The Evolution of Compact Binary Star Systems

Abstract

We review the formation and evolution of compact binary stars consisting of white dwarfs (WDs), neutron stars (NSs), and black holes (BHs). Binary NSs and BHs are thought to be the primary astrophysical sources of gravitational waves (GWs) within the frequency band of ground-based detectors, while compact binaries of WDs are important sources of GWs at lower frequencies to be covered by space interferometers (LISA). Major uncertainties in the current understanding of properties of NSs and BHs most relevant to the GW studies are discussed, including the treatment of the natal kicks which compact stellar remnants acquire during the core collapse of massive stars and the common envelope phase of binary evolution. We discuss the coalescence rates of binary NSs and BHs and prospects for their detections, the formation and evolution of binary WDs and their observational manifestations. Special attention is given to AM CVn-stars - compact binaries in which the Roche lobe is filled by another WD or a low-mass partially degenerate helium-star, as these stars are thought to be the best LISA verification binary GW sources.

Figure 1

Descendants of components of close binaries depending on the radius of the star at RLOF. The boundary between progenitors of He and CO-WDs is uncertain by several 0.1 M_⊙, the boundary between WDs and NSs by ∼ 1 M_⊙, while for the formation of BHs the lower mass limit may be even by ∼ 10 M_⊙ higher than indicated.

Figure 2

Sensitivity limits of GW detectors and the regions of the f-h diagram occupied by some of the potential GW sources. (Courtesy G. Nelemans.)

Figure 3

The maximum initial orbital period (in hours) of two point masses which will coalesce due to gravitational wave emission in a time interval shorter than 10¹⁰ yr, as a function of the initial eccentricity e_o. The lines are calculated for 10 M_⊙ + 10 M_⊙ (BH + BH), 10 M_⊙ + 1.4 M_⊙ (BH + NS), and 1.4 M_⊙ + 1.4 M_⊙ (NS + NS).

Figure 4

Evolutionary scenario for the formation of neutron stars or black holes in close binaries.

Figure 5

Formation of close binary dwarfs and their descendants (scale and colour-coding are arbitrary).

Figure 6

The age of merging pairs of helium WDs. Two components of the distribution correspond to the systems that experienced in the course of formation two or one common envelope episodes, respectively.

Figure 7

Known close binaries with two WD components, or a WD and a sd component. Green circles mark systems known prior to the SPY project. Black filled symbols mark the positions of DDs and WD + sd systems detected in the SPY project. A blue triangle marks the positions of the WD component of the binary planetary nebula nucleus PN G135.9+55.91 detected by Tovmassian et al. [404]. (Courtesy R. Napiwotzki.)

Figure 8

The position of the primary components of known DDs in the “orbital-period-mass” diagram. The underlying gray scale plot is a model prediction from Nelemans et al. [285]. (Figure from [276].)

Figure 9

Mass-loss rate vs. orbital period for “typical” AM CVn-stars: an interacting double degenerate system with initial masses of donor and accretor 0.2 M_∩ and 0.6 M_∩ (red line) and a “semidegenerate” low-mass helium star donor plus white dwarf accretor of the same initial masses (blue line). Black dots on the red curve mark positions of the system at log T (yr) = 5, 6, 7, 8, 9, 10 from Roche-lobe overflow; on the blue curve they mark log T (yr) = 5, 6, 7, 8. Green lines mark lower and upper limits of the disk instability region according to Tsugawa and Osaki [406]. Below the magenta circles q < 0.02 and conventional evolutionary computations may be not adequate for description of mass-transfer process (see the text).

Figure 10

Dependence of the dimensionless strain amplitude for a WD + WD detached system with initial masses of the components of 0.6 M_∩ + 0.6 M_∩ (red line), a WD + WD system with 0.6 M_∩ + 0.2 M_∩ (blue line) and a WD + NS system with 1.4 M_∩ +0.2 M_∩ (green line). All systems have an initial separation of components 1 R_∩ and are assumed to be at a distance of 1 kpc (i.e. the actual strength of the signal has to be scaled with factor 1/d, with d in kpc). For the DD system the line shows an evolution into contact, while for the two other systems the upper branches show pre-contact evolution and lower branches — a post-contact evolution with mass exchange. The total time-span of evolution covered by the tracks is 13.5 Gyr. Red dots mark the positions of systems with mass-ratio of components q = 0.02 below which the conventional picture of evolution with a mass exchange may be not valid. The red dashed line marks the position of the confusion limit as determined in [286].

Figure 11

GWR background produced by detached and semidetached double white dwarfs as it would be detected at the Earth. The assumed integration time is 1 yr. The ‘noisy’ black line gives the total power spectrum, the white line the average. The dashed lines show the expected LISA sensitivity for a S/N of 1 and 5 [210]. Semidetached double white dwarfs contribute to the peak between log f ≃ −3.4 and −3.0. (Figure from [285].)

Figure 12

The number of systems per bin on a logarithmic scale. Semidetached double white dwarfs contribute to the peak between log f ≃ −3.4 and −3.0. (Figure from [285].)

Figure 13

Fraction of bins that contain exactly one system (solid line), empty bins (dashed line), and bins that contain more than one system (dotted line) as function of the frequency of the signals. (Figure from [285].)

Figure 14

Gravitational waves background formed by the Galactic population of white dwarfs and signal amplitudes produced by some of the most compact binaries with white dwarf components ([272]). The green line presents results from [285], the black one from [286], while the red one presents a model with all assumptions similar to [286], hut with the γ-formalism for the treatment of common envelopes [284, 283] (see also Section 3.5). The blue line shows the background derived in [27]. (Figure from [274].)

Figure 15

Strain amplitude h as a function of the frequency for the model populations of resolved DDs (left panel, ≃ 10,700 objects) and AM CVn systems (right panel, ≃ 11,000 objects.). The gray shades give the density distribution of the resolved systems normalized to the maximum density in each panel (1,548 and 1,823 per “cell” for the double white dwarfs and AM CVn-stars panels, respectively). The 200 strongest sources in each sample are shown as dots to enhance their visibility. In the AM CVn panel the periods of several observed short period systems are indicated by the vertical dotted lines. The solid line shows the average background noise due to detached white dwarfs. The LISA sensitivities for an integration time of one year and a signal-to-noise ratio of 1 and 5 are indicated by the dashed lines [210]. (Figure from [286].)

Figure 16

Distribution of short period AM CVn-type systems detectable in soft X-rays and as optical sources as a function of orbital period and distance. Top panel: systems detectable in X-rays only (blue pluses), direct impact systems observable in X-ray and V-band (red filled circles), systems detectable in X-ray with an optically visible donor (green squares), and systems detectable in X-rays and with an optically visible disc (large filled triangles). Bottom panel: direct impact systems (red open circles), systems with a visible donor (green squares), and systems with a visible accretion disc (small open triangles). The overlap of these systems with systems observable in gravitational waves is shown in Figure 17. (Updated figure from [286], see also [273].)

Figure 17

Short-period AM CVn systems, subdivided in different types. Each panel shows the total population as the white histogram. The top left panel shows 11,000 systems that can be resolved by LISA in gray, and they are subdivided into the ones that have optical counterparts (GWR + Opt), X-ray counterparts (GWR + X), and both (GWR + Opt + X). The top right panel shows the systems that are in the direct impact phase of accretion in gray, and they are subdivided in GWR and X-ray sources. The bottom two panels show (again in gray) the populations that are detectable in the optical band (left panel) and the X-ray band (right panel). The distribution of sources detectable both in optical and X-ray bands is shown as hatched bins in both lower panels (Opt + X). (Figure from [286].)

Figure 18

Effects of a finite entropy of donors on the properties of AM CVn-stars. The left panel shows the relation between P_orb and Ṁ along tracks for a system with initial masses of components 0.2 M_⊙ and 0.6 M_⊙ (like in Figure 9). The solid lines show the evolution for donors with T_c = 10⁴ K, 10⁶ K, 5 × 10⁶ K, and 10⁷ K (left to right). The symbols show the positions of models with M₂ =0.01 M_⊙ (triangles), 0.02 M_⊙ (squares), and 0.05 M_⊙ (pentagons). The disk stability criteria (for q = 0.05) are shown by the dashed lines (after [406]). The right panel compares the numbers of systems as a function of P_orb for the model with a T = 0 WD (dot-dashed line) and the model with “realistic” cooling (solid lines, RWDC). The smooth curves show the percentage of each population laying above a given P_orb. (Figures from [76].)