6.5.1 Final mass of the SMS and its fate
Although we have followed the evolution of the protostellar accretion for∼0.1 Myr, it is still way before the SMSs collapse into BHs. In order to determine how massive seed BHs are finally provided, we should further follow the further evolution for the stellar lifetime,
∼ 2 Myr. It is, however, computationally too expensive to accomplish it. We estimate the final stellar and BH masses from the final outputs in our simulations.
Figure 6.17 represents the radial profiles of (a) the enclosed gas mass and (b) the infall velocity at the final epoch of our simulations for the filamentary (green) and the spherical (blue) clouds. Interestingly, the gas is outflowing at R >7–8 pc in both clouds, which is caused by the tidal field due to the nearby massive galaxy. The infalling gas can reach the central star within the stellar lifetime. Panel (a) indicates the infalling gas mass is
≲3×105M⊙ for both clouds.
6.5 Discussion 111
-20 -10 0
2 4 6 8 10
infall velocity [km s-1 ]
R [pc]
4 5 6
log 10 M gas [M ⊙]
Filamentary Spherical
(a)
(b)
Figure 6.17. Radial profiles of (a) enclosed mass and (b) gas infall velocity at 0.1 Myr after the primary protostar formation for the filamentary (green) and the spherical (blue) clouds. The horizontal axis shows the distance (R) from the primary protostars In the panel (a), the dashed lines indicate the outer boundaries of the clouds, above which the gas is outflowing. In the panel (b), the dashed line shows the boundary between the inflow and outflow.
Note that a positive velocity corresponds to the outflow in this plot.
Assuming that no further protostars appear via the disk fragmentation, and that the infalling gas equally accrete onto the existing protostars, we can evaluate the typical masses of final stars: ∼104M⊙ for the filamentary cloud and∼105M⊙ for the spherical cloud. These stars are massive enough to collapse into BHs after exhaustion of nuclear fuel (e.g. Umedaet al., 2016; Woods et al., 2017). During the gravitational collapse, the most of the stellar mass is swallowed up by the BH (e.g. Shibata et al., 2016; Uchida et al., 2017). Therefore, we expect that the filamentary cloud yields around ten BHs with
∼104 M⊙ and the spherical cloud yields several BHs with∼105 M⊙. Hereafter, we will call these BHs as “DCBHs”.
6.5.2 Evolution of the DCBH binaries
We find that stellar binaries are formed in the cloud core. Some of them survive until at the end of the simulation, avoiding the merger with the companion. After the stellar lifetime, they evolve into the massive BH binaries with the mass of 103–105M⊙ and with the separation of 102 – 103 AU (see Figure 6.12). One main process for binaries to lose
angular momentum and to merge is the GW emission. The coalescence time-scale is given by (Peters, 1964);
tGW,merge= 1.25×1011 yr ( a
100 AU )4(
MBH
105 M⊙ )−3
, (6.16)
which is about an order of magnitude larger than the Hubble time, even for the tightest binaries formed in our simulation. Thus, some additional processes are necessary to remove the angular momentum, i.e., the interaction with the surrounding gaseous and stellar components for a BH-BH merger.
If the gas or stars are accreted onto a binary, a fraction of the accreted material is scattered and carries away the angular momentum from the binary. Kashiyama & Inayoshi (2016) estimates the number of stars falling onto the central DCBH, assuming the possible stellar cluster formation. In our simulations, clusters with an order of between several to ten stars appear in the cloud. According to Kashiyama & Inayoshi (2016), the relaxation time of these cluster (trelax,cluster) is,
trelax∼1.6×105yr MBH
105 M⊙
103 M⊙
⟨M∗⟩ ( r
pc )−3/2
, (6.17)
where MBH is the central BH mass, ⟨M∗⟩ is the mean mass of formed stars, ρ∗ is the density of the stars, r is the size of the star cluster. Cluster member stars are scattered into the loss cone orbit with the time-scale of trelax and fall onto the central DCBH. If the central object is a BH binary instead, part of the accreted stars will be ejected and carry away the angular momentum of the binary. If we assume that the accreted stars are ejected at the speed of the escape velocity vesc =√
GMBH/a, then the time-scale of the angular momentum losststar,merge is,
tstar,merge∼ J
⟨M∗⟩vesca/trelax ∼ MBH
⟨M∗⟩trelax
∼1.6×107yr
(103M⊙
⟨M∗⟩ )2(
MBH
105 M⊙ )2(
100 AU a
)
. (6.18)
Thus, the binary can lose the angular momentum by the interaction with the star cluster formed in the same cloud.
After the formation of the DCBH, the host cloud will merge with the nearby massive galaxy, which has been a source of a huge number of LW photons. The stellar masses of the source galaxies are 106 (filamentary) and 107 M⊙ (spherical cloud). The cold gas mass within the source galaxy is an order of magnitude larger than the stellar mass. These stellar and cold gas components interact with the DCBH binaries, which can carry away the angular momentum. If the binary separations become as small as∼10 AU, the binary will merge owing to the GW emission within the Hubble time (eq. 6.16).
The amplitudes of GW peaks at the frequency of 1–10 mHz at the rest frame. Thus the ground based GW detectors are difficult to detect GWs originating from the merging BH binaries. The space GW detectors such as Laser Interferometer Space Antenna (LISA) or
6.5 Discussion 113
10-24 10-22 10-20
-4 -3 -2 -1 0
characteristic strain amplitude
log10 frequency [Hz]
z=30, M=105 M⊙ z=10, M=105 M⊙ z=30, M=104 M⊙ z=10, M=104 M⊙
Figure 6.18. Characteristic strain amplitudes for merging BH binaries and a noise am-plitude of the future space GW detectors. We consider equal mass binaries, with member masses of 104(blue) and 105 M⊙ (red). The solid and dashed lines show the binaries atz= 10 and 30, respectively. The black solid lines indicate the noise amplitudes of LISA (Danzmann et al., 2016 and fitting by Cornish & Robson, 2017) and DECIGO (Kawamuraet al., 2011).
Deci-hertz Interferometer Gravitational wave Observatory (DECIGO) can observe these GWs.
We construct wave forms of the GWs from the merging BH binaries based on Ajith et al.(2011). Wave forms in the Fourier domain at a given frequencyf can be written as h(f)≡A(f)e−iΨ(f), where A(f) is the amplitude and Ψ(f) is the phase of the GW. The amplitudeA(f) can be written as follows,
A(f) = (GM)5/6f1−7/6 dLc3/2π2/3
√5η 24
f′−7/6P1 f < f1, wmf′−2/3P2 f1< f < f2,
wrσ4f2
[σ2+4(f−f2)]2 f2< f < f3,
(6.19)
whereM ≡M1+M2,η ≡M1M2/M2,dLis the luminosity distance,fi(i= 1,2,3) andσ are the frequencies which characterize in-spiral, merger, and ring-down phases,f′≡f /f1, Pi (i = 1,2) are the correction factor coming from the Post-Newtonian correction, and wm and wr are the normalization factors in order to make A(f) continuous across the frequencies f1 and f2. The detectability of the GW is often estimated by the so-called
“characteristic strain amplitude”hc(f), which is defined as,
hc(f) = 2f|h(f)|= 2f A(f). (6.20) Figure 6.18 shows the characteristic strain amplitudehc for merging binaries atz= 10 (solid) and 30 (dashed). We assume equal mass binaries, where the masses of each member are 104 (blue) and 105 M⊙ (red). Black solid lines show the noise amplitudes of LISA (Cornish & Robson, 2017) and DECIGO (Kawamura et al., 2011). Combining the both
1 10 100
0 10 20 30 40 50 60 70 80 90 100
distance / RJ
time / tdyn
nadib = 1011 2* 1012 1013 1014
Figure 6.19. Time evolution of the separations between the most massive protostar and the closest protostar with different nadib, above which the gas particle evolves adiabatically. The separation and the time is normalized by the Jeans length with the fixed temperatureTgas= 8000 K and free-fall time at nadib. The time origin is set to be the moment at which the first fragment appears in the disk.
detectors, binaries with 104–105 M⊙ can be observed at z ≳10. If the seed BHs of the observed SMBHs are mainly provided by the DC model, we expect a large amount of BH binaries with 104–105M⊙and GW signals of merging BH binaries. Still the exact number density of BHs provided by DC model is under debate, the detection of BH binaries in this mass range can place important constraints on the formation scenario of SMBHs.
6.5.3 The impact of FUV radiation inside the circumstellar disk
Since we do not consider the shielding against the external LW radiation, we may over-estimate the photo-dissociation rate of H2. To see whether the shielding effect modifies H2 abundance in the disk, we perform following test simulations. We completely turn off LW radiation once the gas density exceeds ncrit = 106 cm−3, where the gas becomes optically-thick to the external LW radiation (Draine & Bertoldi, 1996). We have followed the evolution in the early accretion phase for 2000 years.
We find that almost no differences between the simulation with and without the shield-ing effect. In the simulation with shieldshield-ing, H2 is completely destroyed with the typical abundances of ∼10−8. The atomic hydrogen cooling dominates in this simulation. Thus we conclude that the external LW radiation is not important for the disk structure at the accretion stage and we can safely ignore the self-shielding of the LW radiation.
Actually, H2is mainly destroyed by the collisional dissociation, not by the LW radiation.
(Inayoshi & Omukai, 2012) indicate that it occurs in the region with n >104 cm−3 and T > 5000 K. In our calculation, most of the disk gas satisfies this condition and the collisional dissociation dominates the H2 dissociation.