Hindawi Publishing Corporation Advances in Astronomy Volume 2011, Article ID 174105, 19 pages doi:10.1155/2011/174105 Research Article Constraining the Stellar Mass Function in the Galactic Center via Mass Loss from Stellar Collisions 1 2 Douglas Rubin and Abraham Loeb Department of Physics, Harvard University, Cambridge, MA 02138, USA Department of Astronomy, Harvard University, Cambridge, MA 02138, USA Correspondence should be addressed to Douglas Rubin, douglas.s.rubin@gmail.com Received 6 September 2011; Accepted 23 November 2011 Academic Editor: Paola Marziani Copyright © 2011 D. Rubin and A. Loeb. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The dense concentration of stars and high-velocity dispersions in the Galactic center imply that stellar collisions frequently occur. Stellar collisions could therefore result in signiﬁcant mass loss rates. We calculate the amount of stellar mass lost due to indirect and direct stellar collisions and ﬁnd its dependence on the present-day mass function of stars. We ﬁnd that the total mass loss rate in the Galactic center due to stellar collisions is sensitive to the present-day mass function adopted. We use the observed diﬀuse X-ray −5 −1 luminosity in the Galactic center to preclude any present-day mass functions that result in mass loss rates > 10 M yr in the vici- −α nity of ∼ 1 . For present-day mass functions of the form, dN/dM ∝ M , we constrain the present-day mass function to have a minimum stellar mass 7M and a power-law slope 1.25. We also use this result to constrain the initial mass function in the Galactic center by considering diﬀerent star formation scenarios. 1. Introduction prevent the giants from becoming bright enough to be obser- ved. The dense stellar core at the Galactic center has a radius of The above studies concentrated on collisions involving 6 −3 ∼0.15–0.4 pc, a stellar density > 10 M pc [1–4], high particular stellar species with particular stellar masses. To −1 ∗ velocity dispersions (≥100 km s ), and Sgr A , the central examine the cumulative eﬀect of collisions amongst an entire supermassive black hole with a mass ≈ 4 × 10 M [5–9]. ensemble of a stellar species with a spectrum of masses, one Due to the extreme number densities and velocities, stellar must specify the present-day stellar mass function (PDMF) collisions are believed to play an important role in shaping for that species. The PDMF gives the current number of stars the stellar structure around the Galactic center and in dis- per unit stellar mass up to a normalization constant. Given rupting the evolution of its stars Frank and Rees [10]. Genzel a certain star formation history, the PDMF can be used to et al. [1] found a paucity of the brightest giants in the galactic determine the initial mass function of stars (IMF), the mass center and proposed that collisions with main sequence (MS) function with which the stars were born. There is currently stars could be the culprit. This hypothesis was found to no consensus as to whether the IMF in the Galactic center be plausible by Alexander [11]. Other investigations of col- deviates from the canonical IMF [15]. lisions between giants and MS, white dwarf and neutron stars First described by Salpeter more than 50 years ago [16], [12] and collisions between giants and binary MS and neu- the canonical IMF is an empirical function which has been tron stars [13] could not account for the dearth of observed found to be universal [17], with the Galactic center as per- giants. The contradictory results were resolved by Dale haps the sole exception. Maness et al. [18] found that models et al. [14], who concluded that the lack of the faintest giants with a top-heavy IMF were most consistent with observa- (but not the brightest giants) could be explained by collisions tions of the central parsec of the Galaxy. Paumard et al. [19] between giants and stellar mass black holes. Signiﬁcant and subsequently Bartko et al. [20] found observational evi- mass loss in the giants’ envelopes after a collision would dence for a ﬂat IMF for the young OB-stars in the Galactic 2 Advances in Astronomy center. On the other hand, Loc ¨ kmann et al. [21]concluded 2. Condition for Mass Loss that models of constant star formation with a canonical IMF Throughout this paper we refer to the star that loses material could explain observations of the Galactic center. as the perturbed star, and the star that causes material to be In this work we use calculated mass loss rates due to lost as the perturber star. Quantities with the subscript or stellar collisions as a method to constrain the PDMF for main superscript “pd” or “pr” refer to the perturbed star and per- sequence stars in the Galactic center. We construct a simple turber star, respectively (Note that for any particular colli- model to estimate the actual mass loss rate in the Galactic sion, it is arbitrary which star we consider the perturber star, center based on observed diﬀuse X-ray emission. PDMFs that and which star the perturbed star. Both stars will lose mass predict mass loss rates from stellar collisions greater than due to the presence of the other, so in order to calculate the the observed rate are precluded. This method allows us to total mass loss, we interchange the labels (pd↔pr), and repeat place conservative constraints on the PDMF, because we the calculation.). We work in units where mass is measured do not include the contribution to the mass loss rate from in the mass of the perturbed star, M , distance in the radius pd stellar winds from massive evolved stars [22]. Speciﬁcally, of the perturbed star, r , velocity in the escape velocity of the pd this method allows us to place a lower limit on the power- pd pd perturbed star, vesc (= 2GM /r ), and time in r /vesc.We pd pd pd law slope and an upper limit on the minimum stellar mass of denote normalization by these quantities (or the appropriate the PDMF in the Galactic center (see Section 5). Inclusion of combination of these quantities) with a tilde: the mass loss rate from stellar winds (or other sources) could further constrain the PDMF of the Galactic center. M ≡ , The work presented in this paper has implications for the pd fueling of active galactic nuclei (AGN). To trigger an AGN, a signiﬁcant amount of matter must be funneled onto the r≡ , pd supermassive black hole in a galactic nucleus. The most com- (1) mon way of channelling gas is through galaxy mergers, which v ≡ , pd has been studied for quite some time (e.g., A.Toomre and esc J.Toomre [23]; Gunn [24] Hernquist and mihos [25]). Even t ≡ . without mergers, AGN can be fed by several processes from pd r /vesc pd stellar residents in a galactic center. The tidal disruption of a star which passes too close to the supermassive black hole We refer to collisions in which b> r + r as “indirect” pd pr can strip mass oﬀ the star. Additionally, it is known that a collisions, and collisions in which b ≤ r + r as “direct” pd pr signiﬁcant amount of gas is ejected into the Galactic center collisions. The impact parameter, b, is the distance of closest due to stellar winds from massive, evolved stars [22, 26, 27]. approach measured from the centers of both stars. Another potential source for the fueling of AGN could be We consider the condition for mass loss at a position, r, from unbound stellar material, ejected in a stellar collision. within the perturbed star to be that the kick velocity due to Since the easiest place to look for such an event (due to its the encounter at rexceeds the escape velocity of the perturber proximity) is the Galactic center, in this paper we theore- star at r, Δv(r) ≥ v (r). The escape velocity as a function esc tically investigate stellar collisions in this environment. By of position within the perturbed star can be found from calculating the cumulative mass loss rate from stellar colli- the initial kinetic and potential energies of a test particle at sions in the Galactic center, we place constraints on the fuel- position r, ing of Sgr A due to this mechanism. We present novel, analytical models to calculate the M (r ) int amount of stellar mass lost due to stellar collisions between v (r) = − dr esc main sequence stars in Section 2 through Section 2.3.In (2) Section 3 we develop the formalism for calculating collision M (r) int rates in the Galactic center. We utilize our calculations of the = +4π ρ(r )r dr , mass loss per collision, and the collision rate as a function of Galactic radius to ﬁnd the radial proﬁle of the mass loss where M is the mass interior at position r and ρ is the den- int rate in Section 4. Since the amount of mass lost is dependent sity proﬁle of the star. on the masses of the colliding stars, the mass loss rate in the Galactic center is sensitive to the underlying PDMF. By com- 2.1. Mass Loss due to Indirect Collisions. To calculate the mass paring our calculations to mass loss rates obtained from the lost due to an indirect collision, we ﬁrst calculate the kick diﬀuse X-ray luminosity measured by Chandra,in Section 5 velocity given to the perturbed star as a function of position we constrain the PDMF of the Galactic center. We derive within the star. We work under the impulse approximation analytic solutions of the PDMF as a function of an adop- [28], valid under the condition that the encounter time ted IMF for diﬀerent star formation scenarios, which allows is much shorter than the characteristic crossing time of a us to place constraints on the IMF in Section 6.In Section 7, constituent of the perturbed system. we estimate the contribution to the mass loss rate from col- Given a mass distribution for the perturbed system, ρ pd lisions involving red giant (RG) stars. and a potential for the perturber system, Φ, the kick velocity Advances in Astronomy 3 0.25 after an encounter under the impulse approximation is given by Binney and Tremaine [29]: 0.2 Δ v r =− ∇Φ r, t − ρ r , t ∇Φ r ,t d r dt. pd −∞ pd (3) 0.15 Equation (3) can be simpliﬁed by expanding the gradient of the potential in a Taylor series, resulting in ⎛ ⎞ −x 0.1 ⎜ ⎟ 2GM pr ⎜ ⎟ 2 Δ v r = y + O r . (4) ⎝ ⎠ b v rel 0.05 The expansion is valid under the “distant tide” approxima- tion which is satisﬁed when r b. The parameter v is pd rel the relative speed between the stars (v ≡| v − v |). We are rel pd pr interested in the magnitude of (4), which when normalized to the units that we have adopted for this paper is n = 3 n = 2 ∼ 2 2 (5) Δv x, y = γ x + y , n = 2.5 n = 1.5 where Figure 1: The fraction of mass lost per collision as a function of γ for several polytropic indices. The lines are third-order polynomial ﬁts, whose coeﬃcients are given in Table 1. pr γ ≡ . (6) b v rel Table 1: Coeﬃcients of polynomial ﬁts for Δ(γ) with varying poly- To solve for the mass lost per encounter as a function of tropic indices. γ, we consider a star within a cubic array, where the star con- tains ∼3 × 10 cubic elements. As a function of γ we compare na a a a 0 1 2 3 the kick velocity in each element to the escape velocity for 1.5 0.395 −0.865 0.559 −0.091 that element and consider the mass within the element to be 2.0 0.210 −0.424 0.246 −0.032 lost to the star if the velocities satisfy the condition given in 2.5 0.105 −0.197 0.102 −0.101 Section 2. We note that by ∼10 elements, the results con- 3.0 0.051 −0.088 0.040 −0.003 verge to within about 2%, and we are therefore conﬁdent that ∼3 × 10 provides adequate resolution. To calculate the amount of mass in each element, the den- ﬁcients of the polynomial ﬁts in Table 1. For each density sity proﬁle for the perturbed star must be speciﬁed. As with proﬁle,nomassislostupuntil γ of about 0.98, and thereafter several previous studies on mass loss due to stellar collisions the mass loss increases monotonically. The increasing trend [30–33] we utilize polytropic stellar proﬁles. Polytropic pro- is due to the fact that larger perturber masses and smaller ﬁles are easy to calculate and yield reliable results for stars of impact parameters result in an increased potential felt by the certain masses. Polytropic proﬁles of polytropic index n = perturbed star. Smaller velocities also cause more mass to 1.5 describe the density structure of fully convective stars, be lost, as this increases the “interaction time” between the and therefore very well describe MS stars with M 0.3M perturber and perturbed stars. (nearly fully convective) and MS stars with M 10M The location of the mass loss within the perturbed star (convective cores). MS stars with M 1M have radiative for ﬁxed γ depends upon the polytropic index, since the envelopes and are therefore well-described by n = 3. For n escape velocity within the star is dependent upon the density for stars with masses of 0.3–1M and 5–10M , we linearly proﬁle, as indicated by (2). In Figure 2,weillustratewhere interpolate between n = 1.5 and 3. We discuss the uncer- mass will be lost in the perturbed star by plotting contours of tainties introduced by this approach in Section 4. Note that the kick velocity (Δv(r)) due to the encounter normalized to this approach is biased towards zero-age main sequence stars, pd since as stars evolve, they are less adequately described by v (r)for n = 1.5and n = 3(bottom andtop rows,resp.). esc We show two diﬀerent cases: a slightly perturbing encounter polytropic proﬁles. with γ = 1.2 in the ﬁrst column, and a severely perturb- We plot the fraction of mass lost from the perturbed star ing encounter with γ = 1.6 in the second column. The per event, Δ, as a function of γ in Figure 1 for several poly- grey region underneath shows where mass is still left after tropic indices. The lines are third-order polynomial ﬁts to pd our results, in the range of 0.98 ≤ γ ≤ 5. We list the coef- the encounter, since Δv/ v (r) within this region is < 1. esc Δ 0.11 0.182 0.182 0.182 0.545 0.545 0.55 0.55 0.11 0.182 0.11 0.11 0.33 0.33 0.545 0.545 0.55 0.55 0.33 0.33 4 Advances in Astronomy γ = 1.2 γ = 1.6 (a) (b) (c) (d) Figure 2: Slices through the perturbed star along the plane parallel to the perturber star’s trajectory. The ﬁrst column ((a) and (c)) corresp- ond to encounters with γ = 1.2, and the second column ((b) and (d)) correspond to γ = 1.6. The ﬁrst row ((a) and (b)) have n = 3, and the second row ((c) and (d)) have n = 1.5. The contours are the kick velocity within the star due to the encounter normalized to the escape pd velocity (as a function of r). The outline of the grey region underneath has Δv/ v (r) = 1, so that the grey region represents the location of esc where mass is still left after the event. The γ = 1.6 encounter results in bigger kick velocities, and lost is substantially diﬀerent (as shown in Figure 1), due to so we see that the mass loss penetrates farther into the star. the diﬀerent density proﬁles. We note that the shape and magnitude of the contours for both polytropic indices at ﬁxed γ converge at large radii. This is due to the fact that regardless of the polytropic index 2.2. Validity of Approach for Indirect Collisions. The impulse pd approximation is valid provided that the time over which the used, v converges to the same value at large radii when esc encounter takes place, t , is much shorter than the time it the second term in (2) becomes negligible. Even though the enc takes to cross the perturbed system, t . To estimate when location of where mass is lost is similar for diﬀerent poly- cross tropic stars at the same value of γ, the amount of the mass our calculations break down, we approximate t as b/v , enc rel 0.909 0.909 0.545 0.545 0.545 0.545 0.909 0.909 0.55 0.55 0.55 0.55 0.76 0.76 0.76 0.76 0.33 0.33 0.33 0.33 0.98 0.98 0.98 0.98 0.182 0.182 1.273 1.273 1.273 1.273 0.182 0.182 0.11 0.11 0.11 0.11 n = 1.5 n = 3 Advances in Astronomy 5 and t as t , the time it takes for a sound wave to cross an 10 cross s object that is in hydrostatic equilibrium: 1 1 t ∼ t ∼ ∼ . cross s 2 (7) Gρ GM /r pd pd pd These approximations lead to the condition that −1 v b 1. (8) rel Aguilar and White [34] ﬁnd that for a large range of collisions, the impulse approximations remains remarkably valid, even when t is almost as long as t . We therefore enc cross assume that the impulse approximation holds until the left hand side of (8)is ∼1. Our calculation of Δ as a function −1 of γ should therefore be valid for γ γ ,where γ ≡ valid valid 1 10 M /b . We plot contours of log(γ ) in the M /M -b/r pr valid pr pd pd b/r pd parameter space in Figure 3, where both the x and y axes span ranges relevant to our calculations. The shaded grey Figure 3: Contours of log(γ )inthe M /M -b/r parameter valid pr pd pd area in the ﬁgure is the region of the parameter space where space, where γ is deﬁned in Section 2. The shaded grey area valid the impulse approximation predicts nonzero mass loss due to indicates where the impulse approximation predicts nonzero mass the encounter. The ﬁgure shows that γ is smaller for low loss. valid M to M ratios at high impact parameters. In fact, most of pr pd the right side of the parameter space has γ less than 0.98 valid (where below this value, the impulse approximation predicts hydrodynamics with various stellar models, mass-radius re- no mass lost). lations, and varying degrees of particle resolution [30–33]. A In our calculations, when, for any particular set of detailed review of the literature can be found in this area in M /M and b/r , γ> γ ,weadopt Δ(γ> γ ) = pr pd pd valid valid Freitag and Benz [38]. Δ(γ = γ ). This approach represents a lower limit on the valid We approach the problem of direct collisions in a highly amount of mass loss that we calculate, since mass loss should simpliﬁed, analytic manner without hydrodynamic consider- increase with increasing γ. We ﬁnd, however, that if we set ations and ﬁnd that for determining the amount of mass lost, Δ(γ> γ ) = 1 (which represents the absolute upper limit valid our method compares well to the complex hydrodynamic in the amount of mass lost) the change in our ﬁnal results simulations. As a ﬁrst-order model, we approximate the en- is negligible at small Galactic radii. At large radii, where the counter as two colliding disks, by projecting the mass of both mass loss from indirect collisions dominates (see Section 4), stars on a plane perpendicular to the trajectory of the pertur- the results change by at most a fact of ∼2. ber star. The problem of calculating mass loss then becomes Equation (4) was derived under the assumption that the easier to handle, as it is two-dimensional. We also assume impact parameter is much bigger than both r and r . pd pr thatmasslosscan only occurinthe geometricalareaofin- −2 Since Δv scales as b , the equation predicts that most mass tersection of the two stars. loss occurs for small impact parameters. However, given the We ﬁnd the kick velocity as a function of position in the assumption that was used to derive the equation, the regime area of intersection by conserving momenta and by assuming of small impact parameters is precisely where (4)breaks that all of the momentum in the perturber star in each area down. Numerical simulations [34, 35] show that for a variety element was transferred to the corresponding area element of perturber mass distributions, the energy input into the in the perturbed star. Working in the frame of the perturbed perturbed system is well described by (4)for b 5r ,where star and with a polar coordinate system at its center (so that r is the half mass radius of the perturber system. For an n = h 2 2 r = x + y ), we ﬁnd 3 polytropic star, 5r = 1.4r . Since for indirect collisions, b/r = 1+ r /r + d/r (where d is the distance between pd pr pd pd Σ (r)v pr rel the surface of both stars), there is only a small region in our Δv(r) = . (9) Σ (r) calculations, 0 ≤ d/r (0.4 − r /r ), for which (4)may pd pd pr pd give unreliable results. The parameters Σ and Σ represent the surface density of pr pd 2.3. Mass Loss due to Direct Collisions. Anumberofpapers the perturber and perturbed stars, respectively, ( ≡ ρdz ). over the past few decades have addressed the outcomes of To ﬁnd the region of intersection, we need to know the stellar collisions where the two stars come so close to each impact parameter and the radii of both stars. To obtain the other that not only gravitational, but also hydrodynamic stellar radii as a function of mass, we use the mass-radius forces must be accounted for. Early studies used one- or two- relation calculated by Kippenhahn and Weigert [39]for aMS dimensional low-resolution hydrodynamic simulations (e.g., star with Z = Z , X = 0.685 and X = 0.294 from a stel- H He [36, 37]). Modern studies typically utilize smooth particle lar evolution model, where X represents the mass fraction. −0.732 −1.707 −2.683 1.220 −0.732 0.244 −3.659 M/M pr pd / 6 Advances in Astronomy 2 stellar masses, relative velocity and impact parameter, our calculations sometimes over- or underpredict the amount of mass lost by of a factor of a few to at most a factor of 10. We discuss the error introduced into our main calculations by this discrepancy at the end of Section 5. 3. Stellar Collision Rates in the Galactic Center To calculate mass loss rates in the Galactic center, we will need to ﬁnd the collision rates as a function of the perturber and perturbed star masses, impact parameter, and relative velocity. Additionally, the collision rate will be a function of distance from the Galactic center, since the stellar densities −1 and relative velocities vary with this distance. In this section, we ﬁrst present the Galactic density proﬁle that we use, and we then derive the diﬀerential collision rate as a function of these parameters. −2 We adopt the stellar density proﬁle of Schodel ¨ et al. [4], −2 −1 012 one of the best measurements of the density proﬁle within log(MM / ) the Galactic center to date. Using stellar counts from high- resolution images of the galactic center, they ﬁnd that the Figure 4: Mass-radius relations used in studies of calculating mass density proﬁle is well-approximated by a broken power law. loss from stars due to stellar collisions. The thin lines are power-law Moreover, they use measured velocity dispersions to con- relations of power-law index 1.0, 0.8, and 0.85 used by Rauch [33], strain the amount of enclosed stellar mass as a function of Lai et al. [32], and Benz and Hills [31], respectively. The dotted line galactic radius, r . Using their density proﬁle, and velocity is the relation used by Freitag and Benz [38], and the thick line is gal the relation used in this work. dispersion measurements, they ﬁnd that −1.2 ⎪ gal 6 −3 2.8 ± 1.3 × 10 M pc , 0.22 pc We ﬁt a polynomial to their [39, Figure 22.2] and extrapolate ⎪ on the high- and low-mass ends so that we have a mass- for r ≤ 0.22 pc, gal ρ r = gal radius relation that spans from about 0.01 to 150M .We ⎪ −1.75 ⎪ r gal compare our mass-radius relation to those used in other 6 −3 ⎪ 2.8 ± 1.3 × 10 M pc , 0.22 pc studies of direct stellar collisions in Figure 4.Rauch [33], Lai ⎪ for r > 0.22 pc. et al. [32], and Benz and Hills [31]all adoptedpower laws gal with power law indices of 1.0, 0.8, and 0.85, respectively, (10) (thin lines). Freitag and Benz [38] (dotted line) use main Their average density can be converted into a local density, sequence stellar evolution codes to obtain a mass-radius ρ(r ), by considering the deﬁnition of ρ, relation for masses > 0.4M and a polytropic mass-radius gal relation of n = 1.5 for masses < 0.4M . gal 4πr ρ r dr Our simple model for calculating mass loss due to direct gal gal gal (11) ρ r ≡ , gal stellar collisions compares surprisingly well to full blown 3 4/3πr gal smooth particle hydrodynamic simulations. We borrow plots of the fractional amount of mass lost as a function of impact from whichwederive parameter for speciﬁc relative velocities and stellar masses dρ r from Freitag and Benz [38] (Figures 5 and 6). They show r gal gal (12) ρ r = ρ r + . gal gal their own work, the best calculations of mass loss due to stel- 3 dr gal lar collisions to date. For comparison, and to show how the calculations have evolved over the years, the results from We use (10)and (12), to ﬁnd ρ(r) and plot the result in older studies are also shown. Our own results are plotted Figure 7. We “smoothed” the unphysical discontinuity in ρ (dashed-dotted black lines) over these previous studies. We arising from the kink of the broken power law ﬁt by ﬁtting a make sure to show results spanning a wide range of stellar polynomial to (10). masses and relative velocities. Note that these plots show the The diﬀerential collision rate, dΓ, between two species, fractional amount of mass lost from both stars normalized “1” and “2” at impact parameter b characterized by distribu- to the initial masses of both stars, and that the impact para- tion functions f and f , and moving with relative velocity 1 2 meter is normalized to the sum of both stellar radii. Our | v − v | in a spherically symmetric system is 1 2 results show the same qualitative trends seen in the Freitag 3 3 and Benz [38] curves, even replicating several “bumps” seen dΓ = f r , v d v f r , v d v 1 gal 1 1 2 gal 2 2 in their curves (see Figures 6(c) and 6(d)). As compared (13) to the Freitag and Benz [38] results, for any speciﬁc set × v − v 2πb db4πr dr . 1 2 gal gal log(RR ) ⊙ Advances in Astronomy 7 ∞ ∞ M = 0.4M M = 0.4M V /V = 1.69 M = 0.5M M = 2.5M V /V = 2.14 1 ⊙ 2 ⊙ ∗ 1 ⊙ 2 ⊙ ∗ rel rel 0.1 0.1 0.01 0.01 −3 −3 −4 −4 −5 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 b/(R + R ) 1 2 b/(R + R ) 1 2 This work LRS93 This work LRS93 FB05 BH87 (V /V = 1.67) FB05 SS66 rel R99 SS66 R99 (a) (b) M = 0.5M M = 0.5M V /V = 1.98 M = 1M M = 19.3M V /V = 1.48 1 ⊙ 2 ⊙ ∗ 1 ⊙ 2 ⊙ ∗ rel rel 0.1 0.1 0.01 0.01 −3 −3 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 b/(R + R ) b/(R + R ) 1 2 1 2 This work LRS93 This work LRS93 FB05 SS66 FB05 SS66 R99 R99 (c) (d) Figure 5: The calculated fractional amount of mass lost as a function of impact parameter from several works. Our results are the black dashed-dotted lines. The acronyms FB05, R99, LRS93, BH87, and SS66 refer to Freitag and Benz [38], Rauch [33], Lai et al. [32], Benz and Hills [30], and Spitzer and Saslaw [40], respectively. The ﬁgures are adopted from Freitag and Benz [38]. δM/(M + M ) 1 2 δM/(M + M ) 1 2 δM/(M + M ) 1 2 δM/(M + M ) 1 2 8 Advances in Astronomy M = 0.6M M = 3.0M V /V = 2.07 1 ⊙ 2 ⊙ ∗ M = 2M M = 19.3M V /V = 3.06 rel 1 ⊙ 2 ⊙ ∗ rel 0.1 0.1 0.01 0.01 −3 10 −3 −4 −4 0 0.2 0.4 0.6 0 0.2 0.4 0.6 b/(R + R ) b/(R + R ) 1 2 1 2 This work LRS93 This work LRS93 ∞ ∞ q = 0.2 α = 10 V /V = 2.v 06 ( = 2.83) q = 0.1 α = 10 V /V = 2.v 76 ( = 3.80) FB05 1 ∗ ∞ FB05 1 ∗ ∞ rel rel R99 SS66 R99 SS66 (a) (b) ∞ ∞ M = 1M M = 4M V /V = 0.66 M = 0.5M M = 2M V /V = 0.07 1 ⊙ 2 ⊙ ∗ 1 ⊙ 2 ⊙ ∗ rel rel 0.1 0.1 0.01 0.01 −3 −3 −4 −4 −5 0 0.2 0.4 0.6 0 0.2 0.4 0.6 b/(R + R ) b/(R + R ) 1 2 1 2 This work LRS93 This work LRS93 FB05 SS66 FB05 SS66 R99 R99 (c) (d) Figure 6: The calculated fractional amount of mass lost as a function of impact parameter from several works. Our results are the black dashed-dotted lines. The ﬁgures are adopted from Freitag and Benz [38]. δM/(M + M ) 1 2 δM/(M + M ) 1 2 δM/(M + M ) 1 2 δM/(M + M ) 1 2 Advances in Astronomy 9 nondimensionalized expression for the diﬀerential collision rate: 2 2 3/2 −3 −v /4σ 2 rel dΓ = 4π σ e v K rel (16) −α −α 2 × M M r bdbdr dv dM dM . gal rel 1 2 1 2 gal The tildes denote normalization by the proper combination of M , r ,and v . The parameter K is the normalization 2 2 esc 10 constant for ξ, which can be solved for by using the density proﬁle of Figure 7 and the following expression: 6 M M max max dn 1−α ρ = MdM = K r M dM gal dM M M min min (17) 2−α 4 2−α 10 M − M max min = K r . gal 2 − α Since the expression for K, which controls the total number −6 −4 −2 0 2 10 10 10 10 10 of stars, has no time dependence, our expression for the r () pc PDMF assumes a constant star formation rate in the Galactic gal center. Figure 7: The stellar density proﬁle that we adopt, based on the Our calculations involve the computation of multidi- average density proﬁle of Schodel ¨ et al. [4]. mensional integrals over a two-dimensional parameter space (see Section 4). Therefore, for the ease of calculation, we ignore the enhancement of the collision rate due to the eﬀects of gravitational focusing. This results in a conservative esti- For simplicity, we adopt Maxwellian distributions, mate of the collision rate. As two projectiles collide with n r each other, their mutual gravitational attraction pulls them 1,2 gal 2 −v /2σ 1,2 (14) f r , v = e , 1,2 gal 1,2 together, resulting in an enhancement of the cross section: 3/2 (2πσ ) 2G(M + M ) where we ﬁnd the velocity dispersion, σ, from the Jean’s 1 2 S −→ S 1+ . (18) equations. Assuming an isotropic velocity dispersion, a sphe- bv rel rical distribution of stars and a power-law density proﬁle −β We discuss the uncertainties in our ﬁnal results due to ignor- with power-law slope β, ρ ∝ r , the Jean’s equations lead to gal ing gravitational focusing at the end of Section 5. 2 6 σ = GM /r (1 + β), where M = 4 × 10 M .From SMBH gal SMBH To illustrate the frequency of collisions in the Galactic Figure 7, it is evident that β is a function of r ,but for gal center, we integrate (16)over v , M ,and M assuming a rel 1 2 simplicity, we adopt an averaged value of β, β = 1.3. Note Salpeter-like mass function (α = 2.35, M = 0.1M and min that we have also assumed that the enclosed mass at position M = 125M )toobtain dΓ/(dlnr db) as a function of max gal r is dominated by the SMBH. This assumptions is valid gal r (Figure 8(a)) (This ﬁgure and subsequent ﬁgures in gal out till ∼1 pc, which is also the point where our impulse this paper with r as the independent variable start from gal approximation starts to break down. −6 r = 10 pc. This value of r corresponds to the tidal gal gal A change of variables allows one to integrate out 3 of the radius for a 1M star associated with a 4 × 10 M SMBH. velocity dimensions and to write the expression in terms of Although stars of diﬀerent masses will have slightly diﬀerent v (see [29]). We can also take into account the fact that both rel tidal radii, the main conclusions of our paper are based oﬀ species have a distribution of masses by introducing, ξ , the 1,2 of distances in r of order 0.1 pc (see Section 5), well gal PDMF, which gives the number density of stars per mass bin above the tidal radius for any particular star.). We plot (ξ ≡ dn/dM). We adopt a power law PDMF, dΓ/(dlnr db) for several diﬀerent impact parameter values. gal −α We calculate dΓ/(dlnr db) with and without the eﬀect of gal ξ ∝ M , (15) gravitational focusing (solid and dashed lines, resp.). The that runs from some minimum mass, M to a maximum latter is obtained by multiplying (16) by the gravitational min mass M . Since most initial mass functions are parameter- focusing enhancement term before the integration. As expec- max ized with a power law, the present-day mass function might ted, gravitational focusing is negligible at small Galactic radii be modiﬁed from a power law due to the eﬀects of collisions since typical stellar encounters involve high relative veloci- and stellar evolution. Although the actual PDMF might have ties. As the typical relative velocities decrease with increasing deviations from a power law, adopting a power law pro- Galactic radius, the enhancement to the collision rate from vides us with a quick and simple way to parameterize the gravitational focusing becomes important. The ﬁgure also PDMF. Taking all of this into account, and assuming that shows that gravitational focusing becomes less important the relative velocities are isotropic, we arrive at the ﬁnal with increasing impact parameter since the gravitational ρ (M /pc ) ⊙ 10 Advances in Astronomy −3 −4 −5 −5 −6 −6 −6 −4 −2 0 2 −6 −4 −2 0 2 10 10 10 10 10 10 10 10 10 10 r () pc r () pc gal gal (a) (b) Figure 8: (a) The diﬀerential collision rate per logarithmic Galactic radius per impact parameter as a function of Galactic radius for several diﬀerent impact parameters. The solid (dashed) lines were calculated ignoring (including) gravitational focusing. The curves were made by made by integrating (16) (with and without the gravitational focusing term) assuming Salpeter values. (b) The cumulative diﬀerential collision rate (integrated over r ) per impact parameter with and without gravitational focusing for the same impact parameter values. gal (a), (b) The vertical line in each panel is placed at r = 0.06 pc, the upper bound in our integration across r as performed in Section 5. gal gal attraction between the stars is weaker. Figure 8(b) shows the the range of r that we consider. Thus, the integral that we gal cumulative diﬀerential collision rate (integrated over r )per evaluate is gal ⎛ ⎞ impact parameter as a function of r . Again, we plot the gal Mmax Mmax vmax dM results with and without gravitational focusing and for the 3/2 −3 3 2 ⎝ ⎠ ∼ = 4π σ r K gal same impact parameters. dlnr gal M M 0 min min pd b (20) 4. Mass Loss Rates in the Galaxy max 3 −α −α × bv Δ γ M M dbdM pd pr rel pr pd 1+r pr To calculate the mass loss rate from stars due to collisions within the Galactic center, we multiply (16) by the fraction of × dM dv . pd rel mass lost per collision, Δ(γ), and compute the multidimen- sional integral. We calculate the total mass loss rate from both For direct collisions, Δ(b, M , M , v ) is calculated pr pd rel the perturbed and perturber stars by simply interchanging given the prescription in Section 2.3. To evaluate the multidi- the “pr” and “pd” labels and reperforming the calculation. mensional integral, we make the approximation of evaluating We ﬁrst compute the diﬀerential mass loss rate for indi- Δ at v = 2σ. The factor of Δ thus comes out of the pd rel pd rect collisions. The mass loss per collision is given by v integral, so that the v integral can be performed analy- rel rel tically: 0for γ< 0.98, ⎛ ⎞ M M 1+r ˙ max max pr dM Δ γ = polynomial for 0.98 ≤ γ ≤ γ , (19) 3/2 3 2 ⎝ ⎠ ∼ valid = 32π σr K gal ⎪ dlnr M M 0 ⎪ gal min min pd Δ γ for γ> γ . valid valid (21) × bΔ b, M , M , v = 2σ r pd pr pd rel gal The coeﬃcients for the polynomial depend on the polytopic index of the perturbed star (and thus on its mass) and are −α −α × M M dbdM dM . pr pd pr pd taken from Table 1. We multiply (19)and (16) and simplify the integration. In principle, b should go to ∞,but we cutoﬀ We evaluate the remaining integrals numerically. the integral at b = 20 as we ﬁnd that the results converge max Once values for α, M and M are speciﬁed, (20)and min max well before this point. The velocity integral is also cut oﬀ at (21) can be integrated to obtain the mass loss rate as a func- v due to the fact that Δ(γ) becomes zero below γ = 0.98. max tion of Galactic radius. To show how the mass loss rate pro- This cut-oﬀ corresponds to v = (M ) /0.98b .We ﬁles vary with M , M ,and α, we plot dM/dlnr for max pr min max gal min max 2 2 may safely throw away the exponential as v σ (r )for direct collisions in Figure 9 and vary these parameters. In the gal max b = 15 b = 5 b = 1 15R b = 1 b = b = 5 −1 −1 dΓ/d dlnr b (yr R ) gal −1 −1 dΓ/db(<r )(yr R ) gal ⊙ Advances in Astronomy 11 −4 −5 −6 −7 −4 −5 −6 −7 −4 −5 −6 −7 −8 −6 −4 −2 0 −6 −4 −2 0 −6 −4 −2 0 1 10 10 10 10 10 10 10 10 10 10 10 10 10 r r r () pc () pc () pc gal gal gal 75 100 125 M (M ) max ⊙ Figure 9: Mass loss rates as a function of Galactic radius due to direct collisions for various parameters of M , M ,and α. The parameter min max M varies in each panel from bottom to top, and M varies from left to right. The power law slope, α, varies within each panel from min max 1.00 (top line) to 2.5 (bottom line) in equal increments of 0.188. The dashed line corresponds to a Salpeter-like mass function values (M = 0.5M , M = 125M , α = 2.29). The arrows indicate the range in the diﬀuse X-ray observations (r < 1.5 ) which we use to min max gal constrain the PDMF (see Section 5). ﬁgure, we have evaluated M at 0.05, 0.5 and 5M , M at the very fast relative stellar velocities. Even though the high min max 75, 100, 125M and α from 1.00 to 2.5 in equal increments. velocities (and high densities) in the Galactic center make The parameter M increases vertically from the bottom collisions more frequent, under the impulse approximation, min panel to the top; M increases horizontally from the left when velocities are very fast, mass loss is minimized. max panel to the right, and in each panel α increases from the To illustrate which mass stars contribute the most to the bottom to the top. We have indicated a Salpeter-like mass total mass loss rate, we plot dM/dlnM as a function of M pd pd function (α = 2.29, M = 0.5M and M = 125M ) in Figure 11 for several diﬀerent PDMFs. The range of inte- min max with the dashed line. Mass loss is extensive and approxi- gration we choose for r is from 0 to 0.06 pc (see Section 5). gal −2 mately constant until about r of 10 pc and then drops We choose M to be 0.05, 0.5 and 5M (in Figures 11(a)– gal min dramatically. This drop reﬂects that fact that collisions are 11(c)), and we use a constant M of 125M .Ineachpanel, max less frequent at larger radii since star densities and relative we vary α from 1.5 to 2.5 in equal increments. The ﬁgure velocities drop. The amount of mass lost for any direct shows that for M = 0.05M , changing α has little eﬀect min collision also decreases with galactic radius since Δ decreases on what mass stars contribute the most to the mass loss with decreasing relative velocities. Note that the proﬁles are rate (although, the total mass loss rate is decreased with in- approximately constant as a function of M , so that the creasing α). For the M = 0.5M and 5M cases, increasing max min choice of M determines the extent of the mass loss rate. α results in lower mass stars contributing more to the mass min In Figure 10 we show the contributions to dM/dlnr loss rate. This trend makes sense, since PDMF proﬁles with gal from both direct and indirect collisions for M = 0.2M , higher values of α have fractionally more lower mass stars. min M = 100M and α = 1.2. We ﬁnd that at small radii the To test how our interpolation between the n = 1.5and max mass loss rate is dominated by direct collisions, and at large 3 polytropic indices aﬀects the main results of this paper, we radii it is dominated by indirect collisions. Mass loss due to consider two extreme cases. The ﬁrst case we consider has indirect collisions is suppressed in the Galactic center, due to n = 1.5for M < 1M and M > 5M ,and n = 3for MM() min 0.05 0.5 5 ˙ ˙ ˙ dM/dlnr (M /yr) dM/dlnr (M /yr) dM/dlnr (M /yr) ⊙ ⊙ ⊙ 12 Advances in Astronomy hydrodynamics (under spherical symmetry) to follow how the gas is accreted onto Sgr A .Quataert[27] ﬁnds that −5 his model agrees with the level of diﬀuse X-ray emission measured by Chandra and predicts an inﬂow of mass at −5 −1 r ∼ 1 at a rate of ∼ 10 M yr . gal Using the 2–10 keV luminosity as measured by Chandra [22], we estimate the totalmasslossrateataradius of r ∼ 1.5 (0.06 pc). We use the word “total” to indicate the gal mass loss rate integrated over Galactic radius. By using this −6 total mass loss rate as an upper limit, we will be able to con- strain the PDMF in the Galactic center by precluding any PDMFs with total mass loss rates greater than this value. We will do this by integrating our calculated mass loss rate pro- ﬁles (e.g., Figures 9 and 10)over r . gal Unbound material at a radius r has a dynamical time- gal scale of −7 1.5 r r gal gal −6 −4 −2 0 2 t r ∼ ≈ 1.1 × 10 yrs , 10 10 10 10 10 dyn gal (22) pc v r char gal r () pc gal Direct collisions where the characteristic velocity at radius r , v (r ), is gal char gal Indirect collisions taken as the velocity dispersion as given in Section 3.The Total electron density at radius r may therefore be estimated by gal Figure 10: Mass loss rates due to direct and indirect stellar collisions Mt r within the Galactic center for M = 0.2M , M = 100M dyn gal min max n r ∼ n r ∼ e gal p gal and α = 1.2. The arrow indicates the range in the diﬀuse X-ray 3 (4/3)πr m gal observations (r < 1.5 ) which we use to constrain the PDMF (see gal (23) Section 5). −1.5 M gal 5 −3 = 1.1 × 10 cm , −1 M yr pc 1M ≤ M ≤ 5M . This approach has n = 1.5for much where m is the proton mass. of the mass spectrum and should result in the highest mass For thermal Bremsstrahlung emission, the volume emis- loss rates since (as is evident from Figure 1) collisions with sivity (dE/dVdtdν)is[41] the perturbed star having n = 1.5 result in the most mass −1/2 lost. This is due to the fact that for n = 1.5 stars, the mass n T ff −38 −1 −3 −1 = 6.8 × 10 erg s cm Hz is less centrally concentrated, and more mass can therefore −3 cm K escape at large radii which receive a stronger velocity kick. −hν/k T × e g , Thesecondcaseweconsiderhas n = 1.5for M < 0.3M ff and M > 10M ,and n = 3for 0.3M ≤ M ≤ 10M . This (24) case should result in the smallest mass loss rates, since it has n = 1.5 for a smaller fraction of the mass spectrum. Since where we set g = 1. The luminosity in the 2–10 keV band, ff diﬀerent mass functions have diﬀerent fractions of the total L , can be found substituting (23) into (24) and integrat- 2−10 mass in the neighborhood of 1M (where we expect the least ing the volume emissivity over volume (assuming spherical mass loss per collision since n = 3), we test the two cases for symmetry) and frequency: several diﬀerent mass functions. We ﬁnd that diﬀerences in 2 −1 0.06 dM/dlnr (r ) for both cases are relatively minor, diﬀering M r gal gal 43 −1 L ∼ 6.7 × 10 erg s 2−10 −1 at most by ∼10% depending on the mass function that we M yr pc min use. (25) r hν −hν/keV × d e d . 5. Constraining the Mass Function in pc keV the Galactic Center We have assumed a constant temperature of 1 keV. A constant It is known through diﬀuse X-ray observations from Chan- valueof1keVshouldsuﬃce for an order of magnitude dra, that the central supermassive black hole in the Galactic estimate as Baganoﬀ et al. [22] ﬁnd that the gas temperature center is surrounded by gas donated from stellar winds (e.g., varies from approximately 1.9 to 1.3 keV from r = 1.5 to gal [22]). The diﬀuse X-ray luminosity is due to Bremsstrahlung 10 (assuming an optically thin plasma model). Quataert’s emission from unbound material supplied at a rate of [27] model also predicts that the temperature varies from −3 −1 ∼ 10 M yr [26]. This unbound material has been studied about 2.5 to 1 keV from r = 0.3 to 10 . By plugging the gal 33 −1 theoretically by Quataert [27], who solved the equations of value of L within 1.5 (2.4× 10 erg s )asmeasuredby 2−10 dM/dlnr (M /yr) gal ⊙ Advances in Astronomy 13 M = 0.05 M M = 0.5 M min ⊙ min −5 α = 1.5 α = 1.5 −6 −7 −6 α = 2.5 α = 2.5 −8 0.1 1 10 100 1 10 100 M (M ) M (M ) pd ⊙ pd ⊙ (a) (b) M = 5 M min ⊙ −5 α = 1.5 α = 2.5 −6 10 100 M (M ) pd ⊙ (c) Figure 11: The amount of mass loss per logarithmic mass interval of the perturbed star as a function of the perturbed star’s mass. Each line was calculated with a diﬀerent PDMF. The titles in each panel indicate the value of M used for that panel. In each panel, α goes from 1.5 min to 2.5 in even increments of 0.167, and for each line M = 125M . max −5 −1 Baganoﬀ et al. [22] into (25), we ﬁnd M ∼ 10 M yr . This from stellar wind should not exist at smaller radii since there value is consistent with the mass inﬂow rate at∼1 calculated are very few stars there to produce it. The value of the integral by Quataert [27]. For clariﬁcation, we again note that even thus ranges from about unity to a few tens. Since M depends thoughour valueagreeswithQuataert[27], the underlying upon the square root of this value, the exact value of r only min physical processes associated with both models are quite aﬀects our calculation at the level of a factor of a few, and we diﬀerent. The model of Quataert [27] takes the source of thus take the square root of the integral to be unity. −5 −1 unbound material to be due to mass ejected by stellar winds, Having established that M ∼ 10 M yr in the vicinity whereas our model uses mass ejected from stellar encounters. of 1.5 , we now calculate the expected mass loss rates due Our results are not sensitive to the choice of the lower to stellar collisions for diﬀerent PDMFs. The value of M that limit in the integral across r . The lower limit should be at contributes to the 2–10 keV ﬂux is given by gal −6 most a few hundred of pcs to at least ∼10 pc. The former 0.06 pc dM value is the tidal radius for the SMBH at the Galactic center ( ) M = ζ r d r , (26) ral gal for a 1M star. Unbound material due to stellar collisions or d r 0 gal dM/dlnM (M /yr) pd ⊙ dM/dlnM (M /yr) pd ⊙ dM/dlnM (M /yr) pd ⊙ 14 Advances in Astronomy −5 −1 0.6 M = 10 M yr . The ﬁgure shows that PDMFs with ﬂat to canonical-like proﬁles are allowed. Very top-heavy proﬁles (α 1.25) are not allowed, as they predict too high of a 0.5 mass loss rate. Mass functions with M 7M are also min not allowed. These results are consistent with measurements 0.4 of the Arches star cluster, a young cluster located about 25 pc from the Galactic center. Recent measurements [42– 44] probing stellar masses down to about 1M show that the 0.3 cluster has a ﬂat PDMF, with α in the range of about 1.2 to 1.9 (depending on the location within the cluster). Since M is a much stronger function of α than of M 0.2 min it is diﬃcult for us to place tight constraints on the allowed range of M . Figure 13 shows that we can, however, place min 0.1 a constraint on the allowed upper limit of M , since very min −5 −1 high values of M result in mass loss rates > 10 M yr . min For α> 1.25, we ﬁt a 3rd degree polynomial (the dashed line −5 −1 −4 −3 −2 −1 0 ˙ in Figure 13) to the M = 10 M yr contour. This ﬁt ana- 10 10 10 10 10 r () pc lytically expresses the upper limit of M as a function of gal min α. We provide the coeﬃcients of this ﬁt in the caption of Figure 12: The fraction of ﬂux emitted from unbound material at Figure 13. radius r that contributes to the 2–10 keV band. gal The small diﬀerence between the solid and dashed lines at r = 0.06 pc in Figure 8(b) suggests that, even for stellar gal where we have shown how to calculate the mass loss rate pro- encounters involving small impact parameters, our integra- ﬁles, dM/d r in the previous section. We account for the tion does not miss many collisions by ignoring gravitational gal fact that not all of the emission from the unbound gas con- focusing. To estimate the contribution to the total mass loss tributes to the 2–10 keV band with ζ(r ), deﬁned as the frac- rate in Figure 13 from gravitational focusing, we take δM , gal typ tion of ﬂux from gas at radius r with 2 keV ≤ hν ≤ 10 keV: the typical amount of mass lost per collision, to be simply a gal function of b. This avoids the multi-dimensional integrations ff 10 keV dν involved in (20)and (21), since for these equations Δ is ν pd resu2 keV −2keV/k T(r ) −10 keV/k T(r ) B gal B gal ζ r ≡ = e − e . gal ∞ ff afunctionof b, M , M ,and v (r ). For simplicity, we pr pd rel gal dν choose δM (b) to decrease linearly from 2M (we assume typ (27) that both stars are completely destroyed) at b = 0to0at b = b .Weﬁnd b by noting from Figure 1 that for all Since the gas at each radius is at a slightly diﬀerent tempera- 0 0 values of the polytropic index, the amount of mass loss for ture, and since ζ is exponentially sensitive to the temperature, an indirect collision goes to zero at around γ = 0.98. By we must estimate T(r ). We do this by setting the thermal gal recalling the deﬁnition of γ (6), we solve for b at γ = 0.98 energy of the unbound material equal to the kinetic energy at 0 aradius r , and ﬁnd that by setting M = 1, and taking v ∼ 2σ(r = 0.06 pc). By gal pr rel gal calculating dΓ/db(<r ) (for Salpeter values) evaluated at gal −1 r 0.06 pc across a range of b, and multiplying by δM (b), we gal typ 2 −2 (28) k T r ≈ m σ r = 7.8 × 10 keV . B gal p gal areabletoestimate dM/db. We do this for dΓ/db(<r )with pc gal and without gravitational focusing and integrate across b. We plot (27)in Figure 12. The value of ζ goes to zero at the Subtracting the two numbers results in our estimate of the highest and smallest radii since, for the former, the gas is cool contribution to the total mass loss rate due to gravitational −7 and emits most of its radiation redward of 2 keV, and for the focusing: 2.3×10 M . This is about twice the mass loss rate latter, the gas is hot and emits mostly blueward of 10 keV. from Figure 13 evaluated at Salpeter values. We perform the Thus, even though the integral in (26) extends to r = 0, the gal same calculation across the M -α parameter space and ﬁnd min contribution to M is suppressed exponentially at the smallest that gravitational focusing contributes a factor of at most radii. ∼2.5 to the total mass loss rate. Since, by (20)and (21), M depends on the parameters of An underestimate of a factor of 2.5 slightly aﬀects the the PDMF, we now constrain these parameters by limiting region of parameter space that we are able to rule out, as −6 −1 the allowed mass loss rate from stellar collisions calculated shown by the line contours in Figure 13. The 4×10 M yr −5 −1 −5 via (26)at10 M yr . We consider changes in M and α, min contour (2.5 times less than the 10 contour) shows that the and keep M set at 125M since (as seen in Figure 9) M is max region of the parameter space that is ruled out is M min approximately independent of M . 1.4M and α 1.4. max We sample the M -α parameter space and use (26)to min compute the total mass loss rate, the results of which are 6. Implications for the IMF shown in Figure 13. The contours represent the calculated M values, where the solid contours are on a logarithmic We now place constraints on the IMF in the Galactic center scale, and where they are limited from above at a value of with a simple analytical approach that connects the IMF to ζ() r gal Advances in Astronomy 15 MM ( /yr) 1e − 05 3.56e − 06 1.27e − 06 4.5e − 07 1.6e − 07 5.7e − 08 0.1 2.03e − 08 0.5 1 1.5 2 2.5 Figure 13: The total mass loss rate contributing the 2–10 keV ﬂux (calculated from (26)) as a function of M and α. The solid contours min −5 −1 are on a logarithmic scale and are limited from above at 10 M yr . The line contours are on a linear scale and are separated by intervals −6 −1 −5 −1 of 1.5× 10 M yr . The thick line denotes the 1× 10 M yr contour. The dashed line is a 3rd-order polynomial ﬁt which represents the absolute allowed upper limit of M as a function of α. The coeﬃcients of this polynomial are 21.71, −42.37, 25.33, and −4.27 for a to a , min 0 3 respectively. 2.5 1.5 1.5 0.5 0.5 0 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 α α (a) (b) Figure 14: (a) The IMF power-law slope as a function of the PDMF power-law slope for the case of constant star formation. (b) The same except for exponentially decreasing star formation with τ = 3, 5, 7, 9 Gyr (bottom to top line, resp.). exp the PDMF, and with the results of the previous section. The and for τ (M) we use the expression given by Mo et al. [45] mass function as a function of time is described by a partial 3 2 2.5 4.5 2.5 × 10 +6.7 × 10 M + M diﬀerential equation that takes into account the birth rate (31) τ (M) = Myr, 1.5 −2 −1 4.5 3.3 × 10 M +3.5 × 10 M and death rate of stars: valid for 0.08M <M < 100M and for solar-type metallic- ∂ξ(M, t) 1 = R (t)Φ(M) − ξ , (29) B ity. ∂t τ (M) In the following paragraphs, we consider diﬀerent star formation history scenarios. For each scenario, we will need where R (t) is the birth rate density of stars (dN /(dtd r )), B B gal to know R (τ ), the star formation rate density in the Φ(M) is the initial mass function normalized such that B MW Galactic center at the age of the Milky Way (which we take to Φ(M)dM = 1, and τ (M) is the main sequence lifetime be 13 Gyr). A rough estimate of this value is given by the of stars as a function of stellar mass. For the initial mass number density of young stars in the Galactic center divided function, we take a power law, by their age: R (τ ) ∼ ρ(r)η/(τM). Here τ and M B MW −γ Φ = M , (30) are the average age and average mass of the young stars in the MM() min γ 16 Advances in Astronomy Galactic center, which we take to be ∼ 10 Myr and ∼10M , R (τ ) at the present-time, R proﬁles with smaller values B MW B respectively. The parameter η is the fraction of stars with of τ have had overall more star formation in the past. exp masses above 10M , which for reasonable mass functions More overall star formation means that the present-day mass is ∼0.1%. For self-consistency, we use ρ evaluated at 0.06 pc function is comprised of fractionally more lower-mass stars 7 −3 (which from (10)is ∼10 M pc ), since this was the radius since the IMF favors lower-mass stars. The constant buildup at which we used to constrain the present-day mass function. of lower-mass stars results in a steeper PDMF, so that for −4 −3 −1 Thesevaluesresultin R (τ ) ∼ 10 pc yr . any given γ, α should be larger. The ﬁgure shows that for B MW For the simple case of a constant star formation rate, exponentially decreasing star formation γ must be 0.6, 0.8, R (t) = R (τ ), and the solution to (29) with the bound- 0.9, and 1.0 for τ = 3, 5, 7, and 9 Gyr, respectively. B B MW exp ary condition that ξ(M, t = 0) = Φ(M)n (t = 0), evaluated The ﬁnal case we consider is episodic star formation, tot at the current age of the Milky Way is where each episode lasts for a duration Δt, where the ending and beginning of each episode is separated by a time, T,and ξ(M, t = τ ) MW where the magnitude of each episode is R (τ ). For such a B MW star formation history, the solution to (34)is −τ /τ (M) MW = Φ(M)e ! " ξ(M, t = τ ) MW τ /τ (M) MW × R (τ )τ (M)e − R (τ )τ (M) + n (0) . B MW B MW tot −τ /τ (M) MW = Φ(M)e (32) max ⎨ & We evaluate the solution at the age of the Milky Way (yielding [(n+1)Δt+nT]/τ (M) × R (τ )τ (M) e B MW the PDMF) because we want to compare with our constraints ⎩ n=0 on the PDMF as found in the previous section. To solve for n (0), we use the known mean density of the Galactic center tot ' ⎬ n(Δt+T)/τ (M) today at 0.06 pc, ρ(τ , r = 0.06 pc), insert (32) into the fol- −e + n (0) , MW tot lowing expression: (36) ( ) ρ τ , r = 0.06 pc = ξ M, t = τ MdM, (33) MW MW where n = ﬂoor{(τ −Δt)/(T+Δt)} and where we again max MW solve for n (0) with (33). We consider 9 cases with Δt and tot and solve for n (0). 6 7 8 tot T = 10 ,10 ,and 10 yrs and show the results in Figure 15. We solve for ξ(M, τ )for arange of diﬀerent IMF 8 MW In each panel the lowest line is Δt = 10 yrs and the highest 6 6 power-law slopes, γ and ﬁt a power law to the solution, with line is Δt = 10 yrs. For T = 10 yrs, γ 0.8 and 0.5 for a power-law slope α. We plot the IMF power-law slope as 6 7 8 Δt = 10 and 10 yrs, respectively, while the Δt = 10 yrs a function of the calculated PDMF power-law slope for case results in constraints on γ that are too low to be realistic. constant star formation in Figure 14(a).Wehaveconstrained 7 6 7 For T = 10 yrs, γ 0.5 and 0.4 for Δt = 10 and 10 yrs, the PDMF in the previous section to have α 1.25, indicated 8 respectively, while again, the Δt = 10 yrscaseresults in in the ﬁgure by the vertical line. The ﬁgure therefore shows 8 unrealistic constraints. Finally, for the T = 10 yrs, γ 0.5 that for the case of constant star formation, the IMF power- 6 7 8 for Δt = 10 , while the Δt = 10 and 10 yrs case result in law slope, γ,mustbe 0.9. unrealistic constraints. We test if when the last star formation For the general case of a star formation rate that varies episode occurs (relative to the present day) it aﬀects our solu- with time, R (t)= R (τ ), and the solution to (29)with B / B MW tion of ξ(M, τ ) by varying the start time of the star MW the same boundary condition and evaluated at τ is: MW formation episodes. By varying the start time and testing all the combinations of Δt and T that we consider, we ﬁnd that −τ /τ (M) MW ξ(M, t = τ ) = Φ(M)e MW the lines in Figure 15 vary by at most about 5%, so that the # $ MW main trends in the ﬁgure are unaﬀected. t /τ(M) × R (t )e dt + n (0) . B tot 7. Contribution from Red Giants (34) Spectroscopic observations have revealed that the central For an exponentially decreasing star formation history, the parsec of the Galaxy harbors a signiﬁcant population of giant star formation rate is given by stars [18, 19]. Due to their large radii (and hence large cross −(t−τ )/τ MW exp sections), it is possible that they could play an important part R (t) = R (τ )e . (35) B B MW in the mass loss rate due to collisions in the Galactic center. Given this star formation history, we solve for ξ(M, τ )(by In assessing their contribution to the mass loss rate, care MW solving for n (0) with (33)) for τ =3, 5, 7 and 9 Gyr. We must be taken when deriving the collision rates, because their tot exp ﬁt power-laws to the resulting PDMFs, and show the results radii, r , are strong functions of time, t. Dale et al. [14] RG in Figure 14(b). The ﬁgure shows that smaller values of τ have already calculated the probability, P(r ), for a red giant exp gal result in larger values of α for any given γ.The trendcan be (RG) in the Galactic center to undergo collisions with main explained by the fact that since a smaller value of τ results sequence impactors. They have taken into account that r (t) exp RG in a steeper R proﬁle, and that all proﬁles must converge to by integrating the collision probability over the time that the B Advances in Astronomy 17 6 7 T = 10 yrs T = 10 yrs 3 3 2 2 0 0 0123 456 0123 456 α α (a) (b) T = 10 yrs 2.5 1.5 0.5 0123 456 (c) Figure 15: The IMF power-law slope as a function of the PDMF power-law slope for the case of episodic star formation. In each panel the 8 6 lowest line is Δt = 10 yrs and the highest line is Δt = 10 yrs. star resides on the RG branch. We use their results to estimate the collision rate averaged over the lifetime of the RG and is the mass loss rate due to RG-MS star collisions. given by To ﬁnd the number density of RGs in the Galactic center, + , P r we weight the total stellar density by the fraction of time the gal ˙ ˙ (38) P r ∼ P r = . gal gal star spends on the RG branch: RG If we deﬁne δM to be the typical amount of mass lost in the RG n r ∼ n r . collision, then the mass loss rate is RG gal gal (37) P r ˙ ˙ gal dM dM 3 3 (39) = 4πr ∼ 4πr n r δM. RG gal This approximation should be valid given a star formation gal gal dlnr d r τ gal gal RG history that is approximately constant when averaged over time periods of order τ . The number of collisions per unit To calculate an upper limit for the contribution of RG time suﬀered by any one red giant, P(r ), shouldbeoforder RG-MS star collisions to the mass loss rate, we assume that gal γ 18 Advances in Astronomy −5 thus possible that for MS-MS collisions, values of M and min −5 −1 α that result in total mass loss rates just below 10 M yr could be pushed past this threshold with the addition of mass −6 loss due to RG collisions. However, we believe that this is unlikely for two reasons. The inclusion of the factor, ζ, when calculating the total mass loss rate (see (26)) will reduce the −7 mass loss by at least a factor of 0.6 (see Figure 12). Also, as noted by the hydrodynamic simulations of Dale et al. [14], for a typical RG-MS star collision, at most ∼10% of the RG envelope is lost to the RG. This will reduce dM/dlnr for gal −8 RG-MS collisions by another factor of 10. −9 10 8. Conclusions We have have derived novel, analytical methods for calcu- −10 lating the amount of mass loss from indirect and direct −4 −3 −2 −1 0 1 stellar collisions in the Galactic center. Our methods com- 10 10 10 10 10 10 pares very well to hydrodynamic simulations and do not r () pc gal require costly amounts of computation time. We have also Figure 16: An upper limit to the mass loss rate due to collisions bet- computed the total mass loss rate in the Galactic center ween RG and MS stars. The arrow indicates the range in the diﬀuse due to stellar collisions. Mass loss from direct collisions X-ray observations (r < 1.5 ) which we use to constrain the gal dominates at Galactic radii below ∼0.1 pc, and thereafter PDMF (see Section 5). indirect collisions dominate the total mass loss rate. Since the amount of stellar material lost in the collision depends upon the masses of the colliding stars, the total mass loss rate all RG and MS stars have masses of 1M and that the entire depends upon the PDMF. We ﬁnd that the calculated mass RG is destroyed in the collision. Collisions involving 1M loss rate is sensitive to the PDMF used and can therefore RGs yield an upper limit, because there is not an appreciable be used to constrain the PDMF in the Galactic center. As amount of RGswithmasseslessthan ∼ 1M due to their MS summarized by Figure 13, our calculations rule out α 1.25 lifetimes being greater than the age of the Galaxy. For RGs and M 7M in the M -α parameter space. Finally, min min with masses greater than 1M , the amount they contribute we have used our constraints on the PDMF in the Galactic to the mass loss rate is a competition between their lifetimes center to constrain the IMF to have a power-law slope 0.4 and radii. Red giant lifetimes decrease with mass (thereby to 0.9 depending on the star formation history of the Galactic decreasing the time they have to collide) and their radii in- center. crease with mass (thereby increasing the cross section). In their Figure 3, Dale et al. [14] clearly show that the number of collisions decreases with increasing RG mass, indicating Acknowledgments that the brevity of their lifetime wins over their large sizes. This work was supported in part by the National Science One solar mass MS impactors should yield approximately an Foundation Graduate Research Fellowship, NSF Grant AST- upper limit to the mass loss rate, since ∼ 1M MS stars are 0907890, and NASA Grants NNX08AL43G and NNA09- the most common for the PDMFs under consideration. DB30A. Since we assume that the entire RG is destroyed in the col- lision δM = 1M . For the case that all impactors are 1M MS stars, we calculate n (r )from(37) by noting that RG gal References n (r ) = ρ (r )/(1M ). 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