Galactic neutron stars I. Space and velocity distributions in the disk and in the halo
aa r X i v : . [ a s t r o - ph . GA ] A ug Astronomy&Astrophysicsmanuscript no. ns˙dynamics˙I c (cid:13)
ESO 2018August 27, 2018
Galactic neutron stars
I. Space and velocity distributions in the disk and in the halo
N. Sartore , E. Ripamonti , , A. Treves and R. Turolla , Dipartimento di Fisica e Matematica, Universit`a dell’Insubria, via Valleggio 11, 22100, Como, Italy e-mail: [email protected] Dipartimento di Fisica, Universit`a di Milano Bicocca, Piazza delle Scienza 3, 20126, Milano, Italy Dipartimento di Fisica, Universit`a di Padova, via Marzolo 8, 35131, Padova, Italy Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UKReceived ...; accepted ...
ABSTRACT
Aims.
Neutron stars (NSs) produced in the Milky Way are supposedly ten to the eighth - ten to the ninth, of which only ∼ × areobserved. Constraining the phase space distribution of NSs may help to characterize the yet undetected population of stellar remnants. Methods.
We perform Monte Carlo simulations of NS orbits, under di ff erent assumptions concerning the Galactic potential and thedistribution of progenitors and birth velocities. We study the resulting phase space distributions, focusing on the statistical propertiesof the NS populations in the disk and in the solar neighbourhood. Results.
It is shown that ∼
80 percent of NSs are in bound orbits. The fraction of NSs located in a disk of radius 20 kpc and width0.4 kpc is .
20 percent. Therefore the majority of NSs populate the halo. Fits for the surface density of the disk, the distribution ofheights on the Galactic plane and the velocity distribution of the disk, are given. We also provide sky maps of the projected numberdensity in heliocentric Galactic coordinates ( l , b ). Our results are compared with previous ones reported in the literature. Conclusions.
Obvious applications of our modelling are in the revisiting of accretion luminosities of old isolated NSs, the issue ofthe observability of the nearest NS and the NS optical depth for microlensing events. These will be the scope of further studies.
Key words. stars: kinematics - stars: neutron - stars: statistics
1. Introduction
Neutron stars are born during core-collapse of massive ( M ≥ M ⊙ ) stars (type Ib, Ic and II supernovae, herafter SNe) or, lessfrequently, by accretion-induced collapse of white dwarves thatreached the Chandrasekhar’s limit (type Ia SNe). A rough es-timate of the total number of NSs generated in the Milky Way(MW) can be obtained from the present-day core-collapse SNrate, β NS , which is of the order of a few per century (Diehl et al.,2006) and assuming β NS constant during the lifetime of theGalaxy ( ∼
10 Gyr). Hence the total number of NSs created inthe MW lies between 10 and 10 . NSs may then represent a nonnegligible fraction of the Galactic stellar content.Up to now only ∼ × NSs have been observed, the ma-jority of which as isolated radio pulsars (PSRs) with ages farshorter ( .
100 Myr) than the MW lifetime. Older NSs havebeen detected only when recycled in binary systems by massand angular momentum transfer from a companion star, thus be-coming millisecond pulsars (see Lorimer 2008 and referencestherein). Isolated old NSs (ONSs) have not been identified sofar because, once their energy reservoir, both thermal and rota-tional is exhausted, they are pretty close to being invisible. Asa consequence little is known about their physical and statisticalproperties.On the other hand the expected phase-space distributionof ONSs can be constrained by means of population synthe-sis models once a realistic set of initial conditions is given.Population synthesis studies of Galactic NS have been per-formed by many authors in the past. Hartmann et al. (1990) and Packzynski (1990, hereafter H90 and P90 respectively) stud-ied the orbits of Galactic NSs, looking for a possible link withgamma-ray bursts. These studies di ff ered in their assumptions.In particular H90 assumed a Gaussian distribution centered at200 km s − for the distribution of NS birth velocities, while P90adopted the distribution of Lyne et al. (1982) which gives a dif-ferent (higher) weight to the low velocity tail of the distribution.NS orbits in the Galactic gravitational potential were alsoinvestigated by Blaes and Rajagopal (1991), Blaes and Madau(1993), Zane et al. (1995), Popov and Prokhorov (1998) andPopov et al. (2000, hereafter BR, BM, Z95, PP98 and P00 re-spectively) in order to constrain the number of nearby NSs ac-creting from the interstellar medium (ISM, see Treves et al. 2000and references therein). BR, BM and Z95 adopted initial condi-tions similar to those of P90 except the distribution of birth ve-locities which, following Narayan and Ostriker (1990), was as-sumed to be Maxwellian with a dispersion of 60 km s − .Popov et al. (2000) explored the observability of accret-ing ONSs for a wide range of initial mean velocities, be-tween 0 and 550 km s − , assuming a Maxwellian distribution.The paucity of observed accretors in the ROSAT catalogue(Neuhauser and Trumper 1999) led to the conclusion that NSsare born with average velocities of at least 200 km s − .This is confirmed by observations of known young NSs.PSRs show in fact spatial velocities of several hundreds km s − ,i.e. of the same order of the escape velocity from the MW (seee.g. Lorimer et al. 1997, Arzoumanian et al. 2002, Brisken et al. No sources have been positively identified so far. N. Sartore, E. Ripamonti, A. Treves and R. Turolla: Galactic neutron stars − . A striking example isPSR B1508 +
55: the proper motion and parallax measurementsobtained from radio observations points to a transverse velocityof ∼ − (Chatterjee et al. 2005).Similar high values of the velocity have been inferred alsofor objects belonging to other classes of isolated NSs. Thanks toChandra observations, Hui and Becker (2006) estimated a veloc-ity of ∼ − for the central compact object RX J0822-4300. Recently Motch et al. (2009) measured the proper motionof one of the ROSAT radio-quiet, thermally emitting NSs (theMagnificent Seven) and found a value of the 3D velocity of600 − − . This is not uncommon in PSRs and hencethey concluded that the velocity distribution of the MagnificentSeven is not statistically di ff erent from that of normal radio pul-sars.The origin of such high velocities is not at all clear.An asymmetric SN explosion is considered one possible ex-planation (e.g. Shklovskii 1970, Dewey and Cordes 1987).Also the e ff ects of binary disruption (e.g. Blaaw 1961,Iben and Tutukov 1996) may contribute to the observed ve-locities. Recently it has been proposed that the fastest NSsare the remnants of runaway progenitors expelled via N-body interactions from the dense core of young star clusters(Gvaramadze, Gualandris and Portegies Zwart, 2008).If all classes of isolated NSs share the same typical birth ve-locities, no matter how these are achieved, a large fraction ofthese objects can escape the potential well of the MW in a rela-tively short time. This fact has consequeces of all observable NSpopulations. In this paper we focus on the e ff ect of high birthvelocity and likely evaporation on the still elusive population ofONS. Constraining the expected phase space distribution is infact crucial to define suitable strategies for thier detection.Based on recent estimates of the birth velocity distribution,in this paper we reconsider the dynamics of isolated NSs. Weperform integration of stellar orbits using our new code PSYCO(Population SYnthesis of Compact Objects), developed for thispurpose. In Section 2 we describe the ingredients of the simu-lation, i.e. the gravitational potential of the Milky Way and thedistributions of progenitors and birth velocities. We present theresults of the simulation in Section 3. In particular we investi-gate the statistical properties of the NS population in the Galacticdisk and at the solar circle. We fit the surface density of the diskand the average height distribution and compute the surface andvolume densities in the solar vicinity. We fit also the velocitydistribution in the disk, both with respect to the Galactic centerand the local rest frame of the ISM. We discuss our results andtheir possible implications in Section 4. The results of this workwill constitute the base for further studies on the observability ofONSs.
2. METHOD
We follow an approach similar to P90. Initial conditions (posi-tion, velocity) are taken randomly from the selected distributionsand assigned to each synthetic NS by means of a Monte Carloprocedure.
The initial positions of NSs in the Galaxy are defined in a galac-tocentric cylindrical coordinates system ( R , φ, z ), where the z axis corresponds to the axis of rotation of the MW. These ini-tial positions reflect the distribution of NS progenitors: accord-ing to Bronfan et al. (2000, hereafter B00), formation of massivestars is currently concentrated in a annular region which followsthe distribution of molecular hydrogen. However, to explore thee ff ects of di ff erent initial conditions on the current phase-spaceconfiguration of NSs, we choose four possible radial distribu-tions of progenitors from the literature.P90 adopted an exponential probability distribution, basedof the observed surface brightness of face-on Sc galaxies(van der Kruit, 1987) p ( R ) dR = a R RR exp exp (cid:18) − RR exp (cid:19) dR , (1)where p ( R ) dR is the probability that a NS is born between R and R + dR , R exp = . a R = . ρ ( R ), has a Gaussian shaped rise until it reaches amaximum at ∼ . p ( R ) dR = R ρ ( R ) dR R ∞ R ρ ( R ) dR . (2)Another possible distribution of NS progenitors can beobtained from the surface density of Galactic SN remnants(Case and Battacharya, 1998, hereafter CB98) ρ ( R ) = (cid:18) RR (cid:19) α exp " − β ( R − R ) R , (3)where α = β = .
53 and R = . p ( R ) = √ πσ exp " − ( R − R peak ) σ , (4)where R peak = .
04 kpc and σ = .
83 kpc. This distributionhas been extrapolated from the observed PSR distribution foundby Yusifov and Kucuk (2004). Finally for all models we assumethat NSs can be born from 0 to 20 kpc.It is our opinion that the distribution proposed by P90, inspite of being obtained from observations of external galaxies,may better represent the long term star formation history of theMW. The other models are based on the present-day distributionof population I objects, which could have been rather di ff erent inpast epochs (see for example Chiappini et at. 2001). The modelsof B00, CB90 and F06 are probably better suited for populationstudies of young / middle-aged NSs (PSRs, magnetars etc.). In th J band.. Sartore, E. Ripamonti, A. Treves and R. Turolla: Galactic neutron stars 3
Fig. 1.
Normalized radial probability distribution of NSs pro-genitors. P90 (solid line), B00 (dotted), CB98 (dashed) and F06(dot-dashed).
Table 1.
Parameters of the spiral arms.
Arm k R ∗ φ [kpc] [radians]Norma 4.25 3.48 1.57Carina-Sagittarius 4.25 3.48 4.71Perseus 4.89 4.90 4.09Crux-Scutum 4.89 4.90 0.95 Massive stars are located in the spiral arms of the MW (they areindeed the ideal tracers of the spiral structure), thus we modelspiral arms in the distribution of NS progenitors adopting thesame prescription of F06, i.e., NS progenitors are distributedalong four logarithmic spirals, each spiral described by the equa-tion φ ( R ) = k ln( R / R ∗ ) + φ . (5)The values of the parameters k , R ∗ and φ for each spiral aregiven in Table 1. Actually, equation 5 describes the position ofarm centroids. A more realistic distribution can be obtained ifthe positions of progenitors are scattered, both in the radial andazimuthal directions, around these centroids (Fig. 2). Details onhow the scatter is added to the initial positions of NSs can befound in F06.The thickness of the starforming region is few tens of parsecs(B00, Maiz-Apellaniz 2001). However, as P90 and Sun and Han(2004) have pointed out, the long term dynamical behavior of aNS population is insensitive to the scale height of its progeni-tors (see also Kiel and Hurley 2009). Following their results weassume that all NSs are born on the Galactic plane ( z = The true form of the distribution of birth velocities is still a hotlydebated issue and few constrains exist (see F06 for an exhaustivediscussion). HP97, L97 and H05 proposed a Maxwellian distri-bution p ( v ) = r π v σ exp (cid:18) − v σ (cid:19) . (6) Fig. 2.
Initial positions of NSs - Radial distribution from P90.The position of the Sun is (8.5, 0.0).Alternatively, Fryer et al. (1998), Cordes and Cherno ff (1998),A02 and B03 proposed a bimodal distribution p ( v ) = r π v (cid:20) w σ exp (cid:18) − v σ (cid:19) + − w σ exp (cid:18) − v σ (cid:19)(cid:21) . (7)where w is the relative weight of the two components of the dis-tribution.Using the same PSR sample of B03, F06 explored, togetherwith Maxwellian and bimodal models, other possible distribu-tion functions like the double-sided exponential p ( v i ) = v exp exp (cid:18) − | v i | v exp (cid:19) , (8)where v i represents a single component of the velocity and v exp is a characteristic velocity; the Lorentzian p ( v i ) = π γ (cid:18) + (cid:16) v i /γ (cid:17)(cid:19) , (9)where γ is a scale parameter defining the half-width at half max-imum, and the distribution proposed by Lyne et al. (1982) andadopted by P90 p ( v ) = π v ∗ (cid:18) + (cid:16) v / v ∗ (cid:17) (cid:19) , (10)where again v represents the tridimensional velocity. F06 con-cluded that the Maxwellian model is less favored. On the otherhand they disfavour also the bimodal distribution and prefere in-stead single parameter models, pointing out that the bimodalityfound by other authors arises if thse alternative single parametermodels are not investigated.To explore the e ff ects of the birth velocities on the fi-nal phase-space distribution of NSs, we adopt the Maxwellianmodel of H05 as well as four of the models proposed by F06, N. Sartore, E. Ripamonti, A. Treves and R. Turolla: Galactic neutron stars
Fig. 3. Di ff erential velocity distributions obtained from sim-ulated velocity vectors. H05 (solid), F06DG (dotted), F06E(dashed), F06L (dot-dashed) and F06P (triple dot-dashed). Table 2.
Velocity distribution models.
Model Parameters < v > [km s − ]H05 σ =
265 km s − σ =
160 km s − σ =
780 km s − w = . v exp =
180 km s − γ =
100 km s − v ∗ =
560 km s − i.e. the bimodal, the double-sided exponential, the lorentzian andthat of P90. From here on we refer to these models as H05, F06B,F06E, F06L and F06P respectively.The value of the mean tridimensional velocity for each dis-tribution is calculated numerically from simulated velocity vec-tors (Table 2). All the velocity distributions refer to the LocalStandard of Rest (LSR) of NS progenitors. Thus, the true 3D ve-locity of a neutron star with respect to the Galactic ReferenceFrame (GRF) is the vector sum of the birth velocity and the cir-cular velocity at the birthplace, v = v birth + v circ . Once the initial conditions have been assigned, the motion ofNSs is described by the equation¨ r = −∇ Φ , (11)where r = r ( R , φ, z ) is the position of the NS and Φ is the grav-itational potential of the MW. We adopt the same 3-componentmodel of Smith et al. (2007, hereafter S07) Φ = Φ B + Φ D + Φ H , (12)where Φ B , Φ D and Φ H represent the bulge, disk and halo contri-butions respectively.The gravitational potential of the bulge is (Hernquist, 1990) Φ B = − GM B r + r B , (13)where M B = . × M ⊙ and r B = . r = √ R + z isthe distance from the GC. The disk potential has instead the following form(Miyamoto and Nagai, 1975) Φ D = − GM D s( R + (cid:20) R D + q z D + z (cid:21) ) , (14)where M D = × M ⊙ is the mass of the disk and the R D = z D = . Φ H = − π G ρ s r vir c r log (cid:18) + c rr vir (cid:19) , (15)where ρ s = ρ cr Ω δ th c ln(1 + c ) − c / (1 + c ) (16)is the characteristic density, c is the concentration parameter, r vir is the virial radius and ρ cr is the critical density of the Universe.The parameters of potential are the same of S07, except forthe concentration parameter c and the virial radius r vir (19.2 and274 kpc respectively), which were adjusted to match the IAUstandard values for the distance of the Sun from the Galacticcenter, R = . v circ ( R ) =
220 km s − , together with the escape velocity fromthe MW at the same distance, v esc ( R ) ≃
544 km s − (S07). Thecorresponding value of the virial mass, M vir , is ∼ M ⊙ .Very recently Reid et al. (2009) gave a new estimate of thecircular velocity, v circ ( R ) ≃
254 km s − with R = . ff ect of the enhanced mass of the Galaxyon NS orbits, we choose a further set of parameters for the po-tential: the masses of the bulge and disk are increased by a fac-tor (254 / , i.e. the ratio of the squared circular velocities inthe two cases. For the halo, the concentration parameter c re-mains the same while the virial radius r vir is in this case 332kpc, which yields an 80 percent increase of the virial mass, M vir ∼ . × M ⊙ .We note that in the model where v circ ( R ) =
254 km s − weget v esc =
664 km s − . This is higher than the central value (544km / s) estimated by S07; however, it is not far from their 90%upper limit (608 km s − ), especially when we consider that v esc was obtained by assuming v circ =
220 km s − , and that modify-ing such assumption introduces further uncertainty in its deter-mination (Smith, private communication). We calculate the orbits of 150000 NSs for each model (Table3), assuming that NSs are born at constant rate during the wholeMW lifetime (10 Gyr). Hence the age of each NS is selectedrandomly from an uniform distribution. The orbit of each NSis then calculated via numerical integration of the system ofequations (11), for a time corresponding to its assigned age. Theaxial symmetry of the potential implies conservation of angularmomentum with respect to the axis of rotation of the MW. Thisallows to reduce the number of equations in (11) to four . Sartore, E. Ripamonti, A. Treves and R. Turolla: Galactic neutron stars 5
Fig. 4.
Rotation curve for our Milky Way model (solid). Dotted,dashed and dot-dashed represent the bulge, disk and halo contri-butions respectively.
Fig. 5.
Escape velocity on the Galactic plane. The circular veloc-ity (dashed) is plotted for comparison dRdt = v R , dzdt = v z , dv R dt = ∂ Φ ∂ R + j z R , (17) dv z dt = ∂ Φ ∂ z , where j z is the angular momentum with respect to the z axis. Integration of equations (17) is performed with a 4th or-der Runge-Kutta algorithm (e.g. Press et al. 1992) with cus-tomized adaptive stepsize. The relative accuracy of integrationsis kept below 10 − using the energy integral E as reference, i.e.( δ E / E ) ≤ − , where E = v + Φ ( r ) . (18)To limit the computation time and avoid lockups of thecode, all NSs reaching 0 . The CPU time for a typical run is about 1 day.
Table 3.
Models for initial conditions. (*) denotes models withupdated potential.
Birth velocity distr.Spatial H05 F06B F06E F06L F06Pdistr.P90 1A 1B 1C 1D 1EB00 2A 2B 2C 2D 2ECB98 3A 3B 3C 3D 3EF06 4A 4B 4C 4D 4EP90 1A* 1B* 1C* 1D* 1E*
Fig. 6.
Surface density of NSs in the disk, obtained from best fitparameters ( N star = ). Models 1B (solid line), 2B (dotted),3B (dashed) and 4B (dot-dashed).
3. RESULTS
Our calculations show that the statistical properties of NSs area ff ected mostly by the distribution of birth velocities while thee ff ects of di ff erent distributions of progenitors are less promi-nent. For this reason we focus on results of models 1A to 1E, i.e.with the distribution of progenitors of P90. Results of modelsdi ff ering only for the distribution of progenitors are quite sim-ilar, the only substantial di ff erence is the shape of the surfacedensity (Fig. 6): in fact, in models based on the P90 progenitordistribution, the density peaks at the center ( R peak =
0) whereasfor other models the density peaks o ff center (2 ≤ R peak ≤ We first compute the fraction of NSs in bound orbits, f bound .Neglecting all those processes that could alter its energetic state(e.g. two body interactions), the final fate of a NS is known onceits initial position and velocity are fixed. A NS star is boundwhen its initial velocity is lower than the escape velocity at thebirthplace, v i < v esc ( r ), with v esc ( r i ) = p − Φ ( r i ) , (19)where r i is the position of the newborn NS. Thus f bound = N ( v < v esc ) N star . (20)The retention fraction is ∼ f bound ∼ N. Sartore, E. Ripamonti, A. Treves and R. Turolla: Galactic neutron stars
Fig. 7.
Distribution of heights f(z) - Model 1A. Dashed line rep-resents the fitting function.
From here on our results are obtained rescaling N star from150000 to 10 , which is our reference value for the total numberof NSs produced in the MW.We study the distribution of NSs, f ( z ), as a function of theheight on the Galactic plane. We adopt a logistic function f ( z ) = b b z + b ) , (21)as fitting function (see e.g. Fig. 7). From these fits we estimatethe average half density half thickness z / of the disk (Table4). The values of the coe ffi cients of the fit for each model, to-gether with the corresponding maximum error, are listed in theAppendix (Table A.2). The half density half thickness showssubstantial variations from model to model, going from 100 to ∼
400 pc for models 1A to 1D: for model 1E in particular, z / is ∼
30 pc, i.e. roughly an order of magnitude smaller than thoseobtained from other models. We will return on this fact later.
Here we define the Galactic disk as the cylindrical volume withradius 20 kpc and height 0.4 kpc (i.e R ≤
20 kpc and | z | ≤ . f disk , goes from ∼ .
05 to ∼ .
20. Hence the majority of NSs born in the MWpopulate the halo (Table 4).We fit the logarithmic surface density of the disk adopting afourth order polynomial as fitting functionlog Σ ( R ) = a + a R + a R + a R + a R . (22)The accuracy of these fits is always better than 5 percent. Thevalues of the coe ffi cients a j are listed in the Appendix (TableA.1).We made a visual check of the final distribution of NSs in thedisk, looking for traces of the spiral arms. We found no evidenceof spiral structure in the evolved distribution (compare Figs. 8and 2).The hypothesis of constant formation rate yields an averageage of ∼ / middle-aged ( <
10 Myr) NSs rep-resenting only ∼ . ∼
10, sinceNSs are born there, and did not have enough time to run away.However the excess of young NSs (Fig. 9) does not alter themean age in the disk, which is also ∼ Fig. 8.
Final distribution of NSs in the disk - Model 1E*.
Fig. 9.
Distribution of ages - Model 1A. Solid an dashed linesrepresent global and disk populations respectively.
The mean velocity of NSs in the disk is roughly the same forall models, < v > ∼ −
230 km s − in the GRF while in theLSR the mean velocity is lower, < v LS R > ∼ −
190 km s − .An exception are models based on the distribution F06P, whichshow mean velocities in the LSR of ∼
80 km s − . This fact canbe easily explained: in the F06P model low birth velocities havehigher probability (see Fig. 3) and thus the main contributor tothe velocity of the star is the circular velocity, v birth + v circ ≃ v circ .The low velocity in the LSR implies also that NSs can notmove too far away from the disk and that is the reason why, formodels F06P, the scale height is considerably lower than in othermodels.Following Z95, we fit the cumulative velocity distributionof NSs, both with respect to the GRF and the LSR, with thefollowing function G ( v ) = (cid:16) v / v (cid:17) m + (cid:16) v / v (cid:17) n , (23) . Sartore, E. Ripamonti, A. Treves and R. Turolla: Galactic neutron stars 7 Table 4.
Statistical properties of NSs in the disk.
Model 1A 1B 1C 1D 1E 1A* 1B* 1C* 1D* 1E* f bound f disk z /
367 225 164 100 33 345 192 149 80 28[pc] < v >
230 220 215 213 213 262 250 249 245 245[km s − ] < v LSR >
180 146 199 164 82 199 156 216 176 89[km s − ] f ≪ .
001 0.002 < .
001 0.002 < . ≪ . f LSR where a v is a characteristic velocity. Fit values for v , m and n are listed in the Appendix (Tables A.3 and A.4). To compare our results with previous works we focus now on thestatistical properties of NSs in the so-called solar region, 7 . ≤ R ≤ . Σ which varies from ∼ . − × ( N star / ) kpc − . The volume density in the solar vicinity, n , also varies by a factor 5 between models, from ∼ − × − ( N star / ) pc − .We can now infer the distance of the nearest NS simply bycomputing the minimum volume around the Sun which containsat least a NS, assuming constant density1 = π n d ⇒ d min = (cid:18) π n (cid:19) / ; (24)typical values of d min are around 10 pc. The distribution of NSs in the halo is shown in Fig. 10. BoundNSs can be found as far as ∼ ∼
270 kpc). The radial dis-tribution clearly shows that unbound NSs start to be dominant at ∼
500 kpc. Accordingly the mean velocity, after an initial drop,starts to rise almost linearly from ∼
500 kpc. The gravitationale ff ects of other galaxies (e.g. Andromeda and the MagellanicClouds) have not been considered. We compute the projected number density of NSs in heliocen-tric coordinates ( l , b , d ) and give the relative sky maps for starswithin 30, 10 and 3 kpc respectively. Our sky maps (Fig. 12)clearly show that the most promising direction to look for NSs istowards the Galactic center, where the density is higher. Movingaway from the center, the density drops rapidly even along theGalactic plane (Fig.11). Fig. 10.
Model 1A - (upper panel) Radial distribution of NSs inthe halo. (lower panel) Mean velocity.In Table 6 we list the inferred values of the projected den-sity towards specific lines of sight (LOS). Tha sampling distancevaries according to the LOS: for example, towards the GC thesampling distance, d max , is equal to R while for the 3 other LOSlying on the Galactic plane d max is equal to the distance at whichthe LOS itself crosses the outer border of the stellar disk (20kpc).For large values of d max the projected density has non-negligible values even at high Galactic latitudes. One intriguingconsequence is that NSs in the halo may contribute to the ob-servable rate of microlensing events, both of Galactic and extra-galactic sources (Sartore et al., in preparation). We thus calculatealso the expected number density of NSs in the direction of the N. Sartore, E. Ripamonti, A. Treves and R. Turolla: Galactic neutron stars
Table 5.
Statistical properties of NSs at the solar circle.
Model 1A 1B 1C 1D 1E 1A* 1B* 1C* 1D* 1E* Σ ( N star ) pc − ] n − ( N star ) pc − ] d min < v >
216 213 204 206 216 248 242 234 238 249[km s − ] < v LSR >
173 140 191 158 72 191 150 203 167 79[km s − ] f ≪ .
001 0.002 < .
001 0.002 < . ≪ . f LSR Table 6.
Projected density of NSs towars di ff erent lines of sight. Line of sight Density [( N star ) deg − ]1A 1B 1C 1D 1E 1A* 1B* 1C* 1D* 1E* l = ◦ , b = ◦ × ] l = ◦ , b = ◦ × ] l = ◦ , b = ◦ × ] l = ◦ , b = ◦ × ] b = + ◦ × b = − ◦ × l = ◦ , b = − ◦ × ] l = ◦ , b = − ◦ × ] Magellanic Clouds, assuming 48 and 61 kpc for distance of theLarge and Small Magellanic Clouds respectively (Table 6).
Calculations made with the updated potential are labelled withan asterisk in the tables and give the following results. The re-tention fraction f bound exhibits a significant increase, especiallyfor models 1A* and 1C*, while for the remaining ones the en- hancement is less conspicuous. In all cases the fraction of NSsretained by the disk is only slightly increased (Table 4).In all models with the updated potential, the scale height ofthe population is lower due to the increased masses of the diskand bulge, the decrease in z / being of the order of 10 - 20 per-cent (see the half density half thickness in Table 4).The higher number of NSs retained by the disk implies alsohigher values of the surface and volume densities, together withthe projected density for LOSs in the Galactic plane. For the . Sartore, E. Ripamonti, A. Treves and R. Turolla: Galactic neutron stars 9 Fig. 11.
Density profile on the Galactic plane ( b =
0, upperpanel) and along a meridian ( l =
0, lower panel) for N star = - Model 1C. Cut-o ff distances are 30 (solid line), 10 (dotted) and3 (dashed) kpc respectively.same reason, even relatively fast NSs can be found in the disk,thus increasing the mean velocities on NSs (see Tables 4 and 5).
4. DISCUSSION
Our calculations show that the distribution of birth velocities isthe main factor driving the dynamics of NSs in the MW. We ob-tain substantially di ff erent values of f bound among di ff erent birthvelocity models, with the shape of the distribution (position ofthe peak, bimodality, etc.) also playing a role in determining thefinal fate of bound NSs.The highest escape fraction, ∼ .
3, are obtained with H05,F06E and F06L distributions. This value is lower than thosefound by Lyne and Lorimer (1994) (hereafter LL) and A02 andsimilar to that inferred by H05. This is probably due to thefact that the velocity distributions proposed by LL and A02were obtained adopting the distribution of free electrons ofTaylor and Cordes (1993). Both H05 and F06 adopted insteadthe revised model of Cordes and Lazio (2002) for free electrons,which reduced estimates for distance, and thus velocity, of youngpulsars. We performed a run with the distribution of A02, obtain-ing f bound ∼ .
54, confirming their results on the escape fraction.The adoption of the updated potential (higher mass of theGalaxy) implies higher escape velocities and hence only thefastest NSs, ∼ −
15 percent, can definitely escape from theMW.Albeit more than 70 percent of the NSs born in the MW arein bound orbits, the present-day number of NSs in the disk is
Fig. 12.
Sky maps of the projected density ( N star = ) - Model1C. The cut-o ff distances are 30 kpc (upper panel), 10 kpc (cen-tral panel) and 3 kpc (lower panel) respectively. The densityscale is normalized to the maximum density at 30 kpc.only a small fraction of the total, . .
20. The remaining onesare found in the halo where they spend most of their life. Thisis a striking result but was not totally unexpected because, giventheir high spatial velocities, our synthetic NSs leave the disk ina short timescale, ∼ −
10 Myr. Another remarkable findingis that the ratio of young to old NSs in the disk is very low: foreach neutron star detected as a young active source there shouldbe still more than 100 old NSs hiding in the disk.Our simulated NSs are born with significantly higher veloc-ities with respect to what is found in other works. In spite ofthis, our results for the half density half thickness show no sig-nificant di ff erences with previous studies (except for the F06Pdistribution). Also, the local spatial density of NSs falls betweenthose found by BR, BM, Z95 and that of P90, i.e. approximatelybetween 1 and 5 × − ( N star / ) pc − . This means that thenearest neutron star lies within ∼
10 pc from the Sun.The mean velocity is higher by at least a factor ∼ v ≤
50 km s − ) in the disk are a tiny fraction, f ∼ . ∼
10 in the LSRwhere f LS R is ∼ .
05. Again results obtained with the F06P dis-tribution show a rather di ff erent behavior, with f ≪ .
01 in the GRF while in the LRF roughly half of NSs in the disk arein the low velocity tail (Tables 4 and 5). However, the e ff ectiveweight of the low velocity tail of the distribution of birth veloci-ties is still matter for debate.Most of NSs, both bound and unbound, run away from theGalactic plane in a short timescale and form a halo which ex-tends well beyond the virial radius of the MW. The phase-spacedistribution of halo NSs clearly shows a separation betweenbound and unbound NSs. Unbound NSs become dominant at r ∼
500 kpc.The results presented in this paper will enable us to revisit anumber of problems concerning isolated old NSs, like the accre-tion luminosity and its observability, the strategies for observingvery close NSs, say within 100 pc and the optical depth of NSsin the perspective of using gravitational lensing to probe the pop-ulation.
Acknowledgements.
We thank the anonymous referee for several helpful com-ments which improved the previous versions of this paper. We thank also M.C. Smith for helpful suggestions on the parameters of the Milky Way potential.NS wishes to thank R. Salvaterra for useful comments on the manuscript and L.Paredi for technical support. The work of RT is partially supported by INAF / ASIthrough grant AAE-I / / / References
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Appendix A: Coefficients of the fits.
We give here the best fit parameters for the surface density ofthe disk (Table A.1), the height distribution (Table A.2) and thecumulative velocity distributions in the disk, both in the GRF(Table A.3) and in the LSR (Table A.4). . Sartore, E. Ripamonti, A. Treves and R. Turolla: Galactic neutron stars 11
Table A.1.
Surface density of the disk.
Model 1A 1B 1C 1D 1E 1A* 1B* 1C* 1D* 1E* a a -2.54 -2.58 -3.08 -2.79 -1.74 -2.47 -2.41 -2.84 -2.68 -1.83[ × − ] a × − ] a -5.69 -8.06 -12.70 -9.20 -2.28 -2.99 -5.25 -10.22 -10.07 0.13[ × − ] a × − ]Error 4 3 3 3 2 5 3 2 3 2[%] Table A.2.
Distribution of heights.
Model 1A 1B 1C 1D 1E 1A* 1B* 1C* 1D* 1E* b b b -136.3 -83.7 -120.0 -91.0 -111.9 -117.1 -70.6 -112.7 -89.9 -106.8Error 20 41 50 63 48 22 46 63 65 56[%] Table A.3.
Cumulative velocity distribution in the disk.
Model 1A 1B 1C 1D 1E 1A* 1B* 1C* 1D* 1E* v − ] n m Table A.4.
Cumulative velocity distribution in the disk (LSR).
Model 1A 1B 1C 1D 1E 1A* 1B* 1C* 1D* 1E* v ′ − ] n ′ m ′′