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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

The dense stellar core at the Galactic center has a radius of ^{−3} [^{−1}), and Sgr A*, the central supermassive black hole with a mass

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 [

First described by Salpeter more than 50 years ago [

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 [

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 [

We present novel, analytical models to calculate the amount of stellar mass lost due to stellar collisions between main sequence stars in Section

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 (pd

We consider the condition for mass loss at a position,

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 [

Given a mass distribution for the perturbed system,

To solve for the mass lost per encounter as a function of ^{6} cubic elements. As a function of ^{5} elements, the results converge to within about 2%, and we are therefore confident that ^{6} 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 [

We plot the fraction of mass lost from the perturbed star per event,

Coefficients of polynomial fits for

1.5 | 0.395 | −0.865 | 0.559 | −0.091 |

2.0 | 0.210 | −0.424 | 0.246 | −0.032 |

2.5 | 0.105 | −0.197 | 0.102 | −0.101 |

3.0 | 0.051 | −0.088 | 0.040 | −0.003 |

The fraction of mass lost per collision as a function of

The location of the mass loss within the perturbed star for fixed

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

The impulse approximation is valid provided that the time over which the encounter takes place,

Aguilar and White [

Contours of log(

In our calculations, when, for any particular set of

Equation (

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., [

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

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 [

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 [

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 [

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 [

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 [

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. [

We use (

The stellar density profile that we adopt, based on the average density profile of Schödel et al. [

The differential collision rate,

A change of variables allows one to integrate out 3 of the velocity dimensions and to write the expression in terms of

Our calculations involve the computation of multidimensional integrals over a two-dimensional parameter space (see Section

To illustrate the frequency of collisions in the Galactic center, we integrate (

(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 (

To calculate the mass loss rate from stars due to collisions within the Galactic center, we multiply (

We first compute the differential mass loss rate for indirect collisions. The mass loss per collision is given by

For direct collisions,

Once values for

Mass loss rates as a function of Galactic radius due to direct collisions for various parameters of

In Figure

Mass loss rates due to direct and indirect stellar collisions within the Galactic center for

To illustrate which mass stars contribute the most to the total mass loss rate, we plot

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

To test how our interpolation between the

It is known through diffuse X-ray observations from

Using the 2–10 keV luminosity as measured by

Unbound material at a radius

For thermal Bremsstrahlung emission, the volume emissivity (^{−1}) as measured by Baganoff et al. [

Our results are not sensitive to the choice of the lower limit in the integral across ^{−6} pc. The former value is the tidal radius for the SMBH at the Galactic center for a

Having established that

The fraction of flux emitted from unbound material at radius

Since, by (

We sample the

The total mass loss rate contributing the 2–10 keV flux (calculated from (

Since

The small difference between the solid and dashed lines at

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

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:

In the following paragraphs, we consider different star formation history scenarios. For each scenario, we will need to know ^{7}^{−3} yr^{−1}.

For the simple case of a constant star formation rate,

We solve for

(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

For the general case of a star formation rate that varies with time,

The final case we consider is episodic star formation, where each episode lasts for a duration

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

Spectroscopic observations have revealed that the central parsec of the Galaxy harbors a significant population of giant stars [

In assessing their contribution to the mass loss rate, care must be taken when deriving the collision rates, because their radii,

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:

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

Since we assume that the entire RG is destroyed in the collision

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 (

The figure shows that by

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

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