5
. Article . SCIENCE CHINA Physics, Mechanics & Astronomy April 2014 Vol. 57 No.4: 791–795 doi: 10.1007/s11433-014-5393-5 c Science China Press and Springer-Verlag Berlin Heidelberg 2014 phys.scichina.com link.springer.com Dark matter heating in strange stars HUANG Xi 1,2* , WANG Wen 1 & ZHENG XiaoPing 1* 1 Institute of Astrophysics, Central China Normal University, Wuhan 430079, China; 2 School ofElectronic and Electrical Engineering, Wuhan Textile University, Wuhan 430073, China Received September 18, 2013; accepted November 12, 2013; published online March 3, 2014 We study the eect of dark matter heating on the temperature of typical strange star (SS hereafter) ( M = 1.4M , R = 10 km) in normal phase (NSS hereafter) and in a possible existing colour-avour locked (CFL)phase (CSS hereafter). For NSS, the inuence of dark matter heating is ignored until roughly 10 7 yr. After 10 7 yr, the dark matter heating is dominant that signicantly delays the star cooling, which maintains a temperature much higher than that predicted by standard cooling model for old stars. Especially for CSS, the emissivity of dark matter will play a leading role after roughly 10 4 yr, which causes the temperature to rise. This leads to the plateau of surface temperature appearing in 10 6.5 yr which is earlier than that of NSS (10 7 yr). stellar evolution, dark matter, quarks PACS number(s): 97.60.Lf, 95.35.+d, 14.65.-q Citation: Huang X, Wang W, Zheng X P. Dark matter heating in strange stars. Sci China-Phys Mech Astron, 2014, 57: 791–795, doi: 10.1007/s11433-014-5393-5 1 Introduction With the discovery of neutron, Landau proposed the conjec- ture of neutron stars. By contrast, does the star composed of quarks absolutely exist? Witten [1] and Farhi & Jae [2] have pointed out that strange quark matter could be the true ground state of hadrons, which probably means the existence of strange quark star (SS). Hereafter, the properties of SS have been studied by several authors [3–6]. So far, as we know the core of SS is made up of roughly equal numbers of up, and down, strange quarks and fewer electrons (maintain- ing electrical neutrality). Based on QCD theories, the quarks near the Fermi sur- face are asymptotically free, with weak interactions between them. The interactions among quarks are very attractive in some channels, and thus the quarks are expected to form Cooper pairs [7].The condensation patterns of Cooper pairs are complex, because quarks not only have spin, but also *Corresponding author (ZHENG XiaoPing, email: [email protected]; HUANG Xi, email: [email protected]) have dierent avors and colors [8,9]. When the densities of strange quark matter are not high and the mass of s quark can not be neglected, u quark and d quark will form zero-spin Cooper pairs. However, s quarks do not participate, which named two-avor color superconductivity (2SC). In the other case, at high densities of strange quark matter the mass of s quark can be ignored, and the color-avor locked (CFL) phase is favored [10], in which quarks of all three colors and avors form conventional zero-momentum spinless Cooper pairs. Therefore, in the calculation of this paper, we only take into account CSS, in which all avor quarks pair. By now, the theoretical study of thermal evolution of SS and the comparison of its predictions with observations of surface emission is an important way to probe the states of matter at supernuclear densities and physical processes. The curves of thermal evolution of SS depend on neutrino emis- sion and surface photon emission. Of course, some inter- nal heating mechanisms may also be present, for example, chemical heating [11], r-mode heating [12] and deconne- ment heating [13]. In recent years, the theoretical and experimental studies on

Dark matter heating in strange stars

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Page 1: Dark matter heating in strange stars

. Article .

SCIENCE CHINAPhysics, Mechanics & Astronomy

April 2014 Vol. 57 No. 4: 791–795doi: 10.1007/s11433-014-5393-5

c© Science China Press and Springer-Verlag Berlin Heidelberg 2014 phys.scichina.com link.springer.com

Dark matter heating in strange starsHUANG Xi1,2*, WANG Wen1 & ZHENG XiaoPing1*

1Institute of Astrophysics, Central China Normal University, Wuhan 430079, China;2School of Electronic and Electrical Engineering, Wuhan Textile University, Wuhan 430073, China

Received September 18, 2013; accepted November 12, 2013; published online March 3, 2014

We study the effect of dark matter heating on the temperature of typical strange star (SS hereafter) (M = 1.4 M�, R = 10 km) innormal phase (NSS hereafter) and in a possible existing colour-flavour locked (CFL)phase (CSS hereafter). For NSS, the influenceof dark matter heating is ignored until roughly 107 yr. After 107 yr, the dark matter heating is dominant that significantly delays thestar cooling, which maintains a temperature much higher than that predicted by standard cooling model for old stars. Especially forCSS, the emissivity of dark matter will play a leading role after roughly 104 yr, which causes the temperature to rise. This leads tothe plateau of surface temperature appearing in ∼ 106.5 yr which is earlier than that of NSS (∼ 107 yr).

stellar evolution, dark matter, quarks

PACS number(s): 97.60.Lf, 95.35.+d, 14.65.-q

Citation: Huang X, Wang W, Zheng X P. Dark matter heating in strange stars. Sci China-Phys Mech Astron, 2014, 57: 791–795, doi: 10.1007/s11433-014-5393-5

1 Introduction

With the discovery of neutron, Landau proposed the conjec-ture of neutron stars. By contrast, does the star composedof quarks absolutely exist? Witten [1] and Farhi & Jaffe [2]have pointed out that strange quark matter could be the trueground state of hadrons, which probably means the existenceof strange quark star (SS). Hereafter, the properties of SShave been studied by several authors [3–6]. So far, as weknow the core of SS is made up of roughly equal numbers ofup, and down, strange quarks and fewer electrons (maintain-ing electrical neutrality).

Based on QCD theories, the quarks near the Fermi sur-face are asymptotically free, with weak interactions betweenthem. The interactions among quarks are very attractive insome channels, and thus the quarks are expected to formCooper pairs [7].The condensation patterns of Cooper pairsare complex, because quarks not only have spin, but also

*Corresponding author (ZHENG XiaoPing, email: [email protected]; HUANG

Xi, email: [email protected])

have different flavors and colors [8,9]. When the densitiesof strange quark matter are not high and the mass of s quarkcan not be neglected, u quark and d quark will form zero-spinCooper pairs. However, s quarks do not participate, whichnamed two-flavor color superconductivity (2SC). In the othercase, at high densities of strange quark matter the mass ofs quark can be ignored, and the color-flavor locked (CFL)phase is favored [10], in which quarks of all three colors andflavors form conventional zero-momentum spinless Cooperpairs. Therefore, in the calculation of this paper, we onlytake into account CSS, in which all flavor quarks pair.

By now, the theoretical study of thermal evolution of SSand the comparison of its predictions with observations ofsurface emission is an important way to probe the states ofmatter at supernuclear densities and physical processes. Thecurves of thermal evolution of SS depend on neutrino emis-sion and surface photon emission. Of course, some inter-nal heating mechanisms may also be present, for example,chemical heating [11], r-mode heating [12] and deconfine-ment heating [13].

In recent years, the theoretical and experimental studies on

Page 2: Dark matter heating in strange stars

792 Huang X, et al. Sci China-Phys Mech Astron April (2014) Vol. 57 No. 4

dark matter have attracted more and more attention [14–16].In this paper, we will pay close attention to another heatingmechanism-dark matter heating. From a theoretical point ofview, weakly interacting massive particles (WIMP hereafter)are one of candidates of dark matter particles [17]. The SS isa massive compact object, thus WIMP in the neighborhoodof SS can be accreted. Once WIMP intersect with the star,it loses enough energy due to scattering to be trapped by thestar. The fraction f of WIMP that undergo one or more scat-tering inside the star would depend strongly on the elasticscattering cross section [18]. It is possible to annihilate be-tween WIMP which depends on the annihilation cross sectionas well as the density of WIMP inside the star. For this rea-son, it releases a huge amount of energy to heat up the star.

This paper is organized as follows: First, the thermal evo-lution equation of NSS and CSS considering the heating dueto WIMP annihilation is given in sect. 2. Then we numeri-cally calculate the effect of WIMP annihilation on the surfacetemperature of NSS and CSS in sect. 3. Finally, we presentthe conclusions in the last section.

2 ModelSS consists of approximately isothermal quark core and thinnuclear crust [3]. For convenience, we assume that the den-sity of the star is constant inside the star. When SS was born,the internal temperature would reach up to 1010 K, for the firstmillion years it cools via neutrino emission. Here we onlyfocus on neutrino emission from the core of the star, withoutconsidering it from the crust which is negligible for thermalevolution of the star. As the internal temperature drops below108 K, the dominant mechanism of cooling is through photonemission from the surface of the star.

Direct Urca processes d → u + e + νe and u + e → d + νeare allowed when electron fraction Ye is larger than criti-cal electron fraction. In addition, modified Urca processesd + q → u + q + e + νe, u + q + e → d + q + νe and quarkbremsstrahlung processes q1 + q2 → q1 + q2 + νe + νe aremain neutrino processes if direct Urca processes are forbid-ded. The emissivities of these neutrino processes, releasedenergy per volume per time, can be written respectively [19]

εqDU = 8.8×1026αc

(nb

n0

)Y1/3

e T 69 erg cm−3 s−1, (1)

εqMU = 2.83×1019α2

c

(nb

n0

)T 8

9 erg cm−3 s−1, (2)

εqB = 2.98×1019

(nb

n0

)T 8

9 erg cm−3 s−1, (3)

where αc is strong coupling constant, nb is the baryon numberdensity and n0 = 0.17 fm−3 is the nuclear saturation density.In our calculation for typical SS of M = 1.4 M� and R = 10km, the average baryon number density nb = 3.9 × 1038 par-ticles per cm3. Ye = ne/nb is the electron fraction mentionedabove. T9 is the interior temperature of the star in units of 109

K.

For CSS, the neutrino emissivity of direct Urca processesis suppressed by the reductive factor exp(−�/T ) due to thepairing gap �, in the same way, the neutrino emissivitiesof modified Urca processes and quark bremsstrahlung pro-cesses are suppressed by the reductive factor exp(−2�/T ) forT < Tc. Tc is characteristic temperature related to the pairinggap �. In our calculations, we take Tc = 0.4� for the CFLphase.

As we know, the surface photon luminosity of the star isLγ = 4πR2σT 4

S, where σ is the Stefan-Boltzmann constantand TS is the surface temperature of the star. For the rela-tionship between the interior temperature T and the surfacetemperature TS, we apply the result given by refs. [20,21],which is valid for a crust with the density at the base that isjust larger than 108 g cm−3,

TS = (0.87 × 106 K)

(gs

1014 cm/s2

)1/4 ( T108 K

)0.55

, (4)

where gs = GM/R2 is the surface gravity, for the typicalSS of M = 1.4 M� and R = 10 km, we can get gs =

1.85 × 1014 cm/s2. It is worth mentioning that the relationbetween T and TS which is applied to SS is the same as inneutron star, because the temperature gradient mainly occursin the outer crust for neutron star. Thus, the emissivity fromthe surface photon emission for the typical SS (M = 1.4 M�,R = 10 km) can now be expressed in terms of the interiortemperature as

εγ =Lγ

(4/3)πR3= 1.8×1014T 2.2

8 erg cm−3 s−1, (5)

where T8 is the interior temperature of the star in units of 108

K.In order to compute the thermal evolution of the star, we

need to know the specific heat which is mainly contributedby quarks and electrons from the core [22]

cq�2.5×1020

(nb

n0

)2/3T9 erg cm−3 K−1, (6)

ce�0.6×1020

(Yenb

n0

)2/3T9 erg cm−3 K−1. (7)

The very small contribution to the specific heat from the crustcan be neglected [23]. Meanwhile, we ignore the contributionof WIMP to the specific heat, because the trapped dark matterrepresents a tiny fraction of the whole mass of the star.

In the calculation for the specific heat of quarks in CFLphase, we should take into account the reductive factor simi-lar to the one that applies for the case of neutrino emissivity.we adopt the consequence in ref. [22]:

csq = 3.2cq(Tc/T )exp(−Δ/T )

×[2.5 − 1.7T/Tc + 3.6(T/Tc)2

]erg cm−3 K−1, (8)

we will use a large gap � = 50 MeV, as suggested for theCFL phase in recent work [8].

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Huang X, et al. Sci China-Phys Mech Astron April (2014) Vol. 57 No. 4 793

Besides SS cools via internal neutrino emission and sur-face photon emission, on the other hand, SS also can beheated by the annihilation of dark matter. Based on com-peting effects of accretion and annihilation, the population ofWIMP inside the star is governed by

dNdt= F − Γannih, (9)

where Γannih is the annihilation rate calculated as Γannih =

〈σannihυ〉∫

n2χdV = CAN(t)2, 〈σannihυ〉 is the thermally av-

eraged annihilation cross section times the velocity, and nχ,the number density of WIMP inside the star, is assumed tobe constant, the constant CA = 〈σannihυ〉/V , where V is thevolume of the star, the accretion rate of WIMP captured by atypical SS in particles per second is given by [18]

F = 3.042 × 1025

mχ(GeV)× A × f , (10)

where mχ is the mass of WIMP, A is a constant that repre-sents the local dark matter density nearby the star in unitsof 0.3 GeV/cm3, which is the standard dark matter densityaround the Earth, f is the fraction mentioned above in sect.1. If the elastic scattering cross section is not smaller than10−45 cm2, we can take f = 1.

The solution of eq. (9) is

N(t) =

√FCA

tanh(t/τ), (11)

the time scale τ = 1/√FCA. Hence, the released energy due

to the annihilation is

E = Γannihmχ = CAN(t)2mχ = F tanh2(t/τ)mχ. (12)

If τ is larger than the age of the star, the influence of WIMPannihilation on the temperature of the star is negligible. Thetime scale τ is given by [18]

τ =2.52 × 105 years√

A f sin4 θ(

T108

) , (13)

where the mixing angle sin θ is defined by a mixing betweenthe left-handed Majorana neutrino and the right-handed Ma-jorana neutrino (Here, WIMP are considered as Majorananeutrinos). In our calculations, we take sin θ = 1 becausethe Majorana neutrino may be exclusively left handed.

The emissivity of WIMP annihilation can be written as

εdm =E

4πR3/3=

3F tanh2(t/τ)mχ4πR3

erg cm−3 s−1, (14)

in terms of eq. (10), for the typical SS of M = 1.4 M� andR = 10 km, the above eq. (14) can be expressed as

εdm = A1.16 × 104 tanh2(t/τ) erg cm−3 s−1. (15)

Thus we can give the thermal evolution equation

dTdt=−εν − εγ + εdm

cV, (16)

where εν is the sum of neutrino emissivities from di-rect Urca processes, modified Urca processes and quarkbremsstrahlung processes, cV is the sum of the specific heatwhich is mainly contributed by quarks and electrons from thecore, T is the interior temperature of the star.

3 Results and discussion

In actual calculations, we consider a typical SS of M =

1.4 M� and R = 10 km, and take the initial interior tempera-ture T0 = 1010 K, strong coupling constant αc = 0.2, electronfraction Ye = 10−5 or Ye = 0 (quark direct Urca processeswill switch on or off) respectively.

The evolution of the surface temperature as a function oftime for typical NSS (M = 1.4 M�, R = 10 km) and CSSwith Δ = 50 MeV are shown in Figure 1. A = 100 is fixed forthree cases. The surface temperature of NSS with Ye = 10−5

is always lower than the one of NSS with Ye = 0, becausethe direct Urca processes are switched off for cooling of NSSwhen Ye = 0, whereas the direct Urca processes, modifiedUrca processes and quark bremsstrahlung processes are allconsidered for NSS with Ye = 10−5. Furthermore, the effectof dark matter heating is neglected until roughly 107 yr. Theplateau of surface temperature will appear as the heating ofdark matter annihilation equilibrates the cooling of surfacephoton emission (see Figure 2). At the extreme, the surfacetemperature of CSS is far lower than that of NSS between∼ 102 and ∼ 104 years due to the rapid cooling process, be-cause the quark specific heat is suppressed by the pairing gap(as discussed by Cheng et al. [24]) and the influence of heat-ing due to WIMP annihilation is ignored. After roughly 104

yr, the emissivity of dark matter will play a leading role (seeFigures 2(c) and 2(d)) which causing the temperature to rise.

logT

S (K

)

8

7

6

5

4

3

2

10 2 4 6 8

log t (yrs)

NSS

NSS

CSS

Ye = 0

Ye = 10−5

Figure 1 The surface temperature as a function of time for typical NSSwith Ye = 0 (dotted curve), NSS with Ye = 10−5 (dashed curve) and CSSwith Δ = 50 MeV (solid curve). A = 100 is fixed for three cases.

Page 4: Dark matter heating in strange stars

794 Huang X, et al. Sci China-Phys Mech Astron April (2014) Vol. 57 No. 4

log ε

(erg

cm

−3 s

−1)

201612840

−4201612840

−4201612840

−4201612840

−4

(a)

(b)

(c)

NSS Ye= 0

NSS Ye = 10−5

CSS

(d)

0 2 4 6 8log t (yrs)

ενεγ

εγ

εν

εγ

εν

εdm

εdm

εdm

εγ

εdm

0.90

0.80

0.70

4.80 4.81 4.82 4.83 4.84 4.85

Figure 2 The emissivities of neutrino (dotted curve), surface photon(dashed curve) and dark matter (solid curve) as a function of time for typicalNSS with Ye = 0 (a), NSS with Ye = 10−5 (b) and CSS with Δ = 50 MeV(c). A = 100 is fixed for three cases.

As it can be seen from Figure 2, the same the plateau of emis-sivities of dark matter and surface photon for NSS and CSS,thus the same the plateau of surface temperature, however,the plateau of CSS emerges at the earliest (∼ 106.5 yr).

Figure 2 shows the evolution of emissivities of neutrino,surface photon and dark matter for typical NSS (M = 1.4 M�,R = 10 km) with Ye = 0 and Ye = 10−5, CSS with Δ = 50MeV. For comparison, we fix A = 100 for above-mentionedcases. It can be seen from Figures 2(a) and 2(b), NSS withYe = 0 appears the performance similar to the one withYe = 10−5. The cooling effect dominates until roughly107 yr. In addition, the emissivity of dark matter increasesmonotonously with increasing time up to saturation state. Asit can be seen from eq. (12), once the annihilation rate ofdark matter equilibrates the rate of accretion for t > 3τ,the released energy due to WIMP annihilation saturates toFmχ and the emissivity of dark matter remains unchanged.However, the saturation state of emissivity of dark matter forNSS with Ye = 0 emerges earlier than the one of NSS withYe = 10−5 due to τ ∝ 1√

T(see eq. (13)). After roughly 107 yr,

the equilibrium between cooling (emission via surface pho-ton) and heating has been reached. For CSS, the emissivityof neutrino is absolutely blocked, thus the cooling of the stardepends largely on the emission of surface photon. Between∼ 102 and ∼ 104 years, the emissivity of dark matter is sup-pressed by the rapid cooling process (as shown in Figure 1).It can be seen from eq. (15), εdm mainly depends on tanh(t/τ)function before rising to the saturated state, which is mono-tone increasing function, and τ ∝ 1√

T, thus the lower the

temperature, the smaller the emissivity of dark matter. Af-ter roughly 104 yr, the emissivity of dark matter is larger thanthe one of surface photon (see Figure 2(d)), which means thestar starts to be heated. Not surprisingly, the plateau of emis-sivity of CSS emerges earlier (∼ 106.5 yr) than that of NSS

(∼ 107 yr).In Figure 3, we have plotted in a logarithmic scale the sur-

face temperature of CSS as a function of time for three dif-ferent local dark matter densities for the vicinity of the star.As we can see from the figure, from ∼ 102 yr to ∼ 104 yr, theCSS has the same behavior for three different cases, whichwill undergo a rapid cooling process. As the temperature ofthe star drops, the WIMP annihilation can heat up the star.At the later time, the equilibrium between the heating of thedark matter annihilation and the cooling of surface photonemission is established, as a result the temperature will re-main flat as a function of time. The platform depends on themass and the radius of the star, and the local dark matter den-sity around the star. Obviously, the platform for A = 100case is higher and appears earlier than the cases for A = 10and A = 1.

4 Conclusion

We have studied the effect of WIMP annihilation on the sur-face temperature of strange stars. For NSS with Ye = 0 orYe = 10−5, the influence of dark matter heating is neglecteduntil roughly 107 yr. For CSS, we find that, it will undergoa rapid cooling process during ∼ 102−4 yrs because the quarkspecific heat is blocked by the pairing gap. As the temper-ature of the star decreases, dark matter heating affects thesurface temperature significantly after roughly 104 yr whichleads to the plateau appears in ∼ 106.5 yr. Once the heatingof dark matter annihilation equilibrates the cooling of surfacephoton emission, the same temperature plateau will appearwhich is independent of the model of the star. In addition,we have also investigated the effect of dark matter heating onCSS with different local dark matter density. The plateau ofsurface temperature for A = 100 case is higher and appearsearlier than the cases for A = 10 and A = 1.

For now, the surface temperature as low as ∼ 104 K is hardto detect, however, as the theoretical investigations we are in-terested to focus on another heating mechanism-dark matter

logT

S (K

)

8

7

6

5

4

3

2

1

00 2 4 6 8

log t (yrs)

CSS

A = 100

A = 10

A = 1

Figure 3 The surface temperature of a isolated CSS with M = 1.4 M�,R = 10 km and Δ = 50 MeV as a function of time for three different localdark matter densities around the star: A = 100 (solid curve); A = 10 (dashedcurve); A = 1 (dotted curve).

Page 5: Dark matter heating in strange stars

Huang X, et al. Sci China-Phys Mech Astron April (2014) Vol. 57 No. 4 795

heating and the effect of WIMP annihilation. Of course, itis a challenge to observe such an effect in the future, whichwould possibly be a signature of WIMP annihilation.

In our analysis, we disregard the effect of stellar rotationwhich leads to chemical heating due to the derivative frombeta equilibrium with the spin-down of the star [11,25]. Thismechanism could possibly be present for old star. We alsoneglect the accretion in a close binary star system as Watts& Andersson did [26] and the accretion of the interstellarmedium [27], which are important for thermal evolution ofthe star.

We thank YU YunWei for the stimulative discussion which motivated this

work. This work was supported by the National Natural Science Founda-

tion of China (Grant Nos. 11103004, 11073008, and 11178001) and by the

Self-Determined Research Funds of CCNU (Grant no. CCNU09A01020)

from the Colleges’ Basic Research and Operation of NO of China.

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