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Advances in Astronomy
Volume 2011 (2011), Article ID 174105, 19 pages
http://dx.doi.org/10.1155/2011/174105
Research Article

Constraining the Stellar Mass Function in the Galactic Center via Mass Loss from Stellar Collisions

1Department of Physics, Harvard University, Cambridge, MA 02138, USA
2Department of Astronomy, Harvard University, Cambridge, MA 02138, USA

Received 6 September 2011; Accepted 23 November 2011

Academic Editor: Paola Marziani

Copyright © 2011 Douglas Rubin and Abraham 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.

Abstract

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 luminosity in the Galactic center to preclude any present-day mass functions that result in mass loss rates >105𝑀yr1 in the vicinity of 1. For present-day mass functions of the form, 𝑑𝑁/𝑑𝑀𝑀𝛼, 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

The dense stellar core at the Galactic center has a radius of ~0.15–0.4 pc, a stellar density >106𝑀 pc−3 [14], high velocity dispersions (≥100 km s−1), and Sgr A*, the central supermassive black hole with a mass 4×106𝑀 [59]. Due to the extreme number densities and velocities, stellar collisions are believed to play an important role in shaping the stellar structure around the Galactic center and in disrupting the evolution of its stars Frank and Rees [10]. Genzel et al. [1] found a paucity of the brightest giants in the galactic center and proposed that collisions with main sequence (MS) stars could be the culprit. This hypothesis was found to be plausible by Alexander [11]. Other investigations of collisions between giants and MS, white dwarf and neutron stars [12] and collisions between giants and binary MS and neutron stars [13] could not account for the dearth of observed giants. The contradictory results were resolved by Dale et al. [14], who concluded that the lack of the faintest giants (but not the brightest giants) could be explained by collisions between giants and stellar mass black holes. Significant mass loss in the giants’ envelopes after a collision would prevent the giants from becoming bright enough to be observed.

The above studies concentrated on collisions involving particular stellar species with particular stellar masses. To examine the cumulative effect of collisions amongst an entire ensemble of a stellar species with a spectrum of masses, one must specify the present-day stellar mass function (PDMF) for that species. The PDMF gives the current number of stars per unit stellar mass up to a normalization constant. Given a certain star formation history, the PDMF can be used to determine the initial mass function of stars (IMF), the mass function with which the stars were born. There is currently no consensus as to whether the IMF in the Galactic center deviates from the canonical IMF [15].

First described by Salpeter more than 50 years ago [16], the canonical IMF is an empirical function which has been found to be universal [17], with the Galactic center as perhaps the sole exception. Maness et al. [18] found that models with a top-heavy IMF were most consistent with observations of the central parsec of the Galaxy. Paumard et al. [19] and subsequently Bartko et al. [20] found observational evidence for a flat IMF for the young OB-stars in the Galactic center. On the other hand, Löckmann et al. [21] concluded that models of constant star formation with a canonical IMF could explain observations of the Galactic center.

In this work we use calculated mass loss rates due to stellar collisions as a method to constrain the PDMF for main sequence stars in the Galactic center. We construct a simple model to estimate the actual mass loss rate in the Galactic center based on observed diffuse X-ray emission. PDMFs that predict mass loss rates from stellar collisions greater than the observed rate are precluded. This method allows us to place conservative constraints on the PDMF, because we do not include the contribution to the mass loss rate from stellar winds from massive evolved stars [22]. Specifically, this method allows us to place a lower limit on the power-law slope and an upper limit on the minimum stellar mass of the PDMF in the Galactic center (see Section 5). Inclusion of the mass loss rate from stellar winds (or other sources) could further constrain the PDMF of the Galactic center.

The work presented in this paper has implications for the fueling of active galactic nuclei (AGN). To trigger an AGN, a significant amount of matter must be funneled onto the supermassive black hole in a galactic nucleus. The most common way of channelling gas is through galaxy mergers, which has been studied for quite some time (e.g., A.Toomre and J.Toomre [23]; Gunn [24] Hernquist and mihos [25]). Even without mergers, AGN can be fed by several processes from stellar residents in a galactic center. The tidal disruption of a star which passes too close to the supermassive black hole can strip mass off the star. Additionally, it is known that a significant amount of gas is ejected into the Galactic center due to stellar winds from massive, evolved stars [22, 26, 27]. Another potential source for the fueling of AGN could be from unbound stellar material, ejected in a stellar collision. Since the easiest place to look for such an event (due to its proximity) is the Galactic center, in this paper we theoretically investigate stellar collisions in this environment. By calculating the cumulative mass loss rate from stellar collisions in the Galactic center, we place constraints on the fueling of Sgr A* due to this mechanism.

We present novel, analytical models to calculate the amount of stellar mass lost due to stellar collisions between main sequence stars in Section 2 through Section 2.3. In Section 3 we develop the formalism for calculating collision rates in the Galactic center. We utilize our calculations of the mass loss per collision, and the collision rate as a function of Galactic radius to find the radial profile of the mass loss rate in Section 4. Since the amount of mass lost is dependent on the masses of the colliding stars, the mass loss rate in the Galactic center is sensitive to the underlying PDMF. By comparing our calculations to mass loss rates obtained from the diffuse X-ray luminosity measured by Chandra, in Section 5 we constrain the PDMF of the Galactic center. We derive analytic solutions of the PDMF as a function of an adopted IMF for different star formation scenarios, which allows us to place constraints on the IMF in Section 6. In Section 7, we estimate the contribution to the mass loss rate from collisions involving red giant (RG) stars.

2. Condition for Mass Loss

Throughout this paper we refer to the star that loses material as the perturbed star, and the star that causes material to be lost as the perturber star. Quantities with the subscript or superscript “pd’’ or “pr’’ refer to the perturbed star and perturber star, respectively (Note that for any particular collision, it is arbitrary which star we consider the perturber star, and which star the perturbed star. Both stars will lose mass due to the presence of the other, so in order to calculate the total mass loss, we interchange the labels (pdpr), and repeat the calculation.). We work in units where mass is measured in the mass of the perturbed star, 𝑀pd, distance in the radius of the perturbed star, 𝑟pd, velocity in the escape velocity of the perturbed star, 𝑣pdesc (=2𝐺𝑀pd/𝑟pd), and time in 𝑟pd/𝑣pdesc. We denote normalization by these quantities (or the appropriate combination of these quantities) with a tilde:𝑀𝑀𝑀pd,𝑟̃𝑟𝑟pd,̃𝑣𝑣𝑣pdesc,̃𝑡𝑡𝑟pd/𝑣pdesc.(1) We refer to collisions in which 𝑏>𝑟pd+𝑟pr as “indirect” collisions, and collisions in which 𝑏𝑟pd+𝑟pr as “direct” collisions. The impact parameter, 𝑏, is the distance of closest approach measured from the centers of both stars.

We consider the condition for mass loss at a position, ̃𝑟, within the perturbed star to be that the kick velocity due to the encounter at ̃𝑟 exceeds the escape velocity of the perturber star at ̃𝑟, Δ̃̃𝑣𝑣(̃𝑟)esc(̃𝑟). The escape velocity as a function of position within the perturbed star can be found from the initial kinetic and potential energies of a test particle at position ̃𝑟,̃𝑣esc(̃𝑟)=̃𝑟𝑀int̃𝑟̃𝑟2𝑑̃𝑟=𝑀int(̃𝑟)̃𝑟+4𝜋1̃𝑟̃𝜌̃𝑟̃𝑟𝑑̃𝑟,(2) where 𝑀int is the mass interior at position ̃𝑟 and ̃𝜌 is the density profile of the star.

2.1. Mass Loss due to Indirect Collisions

To calculate the mass lost due to an indirect collision, we first calculate the kick velocity given to the perturbed star as a function of position within the star. We work under the impulse approximation [28], valid under the condition that the encounter time is much shorter than the characteristic crossing time of a constituent of the perturbed system.

Given a mass distribution for the perturbed system, 𝜌pd and a potential for the perturber system, Φ, the kick velocity after an encounter under the impulse approximation is given by Binney and Tremaine [29]: Δ𝑣𝑟=1Φ𝑟,𝑡𝑀pd𝜌pd𝑟,𝑡Φ𝑟𝑑,t3𝑟𝑑𝑡.(3) Equation (3) can be simplified by expanding the gradient of the potential in a Taylor series, resulting inΔ𝑣=𝑟2𝐺𝑀pr𝑏2𝑣rel𝑦0𝑥+𝑂𝑟2.(4) The expansion is valid under the “distant tide” approximation which is satisfied when 𝑟pd𝑏. The parameter 𝑣rel is the relative speed between the stars (𝑣rel𝑣|pd𝑣pr|). We are interested in the magnitude of (4), which when normalized to the units that we have adopted for this paper isΔ̃𝑣(̃𝑥,̃𝑦)𝛾̃𝑥2+̃𝑦2,(5) where𝑀𝛾pr̃𝑏2̃𝑣rel.(6)

To solve for the mass lost per encounter as a function of 𝛾, we consider a star within a cubic array, where the star contains ~3 × 106 cubic elements. As a function of 𝛾 we compare the kick velocity in each element to the escape velocity for that element and consider the mass within the element to be lost to the star if the velocities satisfy the condition given in Section 2. We note that by ~105 elements, the results converge to within about 2%, and we are therefore confident that ~3 × 106 provides adequate resolution.

To calculate the amount of mass in each element, the density profile for the perturbed star must be specified. As with several previous studies on mass loss due to stellar collisions [3033] we utilize polytropic stellar profiles. Polytropic profiles are easy to calculate and yield reliable results for stars of certain masses. Polytropic profiles of polytropic index 𝑛=1.5 describe the density structure of fully convective stars, and therefore very well describe MS stars with 𝑀0.3𝑀 (nearly fully convective) and MS stars with 𝑀10𝑀 (convective cores). MS stars with 𝑀1𝑀 have radiative envelopes and are therefore well-described by 𝑛=3. For 𝑛 for stars with masses of 0.3–1𝑀 and 5–10𝑀, we linearly interpolate between 𝑛=1.5 and 3. We discuss the uncertainties introduced by this approach in Section 4. Note that this approach is biased towards zero-age main sequence stars, since as stars evolve, they are less adequately described by polytropic profiles.

We plot the fraction of mass lost from the perturbed star per event, Δ, as a function of 𝛾 in Figure 1 for several polytropic indices. The lines are third-order polynomial fits to our results, in the range of 0.98𝛾5. We list the coefficients of the polynomial fits in Table 1. For each density profile, no mass is lost up until 𝛾 of about 0.98, and thereafter the mass loss increases monotonically. The increasing trend is due to the fact that larger perturber masses and smaller impact parameters result in an increased potential felt by the perturbed star. Smaller velocities also cause more mass to be lost, as this increases the “interaction time” between the perturber and perturbed stars.

tab1
Table 1: Coefficients of polynomial fits for Δ(𝛾) with varying polytropic indices.
174105.fig.001
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.

The location of the mass loss within the perturbed star for fixed 𝛾 depends upon the polytropic index, since the escape velocity within the star is dependent upon the density profile, as indicated by (2). In Figure 2, we illustrate where mass will be lost in the perturbed star by plotting contours of the kick velocity (Δ̃𝑣(̃𝑟)) due to the encounter normalized to ̃𝑣pdesc(̃𝑟) for 𝑛=1.5 and 𝑛=3 (bottom and top rows, resp.). We show two different cases: a slightly perturbing encounter with 𝛾=1.2 in the first column, and a severely perturbing encounter with 𝛾=1.6 in the second column. The grey region underneath shows where mass is still left after the encounter, since Δ̃̃𝑣𝑣/pdesc(̃𝑟) within this region is <1. The 𝛾=1.6 encounter results in bigger kick velocities, and so we see that the mass loss penetrates farther into the star. 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 used, ̃𝑣pdesc converges to the same value at large radii when the second term in (2) becomes negligible. Even though the location of where mass is lost is similar for different polytropic stars at the same value of 𝛾, the amount of the mass lost is substantially different (as shown in Figure 1), due to the different density profiles.

fig2
Figure 2: Slices through the perturbed star along the plane parallel to the perturber star’s trajectory. The first column ((a) and (c)) correspond 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 velocity (as a function of ̃𝑟). The outline of the grey region underneath has Δ̃̃𝑣𝑣/pdesc(̃𝑟)=1, so that the grey region represents the location of where mass is still left after the event.
2.2. Validity of Approach for Indirect Collisions

The impulse approximation is valid provided that the time over which the encounter takes place, 𝑡enc, is much shorter than the time it takes to cross the perturbed system, 𝑡cross. To estimate when our calculations break down, we approximate 𝑡enc as 𝑏/𝑣rel, and 𝑡cross as 𝑡𝑠, the time it takes for a sound wave to cross an object that is in hydrostatic equilibrium:tcross𝑡𝑠1𝐺𝜌pd1𝐺𝑀pd/𝑟3pd.(7) These approximations lead to the condition that̃𝑣1rel̃𝑏1.(8)

Aguilar and White [34] find that for a large range of collisions, the impulse approximations remains remarkably valid, even when 𝑡enc is almost as long as 𝑡cross. We therefore assume that the impulse approximation holds until the left hand side of (8) is ~1. Our calculation of Δ as a function of 𝛾 should therefore be valid for 𝛾𝛾valid, where 𝛾valid𝑀pr/̃𝑏3. We plot contours of log(𝛾valid) in the 𝑀pr/𝑀pd-𝑏/𝑟pd parameter space in Figure 3, where both the 𝑥 and 𝑦 axes span ranges relevant to our calculations. The shaded grey area in the figure is the region of the parameter space where the impulse approximation predicts nonzero mass loss due to the encounter. The figure shows that 𝛾valid is smaller for low 𝑀pr to 𝑀pd ratios at high impact parameters. In fact, most of the right side of the parameter space has 𝛾valid less than 0.98 (where below this value, the impulse approximation predicts no mass lost).

174105.fig.003
Figure 3: Contours of log(𝛾valid) in the 𝑀pr/𝑀pd-𝑏/𝑟pd parameter space, where 𝛾valid is defined in Section 2. The shaded grey area indicates where the impulse approximation predicts nonzero mass loss.

In our calculations, when, for any particular set of 𝑀pr/𝑀pd and 𝑏/𝑟pd, 𝛾>𝛾valid, we adopt Δ(𝛾>𝛾valid)=Δ(𝛾=𝛾valid). This approach represents a lower limit on the amount of mass loss that we calculate, since mass loss should increase with increasing 𝛾. We find, however, that if we set Δ(𝛾>𝛾valid)=1 (which represents the absolute upper limit in the amount of mass lost) the change in our final results is negligible at small Galactic radii. At large radii, where the mass loss from indirect collisions dominates (see Section 4), the results change by at most a fact of ~2.

Equation (4) was derived under the assumption that the impact parameter is much bigger than both 𝑟pd and 𝑟pr. Since Δ𝑣 scales as 𝑏2, the equation predicts that most mass loss occurs for small impact parameters. However, given the assumption that was used to derive the equation, the regime of small impact parameters is precisely where (4) breaks down. Numerical simulations [34, 35] show that for a variety of perturber mass distributions, the energy input into the perturbed system is well described by (4) for 𝑏5𝑟, where 𝑟 is the half mass radius of the perturber system. For an 𝑛=3 polytropic star, 5𝑟=1.4𝑟. Since for indirect collisions, 𝑏/𝑟pd=1+𝑟pr/𝑟pd+𝑑/𝑟pd (where 𝑑 is the distance between the surface of both stars), there is only a small region in our calculations, 0𝑑/𝑟pd(0.4𝑟pr/𝑟pd), for which (4) may give unreliable results.

2.3. Mass Loss due to Direct Collisions

A number of papers over the past few decades have addressed the outcomes of stellar collisions where the two stars come so close to each other that not only gravitational, but also hydrodynamic forces must be accounted for. Early studies used one- or two-dimensional low-resolution hydrodynamic simulations (e.g., [36, 37]). Modern studies typically utilize smooth particle hydrodynamics with various stellar models, mass-radius relations, and varying degrees of particle resolution [3033]. A detailed review of the literature can be found in this area in Freitag and Benz [38].

We approach the problem of direct collisions in a highly simplified, analytic manner without hydrodynamic considerations and find that for determining the amount of mass lost, our method compares well to the complex hydrodynamic simulations. As a first-order model, we approximate the encounter as two colliding disks, by projecting the mass of both stars on a plane perpendicular to the trajectory of the perturber star. The problem of calculating mass loss then becomes easier to handle, as it is two-dimensional. We also assume that mass loss can only occur in the geometrical area of intersection of the two stars.

We find the kick velocity as a function of position in the area of intersection by conserving momenta and by assuming that all of the momentum in the perturber star in each area element was transferred to the corresponding area element in the perturbed star. Working in the frame of the perturbed star and with a polar coordinate system at its center (so that 𝑟=𝑥2+𝑦2), we findΔ̃Σ𝑣(̃𝑟)=pr̃𝑣(̃𝑟)relΣpd(̃𝑟).(9) The parameters Σpr and Σpd represent the surface density of the perturber and perturbed stars, respectively, (𝜌𝑑𝑧).

To find the region of intersection, we need to know the impact parameter and the radii of both stars. To obtain the stellar radii as a function of mass, we use the mass-radius relation calculated by Kippenhahn and Weigert [39] for a MS star with 𝑍=𝑍, 𝑋𝐻=0.685 and 𝑋He=0.294 from a stellar evolution model, where 𝑋 represents the mass fraction. 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-radius relation that spans from about 0.01 to 150𝑀. We compare our mass-radius relation to those used in other studies of direct stellar collisions in Figure 4. Rauch [33], Lai et al. [32], and Benz and Hills [31] all adopted power laws with power law indices of 1.0, 0.8, and 0.85, respectively, (thin lines). Freitag and Benz [38] (dotted line) use main sequence stellar evolution codes to obtain a mass-radius relation for masses >0.4𝑀 and a polytropic mass-radius relation of 𝑛=1.5 for masses <0.4𝑀.

174105.fig.004
Figure 4: Mass-radius relations used in studies of calculating mass loss from stars due to stellar collisions. The thin lines are power-law relations of power-law index 1.0, 0.8, and 0.85 used by Rauch [33], Lai et al. [32], and Benz and Hills [31], respectively. The dotted line is the relation used by Freitag and Benz [38], and the thick line is the relation used in this work.

Our simple model for calculating mass loss due to direct stellar collisions compares surprisingly well to full blown smooth particle hydrodynamic simulations. We borrow plots of the fractional amount of mass lost as a function of impact parameter for specific relative velocities and stellar masses from Freitag and Benz [38] (Figures 5 and 6). They show their own work, the best calculations of mass loss due to stellar collisions to date. For comparison, and to show how the calculations have evolved over the years, the results from older studies are also shown. Our own results are plotted (dashed-dotted black lines) over these previous studies. We make sure to show results spanning a wide range of stellar masses and relative velocities. Note that these plots show the fractional amount of mass lost from both stars normalized to the initial masses of both stars, and that the impact parameter is normalized to the sum of both stellar radii. Our results show the same qualitative trends seen in the Freitag and Benz [38] curves, even replicating several “bumps" seen in their curves (see Figures 6(c) and 6(d)). As compared to the Freitag and Benz [38] results, for any specific set 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.

fig5
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].
fig6
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].

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 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.

We adopt the stellar density profile of Schödel et al. [4], one of the best measurements of the density profile within the Galactic center to date. Using stellar counts from high-resolution images of the galactic center, they find that the density profile is well-approximated by a broken power law. Moreover, they use measured velocity dispersions to constrain the amount of enclosed stellar mass as a function of galactic radius, 𝑟gal. Using their density profile, and velocity dispersion measurements, they find that𝜌𝑟gal=2.8±1.3×106𝑀pc3𝑟gal0.22pc1.2,for𝑟gal0.22pc,2.8±1.3×106𝑀pc3𝑟gal0.22pc1.75,for𝑟gal>0.22pc.(10) Their average density can be converted into a local density, 𝜌(𝑟gal), by considering the definition of 𝜌,𝜌𝑟gal𝑟gal04𝜋𝑟2gal𝜌𝑟gal𝑑𝑟gal4/3𝜋𝑟3gal,(11) from which we derive𝜌𝑟gal=𝜌𝑟gal+𝑟gal3𝑑𝜌𝑟gal𝑑𝑟gal.(12)

We use (10) and (12), to find 𝜌(𝑟) and plot the result in Figure 7. We “smoothed” the unphysical discontinuity in 𝜌 arising from the kink of the broken power law fit by fitting a polynomial to (10).

174105.fig.007
Figure 7: The stellar density profile that we adopt, based on the average density profile of Schödel et al. [4].

The differential collision rate, 𝑑Γ, between two species, “1” and “2” at impact parameter 𝑏 characterized by distribution functions 𝑓1 and 𝑓2, and moving with relative velocity |𝑣1𝑣2| in a spherically symmetric system is𝑑Γ=𝑓1𝑟gal,𝑣1𝑑3𝑣1𝑓2𝑟gal,𝑣2𝑑3𝑣2×||𝑣1𝑣2||2𝜋𝑏𝑑𝑏4𝜋𝑟2gal𝑑𝑟gal.(13) For simplicity, we adopt Maxwellian distributions,𝑓1,2𝑟gal,𝑣1,2=𝑛1,2𝑟gal2𝜋𝜎23/2𝑒𝑣21,2/2𝜎2,(14) where we find the velocity dispersion, 𝜎, from the Jean’s equations. Assuming an isotropic velocity dispersion, a spherical distribution of stars and a power-law density profile with power-law slope 𝛽, 𝜌𝑟𝛽gal, the Jean’s equations lead to 𝜎2=𝐺𝑀SMBH/𝑟gal(1+𝛽), where 𝑀SMBH=4×106𝑀. From Figure 7, it is evident that 𝛽 is a function of 𝑟gal, but for simplicity, we adopt an averaged value of 𝛽, 𝛽=1.3. Note that we have also assumed that the enclosed mass at position 𝑟gal is dominated by the SMBH. This assumptions is valid out till ~1 pc, which is also the point where our impulse approximation starts to break down.

A change of variables allows one to integrate out 3 of the velocity dimensions and to write the expression in terms of 𝑣rel (see [29]). We can also take into account the fact that both species have a distribution of masses by introducing, 𝜉1,2, the PDMF, which gives the number density of stars per mass bin (𝜉𝑑𝑛/𝑑𝑀). We adopt a power law PDMF,𝜉𝑀𝛼,(15) that runs from some minimum mass, 𝑀min to a maximum mass 𝑀max. Since most initial mass functions are parameterized with a power law, the present-day mass function might be modified from a power law due to the effects of collisions and stellar evolution. Although the actual PDMF might have deviations from a power law, adopting a power law provides us with a quick and simple way to parameterize the PDMF. Taking all of this into account, and assuming that the relative velocities are isotropic, we arrive at the final nondimensionalized expression for the differential collision rate:𝑑Γ=4𝜋3/2𝜎3𝑒̃𝑣2rel/4𝜎2̃𝑣3rel𝐾2×𝑀1𝛼𝑀2𝛼̃𝑟2gal̃̃𝑏𝑑𝑏𝑑̃𝑟gal𝑑̃𝑣rel𝑑𝑀1𝑑𝑀2.(16) The tildes denote normalization by the proper combination of 𝑀2, 𝑟2, and 𝑣2esc. The parameter 𝐾 is the normalization constant for 𝜉, which can be solved for by using the density profile of Figure 7 and the following expression:𝜌=𝑀max𝑀min𝑑𝑛𝑟𝑑𝑀𝑀𝑑𝑀=𝐾gal𝑀max𝑀min𝑀1𝛼𝑟𝑑𝑀=𝐾gal𝑀2𝛼max𝑀2𝛼min.2𝛼(17) Since the expression for 𝐾, which controls the total number of stars, has no time dependence, our expression for the PDMF assumes a constant star formation rate in the Galactic center.

Our calculations involve the computation of multidimensional 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 estimate of the collision rate. As two projectiles collide with each other, their mutual gravitational attraction pulls them together, resulting in an enhancement of the cross section:𝑀𝑆𝑆1+2𝐺1+𝑀2𝑏𝑣2rel.(18) We discuss the uncertainties in our final results due to ignoring gravitational focusing at the end of Section 5.

To illustrate the frequency of collisions in the Galactic center, we integrate (16) over 𝑣rel, 𝑀1, and 𝑀2 assuming a Salpeter-like mass function (𝛼=2.35, 𝑀min=0.1𝑀 and 𝑀max=125𝑀) to obtain 𝑑Γ/(𝑑𝑙𝑛𝑟gal𝑑𝑏) as a function of 𝑟gal (Figure 8(a)) (This figure and subsequent figures in this paper with 𝑟gal as the independent variable start from 𝑟gal=106 pc. This value of 𝑟gal corresponds to the tidal radius for a 1𝑀 star associated with a 4×106𝑀 SMBH. Although stars of different masses will have slightly different tidal radii, the main conclusions of our paper are based off of distances in 𝑟gal of order 0.1 pc (see Section 5), well above the tidal radius for any particular star.). We plot 𝑑Γ/(𝑑𝑙𝑛𝑟gal𝑑𝑏) for several different impact parameter values. We calculate 𝑑Γ/(𝑑𝑙𝑛𝑟gal𝑑𝑏) with and without the effect of gravitational focusing (solid and dashed lines, resp.). The latter is obtained by multiplying (16) by the gravitational focusing enhancement term before the integration. As expected, gravitational focusing is negligible at small Galactic radii since typical stellar encounters involve high relative velocities. As the typical relative velocities decrease with increasing Galactic radius, the enhancement to the collision rate from gravitational focusing becomes important. The figure also shows that gravitational focusing becomes less important with increasing impact parameter since the gravitational attraction between the stars is weaker. Figure 8(b) shows the cumulative differential collision rate (integrated over 𝑟gal) per impact parameter as a function of 𝑟gal. Again, we plot the results with and without gravitational focusing and for the same impact parameters.

fig8
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 𝑟gal) per impact parameter with and without gravitational focusing for the same impact parameter values. (a), (b) The vertical line in each panel is placed at 𝑟gal=0.06pc, the upper bound in our integration across 𝑟gal as performed in Section 5.

4. Mass Loss Rates in the Galaxy

To calculate the mass loss rate from stars due to collisions within the Galactic center, we multiply (16) by the fraction of mass lost per collision, Δ(𝛾), and compute the multidimensional integral. We calculate the total mass loss rate from both the perturbed and perturber stars by simply interchanging the “pr’’ and “pd’’ labels and reperforming the calculation.

We first compute the differential mass loss rate for indirect collisions. The mass loss per collision is given byΔ(𝛾)=0for𝛾<0.98,polynomialfor0.98𝛾𝛾validΔ𝛾validfor𝛾>𝛾valid.,(19) The coefficients for the polynomial depend on the polytopic index of the perturbed star (and thus on its mass) and are taken from Table 1. We multiply (19) and (16) and simplify the integration. In principle, 𝑏 should go to , but we cut off the integral at ̃𝑏max=20 as we find that the results converge well before this point. The velocity integral is also cut off at ̃𝑣max due to the fact that Δ(𝛾) becomes zero below 𝛾=0.98. This cut-off corresponds to ̃𝑣max𝑀=(pr)max̃𝑏/0.982min. We may safely throw away the exponential as ̃𝑣2max𝜎2(̃𝑟gal) for the range of ̃𝑟gal that we consider. Thus, the integral that we evaluate is𝑑̇𝑀𝑑𝑙𝑛̃𝑟galpd4𝜋3/2𝜎3̃𝑟3gal𝐾2𝑀max𝑀min𝑀max𝑀miñ𝑣max0×̃𝑏max1+̃𝑟pr̃𝑏̃𝑣3relΔpd(𝑀𝛾)𝛼pr𝑀𝛼pd𝑑̃𝑀𝑏𝑑pr𝑀×𝑑pd𝑑̃𝑣rel.(20)

For direct collisions, ̃𝑀Δ(𝑏,pr,𝑀pd,̃𝑣rel) is calculated given the prescription in Section 2.3. To evaluate the multidimensional integral, we make the approximation of evaluating Δpd at ̃𝑣rel=2𝜎. The factor of Δpd thus comes out of the ̃𝑣rel integral, so that the ̃𝑣rel integral can be performed analytically:𝑑̇𝑀𝑑𝑙𝑛̃𝑟galpd32𝜋3/2𝜎̃𝑟3gal𝐾2𝑀max𝑀min𝑀max𝑀min1+̃𝑟pr0×̃𝑏Δpd̃𝑀𝑏,pr,𝑀pd,̃𝑣rel=2𝜎̃𝑟gal×𝑀𝛼pr𝑀𝛼pd𝑑̃𝑀𝑏𝑑pr𝑑𝑀pd.(21) We evaluate the remaining integrals numerically.

Once values for 𝛼, 𝑀min and 𝑀max are specified, (20) and (21) can be integrated to obtain the mass loss rate as a function of Galactic radius. To show how the mass loss rate profiles vary with 𝑀min, 𝑀max, and 𝛼, we plot 𝑑̇𝑀/𝑑𝑙𝑛𝑟gal for direct collisions in Figure 9 and vary these parameters. In the figure, we have evaluated 𝑀min at 0.05, 0.5 and 5𝑀, 𝑀max at 75, 100, 125𝑀 and 𝛼 from 1.00 to 2.5 in equal increments. The parameter 𝑀min increases vertically from the bottom panel to the top; 𝑀max increases horizontally from the left panel to the right, and in each panel 𝛼 increases from the bottom to the top. We have indicated a Salpeter-like mass function (𝛼=2.29, 𝑀min=0.5𝑀 and 𝑀max=125𝑀) with the dashed line. Mass loss is extensive and approximately constant until about 𝑟gal of 102pc and then drops dramatically. This drop reflects that fact that collisions are less frequent at larger radii since star densities and relative velocities drop. The amount of mass lost for any direct collision also decreases with galactic radius since Δ decreases with decreasing relative velocities. Note that the profiles are approximately constant as a function of 𝑀max, so that the choice of 𝑀min determines the extent of the mass loss rate.

174105.fig.009
Figure 9: Mass loss rates as a function of Galactic radius due to direct collisions for various parameters of 𝑀min, 𝑀max, and 𝛼. The parameter 𝑀min varies in each panel from bottom to top, and 𝑀max varies from left to right. The power law slope, 𝛼, varies within each panel from 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 (𝑀min=0.5𝑀, 𝑀max=125𝑀, 𝛼=2.29). The arrows indicate the range in the diffuse X-ray observations (𝑟gal<1.5) which we use to constrain the PDMF (see Section 5).

In Figure 10 we show the contributions to 𝑑̇𝑀/𝑑𝑙𝑛𝑟gal from both direct and indirect collisions for 𝑀min=0.2𝑀, 𝑀max=100𝑀 and 𝛼=1.2. We find that at small radii the mass loss rate is dominated by direct collisions, and at large radii it is dominated by indirect collisions. Mass loss due to indirect collisions is suppressed in the Galactic center, due to the very fast relative stellar velocities. Even though the high velocities (and high densities) in the Galactic center make collisions more frequent, under the impulse approximation, when velocities are very fast, mass loss is minimized.

174105.fig.0010
Figure 10: Mass loss rates due to direct and indirect stellar collisions within the Galactic center for 𝑀min=0.2𝑀, 𝑀max=100𝑀 and 𝛼=1.2. The arrow indicates the range in the diffuse X-ray observations (𝑟gal<1.5) which we use to constrain the PDMF (see Section 5).

To illustrate which mass stars contribute the most to the total mass loss rate, we plot 𝑑̇𝑀/𝑑𝑙𝑛𝑀pd as a function of 𝑀pd in Figure 11 for several different PDMFs. The range of integration we choose for 𝑟gal is from 0 to 0.06 pc (see Section 5). We choose 𝑀min to be 0.05, 0.5 and 5𝑀 (in Figures 11(a)11(c)), and we use a constant 𝑀max of 125𝑀. In each panel, we vary 𝛼 from 1.5 to 2.5 in equal increments. The figure shows that for 𝑀min=0.05𝑀, changing 𝛼 has little effect on what mass stars contribute the most to the mass loss rate (although, the total mass loss rate is decreased with increasing 𝛼). For the 𝑀min=0.5𝑀 and 5𝑀 cases, increasing 𝛼 results in lower mass stars contributing more to the mass loss rate. This trend makes sense, since PDMF profiles with higher values of 𝛼 have fractionally more lower mass stars.

fig11
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 𝑀min used for that panel. In each panel, 𝛼 goes from 1.5 to 2.5 in even increments of 0.167, and for each line 𝑀max=125𝑀.

To test how our interpolation between the 𝑛=1.5 and 3 polytropic indices affects the main results of this paper, we consider two extreme cases. The first case we consider has 𝑛=1.5 for 𝑀<1𝑀 and 𝑀>5𝑀, and 𝑛=3 for 1𝑀𝑀5𝑀. This approach has 𝑛=1.5 for much of the mass spectrum and should result in the highest mass loss rates since (as is evident from Figure 1) collisions with the perturbed star having 𝑛=1.5 result in the most mass lost. This is due to the fact that for 𝑛=1.5 stars, the mass is less centrally concentrated, and more mass can therefore escape at large radii which receive a stronger velocity kick. The second case we consider has 𝑛=1.5 for 𝑀<0.3𝑀 and 𝑀>10𝑀, and 𝑛=3 for 0.3𝑀𝑀10𝑀. This case should result in the smallest mass loss rates, since it has 𝑛=1.5 for a smaller fraction of the mass spectrum. Since different mass functions have different fractions of the total mass in the neighborhood of 1𝑀 (where we expect the least mass loss per collision since 𝑛=3), we test the two cases for several different mass functions. We find that differences in 𝑑̇𝑀/𝑑𝑙𝑛𝑟gal(𝑟gal) for both cases are relatively minor, differing at most by ~10% depending on the mass function that we use.

5. Constraining the Mass Function in the Galactic Center

It is known through diffuse X-ray observations from Chandra, that the central supermassive black hole in the Galactic center is surrounded by gas donated from stellar winds (e.g., [22]). The diffuse X-ray luminosity is due to Bremsstrahlung emission from unbound material supplied at a rate of 103𝑀yr1 [26]. This unbound material has been studied theoretically by Quataert [27], who solved the equations of hydrodynamics (under spherical symmetry) to follow how the gas is accreted onto Sgr A*. Quataert [27] finds that his model agrees with the level of diffuse X-ray emission measured by Chandra and predicts an inflow of mass at 𝑟gal1 at a rate of 105𝑀yr1.

Using the 2–10 keV luminosity as measured by Chandra [22], we estimate the total mass loss rate at a radius of 𝑟gal1.5 (0.06 pc). We use the word “total’’ to indicate the mass loss rate integrated over Galactic radius. By using this total mass loss rate as an upper limit, we will be able to constrain 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 profiles (e.g., Figures 9 and 10) over 𝑟gal.

Unbound material at a radius 𝑟gal has a dynamical timescale of𝑡dyn𝑟gal𝑟gal𝑣char𝑟gal1.1×104𝑟yrsgalpc1.5,(22) where the characteristic velocity at radius 𝑟gal, 𝑣char(𝑟gal), is taken as the velocity dispersion as given in Section 3. The electron density at radius 𝑟gal may therefore be estimated by𝑛𝑒𝑟gal𝑛𝑝𝑟gal̇𝑀𝑡dyn𝑟gal(4/3)𝜋𝑟3gal𝑚𝑝=1.1×105cm3̇𝑀𝑀yr1𝑟galpc1.5,(23) where 𝑚𝑝 is the proton mass.

For thermal Bremsstrahlung emission, the volume emissivity (𝑑𝐸/𝑑𝑉𝑑𝑡𝑑𝜈) is [41]𝜖𝜈𝑓𝑓=6.8×1038ergs1cm3Hz1𝑛𝑒cm32𝑇𝐾1/2×𝑒𝜈/𝑘𝐵𝑇𝑔𝑓𝑓,(24) where we set 𝑔𝑓𝑓=1. The luminosity in the 2–10 keV band, 𝐿210, can be found substituting (23) into (24) and integrating the volume emissivity over volume (assuming spherical symmetry) and frequency:𝐿2106.7×1043ergs1̇𝑀𝑀yr12𝑟0.06min𝑟pc1𝑟×𝑑pc210𝑒𝜈/keV𝑑𝜈.keV(25) We have assumed a constant temperature of 1 keV. A constant value of 1 keV should suffice for an order of magnitude estimate as Baganoff et al. [22] find that the gas temperature varies from approximately 1.9 to 1.3 keV from 𝑟gal= 1.5 to 10 (assuming an optically thin plasma model). Quataert’s [27] model also predicts that the temperature varies from about 2.5 to 1 keV from 𝑟gal= 0.3 to 10. By plugging the value of 𝐿210 within 1.5 (2.4×1033 erg s−1) as measured by Baganoff et al. [22] into (25), we find ̇𝑀105𝑀yr1. This value is consistent with the mass inflow rate at ~1′′ calculated by Quataert [27]. For clarification, we again note that even though our value agrees with Quataert [27], the underlying physical processes associated with both models are quite different. The model of Quataert [27] takes the source of unbound material to be due to mass ejected by stellar winds, whereas our model uses mass ejected from stellar encounters.

Our results are not sensitive to the choice of the lower limit in the integral across 𝑟gal. The lower limit should be at most a few hundred of pcs to at least ~10−6 pc. The former value is the tidal radius for the SMBH at the Galactic center for a 1𝑀 star. Unbound material due to stellar collisions or from stellar wind should not exist at smaller radii since there are very few stars there to produce it. The value of the integral thus ranges from about unity to a few tens. Since ̇𝑀 depends upon the square root of this value, the exact value of 𝑟min only affects our calculation at the level of a factor of a few, and we thus take the square root of the integral to be unity.

Having established that ̇𝑀105𝑀yr1 in the vicinity of 1.5, we now calculate the expected mass loss rates due to stellar collisions for different PDMFs. The value of ̇𝑀 that contributes to the 2–10 keV flux is given bẏ𝑀=0.06pc0𝑑̇𝑀𝑑3𝑟gal𝜁𝑟ral𝑑3𝑟gal,(26) where we have shown how to calculate the mass loss rate profiles, 𝑑̇𝑀/𝑑3𝑟gal in the previous section. We account for the fact that not all of the emission from the unbound gas contributes to the 2–10 keV band with 𝜁(𝑟gal), defined as the fraction of flux from gas at radius 𝑟gal with 2keV𝜈10keV:𝜁𝑟gal10keVresu2keV𝜖𝜈𝑓𝑓𝑑𝜈0𝜖𝜈𝑓𝑓𝑑𝜈=𝑒2keV/𝑘𝐵𝑇(𝑟gal)𝑒10keV/𝑘𝐵𝑇(𝑟gal).(27) Since the gas at each radius is at a slightly different temperature, and since 𝜁 is exponentially sensitive to the temperature, we must estimate 𝑇(𝑟gal). We do this by setting the thermal energy of the unbound material equal to the kinetic energy at a radius 𝑟gal, and find that𝑘𝐵𝑇𝑟gal𝑚𝑝𝜎2𝑟gal=7.8×102𝑟keVgalpc1.(28) We plot (27) in Figure 12. The value of 𝜁 goes to zero at the highest and smallest radii since, for the former, the gas is cool and emits most of its radiation redward of 2 keV, and for the latter, the gas is hot and emits mostly blueward of 10 keV. Thus, even though the integral in (26) extends to 𝑟gal=0, the contribution to ̇𝑀 is suppressed exponentially at the smallest radii.

174105.fig.0012
Figure 12: The fraction of flux emitted from unbound material at radius 𝑟gal that contributes to the 2–10 keV band.

Since, by (20) and (21), ̇𝑀 depends on the parameters of the PDMF, we now constrain these parameters by limiting the allowed mass loss rate from stellar collisions calculated via (26) at 105𝑀yr1. We consider changes in 𝑀min and 𝛼, and keep 𝑀max set at 125𝑀 since (as seen in Figure 9) ̇𝑀 is approximately independent of 𝑀max.

We sample the 𝑀min-𝛼 parameter space and use (26) to compute the total mass loss rate, the results of which are shown in Figure 13. The contours represent the calculated ̇𝑀 values, where the solid contours are on a logarithmic scale, and where they are limited from above at a value of ̇𝑀=105𝑀yr1. The figure shows that PDMFs with flat to canonical-like profiles are allowed. Very top-heavy profiles (𝛼1.25) are not allowed, as they predict too high of a mass loss rate. Mass functions with 𝑀min7𝑀 are also not allowed. These results are consistent with measurements of the Arches star cluster, a young cluster located about 25 pc from the Galactic center. Recent measurements [4244] probing stellar masses down to about 1𝑀 show that the cluster has a flat PDMF, with 𝛼 in the range of about 1.2 to 1.9 (depending on the location within the cluster).

174105.fig.0013
Figure 13: The total mass loss rate contributing the 2–10 keV flux (calculated from (26)) as a function of 𝑀min and 𝛼. The solid contours are on a logarithmic scale and are limited from above at 105𝑀yr1. The line contours are on a linear scale and are separated by intervals of 1.5×106𝑀yr1. The thick line denotes the 1×105𝑀yr1 contour. The dashed line is a 3rd-order polynomial fit which represents the absolute allowed upper limit of 𝑀min as a function of 𝛼. The coefficients of this polynomial are 21.71, −42.37, 25.33, and −4.27 for 𝑎0 to 𝑎3, respectively.

Since ̇𝑀 is a much stronger function of 𝛼 than of 𝑀min it is difficult for us to place tight constraints on the allowed range of 𝑀min. Figure 13 shows that we can, however, place a constraint on the allowed upper limit of 𝑀min, since very high values of 𝑀min result in mass loss rates >105𝑀yr1. For 𝛼>1.25, we fit a 3rd degree polynomial (the dashed line in Figure 13) to the ̇𝑀=105𝑀yr1 contour. This fit analytically expresses the upper limit of 𝑀min as a function of 𝛼. We provide the coefficients of this fit in the caption of Figure 13.

The small difference between the solid and dashed lines at 𝑟gal=0.06 pc in Figure 8(b) suggests that, even for stellar encounters involving small impact parameters, our integration does not miss many collisions by ignoring gravitational focusing. To estimate the contribution to the total mass loss rate in Figure 13 from gravitational focusing, we take 𝛿𝑀typ, the typical amount of mass lost per collision, to be simply a function of 𝑏. This avoids the multi-dimensional integrations involved in (20) and (21), since for these equations Δpd is a function of 𝑏, 𝑀pr, 𝑀pd, and 𝑣rel(𝑟gal). For simplicity, we choose 𝛿𝑀typ(𝑏) to decrease linearly from 2𝑀 (we assume that both stars are completely destroyed) at 𝑏=0 to 0 at 𝑏=𝑏0. We find 𝑏0 by noting from Figure 1 that for all values of the polytropic index, the amount of mass loss for an indirect collision goes to zero at around 𝛾=0.98. By recalling the definition of 𝛾 (6), we solve for 𝑏0 at 𝛾=0.98 by setting 𝑀pr=1, and taking 𝑣rel2𝜎(𝑟gal=0.06pc). By calculating 𝑑Γ/𝑑𝑏(<𝑟gal) (for Salpeter values) evaluated at 0.06 pc across a range of 𝑏, and multiplying by 𝛿𝑀typ(𝑏), we are able to estimate 𝑑̇𝑀/𝑑𝑏. We do this for 𝑑Γ/𝑑𝑏(<𝑟gal) with and without gravitational focusing and integrate across 𝑏. Subtracting the two numbers results in our estimate of the contribution to the total mass loss rate due to gravitational focusing: 2.3×107𝑀. This is about twice the mass loss rate from Figure 13 evaluated at Salpeter values. We perform the same calculation across the 𝑀min-𝛼 parameter space and find that gravitational focusing contributes a factor of at most ~2.5 to the total mass loss rate.

An underestimate of a factor of 2.5 slightly affects the region of parameter space that we are able to rule out, as shown by the line contours in Figure 13. The 4×106𝑀yr1 contour (2.5 times less than the 105 contour) shows that the region of the parameter space that is ruled out is 𝑀min1.4𝑀 and 𝛼1.4.

6. Implications for the IMF

We now place constraints on the IMF in the Galactic center with a simple analytical approach that connects the IMF to the PDMF, and with the results of the previous section. The mass function as a function of time is described by a partial differential equation that takes into account the birth rate and death rate of stars:𝜕𝜉(𝑀,𝑡)𝜕𝑡=𝑅𝐵(1𝑡)Φ(𝑀)𝜉𝜏,(𝑀)(29) where 𝑅𝐵(𝑡) is the birth rate density of stars (𝑑𝑁𝐵/(𝑑𝑡𝑑3𝑟gal)), Φ(𝑀) is the initial mass function normalized such that Φ(𝑀)𝑑𝑀=1, and 𝜏(𝑀) is the main sequence lifetime of stars as a function of stellar mass. For the initial mass function, we take a power law,Φ=𝑀𝛾,(30) and for 𝜏(𝑀) we use the expression given by Mo et al. [45]𝜏(𝑀)=2.5×103+6.7×102𝑀2.5+𝑀4.53.3×102𝑀1.5+3.5×101𝑀4.5𝑀yr,(31) valid for 0.08𝑀<𝑀<100𝑀 and for solar-type metallicity.

In the following paragraphs, we consider different star formation history scenarios. For each scenario, we will need to know 𝑅𝐵(𝜏𝑀𝑊), the star formation rate density in the Galactic center at the age of the Milky Way (which we take to be 13 Gyr). A rough estimate of this value is given by the number density of young stars in the Galactic center divided by their age: 𝑅𝐵(𝜏𝑀𝑊)𝜌(𝑟)𝜂/(𝜏𝑀). Here 𝜏 and 𝑀 are the average age and average mass of the young stars in the Galactic center, which we take to be 10𝑀yr and ~10𝑀, respectively. The parameter 𝜂 is the fraction of stars with masses above 10𝑀, which for reasonable mass functions is ~0.1%. For self-consistency, we use 𝜌 evaluated at 0.06 pc (which from (10) is ~107𝑀pc3), since this was the radius at which we used to constrain the present-day mass function. These values result in 𝑅𝐵(𝜏𝑀𝑊)104 pc−3 yr−1.

For the simple case of a constant star formation rate, 𝑅𝐵(𝑡)=𝑅𝐵(𝜏𝑀𝑊), and the solution to (29) with the boundary condition that 𝜉(𝑀,𝑡=0)=Φ(𝑀)𝑛tot(𝑡=0), evaluated at the current age of the Milky Way is𝜉𝑀,𝑡=𝜏𝑀𝑊=Φ(𝑀)𝑒𝜏𝑀𝑊/𝜏(𝑀)×𝑅𝐵𝜏𝑀𝑊𝜏(𝑀)𝑒𝜏𝑀𝑊/𝜏(𝑀)𝑅𝐵𝜏𝑀𝑊𝜏(𝑀)+𝑛tot.(0)(32) We evaluate the solution at the age of the Milky Way (yielding the PDMF) because we want to compare with our constraints on the PDMF as found in the previous section. To solve for 𝑛tot(0), we use the known mean density of the Galactic center today at 0.06 pc, 𝜌(𝜏𝑀𝑊,𝑟=0.06pc), insert (32) into the following expression:𝜌𝜏𝑀𝑊=𝜉,𝑟=0.06pc𝑀,𝑡=𝜏𝑀𝑊𝑀𝑑𝑀,(33) and solve for 𝑛tot(0).

We solve for 𝜉(𝑀,𝜏𝑀𝑊) for a range of different IMF power-law slopes, 𝛾 and fit a power law to the solution, with a power-law slope 𝛼. We plot the IMF power-law slope as a function of the calculated PDMF power-law slope for constant star formation in Figure 14(a). We have constrained the PDMF in the previous section to have 𝛼1.25, indicated in the figure by the vertical line. The figure therefore shows that for the case of constant star formation, the IMF power-law slope, 𝛾, must be 0.9.

fig14
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 𝜏exp= 3, 5, 7, 9 Gyr (bottom to top line, resp.).

For the general case of a star formation rate that varies with time, 𝑅𝐵(𝑡)𝑅𝐵(𝜏𝑀𝑊), and the solution to (29) with the same boundary condition and evaluated at 𝜏𝑀𝑊 is:𝜉𝑀,𝑡=𝜏𝑀𝑊=Φ(𝑀)𝑒𝜏𝑀𝑊/𝜏(𝑀)×𝜏𝑀𝑊0𝑅𝐵𝑡𝑒𝑡/𝜏(𝑀)𝑑𝑡+𝑛tot.(0)(34) For an exponentially decreasing star formation history, the star formation rate is given by𝑅𝐵(𝑡)=𝑅𝐵𝜏𝑀𝑊𝑒(𝑡𝜏𝑀𝑊)/𝜏exp.(35) Given this star formation history, we solve for 𝜉(𝑀,𝜏𝑀𝑊) (by solving for 𝑛tot(0) with (33)) for 𝜏exp=3, 5, 7 and 9 Gyr. We fit power-laws to the resulting PDMFs, and show the results in Figure 14(b). The figure shows that smaller values of 𝜏exp result in larger values of 𝛼 for any given 𝛾. The trend can be explained by the fact that since a smaller value of 𝜏exp results in a steeper 𝑅𝐵 profile, and that all profiles must converge to 𝑅𝐵(𝜏𝑀𝑊) at the present-time, 𝑅𝐵 profiles with smaller values of 𝜏exp have had overall more star formation in the past. More overall star formation means that the present-day mass function is comprised of fractionally more lower-mass stars since the IMF favors lower-mass stars. The constant buildup of lower-mass stars results in a steeper PDMF, so that for any given 𝛾, 𝛼 should be larger. The figure shows that for exponentially decreasing star formation 𝛾 must be 0.6, 0.8, 0.9, and 1.0 for 𝜏exp= 3, 5, 7, and 9 Gyr, respectively.

The final case we consider is episodic star formation, where each episode lasts for a duration Δ𝑡, where the ending and beginning of each episode is separated by a time, 𝑇, and where the magnitude of each episode is 𝑅𝐵(𝜏𝑀𝑊). For such a star formation history, the solution to (34) is𝜉𝑀,𝑡=𝜏𝑀𝑊=Φ(𝑀)𝑒𝜏𝑀𝑊/𝜏(𝑀)×𝑅𝐵𝜏𝑀𝑊𝜏(𝑀)𝑛max𝑛=0𝑒[(𝑛+1)Δ𝑡+𝑛𝑇]/𝜏(𝑀)𝑒𝑛(Δ𝑡+𝑇)/𝜏(𝑀)+𝑛tot,(0)(36) where 𝑛max=oor{(𝜏M𝑊Δ𝑡)/(𝑇+Δ𝑡)} and where we again solve for 𝑛tot(0) with (33). We consider 9 cases with Δ𝑡 and 𝑇=106, 107, and 108yrs and show the results in Figure 15. In each panel the lowest line is Δ𝑡=108yrs and the highest line is Δ𝑡=106yrs. For 𝑇=106yrs, 𝛾 0.8 and 0.5 for Δ𝑡=106 and 107yrs, respectively, while the Δ𝑡=108yrs case results in constraints on 𝛾 that are too low to be realistic. For 𝑇=107yrs, 𝛾 0.5 and 0.4 for Δ𝑡=106 and 107yrs, respectively, while again, the Δ𝑡=108yrs case results in unrealistic constraints. Finally, for the 𝑇=108yrs, 𝛾 0.5 for Δ𝑡=106, while the Δ𝑡=107 and 108yrs case result in unrealistic constraints. We test if when the last star formation episode occurs (relative to the present day) it affects our solution of 𝜉(𝑀,𝜏𝑀𝑊) by varying the start time of the star formation episodes. By varying the start time and testing all the combinations of Δ𝑡 and 𝑇 that we consider, we find that the lines in Figure 15 vary by at most about 5%, so that the main trends in the figure are unaffected.

fig15
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 lowest line is Δ𝑡=108yrs and the highest line is Δ𝑡=106yrs.

7. Contribution from Red Giants

Spectroscopic observations have revealed that the central parsec of the Galaxy harbors a significant population of giant stars [18, 19]. Due to their large radii (and hence large cross sections), it is possible that they could play an important part in the mass loss rate due to collisions in the Galactic center.

In assessing their contribution to the mass loss rate, care must be taken when deriving the collision rates, because their radii, 𝑟RG, are strong functions of time, 𝑡. Dale et al. [14] have already calculated the probability, 𝑃(𝑟gal), for a red giant (RG) in the Galactic center to undergo collisions with main sequence impactors. They have taken into account that 𝑟RG(𝑡) by integrating the collision probability over the time that the star resides on the RG branch. We use their results to estimate the mass loss rate due to RG-MS star collisions.

To find the number density of RGs in the Galactic center, we weight the total stellar density by the fraction of time the star spends on the RG branch:𝑛RG𝑟gal𝑛𝑟gal𝜏RG𝜏.(37) This approximation should be valid given a star formation history that is approximately constant when averaged over time periods of order 𝜏RG. The number of collisions per unit time suffered by any one red giant, ̇𝑃(𝑟gal), should be of order the collision rate averaged over the lifetime of the RG and is given bẏ𝑃𝑟gal̇𝑃𝑟gal𝑡=𝑃𝑟gal𝜏RG.(38) If we define 𝛿𝑀 to be the typical amount of mass lost in the collision, then the mass loss rate is𝑑̇𝑀𝑑𝑙𝑛𝑟gal=4𝜋𝑟3gal𝑑̇𝑀𝑑3𝑟gal4𝜋𝑟3gal𝑛RG𝑟gal𝑃𝑟gal𝜏RG𝛿𝑀.(39)

To calculate an upper limit for the contribution of RG-MS star collisions to the mass loss rate, we assume that all RG and MS stars have masses of 1𝑀 and that the entire RG is destroyed in the collision. Collisions involving 1𝑀 RGs yield an upper limit, because there is not an appreciable amount of RGs with masses less than 1𝑀 due to their MS lifetimes being greater than the age of the Galaxy. For RGs with masses greater than 1𝑀, the amount they contribute to the mass loss rate is a competition between their lifetimes and radii. Red giant lifetimes decrease with mass (thereby decreasing the time they have to collide) and their radii increase 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 that the brevity of their lifetime wins over their large sizes. One solar mass MS impactors should yield approximately an upper limit to the mass loss rate, since 1𝑀 MS stars are the most common for the PDMFs under consideration.

Since we assume that the entire RG is destroyed in the collision 𝛿𝑀=1𝑀. For the case that all impactors are 1𝑀 MS stars, we calculate 𝑛RG(𝑟gal) from (37) by noting that 𝑛(𝑟gal)=𝜌(𝑟gal)/(1𝑀). For self-consistency, we must truncate 𝑃(𝑟gal) at 1 for all 𝑃(𝑟gal)>1 since we are considering the case where one collision destroys the entire star. We plot (39) for this calculation in Figure 16. The discontinuity is due to our truncating 𝑃(𝑟gal) at 1. The figure shows that the mass loss rate for RG-MS star collisions never exceeds 105𝑀yr1, well below typical 𝑑̇𝑀/𝑑𝑙𝑛𝑟gal for values for MS-MS collisions (see Figures 9 and 10). Moreover, in their hydrodynamic simulations, Dale et al. [14] note that in a typical RG-MS star collision, at most ~10% of the RG envelope is lost to the RG. We therefore conclude that the contribution of RGs to the total mass loss rate in the central parsec of the Galaxy is negligible.

174105.fig.0016
Figure 16: An upper limit to the mass loss rate due to collisions between RG and MS stars. The arrow indicates the range in the diffuse X-ray observations (𝑟gal<1.5) which we use to constrain the PDMF (see Section 5).

The figure shows that by 𝑟gal=0.06 pc, the mass loss rate for RG-MS star collisions is at most about 106𝑀yr1. It is thus possible that for MS-MS collisions, values of 𝑀min and 𝛼 that result in total mass loss rates just below 105𝑀yr1 could be pushed past this threshold with the addition of mass 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 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 𝑑̇𝑀/𝑑𝑙𝑛𝑟gal for RG-MS collisions by another factor of 10.

8. Conclusions

We have have derived novel, analytical methods for calculating the amount of mass loss from indirect and direct stellar collisions in the Galactic center. Our methods compares very well to hydrodynamic simulations and do not require costly amounts of computation time. We have also computed the total mass loss rate in the Galactic center due to stellar collisions. Mass loss from direct collisions dominates at Galactic radii below ~0.1 pc, and thereafter 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 depends upon the PDMF. We find that the calculated mass loss rate is sensitive to the PDMF used and can therefore be used to constrain the PDMF in the Galactic center. As summarized by Figure 13, our calculations rule out 𝛼1.25 and 𝑀min7𝑀 in the 𝑀min-𝛼 parameter space. Finally, we have used our constraints on the PDMF in the Galactic center to constrain the IMF to have a power-law slope 0.4 to 0.9 depending on the star formation history of the Galactic center.

Acknowledgments

This work was supported in part by the National Science Foundation Graduate Research Fellowship, NSF Grant AST-0907890, and NASA Grants NNX08AL43G and NNA09DB30A.

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