutron stars makes it difficult to discriminate these two internally different compact stars observationally [5]. Anyway, a great effort has been made to try to reveal the difference between them. For example, they may have different mass-radius (M − R) relations, cooling rates, maximum masses, gravitational wave patterns, etc. It was even suggested that strange stars can host very close-in planets so that pulsar planets with an orbit period less than ∼ 6100 s can be safely regarded as an indication of the existence of SQM planetary systems [12, 13]. Recent discovery of several 2 M pulsars [14–16] greatly stimulated the study of the internal composition of these enigmatic compact stars. Especially, an interesting compact object of 2.6 M was reported to be associated with the gravitational wave event GW190814 [17]. If it was a “neutron star” instead of a black hole, then the equation of state (EOS) of hadronic matter will be significantly constrained so that it will greatly benefit the study of dense matter. More interestingly, a recent study by [18] suggested some evidence for the existence of quark-matter cores in massive neutron stars. It further arizes researchers’ interests of trying to find SQM in compact stars. Here we will mainly focus our attention on strange dwarfs (SDs). The normal matter crust of strange dwarfs makes them similar to normal white dwarfs (WDs), but they still

– 1 – have different features in the M − R relation. As mentioned above, [6,7] and many other authors have investigated the properties of strange dwarfs and tried to differentiate them from white dwarfs in both theoretical [19–28] and observational aspects [29–33]. Especially, [24, 30] identified eight objects as candidates of strange dwarfs because their measured masses and radii are consistent with the expected mass-radius relation of strange quark dwarfs with a carbon crust. The progress in observational technology leads to a drastic increase in the number of white dwarfs being detected in the past decades. These vast amounts of data inspire us to re-examine all of them to identify more strange dwarf candidates. In this study, we will study the structural parameters of the dwarf objects systematically and try to identify possible strange dwarfs by comparing the observational data with theoretical modeling. Our paper is organized as follows. In Section2, the theoretical background relevant to strange dwarf is briefly introduced. In Section3, we describe the data source of our sample. In Section4, strange dwarf candidates are selected according to the theoretical modeling in Section2. In Section5, we compare our strange dwarf candidates with previously published samples and determine the SQM fraction. Finally, Section6 presents our conclusions and discussion.

2 Structure of strange dwarf

The internal structure of compact objects can be inferred from the general relativistic form of Tolman-Oppenheimer-Volkoff (TOV) equation [34], dp(r) −G[ρ(r) + p(r)/c2][m(r) + 4πr3p(r)/c2] = , (2.1) dr r2[1 − 2Gm(r)/rc2] dm(r) = 4πr2ρ(r), (2.2) dr where m(r), ρ(r), and p(r) are mass, mass density, and pressure at radial coordinate r, respectively; G is the gravitational constant, and c is the speed of light. The relation between p(r) and ρ(r) is determined by the EOS which itself depends on the internal composition of the compact object. As described in [6,7], a strange dwarf can be composed of SQM in the core and normal matter in the crust. Consequently, the property of the strange dwarf is determined by two parameters, the central density and the crust bottom density (ρcb). The central density at the exact center of the whole star can be as high as ∼ 4.0 × 1014 g cm−3. It determines the mass (Mcore) and radius of the SQM core. For the normal matter at the bottom of the crust, the maximum density should definitely be less than the so called neutron drip density 11 −3 (ρdrip = 4.3 × 10 g cm ). If ρcb & ρdrip, neutrons will drop out of the nuclei and form a neutron reach crust. These free neutrons cannot be supported by the outwardly directed electric field near the SQM surface. They will fall onto the SQM core and finally be converted into strange quark matter. Note that a further study by [35, 36] indicates that the maximum density at the crust bottom actually should be significantly smaller, i.e. no larger than 10 −3 ∼ ρdrip/5 ≈ 8.3 × 10 g cm . The SQM core and the normal matter crust are two distinct regions. Each region requires a different EOS. For SQM in the core, we employe the EOS derived from the simple MIT bag model [4,5], 1 p(r) = (ρ(r)c2 − 4B), (2.3) 3

– 2 – 101

10 3 ) M ( e r o c 10 7 M

10 11

14.60 14.65 14.70 14.75 14.80 3 log core (g cm )

Figure 1. Mean-density vs. the total mass for pue SQM cores. where B is the so called bag constant. In our study, we take a typical value of B = −3 57 MeV fm . For normal matter in the crust, we employe the EOS (BPS, with ρ < ρdrip) of [37]. Figure1 illustrates the mean-density vs. the total mass for pure SQM cores. From the figure, we see that the mean-density almost remains constant when the SQM core is less −2 massive than 10 M . For strange dwarfs, the SQM core is usually not massive so that the mean core density is almost fixed as ∼ 4 × 1014 g cm−3. On the other hand, the pressure versus density in the crust are plotted in figure2, with various ρcb being assumed. Having specified the EOS, we can solve the TOV equation by using the Runge-Kutta method to derive the properties of SQM objects with a normal matter crust. Our calculations will be carried out by assuming different initial values for ρcb to investigate the effect of crusts 11 −3 10 −3 9 −3 8 −3 with various thicknesses (ρcb = 4.3 × 10 g cm , 8.3 × 10 g cm , 10 g cm , 10 g cm , −12 etc). For each ρcb value, the Mcore will further vary from 10 M to 0.1 M . The results of our calculations will be presented in Section4.

3 Data Collection

Mass and radius are important parameters that can be eﬀectively used to probe the intrinsic composition of stars. Especially, the mass-radius relations should be diﬀerent for strange dwarfs and normal white dwarfs. To search for possible strange dwarf candidates, we will systematically examine all the white dwarfs listed in the popular Montreal White Dwarf Database (hereafter, MWDD 1)[38]. All together, there are about 55900 objects listed as white dwarf in MWDD. Among them, 39041 are available with the parameters of surface gravity (g) and mass (M). From these two parameters, we can easily derive the stellar radius

1http://www.montrealwhitedwarfdatabase.org/tables-and-charts.html

– 3 – 1035

4.3e11

8.3e10 1027 1e9 EOS for SQM ) 3 2 cb = 1e8 g cm

Density gap between core surface and EOS for crust (BPS) 19 crust bottom (dyn cm 10 p

1011

102 106 1010 1014 (g cm 3)

Figure 2. Equation of state in the crust region of strange dwarfs, with various ρcb being assumed. as GM 1/2 R = . (3.1) g In this study, we will examine the masses and radii of these 39041 objects to search for possible strange dwarfs.

4 Identifying strange dwarf candidates

4.1 Mass-Radius Relation Following the procedure described in Section2, we have calculated the theoretical M − R relations for both strange dwarfs and white dwarfs. The results are plotted in figure3. In this figure, the solid curves illustrate the mass-radius relations of white dwarfs, under various approximation. The solid magenta curve represents the M − R relation in Chandrasekhar’s model [39]. The solid orange line and the solid lime curve represent the M − R relation in the zero temperature model for pure He and pure Mg white dwarfs [40], respectively. The solid red curve represents white dwarfs in the BPS approach. On the contrary, the dashed curves in figure3 are plotted for strange dwarfs with various crust-bottom densities. The detailed density value is marked near each curve (in units of g cm−3). The vertical bar “|” marked with a letter “b” in each of the sequences represents the lightest object, and the cross symbol “×” marked with “c” indicates the most massive strange dwarf. The cross symbol “×” marked with “d” is the endpoint of the sequence in the case 11 −3 of ρcb = 4.3 × 10 g cm . The SQM core shrinks to almost zero at the end point of each curve. According to various stability analyses [6,7, 21, 22, 32], all the strange dwarfs with 9 −3 ρcb 6 10 g cm are absolutely stable against radial oscillation. Additionally, for the SQM 9 −3 objects with ρcb > 10 g cm , only the segment between “c” and “d” is unstable [6,7, 21, 22], and strange dwarfs in all other segments can stably exist. Comparing with the white dwarf

– 4 – 100 c d

10 1 )

M 4.3e11

( b

M 2 10 8.3e10 Ch He Mg 1e9 10 3 WD (BPS) WD = 1e8 g cm 3 cb SD

103 104 105 R (km)

Figure 3. The mass-radius relations for the white dwarf and strange dwarf sequences. Different curves represent the mass-radius relations for compact dwarfs with different compositions. Observational data points corresponding to the 39041 white dwarfs in the MWDD database are also plotted. While the black points and red points represents normal white dwarfs, the blue points represents candidates of strange dwarfs identified in this study. See the main text in Section4 for more details of the curves and symbols. sequence, the strange dwarf sequence is generally more compact. For example, for an object with a typical mass of 0.8 M , the radius of strange dwarf is significantly less than that of white dwarf. It provides us a useful clue to search for candidates of strange dwarfs.

4.2 Comparison with Observations In ﬁgure3, we have also plotted the observational data points which represents the 39041 white dwarfs in the MWDD database. For the sake of clarity, we did not show the error bar of each point. Note that 32 data points are plotted in red. For these white dwarfs, the data directly extracted from the MWDD database are actually incorrect. For example, the mass of SDSS J235338.87+351346.3 is 1.4(0.81) M as reported in [41], but it is recorded as 0.81(0) M in MWDD. For other 31 white dwarfs, both the mass and the surface gravity are reported in [42]. However, the MWDD website takes only the mass data from this document, but adopts the gravity data from [43] and [44]. This will lead to obvious inconsistency in the calculation of stellar radius. In our plot, we have corrected the above problems. Especially, we have taken both the mass and gravity data from [42]. We see that the majority of the data points generally comply with the conventional white dwarf theory [39, 45]. But there are still some data points deviating from the conventional M − R relation. For example, some massive white dwarfs exceed the Chandrasekhar mass limit of 1.44 M . There are also some white dwarfs having a large radius. These phenomena

– 5 – 101 Ch WD (BPS) WD SD 100 c d

) 10 M (

M b 4.3e11

2 10 8.3e10

1e9 10 3 3 cb = 1e8 g cm

5 7 9 11 log g (cm s 2)

Figure 4. Mass versus the surface-gravity for white dwarfs and strange dwarfs. Observational data points representing the 39041 white dwarfs in the MWDD database are also plotted. Line styles and symbols are the same as those in ﬁgure3.

MWDD ID Teﬀ log g MR References −2 (K) (cm s )(M )(km) SDSS J165143.45 +364647.6 12818 ± 153 7.04 ± 0.05 0.122 ± 0.012 12012.58 ± 209.14 [42] LSPM J0815+1633 4655 ± 35 6.772 ± 0.076 0.082 ± 0.031 13563.23 ± 1024.76 [51] LP 240-30 4680 ± 25 6.768 ± 0.039 0.081 ± 0.016 13542.5 ± 626.6 [51] BD+20 5125B 4395 ± 90 6.795 ± 0.097 0.08 ± 0.038 13046.72 ± 1124.18 [51] LP 462-12 4800 ± 20 6.697 ± 0.054 0.054 ± 0.024 11999.23 ± 1552.78 [51] WD J1257+5428 7485 ± 85 6.441 ± 0.068 0.032 ± 0.03 12403.13 ± 3561.25 [51] 2MASS J13453297 +4200437 4270 ± 75 6.688 ± 0.086 0.031 ± 0.04 9186.23 ± 3405.51 [51] SDSS J085557.46 +053524.5 10670 ± 1677 6.09 ± 1.078 0.02 ± 0.245 14688.29 ± 767.07 [52]

Table 1. Strange dwarf candidates in our sample. may be caused by diﬀerent composition inside the objects [40, 46], or by strong magnetic ﬁeld [47, 48] and fast rotation [49, 50]. However, among the observed 39041 “white dwarfs”, we notice that eight stars appear to be quite special. They obviously deviate from the mass-radius curves for white dwarfs, but well match the relation of strange dwarfs. Their masses are in the range of ∼ 0.02 – 0.12 M and their radii are ∼ 9,000 km – 15,000 km. In other words, they are too compact

– 6 – to be normal white dwarfs. We argue that these eight objects are good candidates for strange dwarfs. In ﬁgure3, we have specially marked these strange dwarf candidates with blue color. Some key parameters of them are listed in table1. Figure4 further illustrates the mass as a function of the surface-gravity. The currently available 39041 white dwarfs in the MWDD database are also plotted for comparison. Again, we see that the eight candidates deviate from the white dwarf sequence, but are better matched by the strange dwarf sequence.

5 Comparison with previous studies

It is interesting to note that previous study [30] have also suggested a few compact objects as candidates of strange dwarfs. [30] is similarly based on the mass-radius relation. To give a direct comparison with their results, we present a zoomed-in plot of the mass-radius relation for strange dwarfs in figure5. From this figure, we see that candidates in [30] are mainly in the mass range of 0.4 – 0.8 M and the radii are 6,000 – 9,000 km. In such a high-mass region, the difference between strange dwarfs and white dwarfs on the mass-radius plot is actually very small. In fact, all of the [30]’s candidates lie slightly above the dashed curves, which means they still seem to be not compact enough. On the contrary, our candidates are in the low-mass region, with masses between 0.02 – 0.12 M and radii between 9,000 – 15,000 km. In this region, the difference between strange dwarfs and white dwarfs is quite significant. So, it is a more appropriate region for discriminating between these two kinds of dwarfs. From figure5, we clearly see that our candidates can only be fitted by the strange dwarf curves. They obviously deviate from the white dwarf sequence even when the error bars are considered (c.f. figure3). Figure5 also shows that these strange dwarfs generally 9 −3 11 −3 should have a crust bottom density of 1.0 × 10 g cm ≤ ρcb ≤ 1.0 × 10 g cm . From figure5, we believe that our sample are more credible strange dwarf candidates. The lower panel of figure5 shows the SQM fraction versus the stellar radius of strange dwarfs. Here, the SQM fraction is defined as the mass percentage of the SQM core with respect to the whole dwarf star. From this plot, we see that the SQM fraction (Mcore/M) of low-mass strange dwarfs are higher than that of high-mass ones on each curve (see 4.1 for −3 the variation of Mcore). The SQM fractions is about 3.58% (M = 9.1 × 10 M , Mcore = −4 8 −3 3.26 × 10 M ) for the largest radius object in the case of ρcb = 10 g cm , and is about −2 −2 19.2% (M = 8.75×10 M , Mcore = 1.68×10 M ) for the largest radius object in the case of 4.3 × 1011 g cm−3. Since candidates in [30] are generally more massive, their SQM fractions are typically much smaller than 0.001. However, the SQM fraction of our candidates are larger than 0.001, which means the SQM core plays a more important role inside the star so that its observational effects could be more significant. For example, it may affect the cooling process of the star (see discussion in the next section). In fact, the strange dwarf candidates −4 −2 in our sample generally have a core mass of Mcore ∼ 10 – 10 M . The relatively larger SQM fraction of less massive strange dwarfs further supports our scheme of trying to search for SQM objects among smaller white dwarfs.

6 Discussion and Conclusions

In this study, we have tried to search for strange quark dwarf candidates among the observed white dwarfs. The selection of candidates is conducted by comparing the mass and radius of observed dwarfs with the theoretical mass-radius relation of strange dwarfs. It is found that

– 7 – 100 Ours Mathews

10 1 )

4.3e11 ( M 2 M 10 8.3e10

1e9 10 3 3 cb = 1e8 g cm

6 × 103

cb = 1e8 g cm R (km) 10 3 3 1e9

2 M 10 / 8.3e10 core 4.3e11 M

10 1

100 104 R (km)

Figure 5. A zoomed-in plot of the mass-radius relation of strange dwarfs (upper panel), and the SQM fraction versus the total radius (lower panel). In the upper panel, the blue circles represent the strange dwarf candidates identiﬁed in this study, and the blue triangles correspond to the candidates suggested by [30]. In both panels, the curves are plotted by assuming diﬀerent crust bottom densities (marked near each curve, in units of g cm−3).

– 8 – the masses and radii of eight objects are consistent with that of strange dwarfs. The mass −4 −2 of the SQM core of these candidates ranges in ∼ 10 – 10 M . The eight objects are SDSS J165143.45+364647.6, LSPM J0815+1633, LP 240-30, BD+20 5125B, LP 462-12, WD J1257+5428, 2MASS J13453297+4200437, and SDSS J085557.46+053524.5. Note that SDSS J085557.46+053524.5 is in a binary system [52] and the other seven are single according to currently available observations. How strange dwarfs are produced is an interesting question. It has been suggested that strange dwarfs may be formed either from the capture of strange nuggets by stellar objects (main-sequence stars, brown dwarfs, etc.), or from the growth of strange clumps by accreting from ambient medium. Below are several detailed formation channels for strange dwarfs. Firstly, the Universe once went through a quark era in its expansion history according to the big bang theory. At that time, a large amount of SQM clumps may be formed and survive up to now [53]. They can be captured by small main-sequence stars and fall into the center of the star, and convert the normal matter into SQM [6]. Secondly, explosive events such as supernovae, merging of two compact stars, phase transition of massive neutron stars may be responsible for the formation of strange stars, and a large amount of SQM nuggets may be ejected during these processes. The flux of SQM nuggets produced in this way in a typical galaxy is estimated as ∼ 0.1 cm−2 s−1 [54–56]. These SQM nuggets can contaminate surrounding brown dwarfs and low-mass main-sequences, and convert them into strange dwarfs [57]. Thirdly, a newborn strange star is quite hot and highly turbulent. It may directly eject a large clump of SQM due to the joint effect of fast-spinning and turbulence [9, 10, 58]. The SQM clump then can accrete ambient matter throughout its life and evolve to its current form. A remarkable feature of the strange dwarf candidates in our sample is that their temperatures and masses are very low (see table1). Generally, their temperatures are cooler than 13,000 K and their masses are in the range of 0.02 – 0.12 M . Many authors have discussed this feature in the framework of white dwarfs. Low-mass white dwarfs are likely formed in binary systems because it is impossible for a star with a mass below 0.45 M to become a white dwarf through single-star evolution within the age of the universe [59, 60]. On the contrary, in a binary system, the progenitor star can lose mass through binary interactions [26, 61, 62]. This opinion is supported by the observations of several extremely low-mass white dwarfs in compact binaries involving NSs/pulsars [26, 61, 63, 64] and WDs [65, 66]. However, our samples are not appearing in these studies as binary. As for the low temperature, [51] suggested that these ultra-cool objects may be polluted by rocky debris, but he did not provide any information about the origion of these rocky debris around such low-mass objects. In this study, we have identified these objects as strange dwarf candidates. The low temperature can then be self-consistently explained as due to the effect of the SQM core inside them. In fact, SQM objects generally cool faster so that their surface temperature is usually lower than their hadronic matter counterparts [67]. The low-mass SQM dwarfs may form from contamination of stellar objects (small main-sequence stars, brown dwarfs, etc.) by strange nuggets, or accretion of ambient matter by a large clump of SQM. They may also have experienced binary interactions during this process. It is interesting to note that more candidates of small SQM stars up to planetary SQM objects have been suggested recently by [12, 13, 68]. Especially, the planet PSR 1719-14 b is generally regarded as an appropriate candidate because it has a density larger than 23 g cm−3 [12, 13, 68, 69]. Current observational data show that only one strange dwarf candidate in our sample is clearly accompanied by a main-sequence star. For the other seven candidates, no clues

– 9 – supporting the existence of any companion star have been reported. However, it should be noted that the possibility that they are actually in binary systems still could not be completely expelled, because an accompanying pulsar or quiecent neutron star is not easy to detect. In the future, these objects deserve being extensively examined by using large optical/infrared telescopes and radio facilities.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11873030, U1938201, 11535005, 11690030, 11903019, and 12041306), by Nation Major State Basic Research and Development of China (2016YFE0129300), and by the Strategic Priority Research Program of the Chinese Academy of Sciences (“multi-waveband Gravitational-Wave Universe”, Grant No. XDB23040000).

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