ORBITAL ECCENTRICITY OF BINARY RADIO PULSARS IN GLOBULAR CLUSTERS AND THE INTERACTION BETWEEN STARS

Sustained high-sensitivity searches of globular clusters (GCs) for radio pulsars with improved pulsar-search algorithms have yielded 140 pulsars so far in 26 GCs¹ (Camilo & Rasio 2005 and references therein). Radio pulsars can be timed more easily and accurately by ground-based telescopes over well-separated epochs, as the underlying neutron stars (NSs) are less prone to noise from episodically varying accretion torques (as for X-ray pulsars). These lead to easier measurement of orbital parameters. Binary systems can provide an important source of energy for GCs, since the binding energies of a few, very close binaries can approach that of a moderately massive host GC (Hut et al. 2003) and therefore can have dynamical effects on the cluster's evolution. On the other hand, the observable parameters of the neutron stars and their binary companions in the clusters, such as spin, orbital period and eccentricity, projected radial position in the cluster, companion mass, and their distributions provide a valuable test bed to examine the theoretical scenarios of formation and evolution of recycled pulsars. These parameters can provide a tracer of the past history of dynamical interactions of the binary NSs in individual GCs. We discuss below the distribution of orbital eccentricities and periods of GC pulsars in light of interaction of stars with binaries already formed or in the process of formation inside GCs, due to the high stellar density in their cores. The observed distributions are examined with analytical results and our numerical experiments of scattering of stars simulated by direct N-body integration tools.

Although there are a number of low-mass X-ray binaries (LMXBs) in the GCs, their orbital period and eccentricity information is poorly known compared to that for radio pulsars; moreover the initial eccentricity of formation is quickly quenched due to tidal coupling with the Roche lobe-filling companion. Hence we concentrate entirely on binary radio pulsars whose companions are usually smaller than their Roche lobes, and less well coupled tidally. Of the 140 radio pulsars in the GCs, 74 are binaries (a large fraction of which are millisecond pulsars), 59 are isolated and seven others have no published timing/orbital solutions. Orbital parameters are not well determined for one binary (PSR J2140−2310B in M30), only the lower limit of P_orb and e are known as P_orb > 0.8 days and e > 0.52, and thus it is excluded from the present study. In Figure 1, we plot e versus P_orb for 73 binaries in GCs with known orbital solutions. Logarithmic scales in both eccentricity and orbital periods are chosen, as the enormous range of both variables and the regions occupied by observed pulsars are less obvious in linear scales. Observed binary radio pulsars in GCs can be categorized into three groups: (I) 21 pulsars with large eccentricity (1>e ⩾ 0.01); (II) 20 pulsars with moderate eccentricity (0.01>e ⩾ 2 × 10⁻⁶); (III) 32 pulsars with small eccentricity (e ∼ 0). Using the "R" statistical package² k-means test, we show that these groups are distinct at a statistically significant level.³

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In the database, several pulsars' orbital eccentricities have been listed as zero, but we assign them an arbitrarily small value of e = 3 × 10⁻⁷. Note that the smallest eccentricity measurable is determined by how well the orbit of the binary is sampled and the overall timing accuracy achieved for radio pulse arrivals (Phinney 1992). The timing accuracy translates into an upper limit on the smallness of the eccentricity. This is displayed as the brown solid line in the lower left corner of Figure 1 following the functional relation: e_min = (δt)/(asin i/c) = 4π²cδt/[sin i(G(m_p + m_c)^1/3)P^2/3_orb]. Here i is the inclination angle of the binary, m_p is the pulsar mass, m_c is the companion mass, and δt is the timing accuracy. We take m_p = 1.4 M_☉, m_c = 0.35 M_☉, i = 60°, δt = 1 μsec for the "limit of timing sensitivity" line.

Observational selection effects may be influencing the distribution of GC pulsars seen in Figure 1 (Camilo & Rasio 2005), the most important selection effect operates toward the left of the diagram: it is more difficult to detect pulsars with larger DM and/or shorter spin periods, especially millisecond pulsars in short orbital periods and highly eccentric binaries. Another important selection effect is due to distance, since only the brightest pulsars can be observed at large distance.

The presence of a large number of LMXBs in GCs compared to the galactic field had led to the suggestion (Fabian et al. 1975) that a binary is formed by tidal capture of a noncompact star by a neutron star in the dense stellar environment of the GC cores. If stable transfer of mass and angular momentum ensued from the companion star, this could lead to recycled pulsars in binaries or as single millisecond pulsars e.g., Alpar et al. (1982). However tidal capture of a neutron star by a low-mass main-sequence star can lead to large energies being deposited in tides. The resultant structural readjustments of the star in response to the dissipation of the modes could be very significant in stars with either convective or radiative damping zones (Ray et al. 1987; McMillan et al. 1987) and the companion star can undergo size "inflation" due to its high tidal luminosity which may be much larger than that induced by nuclear reactions in the core. The efficiency of viscous dissipation and orbit evolution is crucial to the subsequent evolution of the system as viscosity regulates the growth of oscillations and also the extent to which the extended star is bloated and shed. A significant fraction of the encounters lead to binaries that either become unbound as a result of de-excitation or heating from other stars in the vicinity or they are scattered into orbits with large pericenters (compared to the size of the noncompact star) due to angular momentum transfer from other stars (Kochanek 1992). For a recent summary of the formation channels of retained neutron star binaries in GCs obtained via population synthesis, see Ivanova et al. (2008).

If the local binary fraction is substantial, the binary–single-star interaction can exceed the encounter rate between single stars by a large factor (Sigurdsson & Phinney 1993). The existence of a significant population of primordial binaries in GCs (Yan & Mateo 1994; Pryor et al. 1989) indicate that three-body processes have to be accounted for in any dynamical study of binaries involving compact stars. For a literature summary of the constraints on the binary fraction in GCs see Davis et al. (2008).

An encounter between a field star and a binary may lead to a change of state of the latter, e.g., (1) the original binary may undergo a change of eccentricity and orbital period but otherwise remain intact—a "fly-by" interaction; (2) a member of the binary may be exchanged with the incoming field star, forming a new binary—an "exchange" process; (3) two of the stars may collide and merge into a single object, and may or may not remain bound to the third star—a "merger" process; or (4) all three stars become unbound—an "ionization" process.

The formation scenario of millisecond pulsars in primordial binaries suggests that their eccentricity should be very small, ∼10⁻⁶–10⁻³ (Phinney 1992). But inside GCs, they may acquire eccentricity through interactions with single stars. Rasio & Heggie (1995); Heggie & Rasio (1996) studied the change of orbital eccentricity (δe) of an initially circular binary following a distant encounter with a third star in a parabolic orbit. They used the secular perturbation theory, i.e., averaging over the orbital motion of the binary for sufficiently large values of the pericenter distance r_p, where the encounter is quasi-adiabatic and used nonsecular perturbation theory for smaller values of r_p where the encounter is nonadiabatic. In the first case δe varies as a power law with r_p/a and in the second case δe varies exponentially with r_p/a (a is the semimajor axis of the binary). The power law dominates for e ≲ 0.01 and the exponential dominates for e ≳ 0.01. They estimated the cross sections (σ) for eccentricity changes and calculated timescales for eccentricity changes as t = 1/(rate) = 1/〈nσv〉 where n is the number density of the stars and v is the velocity of the incoming star. The expressions of the timescales for fly-by are (see Rasio & Heggie 1995):

$$\begin{equation*} \begin{array}{cc} t_{\rm fly}=4\times 10^{11} n_4^{-1}v_{10}P_{\rm orb}^{-2/3}e^{2/5} \quad {\rm for\quad } e \lesssim 0.01\vspace*{6pt}\\ t_{\rm fly}=2\times 10^{11} n_4^{-1}v_{10}P_{\rm orb}^{-2/3}\left[- \ln (e/4) \right]^{-2/3} \quad {\rm for\quad } e \gtrsim 0.01 \end{array}\vspace*{2pt} \end{equation*}$$

where n₄ is the number density (v) of single stars in units of 10⁴pc^-3 and v₁₀ is the velocity dispersion (v) in units of 10 km s⁻¹ in GCs; P_orb is the orbital period in days giving t_fly in years.

The eccentricities of the binary pulsars in GCs are likely to be due to binary single star interactions when the interaction timescale is less than the binary age. We take the maximum age of the binaries in a GC to be the GC ages which are ∼10¹⁰ yr. The value of v₁₀/n₄ which determines t_fly varies from 0.0024 to 2.167 for different GCs (Webbink 1985). We grouped them according to the values of v₁₀/n₄ and calculated t_fly with the mean values of v₁₀/n₄ for each of six groups. In Figure 1, we plot the isochrones of fly-by encounters (t_fly) in the e–P_orb plane for all six groups using the above expressions. Pulsars which lie inside or outside the GC core in the projected image are depicted using different symbols and the colors are the same as those used to draw corresponding t_fly contours. If t_fly > 10¹⁰ yr for a particular binary, then it would not have interacted and it would preserve its original eccentricity. If t_fly < 10¹⁰ yr for a particular binary, then it could be eccentric due to fly-by interactions which is the case for most of the eccentric GC binaries (Figure 1). So, many GC pulsars' orbital eccentricities can be explained by fly-bys. However the local stellar densities at the pulsar positions (especially with positional offset from cluster cores) may indicate a higher v₁₀/n₄ value from central values used to calculate the isochrones. Thus these pulsars might be in regions where the effective timescale for fly-by induced eccentricity is larger than the Hubble time. Moreover, for pulsars with high eccentricities (e > 0.1, a majority of the group I pulsars) a very close fly-by or multiple fly-bys are necessary if the initial binary was circular (see Heggie & Rasio 1996 and Camilo & Rasio 2005 for a discussion). In these cases, other processes, e.g., exchange and merger, may produce these systems more naturally. We discuss this below with numerical simulations.

We have used the STARLAB⁴ software (which reports the results for both resonant and nonresonant encounters for exchange and merger processes) to perform numerical simulations. We discuss the results for pulsar binaries found in Ter 5 since it has the lowest value of the parameter v₁₀/n₄ where all classes of encounters can take place. We find that the "phase space" of the encounter reactions of different kind in 47 Tuc and other clusters, is not as large as that of Ter 5. In Figure 2, we plot P_orb,in of the initial binary along the top x-axis and P_orb,fin along the bottom x-axis. P_orb,fin is obtained from P_orb,in putting Δ = 0 in the relation a_fin = [a_inm_am_b/m₁m₂(1 − Δ)] where m₁ and m₂ are masses of the members of the initial binary, m₃ is the mass of the incoming star, m_a and m_b are masses of the members of the final binary, Δ is the fractional change of binary binding energy. For exchange, m_a = m₁, m_b = m₃; for merger, m_a = m₁, m_b = m₂ + m₃.

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It is clear from the scatter plots (Figure 2) that the final binaries will most probably have e > 0.1 if they undergo either exchange or merger events. Six high-eccentricity (e > 0.1) binaries in Ter 5 are also shown (black in color, symbols same as used in Figure 1) in this plot. All of them may result from exchange interactions with either a normal-mass companion (∼0.40 M_☉) or with a low-mass companion (∼0.16 M_☉). PSR U, X, and Z may even come from exchange with ultra-low-mass companion (∼0.024 M_☉). Note that all the exchange conclusions here are only guidelines, since they are based on stars of a single mass (0.33 M_☉) exchanging into the systems, and some of the minimum inferred companion masses are quite different from 0.33 M_☉. Mergers with 0.40 M_☉ (initial) companions and incoming 0.33 M_☉ stars are problematic because of the high final companion masses. Q is the only system for which this might be possible, and that would require a fairly small orbital inclination angle. Similar problems apply to initial companion masses of 0.16 M_☉ for all except Q and U. Finally, while the mass restriction is not really a problem for any of the lower-mass systems in the ultra-low-mass case, the small orbital periods imply that too long a time would have to pass before a suitable encounter took place.

In Figure 1, there is a cluster of three pulsars with 0.01 < e < 0.1 and 60 < P_orb < 256 d—NGC1851A, M3(D), and Ter 5(E), all with companions of mass in the range m_c = 0.21–0.35  M_☉ assuming i = 60° (except NGC 1851A which has a 1.12 M_☉ companion). These are possibly white dwarf (WD) cores of red giant companions that overflowed from the Roche lobe (Webbink et al. 1983). Such binaries would normally have the "relic" eccentricities ∼10⁻⁴ (Phinney 1992). The above binary pulsars with their present mildly high eccentricities, have undergone fly-by encounters with field stars, rather than exchange reactions, which would produce very high eccentricities e > 0.1. Ter 5 E lies outside the core but even then, the density may have been high enough to allow a strong fly-by interaction. It could also have been ejected out of the high-density core after a strong interaction. In addition, the pulsar B1620−26 (in M4) occupies approximately the same region of the phase space. However, this system is most likely a triplet system with a planet sized third body and their interactions lead to the characteristics of the inner orbit (Thorsett et al. 1999, Ford et al. 2000, Sigurdsson et al. 2003 and references there in).

Another set of 3 ms pulsars have 0.01 < e < 0.1, 2 < P_orb < 10 d; Ter 5 (W), 47 Tuc (H) and NGC6440 (F). These clusters have low values of v₁₀/n₄, and so fly-by encounters in these clusters would be efficient and could generate these eccentricities in GCs, even if their progenitor binaries had short orbital periods and had sub-giant companions of the NSs. Alternately, these binaries could also have been formed by fly-by interactions from a presently less abundant longer period 2 < P_orb < 10 d cluster of "intermediate eccentricity" binaries to the right of the pulsars seen in the middle of Figure 1.

The "intermediate eccentricity" binaries (group II: 0.01>e ⩾ 2 × 10⁻⁶), could have been generated by fly-by encounters with low- (or "zero") eccentricity progenitor pulsars below the line of "timing sensitivity limit" (group III pulsars). Some of the shorter P_orb binaries would again be circularized by gravitational radiation (see t_gr contours in Figure 1 calculated using the formalism of Peters & Mathews 1963). The progenitor group III pulsars, themselves occur in regions of favorable fly-by encounters inducing higher eccentricities. These nearly circular binaries have their P_orb in the range of 0.06–4 days among which the tighter binaries cannot be progenitors of exchanges/mergers whenever interaction timescales are greater than 10¹⁰ yr (see Figure 2). The minimum P_orb,in above which they can undergo exchange/merger interactions decreases as m₂ decreases. About the origin of the group III pulsars themselves, we note that Camilo & Rasio (2005) discuss the dynamical formation of ultracompact binaries involving intermediate mass main sequence stars in the early life of the GC. These companions must have been massive enough (beyond the present-day cluster turn-off mass of 0.8 M_☉) so that the initial mass transfer became dynamically unstable, leading to tight NS–WD binaries through common envelope evolution. Alternately, present-day red giant and NS collisions lead to a prompt disruption of the red giant envelope and the system ends up as eccentric NS–WD binary (Rasio & Shapiro 1991). NS–WD binaries can be circularized to group III by gravitational wave radiation if P_orb < 0.2 days (see Figure 1 here and also discussions by Camilo & Rasio 2005).

In conclusion, we find that the presently observed orbital eccentricity and period data of GC binary pulsars are largely consistent with numerical scattering experiments on stellar interaction scenarios of fly-bys, exchanges, and mergers with typical field stars characterized by the central regions of the GCs. The efficiency of fly-by interaction is subject to the local stellar density at the location of the initially circular binaries. Exchange and merger interactions induce the highest range of eccentricities (1>e > 0.1) and may be operative in different GCs.

We thank Roger Blandford, Avinash Deshpande, and Shri Kulkarni for discussions, Sayan Chakraborti for comments on the manuscript and the anonymous referee for constructive suggestions. We thank the STARLAB development group for the software and the ATNF pulsar group and Paulo Freire for pulsar data bases. This research is a part of the 11th plan project 11P-409 at TIFR.