Asymmetrical mass ejection from proto-white dwarfs and the formation of eccentric millisecond pulsar binaries
aa r X i v : . [ a s t r o - ph . H E ] J a n Draft version February 1, 2021
Typeset using L A TEX default style in AASTeX62
Asymmetrical mass ejection from proto-white dwarfs and the formation of eccentric millisecond pulsar binaries
Qin Han and Xiang-Dong Li
1, 2 Department of Astronomy, Nanjing University, Nanjing 210023, China; [email protected] Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Ministry of Education, Nanjing 210023, China
Submitted to ApJ
ABSTRACTBinary millisecond pulsars (MSPs) are believed to have descended from low-mass X-ray binaries(LMXBs), which have experienced substantial mass transfer and tidal circularization. Therefore,they should have very circular orbits. However, the discovery of several eccentric binary MSPs (witheccentricity e ∼ . − .
1) challenges this standard picture. Three models have been proposed thusfar based on accretion-induced collapse of massive white dwarfs (WDs), neutron star-strange startransition, and formation of circumbinary disks. All of them are subject to various uncertainties, andare not entirely consistent with observations. Here we propose an alternative model taking into accountthe influence of thermonuclear flashes on proto-WDs. We assume that the flashes lead to asymmetricalmass ejection, which imparts a mild kick on the proto-WDs. By simulating orbital changes of binaryMSPs with multiple shell flashes, we show that it is possible to reproduce the observed eccentricities,provided that the kick velocities are around a few kms − . Keywords: pulsars: general - pulsars: evolution - white dwarfs INTRODUCTION
Millisecond pulsars (MSPs) are radio pulsars with spin periods shorter than 20 ms and period deriva-tive ˙ P less than 10 − ss − . The majority of them are in binaries. In the standard formation theory,they are neutron stars (NSs) that have been recycled by accretion from their companion stars inthe previous low-mass X-ray binary (LMXB) phase (Alpar et al. 1982; Radhakrishnan & Srinivasan1982). The companion star finally evolves to be a white dwarf (WD) or a main-sequence (MS) star ofvery low mass (Bhattacharya & van den Heuvel 1991). For LMXBs with initial orbital periods longerthan the so-called bifurcation period (Pylyser & Savonije 1988), mass transfer causes the orbit to ex-pand, and there exists a relationship between the mass of the companion and the orbital period atthe end of the mass transfer (Phinney & Kulkarni 1994; Rappaport et al. 1995; Tauris & Savonije1999, hereafter TS99). Meanwhile, tidal torques should bring the binary eccentricities down to e ∼ − − − (Phinney 1992).However, several eccentric MSPs (eMSPs) were discovered in recent years, including PSRJ1903+0327, which is a 2.15 ms MSP in a 95 d eccentric (with e = 0 .
44) orbit around a 1 . M ⊙ dwarfcompanion star (Barr et al. 2013; Freire et al. 2011). The other eMSPs (with e ∼ . − .
1) form agroup with similar orbital periods ( P orb ∼ −
30 d) and (possible) helium WD (He WD) companionsof mass M WD ∼ . − . M ⊙ (Bailes 2010; Barr et al. 2013; Deneva et al. 2013; Camilo et al. 2015;Knispel et al. 2015). In addition, this group of binaries seem to obey the WD mass - orbital periodrelation mentioned above. Their locations in the Galaxy rule out a dynamical origin in a dense stellarenvironment, so a natural presumption arises that some internal process(es) in the binaries followingthe mass transfer phase have induced the observed eccentricities. There are three existing models regarding the origin of the eccentricities. Firstly, Freire & Tauris(2014) proposed a model based on the accretion-induced collapse of super-Chanderasekhar mass WDs.If the collapse is delayed because of the rapid rotation of the WD, the instant loss of gravitationalenergy during the collapse imposes an eccentricity on the detached binary. They predicted a massrange of 1 . − . M ⊙ for the MSPs along with small peculiar space velocities of the binaries( ∼
10 km s − ). But the predicted MSP mass and space velocity seem to be in tension with currentobservations of some eMSPs (see e.g., Antoniadis et al. 2016).Following the similar idea, Jiang et al. (2015) suggested that this sudden loss of mass could happenin the transition from a rapidly rotating NS into a strange star after the mass transfer. Their modelconstrains the masses of the WD companions to be 0 . − . M ⊙ . The newborn MSP should bemassive enough ( & . M ⊙ ) to trigger a phase transition, but the mass of the strange star can lie ina wide range. The uncertainties in this model stem from the undetermined physics of compact stars,including the equation of state and how much binding energy is lost during the phase transition.Antoniadis (2014) proposed a model from a different point of view, based on the possible formationof a circumbinary (CB) disk after the mass transfer. When the detached companion, a proto-WD ,experiences thermonuclear flashes due to unstable CNO shell burning, the shell expands and re-fills the Roche-lobe (RL). The rate of the resultant mass transfer typically exceeds the Eddingtonaccretion rate of the NS, so part of the transferred matter may leave from the second Lagrangian pointto feed a CB disk. Tidal torques exerted by the CB disk on the binary then excite the orbit to becomeeccentric, if the disk is sufficiently massive and long-lived. This model suffers from uncertainties in theinteraction between the CB disk and the inner binary, and the growth in eccentricities is dependenton the assumptions regarding the magnitude of the viscosity in the disk and possible re-accretionfrom the disk (Rafikov 2016).In this paper, we suggest that the shell flashes on the proto-WDs themselves may considerablyinfluence the orbital evolution by driving asymmetrical mass-loss. We describe the link between theflashes of proto-WDs and the possible kicks imparted on the WDs in Sec. 2. We then introduce thedetails of the kick model and present the calculated results on the related orbital characteristics inSec. 3. The uncertainties in our model are also discussed. Finally we summarize in Sec. 4. ASYMMETRICAL MASS-EJECTION DURING SHELL FLASHES ON PROTO-WDS AND THERESULTANT KICKS
We describe the conventional picture of shell flashes in MSP-WD binarie in Sec 2.1, and evaluatethe orbital changes they may introduce in MSP-WD binaries in Sec 2.2.
Shell flashes
There are numerous studies using 1D stellar evolution code to simulate the evolution of LMXBsand the descendent MSP-WD systems. These studies show that the proto-WDs are able to retain athin shell of hydrogen (H) outside the core. Moreover, it has been shown that proto-WDs within acertain mass range experience shell flashes due to unstable CNO burning (Driebe et al. 1998). Themass interval given in the literature varies from ∼ . − . M ⊙ to ∼ . − . M ⊙ (Sarna et al. 2000;Nelson et al. 2004; Althaus et al. 2001, 2013; Istrate et al. 2014, 2016). In the previous simulationstudies, it was implicitly assumed that part of the thermonuclear energy generated in the flashes isconsumed during the spherically symmetrical expansion of the shell. The subsequent RL-overflowepisode ensues when the size of the shell exceeds the WD’s RL and the matter outside the RL is We use this term for the detached companions formed after the LMXB phase and prior to the WD cooling stage. For a more precisedefinition of proto-WD, the readers may refer to, for example, Webbink (1975).
D kicks cause the eccentricities of MSP-He WD binaries artificially stripped away. This mass-loss process is considered to exert no direct influence on theorbital eccentricity.However, the mass-loss processes may be much more complicated than the predictions by the1D stellar simulation studies. And it will be shown that asymmetrical mass-ejection events mayconsiderably influence the binary orbit. Asymmetrical mass ejection
This issue was addressed in a series of studies by Schaefer and his coworkers (Schaefer et al. 2019;Schaefer 2020a,b). By analyzing the archival data of nova systems, Schaefer (2020b) showed thatfive out of the six examined systems experienced decrease in the orbital periods (with | ∆ P orb /P orb | ∼ − ) after a classical nova outburst. This is contradictory to the conventional understanding thatnova systems should experience orbital expansion as a consequence of spherically symmetrical massloss from the surface of the WDs. The authors therefore investigated the case of asymmetrical massloss and demonstrated that it can account for the decrease in the orbital periods .Apart from causing the orbital period changes, asymmetrical mass loss can also introduce orbitaleccentricities. If part of the nuclear energy of shell flash(es) is used to accelerate and eject matterasymmetrically, a mild kick could be imparted on the proto-WD due to momentum conservation(we refer to this kick as the WD-kick hereafter). Since orbital circularization in detached NS-WDbinaries usually takes an extremely long time (Antoniadis 2014), the kick-induced eccentricity canthen be preserved.Based on this idea, we examine the influence of the WD-kicks on the orbital characteristics ofMSP-WD binaries. We quantify the physical processes during a thermonuclear shell flash with asmall amount of instantaneous mass-loss ∆ M along with a WD-kick velocity V k . We first show thatthe energy E nuc generated during a shell flash is sufficient to accelerate at least part of the shell toa high velocity. The amount of H burned during the flash is typically ∼ − . M ⊙ , approximatelythe mass increase in the He core mass (we use the characteristic values in Sarna et al. (2000) for thephysical quantities of proto-WDs). This gives E nuc ∼ × erg. This energy can accelerate theentire shell (of mass ∼ − M ⊙ ) to a velocity of 5 × kms − and potentially lead to a sufficientlylarge WD-kick. Therefore the kick velocity is not severely constrained by energy conservation.We then estimate the WD-kick velocity V k via momentum conservation M WD V k = ∆ M V ej , where V ej is the velocity of the ejecta. It is noted that ∆ M should be smaller than the mass ( ∼ × − − − M ⊙ ) of the shell when the flash occurs (Sarna et al. 2000; Istrate et al. 2016). If massejection takes a similar way during thermonuclear flashes on WDs, we can approximate V ej withthe velocity of ejecta in nova systems. Observations show that the ejected shells of novae havea characteristic velocity ∼ kms − (Della Valle & Izzo 2020), and the maximum velocities of theejecta in nova systems are ∼ × kms − (Bode & Evans 2008), comparable to the escape velocitiesof the WDs. Assuming that around 10% of the material in the shell is ejected with the escape velocityin the opposite direction of the resultant WD-kick, and that the rest of the ejecta has a negligiblecontribution to the WD-kick, we can approximate the maximum kick velocity as follows, V k ≃ − (cid:18) ∆ M − M ⊙ (cid:19) (cid:18) V ej . × kms − (cid:19) (cid:18) . M ⊙ M WD (cid:19) . (1)In Eq. (1), ∆ M and V ej are also dependent on M WD , but the overall change in V k is small. To achievea larger kick velocity would require a larger fraction of the WD’s shell to be ejected in a collimatedform. Schaefer et al. (2019) considered the case of forward asymmetry (along the binary orbit) and showed that an asymmetry parameter of ǫ ≈ .
38 with an ejected velocity of v ≈ − is needed to account for the orbital changes of the Nova system QZ Aur. They alsopointed out that this degree of asymmetry can be easily achieved in Nova systems. THE ORBITAL PROPERTIES OF THE POST-KICK BINARY POPULATIONS
We describe the influence of the WD-kicks and analyze the properties of the post-kick binarypopulations in this section.We consider the binaries consisting of an MSP and a proto-WD. Since their orbits are expected tobe circular ( e < − ), we set the initial eccentricities to be zero for simplicity. We take the logarithmof the initial binary orbit periods log( P orb / d) to be evenly distributed in the range of [ − , M WD of the proto-WD is calculated from P orb by using the M WD − P orb relation with Solarmetallicity in TS99, and the mass of the NS is taken to be M NS = 1 . M ⊙ . The initial orbital period P orb is, therefore, the only parameter specifying the initial conditions of an individual binary underthese assumptions. We draw a sample of 10 such binaries as the initial population.The orbital changes in our WD-kick model are computed as follows. The changes in the semi-majoraxis a and the eccentricity e are governed by the following equations, aa = [ 1 − (∆ M/M )1 − ∆ M/M − ( V k /V ) − V k /V ) cos θ ] , (2)and 1 − e = (cid:18) M M − ∆ M (cid:19) (cid:16) a a (cid:17) , (3)where the quantities with and without subscript 0 correspond to the pre-kick and post-kick binaries,respectively; M = M WD + M NS is the mass of the binary, V = ( GM /a ) / is the orbital velocity,and θ is the angle between the direction vectors of V k and V .We define two dimensionless factors, f m = ∆ MM , and f k = V k V . Using the TS99 relation we can parameterise f k with V k and M WD , f k = ( V k .
32 kms − )( M WD M ⊙ − . / . (4)Then the eccentricity of the post-kick orbit can be expressed as, e = p f − f m + f + 2 f k cos θ − f m . (5)We use a subroutine in the binary population synthesis (BPS) code developed by Hurley et al.(2002) to compute the orbital evolution. Eccentricities caused by single kick
We start by imparting one kick on the proto-WDs and investigate the orbital properties of the post-kick population. We tentatively assume that the magnitude of the kick velocity follows a Maxwelldistribution specified with a parameter σ k . We then construct three scenarios with the same amountof (asymmetric) mass-loss ∆ M = 10 − M ⊙ , but with different values of σ k , i.e., σ k = 1 ,
2, and3 kms − . In the following we use the properties of the kick(s) to name the post-kick populations,such as K σ , K σ and K σ . Here the subscripts of K and σ denote the number of kicks and themagnitude of σ (in units of kms − ), respectively. D kicks cause the eccentricities of MSP-He WD binaries The eccentricities of the post-kick systems are plotted against the masses of the WDs in Fig. 1.The left, middle, and right panels correspond to populations K σ , K σ and K σ , respectively. Thecolored dots mark the simulated post-kick systems. The magnitude of the kick velocity can be readfrom the right color bar. We plot the five eccentric MSPs in green dots with error bars, which coverthe 90% probability mass range for randomly oriented orbits (the measured and derived parametersfor the five binaries are listed in Table 1). The black dotted line represents the eccentricity causedby 10 − M ⊙ mass loss without any kick.From Fig. 1 we can see that the kick can substantially affect the distribution of the eccentricity.While population K σ covers the parameter space where eMSPs occupy, populations K σ and K σ with relatively smaller kick velocities can only reproduce some of the observed eMSPs. In all thethree panels, there is a trend that the magnitude of the induced eccentricities increases with theorbital period. This is because with the same V k , f k is larger for systems with larger M WD (seeEq. [4]) or longer P orb .However, we need to point out that, to reproduce the eccentricities of ∼ . P orb ∼
30 d, the required kick velocities are ∼ . − . − . These values are close to oreven exceed the maximum achievable kick velocity we calculated above. So it is probably difficult toattribute the origin of the observed eccentricities to single shell flash event. Eccentricities caused by multiple kicks
Multiple flashes have been often observed in the theoretical studies of the evolution of proto-WDs. The majority of proto-WDs are expected to experience less than 5 kicks (Sarna et al. 2000;Nelson et al. 2004; Althaus et al. 2001, 2013), and the maximum number of kicks is 26 in Istrate et al.(2016). We therefore conduct simulations with multiple kicks to investigate whether they can repro-duce the observed eccentricities and to constrain the magnitude of the kick velocity.We start from one up to 15 kicks, and plot the results with 1, 2, 3, 5, 10, 15 kicks on the e − M WD plane in panels A to E of Fig. 2, respectively. To avoid the total ejecta mass getting too large, weset ∆ M = 1 × − M ⊙ and σ k = 1 kms − in each flash. The colors of the scattered points mark themagnitude of the latest kick velocity, that is, the 1st, 2nd, 3rd, 5th, 10th and 15th kick velocity forpanels A to F, respectively. We can see from Fig. 2 that the K σ and K σ populations cover theparameter space where the observed eMSPs occupy. Therefore, we need approximately at least 10kicks to reproduce the eMSPs.In order to compare the influence of the kick numbers in more detail, we plot the number densitydistributions of populations K σ and K σ on the P orb − e plane in the left panels (A and Crespectively) of Fig. 3. In both cases we have similar number density distributions. Then, weselect binaries with eccentricities larger than a threshold value, e crit , and plot the cumulative numberdistributions in the right panels (B and D respectively) for the two populations (the distributions ofbinaries from populations K σ and K σ are also plotted in panel D). The dashed and solid linescorrespond to e crit = 0 .
01 and 0.1, respectively. Comparing the lines with N k = 5, 10, and 15, wecan see that the e distributions become more and more similar when N k increases.To further examine this trend, we display the e distributions of populations K i σ (where i rangesfrom 1 to 15) in Fig. 4. They are plotted with different colors which represent the number of kicks.The vertical lines in the same color as the distribution curve represents its mean value, and the twoblack vertical lines represent e = 0 .
01 and 0.1. We also see that the distributions become moreand more similar from the top panel to the bottom panel. In order to check whether they have(statistically) evolved into a “saturated” state, we conduct two sample Kolmogorov-Smirnov (KS)and Anderson-Darling (AD) tests of the e distributions for populations K i σ and K i +1 σ ( i =1 to 14).The null hypothesis is that the eccentricities of populations K i σ and K i +1 σ come from the same distribution function. For the two sample KS test, we choose the critical KS value as S ks 2samp = 0 . P value as P ks 2samp = 0 .
01; and for the AD test, we choose a significance level of 5%. Bothtests show that the null hypothesis cannot be rejected when the number of kicks is larger than 11,that is, the e distribution has reached a “saturated” state after 12 kicks.We also adopt other values of σ k to examine its influence. When σ k = 0 .
75 kms − , approximately10 kicks can still reproduce the e distribution of the eMSPs; when σ k = 0 .
50 kms − , approximately15 kicks are needed for the observed eMSPs. Comparison with the MSP-He WD binaries
In this subsection, we compare the simulated post-kick populations with the entire population ofthe observed MSP-He WD binaries. Since the orbital periods are more accurately determined thanthe WD masses, we choose to compare our results with the observations on the P orb − e plane.As shown in Figs. 1 and 2, populations K σ and K σ contain eMSPs similar to the observedones, so we first compare them with the observed MSP-He WD binaries in panels A and C of Fig. 5,respectively. One can see that the distributions of the post-kick populations clearly deviate from thatof the observed MSP-He WD binaries. The small group of eMSPs are the only ones that overlapwith our simulated populations. This discrepancy suggests that if the kick scenario applies, a certainmass interval for the WD-kick is required.We therefore add a mass interval [ M min , M max ] for shell flashes to our model. Currently, there isno consensus on the exact values of M min and M max , so we arbitrarily set them to be 0.268 M ⊙ and 0.281 M ⊙ , which are converted from the minimum and maximum P orb (22 and 32 d) of knowneMSPs. We then redo the simulation assuming that kick(s) are only imparted on the proto-WDswith masses within that interval, and other proto-WDs only experience spherically ejected mass loss.The simulated K σ and K σ populations are plotted in panels B and D of Fig. 5, respectively.Comparing panels B and D with panels A and C, we can see that the mass interval is translated intoan “orbital interval” due to the M WD − P orb relation, producing a distinct group with relatively higheccentricities on the P orb − e plane. Note that the kick(s) also cause the orbital period to slightlychange, so the range of the orbital periods of eMSPs (with e & − ) are somewhat broadened to be[18 ,
45] d and [19 ,
39] d for populations K σ and K σ , respectively.Therefore, the mass interval is of great importance for making predictions. However, the conditionsfor establishing shell flashes are still uncertain, which closely depend on the mass retained in theshell at the moment of detachment and its chemical structure. So it is not surprising that differentcombinations of the related physical ingredients in the recipes of stellar models, as well as the initialconditions such as metallicity and the WD mass, lead to different boundary values of the mass interval(e.g. Istrate et al. 2014).Our simulated population does not match the observations of normal MSP-He WD binaries. Thatis not unexpected, since we have adopted an initial zero eccentricity for simplicity while the actualeccentricities of MSP-He WD binaries should be largely determined by the tidal interactions betweenthe NS and its low-mass companion (Phinney 1992). In panel B of Fig. 5, the simulated eccentricitiesare distributed within a strip because there is only one mass loss event with fixed amount of the ejectamass, while in real situations, the ejecta mass is likely to be diverse and dependent on the mass of theproto-WD. In comparison, the simulated eccentricities are distributed over a wider range in panel D,suggesting that multiple mass-loss events could also be partly responsible for the small eccentricitiesof normal MSP-He WD binaries.To investigate the influence of the ejecta mass and the kick in some detail, we conduct simulationstaking into account other distribution laws for these two factors. We assume that all proto-WDsexperience WD-flashes, and that both the ejecta mass and the magnitude of the kick velocity fol- D kicks cause the eccentricities of MSP-He WD binaries low a power-law distribution. We define a factor f a to quantify the asymmetry in the mass-lossprocesses during shell flashes, and use this factor to parameterize the kick imparted on the proto-WD. Specifically, we let log f a to be randomly chosen from a flat distribution between − M/M ⊙ ) = − . f a andlog( V k / kms − ) = log(8) + log f a , respectively. The simulated post-kick populations are plotted inFig. 6. The left panel shows the number distribution of the simulated population, and the right panelcompares the simulated and observed distributions. We can see that if the degree of asymmetry inthe mass loss during a shell flash can vary over several orders of magnitude, the distribution of theinduced eccentricities is able to cover the entire eccentricity range of the observed systems. SUMMARY
To summarize, we propose a model based on the shell flashes of proto-WDs to explain the abnor-mal eccentricities of some MSP-He WD binaries. Assuming that thermonuclear shell flashes powerasymmetrical mass loss and result in mild kicks on the proto-WDs, we simulate the eccentricity dis-tribution of the post-kick populations under various situations. Our results show that it is possible toaccount for the eccentricities of eMSPs, if reasonable choices of mass loss and kick velocities duringshell flashes are adopted. On one hand, if the occurrences of shell flashes are distinct for proto-WDsof mass within a specific interval, then we would expect the orbital periods of eMSP binaries to bedistributed within a particular range, as the current sample have shown. If, on the other hand, themass interval is quite large, then we would expect to find eMSP binaries over a wide orbital periodrange in the future.We are grateful to the referee for valuable comments. We also thank Prof. Kinwah Wu for helpfulsuggestions. This work was supported by the National Key Research and Development Programof China (2016YFA0400803), the Natural Science Foundation of China under grant No. 11773015,10241301, and Project U1838201 supported by NSFC and CAS.
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D kicks cause the eccentricities of MSP-He WD binaries WD (M ⊙ )10 −4 −3 −2 −1 e σ k =1⊙km s −1 simulated⊙systemsobserved⊙systems 0.20 0.25 0.30 0.35 0.40 0.45M WD (M ⊙ )σ k =2⊙km s −1 WD (M ⊙ )σ k =3⊙km s −1 l o g ( V k ( k m s − )) Figure 1.
Simulated distributions of MSP-He WD binaries on the e − M WD plane. We assume that the proto-WD was impartedwith a single kick and use different colors to indicate the magnitude of the kick velocity V k . The amount of mass loss is takento be 10 − M ⊙ . In the left, middle, and right panels the Maxwellian distribution parameter σ k equals to 1, 2, and 3 kms − ,respectively. The black dotted line represent the results with no kick considered. The green dots with error bars represent thefive eMSPs. e A: 1 kicksimulated s stemsobserved s stems B: 2 kicks C: 3 kicks0.20 0.25 0.30 0.35 0.40 0.45M WD (M ⊙ ⊙0.000.050.100.150.200.250.30 e D: 5 kicks 0.20 0.25 0.30 0.35 0.40 0.45M WD (M ⊙ ⊙E: 10 kicks 0.20 0.25 0.30 0.35 0.40 0.45M WD (M ⊙ ⊙F: 15 kicks 1234 V k ( k m s − ⊙ Figure 2.
Same as Fig. 1 but for multiple kicks. Panel A - E corresponds to populations that have experienced 1, 2, 3, 5, 10,and 15 kicks with σ k = 1 kms − , respectively. −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0−0.50.00.51.01.52.02.53.0 l o g ( P o r b ( d )) A −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0log(e)−0.50.00.51.01.52.02.53.0 l o g ( P o r b ( d )) C −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.00100200300400500 N ( σ k = k m s − ) B e>0.01e>0.1−1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0log(P orb (d))0100200300400500600700 N ( σ k = k m s − ) D e>0.01e>0.15 kicks10 kicks15 kicks0.00.51.01.52.02.5 log(N) log(N) Figure 3.
The number distributions of post-kick populations on the P orb − e plane (left panels) and the cumulative numberdistribution of eMSP binaries (right panel). The upper and lower panels correspond to populations K σ and K σ , respectively. D kicks cause the eccentricities of MSP-He WD binaries Figure 4.
The Probability Distribution Function (PDF) of the eccentricities after 1-15 kicks. The vertical lines with the samecolor as the PDF lines represent the mean value of the specific population, and the two black dashed vertical lines mark e = 0 . . P o r b ( d ) A simulated systems, 1 kick σ k =3 km s −1 observed systemsobserved eccentric systems B −6 −5 −4 −3 −2 −1 e10 P o r b ( d ) C sim lated systems, 15 kicks σ k =1 km s −1 observed systemsobserved eccentric systems 10 −6 −5 −4 −3 −2 −1 e D V k ( k m s − ) V k ( k m s − ) Figure 5.
Distributions of the simulated population on the P orb − e plane, overlaid with the observed MSP-He WD binaries.The MSP data are taken from Hui et al. (2018) and the ATNF pulsar catalog (Manchester et al. 2005). The upper and lowerpanels correspond to populations K σ and K σ , respectively. In the left and right panels, all proto-WDs and only thosewithin the mass interval ([0 . , . M ⊙ ) are assumed to be subject to kicks, respectively. The grey dashed horizontal linesmark the “period interval” translated from the mass interval with the TS99 relation. −7 −6 −5 −4 −3 −2 −1log(e)−0.50.00.51.01.52.02.53.0 l o g ( P o r b ( d )) −7 −6 −5 −4 −3 −2 −1 e10 −1 P o r b ( d ) simulated systemsobserved systems0255075100125150175200 N −4−3−2−10 l o g ( V k ( k m s − )) Figure 6.
The left panel shows the number distribution of the simulated population on the P orb − e plane, and the right panelcompares the simulated and observed distributions on the P orb − e plane. We assume that all proto-WDs experience a kickduring the shell flash, and the ejecta mass and kick velocity obey a power-law distribution. D kicks cause the eccentricities of MSP-He WD binaries Table 1.
Physical parameters of the five observed eccentric MSP binariesPSR name J1950+0327 J2234+0611 J1946+3417 J1618 − − P orb (d) 22.2 32.0 22.2 22.7 24.6the WD mass M WD ( M ⊙ ) 0.28 0.30 0.31 0.20( M med ) 0.25( M med )the pulsar mass M PSR ( M ⊙ ) 1.50 1.38 1.78 − − eccentricity e P (ms) 4.30 3.58 3.17 12.0 2.00first derivative of P , ˙ P ((10 − ss − )) 1.88 1.20 0.314 5.41 − transverse velocity V T (km s − ) −
145 200 ±
60 80, 160 −−