On the minimum mass of neutron starssuwa/presentation/190123_DTA10.pdf · majority of the fully...

Yudai Suwa （Kyoto Sangyo University）

collaboration with T. Yoshida (Tokyo), M. Shibata (Kyoto/AEI), H. Umeda (Tokyo), K. Takahashi (Bonn)

On the minimum mass of neutron stars

Yudai Suwa @ Stellar deaths and their diversity, NAOJ /1623/1/2019

Kyoto Sangyo University（京都産業大学）

2

Kyoto Station

Kyoto University

Kyoto Sangyo University

~10km (~40min w/

metro & bus)

Long-term workshop “Multi-messenger astrophysics

in the GW era” (24 Sep.-25 Oct. 2019)

incl. YKIS conference (7 Oct.-11 Oct. 2019)


Pulsar number is increasing

3

pulsar

millisecond pulsar pulsar w/ precise mass

compiled data from ATNF pulsar catalog and P. Freire’s table


Mass measurements of NSs

>2600 pulsars have been found in the Galaxy 10% in the binary system →mass measurement possible

15 double NSs so far [Tauris+ 2017]

4

AA54CH11-Ozel ARI 25 August 2016 16:41

J0453+1559

J0453+1559c

J0737-3039A

J0737-3039B

B1534+12

B1534+12cJ1756-2251

J1756-2251c

J1906+0746J1906+0746c

B1913+16B1913+16cB2127+11C

B2127+11Cc

J1518+4904

J1518+4904c

J1811-1736

J1811-1736c

J1829+2456J1829+2456cJ1930-1852

J1930-1852c

J0337+1715

J0348+0432

J0437-4715

J0621+1002

J0751+1807

J1012+5307

J1614-2230

J1713+0747

J1738+0333

J1802-2124

J1807-2500B

B1855+09J1903+0327

J1909-3744

J1910-5958A

J1946+3417

J2222-0137

J2234+0611J0024-7204H

J0514-4002A

B1516+02B

J1748-2021B

J1748-2446I

J1748-2446J

J1750-37A

B1802-07

J1824-2452C

EXO 1745-248

4U 1608-52

4U 1820-30

KS 1731-260

SAX J1748.9-20214U 1724-207

Cyg X-2

Vela X-1

SMC X-1

LMC X-4

Cen X-3SAX J1802.7-2017

XTE J1855-026

EXO 1722-363OAO 1657-415

Her X-14U1538-52

B2303+46J1141-6545

Doubleneutron stars

Recycledpulsars

Bursters

Slowpulsars

Mass (M☉)0.0 1.0 2.0 3.0

Figure 2The most recent measurements of neutron-star masses. Double neutron stars (magenta), recycled pulsars( gold ), bursters ( purple), and slow pulsars (cyan) are included.

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00.8 1.0 1.2

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Recycledpulsars

Slowpulsars

1.4 1.6 1.8 2.0

Mass (M☉)

Figure 3The inferred mass distributions for the different populations of neutron stars.

parameters for these distributions are the following: M 0 = 1.33 M⊙ and σ = 0.09 M⊙ forthe DNSs, M 0 = 1.54 M⊙ and σ = 0.23 M⊙ for the recycled NSs, and M 0 = 1.49 M⊙ andσ = 0.19 M⊙ for the slow pulsars. A recent study also raised the possibility of two peaks withinthe recycled MSP population, with the first peak at M = 1.39 M⊙ and a dispersion σ = 0.06 M⊙

and a second peak appearing at M = 1.81 M⊙ with a dispersion of σ = 0.18 M⊙ (Antoniadiset al. 2016).

Among these inferred distributions, the narrowness of the DNS distribution stands out.Although clearly not representative of NSs as a whole, as it was once thought (Thorsett &Chakrabarty 1999), it probably points to a particular evolutionary mechanism that keeps themasses of NSs in these systems in a narrow range. Recent discoveries, such as the DNS J0453+1559(Deneva et al. 2013), indicate that the range of masses in DNS systems may also be wider thanpreviously believed: the recycled pulsar has a mass of 1.559 (5) M⊙, the heaviest known in any DNS(Martinez et al. 2015), whereas the companion has a mass of 1.174 (4) M⊙, the smallest preciselymeasured mass of any NS (we infer that the companion is an NS from the orbital eccentricity ofthe system, e = 0.11251837(5), which would not arise if it had slowly evolved to a massive WDstar).

2.6. Maximum Mass of Neutron StarsFinding the maximum mass of NSs is of particular interest in mass measurements because of itsdirect implications for the NS EoSs and NS evolution. The largest NS mass can rule out the EoSsthat have maximum masses and fall below this value. The current record holder on this front isJ0348+0432 with a mass of 2.01 ± 0.04 M⊙ (Antoniadis et al. 2013).

There are also some studies of a particular class of MSPs called black widows (and their cousinsredbacks) that have suggested higher NS masses (e.g., van Kerkwijk et al. 2011). These MSPsirradiate and ablate their very low-mass companions. Although the pulsar timing provides theKeplerian parameters for the orbit, all other information about the masses in these systems isobtained from the modeling of the optical light curves (to determine orbital inclination) and thespectroscopy (to measure the mass ratio) of the companion star. Unfortunately, there are manydifficulties in obtaining accurate measurements from these ablated companions. Even when usinga model of an irradiated companion, the short timescale variability, the unevenly heated surface,

www.annualreviews.org • Neutron-Star Masses and Radii 415

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Massive NSs tell us nuclear physics

5

common feature of models that include the appearance of ‘exotic’hadronic matter such as hyperons4,5 or kaon condensates3 at densitiesof a few times the nuclear saturation density (ns), for example modelsGS1 and GM3 in Fig. 3. Almost all such EOSs are ruled out by ourresults. Our mass measurement does not rule out condensed quarkmatter as a component of the neutron star interior6,21, but it stronglyconstrains quark matter model parameters12. For the range of allowedEOS lines presented in Fig. 3, typical values for the physical parametersof J1614-2230 are a central baryon density of between 2ns and 5ns and aradius of between 11 and 15 km, which is only 2–3 times theSchwarzschild radius for a 1.97M[ star. It has been proposed thatthe Tolman VII EOS-independent analytic solution of Einstein’sequations marks an upper limit on the ultimate density of observablecold matter22. If this argument is correct, it follows that our mass mea-surement sets an upper limit on this maximum density of(3.74 6 0.15) 3 1015 g cm23, or ,10ns.

Evolutionary models resulting in companion masses .0.4M[ gen-erally predict that the neutron star accretes only a few hundredths of asolar mass of material, and result in a mildly recycled pulsar23, that isone with a spin period .8 ms. A few models resulting in orbital para-meters similar to those of J1614-223023,24 predict that the neutron starcould accrete up to 0.2M[, which is still significantly less than the>0.6M[ needed to bring a neutron star formed at 1.4M[ up to theobserved mass of J1614-2230. A possible explanation is that someneutron stars are formed massive (,1.9M[). Alternatively, the trans-fer of mass from the companion may be more efficient than currentmodels predict. This suggests that systems with shorter initial orbitalperiods and lower companion masses—those that produce the vastmajority of the fully recycled millisecond pulsar population23—mayexperience even greater amounts of mass transfer. In either case, ourmass measurement for J1614-2230 suggests that many other milli-second pulsars may also have masses much greater than 1.4M[.

Received 7 July; accepted 1 September 2010.

1. Lattimer, J. M. & Prakash, M. The physics of neutron stars. Science 304, 536–542(2004).

2. Lattimer, J. M. & Prakash, M. Neutron star observations: prognosis for equation ofstate constraints. Phys. Rep. 442, 109–165 (2007).

3. Glendenning, N. K. & Schaffner-Bielich, J. Kaon condensation and dynamicalnucleons in neutron stars. Phys. Rev. Lett. 81, 4564–4567 (1998).

4. Lackey, B. D., Nayyar, M. & Owen, B. J. Observational constraints on hyperons inneutron stars. Phys. Rev. D 73, 024021 (2006).

5. Schulze, H., Polls, A., Ramos, A. & Vidana, I. Maximum mass of neutron stars. Phys.Rev. C 73, 058801 (2006).

6. Kurkela, A., Romatschke, P. & Vuorinen, A. Cold quark matter. Phys. Rev. D 81,105021 (2010).

7. Shapiro, I. I. Fourth test of general relativity. Phys. Rev. Lett. 13, 789–791 (1964).8. Jacoby, B.A., Hotan, A., Bailes,M., Ord, S. & Kulkarni, S.R. The massof a millisecond

pulsar. Astrophys. J. 629, L113–L116 (2005).9. Verbiest, J. P. W. et al. Precision timing of PSR J0437–4715: an accurate pulsar

distance, a high pulsar mass, and a limit on the variation of Newton’s gravitationalconstant. Astrophys. J. 679, 675–680 (2008).

10. Hessels, J.et al. inBinaryRadio Pulsars (edsRasio, F. A.& Stairs, I. H.)395 (ASP Conf.Ser. 328, Astronomical Society of the Pacific, 2005).

11. Crawford, F. et al. A survey of 56 midlatitude EGRET error boxes for radio pulsars.Astrophys. J. 652, 1499–1507 (2006).

12. Ozel, F., Psaltis, D., Ransom, S., Demorest, P. & Alford, M. The massive pulsar PSRJ161422230: linking quantum chromodynamics, gamma-ray bursts, andgravitational wave astronomy. Astrophys. J. (in the press).

13. Hobbs, G. B., Edwards, R. T. & Manchester, R. N. TEMPO2, a new pulsar-timingpackage - I. An overview. Mon. Not. R. Astron. Soc. 369, 655–672 (2006).

14. Damour, T. & Deruelle, N. General relativistic celestial mechanics of binarysystems. II. The post-Newtonian timing formula. Ann. Inst. Henri Poincare Phys.Theor. 44, 263–292 (1986).

15. Freire, P.C.C.&Wex,N.Theorthometricparameterisationof theShapiro delay andan improved test of general relativity with binary pulsars. Mon. Not. R. Astron. Soc.(in the press).

16. Iben, I. Jr & Tutukov, A. V. On the evolution of close binaries with components ofinitial mass between 3 solar masses and 12 solar masses. Astrophys. J Suppl. Ser.58, 661–710 (1985).

17. Ozel, F. Soft equations of state for neutron-star matter ruled out by EXO 0748 -676. Nature 441, 1115–1117 (2006).

18. Ransom, S. M. et al. Twenty-one millisecond pulsars in Terzan 5 using the GreenBank Telescope. Science 307, 892–896 (2005).

19. Freire, P. C. C. et al. Eight new millisecond pulsars in NGC 6440 and NGC 6441.Astrophys. J. 675, 670–682 (2008).

20. Freire, P. C. C., Wolszczan, A., van den Berg, M. & Hessels, J. W. T. A massive neutronstar in the globular cluster M5. Astrophys. J. 679, 1433–1442 (2008).

21. Alford,M.etal.Astrophysics:quarkmatterincompactstars?Nature445,E7–E8(2007).22. Lattimer, J. M. & Prakash, M. Ultimate energy density of observable cold baryonic

matter. Phys. Rev. Lett. 94, 111101 (2005).23. Podsiadlowski, P., Rappaport, S. & Pfahl, E. D. Evolutionary sequences for low- and

intermediate-mass X-ray binaries. Astrophys. J. 565, 1107–1133 (2002).24. Podsiadlowski, P. & Rappaport, S. Cygnus X-2: the descendant of an intermediate-

mass X-Ray binary. Astrophys. J. 529, 946–951 (2000).25. Hotan, A. W., van Straten, W. & Manchester, R. N. PSRCHIVE and PSRFITS: an open

approach to radio pulsar data storage and analysis. Publ. Astron. Soc. Aust. 21,302–309 (2004).

26. Cordes, J. M. & Lazio, T. J. W. NE2001.I. A new model for the Galactic distribution offree electrons and its fluctuations. Preprint at Æhttp://arxiv.org/abs/astro-ph/0207156æ (2002).

27. Lattimer, J. M. & Prakash, M. Neutron star structure and the equation of state.Astrophys. J. 550, 426–442 (2001).

28. Champion, D. J. et al. An eccentric binary millisecond pulsar in the Galactic plane.Science 320, 1309–1312 (2008).

29. Berti, E., White, F., Maniopoulou, A. & Bruni, M. Rotating neutron stars: an invariantcomparison of approximate and numerical space-time models. Mon. Not. R.Astron. Soc. 358, 923–938 (2005).

Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements P.B.D. is a Jansky Fellow of the National Radio AstronomyObservatory. J.W.T.H. is a Veni Fellow of The Netherlands Organisation for ScientificResearch. We thankJ. Lattimer for providing the EOSdataplotted inFig. 3, and P. Freire,F. Ozel and D. Psaltis for discussions. The National Radio Astronomy Observatory is afacility of the US National Science Foundation, operated under cooperative agreementby Associated Universities, Inc.

Author Contributions All authors contributed to collecting data, discussed the resultsand edited the manuscript. In addition, P.B.D. developed the MCMC code, reduced andanalysed data, and wrote the manuscript. T.P. wrote the observing proposal andcreated Fig. 3. J.W.T.H. originally discovered the pulsar. M.S.E.R. initiated the survey thatfound the pulsar. S.M.R. initiated the high-precision timing proposal.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of this article atwww.nature.com/nature. Correspondence and requests for materials should beaddressed to P.B.D. ([email protected]).

0.07 8 9 10 11

Radius (km)12 13 14 15

0.5

1.0

1.5

2.0

AP4

J1903+0327

J1909-3744

systemsDouble neutron sDouble neutron star sysy

J1614-2230

AP3

ENG

MPA1

GM3

GS1

PAL6

FSUSQM3

SQM1

PAL1

MS0

MS2

MS1

2.5 GR

Causality

Rotation

P < ∞

Mas

s (M

()

Figure 3 | Neutron star mass–radius diagram. The plot shows non-rotatingmass versus physical radius for several typical EOSs27: blue, nucleons; pink,nucleons plus exotic matter; green, strange quark matter. The horizontal bandsshow the observational constraint from our J1614-2230 mass measurement of(1.97 6 0.04)M[, similar measurements for two other millisecond pulsars8,28

and the range of observed masses for double neutron star binaries2. Any EOSline that does not intersect the J1614-2230 band is ruled out by thismeasurement. In particular, most EOS curves involving exotic matter, such askaon condensates or hyperons, tend to predict maximum masses well below2.0M[ and are therefore ruled out. Including the effect of neutron star rotationincreases the maximum possible mass for each EOS. For a 3.15-ms spin period,this is a =2% correction29 and does not significantly alter our conclusions. Thegrey regions show parameter space that is ruled out by other theoretical orobservational constraints2. GR, general relativity; P, spin period.

LETTER RESEARCH

2 8 O C T O B E R 2 0 1 0 | V O L 4 6 7 | N A T U R E | 1 0 8 3

Macmillan Publishers Limited. All rights reserved©2010

Demorest+ 2010

←1.97±0.04M⊙NB) mass estimation was

updated by Arzoumanian+ 2018 as 1.908±0.016M⊙

Another massive NS was reported by Antoniadis+ (2013),

J0348+0432, 2.01±0.04M⊙


So, what does a small NS tell?

6


Double NSs

7


J0453+1559

J0453+1559c

J0737-3039A

J0737-3039B

B1534+12

B1534+12cJ1756-2251

J1756-2251c

J1906+0746J1906+0746c

B1913+16B1913+16cB2127+11C

B2127+11Cc

J1518+4904

J1518+4904c

J1811-1736

J1811-1736c

J1829+2456J1829+2456cJ1930-1852

J1930-1852c

J0337+1715

J0348+0432

J0437-4715

J0621+1002

J0751+1807

J1012+5307

J1614-2230

J1713+0747

J1738+0333

J1802-2124

J1807-2500B

B1855+09J1903+0327

J1909-3744

J1910-5958A

J1946+3417

J2222-0137

J2234+0611J0024-7204H

J0514-4002A

B1516+02B

J1748-2021B

J1748-2446I

J1748-2446J

J1750-37A

B1802-07

J1824-2452C

EXO 1745-248

4U 1608-52

4U 1820-30

KS 1731-260

SAX J1748.9-20214U 1724-207

Cyg X-2

Vela X-1

SMC X-1

LMC X-4

Cen X-3SAX J1802.7-2017

XTE J1855-026

EXO 1722-363OAO 1657-415

Her X-14U1538-52

B2303+46J1141-6545

Doubleneutron stars

Recycledpulsars

Bursters

Slowpulsars

Mass (M☉)0.0 1.0 2.0 3.0


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J0453+1559

J0453+1559c

J0737-3039A

J0737-3039B

B1534+12

B1534+12cJ1756-2251

J1756-2251c

J1906+0746J1906+0746c

B1913+16B1913+16cB2127+11C

B2127+11Cc

J1518+4904

J1518+4904c

J1811-1736

J1811-1736c

J1829+2456J1829+2456cJ1930-1852

J1930-1852c

J0337+1715

J0348+0432

J0437-4715

J0621+1002

J0751+1807

J1012+5307

J1614-2230

J1713+0747

J1738+0333

J1802-2124

J1807-2500B

B1855+09J1903+0327

J1909-3744

J1910-5958A

J1946+3417

J2222-0137

J2234+0611J0024-7204H

J0514-4002A

B1516+02B

J1748-2021B

J1748-2446I

J1748-2446J

J1750-37A

B1802-07

J1824-2452C

EXO 1745-248

4U 1608-52

4U 1820-30

KS 1731-260

SAX J1748.9-20214U 1724-207

Cyg X-2

Vela X-1

SMC X-1

LMC X-4

Cen X-3SAX J1802.7-2017

XTE J1855-026

EXO 1722-363OAO 1657-415

Her X-14U1538-52

B2303+46J1141-6545

Doubleneutron stars

Recycledpulsars

Bursters

Slowpulsars

Mass (M☉)0.0 1.0 2.0 3.0


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concentrating within a small range


First asymmetric DNS system

8

4. DISCUSSION AND CONCLUSIONS

The accretion episode in DNS systems is very short-livedand therefore the mass of the recycled pulsar is only slightly

larger than its mass at birth (Tauris et al. 2015). Until now,most well-measured NS masses in DNS systems fell in anarrow range between 1.23 and 1.44 Me (Weisberg et al. 2010and Faulkner et al. 2005, see Table 1). This has led tospeculation that all NSs might be born within this narrow band,and that the large masses observed in some MSPs like PSRJ1903+0327 (Freire et al. 2011), PSR J1614−2230 (Demorestet al. 2010), and PSR J0348+0432 (Antoniadis et al. 2013) aredue to accretion. However, from an analysis of the evolution ofPSR J1614–2230, Tauris et al. (2011) had already suggestedthat at least some NSs must be born more massive than 1.44Me. The mass of PSR J0453+1559—the largest ever measuredin a DNS system (see Table 1)—and that of its companion—the smallest precisely measured for any NS—shows that therange of NS birth masses is indeed substantially wider thanearlier studies indicated (Thorsett & Chakrabarty 1999 andÖzel et al. 2012).It is interesting to speculate on how the companion might

have formed. Its mass is lower than the 1.24 M: measured forthe companion of J1756−2251 (Ferdman et al. 2014) and PSRsJ1802−2124 (Ferdman et al. 2010) and J0737−3039B(Kramer et al. 2006) that are thought to have formed inelectron capture supernovae (ECSN). It is possible that thecompanion formed instead in an iron core collapse SN, wherethe core of the progenitor of the companion was stripped of itsenvelope (Tauris et al. 2015).

Figure 3. Current constraints from the timing of PSR J0453+1559. Each triplet of lines corresponds to the nominal and ±1σ uncertainties of the post-Keplerianparameters measured using the DDH model in TEMPO2 (see Table 2), which are the rate of advance of periastron ,w the orthometric ratio of the Shapiro delay ς, and theorthometric amplitude of the Shapiro delay, h3 (Freire & Wex 2010). The contour levels contain 68.27% and 95.45% of the 2D probability density functions (pdfs)derived from the quality of the timing solution at each point of the Mc– icos plane using only the Shapiro delay (black) and Shapiro delay plus the assumption that thew is due only to the effects of GR (red). Left: Mc– icos plane. The gray region is excluded by the physical constraint M 0.p > Right: Mc–Mp plane. The gray region isexcluded by the mathematical constraint isin 1.- Top and right panels: pdfs for icos , Mp, and (on the right) Mc, derived from marginalizing the 2D pdf in the mainpanel for these quantities. When w is taken into account (red), the precision of the mass estimates improves by two orders of magnitude.

Figure 4. Estimated minimum mean flux density of the companion ifdetectable with an S/N = 10, as a function of its spin period Pcom at 1.4 GHz(black lines) and 327 MHz (blue lines). For each frequency, the cases of anintrinsic duty cycle wcom of 1% (solid line), 5% (dashed line), and 10% (dotted–dashed line) of Pcom are shown. In the L band, the parameters used weret 6300 s,obs = T 30sys = K, G = 10 K Jy−1, f 600 MHz;D = at 327 MHz theywere t 240 s,obs = Tsys = 113 K, G = 11 K Jy−1, Δ f = 60 MHz; at bothfrequencies β = 1 and np = 2. Because the companion was not detected, itsmean flux density must be out of our line of sight.

6

The Astrophysical Journal, 812:143 (8pp), 2015 October 20 Martinez et al.Martinez+ 2015Mp=1.559±0.005M⊙

Mc=1.174±0.004M⊙


A low-mass NS

MNS=1.174M⊙! (NB, it ‘s gravitational mass, baryonic mass is ~1.28M⊙)

Is it a white dwarf? Maybe no a large eccentricity (e=0.112) is difficult to explain by slow evolution into a WD

How to make it? a small iron core of massive star?(typically MFe~1.4–1.8M⊙) getting rid of mass from a NS?

9

11

3.2. Core Masses

Fig. 5 shows the helium, carbon-oxygen (CO), andiron-core masses for all our presupernova models withMN and MN/2. The dispersion in helium and COcore masses for a given main sequence mass is small,much less than the hydrogen envelope masses inferredin Fig. 3. This implies that the helium and CO coremasses are only slightly a↵ected by the assumed massloss rate and are probably not very sensitive to metal-licity, at least for non-trivial values of metallicity. Thehelium core mass as a function of main sequence mass isapproximately MHe ⇡ 6.46 (MZAMS/20)1.27 which canbe combined with the previous relation in §3.1 to giveLpreSN ⇡ 6.5 ⇥ 1038 (MZAMS/20 M�)1.92 ergs s�1. Sincepresupernova core compactness is chiefly a function ofhelium core mass (Sukhbold & Woosley 2014), these re-sults suggest a near universal dependence of presuper-nova core structure on initial mass, provided mass lossdoes not remove the entire hydrogen envelope.

Iron core masses increase, on the average with increas-ing stellar mass reaching maximum of about 2.0 M� forthe most massive stars studied (> 40 M�). Still largeriron cores, up to about 2.5 M� characterize more mas-sive stars in the pulsational-pair instability range (70 -140 M�; Woosley 2017). For stars below 23 M�, theiron core mass is markedly multi-valued for stars withnearly the same initial mass. This reflects the operationof multiple shells of carbon and oxygen burning as willbe discussed further in §4. The two major branches ofiron core masses below 20 M�, which are most of thestars that leave neutron star remnants might result inbimodality in the neutron star mass function.

3.3. Core Structure

3.3.1. Measures of “Explodability”

Early theoretical studies of supernovae noted a strongcorrelation of a rapidly declining density outside the ironcore with the degree of di�culty encountered in tryingto explode the star using the neutrino energy transport(e.g., Burrows & Lattimer 1987; Fryer 1999). O’Connor& Ott (2011) introduced a simple, single parameter mea-sure of this density decline called the “compactness pa-rameter”:

⇠M =M/M�

R(Mbary = M)/1000 km

��tbounce

, (3)

where R(Mbary = M) is the radius of the Lagrangianmass shell enclosing mass M in the presupernova star.The fiducial mass is often chosen as the innermost 2.5M� so that for a wide range of initial masses it notonly encloses the iron-core but samples enough of theoverlying shell material around it. Though the param-eter is defined to be evaluated at the time of bounce,it is more often measured at the time of presupernova(when the collapse begins), since the systematics are in-sensitive to slightly di↵erent fiducial choices (Sukhbold& Woosley 2014). Subsequent studies by Ugliano et al.(2012), O’Connor & Ott (2013), and Sukhbold et al.(2016) showed strong correlation between the ease withwhich model stars exploded and the ⇠ parameter, in thesense that stars with small ⇠, i.e., steep density gradientsoutside the iron core were easier to explode using a stan-

15 20 25 30 35 40MZAMS [M

�

]

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Iron

Cor

eM

ass

[M�

]

MN

MN/2

Fig. 5.— Helium core, CO core, and Fe core masses for all pre-supernova stars from sets with MN and MN/2. Despite significantvariations in mass loss (Fig. 3), the final helium and carbon-oxygencores are well determined by the star’s initial mass and a standardchoice of stellar physics. Note multiple branches for the iron coremass below 19 M�.

dard, albeit approximate, set of supernova physics. Al-though this is a useful starting point, a single parameterconveys limited information about the structure of thecore and more physics-based representations followed.

In particular, Ertl et al. (2016) suggested an alternativetwo-parameter characterisation based upon M4, the masscoordinate, in solar masses, where the entropy per baryonreaches 4.0 kB , and the radial gradient, µ4, of the densityat that point. In practice, µ4, was obtained by evaluatingthe change in mass over the change in radius betweenseveral mass shells separated by 0.3 M� in the vicinityof M4, i.e.,

µ4 =dm/M�

dr/1000 km(4)

where dm = 0.3 M�. Smaller values of µ4 thus implysteeper density gradients (less change in enclosed masswhen the radius changes). The quantity M4 has longbeen used to locate the the steep density decline oftenassociated with a strong oxygen-burning shell in the pre-supernova star (e.g., Woosley & Heger 2007). Where theentropy per baryon abruptly rises at nearly constant tem-

Sukhbold+ 2018


A path toward a low mass NS?: Ultra-stripped SN

10

[Suwa+, MNRAS, 454, 3073 (2015); Yoshida+, MNRAS, 471, 4275 (2017)]

1.4. Résumé of DNS Formation

Previous theoretical works on the physics of DNS formationincludes (here disregarding general population synthesis studies)Bisnovatyi-Kogan & Komberg (1974), Wheeler et al. (1974),Flannery & van den Heuvel (1975), Srinivasan & van den Heuvel(1982), van den Heuvel (1994), Ivanova et al. (2003), Dewi &Pols (2003), Podsiadlowski et al. (2004), van den Heuvel (2004),and Dewi et al. (2005). From these papers, a standard scenario12

has emerged (e.g., Bhattacharya & van den Heuvel 1991; Tauris& van den Heuvel 2006), which we now summarize in moredetail.

In Figure 1, we show an illustration of the formation of a DNSsystem. The initial system contains a pair of OB-stars that aremassive enough13 to terminate their lives in a core-collapse SN(CCSN). To enable the formation of a tight DNS system in the end,the two stars must initially be in a binary system close enough toensure interactions via either stable or unstable mass transfer. If thebinary system remains bound after the first SN explosion (which isof Type Ib/c; Yoon et al. 2010), the system eventually becomesobservable as a HMXB. Before this stage, the system may also bedetectable as a radio pulsar orbiting an OB-star, e.g., as inPSRsB1259−63 (Johnston et al. 1992) and J0045−7319 (Kaspiet al. 1994). When the secondary star expands and initiates full-blown Roche-lobe overflow (RLO) during the HMXB stage, thesystem eventually becomes dynamically unstable. For widesystems, where the donor star has a deep convective envelope atthe onset of mass transfer (i.e., during the so-called Case B RLO,following the termination of core hydrogen burning), the timescaleon which the system becomes dynamically unstable might be asshort as a few 100yr (Savonije 1978). This leads to the formationof a CE (Paczyński 1976), where the dynamical friction of themotion of the NS inside the giant star’s envelope often causes

extreme loss of orbital angular momentum and (in some cases)ejection of the hydrogen-rich envelope. If the binary systemsurvives the CE phase, it consists of a NS orbiting a helium star(the naked core of the former giant star). Depending on the orbitalseparation and the mass of the helium star, an additional phase ofmass transfer (Case BB RLO; Habets 1986; Tauris et al. 2015)maybe initiated. This stage of mass transfer is important since it enablesa relatively long phase of accretion onto the NS, whereby the NS isrecycled, and it allows for extreme stripping of the helium star priorto its explosion (as a so-called ultra-stripped SN; Tauris et al. 2013,2015; Suwa et al. 2015; Moriya et al. 2017). Whether or not thesystem survives the second SN depends on the orbital separationand the kick imparted onto the newborn NS (Flannery & van denHeuvel 1975; Hills 1983; Tauris & Takens 1998). As we shallargue in this paper, we expect most systems to survive the secondSN explosion. If the post-SN orbital period is short enough (andespecially if the eccentricity is large), the DNS system willeventually merge due to GW radiation and produce a strong high-frequency GW signal and possibly a shortGRB (e.g., Eichler et al.1989). The final remnant is most likely a BH, although, dependingon the EoS, a NS (or, at least, a metastable NS) may be left behindinstead (Vietri & Stella 1998).

1.5. Major Uncertainties in DNS Formation

Aside from the pre-HMXB evolution, which is discussed inSection 3.1, the most important and uncertain aspects of ourcurrent understanding of DNS formation are related to

Table 1Observed Ranges of Key Properties of DNS Systems

Properties of Recycled (Old) NSs:Spin period, P 23 185 ms–Period derivative, P 0.027 18 10 s s18 1´ - -( – )Surface dipole B-field, B 0.29 18 10 G9´( – )Mass, MNS,1 1.32–1.56 Me

a

Properties of Young NSs:Spin period, P 144 2773 ms–Period derivative, P 0.89 20 10 s s15 1´ - -( – )Surface dipole B-field, B 2.7 5.3 10 G11´( – )Mass, MNS,2 M1.17 1.39 :–

Orbital Properties:Orbital period, Porb 0.10 45 days–Eccentricity, e 0.085 0.83–Merger time, gwrt 86 Myr ¥Systemic velocity, vsys 25 240 km s 1-–

Note. Data taken from the ATNF Pulsar Catalogue (Manchester et al. 2005)—see Table 2 for further details. Only DNS systems in the Galactic disk arelisted. The systemic recoil velocity, v vsys

LSR= , is quoted with respect to thelocal standard of rest (Section 2.2).a 1.32 Me Mark an upper limit to the lowest mass of the first-born NS.

Figure 1. Illustration of the formation of a DNS system that merges within aHubble time and produces a single BH, following a powerful burst of GWs anda shortGRB. Acronyms used in this figure—ZAMS: zero-age main sequence;RLO: Roche-lobe overflow (mass transfer); He-star: helium star; SN:supernova; NS: neutron star; HMXB: high-mass X-ray binary; CE: commonenvelope; BH: black hole.

12 See brief discussion given in Section 4.2 for an alternative “double corescenario” (Brown 1995; Dewi et al. 2006) in which CE evolution with a NS isavoided.13 The secondary (initially less massive) star may be a M5 7 :– star whichaccretes mass from the primary (initially more massive) star to reach thethreshold limit for core collapse at M8 12~ :– (Jones et al. 2013; Woosley &Heger 2015; see also Section 3.1).

3

The Astrophysical Journal, 846:170 (58pp), 2017 September 10 Tauris et al.

Tauris+ 2017

Explosions of ultra-stripped SNe 3079

Figure 7. Evolutions of the radius of shocks.

Figure 8. Time evolutions of PNS mass (defined by ρ > 1011 g cm−3).

detailed nucleosynthesis calculation, which is beyond the scope ofthis paper. The NS kick velocity is estimated by assuming the linearmomentum conservation of the whole progenitor star, i.e. assumingthat anisotropic mass ejection leads to NS kick (e.g. Wongwatha-narat, Janka & Muller 2013). The linear momentum of ejecta iscalculated by

Pej =!

ρ<1011 g cm−3,vr>0ρvdV , (3)

where v is the velocity vector and vr is its radial component. The NSkick velocity is then given by vkick = −Pej/MNS,baryon. Since theaxial symmetry is assumed in our simulations, the kick velocity maybe overestimated due to the existence of preferable direction of NSkick, i.e. symmetry axis. Additionally, the stochastic nature of post-shock turbulent flow would also change the degree of asymmetryof ejecta so that the initial small perturbation could change the kickvelocity significantly (Scheck et al. 2006). More statistical study isneeded to pin down this issue. It can be argued that small envelope,not small iron core itself, which can rapidly accelerate shock, wouldgenerally lead to small kick velocity due to too short time for SASIto build up (see also, e.g., Podsiadlowski et al. 2004; Bogomazov,Lipunov & Tutukov 2007).

4 SU M M A RY A N D D I S C U S S I O N

We have performed both stellar evolution simulations of bare COcores and explosion simulations for the end product of the CO coresfor modelling ultra-stripped Type Ic SNe. We have found that allCO cores with mass from 1.45 to 2.0 M⊙ resulted in explosionwith energy of O(1050) erg, which left NSs with gravitational massfrom ∼1.24 to 1.44 M⊙ and ejecta from ∼0.1 to 0.4 M⊙ withsynthesized 56Ni of O(10−2) M⊙. These values are compatible withobservations of ultra-stripped SN candidates (Drout et al. 2013;Tauris et al. 2013, 2015). For SN 2005ek, Mej ≈ 0.2–0.7 M⊙ andMNi ≈ 0.02–0.05 M⊙ are appropriate to fit its light curve. Theevent rate of these SNe is estimated as ∼1 per cent of core-collapseSN rate (Drout et al. 2013, 2014), which is also compatible with anNS merger rate estimation (Abadie et al. 2010).

We took a different approach from previous studies on ultra-stripped SNe (Tauris et al. 2013, 2015). In previous works, theyself-consistently performed stellar evolutionary simulations untiloxygen burning phase with self-consistent mass-loss driven by windbut explosion calculations were based on phenomenological mod-elling with three free parameters: kinetic energy of SN, Ni mass,and mass cut (i.e. NS mass). Based on this model, they found thatultra-stripped SN model could account for the light curve of SN2005ek quite well. In our work, on the other hand, we performedstellar evolutionary simulations until the last phase of evolution, i.e.iron core collapse, but for initially bare CO cores without mass-loss.For the explosion phase, we performed neutrino-radiation hydrody-namics simulations to calculate explosion energy, Ni mass, and NSbaryon mass in self-consistent manner. In this sense, this work is

Table 2. Summary of simulation results.

Model tfinala Rsh

b Eexpc MNS, baryon

d MNS, grave Mej

f MNig vkick

h

(ms) (km) (B) (M⊙) (M⊙) (10−1 M⊙) (10−2 M⊙) (km s−1)

CO145 491 4220 0.177 1.35 1.24 0.973 3.54 3.20CO15 584 4640 0.153 1.36 1.24 1.36 3.39 75.1CO16 578 3430 0.124 1.42 1.29 1.76 2.90 47.6CO18 784 2230 0.120 1.49 1.35 3.07 2.56 36.7CO20i 959 1050 0.0524 1.60 1.44 3.95 0.782 10.5

Notes. aThe final time of simulations measured by post-bounce time.bThe angle-averaged shock radius at tfinal.cThe explosion energy in units of B (=1051 erg) at tfinal, which is still increasing.dThe baryonic mass of NS at tfinal.eThe gravitational mass of NS computed by equation (2) at tfinal.fThe ejecta mass at tfinal.gThe Ni mass at tfinal.hThe kick velocity at tfinal.iNote that this model is marginally exploding.

MNRAS 454, 3073–3081 (2015)

at MPI A

strophysics on October 19, 2015

http://mnras.oxfordjournals.org/

Dow

nloaded from

Nucleosynthesis of ultra-stripped SNe 4279

Figure 4. Abundance ratios of elements in the ejecta of ultra-stripped TypeIc SNe to the solar abundance. The red and black lines denote the ratiosof the CO145 and CO15 models, respectively. The dashed lines denote themaximum ratios and the ratios of the 10 per cent of the maximum ratios.

Figure 5. Abundance ratios of isotopes in the ejecta of the (a) CO145 and(b) CO15 model to the solar abundance. The red and black lines correspondto odd-Z and even-Z nuclei. The green-hatched region denotes the range ofthe abundance ratio between the maximum and one-tenth of that.

is ignored, the peak luminosity is approximately halved. The mainenergy source other than 56Ni and 56Co is 57Ni and 66Cu for theCO145 and the CO15 model, respectively. The decay time of theluminosity from these elements is about 4 and 8 d in the CO145 andthe CO15 model, respectively. The difference of the contribution

-17

-16

-15

-14

-13

-12

-11 0 5 10 15 20 25 30 35 40

1040

1041

1042

Abso

lute M

agnit

ude Luminosity [erg s -1]

Days since explosion

CO145CO15

CO145: Ye-BSN2005ek

Figure 6. Light-curves of the ultra-stripped Type Ic SNe. The red and bluecurves denote the CO145 and CO15 models, respectively. The circles denotethe light-curve of SN 2005ek. The orange dashed line denotes the CO145Ye-B model (see Section 4.2).

from these elements is mainly the result of the difference in the 56Niyield.

Recently, a variety of fast-decaying SNe have been found insurvey programs for transient objects. Subluminous SNe have alsobeen observed as Types Ia and Ib/c SNe (e.g. Foley et al. 2013;Drout et al. 2014). Some subluminous fast-decaying SNe could beultra-stripped SNe. These observed SNe showed spectral featuresdifferent from those of normal Types Ia and Ib/c SNe. The ejecta ofthe ultra-stripped SNe in our models indicate a higher abundanceratio of intermediate elements to oxygen compared with the casefor more massive CO cores. These compositional differences couldgive distinctive spectral features. The identification of ultra-strippedSNe from Type I SNe is important for the evaluation of the ultra-stripped SN rate. Future observations of ultra-stripped SNe couldconstrain the rates of ultra-stripped SNe.

We note, as pointed out in Suwa et al. (2015), that it is safe toconsider that our results give a lower limit of the explosion energyof an ultra-stripped SN. In the case of a stronger explosion of anultra-stripped SN, the ejected 56Ni mass could be larger. If so, ultra-stripped SNe could be observed as fast-decaying SNe such as theType Ic SN 2005ek. We also note that the 56Ni mass would beunderestimated because of the missing proton-rich component inthe neutrino-irradiated ejecta. This will be discussed in Section 4.2.

4.2 Uncertainties of the yield of light trans-iron elements inultra-stripped SN models

We obtained light trans-iron elements in the ultra-stripped SN mod-els. However, the production efficiency of the elements dependson the Ye distribution of the SN ejecta, which in turn depends onthe detailed treatment of neutrino transport. Indeed, Muller (2016)showed that an approximate treatment of neutrino transport intro-duces a broader Ye distribution than in a more stringent modelincluding sophisticated microphysics. On the other hand, an updateof the code can even lead to a smaller Ye distribution. For instance,an ECSN simulation performed by the Garching group with an up-dated code showed a smaller minimum value of Ye (0.34) than theprevious result (0.404; Wanajo et al. 2011) (Janka 2016, private

MNRAS 471, 4275–4285 (2017)

MC/O=1.45M⊙

MC/O=2.0M⊙

lower MC/O


When does a core collapse?

11

Time till collapse

Mass

MChandra

Mcore

collapse!


Modified Chandrasekhar mass

Chandrasekhar mass without temperature correction

Chandrasekhar mass with temperature correction

To make a small core, low Ye and low entropy are necessary

12

Baron+ 1990; Timmes+ 1996

MCh0(Ye) = 1.46M⊙ ( Ye

0.5 )2

MCh(T ) = MCh0(Ye)[1 + ( se

πY2 )2

] se = 0.5ρ−1/310 (Ye/0.42)2/3TMeV


Mch vs. Mcore

13

0.6

0.8

1

1.2

1.4

1.6

1.8

2

10-710-610-510-410-310-210-1100101102

Mas

s [M

⊙]

Time until ρc=1010g cm-3 [year]

CO138CO14515M⊙ 0.6

0.8

1

1.2

1.4

1.6

1.8

2

10-710-610-510-410-310-210-1100101102

Mas

s [M

⊙]


CO138CO14515M⊙ 0.6

0.8

1

1.2

1.4

1.6

1.8

2

10-710-610-510-410-310-210-1100101102

Mas

s [M

⊙]


CO138CO14515M⊙

MCh0

MCh

MFe

[Suwa, Yoshida, Shibata, Umeda, Takahashi, MNRAS, 481, 3305 (2018)]


Explosion simulations and NS masses

14

Model MCO (M⊙) MZAMS (M⊙) MFe (M⊙) MNS,b (M⊙) MNS,g (M⊙)

CO137 1.37 9.35 1.280 1.289 1.174

CO138 1.38 9.4 1.274 1.296 1.179

CO139 1.39 9.45 1.258 1.302 1.184

CO140 1.4 9.5 1.296 1.298 1.181

CO142 1.42 9.6 1.265 1.287 1.172

CO144 1.44 9.7 1.234 1.319 1.198

CO145 1.45 9.75 1.277 1.376 1.245

MNS,b-MNS,g=0.084M⊙(MNS,g/M⊙)2

(Lattimer & Prakash 2001)



Discussion

15

MNS,g (M⊙)

MCO (M⊙)~1.37 ~1.42

~1.20

~1.17

ONeMg core -> electron-capture SN

Fe core -> core-collapse SN

MNS,b (M⊙)

~1.28

~1.32



Summary

A low-mass NS of MNS,g=1.174M⊙ was found

Q: Is it possible to make such a low-mass NS with standard modeling of SN?

A: Yes, it is. The minimum mass is ~1.17M⊙. If a new observation finds even lower mass NS, we cannot make it. Something wrong.

16

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