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the milky way contains a supermassive black hole at its very center , sagittarius a * , whose mass is @xmath0m@xmath1 ( e.g. * ? ? ? a dense stellar cluster surrounds this black hole , with a central density at least comparable to that seen in the cores of the densest globular clusters . at least one , and possibly two , discs containing young stars at distances between @xmath2 and @xmath3pc from the black hole are also seen @xcite . the latter stellar population is thought to have an unusually flat imf @xcite . it has long been known that the central 0.2pc or so of the galactic center is deficient in bright red giants @xcite . since the center of the galaxy has a high number density of stars , it is natural to suggest that stellar collisions may explain the observed depletion . a recent crop of papers have studied the stellar population in the central regions @xcite . they report that the early - type stars ( bright main - sequence stars ) follow a cusp - like profile whilst the surface density of late - type ( red - giant ) stars that are brighter than a k magnitude of 15.5 is rather flat out to 0.4pc or 10arcsec . this surprising result implies that the red - giant population is depleted out to about 0.4pc from the central supermassive black hole . we will consider here whether stellar collisions could be responsible for this observed depletion . we will consider two different cases : 1 ) where the stellar population in the galactic center is drawn from a miller - scalo imf , and 2 ) where the imf is much flatter . in each case , we calculate collision probabilities between the various stellar species , and produce , via monte carlo techniques , a stellar population . from this we measure the surface density of the early and late - type stars ( to compare directly with what is observed ) making some reasonable assumptions concerning the effects of collisions on the population . the masses of stars contributing to the observed early and late - type populations are rather different , as shown in fig . [ mdavies_figure1 ] , where we have produced a synthetic population with a flat imf . we see that the early - type stars are virtually all between 12 and 27m@xmath1 whereas stars contributing to the observed late - type population have much lower masses , between 1 and 5m@xmath1 . the galactic center is a crowded environment , with number densities of stars probably of order @xmath4starspc@xmath5 or more . in such dense environments , collisions between stars will be frequent , and thus affect the stellar population . the cross section for two stars , having a relative velocity at infinity of @xmath6 , to pass within a distance @xmath7 is given by    @xmath8    where @xmath9 is the relative velocity of the two stars at closest approach in a parabolic encounter ( _ i.e. _  @xmath10 , where @xmath11 and @xmath12 are the masses of the two stars ) . the second term is due to the attractive gravitational force between the two stars , and is referred to as the gravitational focusing term . in the very center of the galaxy , where the supermassive black hole dominates , and stars may be assumed to move on keplerian orbits around the black hole at speeds exceeding 1000km / s , @xmath13 and we recover the result , @xmath14 . in this regime collisions involving larger red giants will be relatively more frequent despite their short lifetimes compared to main - sequence stars . this is not the case in globular clusters where @xmath15 and thus @xmath16 . one may estimate the timescale for a given star to undergo an encounter with another star , @xmath17 . the collision timescale will therefore be a function of both the number density of stars and the makeup of the stellar population , _ i.e. _  the distribution of stellar masses and types .    the effects of collisions will depend on the types of stars involved and on the relative speed of the two stars when they collide . collisions involving two main - sequence stars occurring at relatively low speed ( less than the surface escape speed of the stars ) are likely to result in the merger of the two objects with relatively low amounts of mass loss . collisions occurring at much higher speeds are likely to lead to significant mass loss and even to the destruction of the stars involved . collisions between main - sequence stars and compact objects ( _ i.e. _  black holes , white dwarfs or neutron stars ) are likely to be destructive ( dale et al . , in preparation ) . in the case of a black hole impactor , the main - sequence star is often torn apart by tidal forces , with some material being accreted by the black hole and rest being ejected . this is illustrated in fig . [ mdavies_figure2 ] where we show snapshots of a collision between a 1m@xmath18 main - sequence star and a 10m@xmath18 black hole . the close passage of the black hole results in the tidal disruption of the main - sequence star . a small fraction of the material is accreted by the black hole and the rest is dispersed . for neutron star and white dwarf impactors , a larger fraction of the material from the main - sequence star may form an envelope around the compact object . however such an object is likely to be relatively short lived , perhaps appearing as a bright red supergiant , before the envelope is ejected . thus all collisions between main - sequence stars and compact objects will act to reduce the population of luminous stars . even though they are very frequent , collisions between red giants and main - sequence stars in fact have very little effect , as the main - sequence star passes through the envelope and effects very little mass loss @xcite . the same is also true for white dwarf or neutron star impactors . more interesting are encounters involving black holes . in fig . [ mdavies_figure3 ] we show snapshots of an encounter between a 1m@xmath18 red giant and and 10m@xmath18 black hole ( @xmath19 km / s , @xmath20 r@xmath18 ) . as the black hole passes close to the red - giant core , it gives it a jerk and the core is ejected at high speed , retaining only a small fraction of the envelope ( in this case about 13% ) . such excessive mass loss will prevent this red giant from becoming brighter : we have thus removed it from the pool of brighter red giants .    thus we see that in order to deplete the stellar population of red giants ( as indicated by the observations ) , we must consider collisions of three types : 1 ) encounters between red giants and black holes , 2 ) encounters between two main - sequence stars , and 3 ) encounters between main - sequence stars and compact objects ( _ i.e. _  black holes , white dwarfs and neutron stars ) . the relative frequencies of encounters will depend on the imf of the underlying stellar population . with a miller - scalo type imf , where most stars are of low mass , the majority of stars formed will still be on the main sequence today , with relatively few stars having evolved off the main - sequence to ultimately form compact remnants . thus collisions involving two main - sequence stars will be more frequent than collisions between main - sequence stars and compact objects . collisions between red giants and black holes may be relatively frequent , at least in the very center , owing to the larger size ( and thus collisional cross section ) of red giants . we consider first that the stellar population in the galactic center has been drawn from a miller - scalo imf @xcite where we also assume the stars have a uniform spread in ages from 0 to 14gyr . using a density distribution for the stars given by @xmath21 and a density distribution for 10m@xmath18 black holes of @xmath22 @xcite , we compute the collision probabilities between the various stellar types as a function of galactocentric radius . the results are shown in fig . [ mdavies_figure4 ] . we see that collisions between red giants and black holes are the most frequent in the very central regions whilst collisions between two main - sequence stars dominate further out . for stars following a miller - scalo imf , collisions between main - sequence stars and black holes ( or other compact remnants ) are relatively less frequent and are unlikely to have a significant effect on the stellar population apart from perhaps in the very central regions . even though collisions between main - sequence stars are relatively frequent , in the case of a miller - scalo imf with a low - mass cutoff of 0.2m@xmath18 the number of main - sequence stars in the mass range 14m@xmath18 will not be reduced . the reason is that although some stars in this mass range will be removed as they merge with other stars to produce more massive stars , they will be replaced by stars produced by the merger of lower - mass stars . this process will be investigated in detail in dale et al . ( in preparation ) . it could be that a stellar population with a miller - scalo imf but with a larger low - mass cutoff may lead to a depletion of the lower - mass main - sequence stars which later evolve to produce the observed population of red giants . collisions between red giants and stellar - mass black holes are able to deplete the red - giant population within the inner 0.08pc or so but not further out . collisions can plausibly explain the depletion of red giants of middle brightness ( @xmath23 ) but not those in the brightest band ( @xmath24 ) which are seen to be depleted to about 0.2pc @xcite . collisions are not able to explain the observed flattening of the red - giant population seen out to 0.4pc @xcite . in this section we consider the case where the stellar population in the galactic center is drawn from a much flatter imf than a miller - scalo imf . observations of the young disc of stars in the galactic center suggest that they may have been drawn from such an imf with a power - low slope as low as @xmath25 @xcite . a stellar population drawn from such a flat imf would be quite different from one drawn from a miller - scalo imf . a much larger fraction of stars would be massive , and explode as core - collapse supernovae producing either black holes or neutron stars . in addition , assuming stars have an approximately uniform spread in ages , a very large fraction of all stars will have evolved to become compact remnants . ( _ i.e. _  black holes , white dwarfs , or neutron stars ) . in other words , the majority of stellar objects will in fact be compact objects . the most common flavor of collision will then be collisions involving compact objects .    in order to produce a stellar population , we assume a model where a stellar disc , similar to the one seen today , is produced about every @xmath26 years up to 14gyr ago , with the most recent disc being produced only 6 million years ago to be consistent with observations . the disc mass is set to match the observations of the early - type stars which come entirely from the most - recent disc , whilst the total number of discs produced is set to match the observed late - type population 10arcsec from the supermassive black hole . one should note that the formation mechanism for such discs of stars is unclear , one suggestion being that gas clouds interact tidally with the supermassive black hole @xcite . we consider here for purposes of illustration the case of @xmath27 . for this imf ( which is flat in log mass ) , the mass is each disc is @xmath28m@xmath18 , with the stars having a range of masses between 1.0 and 32.5m@xmath18 . it is entirely possible that the star formation rate was higher in the past . as a simple first model , we assume that in addition to the discs produced at a steady rate , there was an excess of discs produced sufficiently early in the history of the galaxy that essentially all the stars from these earlier discs have evolved into compact remnants . the exact excess of compact objects will depend on the history of the disc formation rate , but here we assume that two thirds of all star formation occurred at this earlier epoch . this will increase the number of compact objects present in the galactic center population . as collisions involving compact remnants destroy main - sequence stars , increasing the compact object population will enhance the red - giant depletion . the collision probability for collisions between stars of various types for the stellar population produced as described above are shown in fig . [ mdavies_figure5 ] . we see here that collisions between main - sequence stars and compact objects are most likely ( at least further out from the center ) . main - sequence stars are likely to undergo such a collision , and thus be destroyed , out to a distance of about 0.2pc . as discussed earlier , collisions between compact objects and main - sequence stars are likely to be destructive . therefore with a flat imf , and a suitably dense stellar population , the low - mass main - sequence stars which would ordinarily have evolved to become the observed late - type population are depleted within a distance of about 0.2pc of the supermassive black hole . therefore a stellar population produced from a flat imf may be able to explain the flat surface density profile which has been observed for the late - type stars ( red giants ) as stars in the mass range 1 - 5m@xmath18 within 0.2pc have simply been destroyed before they could evolve into red giants . equipped with the collision probabilities as a function of radius , we are now able to synthesize a stellar population for both miller - scalo and flat imfs allowing for the collisional depletion of the red - giant population . in fig .  [ mdavies_figure6 ] we show the calculated surface density profiles for early - type stars and late - type stars having k magnitudes brighter than 15.5 for both imfs . in both cases , the stellar population has been normalised to the surface brightness of late - type stars 10arcsec from the galactic center , seen to be about 3 sources per square arcsec ( this normalization was also applied to the collision probability calculations and figures shown earlier ) . in both plots we do not include the effect of stellar collisions on the population of early - type stars as the collisional depletion of these massive stars will be very small . however it should be noted that the existence of the so - called s stars in the very center of the galaxy ( distances less than 1 arcsecond from the central black hole ) may place limits on the number density of compact objects . the plots shown in fig . [ mdavies_figure6 ] should be compared to fig . 11 of @xcite . one can see from the plots that the early - type population only matches the observations in the case of the flat imf . a miller - scalo population is clearly not able to consistently match both the observed early and late populations . it is also clear from these plots that in the case of the miller - scalo imf , the effect of stellar collisions on the red - giant population is negligible . however for the flat imf , appreciable red - giant depletion occurs out to 10arcseconds . thus a flat imf would seem to have the potential to explain the observed ( flat ) surface density profile of the red giants . one problem with the model used here is that the total stellar mass out to 0.4pc is about @xmath29m@xmath18 . this is much higher than suggested by observations @xcite . however , in our calculations using a flat imf here we have made the unrealistic assumption that stars remain at the radius at which they were formed . in other words , we have not allowed for the effects of mass segregation . in reality , the large population of heavier stellar - mass black holes will sink in the potential , forming a central sub - cluster . this sub - cluster may then collapse to form a supermassive black hole in the manner envisaged by @xcite if a supermassive black hole is not there already , or the stellar - mass black holes may simply be fed into an existing supermassive black hole . in either case , a large fraction of the stellar - mass black holes may end up inside the supermassive black hole . it is interesting in this context to note that the total mass contained in stellar - mass black holes for our flat imf ( about @xmath30m@xmath18 ) is close to the observed value for the supermassive black - hole mass . a larger population of black holes produced by a flat imf would also lead to a much higher rate of emri  type events , where stellar - mass black holes spiral in to the central , supermassive black hole , emitting gravitational radiation in the process . emris could be an important source for lisa . our calculations here have been a simplification . we have made the unrealistic assumption that stars remain at the radius they were formed . in addition we have not allowed for the growth of the supermassive black hole . early on , before the supermassive black hole has grown , the disc - mode of star formation may be different or not occur at all owing to a lack of a supermassive black hole . a natural next step in these calculations will be inclusion of the dynamical evolution of the stellar cluster and for the growth of the central black hole ( nzoke et al . , in preparation ) . rpc acknowledges support from the wenner gren foundation . jed is supported by a marie curie fellowship as part of the european commission fp6 research training network ` constellation ' under contract mrtn  ct2006035890 and the institutional research plan av0z10030501 of the academy of sciences of the czech republic and project lc06014centre for theoretical astrophysics of the ministry of education , youth and sports of the czech republic .
we consider whether stellar collisions can explain the observed depletion of red giants in the galactic center . 
 we model the stellar population with two different imfs : 1 ) the miller - scalo and 2 ) a much flatter imf . in the former case , 
 low - mass main - sequence stars dominate the population , and collisions are unable to remove red giants out to 0.4pc although brighter red giants much closer in may be depleted via collisions with stellar - mass black holes . for a much flatter imf , 
 the stellar population is dominated by compact remnants ( _ i.e. _  black holes , white dwarfs and neutron stars ) . 
 the most common collisions are then those between main - sequence stars and compact remnants . 
 such encounters are likely to destroy the main - sequence stars and thus prevent their evolution into red giants . in this way 
 , the red - giant population could be depleted out to 0.4pc matching observations . 
 if this is the case , it implies the galactic center contains a much larger population of stellar - mass black holes than would be expected from a regular imf . 
 this may in turn have implications for the formation and growth of the central supermassive black hole .