Bao-Minh Hoang

Black Hole Mergers in

Galactic Nuclei

MODEST-18, June 2018

Background Image: LIGO, Caltech, MIT, Aurore Simonet Sonoma State

Bao-Minh Hoang Smadar Naoz, Bence Kocsis, Fred Rasio, Fani Dosopoulou

https://www.ligo.caltech.edu/system/media_files/binaries/382/original/BlackHoleArt.jpg?1496281274

PRED ICTED MERGER RATES

Extrapolated rate from

observations

~12-240 Gpc-3 yr-1

(LIGO Scientific and Virgo Collaboration 2017 )

Abbott et. al. 2016



observations

~12-240 Gpc-3 yr-1


Isolated Binary Evolution

Abbott et. al. 2016



observations

~12-240 Gpc-3 yr-1



~0.7 Gpc-3 yr-1

(e.g. Tutukov & Yungelson 1993,Portegies Zwart &

Yungelson 1998, Bethe & Brown 1999)

Abbott et. al. 2016

The Hubble Heritage Team

(AURA/STScI/NASA)(Not to scale)

E X T R E M E G R A V I T Y


(AURA/STScI/NASA)(Not to scale)

E X T R E M E G R A V I T Y


(AURA/STScI/NASA)

There’s a supermassive black hole (SMBH) at the center of almost every galaxy! (e.g. Ghez et. al. 2000)

(Not to scale)

EV IDENCE FOR B INAR I ES I N GALACT IC NUCLE I

Three known stellar binaries:

• IRS 16SW: period ~ 19.5 days, mass ~ 50 M⊙, at ~0.05 pc (Ott et al. 1999, Martins et al. 2006)

• IRS 16NE: period ~ 224 days, e~0.3, at ~0.1 pc (Pfuhl et al. 2014)

• E60: period ~ 2.3 days, mass ~ 30 M⊙, at ~ 0.1 pc (Pfuhl et al. 2014)


L52 LU ET AL. Vol. 625

Fig. 1.—(a) Positions of all stars in our sample (asterisks) overlaid on a map of the two-dimensional velocity dispersion (gray scale). The sizes of the asterisksrepresent the stars’ 2.2 mm brightness. The black region is a minimum in the velocity dispersion, located ∼2! from Sgr A* (black plus sign), and is caused byfive comoving stars (blue), which define the newly identified IRS 16 SW comoving group. (b) A K-band (2.2 mm) speckle image showing the clustering of brightsources at the position of the IRS 16 SW comoving group. Group members are marked with blue crosses. Bottom: Proper motions of the IRS 16 SW comovinggroup members. In each 0!.1#0!.1 panel, the stellar positions are plotted with different years’ data labeled with different colors.

the main maps and above 0.5 in the submaps. Only sourcesdetected in all three submaps are included in the final sourcelist for each observation. The coordinate system for each listis transformed to a common local reference frame by mini-mizing the net offsets of all stars as described in Ghez et al.(1998, 2005). Centroiding uncertainties are ∼1 mas, while align-ment uncertainties range from ∼1 to 5 mas. The final relativepositional uncertainty is the quadrature sum of the centroid-ing and alignment uncertainties and is ∼2 mas for the bright(K " 13.5) stars near IRS 16 SW. Proper motions are derivedby fitting lines to the positions as a function of time, weightedby the positional uncertainties. We conservatively require thatonly sources detected in nine or more epochs, out of 22 totalepochs, are included in the final sample. This results in a finalsample of 180 stars, which have an average total proper-motionuncertainty of 0.53 mas yr!1 for all sources located beyond 1!of the central SBH. All proper motions were converted to linearvelocities using a distance of 8 kpc, and the uncertainty in thisdistance is not included in the velocity uncertainties (Reid1993).

3. RESULTS

A two-dimensional velocity dispersion map of the stars inthe sample reveals a minimum located between IRS 16 SWand IRS 16 SW-E (Fig. 1a, gray scale). The velocity dispersionmap is produced by calculating, at each position separated by0!.1, the following quantity for the nearest six stars: p2jintrinsic

! [error2( x, i)" error2( y, i)]/[2(N ! 1)], where theN2j ! v vmeasured ip0first term is the dispersion of the measured proper motions andthe second term removes the bias introduced by the uncertain-ties in the proper-motion measurements. The minimum in thevelocity dispersion map is insensitive to the number of starsused in the calculation; using the nearest five to the nearest

eight stars produces a similar result. The significance of thevelocity dispersion minimum is determined by comparing itwith the velocity dispersion of stars in the sample that are atcomparable radii (1! ≤ r2D ≤ 2!.6). Because the young stars areknown to show some level of dynamical anisotropy due tocoherent rotation about the SBH (Genzel et al. 2003), we re-strict the comparison sample to known late-type stars (Figer etal. 2003; Ott 2003). The minimum in the velocity dispersionis significantly lower (4.6 j) than the field velocity dispersion.The velocity dispersion minimum arises from a comoving

group of stars; to formally define the members of the comov-ing group, we must first eliminate those stars that appear nearthe group as a result of projection effects. Formal membershipis determined by considering the difference between the ve-locity of each individual star and the group’s average velocity.Within the region of the velocity dispersion minimum, onlyfive stars have velocity offsets that are consistently ≤2j. Usingonly these five stars to redefine the values of group velocityand velocity dispersion, we find no additional stars with a totalvelocity offset less than 3.5 j within a 1!.1 search radius. Wetherefore define these five stars, which include IRS 16 SW,as the members of the comoving group (Table 1 and Fig. 1,bottom). The IRS 16 SW comoving group has an average andrms distance from Sgr A* of 1!.92 and 0!.43, respectively, witha velocity dispersion of 36!13 km s!1 in right ascension and38!13 km s!1 in declination.There are two additional, independent lines of evidence sup-

porting the existence of this comoving group. First, two of thegroup members, IRS 16 SW and IRS 16 SW-E, have identicalradial velocities (Ott 2003); the other members, unfortunately,have no measured radial velocities. Second, the stellar numberdensity counts show an enhancement at the position of theIRS 16 SW comoving group (see Fig. 1b). Since the stellar

Binary system

Lu et al 2005

Three known stellar binaries:

• IRS 16SW: period ~ 19.5 days, mass ~ 50 M⊙, at ~0.05 pc (Ott et al. 1999, Martins et al. 2006)

• IRS 16NE: period ~ 224 days, e~0.3, at ~0.1 pc (Pfuhl et al. 2014)

• E60: period ~ 2.3 days, mass ~ 30 M⊙, at ~ 0.1 pc (Pfuhl et al. 2014)


Each observation has been processed using the techniquesdescribed inMuno et al. (2003). In brief, for each observation wecorrected the pulse heights of the events for position-dependentcharge-transfer inefficiency (Townsley et al. 2002a), excludedevents that did not pass the standard ASCA grade filters andChandraX-ray Center (CXC) good-time filters, and removed in-tervals during which the background rate flared to !3 ! above

the mean level. Finally, we applied a correction to the absoluteastrometry of each pointing using three Tycho sources detectedstrongly in eachChandra observation (Baganoff et al. 2003).Weestimated combined accuracy of our astrometric frame and of thepositions of the individual X-ray sources by comparing the off-sets between 36 foregroundX-ray sources that were locatedwithin50 of Sgr A" (Muno et al. 2003) and their counterparts from the

TABLE 1

Observations of the Inner 20 pc of the Galaxy

Aim Point (deg)

Start Time(UT) Sequence

Exposure(s)

R.A.

(J2000.0)

Decl.

(J2000.0)

Roll(deg)

1999 Sep 21 02:43:00 ................. 0242 40,872 266.41382 #29.0130 268

2000 Oct 26 18:15:11.................. 1561 35,705 266.41344 #29.0128 265

2001 Jul 14 01:51:10................... 1561 13,504 266.41344 #29.0128 265

2002 Feb 19 14:27:32 ................. 2951 12,370 266.41867 #29.0033 91

2002 Mar 23 12:25:04................. 2952 11,859 266.41897 #29.0034 88

2002 Apr 19 10:39:01 ................. 2953 11,632 266.41923 #29.0034 85

2002 May 7 09:25:07 .................. 2954 12,455 266.41938 #29.0037 82

2002 May 22 22:59:15 ................ 2943 34,651 266.41991 #29.0041 76

2002 May 24 11:50:13 ................ 3663 37,959 266.41993 #29.0041 76

2002 May 25 15:16:03 ................ 3392 166,690 266.41992 #29.0041 76

2002 May 28 05:34:44 ................ 3393 158,026 266.41992 #29.0041 76

2002 Jun 3 01:24:37.................... 3665 89,928 266.41992 #29.0041 76

2003 Jun 19 18:28:55.................. 3549 24,791 266.42092 #29.0105 347

2004 Jul 5 22:33:11..................... 4683 49,524 266.41605 #29.0124 286

2004 Jul 6 22:29:57..................... 4684 49,527 266.41597 #29.0124 285

2004 Aug 28 12:03:59 ................ 5630 5,106 266.41477 #29.0121 271

2005 Feb 27 06:26:04 ................. 6113 4,855 266.41870 #29.0035 91

Fig. 1.—Images of the 1000 around the super-massive black hole Sgr A", which illustrate the appearance of CXOGC J174540.0#290031. Left: Image created fromthe average of 13 observations (650 ks exposure) taken between 1999 September and 2003 June, demonstrating the quiescent state of the region. Right: Image createdfrom 2 observations (99 ks) taken on 2004 July 5–7, in which a new transient X-ray source is evident 2B5 south of Sgr A" (circle). The location of twin lobes of the radiotransient identified with the VLA are indicated by diamonds. Finally, a portion of the diffuse emission brightened coincident with the transient outburst (ellipse). Bothimages are displayed at the 0B5 resolution of the detector. They were generated from the raw counts and then scaled to correct for the relative exposures and the spatiallyvarying effective area of the detector.

REMARKABLE GALACTIC CENTER TRANSIENT 229

Muno et al 20055 arcsec from Sgr A*

•X-ray Binaries: Many in the inner 1 pc (e.g., Muno et al. 2005a, b,

Hailey et al. 2018)

•Hypervelocity stars (e.g., Brown et al

2005,2006,2007,2008, Hills 1988; Miralda-Escude & Gould 2000; Quillen & Gould 2003; Yu and Tremaine 2003; O’Leary & Loeb 2007, Perets et al. 2009; Perets 2009)

•Stellar disk properties suggest large binary fraction (Naoz et al. 2018)

•G2 has been suggest to be a binary star merger (Phifer et al.

2013, Witzel et al. 2014, Prodan et al. 2015, Stephan et al. 2016)

THE KOZA I - L IDOV EFFECT

(Not drawn to scale)



m1

m2

m3


Inner Orbit(Not drawn to scale)

a1

m1

m2

m3


Outer orbit


a2

a1

m1

m2

m3


e.g., Naoz 2016


e.g., Naoz 2016

a1/a2 << 1


e.g., Naoz 2016

a1/a2 << 1Quadrupole order approximation (n = 2):

Assuming m2 is test particle & circular outer orbit: e.g. Lidov 1962, Kozai 1962


e.g., Naoz 2016

a1/a2 << 1Quadrupole order approximation (n = 2):

Assuming m2 is test particle & circular outer orbit: e.g. Lidov 1962, Kozai 1962

Lz /p1 e

2cos(i) = constant


Outer orbit


a2

a1

m1

m2

m3


Outer orbit


a2

a1

m1

m2m3

THE ECC EN TR I C KOZA I - L IDOV EFFECT

• Need to go up to the octupole level of approximation for eccentric outer orbit (e.g. Ford et. al. 2000, Blaes et. al. 2002, Naoz et. al. 2011,2013)

• Lz is not conserved

Naoz 2016

Octupole

Quadrupole

Lz /p

1 e

2cos(i)

6= constant

THE B I G P ICTURE

Outer orbit

Inner Orbit


SMBH

Stellar-mass BBH

Inside 0.1 pc

PHYS ICS I NCLUDED IN MONTE CARLO S IMULAT IONS


Long term evolution equations up to the octupole level of approximation



General Relativity (GR) precession of the inner and outer orbits (PN1)




Gravitational wave emission for the inner orbit (PN2.5)




Gravitational wave emission for the inner orbit (PN2.5)

Long term interaction with background stars

O BLACK HOLES , WHERE ART THOU?

Hoang et. al. 2018

How do we distribute a2?

Will different black hole

binary number density profiles

make a difference?

n(r) ~ r−2 Bahcall & Wolf (1976)n(r) ~ r−3 de-projection of a disk

After 3-body stability

WHAT ARE THE POSS I BLE OUTCOMES?


1) Fly-bys from other stars cause the BH binary to become

unbound



unbound

Soft binaries



unbound

2) The BH binary merges without EKL’s help (GW-only mergers). Average merger time ~ 100 Myr

Soft binaries



unbound

2) The BH binary merges without EKL’s help (GW-only mergers). Average merger time ~ 100 Myr

~ 9% of all GC systems ~9.1% of all BW systems

Soft binaries


3) EKL causes the BH binary to merge much more quickly! (EKL mergers).

Average merger time ~ 10 Myr




SMBH

Now Merge!




SMBH

Now Merge!

~7% of all GC systems ~1.7% of all BW systems

EXAMPLE EVOLUT ION

Hoang et. al. 2018

Without EKL, this system would take ~ 6.8 x 1012 years

to merge!

MOVE A L I TTLE CLOSER , B INARYHoang et. al. 2018

MOVE A L I TTLE CLOSER , B INARY

EKL mergers are more likely to take place closer to the SMBH

Hoang et. al. 2018

MERGERS ACROSS THE UN IVERSE

total = ng fSMBH


total = ng fSMBH

= Nsteady fmerge


? ??

? ??

?

total = ng fSMBH

= Nsteady fmerge


Hoang et. al. 2018

1 14 Gpc3yr1


Hoang et. al. 2018

1 14 Gpc3yr1

=a1a2

e21 e22

I S TH I S MERGER MECHAN ISM D I ST INGU I SHABLE FROM OTHERS?

I S TH I S MERGER MECHAN ISM D I ST INGU I SHABLE FROM OTHERS?The supermassive black hole causes a phase shift in the GW

signal that may be detectable by LISA

(e.g. Meiron et. al. 2017, Inayoshi et. al. 2017)




Blueshifted




Redshifted

Blueshifted

SUMMARY1) The SMBH results in an enhanced number of BH binary

mergers

2) The number of steady state binaries is the key parameter in determining the total merger rate

3) EKL produces merger rates comparable to other dynamical mechanisms

4) We may be able to distinguish binary BH mergers in the vicinity of SMBH from other channels of mergers (e.g.

Inayoshi et. al. 2017, Meiron et. al. 2017)

EXTRA SL IDES

T IMESCALES

unbinding

timescale

vector resonant

relaxation

GR precession (inner orbit)

quad

times

cale

For a nominal system:

a1 = 1AU e2 = 0.5

Hoang et al. 2018

E C C E N T R I C I T Y S P I K E S

The Astrophysical Journal, 785:116 (8pp), 2014 April 20 Li et al.

0

100

180

degr

ee

10−2

100

AU

0 2 4 6 8 10 1230

35

40

time (Myr)

a 1(AU

)

iψ

rp

rL

Figure 9. Example illustrating a tidal disruption event. The initial conditionis the same as in Figure 7, except a1 = 39 AU. Similar to Figure 7, bothtidal dissipation and general relativity precession effects are included (see text).During the flip, e1 ∼ 1 and the tidal dissipation forces the orbit to decay (asshown in the bottom panel). However, the tidal circularization is outrun by theeccentricity excitation during the flip, and the object is disrupted before reaching180 when rp < rL, where rL is the Roche limit of the object to m1.(A color version of this figure is available in the online journal.)

0 0.5 1e1, 0

−6

−5

−4

−3

−2

−1

0

0

20

40

60

80

i 0

0 0.5 10

20

40

60

80

e1, 0

i 0

t = 3 tKozai

t = 10 tKozai t = 30 t

Kozai

t = 5 tKozai

log[min(1−e1)], ω = 0, ε = 0.03

Figure 10. Maximum eccentricity. The maximum eccentricity reached duringthe secular evolution in time 3tKozai (upper left panel), 5tKozai (upper right panel),10 tKozai (lower left panel), and 30 tKozai (lower right panel) as a function of theinitial eccentricity (horizontal axis) and inclination (vertical axis). Tides are notincluded in the simulation. The initial conditions of the runs are m1 = 1 M⊙,m2 = 0.1 M⊙, a1 = 1 AU, a2 = 45.7 AU, e2 = 0.7, ω1 = 0, andΩ1 = 180. The typical eccentricity reached at the first flip is ∼1–10−4, and theeccentricity may increase to ∼1–10−6 after several flips. The HiLe case reachesthe maximum eccentricity later than the LiHe case. The inner orbit flips abovethe black solid lines.(A color version of this figure is available in the online journal.)

A very large eccentricity does not immediately imply a tidaldissipation event, since this depends on the initial separation ofthe orbit. We map the maximum eccentricity that can be reachedduring the evolution, which may then be useful to examine thelikelihood of tidal disruption for specific systems.

Specifically, we study the maximum eccentricity reachedduring the evolution for ϵ = 0.03. Since this depends on thetime the integration stops, we record the respective maximumeccentricity of the inner orbit for integration times 3tKozai,5 tKozai, 10 tKozai, and 30 tKozai. As shown in Figure 10, theeccentricity of the inner orbit can be very close to 1, with

1 − e1,max ∼ 10−4 during the first flip, and 10−6 over longertime periods.

This process is relevant for estimating the rates of planet–starcollisions (Hellier et al. 2009; Bear et al. 2011), stellar tidaldisruptions due to black hole binaries (Ivanov et al. 2005; Colpi& Dotti 2011; Chen et al. 2011; Wegg & Bode 2011; Bode& Wegg 2013; Stone & Loeb 2012; G. Li et al. 2014b, inpreparation), Type 1a supernovae (Katz & Dong 2012), star–starcollisions (e.g., Perets & Fabrycky 2009; Thompson 2011;Katz & Dong 2012; Shappee & Thompson 2013; Naoz et al.2013a; Naoz & Fabrycky 2014), and gravitational wave sources(O’Leary et al. 2009; Kocsis & Levin 2012).

5. CONCLUSION

We have presented a new mechanism that flips an eccentricinner orbit by 180 starting with a near-coplanar configurationin a hierarchical three-body system with an eccentric outer per-turber. We use the secular approximation to study the dynamics,and show the agreement between the secular treatment and theN-body simulation in Figure 2.

The HeLi flip is a different mechanism from the LeHiflip discussed by Naoz et al. (2011, 2013a). The underlyingresonances causing the large oscillation in the inclination andthe flip are different: the LeHi flip is caused by both thequadrupole and the octupole interactions. However, in the HeLicase, only octupole resonances are in play (see G. Li et al. 2014a,in preparation for further discussion). Moreover, for the lowinclination case, the orbital evolution is regular, which admits asimple analytic flip criterion and timescale (which were shownto agree with the numerical results in Figure 5). Specifically,the flip criterion is shown in Equation (14). In addition, thedifference can be seen through the evolution of the orbit: theeccentricity increases monotonically and the inclination remainslow before the flip, and the flip timescale of the coplanar caseis shorter compared with the high inclination case (see Figure 3and movies). Finally, we explored the entire e1 and i0 parameterspace, including both the high inclination and low inclinationflips. We studied the flip condition for the initial condition inFigure 6. The evolution of the near-coplanar systems is distinctfrom the exact coplanar systems, because in the exact coplanarsystems the net force normal to the orbital plane is zero andthus the orbit cannot flip. Therefore, the N-body simulations thatassume exactly zero inclination may miss some of the dynamicalbehavior arises, even for small deviations from coplanarity.

Observations of the sky-projected obliquity angle of hotJupiters shows that their orbital orientation ranges from almostperfectly aligned to almost perfectly anti-aligned with respectto the spin of the star (Albrecht et al. 2012). We showedin the hierarchal, nearly coplanar, three-body framework aninitial eccentric inner orbit can flip its orientation by almost180 in the presence of an eccentric companion (Figures 5and 6). During the planet’s evolution, its eccentricity is increasedmonotonically, and thus tides are able to shrink and circularizethe orbit. If the planet has flipped by ∼180 before the tidalevolution dominates, a counter-orbiting close-in planet can beformed.

Figure 7 demonstrated this behavior. Not only does the fi-nal planet inclination reach 180 with respect to the total an-gular momentum, but also the obliquity. This is because thetimescale to torque the spin of the star is much longer than theorbital flip timescale, the spin–orbit angle is similar to the incli-nation at ∼180. Therefore, starting with an initially aligned

7


0

100

180

degr

ee

10−2

100

AU

0 2 4 6 8 10 1230

35

40

time (Myr)

a 1(AU

)

iψ

rp

rL


0 0.5 1e1, 0

−6

−5

−4

−3

−2

−1

0

0

20

40

60

80

i 0

0 0.5 10

20

40

60

80

e1, 0

i 0

t = 3 tKozai


Kozai

t = 5 tKozai

log[min(1−e1)], ω = 0, ε = 0.03






5. CONCLUSION





7


0

100

180

degr

ee

10−2

100

AU

0 2 4 6 8 10 1230

35

40

time (Myr)

a 1(AU

)

iψ

rp

rL


0 0.5 1e1, 0

−6

−5

−4

−3

−2

−1

0

0

20

40

60

80

i 0

0 0.5 10

20

40

60

80

e1, 0

i 0t = 3 t

Kozai


Kozai

t = 5 tKozai

log[min(1−e1)], ω = 0, ε = 0.03






5. CONCLUSION





7


0

100

180

degr

ee

10−2

100

AU

0 2 4 6 8 10 1230

35

40

time (Myr)

a 1(AU

)

iψ

rp

rL


0 0.5 1e1, 0

−6

−5

−4

−3

−2

−1

0

0

20

40

60

80

i 0

0 0.5 10

20

40

60

80

e1, 0

i 0

t = 3 tKozai


Kozai

t = 5 tKozai

log[min(1−e1)], ω = 0, ε = 0.03






5. CONCLUSION





7


0

100

180

degr

ee

10−2

100

AU

0 2 4 6 8 10 1230

35

40

time (Myr)

a 1(AU

)

iψ

rp

rL


0 0.5 1e1, 0

−6

−5

−4

−3

−2

−1

0

0

20

40

60

80

i 0

0 0.5 10

20

40

60

80

e1, 0

i 0

t = 3 tKozai


Kozai

t = 5 tKozai

log[min(1−e1)], ω = 0, ε = 0.03






5. CONCLUSION





7

Maximum eccentricity at the test particle regime:

Li, Naoz et al, (2014), ApJ 785, 116 + ApJ 791, 86

The Astrophysical Journal, 779:166 (14pp), 2013 December 20 Teyssandier et al.

40 50 60 70 80 90initial mutual inclination

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

e 2

10-6

10-5

10-4

10-3

10-2

10-1

100

1-e 1

,max

Figure 11. Maximum inner eccentricity (given as 1 − e1,max in logarithmicscale) for the run in Figure 10. Very high eccentricities are associated with flipsof the inner orbit.(A color version of this figure is available in the online journal.)

our integration time (8 Gyr) most of the runs already converged(see Appendix A.2). Another interesting regime that arises fromthe parameter maps is a “transition zone” where the inner planetspends only about 10%–20% of its time on a retrograde orbit(colored pale blue in the figures).

Also important are the behaviors of the inner and outerorbits’ eccentricities. In Figure 11, we show the maximum e1reached in the corresponding run of Figure 10, and in Figure 12we show the (relative) maximum e2 for the same run. Notsurprisingly, the behavior closely resembles that of the testparticle approximation. The probability of flipping the orbitsmatches the maximum value of e1: flips are associated withexcursions to very high eccentricities, which, in fact, happen justbefore the flip. We find excursions of at least 1 − e1,max ! 10−4

when f ≃ 0.5. Furthermore, in our case, the outer orbit’sangular momentum is changing too, as can be seen in Figure 12,where we show the maximal relative value reached by the outereccentricity. This plot shows that the suppression of flips at highinitial mutual inclinations is highly related to the outer orbit’sevolution. When the outer orbit’s eccentricity almost does notchange (marked in pale blue), the inner orbit is more likely toflip.

These numerical results suggest that HJs that formedthrough planet–planet secular interactions should have a massive("3 MJ ), eccentric ("0.25) companion with a SMA between 50and 100 AU and a mutual inclination between 55 and 85. Aplanetary companion like this can drive a Jupiter-like planet in5 AU to a large eccentricity, which in the presence of dissipationcan result in shrinking the orbit to form a HJ (see Naoz et al.2011).

In Appendix A.1, we study the distribution of another variableof interest, the maximum mutual inclination reached by the samesystems as the ones studied in this section. We show that systemsfor which f > 0 all reach the same maximum inclination ofabout 140, which is one of the critical Kozai angles.

3.2. Inner Orbit Eccentricity Distribution

As noted before, we focus on the dynamical evolutionand neglect dissipation throughout the paper. However, tidaldissipation will become important when the inner planet reachesvery high eccentricities. Therefore, in this section, we focus

40 50 60 70 80 90initial mutual inclination

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

e 2

0

0.05

0.1

0.15

0.2

0.25

0.3

(e2,

max

-e2,

0) /

e 2,0

Figure 12. Variation of the outer eccentricity e2 for the run in Figure 10. Thecolor scale shows (e2,max −e2,0)/e2,0, where e2,0 is the initial outer eccentricity.This map indicates that the back reaction from the inner planet on the outerplanet is more important at high mutual inclination and low eccentricities.(A color version of this figure is available in the online journal.)

specifically on the inner orbit’s eccentricity distribution for thesesystems. In Figure 13, we show the cumulative distribution of theinner orbit’s eccentricity for different outer orbit configurations.Although a flip (itot > 90) happens when the inner orbit’seccentricity reaches a minimum, it also happens right after alarge-eccentricity peak (see Lithwick & Naoz 2011; Naoz et al.2013a for discussion); thus, the large-eccentricity peaks are agood proxy for a flip and vice versa (it is certainly the case forthe test particle scenario, as shown in Naoz et al. 2012, and weshow here that it remains true when this approximation breaksdown.)

As shown in Figure 13, a systematically low inner orbiteccentricity excitation is achieved for a combination of oneor more of the following conditions for the outer orbit: lowmass, low eccentricity, large orbital separation, and low mutualinclinations. However, for high mutual inclinations (#50),high outer orbit eccentricities (#0.25), and a massive perturber(#5 MJ ), the cumulative distribution is insensitive to the initialconditions. For these cases, as soon as the octupole effectsare triggered, the inner eccentricity reaches extreme values(e1 # 0.99). As a consequence, a counterplay may take placebetween the nearly radial orbit, which drives the planet to thestar, and tidal dissipation, which can shrink and circularize theplanet’s orbit. As shown in Naoz et al. (2011), a fairly highpercentage of planets formed by this mechanism end up as HJs.

4. STATISTICAL ESTIMATION THROUGH A MONTECARLO EXPERIMENT

We explore the statistical properties of two representativescenarios of systems that are not only significantly differentfrom the test particle approximation but also distinct from oneanother. In the first scenario, we consider a perturber with amass of 2 MJ (comparable to that of the inner planet, 1 MJ) ata2 = 61 AU. Such a system was shown in the previous section tosuppress the EKL behavior. In the second scenario, we considera system with a perturber with a mass of 6 MJ at a2 = 61 AU.We showed that such a system can undergo large inclinationand eccentricity oscillations but still significantly differs fromthe test particle approximation since the EKL mechanism issuppressed near initial perpendicular configurations. As shownin Figure 24 in Appendix A.2, most of these systems have

8

Maximum eccentricity outside the test particle regime

Teyssandier, Naoz, Lizarraga Rasio (2013), ApJ 779, 166

I N I T IAL COND IT IONS

I NCL INED TO MERGE

B INAR I ES I N GALACT IC NUCLE I

Frac

tion

of S

tars

with

Co

mp

anio

ns (%

)

Spectral Type

Raghavan et. al. 2010

B INAR I ES I N GALACT IC NUCLE I

Frac

tion

of S

tars

with

Co

mp

anio

ns (%

)

Spectral Type

Raghavan et. al. 2010

Most massive stars are in binaries or higher multiples!

M E R G E R T I M E S C A L E D I S T R I B U T I O N

QUADRUPOLE VS . OCTUPOLE

m1 = 12.8 M

m2 = 63.3 M

a1 = 5.1 AU

a2 = 936 AU

e1 = 0.014

e2 = 0.4

i = 92.8

degrees


GC EKL Mergers: BW EKL Mergers:

m1 = 12.8 M

m2 = 63.3 M

a1 = 5.1 AU

a2 = 936 AU

e1 = 0.014

e2 = 0.4

i = 92.8

degrees


GC EKL Mergers: BW EKL Mergers:-EKL important in 16% -2 non-mergers using

only quadrupole

m1 = 12.8 M

m2 = 63.3 M

a1 = 5.1 AU

a2 = 936 AU

e1 = 0.014

e2 = 0.4

i = 92.8

degrees


GC EKL Mergers: BW EKL Mergers:-EKL important in 16% -2 non-mergers using

only quadrupole

-EKL important in 40% -6 non-mergers using

only quadrupole

m1 = 12.8 M

m2 = 63.3 M

a1 = 5.1 AU

a2 = 936 AU

e1 = 0.014

e2 = 0.4

i = 92.8

degrees

ADD IT IONAL TESTS

P R E D I C T E D M E R G E R R AT E S


observations

~12-240 Gpc-3 yr-1



~0.7 Gpc-3 yr-1

(e.g., Tutukov & Yungelson 1993,Portegies Zwart & Yungelson 1998, Bethe & Brown 1999)

Globular clusters ~5 Gpc-3 yr-1

(Rodriguez et. al. 2016)

Nuclear star clusters

~1.5 Gpc-3 yr-1

(Antonini & Rasio 2016)

Special modes of binary evolution

~10 Gpc-3 yr-1

(Mandel & de Mink 2016)

A TALE OF TWO S IMULAT IONS

Two sets of Monte Carlo simulations

MSMBH 1 107M 4 106M

n(r) / a2 = 2 = 3



MSMBH

Bahcall-Wolf (BW)

1 107M 4 106M

n(r) / a2 = 2 = 3



MSMBH

Galactic Center (GC)

Bahcall-Wolf (BW)

1 107M 4 106M

n(r) / a2 = 2 = 3

SOFT B INAR I ES SOFTEN , HARD B INAR I ES HARDEN

• Heggie’s Law

• We define hard binaries as (Quinlan 1996):

• We only take soft binaries into account in our rates calculation

Vbin

>

p1 +m1/m2

TYPES OF MERGER

• tGR < tKL: GR precession of the inner orbit suppresses EKL eccentricity excitations

• tGW < tEvap: The BHB still merges, even without eccentricity excitations

GW-only Mergers

TYPES OF MERGER

• tGR < tKL: GR precession of the inner orbit suppresses EKL eccentricity excitations

• tGW < tEvap: The BHB still merges, even without eccentricity excitations

GW-only Mergers

~ 9% of all GC systems ~9.1% of all BW systems

TYPES OF MERGER

• tGR > tKL: GR precession of the inner orbit is small and does not interfere with the EKL mechanism. Eccentricity is excited!

• tmerge < tEvap: the binary merges before it becomes unbound due to interactions with background stars

EKL-induced Mergers

TYPES OF MERGER



EKL-induced Mergers

This BHB system would take ~ 1 Gyr to merge withoutEKL!

TYPES OF MERGER



EKL-induced Mergers

~7% of all GC systems ~1.7% of all BW systems

ADD IT IONAL TESTS

DOES THE MERGER RATE DEPEND ON D I STANCE FROM SMBH?

B INARY BLACK HOLES MERGERS DETECTED BY L I GO/V I RGO !

Abbott et. al. 2016

Five mergers so far! GW150914:

29 and 36 solar massesGW151226:

14 and 8 solar massesGW170104:

32 and 19 solar massesGW170608

12 and 7 solar masses GW170814

31 and 25 solar masses

(+ GW170817: NS-NS merger)

Bao-Minh Hoang

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