Cooling of Compact Stars With Color Superconducting Phase in Quark Hadron Mixed Phase


We present a new scenario for the cooling of compact stars considering the central source of Cassiopeia A (Cas A). The Cas A observation shows that the central source is a compact star that has high effective temperature, and it is consistent with the cooling without exotic phases. The observation also gives the mass range of , which may conflict with the current plausible cooling scenario of compact stars. There are some cooled compact stars such as Vela or 3C58, which can be barely explained by the minimal cooling scenario, which includes the neutrino emission by nucleon superfluidity (PBF). Therefore, we invoke the exotic cooling processes, where a heavier star cools faster than lighter one. However, the scenario seems to be inconsistent with the observation of Cas A. Therefore, we present a new cooling scenario to explain the observation of Cas A by constructing models that include a quark color superconducting (CSC) phase with a large energy gap; this phase appears at ultrahigh density region and reduces neutrino emissivity. In our model, a compact star has CSC quark core with a low neutrino emissivity surrounded by high emissivity region made by normal quarks. We present cooling curves obtained from the evolutionary calculations of compact stars: while heavier stars cool slowly, and lighter ones indicate the opposite tendency without considering nucleon superfluidity. Furthermore, we show that our scenario is consistent with the recent observations of the effective temperature of Cas A during the last 10 years, including nucleon superfluidity.

Subject headings:
dense matter – stars: neutron

Feb. 7, 2013

1. Introduction

The cooling of compact stars has been discussed mainly in the context of neutron stars for decades (Tsuruta, 1998; Becker, 2009). It has been believed that some stars require exotic cooling to explain the observed effective temperature and others can be explained by the modified URCA and Bremsstrahlung processes, where the central density of the star determines which cooling process works: an exotic cooling phase appears at higher density above a threshold density (e.g., Yakovlev et al., 2005; Gusakov et al., 2005). As a consequence, the heavier star which has higher central density cools faster than lighter one (Lattimer et al., 1991). However, as described below this scenario becomes inconsistent when we consider the recent observation of the effective temperature of Cas A whose mass has been found to be unexpectedly large.

Cas A is the youngest-known supernova remnant in the Milky Way and it is located from the solar system (Reed et al., 1995). The supernova explosion occurred about 330 years ago, but due to absorption by the interstellar medium, there are no exact historical records except for an unclear detection by J. Flamsteed in 1680 (Ashworth, 1980). Recently, Ho & Heinke (2009) and Heinke & Ho (2010) have analyzed the X-ray spectra of Cas A. They give the effective temperature and possible regions occupied by mass and radius relations. Since Cas A is the isolated remnant, the uncertainty of mass–radius relation could be large. The lowest mass obtained from the fitting is about . Considering the age of  yr, of Cas A must occupy a point of a cooling curve due to the scenario with modified URCA and Bremsstrahlung processes included on the plane. This gives strong constraint on the equation of state (EoS) and cooling processes. Furthermore, Heinke & Ho (2010) reported the observation of for Cas A in the past 10 years. Yakovlev et al. (2011), Page et al. (2011a), and Shternin et al. (2011) insist that the rapid decrease in over time shows that the transition to nucleon superfluidity occurs.

On the other hand, there are some cooled stars whose effective temperature cannot be explained by the neutrino emission processes without nucleon superfluidity, including the modified URCA and Bremsstrahlung. It needs stronger cooling process as in the case of J0205+6449 in 3C58 (hereafter “3C58”) or Vela pulsar (B0833–45). Also an accreting neutron star SAX J1808 requires strong cooling. 3C58 and Vela may be explained by the minimal cooling model which includes nucleon superfluidity (Page et al., 2009). However, SAX J1808 needs stronger cooling than the minimal cooling (Heinke et al., 2008). If we consider the strong cooling process according to the conventional scenario, their masses should become larger than that of Cas A, which may be inconsistent with the mass observations of double neutron stars; the mass of each neutron star is nearly (e.g., Kaspi et al., 2006). Isolated stars should have smaller (similar) masses compared with the case of NS-WD (NS-NS) binaries, respectively. The long-standing accretion from companions make the primaries heavier in the case of NS-WD binary systems (Bogomazov et al., 2005). Although a single EoS must be applied to all the compact stars, the existing phase of matter depends on the density. Therefore, the location of the Cas A observation on the plane becomes very difficult to interpret if we believe the models with strong cooling mechanisms explain all other the observations of .

In this paper, we present models that satisfy both cases of Cas A and other cooled stars such as 3C58 and/or Vela, by considering hybrid stars composed of quark matter, hadron matter, and their mixed phase (MP), where a characteristic property of color superconducting (CSC) phase is utilized. In addition, we also show cooling curves of Cas A for observations over the past 10 years and indicate that the phase transition to the superfluidity is consistent with the observations.

2. Cooling curve models

We construct a model that includes both quark–hadron MP and its CSC phase. Considering the first-order phase transition between hadron and quark phases, it would be plausible that both phases coexist and form some kind of MP. Similar to the “nuclear pasta” phase in the crust of a neutron star (Ravenhall et al., 1983; Hashimoto et al., 1984), it has been shown that MP could form geometrical structures (Maruyama et al., 2008); Yasutake et al. (2009) have made EoS of an MP under a Wigner-Seitz (hereafter “WS”) approximation using an MIT Bag model for a quark phase in finite temperature. In the present study, we employ an EoS with the same framework using the bag constant , the coupling constant , and the surface tension parameter . For hadron phase, we adopt the results of the Bruekner–Hartree–Fock (BHF) theory including hyperons, , and  (Schulze et al., 1995; Baldo et al., 1998; Baldo, 1999). However, the hyperons do not appear for the EoS calculation including geometrically structured MP (Yasutake et al., 2009); therefore, we do not include the effects of hyperons. Although this does not occur in our model, if hyperons appear in other models, the hyperon mixed matter has large neutrino emissivity called hyperon direct URCA process (e.g., ), and causes the rapid cooling of compact stars (Takatsuka & Tamagaki, 1997). Since the BHF results are inappropriate for low-density matter in the crust, we apply EoS of BPS (Baym et al., 1971) for the crust. The EoS gives a maximum mass of with a radius of , and the mass lies within the limits of the observation of Cas A. Although our EoS is inconsistent with the recent observation of the mass of pulsar J1614–2230 (Demorest et al., 2010), we could overcome this issue by adopting other EoS models (e.g., Alford et al., 2005).

Using the WS approximation, we obtain a cell radius of each phase and calculate the volume fraction of quark matter in MP as seen in Figure 1. It is difficult to calculate the neutrino emissivity in MP. Therefore, the volume fraction is multiplied by the original quark neutrino emissivity  (Iwamoto, 1980); the total emissivity by quarks is set to be . We adopt well-known neutrino emission processes without nucleon superfluidity for hadronic matter (Friman & Maxwell, 1979): modified URCA process for the higher density region and Bremsstrahlung process for the crust. We note that the special case of Cas A during the past 10 years is discussed in Section 3.

Figure 1.— Volume fractions of quark matter phase having particular geometrical structures with a bag constant and a coupling constant  (Maruyama et al., 2007; Yasutake et al., 2009).

The color superconductivity is the key of the present study. There are different kinds of quark pairings such as CFL (Color Flavor Locking) or 2SC (Two-Flavor Color Superconducting) according to the degrees of freedom of quark flavor and color. It is considered that the energy gap of is very large compared with the temperature of the center of compact stars ,  (e.g., Schmitt, 2010). Once matter becomes superconducting, neutrino emissivity must be suppressed due to the large energy gap and it could be proportional to , where is the temperature at the relevant layer and is the Boltzmann constant (Negreiros et al., 2012). Therefore, in the CSC phase with a large energy gap (), neutrino emissivity by quarks is almost negligible (Alford et al., 2008). We note that if we adopt a large energy gap we do not need to consider which kinds of CSC pairing appear. In particular, the difference between CFL and 2SC is unimportant for the cooling. In fact, the pseudo NG bosons may exist and decay in the CFL phase and make changes to neutrino emissivity (Jaikumar et al., 2002). However, it is known that it is not quantitatively affected in low temperature region (Alford et al., 2008).

We assume that the energy gap in the CSC phase is very large and the phase appears in the density region above the threshold density . For simplicity, we use the critical volume fraction instead of . If the matter has a volume fraction at the density in the layer of MP, quarks change into a CSC state. As a consequence, the neutrino emission process due to the quark decay works only in the lower density region of the quark–hadron MP.

Figure 2.— Cooling curves with color superconducting quark phases. Solid, dotted, and dashed lines denote the models with the masses of , , and , respectively. Thick gray line on the middle panel denotes a model with nucleon superfluidity and carbon envelope. Dot-dashed lines with marks on the middle panel indicate the model of the mass except for the neutrino emissivity in normal quark phase multiplied by one-tenth and one-one-hundredth for the lines with marks of triangle and circle, respectively.

We select stellar masses of compact stars to be , , and whose central densities are 1.48, 1.82, and 2.67 , respectively. We take the critical fraction to be 0.1, 0.125, and 0.2 which are appropriate to explain the observations with our scenario. The results are shown in Figure 2, with available observational values. Most of the observational data are taken from the Table 7.2 in Kaspi et al. (2006)Slane et al. (2002) for 3C58; Weisskopf et al. (2004) for Crab; Pavlov et al. (2001) for Vela; Brisken et al. (2003) for 0656+14; Gotthelf et al. (2002) for 1706–44; Kaspi et al. (2006) for 1811–1925; Gaensler et al. (2003) for 1823–13; Becker et al. (1996) for 2334+61. Other data are taken from Page (2002). The data of Cas A are attached (Heinke & Ho, 2010) as the youngest compact star. We find that the cooling curves transit from hotter regions to cooler regions for the parameter between . As indicated in the middle panel of Figure 2, the cooling curves split into two regions for and stars with larger masses cool more slowly than those with lighter masses. Since lighter mass stars cool faster, they are suitable for the 3C58 case which does not have lower limit of the effective temperature observation. However, the calculated cooling curves are inconsistent with the Vela case which has lower limit. Also, the quark cooling is still too strong to explain this case.

Since the neutrino emissivity of a quark phase involves large uncertainty, we have calculated the additional cooling curves for the mass in the case of with the neutrino emissivity reduced by a factor of and . There are some possible factors of this reduction for neutrino emissivity accompanying quark -decay, such as an increase of the abundance of strange quarks; a decrease in electron numbers inside MP leads to a reduction of neutrino emissivity (Iwamoto, 1980). The presence of 2SC at low density also suppresses the emissivity; Maruyama et al. (2008) discussed that the abundance of quarks in MP changes and may cause CSC phase. We suppose that the reduction of emissivity originates from the above physical processes. If the emissivity of quarks is reduced by these factors, the observation of Vela can be explained as shown in the middle panel of Figure 2.

3. Discussions

We demonstrate the effect of color superconductivity in quark–hadron MP on the cooling curve: the larger the masses of the compact stars, the slower the speed of the cooling. This situation is caused by the layer that emits a large number of neutrinos through quark decay processes, which encircles the center of the star. The thickness of this layer decreases as stellar mass increases. Although the maximum mass of our model is at most that would be near the lower limit for the recent observation of Cas A, our cooling scenario can be applied for the cases of if more concrete EoS are devised. Some problems also remain concerning the fundamental physics: uncertain physical properties of quark–hadron MP, indefinite tuning of the threshold density of CSC, unknown values of the energy gap . Nonetheless, to explain the observed effective temperature of Cas A, our model incorporating the color superconductivity with a large energy gap is compatible with available observational data. The cooling mechanism associated with CSC quarks could be plausible because, as shown by the existence of CSC, it may be quite natural from the recent study of phase diagram between quarks and hadrons (Rüster et al., 2005).

The fundamental physics associated with compact stars is still largely uncertain. In the quark phase, the abundance of each quark is still unknown, and therefore the neutrino emissivity is not determined from the fractions of , , and quarks and/or the chemical potential of electrons (Maruyama et al., 2008). Although the existence of the CSC phase has been well studied, it is still open to debate which type of CSC appears, and the quantitative values of the critical density and the energy gap should be clarified. Since the physical properties of quarks in the MP are different from those in the uniform phase, there would be many factors to change the emissivity. Further theoretical study in collaboration with the observations is required to constrain the physics of compact stars.

Considering the above uncertainties, the observations of the rapid cooling for Cas A  (Heinke & Ho, 2010) would give insight for constraining some properties of high density matter. We have tried to fit the observational data using a model with nucleon superfluidity. We adopt the neutrino emissivity accompanied by the phase transition from the normal state to that of the nucleon superfluidity (Yakovlev et al. (2004)), where we tune the critical temperature of the neutron superfluidity and the associated neutrino emissivity. For simplicity, we adopted only the superfluidity effect of the neutron , not of the neutron/proton . The singlets affect the cooling of a compact star, but the neutron singlet works in a lower density than the triplet, and the proton singlet is still speculative. The most effective nucleon superfluidity is caused by the neutron triplet state (Page et al., 2011b). Since  Ho & Heinke (2009) concluded that in order to reproduce the observations in the X-ray spectrum the surface composition of Cas A must be carbon and/or helium, we set the surface composition to be of carbon and a small amount of helium. This is because the existence of carbon results in a rather high effective temperature at the beginning of cooling phase. To calculate the cooling curves, we assume the functional form of the critical temperature which is a phenomenological extension of that for the neutron superfluidity as seen in the left panel of Figure 3. This approach is similar to the method used by Shternin et al. (2011) except for the profile of the critical temperature. Considering that the cooling curve sensitively depends on the critical temperature (left panel in Figure 3) and the neutrino emissivity, we have fine tuned the two quantities to fit the observational data() of Cas A over the past 10 years (right panel in Figure 3).

Figure 3.— Left panel: density dependence of the critical temperature for neutron superfluidity used by this study (solid and thick line) and previous works. Dot-dashed (labeled “1”), long dashed (labeled “2”), and short dashed (labeled “3”) denote the critical temperature adopted by Shternin et al. (2011), Page et al. (2009), and Kaminker et al. (2006), respectively. Right panel: cooling curve with the nucleon superfluidity which crosses the observational data of in Cas A (Heinke & Ho, 2010).

There are some experimental projects of hadron colliders with intermediate energy, such as J-PARC or GSI, that may help us to understand the state of ultrahigh density. They are very useful for examining high density character composed of hyperons, mesons, transition to normal quark matter and CSC phase (Andronic et al., 2010). However, it is still difficult to reproduce the same phase as in the core of compact stars by the colliders, where they produce high temperature ( MeV) in the high density region of compared with the core of compact stars (Andronic et al., 2009). Therefore, it is worthwhile to check the theories by both observing as many compact stars as possible and comparing theoretical predictions with observations such as the effective temperature. From this viewpoint, further observations of Cas A and central sources of other supernova remnants such as SN1987A are necessary to understand the fundamental physics in these extreme conditions.

There are some recent studies of hybrid star cooling, such as Negreiros et al. (2012), Yin et al. (2011) or Schramm et al. (2012). Negreiros et al. (2012) employed a similar model to our model, but the assumed energy gap is in the range of , and the pair breaking and formation (PBF) process is not included; thus resulting rapid decrease in the effective temperature of Cas A is not compatible with their model. Yin et al. (2011) considered the quark–hadron mixture and direct URCA process in hadronic phase, but did not included quark CSC, and resulting the cooling with the direct URCA is too strong to explain the observational data of older compact stars. Schramm et al. (2012) adopted the rotation of compact stars which delays the isothermal relaxation, and this effect would help us to understand the temperature drop of Cas A. These effects should be included in our further study.

Even if the quark–hadron MP does not exist, our scenario of color superconducting core surrounded by exotic phase could be applied to a cooled object such as Vela. Considering a meson condensed phase, a nucleon superfluidity, which reduces the strong neutrino emission by mesons, is expected to explain the Vela data consistent with cooling curves with some mass ranges. The cooling scenario of compact stars not only affects the cooling of isolated stars, but also the stars in binary systems. There are some observational data of X-ray transients (e.g., Rutledge et al., 2002) that have gravitational energy supply on the surface due to accretion from companions, and X-ray bursts could result from thermonuclear reactions on the surface. These systems are worthwhile to re-examine from the point of view of exotic cooling as was partly done in Yakovlev et al. (2004) for a quiescence period. Since binary systems have larger luminosity and/or orbital information such as inclination angle, rotational period, and a signal of the gravitational wave, more accurate mass detection than isolated compact stars would be possible (e.g., Kramer et al., 2006; Weisberg et al., 2010; Lattimer, 2010).

This work has been supported in part by the Grant-in-Aid for Scientific Research (19104006, 21105512, 21540272, and 24540278) and the Grant-in-Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.


  1. slugcomment: Published in ApJ.


  1. Alford, M. G., Braby, M., Paris, M. & Reddy, S. 2005, ApJ, 629, 969
  2. Alford, M. G., Schmitt, A., Rajagopal, K. & Schäfer, T. 2008, Rev. Mod. Phys., 80, 1455
  3. Andronic, A., Blaschke, D., Braun-Munzinger, P., et al. 2010, Nucl. Phys. A, 837, 65
  4. Andronic, A., Braun-Munzinger, P. & Stachel, J. 2009, Phys. Lett. B, 673, 142
  5. Ashworth, W. B. 1980, J. Hist. Astron., 11, 1
  6. Baldo, M. “Nuclear Methods and the Nuclear Equation of State”, 1999, World Scientific
  7. Baldo, M., Burgio, G. F. & Schulze, H. -J. 1998, Phys. Rev. C, 58, 3688
  8. Baym, G., Pethick, C. J. & Sutherland, P. 1971, ApJ, 170, 299
  9. Becker, W. ed., “Neutron Stars and Pulsars”, 2009, Springer
  10. Becker, W., Brazier, K. T. S. & Truemper, J. 1996, A&A, 306, 464
  11. Bogomazov, A. I., Abubekerov, M. K., Lipunov, V. M. & Cherepashchuk, A. M. 2005, Astron. Rep., 49, 295
  12. Brisken, W. F., Thorsett, S. E., Golden, A. & Goss, W. M. 2003, ApJ, 593, L89
  13. Demorest, P. B., Pennucci, T., Ransom, S. M., Roberts, M. S. E. & Hessels, J. W. T. 2010, Nature, 467, 1081
  14. Friman, B. L. & Maxwell, O. V. 1979, ApJ, 232, 541
  15. Gaensler, B. M., Schulz, N. S., Kaspi, V. M., Pivovaroff, M. J. & Becker, W. E. 2003, ApJ, 588, 441
  16. Gotthelf, E. V., Halpern, J. P. & Dodson, R. 2002, ApJ, 567, L125
  17. Gusakov, M. E., Kaminker, A. D., Yakovlev, D. G. & Gnedin, O. Y. 2005, MNRAS, 363, 555
  18. Hashimoto, M., Seki, H. & Yamada, M. 1984, Prog. Theor. Phys., 71, 320
  19. Heinke, C. O. & Ho, W. C. G. 2010, ApJ, 719, L167
  20. Heinke, C. O., Jonker, P. G., Wijnands, R., Deloye, C. J. & Taam, R. E. 2009, ApJ, 691,1035
  21. Ho, W. C. G. & Heinke, C. O. 2009, Nature, 462, 71
  22. Iwamoto, N. 1980, Phys. Rev. Lett., 44, 1637
  23. Jaikumar, P., Prakash, M. & Schäfer, T. 2002, Phys. Rev. D, 66, 063003
  24. Kaminker, A. D., Gusakov, M. E., Yakovlev, D. G. & Gnedin, O. Y. 2006, MNRAS, 365, 1300
  25. Kaspi, V. M., Roberts, M. S. E. & Harding, A. K. 2006, in Compact Stellar X-Ray Sources, eds. W. H. G. Lewin & M. van der Klis (Cambridge University Press), 279-340
  26. Kramer, M., Stairs, I. H., Manchester, R. N., et al. 2006, Science, 314, 97
  27. Lattimer, J. M. 2010, Prog. Theor. Phys. Suppl., 186, 1
  28. Lattimer, J. M., Pethick, C. J., Prakash, M. & Haensel, P. 1991, Phys. Rev. Lett., 66, 2701
  29. Maruyama, T., Chiba, S., Schulze, H. -J. & Tatsumi, T. 2007, Phys. Rev. D, 76, 123015
  30. Maruyama, T., Chiba, S., Schulze, H. -J. & Tatsumi, T. 2008, Phys. Lett. B, 659, 192
  31. Negreiros, R., Dexheimer, V. A. & Schramm, S. 2012, Phys. Rev. C, 85, 035805
  32. Page, D. 2002,
  33. Page, D., Lattimer, J. M., Prakash, M. & Steiner, A. W. 2009, ApJ, 707, 1131
  34. Page, D., Prakash, M., Lattimer, J. M. & Steiner, A. W. 2011a, Phys. Rev. Lett., 106, 081101
  35. Page, D., Prakash, M., Lattimer, J. M. & Steiner, A. W. 2011b, PoS(XXXIV BWNP)005, (arXiv:1110.5116)
  36. Pavlov, G. G., Zavlin, V. E., Sanwal, D., Burwitz, V. & Garmire, G. P. 2001, ApJ, 552, L129
  37. Ravenhall, D. G., Pethick, C. J. & Wilson, J. R. 1983, Phys. Rev. Lett., 50, 2066
  38. Reed, J. E., Hester, J. J., Fabian, A. C. & Winkler, P. F. 1995, ApJ, 440, 706
  39. Rüster, S. B., Werth, V., Buballa, M., Shvkovy, I. A. & Rischke, D. H. 2005, Phys. Rev. D, 72, 034004
  40. Rutledge, R. E., Bildsten, L., Brown, E. F., Pavlov, G. G. & Zavlin, V. E. 2002, ApJ, 577, 346
  41. Schmitt, A. “Dense matter in compact stars”, 2010, Springer
  42. Schramm, S., Dexheimer, V., Negreiros, R., Schürhoff, T. & Steinheimer, J. 2012, arXiv:1202.5113
  43. Schulze, H. -J., Lejeune, A., Cugnon, J. & Lombardo, U. 1995, Phys. Lett. B, 355, 21
  44. Shternin, P. S., Yakovlev, D. G., Heinke, C. O., Ho, W. C. G. & Patnaude, D. J. 2011, MNRAS, 412, L108
  45. Slane, P. O., Helfand, D. J. & Murray, S. S. 2002, ApJ, 571, L45
  46. Takatsuka, T. & Tamagaki, R. 1997, Prog. Theor. Phys., 97, 345
  47. Tsuruta, S. 1998, Phys. Rep., 292, 1
  48. Weisberg, J. M., Nice, D. J. & Taylor, J. H. 2010, ApJ, 722, 1030
  49. Weisskopf, M. C., O’Dell, S. L., Paerels, F., et al. 2004, ApJ, 601, 1050
  50. Yakovlev, D. G., Gnedin, O. Y., Gusakov, M. E., Kaminker, A. D., Levenfish, K. P. & Potekhin, A. Y. 2005, Nucl. Phys. A, 752, 590
  51. Yakovlev, D. G., Gnedin, O. Y., Kaminker, A. D., Levenfish, K. P. & Potekhin, A. Y. 2004, Adv. Space Res., 33, 523
  52. Yakovlev, D. G., Ho, W. C. G., Shternin, P. S., Heinke, C. O. & Potekhin, A. Y. 2011, MNRAS, 411, 1977
  53. Yasutake, N., Maruyama, T. & Tatsumi, T. 2009, Phys. Rev. D, 80, 123009
  54. Yin, S., van Heugten, J. J. R. M., Diederix, J., et al. 2011, arXiv:1112.1880
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description