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compact remnants of massive stars in star clusters , which are neutron stars ( ns ) and black holes ( bh ) , form a dynamically interesting sub - population due to their tendency of segregating towards the cluster s center and augmenting their population density therein . in this respect , the bhs are special in that they undergo a `` runaway '' mass segregation . these remnant bhs are typically several 10s of @xmath0 heavy , enough to form a spitzer - unstable sub - system , provided a significant number of them are retained in their parent cluster . due to this instability ( also called the mass - stratification instability , @xcite ) , the continually sinking bhs can not come to an energy equipartition with the local surrounding stars and finally end - up in a central , highly concentrated sub - cluster made purely of bhs , which is self - gravitating and dynamically nearly isolated from the rest of the stellar cluster @xcite . such a dense environment of bhs is dynamically very active due to the formation of bh - bh binaries via 3-body encounters @xcite and their hardening by super - elastic encounters @xcite with their surrounding bhs . studies of the dynamics of pure bh sub - clusters using monte - carlo and direct n - body integration methods indicate that the dynamical bh - bh merger events they generate are likely to contribute a significant gravitational wave ( gw ) detection rate to the future `` advanced ligo '' ( adligo ) and `` lisa '' gw observatories @xcite . such studies show that a bh sub - cluster is typically self - depleted in a few gyr due to the super - elastic dynamical encounters and the resulting escape of the bhs @xcite . the energy extracted from the tight bh - bh binaries heats - up and expands the cluster s core @xcite , which can be detectable by the future optical missions such as the `` gaia '' mission . furthermore , the bhs can be important for dynamically formed bh x - ray sources due to their encounters with the surrounding stars @xcite . x - ray observations have indicated the presence of bh x - ray binary candidates in gcs @xcite . the presence of a bh sub - cluster within a star cluster , therefore , has the potential to give rise to a plethora of physical phenomena , all of which have significance to upcoming prime missions such as the gaia , adligo and the present and future x - ray missions . is it possible to obtain any direct observational signature of the presence of a bh sub - cluster within a star cluster ? we predict here , for the first time , that within a few kpc from the galactic center , rapid tidal stripping of star clusters by the strong tidal field can expose its bh sub - cluster . this would happen when the timescale of the preferential removal of stars from the outer regions of the cluster is shorter or comparable to the encounter - driven self - depletion timescale of its central bh sub - cluster ( see above ) . such a dissolved phase of the cluster would consist of a few stars orbiting around a cluster of bhs and would observationally appear as a highly super - virial star cluster with a large mass - to - light - ratio .    as we discuss here , a number of direct n - body computations of model star clusters indeed support the formation of such systems . these objects comprise a predicted , new class of compact stellar populations which we name `` dark star clusters '' . the importance of dark star clusters ( hereafter dsc ) is twofold : on one hand , if they are found to exist , then they guarantee that star clusters are potential sites for gw emission and formation of bh x - ray binaries and on the other hand , they naturally constrain the uncertain natal kicks of bhs @xcite , as dscs can form only if a significant number of bhs retain in the cluster following their progenitor supernovae . this , in turn , restricts the theoretical models of core collapse supernovae @xcite . we compute the evolution of model star clusters subjected to the galactic tidal field using the direct n - body integration method . for our purposes , we use the state - of - the - art n - body integration code `` nbody6 '' @xcite , which , apart from utilizing a highly sophisticated numerical integration scheme @xcite , also follows the evolution of the individual stars until their remnant phases , using an analytic but well - tested stellar evolution recipe @xcite . a unique feature of nbody6 is its use of highly accurate regularization methods in resolving close encounters @xcite . furthermore , the code exploits the remarkable hardware - accelerated computing capacity of graphical processing units ( gpus ) in integrating the centers of masses . nbody6 currently incorporates general - relativistic effects only through analytic prescriptions of gw energy - loss . we follow the evolution of initial plummer clusters @xcite of single stars , having masses between @xmath1 and half - mass radii between @xmath2 . all the clusters initially consist of zero - age - main - sequence ( zams ) stars with their masses @xmath3 chosen from the canonical initial mass function ( imf ; @xcite ) @xmath4 , where @xmath5 for @xmath6 and @xmath7 ( salpeter index ) for @xmath8 . their metallicities are chosen to be solar , as suitable for our galaxy s disk . we assume for now that all the supernova remnants ( _ i.e. _ , bhs and nss ) receive low natal kicks in general such that they remain bound to their parent clusters at their formation . such models follow circular orbit around a point mass of @xmath9 representing the milky way bulge . [ fig : qevol ] ( top panel ) shows an example of the evolution of the virial coefficient for one of our computed model clusters with initially @xmath10 stars and @xmath11 pc , located at @xmath12 kpc galactocentric distance . the orange curve shows the time - evolution of the virial coefficient , @xmath13 , for all the members bound to the cluster including the bhs and the nss , which mostly remains constant at @xmath14 , as it should be for the quasi - static relaxation of a self - gravitating system through two - body encounters @xcite .    the green curve in fig .  [ fig : qevol ] ( top ) shows the time - evolution of the virial coefficient taking into account only the luminous objects , _ i.e. _ , the nuclear burning stars and the white dwarfs ( hereafter wd ) , which are those an observer sees . the bhs formed have masses @xmath15 and the nss are typically of @xmath16 , which are significantly more massive than the majority of the remaining luminous members and hence are segregated to the cluster s center . as the lower mass luminous stars in the outer regions are stripped by the external field , the gravitational potential of these central invisible remnants becomes increasingly important . therefore , the kinetic energy of the luminous sub - system increasingly exceeds the corresponding self - equilibrium ( or quasi - static ) value as the constituents perceive a potential that becomes increasingly deeper than their self - potential . this correspondingly raises their exclusive virial coefficient , @xmath17 , above 0.5 as in fig . [ fig : qevol ] ( top panel , green curve ) . the cluster thus evolves to a dsc state ( see sec . [ dcpop ] ) . note that while @xmath17 reaches a very high value , the system as a whole remains bound since @xmath18 throughout ( except at the final dissolved state , not shown in the figure ) . notably , nss form a few factors more often than the bhs and contribute significantly in elevating @xmath17 . however , the nss being lighter than the bhs , their sub - population occupies a more extended zone in the cluster s center . this causes them to get stripped earlier than the bhs ( _ c.f . _ [ fig : qevol ] , bottom panel ) so that in the late evolutionary phase , typically when @xmath19 , it is mostly the bh population that contributes to the augmented @xmath17 , thereby determining the lifetime of the dsc phase ( see sec . [ dcpop ] ) . nevertheless , at large enough distances from the galactic center , where slower tidal stripping causes the dsc phase to appear later than the self - depletion of the bh sub - cluster ( see sec .  [ intro ] ; @xcite ) , the nss constitute the primary dark component of the corresponding dsc state ( see sec .  [ dcpop ] ) . this self - depletion process , although operative for both the ns and the bh sub - populations , is more efficient for the latter as it is significantly more concentrated . [ fig : tfram ] ( top ) shows the expected increasing trend in the lifetime of the dscs with initial cluster mass @xmath20 ( @xmath21 , @xmath22 ) . the dsc phase can be defined when the cluster appears unbound , _ i.e. _ , @xmath23 or when it appears significantly super - virial , which we take when @xmath24 , and denote the corresponding lifetimes by @xmath25 with @xmath26 and 0.75 respectively . fig . [ fig : tfram ] ( bottom ) shows @xmath27 against @xmath28 ( @xmath29 , @xmath22 ) . @xmath25 also increases with increasing @xmath28 since the dsc takes a longer time to get depleted in a weaker external field . notably , for @xmath30 kpc , the dsc state becomes ns - dominated . beyond @xmath31 kpc , it takes more than the age of our galaxy ( @xmath32 gyr ) for the representative cluster to evolve to its dsc phase ( _ i.e. _ , @xmath24 ) . it then follows that for the present - day galaxy the dscs are formed from clusters with @xmath33 within @xmath34 kpc , to be taken as representative numbers , and have lifetimes @xmath35 myr and @xmath36 myr as conservative estimates . although these estimates are based on a point - mass tidal field , note that the dscs progenitor clusters form in the galactic disk and orbit on nearly circular paths ( in the equatorial plane ) and hence would experience the same external field with an axisymmetric disk - like distribution of the same mass . the @xmath37 dependence can however be moderately modified as the clusters see a mass increasing moderately with @xmath28 .    to estimate the galactic population of dscs , we take the average star cluster formation rate ( hereafter cfr ) over the entire galactic disk ( @xmath38 kpc ) to be @xmath39 @xcite following a schechter initial cluster mass function @xcite over the mass range @xmath40 @xcite which is assumed to remain invariant over the last few hundred myr . this implies @xmath41 clusters form per myr which have properties ( @xmath42 and @xmath43 kpc ) that must have them evolve to dscs if a sufficiently high fraction of bhs are retained . assuming a steady state conversion to the dsc phase with lifetime @xmath35 myr ( see above ) , the expected number of @xmath19 dscs in the galaxy within @xmath43 kpc is @xmath44 . for dscs with @xmath45 , the corresponding number is @xmath46 . hence , a significant number of dscs can be expected in the inner galactic zone .      there has been recent concern with the widely debated irs 13e ( hereafter irs13e ) ; an extremely compact stellar association of a few young , massive stars at a close projection to the galactic center that apparently survives the extreme tidal field by being bound by an invisible mass @xcite . while a @xmath47 intermediate mass black hole ( imbh ) was widely believed to be this invisible mass @xcite , this possibility has recently been ruled out with a significant confidence by @xcite through their newer proper motion measurements , leaving the nature of the invisible component of irs13e currently ambiguous . this status of irs13e prompts us to consider whether its dark component can be an ensemble of stellar - mass bhs instead . to that end , we perform preliminary n - body calculations to determine whether a dsc configuration resembling irs13e can be a possible fate of a star cluster very close to the galactic center . these computed clusters follow circular orbits around the galactic smbh ( a central mass of @xmath48 which is a combination of the mass of the smbh and that of the nuclear star cluster within @xmath49 pc ) within @xmath50 few pc , where star formation has been shown to possibly lead to a top - heavy stellar imf @xcite . since we are primarily interested in the final state , we begin the n - body calculations from an evolved phase of the cluster , for computational ease . therefore , we initiate the computations with plummer clusters made of stars which are pre - evolved until @xmath51 myr age . this age is slightly earlier than when the most massive star ( of @xmath52 ) evolves to a bh . all the stars are thus still on their main sequence . observations indicate a very flat imf ( @xmath53 ; see @xcite ) for stars close to the galactic center and also a dearth of low - mass stars . the latter is evident from a declining density of b - stars away from the central smbh @xcite and a significant lack of coronal x - ray emission @xcite from the sgra@xmath54 field . it is currently unclear from the literature what would be the lower - mass limit of such an imf , which , in turn , determines the @xmath55 of the mass function ( mf ) and its lower limit @xmath56 at cluster - age @xmath57 . in this preliminary study , we simply take the total number of stars in the cluster @xmath58 , @xmath55 and @xmath56 at age @xmath57 as free parameters . the upper mf limit is chosen to be the canonical @xmath59 and the stellar metallicity is solar . these limits are the zams values and they are appropriately reduced during the @xmath51 myr pre - evolution by the nbody6 s built - in stellar evolution prescription ( see above ) . like the computations in sec . [ darkcl ] , we retain all the bhs ( of @xmath60 as obtained from within nbody6 ) in the cluster . at the beginning of the computations , the models are taken compact to a similar extent as irs13e @xcite , with half - mass radii between @xmath61 and they orbit within @xmath62 . we find that to reach a irs13e - like state , _ i.e. _ , a state where typically @xmath32 luminous , young stars are very tightly bound to a cluster of @xmath63 bhs , it takes @xmath64 and a rather extreme lower cutoff of @xmath65 for mf index @xmath66 ( at age @xmath57 ) , this @xmath55 being close to the upper limit of the @xcite index . a steeper mf requires even higher @xmath56 . as a demonstration , it can be seen in fig . [ fig : irs13 ] ( top panel ) that this cluster eventually evolves to a configuration consisting of @xmath67 bhs , comprising a @xmath47 dark component , orbited by @xmath68 young stars . although the half - mass radius of the system expands at the beginning of the evolution , primarily driven by the mass loss through the massive stars winds and their supernovae , it finally collapses to @xmath69 pc , as shown in fig . [ fig : irs13 ] ( middle panel ) , by the time the system arrives at the above configuration . the luminous members in this state include 1 - 2 o - stars ( those still on the main sequence ) and a few helium and wolf - rayet stars , thereby being of similar variety as observed in irs13e . the final state of the system , therefore , resembles irs13e in terms of compactness and stellar content @xcite . [ fig : irs13 ] ( lower panel ) shows that the luminous sub - system becomes highly super - virial , _ i.e. _ , a dsc , as the irs13e - like state occurs . the calculations presented in this section are preliminary which suggest an intermediate state of a cluster , close to the galactic center , that would evolve to an irs13e - like configuration . it remains an open question whether any reasonable initial cluster evolves to such an intermediate phase , which depends on the low - mass limit of the galactic - central imf , its index and the initial mass and compactness of the cluster , at a given @xmath28 . a scan over these parameters , beginning with much larger and more compact clusters , is necessary to determine such possible initial configuration(s ) , which is much more compute - intensive . although the above @xmath56 appears too high for a lower cut - off , its progenitor cluster would have a smaller limit taking into account the rapid tidal stripping over the 3.5 myr pre - evolution . tidal stripping also implies a steeper initial index , in better agreement with the observed one . although we over - simplify by effectively ignoring the dynamical evolution of the cluster during the stellar pre - evolution , the dynamical history is not instrumental in determining the occurrence of a irs13e - type state which happens merely due to the competition between stellar evolution and tidal dissolution of the cluster . it is enough that the cluster remains bound by the time most of the stars become bhs quenching the wind mass loss ; the system would then core - collapse to a compact configuration irrespective of its history . this is in contrast with the `` classical '' dscs discussed in sec . [ darkcl ] whose formation depends crucially on mass segregation .    a potential drawback of this `` in - situ '' model is that it is likely to significantly over - produce the number of massive stars seen in the galactic center .    given these drawbacks , the calculations in this section are only to suggest that irs13e might be a dsc but are by no means conclusive . our calculations ( sec .  [ darkcl ] ) signify that a gravitationally bound star cluster naturally evolves to an apparent super - virial state , while remaining bound as a whole , as a consequence of the interplay between the dynamics and the evolution of stars , provided a significant number of stellar remnants survive in the bound system after their formation via supernovae . the existence of such dark star clusters is a first - time prediction and serves as an excellent cross - check of the retention of supernova remnants in star clusters , the effects of which is widely explored . the very presence of dscs would kill two birds with one stone by having consequences on the widely debated theoretical stellar collapse models @xcite due to the implied direct constraints on natal kicks , and by securing star clusters as potential sources for the forthcoming `` advanced ligo '' @xcite gw detector @xcite . given that the dscs must have a significant population in the inner region of our galaxy , it is tempting to conduct a survey of intermediate - aged ( see below ) stellar assemblies to identify them . dark star clusters can be observationally distinguished from actually dissolving clusters ( which are also super virial and hence expanding and are generally young ) through their compact sizes in spite of their intermediate ages , as shown in fig . [ fig : lagrev ] . that the apparent super - virial state of a dsc is not due to an imbh , would be indicated by the absence of a central cusp in its velocity - dispersion profile , which would be there otherwise @xcite the present studies justify dscs as predicted , new type of compact stellar populations . an immediate improvement over this study is to explore any effects of varying orbital eccentricity and different initial profile types ( _ e.g. _ , using king instead of plummer profiles ) . also , the effect of a varying star ( and hence cluster ) formation rate over the galactic disc needs to be incorporated to determine the predicted galactic dsc population , as an improvement over our assumption of a uniform average cluster formation . another development would be to consider a disk - like mass distribution instead of a central point - mass ( but see sec . [ dcpop ] ) . the preliminary computations in sec .  [ irs13e ] suggest that the dark component of irs13e can be an ensemble of stellar mass bhs and irs13e may therefore perhaps be a dsc . however , this conclusion should , for now , be taken as being suggestive rather than conclusive due to the drawbacks discussed ( sec . [ irs13e ] ) . we postpone a more detailed and self - consistent study on this issue in a future paper .                                                                            , @xmath11 pc and @xmath12 kpc . the lower ( orange ) curve represents the virial coefficient , @xmath13 , for the whole cluster , _ i.e. _ , including the luminous stars , nss and bhs bound to the cluster . the upper ( green ) curve represents the virial coefficient , @xmath17 , of the luminous objects exclusively . the monotonic growth of @xmath17 implies the approach towards an apparent highly super - virial state  the `` dark star cluster '' phase . * bottom : * the corresponding evolution of the numbers of the luminous stars , @xmath70 , the nss , @xmath71 and of the bhs , @xmath72.,title="fig:",width=434,height=264 ] , @xmath11 pc and @xmath12 kpc . the lower ( orange ) curve represents the virial coefficient , @xmath13 , for the whole cluster , _ i.e. _ , including the luminous stars , nss and bhs bound to the cluster . the upper ( green ) curve represents the virial coefficient , @xmath17 , of the luminous objects exclusively . the monotonic growth of @xmath17 implies the approach towards an apparent highly super - virial state  the `` dark star cluster '' phase . * bottom : * the corresponding evolution of the numbers of the luminous stars , @xmath70 , the nss , @xmath71 and of the bhs , @xmath72.,title="fig:",width=434,height=264 ]    , of the dscs with increasing initial plummer cluster mass @xmath20 ( @xmath73 kpc , @xmath11 pc ) . * bottom : * @xmath25 ( @xmath74 ) as a function of @xmath28 for a plummer cluster with @xmath29 and @xmath22.,title="fig:",width=434,height=302 ] , of the dscs with increasing initial plummer cluster mass @xmath20 ( @xmath73 kpc , @xmath11 pc ) . * bottom : * @xmath25 ( @xmath74 ) as a function of @xmath28 for a plummer cluster with @xmath29 and @xmath22.,title="fig:",width=434,height=302 ]    , and of the bhs , @xmath72 , for a model star cluster computation , leading to an irs13e solution . the direct n - body computation starts with a plummer cluster of pre - evolved stars having @xmath75 , @xmath76 pc and @xmath77 pc , where the stars are taken from a @xmath78 mf within @xmath79 zams mass ( see sec .  [ irs13e ] ) . the evolution of the lagrangian radii of the system is shown in the middle panel . the final state of the cluster resembles that of irs13e . the evolution of the viral coefficient @xmath13 , for the whole system , and @xmath17 , for the luminous sub - system is shown in the bottom panel.,title="fig:",width=415,height=245 ] , and of the bhs , @xmath72 , for a model star cluster computation , leading to an irs13e solution . the direct n - body computation starts with a plummer cluster of pre - evolved stars having @xmath75 , @xmath76 pc and @xmath77 pc , where the stars are taken from a @xmath78 mf within @xmath79 zams mass ( see sec . [ irs13e ] ) . the evolution of the lagrangian radii of the system is shown in the middle panel . the final state of the cluster resembles that of irs13e . the evolution of the viral coefficient @xmath13 , for the whole system , and @xmath17 , for the luminous sub - system is shown in the bottom panel.,title="fig:",width=423,height=245 ] , and of the bhs , @xmath72 , for a model star cluster computation , leading to an irs13e solution . the direct n - body computation starts with a plummer cluster of pre - evolved stars having @xmath75 , @xmath76 pc and @xmath77 pc , where the stars are taken from a @xmath78 mf within @xmath79 zams mass ( see sec . [ irs13e ] ) . the evolution of the lagrangian radii of the system is shown in the middle panel . the final state of the cluster resembles that of irs13e . the evolution of the viral coefficient @xmath13 , for the whole system , and @xmath17 , for the luminous sub - system is shown in the bottom panel.,title="fig:",width=415,height=245 ]    . the half - mass radius ( black curve , top panel ) remains nearly constant throughout and the core also remains compact following an initial brief contraction phase.,title="fig:",width=434,height=283 ] . the half - mass radius ( black curve , top panel ) remains nearly constant throughout and the core also remains compact following an initial brief contraction phase.,title="fig:",width=434,height=283 ]
among the most explored directions in the study of dense stellar systems is the investigation of the effects of the retention of supernova remnants , especially that of the massive stellar remnant black holes ( bh ) , in star clusters . by virtue of their eventual high central concentration , these stellar mass bhs potentially invoke a wide variety of physical phenomena , the most important ones being emission of gravitational waves ( gw ) , formation of x - ray binaries and modification of the dynamical evolution of the cluster . 
 here we propose , for the first time , that rapid removal of stars from the outer parts of a cluster by the strong tidal field in the inner region of our galaxy can unveil its bh sub - cluster , which appears like a star cluster that is gravitationally bound by an invisible mass . 
 we study the formation and properties of such systems through direct n - body computations and estimate that they can be present in significant numbers in the inner region of the milky way . 
 we call such objects `` dark star clusters '' ( dscs ) as they appear dimmer than normal star clusters of similar mass and they comprise a predicted , new class of entities 
 . the finding of dscs will robustly cross - check bh - retention ; they will not only constrain the uncertain natal kicks of bhs , thereby the widely - debated theoretical models of bh - formation , but will also pin - point star clusters as potential sites for gw emission for forthcoming ground - based detectors such as the `` advanced ligo '' . 
 finally , we also discuss the relevance of dscs for the nature of irs 13e .