tst ss s - - PowerPoint PPT Presentation
tst ss s ts r t t tr tst ss s
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r
log10 a/mpc log10 j LSO GW timescales GW Separatrix aGW RR>NR contour AI contour MW S-stars
1 2 3
RR > NR RR < NR MBH GW Plunge
Bar-Or & TA 2015
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r
Test star "Loss cone" BH and "strong interaction zone"
1:TA & Hopman 2003 2:Bar-Or & TA 2015 3: TA 2005 4:TA & Livio 2001 5:Hopman & TA 2005, 2006a, 2006b 6:TA & Morris 2003 7:TA 1999 8:TA & Livio 2004 9:Perets, Hopman & TA 2007 10:TA & Hopman 2009 11:TA & Kumar 2001 12:Bar-Or & TA 2014 13:Bregman & TA 2012
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r
❘❛♥❞♦♠✐③❛t✐♦♥ ❜② r❡❧❛①❛t✐♦♥
Test star "Loss cone" BH and "strong interaction zone"
Stationary ellipses in point mass potential Planar rosettes in spherical potential
Effect on perturbed star Perturbing stars Scalar resonant relaxation Vector resonant relaxation
Rauch & Tremaine 1996
◗ = ▼•/▼⋆
✶/ ❧♦❣ ◗✿ r❡❧❛①❛t✐♦♥ ❜♦♦st ❢r♦♠ ❝❧♦s❡ ❡♥❝♦✉♥t❡rs P/t❝♦❤✿ r❡❧❛①❛t✐♦♥ ❜♦♦st ❢r♦♠ ❧♦♥❣ ❝♦❤❡r❡♥❝❡
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r
❘❛♥❞♦♠✐③❛t✐♦♥ ❜② r❡❧❛①❛t✐♦♥
❥ = ❏/❏❝ (❊) short P long P
c isco
log a log a Plunge log J/J
c
log J /J
lc J−scattering E,J−scattering
short P long P
crit isco
log J/J log a log a log a
c c
log J /J
lc
Inspiral Plunge
Detectable GW J−scattering Dissipation (Lightman & Shapiro 1977; Cohn & Kulsrud 1978) (TA & Hopman 2003; Hopman & TA 2005)
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r
❘❛♥❞♦♠✐③❛t✐♦♥ ❜② r❡❧❛①❛t✐♦♥
❚❤❡ ✏❢♦rt✉♥❛t❡ ❝♦✐♥❝✐❞❡♥❝❡✑ ❝♦♥❥❡❝t✉r❡✿
(Hopman & TA 2006)
◮ ❯♥q✉❡♥❝❤❡❞✱ ❘❘ ❞r✐✈❡s ❛❧❧ st❛rs t♦ ♣❧✉♥❣❡ ♦r❜✐ts ✭♥♦ ❊▼❘■s✦✮✳ ◮ O(β✷❥−✷✮ ●❘ ✐♥✲♣❧❛♥❡ ❙❝❤✇❛r③s❝❤✐❧❞ ♣r❡❝❡ss✐♦♥ ❜❡❝♦♠❡s s✐❣♥✐✜❝❛♥t ❜❡❢♦r❡ O(β✺/✷❥−✼◗−✶) ●❲ ❞✐ss✐♣❛t✐♦♥✳ ◮ ●❘ ♣r❡❝❡ss✐♦♥ q✉❡♥❝❤❡s ❘❘ ❛♥❞ ❛❧❧♦✇s ❊▼❘■s t♦ ♣r♦❝❡❡❞ ✉♥♣❡rt✉r❜❡❞✱ ❞❡❝♦✉♣❧❡❞ ❢r♦♠ t❤❡ ❜❛❝❦❣r♦✉♥❞ st❛rs✳
(Bar-Or & TA 2015)
log10(a/mpc) log10(j)
log10(d2n/dlgj dlga)
<-5.5 LSO GW inspiral plunge
0.5 1 1.5 2 2.5 3
nsnap=24982944 np=10469 ni=0
Rp= 8.0e-04 1/yr Ri= 0.0e+00 1/yr −2.5 −2.0 −1.5 −1.0 −0.5
log10(J/Jc)
−2 −1 1 2 3
log10(a/mpc)
RR > NR Last Stable Orbit
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
log10 a/mpc log10 j LSO GW timescales GW Separatrix aGW RR>NR contour AI contour MW S-stars
1 2 3
RR > NR RR < NR MBH GW Plunge
❙t❡❧❧❛r ❞②♥❛♠✐❝s ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡ ✭❇❡♥ ❇❛r✲❖r✬s t❛❧❦✮
❊✛❡❝t✐✈❡ ❘❘ ❞✐✛✉s✐♦♥ t❤❛t ❡①♣r❡ss ❝♦rr❡❧❛t❡❞ ♥♦✐s❡ ❛♥❞ s❡❝✉❧❛r ♣r❡❝❡ss✐♦♥s✱ t♦❣❡t❤❡r ✇✐t❤ ◆❘ ❞✐✛✉s✐♦♥ ❛♥❞ ●❲ ❞✐ss✐♣❛t✐♦♥✱ ♣r♦✈✐❞❡ ❛ ♣♦✇❡r❢✉❧ s❝❛❧❛❜❧❡ ▼♦♥t❡ ❈❛r❧♦ t♦♦❧ ❢♦r ♠♦❞❡❧✐♥❣ ❧♦♥❣✲t❡r♠ ❞②♥❛♠✐❝s ❛♥❞ ❧♦ss✲r❛t❡s ♦❢ ❣❛❧❛❝t✐❝ ♥✉❝❧❡✐ ✐♥ t❤❡ r❡❛❧✐st✐❝ ◆⋆ ≫ ✶ ❧✐♠✐t(†)✳
⋆ ❈♦rr❡❝t ❢♦r♠ ❛♥❞ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ✏❙❝❤✇❛r③s❝❤✐❧❞ ❇❛rr✐❡r✑ ✭▼❡rr✐tt✱ ❚❆✱ ▼✐❦❦♦❧❛ ✫ ❲✐❧❧ ✷✵✶✶✮✳ † ❱❛❧✐❞❛t❡❞ ❛❣❛✐♥st ◆✲❜♦❞② r❡s✉❧ts ✐♥ t❤❡ ❧♦✇✲◆⋆r❡❣✐♠❡✳
✭❇❛r✲❖r ✫ ❚❆ ✷✵✶✺✮
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
log10(a/mpc) log10(j)
log10(d2n/dlgj dlga)
<-3.5 LSO GW AI contour DC contour inspiral plunge
0.5 1 1.5 2 2.5 3
nsnap=24984314 np=9625 ni=29
Rp= 7.4e-04 1/yr Ri= 2.2e-06 1/yr
max a/a R (NR+RR) / R (NR) p p
102 103 104 105 106 107 108 109 1010
M•/M⊙
10−7 10−6 10−5 10−4 10−3 10−2
Rate [1/yr]
Plunges Inspirals MC data without RR MC data with RR
✑❋♦rt✉♥❛t❡ ❝♦✐♥❝✐❞❡♥❝❡✑ ✈❛❧✐❞❛t❡❞✳
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Incoming scattered binary Captured star Ejected Hyper−Velocity Star Efficient scattering by massive perturbers
(Perets, Hopman & TA 2007) (Hills 1991)
log10 a/mpc log10 j LSO Tidal disruption RR>NR contour AI contour MW S-stars (observed) Binary capture model ji Disk origin model ji
1 2 3
RR > NR RR < NR MBH TD 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 CDF[P(>jobs)] P(>jobs)
Cusp TBC PKS=0.40 ± 0.10 Cusp Disk PKS=0.10 ± 0.05 Core TBC PKS=6e-9 ± 1e-7 Core Disk PKS=1e-3 ± 1e-2
Blue giants Compact remnants Red giants B dwarfs Faint low−mass stars Warped disk(s) GMC [Binaries?] [Disrupted cluster + IMBH?] S−cluster [Dark cluster?] Molecular clumps ~0.5 pc ~0.04 pc 10−100 pc r_infl ~2 pc ~1.5 pcSabsovich, Bar-Or & TA 2016, in prep.
❈♦♥❝❧✉s✐♦♥s✿ ✶✳ ❘❘ ✐s ❡ss❡♥t✐❛❧ ❢♦r ❙✲st❛rs ♣♦st✲❝❛♣t✉r❡✴♠✐❣r❛t✐♦♥ r❛♥❞♦♠✐③❛t✐♦♥✳
Disk formation and migration (Levin 2007)
✷✳ ❇❡st ✜t ♠♦❞❡❧✿ ❚✐❞❛❧ ❝❛♣t✉r❡ ✐♥ ❞❡♥s❡ ♦❧❞ ❝✉s♣ ♦❢ st❡❧❧❛r r❡♠♥❛♥ts✳
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
◮ ❆❧❧ ②♦✉♥❣ st❛rs ✐♥s✐❞❡ ✵.✵✶ ♣❝ ♦❜s❡r✈❡❞✖♥♦♥❡ str♦♥❣❧② r❡❧❛t✐✈✐st✐❝✳ ◮ ▲♦✇ st❡❧❧❛r ❞❡♥s✐t② ⇒ ✈❡r② s❧♦✇ ❞②♥❛♠✐❝❛❧ ❡✈♦❧✉t✐♦♥✳ ◮ ⇒ ◆♦ ❧♦❝❛❧ st❡❧❧❛r t❛r❣❡ts ❢♦r t❡sts ♦❢ str♦♥❣ ●❘✳
◮ ❘❛♣✐❞ ❡✲❡✈♦❧✉t✐♦♥ ✭❘❘✮✱ ❜✉t s❧♦✇❡r ❛✲❡✈♦❧✉t✐♦♥ ✭◆❘✮✳ ◮ ❙t❡❧❧❛r ❞❡str✉❝t✐♦♥ ✭❚ ⋆ ∼ ✺ × ✶✵✼ ②r✮ ❢❛st❡r t❤❛♥ ❚❊ ❛t O(✶ ♣❝)✳ ◮ ❙tr♦♥❣ ❞❡♣❧❡t✐♦♥ ♦❢ ❧♦❝❛❧ str♦♥❣❧② r❡❧❛t✐✈✐st✐❝ st❛rs✳
◮ ▲♦✇✲♠❛ss ❡q✉✐✈❛❧❡♥ts ♦❢ ❙✲st❛rs ✴ ❍②♣❡r✲✈❡❧♦❝✐t② ❇✲st❛rs✳ ◮ ❋❛st✱ ❝♦♥t✐♥✉♦✉s s✉♣♣❧② r❛t❡ ✐❢ s❝❛❧❡❞ ❜② ❙✲st❛rs✳
1:Zucker, TA, Gillessen et al 2006 2:Will 2008 3:Merritt, TA, Mikkola & Will 2010 4:Buchholz et al 2009 5:Do et al 2009 6:Bartko et al 2010
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 3.0e+02 yr µas/yr 1 5 10 S 0.0004 0.0003 0.0002 J 0.0002 0.0001 8e-05 Q 4e-05 2e-05 6e-06 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 1.0e+03 yr µas/yr 1 5 10 S 0.003 0.002 0.001 J 0.001 0.0006 0.0004 Q 0.0002 9e-05 3e-05 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 3.0e+03 yr µas/yr 1 5 10 S 0.008 0.005 0.004 J 0.004 0.002 0.001 Q 0.0008 0.0003 0.0001 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 1.0e+04 yr µas/yr 1 5 10 S 0.03 0.02 0.01 J 0.01 0.007 0.005 Q 0.003 0.0009 0.0003 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 3.0e+04 yr µas/yr 1 5 10 S 0.09 0.05 0.04 J 0.04 0.02 0.01 Q 0.008 0.002 0.0007 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 1.0e+05 yr µas/yr 1 5 10 S 0.3 0.2 0.1 J 0.1 0.07 0.05 Q 0.03 0.006 0.001 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 3.0e+05 yr µas/yr 1 5 10 S 0.7 0.5 0.4 J 0.3 0.2 0.1 Q 0.06 0.01 0.002 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 1.0e+06 yr µas/yr 1 5 10 S 2 1 1 J 0.9 0.5 0.3 Q 0.1 0.02 0.003 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 3.0e+06 yr µas/yr 1 5 10 S 5 3 2 J 2 1 0.6 Q 0.2 0.03 0.004 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 1.0e+07 yr µas/yr 1 5 10 S 1e+01 5 4 J 3 2 0.9 Q 0.3 0.03 0.004 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 3.0e+07 yr µas/yr 1 5 10 S 2e+01 8 6 J 5 2 1 Q 0.3 0.04 0.005 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 1.0e+08 yr µas/yr 1 5 10 S 4e+01 1e+01 8 J 6 2 1 Q 0.4 0.04 0.005 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 3.0e+08 yr µas/yr 1 5 10 S 5e+01 2e+01 9 J 7 2 1 Q 0.4 0.04 0.005 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❘❡s✉❧ts
Orbital evolution of captured stars with NR+RR+GW+tides+collisions
1 5 10 1 5 10 1 5 µas/yr 10
log10(1-e)
0.5 1 1.5 log10(P/yr)
TD TH Q2 J S
t = 1.0e+09 yr µas/yr 1 5 10 S 5e+01 2e+01 9 J 7 2 1 Q 0.4 0.04 0.005 Sabsovich, Bar-Or & TA, 2016, in prep.
❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ❞②♥❛♠✐❝s✿ ■♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ●❛❧❛❝t✐❝ ❈❡♥t❡r ❙✉♠♠❛r②
◮ ◆❘✱ ❘❘✱ ●❲ ❞✐ss✐♣❛t✐♦♥ ❛♥❞ s❡❝✉❧❛r ♣r❡❝❡ss✐♦♥ ❝❛♥ ❜❡ tr❡❛t❡❞
◮ ❚❤❡ st❡❛❞② st❛t❡ ❞❡♣❡♥❞s ♠♦st❧② ♦♥ ◆❘✱ ✇❤✐❝❤ ❡r❛s❡s ❆■✳
◮ ❘❡❧❛t✐✈✐st✐❝ ❧♦ss✲❝♦♥❡ ♠♦❞❡❧✐♥❣ ♦❢ ❣❛❧❛❝t✐❝ ♥✉❝❧❡✐ ✐♥ ◆⋆ ≫ ✶ ❧✐♠✐t✳ ◮ P❧✉♥❣❡ ✴ ❊▼❘■ r❛t❡s ❛♥❞ ❜r❛♥❝❤✐♥❣ r❛t✐♦s✳
◮ ❖r✐❣✐♥ ♦❢ ❙✲st❛rs✿ ❉❛r❦ ❝✉s♣ ✰ ❜✐♥❛r② ❝❛♣t✉r❡ ❢❛✈♦r❡❞✳ ◮ O(✶✵✵) ❝❛♣t✉r❡❞ ❧♦✇✲♠❛ss r❡❧❛t✐✈✐st✐❝ st❛rs ♠❛② ❡①✐st ✐♥ ●❈✱ ❜✉t