"The Evolution of Compact Binary Star Systems"
Konstantin A. Postnov and Lev R. Yungelson 
1 Introduction
1.1 Formation of stars and end products of their evolution
1.2 Binary stars
2 Observations of Double Compact Stars
2.1 Compact binaries with neutron stars
2.2 How frequent are NS binary coalescences?
2.3 Black holes in binary systems
2.4 A model-independent upper limit on the BH-BH/BH-NS coalescence rate
3 Basic Principles of the Evolution of Binary Stars
3.1 Keplerian binary system and radiation back reaction
3.2 Mass exchange in close binaries
3.3 Mass transfer modes and mass and angular momentum loss in binary systems
3.4 Supernova explosion
3.5 Kick velocity of neutron stars
3.6 Common envelope stage
3.7 Other notes on the CE problem
4 Evolutionary Scenario for Compact Binaries with Neutron Star or Black Hole Components
4.1 Compact binaries with neutron stars
4.2 Black-hole–formation parameters
5 Formation of Double Compact Binaries
5.1 Analytical estimates
5.2 Population synthesis results
6 Detection Rates
7 Short-Period Binaries with White-Dwarf Components
7.1 Formation of compact binaries with white dwarfs
7.2 White-dwarf binaries
7.3 Type Ia supernovae
7.4 Ultra-compact X-ray binaries
8 Observations of Double-Degenerate Systems
8.1 Detached white dwarf and subdwarf binaries
9 Evolution of Interacting Double-Degenerate Systems
9.1 “Double-degenerate family” of AM CVn stars
9.2 “Helium-star family” of AM CVn stars
9.3 Final stages of evolution of interacting double-degenerate systems
10 Gravitational Waves from Compact Binaries with White-Dwarf Components
11 AM CVn-Type Stars as Sources of Optical and X-Ray Emission
12 Conclusions

2 Observations of Double Compact Stars

2.1 Compact binaries with neutron stars

NS binaries have been discovered when one of their components is observed as a radio pulsar [292*]. The precise pulsar timing allows one to search for a periodic variation due to the binary motion. This technique is reviewed in detail by Lorimer [444]; applications of pulsar timing for general relativity tests are reviewed by Stairs [733]. Techniques and results of measurements of NS mass and radii in different types of binaries are summarized by Lattimer [411].

Basically, pulsar timing provides the following Keplerian orbital parameters of the binary system: the binary orbital period Pb as measured from periodic Doppler variations of the pulsar spin, the projected semi-major axis x = asini as measured from the semi-amplitude of the pulsar radial velocity curve (i is the binary inclination angle defined such that i = 0 for face-on systems), the orbital eccentricity e as measured from the shape of the pulsar radial velocity curve, and the longitude of periastron ω at a particular epoch T 0. The first two parameters allow one to construct the mass function of the secondary companion,

2 3 3 f(M ,M ) = 4π-x--= -(Mc-sini)--. (2 ) p c P 2bT⊙ (Mc + Mp )2
In this expression, x is measured in light-seconds, 3 T⊙ ≡ GM ⊙ ∕c = 4.925490947 μs, and Mp and Mc denote masses of the pulsar and its companion, respectively. This function gives the strict lower limit on the mass of the unseen companion. However, assuming the pulsar mass to have the typical value of a NS mass (for example, confined between the lowest measured NS mass 1.25 M ⊙ for PSR J0737–3039B [451*] and the maximum measured NS mass of 2.1 M ⊙ in the NS–WD binary PSR J0751+1807 [527]), one can estimate the mass of the secondary star even without knowing the binary inclination angle i.

Long-term pulsar timing allows measurements of several relativistic phenomena: the advance of periastron ˙ω, the redshift parameter γ, the Shapiro delay within the binary system qualified through post-Keplerian parameters r, s, and the binary orbit decay P˙b. From the post-Keplerian parameters the individual masses Mp, Mc and the binary inclination angle i can be calculated [74].

Of the post-Keplerian parameters of pulsar binaries, the periastron advance rate is usually measured most readily. Assuming it to be entirely due to general relativity, the total mass of the system can be evaluated:

( 2π )5∕3 T2∕3(M + M )2∕3 ω˙= 3 --- -⊙-----c--2--p---. (3 ) Pb (1 − e )
The high value of the derived total mass of a system (≳ 2.5M ⊙) suggests the presence of another NS or even a BH.11

If the masses of components, binary period, and eccentricity of a compact binary system are known, it is easy to calculate the time it takes for the binary companions to coalesce due to GW emission using the quadrupole formula for GW emission [579*] (see Section 3.1.4 for more detail):

( )8 ∕3 ( ) −1 ( )− 2∕3 τGW ≈ 4.8 × 1010 yr Pb- -μ-- Mc-+-Mp-- (1 − e2)7∕2. (4 ) d M ⊙ M ⊙
Here μ = MpMc ∕(Mp + Mc ) is the reduced mass of the binary. Some observed and derived parameters of known compact binaries with NSs are collected in Tables 2 and 3.

Table 2: Observed parameters of neutron star binaries.
PSR P Pb a1sini e ˙ω ˙ Pb Ref.
[ms] [d] [lt-s] [deg yr–1] [× 10–12]
J0737–3039A 22.70 0.102 1.42 0.088 16.88 –1.24 [77*]
J0737–3039B 2773 [451]
J1518+4904 40.93 8.634 20.04 0.249 0.011 ? [526]
B1534+12 37.90 0.421 3.73 0.274 1.756 –0.138 [844, 735]
J1756–2251 28.46 0.320 2.75 0.181 2.585 ? [193]
J1811–1736 104.18 18.779 34.78 0.828 0.009 < 30 [450]
J1829+2456 41.00 1.176 7.236 0.139 0.28 ? [96]
J1906+0746 144.07 0.116 1.42 0.085 7.57 ? [445]
B1913+16 59.03 0.323 2.34 0.617 4.227 –2.428 [292*]
B2127+11C 30.53 0.335 2.52 0.681 4.457 –3.937 [12, 615]

Table 3: Derived parameters of neutron star binaries.
PSR f(m ) Mc + Mp ˙ τc = P ∕(2P ) τGW
[M ⊙] [M ⊙] [Myr] [Myr]
J0737–3039A 0.29 2.58 210 87
J0737–3039B 50
J1518+4904 0.12 2.62 9.6 × 106
B1534+12 0.31 2.75 248 2690
J1756–2251 0.22 2.57 444 1690
J1811–1736 0.13 2.6 1.7 × 106
J1829+2456 0.29 2.53 60 × 103
J1906+0746 0.11 2.61 0.112 300
B1913+16 0.13 2.83 108 310
B2127+11C 0.15 2.71 969 220

2.2 How frequent are NS binary coalescences?

As is seen in Table 3, only six NS binary systems presently known will merge over a time interval shorter than ≈ 10 Gyr: J0737–3039A, B1534+12, J1756–2251, J1906+0746, B1913+16, and B2127+11C. Of these six systems, one (PSR B2127+11C) is located in the globular cluster M15. This system may have a different formation history, so usually it is not included in the analysis of the coalescence rate of Galactic compact binaries. The formation and evolution of relativistic binaries in dense stellar systems is reviewed elsewhere [39*]. For a general review of pulsars in globular clusters see also [82].

Let us try to estimate the NS-binary merger rate from pulsar binary statistics, which is free from many uncertainties of stellar evolution. Usually, the estimate is based on the following extrapolation [505, 586*]. Suppose, we observe i classes of Galactic pulsar binaries. Taking into account various selection effects of pulsar surveys (see, e.g., [504, 361*]), the Galactic number of pulsars N i in each class can be evaluated. To compute the Galactic merger rate of NS binaries, we need to know the time since the birth of the NS observed as a pulsar in the given binary system. This time is the sum of the observed characteristic pulsar age τc and the time required for the binary system to merge due to GW orbit decay τGW. With the exception of PSR J0737–3039B and the recently discovered PSR J1906+0746, pulsars that we observe in NS binary systems are old recycled pulsars that were spun-up by accretion from the secondary companion to the period of several tens of ms (see Table 2). Thus, their characteristic ages can be estimated as the time since termination of spin-up by accretion (for the younger pulsar PSR J0737–3039B this time can also be computed as the dynamical age of the pulsar, P ∕(2P˙), which gives essentially the same result).

Then the merger rate ℛi can be calculated as ℛi ∼ Ni∕ (τc + τGW ) (summed over all pulsar binaries). The detailed analysis [361] indicates that the Galactic merger rate of NS binaries is mostly determined by pulsars with faint radio luminosity and short orbital periods. Presently, it is the nearby (600 pc) pulsar-binary system PSR J0737–3039 with a short orbital period of 2.4 hr [77*] that mostly determines the empirical estimate of the merger rate. According to Kim et al. [362], “the most likely values of DNS merger rate lie in the range 3 – 190 per Myr depending on different pulsar models”. This estimate has recently been revised in [363*] based on the analysis of binary pulsars in the Galactic disk: PSR 1913+16, PSR 1534+12, and pulsar binary PSR J0737–3039A and J0737–3039B, giving the Galactic NS + NS coalescence rate +28+40 ℛG = 21−14−17 per Myr (95% and 90% confidence level, respectively). The estimates by population synthesis codes are still plagued by uncertainties in the statistics of binaries, in modeling binary evolution and supernovae. The most optimistic “theoretical” predictions amount to ≃ 300 Myr–1 [788*, 34*, 157*].

An independent estimate of the NS-binary merger rate can also be obtained using another astrophysical argument, originally suggested by Bailes [20]. 1) Take the formation rate of single pulsars in the Galaxy ℛ ∼ 1∕50 yr−1 PSR (e.g., [190]; see also the discussion on the NS formation rate of different types in [351*]), which appears to be correct to within a factor of two. 2) Take the fraction of NS binary systems in which one component is a normal (not recycled) pulsar and which are close enough to merge within the Hubble time, fDNS ∼ (a few) × 1/2000). Assuming a steady state, this fraction yields the formation rate of coalescing NS binaries in the Galaxy ℛ ∼ ℛPSRfDNS ∼ (a few 10s)× Myr–1, in good correspondence with other empirical estimates [363*]. Clearly, the Bailes limit ignores the fact that the NS formation rate can actually be higher than the pulsar formation rate [351] and possible selection effects related to the evolution of stars in binary systems, but the agreement with other estimates seems to be encouraging.

Extrapolation beyond the Galaxy is usually done by scaling the Galactic merger rate to the volume over which the merger events can be detected for a given GW detector’s sensitivity. The scaling factor widely used is the ratio between the B-band luminosity density in the local Universe, correlating with the star-formation rate (SFR), and the B-band luminosity of the galaxy [586, 333, 376*]. For this purpose one can also use the direct ratio of the galactic star formation rate SFR ≃ 2 M yr−1 G ⊙ [108, 353, 537] to the (dust-corrected) star formation rate in the local Universe − 1 −3 SFRloc ≃ 0.03M ⊙ yr Mpc [577, 682]. These estimates yield the (Euclidean) relation

−3 ℛV ≃ 0.01ℛG [Mpc ]. (5 )
This estimate is very close to the local density of equivalent Milky-Way type galaxies found in [376*]: 0.016 per cubic Mpc.

Bear in mind that since NS-binary coalescences can be strongly delayed compared to star formation, the estimate of their rate in the nearby Universe based on present-day star formation density may be underevaluated. Additionally, these estimates inevitably suffer from many other astrophysical uncertainties. For example, a careful study of local SFR from analysis of an almost complete sample of nearby galaxies within 11 Mpc using different SFR indicators and supernova rate measurements during 13 years of observations [63*] results in the local SFR density −1 −3 SFRloc ≃ 0.008 M ⊙ yr Mpc. However, Horiuchi et al. [288] argue that estimates of the local SFR can be strongly affected by stellar rotation. Using new stellar evolutionary tracks [173], they derived the local SFR density SFRloc ≃ 0.017M ⊙ yr−1 Mpc −3 and noted that the estimate of the local SFR from core-collapse supernova counts is higher by a factor of 2 – 3.

Therefore, the actual value of the scaling factor from the galactic merger rate is presently uncertain to within a factor of at least two. The mean SFR density steeply increases with distance (e.g., as 3.4±0.4 ∼ (1 + z ) [127]) and can be higher in individual galaxies. Thus, for the conservative galactic NS + NS merger rate ℛG ∼ 10− 5 yr− 1 using scaling relation (5*) we obtain a few NS-binary coalescences per year of observations within the assumed advanced GW detectors horizon Dhor = 400 Mpc.

However, one should differentiate between the possible merging rate within some volume and the detection rate of certain types of compact binaries from this volume (see Section 6 below and [2*] for more detail) – the detection rate by one detector can be lower by a factor of (2.26)3. The correction factor takes into account the averaging over all sky locations and orientations.

The latest results of the search for GWs from coalescing binary systems within 40 Mpc volume using LIGO and Virgo observations between July 7, 2009, and October 20, 2010 [3*] established an observational upper limit to the 1.35M ⊙ + 1.35 M ⊙ NS binary coalescence rate of < 1.3 × 10 −4 yr− 1 Mpc −3. Adopting the scaling factor from the measured local SFR density [63], the corresponding galactic upper limit is ℛ < 0.05 yr−1 G. This is still too high to put interesting astrophysical bounds, but even upper limits from the advanced LIGO detector are expected to be very constraining.

2.3 Black holes in binary systems

Black holes (BH) in binary systems remain on the top of astrophysical studies. Most of the experimental knowledge on BH physics has so far been obtained through electromagnetic channels (see [618] for a review), but the fundamental features of BHs will be studied through GW experiments [673*]. Stellar-mass black holes result from gravitational collapse of the cores of the most massive stars [851] (see [216] for the recent progress in the physics of gravitational collapse).

Stellar-mass BHs can be observed in close binary systems at the stage of mass accretion from the secondary companion as bright X-ray sources [690]. X-ray studies of BHs in binary systems (which are usually referred to as ‘Black Hole Candidates’, BHC) are reviewed in [633*]. There is strong observational evidence of the launch of relativistic jets from the inner parts of accretion disks in BHC, which can be observed in the range from radio to gamma-rays (the ‘galactic microquasar’ phenomenon, [486, 197, 756]).

Optical spectroscopy and X-ray observations of BHC allow measurements of masses and, under certain assumptions, spins of BHs [476]. The measured mass distribution of stellar mass BHs is centered on ∼ 8M ⊙ and appears to be separated from NS masses [550] by a gap (the absence of compact star masses in the ∼ 2 –5 M ⊙ range) [22, 100, 549, 189, 580] (see, however, the discussion of possible systematic errors leading to overestimation of dynamical BH masses in X-ray transients in [386]). The mass gap, if real, can be indicative of the supernova-explosion–mechanism details (see, for example, [610, 37, 213]). It is also possible that the gap is due to “failed core-collapse supernovae” from red supergiants with masses 16.5 M < M < 25M ⊙ ⊙, which would produce black holes with a mass equal to the mass of the helium core (5– 8M ⊙) before the collapse, as suggested by Kochanek [371*]. Adopting the latter hypothesis would increase by ∼ 20% the stellar-mass BH formation rate. However, it is unclear how binary interaction can change the latter possibility. We stress here once again that this important issue remains highly uncertain due to the lack of “first-principles” calculations of stellar-core collapses.

Most of more than 20 galactic BHC appear as X-ray transients with a rich phenomenology of outbursts and spectral/time variability [633*], and only a few (Cyg X-1, LMC X-3, LMC X-1 and SS 433) are persistent X-ray sources. Optical companions in BH transients are low-mass stars (main-sequence or evolved) filling Roche lobes, and the transient activity is apparently related to accretion-disk instability [409*]. Their evolution is driven by orbital angular momentum loss due to GW emission and magnetic stellar wind [619]. Persistent BHC, in contrast, have early-type massive optical companions and belong to the class of High-Mass X-ray Binaries (HMXB) (see [755*, 797] and references therein).

Like in the case of pulsar timing for NSs, to estimate the mass of the BH companion in a close binary system one should measure the radial velocity curve of the companion (usually optical) star from spectroscopic observations. The radial velocity curve (i.e., the dependence of the radial velocity of the companion on the binary phase) has a form that depends on the orbital eccentricity e, and the amplitude K. In the Newtonian limit for two point masses, the binary mass function can be expressed through the observed quantities Pb (the binary orbital period), orbital eccentricity e and semi-amplitude of the radial velocity curve K as

( )3 ( ) --Mx-sin3i-- -Pb-- 3 23∕2 −7 -Kv--- -Pb 2 3∕2 f(Mv ) = (Mx + Mv )2 = 2πG K v(1 − e ) ≈ 1.038 × 10 [M ⊙] km/s 1d (1 − e ) . (6 )
From here one readily finds the mass of the unseen (X-ray) companion to be
( )2 M = f(M ) 1 + Mv-- -1---. (7 ) x v Mx sin3 i
Unless the mass ratio Mv ∕Mx and orbital inclination angle i are known from independent measurements (for example, from the analysis of optical light curve and the duration of X-ray eclipse), the mass function of the optical component gives the lower limit of the BH mass: Mx ≥ f(Mv ).

When using the optical mass function to estimate the BH mass as described above, one should always check the validity of approximations used in deriving Eq. (6*) (see a detailed discussion of different effects related to the non-point-like shape of the optical companion in, e.g., [99]). For example, the optical O-B or Wolf–Rayet (WR) stars in HMXBs have a powerful high-velocity stellar wind, which can affect the dynamical BH-mass estimate based on spectroscopic measurements (see the discussion in [452] for BH + WR binaries).

Parameters of known Galactic and extragalactic HMXB with black holes are summarized in Table 4.

No PSR + BH system has been observed so far, despite optimistic expectations from the early population synthesis calculations [431*] and recent examination of the possible dynamical formation of such a binary in the Galactic center [191]. Therefore, it is not possible to obtain direct experimental constraints on the NS + BH coalescence rate based on observations of real systems. Recently, the BH candidate with Be-star MWC 656 in a wide 60-day orbit was reported [93] (see Table 4). It is a very weak X-ray source [498], but can be the counterpart of the gamma-ray source AGL J2241+4454. The BH mass estimation in this case depends on the spectral classification and mass of the Be star. This system can be the progenitor of the long-sought BH + NS binary. Whether or not this binary system can become a merging BH + NS binary depends on the details of the common envelope phase, which is thought to happen after the bright accretion stage in this system, when the Be-star will evolve to the giant stage.

Table 4: Observed parameters of HMXB with black holes.
Source Pb (hr) f(Mv ) [M ⊙] Mx [M ⊙] Mv [M ⊙] Sp. type Ref.
Galactic binaries
Cyg X-1 134.4 0.244 ∼ 15 ∼ 19 O9.7Iab [543]
SS 433 313.9 0.268 ∼ 5 ∼ 18 A7I: [101*]
Cyg X-3 4.8 0.027 ∼ 2.4 ∼ 10 WN [883]
MWC 656 1448.9 0.205 3.8 – 6.9 10 – 16 B1.5–B2 IIIe [93]
Extragalactic binaries
LMC X-1 93.8 0.13 ∼ 11 ∼ 32 O7III [545]
M33 X-7 82.8 0.46 ∼ 16 ∼ 70 O7-8III [544]
IC10 X-1 34.4 7.64 ∼ 23 – 33 ∼ 17 – 35 WR [611, 710]
NGC 300 X-1 32.3 2.6 ∼ 14.5 ∼ 15 WR [124*]
Extragalactic BH + WR candidates
CXOU J123030.3+413853 6.4 ? ≳ 14.5 ? WR [179]
(in NGC 4490)            
CXOU J004732.0–251722.1 14 – 15 ? ? ? WR [452]
(in NGC 253)            

2.4 A model-independent upper limit on the BH-BH/BH-NS coalescence rate

Even without using the population synthesis tool, one can search for NS + BH or BH + BH progenitors among known BH in HMXB. This program has been pursued in [75, 32*], [33]. In these papers, the authors examined the future evolution of two bright HMXB IC10 X-1/NGC300 X-1 found in nearby low-metallicity galaxies. Both binaries consist of massive WR-stars (about 20 M ⊙) and BH in close orbits (orbital periods about 30 hours). Masses of WR-stars seem to be high enough to produce the second BH, so these system may be immediate progenitors of coalescing BH-BH systems. Analysis of the evolution of the best-known Galactic HMXB Cyg X-1 [32], which can be a NS + BH progenitor, led to the conclusion that the galactic formation rate of coalescing NS + BH is likely to be very low, less than 1 per 100 Myrs. This estimate is rather pessimistic, even for advanced LIGO detectors. Implications of the growing class of short-period BH + WR binaries (see Table 4) for the (BH + BH)/(BH + NS) merger rate are discussed in [452].

While for NS-binary systems it is possible to obtain the upper limit for the coalescence rate based on observed pulsar binary statistics (the Bailes limit, see Section 2.2 above), it is not so easy for BH + NS or BH + BH systems due to the (present-day) lack of their observational candidates. Still, a crude estimate can be found from the following considerations. A rough upper limit on the coalescence rate of BH + NS(BH) binaries is set by the observed formation rate of high-mass X-ray binaries, their direct progenitors. The present-day formation rate of galactic HMXBs is about ℛHMXB ∼ NHMXB ∕tHMXB ∼ 10− 3 per year (here we conservatively assumed NHMXB = 100 and 5 tHMXB = 10 yr). This estimate can be made more precise considering that only very close compact binary systems can coalesce in the Hubble time, which requires a common envelope stage after the HMXB stage to occur (see Figure 7*). The CE stage is likely to happen in sufficiently close binaries after the bright X-ray accretion stage.12 The analysis of observations of different X-ray–source populations in the galaxies suggests ([243] and references therein) that only a few per cent of all BHs formed in a galaxy can pass through a bright accretion stage in HMXBs. For a galactic NS formation rate of once per several decades, a minimum BH progenitor mass of 20 M ⊙ and the Salpeter initial mass function, this yields an estimate of a few ×10 −5 per year. The probability of a given progenitor HMXB becoming a merging NS + BH or BH + BH binary is very model-dependent. For example, recent studies of a unique galactic microquasar SS 433 [58, 101] suggest the BH mass in such a system to be at least ∼ 5M ⊙ and the optical star mass to be above 15M ⊙. In SS 433, the optical star fills the Roche lobe and forms a supercritical accretion disk [183]. The mass transfer rate onto the compact star is estimated to be about 10− 4M ⊙ yr− 1. According to the canonical HMXB evolution calculations, a common envelope must have been formed in such a binary on a short time scale (thousands of years) (see Section 3.5 and [319]), but the observed stability of binary-system parameters in SS433 over 30 years [132] shows that this is not the case. This example clearly illustrates the uncertainty in our understanding of HMXB evolution with BHs.

Thus, we conclude that in the most optimistic case where the (NS + BH)/(BH + BH) merger rate is equal to the formation rate of their HMXB progenitors, the upper limit for Galactic ℛNS+BH is a few ×10 − 5 yr−1. The detection or non-detection of such binaries within ∼ 1000 Mpc distance by the advanced LIGO detectors can therefore very strongly constrain our knowledge of the evolution of HMXB systems.

  Go to previous page Scroll to top Go to next page