THE ASTROPHYSICAL JOURNAL, 487 :290È303, 1997 September 201997. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

EVOLUTION OF STELLAR COLLISION PRODUCTS IN GLOBULAR CLUSTERS.I. HEAD-ON COLLISIONS

ALISON SILLS

Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520-8101

JAMES C. LOMBARDI, JR.1Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853

CHARLES D. AND PIERREBAILYN DEMARQUE

Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520-8101

FREDERIC A. RASIO

Department of Physics, Massachusetts Institute of Technology 6-201, Cambridge, MA 02139

AND

STUART L. SHAPIRO2Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801

Received 1996 December 19 ; accepted 1997 April 24

ABSTRACTWe explore the evolution of collisionally merged stars in the blue straggler region of the H-R diagram.

The starting models for our stellar evolution calculations are the results of the smoothed particle hydro-dynamics (SPH) simulations of parabolic collisions between main-sequence stars performed by Lom-bardi, Rasio, & Shapiro. Since SPH and stellar evolution codes employ di†erent and often contradictoryapproximations, it is necessary to treat the evolution of these products carefully. The mixture and dis-parity of the relevant timescales (hydrodynamic, thermal relaxation, and nuclear burning) and of theimportant physical assumptions between the codes makes the combined analysis of the problem chal-lenging, especially during the initial thermal relaxation of the star. In particular, the treatment of convec-tion is important and semiconvection must be modeled in some detail.

The products of seven head-on collisions are evolved through their initial thermal relaxation and thenthrough the main-sequence phase to the base of the giant branch. Their evolutionary tracks are present-ed. In contrast to what was assumed in previous work, these collision products do not develop substan-tial convective regions during their thermal relaxation and therefore are not mixed signiÐcantly after thecollision.Subject headings : blue stragglers È globular clusters : general È hydrodynamics È stars : evolution È

stars : kinematics

1. INTRODUCTION

Blue stragglers are main-sequence stars that are moremassive than the main-sequence turno† stars in clusters.Since massive stars evolve more quickly than low-massstars, blue stragglers must have been born (or reborn) on themain sequence more recently than the majority of the starsin the cluster. Numerous formation mechanisms have beenproposed (for a recent review, see The currentStryker 1993).observations suggest that stellar collisions could be a viablemechanism for some blue stragglers (Bailyn 1995).

In most environments, collisions between stars are veryrare events. However, the cores of some globular clustersare dense enough that collisions between single stars areexpected to happen at dynamically signiÐcant rates &(HillsDay Also, binary star systems have a much larger1976).collisional cross section than single stars and can enhancethe number of stellar collisions (Leonard 1989 ; Sigurdsson& Phinney 1993).

It is important to include the e†ects of physical collisionsbetween stars when studying the dynamics and overall evol-ution of globular clusters. These collisions produce starsthat are more massive than most of the stars in clusters. The

1 Also at the Department of Astronomy, Cornell University.2 Also at the Department of Astronomy and National Center for Super-

computing Applications, University of Illinois at Urbana-Champaign.

relative proportions of stars of di†erent masses a†ect therate at which various dynamical processes occur, such ascore collapse and mass segregation Hut, & Inagaki(Elson,

The exchange of energy between the cluster kinetic1987).energy and the collisional energy can also a†ect dynamicalprocesses. Therefore, it is important to determine how manymassive stars are created by collisions. Stellar collisions arealso important tracers of the dynamical state of globularclusters. The collision rates and participants are determinedby global cluster properties such as density, velocity disper-sion, mass function, and mass segregation. Therefore, bystudying the products of stellar collisions, we will be able toprobe the dynamics, both past and present, of globular clus-ters.

To study the consequences of stellar collisions in globularclusters fully, many di†erent lines of investigation need tobe combined. We need to know how many collisionshappen in a particular cluster, as well as what kinds ofcollisions occur, between which kinds of stars, and withwhat frequency. These questions can be answered withmultimass King-Mitchie models (see, e.g., & BenzDavies

that give the density and velocity dispersion of the1995)clusters. Two- and three-body scattering experiments, e.g.,those of & Hut and Benz, & HillsMcMillan (1996) Davies,

can determine the probabilities that di†erent kinds(1994),of collision products will be produced.

290

STELLAR COLLISION PRODUCTS. I. 291

In addition to determining the global collision propertiesof the cluster, we need to consider individual collisions.Smoothed particle hydrodynamic (SPH) calculations ofstellar collisions relevant to stars in globular clusters (see,e.g., & Hills Lombardi, Rasio, & ShaprioBenz 1987 ; 1995,

[hereafter Bolte, & Hernquist1996 LRS] ; Sandquist, 1997)can provide a detailed description of the structure and com-position of the collision product. Stellar evolution calcu-lations can then follow the evolution of the collisionproduct to determine its track in a color-magnitudediagram, whether or not it has surface abundance anom-alies, and other observable properties (e.g., rotation rate orpulsation period).

Previous evolution studies & Pinsonneault(Bailyn 1995 ;Bailyn, & Demarque & PritchetSills, 1995 ; Ouellette 1996 ;

et al. used only a portion of the SPH resultsSandquist 1997)to model collision products. These groups imposed thechemical proÐle, but not the structure, of a collision producton an otherwise normal stellar model and compared theevolution of these stars to that of fully mixed models. Theresults of these studies are inconsistent. & Pinson-Bailynneault and et al. demonstrated that the(1995) Sills (1995)blue stragglers in 47 Tucanae (NGC 104) and the brightestblue stragglers in NGC 6397 could be explained only byfully mixed models. However, the SPH simulations showthat the collision product is not thoroughly mixed by thecollision. Therefore, the blue stragglers would have to bemixed by some process that occurs after the collision, notduring it.

& Livio proposed such a mechanism.Leonard (1995)They suggested that the collision products will contain alarge amount of thermal energy deposited by the collisionand consequently will swell up into something resembling apreÈmain-sequence star with a large convective envelope.Under this assumption, blue stragglers will be largely orfully mixed owing to convection when they arrive on themain sequence despite the fact that the collision itself doesnot result in a signiÐcant amount of mixing. This scenario isbased on the speculation that stellar collision productscontain enough thermal energy to become preÈmain-sequence stars. It was not clear whether this situation wouldactually occur or not. et al. modeled theSandquist (1997)Leonard & Livio scenario by adding enough energy to theirinitial stellar models that the stars were on the Hayashitrack and then began their evolutionary runs. As a result,their stars developed large surface convective zones as wellas small, short-lived convective cores. A signiÐcant fractionof the more massive collision products were mixed as aresults of the convection.

In this paper, we present a set of evolution calculationsbased on more rigorous initial conditions than used in theprevious studies. Instead of imposing the collisional chemi-cal composition proÐle on an otherwise normal stellarmodel, we use all the stellar structure information from theSPH results in the starting models for our evolution calcu-lations. We do not add energy to our models since we donot want to make any assumptions about where in the H-Rdiagram the collision models should reside. In particular,we wish to investigate the apparent discrepancy betweenthe negligible amount of mixing that is caused by the colli-sion itself and the substantial amount of mixing that seemsnecessary to explain at least some blue stragglers in someclusters. We concentrate on the head-on collisions in thispaper. Future work will incorporate the results from nonÈ

head-on collisions, and we will use population synthesistechniques to compare our theoretical predictions withobservations.

In we describe the di†erent tools used in this work. In° 2we discuss the details of converting an SPH collision° 3

product to a valid evolutionary starting model, and in ° 4we present the results of the evolutionary calculations. In ° 5we discuss the implications of these models to the origins ofblue stragglers and detail future work to be done.

2. NUMERICAL METHODS

This section brieÑy describes the SPH and stellar evolu-tion codes used in this study. We highlight the di†erences inunderlying physical assumptions, which must be dealt withcarefully in using the output of the SPH code as input forstellar evolution calculations.

2.1. T he Smoothed Particle Hydrodynamics CodeSPH is a Lagrangian method and therefore ideally suited

to follow chemical mixing during the stellar collision. LRSused a modiÐed version of the code developed by Rasio

speciÐcally for the study of stellar interactions. Fluid(1991)elements are represented by SPH ““ particles, ÏÏ and associ-ated with each particle is its position r, velocity mass m,¿,entropy s, and a purely numerical ““ smoothing length ÏÏ hspecifying the local spatial resolution. Local quantities arecalculated by weighted sums over nearby particles. TheLRS collision simulations solve the equations of motion ofa large number (typically 3] 104) of particles, which areinteracting by both gravitational and hydrodynamic forces,and yield accurate structural and chemical compositionproÐles for the merger remnant.

performed SPH calculations of 23 parabolic colli-LRSsions with various mass combinations and impact param-eters. The parent stars were main-sequence starsappropriate for globular clusters, with chemical composi-tion proÐles based on evolutionary models evolved to 15Gyr. The M \ 0.8 star was modeled as an n \ 3 poly-M

_trope, the M \ 0.4 star as an n \ 1.5 polytrope, and theM_M \ 0.6 star as a composite polytrope with an n \ 3M

_core and an n \ 1.5 envelope. In all cases, the Ñuid wastreated as an ideal gas with adiabatic index The!1 \ 5/3.parent models and Ðnal results are described in detail inLRS.

SPH codes, by deÐnition, are hydrodynamic and can dealwith three-dimensional Ñuids that are far from hydrostaticequilibrium. They follow the system over dynamical time-scales, which are typically on the order of an hour for colli-sions involving main-sequence stars in globular clusters.The adopted SPH code includes the e†ects of shocks but isadiabatic and otherwise neglects all radiative and heattransport between particles, which is not signiÐcant ondynamical timescales. Therefore, convective energy trans-port is not modeled, but the mixing e†ects of convectiveÑuid motions occurring on a dynamical timescale areincluded.

2.2. T he Stellar Evolution CodeWe used the Yale Rotating Evolution Code (YREC) in its

nonrotating mode for our stellar evolution calculations.The code solves the equations of stellar structure in onedimension using the Henyey technique. A detailed descrip-tion of the physics included in the code can be found in

et al. We have since updated the opacitiesGuenther (1992).

292 SILLS ET AL. Vol. 487

to include OPAL opacities for log T º 4.00 &(IglesiasRogers and Alexander low-temperature opacities for1996)log T ¹ 4.00 & Ferguson We chose a(Alexander 1994).mixing length of 1.9 pressure scale heights, which is thevalue that gives a standard solar model using this code. Forthe head-on collisions, the assumption of nonrotation isjustiÐed since the initial conditions have zero angularmomentum. However, the products of grazing collisions arerotating rapidly and consequently aspherical. In futurework involving nonÈhead-on collisional remnants, thestructural and mixing e†ects of rotation will be included.

YREC assumes that the star is in hydrostatic equilibriumand uses a one-dimensional treatment where the star ismodeled as a series of concentric shells. One of the mainconcerns is to follow energy transport closely. The time-scales for evolution codes range from thermal (D106 yr) tonuclear (D109 yr) timescales. All of these assumptions aredrastically di†erent from those used by the SPH code.

3. PREPARING THE COLLISION MODELS FOR EVOLUTION

The di†erent assumptions about the important physics,the timescales and geometry treated by SPH, and the stellarevolution codes create difficulties in converting the SPH

results into an acceptable starting model for use in YREC.This section describes our approach for dealing with each ofthese assumptions. Figures and show the structure of1, 2, 3three collision remnants immediately after their conversionto YREC format.

3.1. Spherical SymmetryThe Ðrst step is to turn the three-dimensional particle

data from SPH into one-dimensional radial proÐles. TheSPH particles that remained gravitationally bound to thecollision product were divided into 32È45 spherical shells.The entropy and composition of the SPH particles in eachshell were averaged to give a value for that shell. SinceYREC is more accurate with more shells, the number ofshells was increased to 1500 by interpolating each quantityusing a cubic spline. A visual comparison of the resultingentropy and composition proÐles with the proÐles of LRSshows that no signiÐcant spurious oscillations were intro-duced by this interpolation technique. To ensure smooth-ness in the derivatives of the proÐles, we interpolated o† ofcarefully chosen mass shells not evenly spaced in mass frac-tion ; however, the evolutionary tracks are virtually unaf-fected by the particular choice of mass shells used.

FIG. 1.ÈProÐles of luminosity L , helium abundance Y , temperature T , radius R, pressure P, and density o vs. mass fraction for the collision product ofcase A. This collision is a head-on collision between two 0.8 stars, resulting in a product with a total mass of 1.498 The large increase in luminosityM

_M

_.

in the outer 20% of the star is caused by the released gravitational energy of gas that was ejected from the star by the collision and then fell back onto thecollision product.

No. 1, 1997 STELLAR COLLISION PRODUCTS. I. 293

FIG. 2.ÈSame as for case G. This collision is a head-on collision between a 0.8 star and a 0.4 star, resulting in a product with a total massFig. 1 M_

M_of 1.133 Notice the discontinuity in helium abundance at a mass fraction of 0.2. Interior to this point lies the majority of the smaller parent star, whichM

_.

has lower entropy and so sank to the center of the collision product. The evolved core of the turno†-mass star has been displaced by this low-entropymaterial, so the maximum helium abundance occurs away from the stellar core.

Since YREC solves the equations of stellar evolution inone dimension, all stars are assumed to be spherically sym-metric. This constraint poses two problems. The Ðrst is thatthe head-on collision products are not spherically sym-metric in their chemical composition. We therefore took theaverage of all particles in given shells. It can be shown (see

that this is a plausible approximation. TheAppendix)second problem caused by the requirement of sphericalsymmetry is that, for nonÈhead-on collisions (the generalcase), the collision products are rotating rapidly. However,the products of head-on collisions are not rotating, so wetreat only spherically symmetric models in this paper. Ascan be seen in Figure 8 of this assumption of spher-LRS,ically symmetric structure for head-on collisions is quiterealistic.

3.2. Hydrostatic EquilibriumStarting with the entropy and composition proÐles pro-

vided by the SPH simulations of we calculated theLRS,pressure and density proÐles of each collision product byintegrating the equation of hydrostatic balance and impos-ing the boundary condition P(M) \ 0. This procedureensures that the initial evolutionary models are in hydro-static equilibrium. The pressure and density proÐles were

not taken directly from the SPH results since these proÐlesare not accurate enough to guarantee hydrostatic equi-librium at the level of precision needed for YREC.

Since, at the termination of the SPH simulation, the out-ermost 2%È4% of the gravitationally bound Ñuid is still farfrom dynamical equilibrium, we do not use the SPHentropy proÐle in this region but rather use the cubic splineto extrapolate from the interior so that entropy increaseswith mass fraction throughout the star. We tested a numberof schemes for approximating the proÐles in the outer fewpercent and found that the tracks converged after a fewevolutionary time steps : the stellar interior is vastly moreimportant in determining long-lived global characteristics.

These starting models have therefore been constructed tobe in precise hydrodynamical equilibrium, although theyare typically far from thermal equilibrium. The initialdensity and pressure proÐles input to YREC di†er some-what from the Ðnal SPH proÐles of for two reasons.LRSFirst, averaging over nearest neighbors in the SPH schememakes the central density and pressure smaller than theyshould be for hydrostatic balance (these errors in the SPHscheme vanish in the limit of perfect spatial resolution).Second, the SPH density and pressure depend on when thecollision simulation is terminated, since they continually

294 SILLS ET AL. Vol. 487

FIG. 3.ÈSame as for case J. This collision is a head-on collision between two 0.6 stars, resulting in a product with a total mass of 1.142Fig. 1 M_

M_

.

change as Ñuid falls back to the stellar surface. Note,however, that the entropy proÐle is virtually una†ected bythe hydrodynamics of the outermost region and is thereforeappropriate for determining the proÐles of the collisionproduct in complete hydrostatic equilibrium.

3.3. Equation of StateWe used the equation of state and the previously deter-

mined proÐles of pressure and density to calculate the tem-perature proÐle of the collision product. While SPHassumes a fully ionized ideal gas equation of state, YRECuses a much more sophisticated equation of state thatincludes radiation pressure, electron degeneracy, and thee†ects of partial ionization. For the stars we are consider-ing, the ideal gas equation of state is a very good approx-imation, with the only signiÐcant correction being radiationpressure. We used the YREC implementation of the Sahaequation to determine the number of free electrons pernucleon in each shell for temperatures less thanlog T \ 6.3 and assumed that the gas was fully ionized forhotter regions.

3.4. Energy T ransportSince the SPH code did not include energy transport, it

gives no direct information about luminosity. We used the

equation of radiative transport

dTdm

\ [364n2ac

iL (m)r4T 3 (1)

to give us L (m), the luminosity for each shell. Here T is thetemperature and r is the radius of that shell. The opacity iincludes both radiative and conductive opacities. Equation

is a good approximation at t \ 0 since none of the initial(1)models contain convective regions. The unusual luminosityproÐles obtained reÑect the strongly nonequilibrium dis-tribution of internal energy caused by the collision andconstitute one of the major di†erences from more straight-forward approximations (see Figs. and For example,1, 2, 3).the luminosity in case G is initially negative near m/M B 0.25, which is a direct consequence of dT /dm[ 0 inthis region.

4. EVOLUTION OF THE COLLISION PRODUCTS

Here we discuss the results of evolving the seven head-oncollision products. A summary of the collision pro-LRS

ducts is given in We emphasize the results of threeTable 1.particular collisions that demonstrate important features ofthe evolution calculations : case A (0.8 caseM

_] 0.8 M

_),

G (0.8 and case J (0.6 CaseM_

] 0.4 M_

), M_

] 0.6 M_).

A was chosen to represent the most massive collision

No. 1, 1997 STELLAR COLLISION PRODUCTS. I. 295

TABLE 1

SUMMARY OF COLLISIONS

Local Thermal Local ThermalGlobal Thermal Timescale Timescale

M1 M2 Mtotal Helium Timescale Center m/M \ 0.95Case (M

_) (M

_) (M

_) Fraction (yr) (yr) (yr)

A . . . . . . 0.8 0.8 1.498 0.422 7.0] 105 7.7] 106 2.0] 103D . . . . . . 0.8 0.6 1.320 0.362 1.2] 106 1.1] 109 4.5] 103G . . . . . . 0.8 0.4 1.133 0.363 8.5] 105 1.7] 109 6.8] 103J . . . . . . . 0.6 0.6 1.142 0.284 6.8] 105 2.1] 108 4.9] 103M . . . . . . 0.6 0.4 0.946 0.270 3.3] 105 2.8] 1010 3.2] 104P . . . . . . 0.4 0.4 0.770 0.250 2.2] 106 2.2] 1010 2.3] 104U . . . . . . 0.8 0.16 0.935 0.385 2.5] 104 3.0] 1010 2.0] 104

product. We also chose to discuss two collision products(cases G and J) with similar masses but very di†erent col-lisional histories to demonstrate that collisional history is atleast as important as Ðnal mass in determining the evolu-tionary track of the collision product.

4.1. ProcedureAfter converting each of the SPH models into a starting

model for YREC, the stars were evolved with time steps of1 ] 102 to 5 ] 102 yr (about 10~4 of a thermal timescale)through their initial thermal relaxation phase. This valuewas chosen to ensure that the evolutionary time step wasshorter than the local thermal timescale for most of the star.When the star reached the main sequence, the short timestep constraint was lifted and time steps were chosen auto-matically to maximize the efficiency of the computing. Theevolution was stopped when the star reached the giantbranch.

For comparison, we also evolved normal stars withmasses similar to the collision products. To create the start-ing models for these stars, we calculated fully convectivepolytropes (n \ 1.5) with the same initial composition as thecollision products (Y \ 0.25, Z\ 0.001). We then evolvedthese models from the preÈmain-sequence to the mainsequence and beyond to the giant branch.

To determine the maximum e†ect mixing could have onthe evolution, we also evolved fully mixed collision pro-ducts with helium abundances higher than a normal globu-lar cluster star. The chemical composition of the SPHmodel was averaged over the entire star, keeping the totalhelium mass the same but spreading the hydrogen andhelium evenly throughout the star. These stars were alsoevolved from n \ 1.5 polytropes.

4.2. Convection and SemiconvectionConvection is a hydrodynamical process, since it involves

the motion of Ñuid elements. It is also a thermal process,since convection transports energy. Since the SPH codedoes not model energy transport and the evolution codedoes not model dynamical processes, both codes have totreat convection as an approximation. The convective turn-over timescale for stars like the Sun is typically on the orderof a month. This timescale is between the hour-longdynamical timescale followed by the SPH code and thecentury-long timescale that is the shortest numericallystable time step in YREC. Therefore, convection is the mostimportant process that spans all time steps and both codesand must be treated with particular care.

YREC, in its original form, uses the Schwarzschild cri-terion for stability,

+rad\ +ad , (2)

where is the radiative temperature+rad\ (d ln T /d ln P)radgradient and is the adiabatic temperature gradient,+adwithout regard for the presence of composition gradients.However, a gradient in the mean molecular weight k withinthe star can stabilize the Ñuid against convection, so it maybe necessary to use the more general Ledoux criterion,

+rad \ +L4 +ad] r

d+k, (3)

where is the mean molecular weight+k \ (d ln k)/(d ln P)gradient, and d \ [[(d ln o)/r\ [(d ln o)/(d ln k)]

P,T,are quantities that depend on the equation(d ln T )]

P,kof state. It is possible that a particular shell is stableagainst convection according to the Ledoux criterion butunstable according to the Schwarzschild criterion. Thissituation is called semiconvection. Fluid elements in theseregions are vibrationally unstable and oscillate slowlyabout their initial position with increasing amplitude (see,e.g., & WeigertKippenhahn 1994).

Semiconvective regions mix on a thermal timescale untilthe mean molecular weight gradient becomes small enoughthat the region is stable against semiconvection. Using onlythe Schwarzschild criterion is an acceptable approximationfor most evolution calculations, since the evolution timesteps are longer than the thermal timescale. However, this isnot the case in our postcollision scenario. One of the mostimportant epochs that we must calculate accurately is theinitial readjustment of the star to thermal equilibrium. BydeÐnition, this readjustment occurs on a thermal timescale.We must therefore take time steps that are small comparedto the thermal time. Semiconvection must be modeledexplicitly.

We use the approach of Langer Fricke, & Sugi-(Langer,moto El Eid, & Fricke to add a full1983 ; Langer, 1985)treatment of semiconvection to YREC. First, we determineif each shell is radiative, convective, or semiconvectiveaccording to the Ledoux and Schwarzschild criteria above.The chemical composition in each semiconvective shell isupdated according to the di†usion equation

LXi

Lt\ALX

iLtBnuc

] LLM

r

C(4nr2o)2D LX

iLM

r

D, (4)

where is the abundance of the ith element,Xi

[(LXi)/(Lt)]nucis the change in abundance due to nuclear burning, and D is

296 SILLS ET AL. Vol. 487

the di†usion coefficient. This coefficient is given by

D\ a2acT 39io2c

p

+[ +ad+

L[ +

, (5)

where +\ (d ln T )/(d ln P) is the actual temperature gra-dient in that shell and is the heat capacity as calculatedc

pfrom the equation of state. et al. determinedLanger (1985)the efficiency factor a to be of order 0.1, based on theexpected values of from mixing length theory and+[ +adon an analysis of the Ñuid dynamics by Equat-Unno (1962).ing the eddy viscosity with the di†usion coefficient alsogives a \ 0.1.

The temperature gradient also depends on the convec-tive state of each shell of the star. This gradient, in thesemiconvective region, is determined by solving +\

for +, where+rad/(1] L sc/L rad)

L scL rad\ a

+[ +ad2+(+

L[ +)

C(+[ +ad)[

b(8 [ 3b)+k32 [ 24b [ 3b2

D. (6)

The two luminosities, L sc and L rad, represent the amount ofluminosity carried by the two energy transport mechanisms,semiconvection and radiation, respectively. Here b is theratio of gas pressure to total pressure in that shell of thestar.

It can be shown that the condition for dynamical stabil-ity, ds/dr [ 0, is the same as the Ledoux criterion for sta-bility against convection for any nonrotating Ñuid thatexpands when heated (see, e.g., & Lifshitz If aLandau 1959).star has an entropy gradient that increases outward, it willnot be convective. This stability condition was used by LRSto determine when it was acceptable to stop the SPH simu-lations. Therefore, our collision products are radiative orsemiconvective when they begin their thermal contraction.Our evolution calculations show that they do not develop aconvective envelope until they reach the giant branch.

4.3. ResultsThe evolutionary tracks are plotted in Figures and4 5.

Distinguishable points are marked on each track, and thetimes from the collision to that point are given in Table 2.We describe the details of the evolution of case A in thefollowing paragraphs ; the underlying physics of the othercollisions is similar. Other speciÐc cases will be discussedlater in this section.

TABLE 2

AGE OF STAR (IN YR) AT POSITIONS IN THE H-R DIAGRAM

Zero-Age Terminal-AgeCase Main Sequence Main Sequence Giant Branch

A collision . . . . . . . . . 1.12] 106 4.47] 106 8.85] 107D collision . . . . . . . . . 3.51] 106 3.78] 108 6.19] 108G collision . . . . . . . . . 2.82] 106 1.42] 109 1.99] 109J collision . . . . . . . . . . 1.41] 107 1.80] 109 2.95] 109M collision . . . . . . . . 7.11] 106 5.90] 109 7.00] 109P collision . . . . . . . . . 1.42] 107 1.60] 1010 1.75] 1010U collision . . . . . . . . . 4.31] 106 1.84] 109 3.06] 109A normal . . . . . . . . . . 3.86] 106 a 1.24] 109 1.58] 109G normal . . . . . . . . . . 7.85] 106 a 3.38] 109 4.13] 109J normal . . . . . . . . . . . 7.63] 106 a 3.28] 109 4.02] 109A fully mixed . . . . . . 1.80] 106 a 4.83] 108 5.62] 108G fully mixed . . . . . . 4.60] 106 a 1.52] 109 2.02] 109J fully mixed . . . . . . . 6.57] 106 a 2.62] 109 3.26] 109

a Global thermal timescale.

FIG. 4.ÈEvolutionary tracks for three head-on collisions : cases A, G,and J. Each track is labeled near the main sequence. Important evolution-ary stages are marked and discussed in the text.

Position 0 is the position of the Ðrst model after thecollision. Since the collision product is not in thermal equi-librium, it has a larger radius and is less concentrated than amain-sequence star of the same mass. Its core is not yet hotor dense enough for any signiÐcant fusion to occur, so thestar cannot support itself against gravitational collapse on aKelvin-Helmholtz timescale. As the starÏs radius shrinks,the amount of available gravitational energy is reduced sothe luminosity also decreases. The starÏs e†ective tem-perature increases, and the star moves toward the lowerleft-hand corner of the H-R diagram.

As the amount of gravitational energy that can be rel-eased by further contraction decreases, the contractionslows. At the same time, the temperature of the core con-tinues to increase so that the opacity in the core is reducedand energy Ñows more rapidly. This causes the luminosityand the surface temperature of the star to increase, and thetrack turns toward the upper left-hand corner of thediagram. At the maximum luminosity of this loop, nuclearreactions begin and the gravitational contraction slows.When the starÏs luminosity is supplied by nuclear reactionsalone, the star has reached the main sequence. The thermalrelaxation phase of the evolution of these collision productsoccurs on a global thermal timescale, tth\ (GM2)/(2RL ),where M is the mass of the star and R and L are the radiusand luminosity at the start of the collapse phase.

The equivalent of the zero-age main sequence is at posi-tion 1, and the turno†, or terminal-age main sequence, is atposition 2. Between positions 1 and 2, the star is burning

No. 1, 1997 STELLAR COLLISION PRODUCTS. I. 297

FIG. 5.ÈEvolutionary tracks for four head-on collisions : cases D, M,U, and P. Each track is labeled as in Fig. 4.

hydrogen to helium in its core and is in both hydrostaticand thermal equilibrium. At position 2, the central hydro-gen abundance has dropped below X \ 0.001. Betweenpositions 2 and 3, the star is on the subgiant branch, wherethe energy source for the star is gravitational contraction ofthe core and a small hydrogen burning shell. Position 3 isthe base of the giant branch, where the track turns andbegins its ascent up the giant branch. Case G shows a hookin its main-sequence evolution, since this star has a discon-tinuity in the hydrogen abundance of the star created by thecollision. It also develops a very small convective coreD5000 yr after the collision, which lasts for its entire main-sequence lifetime. When the convective core moves outwardduring the main-sequence evolution and reaches this dis-continuity, a large amount of new hydrogen is mixed intothe core, so the star brieÑy turns back away from the sub-giant branch and toward the main sequence. However, thatnew fuel is quickly consumed, and the star must end itsmain-sequence life.

Case J shows an interesting kink in its track at the begin-ning of its evolution. Instead of collapsing immediately, thestar increases in luminosity for D1200 yr and then col-lapses. This is caused by the large hump in luminosity at

which radiates outward and leaves the star.m/M*

D 0.9,This hump is created when the outer few percent of the star,which was thrown o† to large distances by the collision,rains back down on the collision product and converts itsgravitational energy to thermal energy when it hits thesurface. After that energy has been radiated away, the starcan collapse and become fainter, in the same manner as theother stars. The other cases do not show this particular

feature since the initial outermost luminosity is themaximum, so the star can only become fainter. Despite thelarge initial luminosities in the outer regions of these colli-sion products, the structure of their outer 20% by mass is sorareÐed that the luminosity can be carried by radiativetransport. Therefore, none of the cases we studied devel-oped convective envelopes during the thermal collapsephase.

shows the tracks from the SPH models (solidFigure 6line), along with evolutionary tracks for normal stars (dottedline) and fully mixed stars (dashed line) that have the samemasses as our collision products. The normal tracks showthe expected evolution for an initially chemically homoge-neous star with a normal composition for a globular cluster(Z\ 0.001, Y \ 0.25). The fully mixed tracks also show theexpected evolution for chemically homogeneous stars witha composition given for a star with the same total heliummass as the collision product.

The SPH models are brighter than the normal stars and,with a signiÐcant main sequence, are hotter as well. Theincreased helium content of the collision product, especiallythe higher mass stars, has a†ected the opacities and thenuclear reaction rates in the stellar core. A higher heliumcontent reduces the opacity in the core, which makes the

FIG. 6.ÈEvolutionary tracks for the collision product (solid line), a starwith the same mass and the composition of a normal globular cluster star(dotted line), and a version of the collision product that was fully mixedbefore evolution was started (dashed line). The normal and fully mixedevolutionary tracks are shown only from the zero-age main sequenceonward. Notice that case J follows the evolutionary path of a normal starof the same mass but starts its main-sequence evolution about half-way upthe main sequence. This is expected since the star has a composition proÐlequite similar to the composition proÐle of the parent stars, which arepartially evolved main-sequence stars.

298 SILLS ET AL. Vol. 487

star brighter. Also, this increase in luminosity coming fromthe core heats up the star. Case A shows a star that hasdepleted almost all of its hydrogen fuel and thus starts its““ main-sequence ÏÏ life well up the hydrogen exhaustionphase, never truly living on the main sequence. If most ofthe massive blue stragglers evolve with very short lifetimes,their birthrate must be very large to account for theirobserved numbers.

The fully mixed tracks are signiÐcantly di†erent fromthose of the collision products, as expected. Since the mixedstars have more helium in their envelopes, their tracks arebrighter and hotter on the main sequence than on the colli-sion tracks. These stars also live on the main sequence for alonger period of time since they have more hydrogen mixedinto their interiors. The ages for the normal and fully mixedtracks are given in The thermal timescale given inTable 2.

is the global thermal timescale at the zero-age mainTable 2sequence and represents an order of magnitude estimate forthe preÈmain-sequence lifetime of these stars.

The e†ects of collisional history are demonstrated in thelarge di†erences in the tracks of case G and case J. Thesetwo stars have similar masses (1.133 and 1.142 M

_,

respectively), but their evolution di†ers greatly owing totheir very di†erent initial composition and stellar structure.Case G is a head-on collision between a turno† mass starand a star with half this mass. Since the lower mass starbegins with a lower entropy than the higher mass star, thelower mass star essentially sinks to the center of the colli-sion product, which creates a near-discontinuity in the com-position and temperature proÐles. These anomalous initialproÐles cause the star to live on the main sequence longerthan case J, which begins its life with a core only slightlydepleted in hydrogen. Case G is hotter and brighter thancase J on the main sequence because case G has a densercore and therefore burns its fuel more quickly. Since itsluminosity is increased while its radius has remained thesame, its e†ective temperature also increases. The remnantfrom case J, a head-on collision between two stars,0.6M

_behaves much more like a normal star of the appropriatemass than the case G remnant since the initial temperatureand composition proÐles in case J mimic those of the parentstars. The increased helium content of the core of case Jreduces its lifetime on the main sequence.

The four other head-on collision products that we haveevolved show similar characteristics to the three discussedabove. Cases M and U are another pair of collision pro-ducts with similar masses but with di†erent collision his-tories. Case M is the product of a collision between a 0.6

star and a 0.4 star, which results in a product withM_

M_a total mass of 0.946 Case U is the remnant of aM

_.

collision between a turno† mass star (0.8 and a veryM_)

low mass star (0.16 Again, the collision product withM_

).the larger mass ratio (case U) is hotter and brighter. Thedi†erences between cases U and M are less than thosebetween cases G and J since the parent stars of the lowermass pair (U and M) had created less helium in their precol-lision lifetime.

Case P is not really a blue straggler, even though it is acollision product. This star is the product of a collisionbetween two 0.4 stars, so it has a mass slightly less thanM

_the turno† mass of 0.8 The parent stars of this mergerM_

.remnant have evolved through only a small fraction of theirmain-sequence lifetime and so have a large amount of theirhydrogen fuel left to burn. For these two reasons, the case P

star evolves on a normal track for a turno† mass star andwill be hidden among the normal main-sequence stars in aglobular cluster.

The track for case D lies between those for case A andcase G. This is expected since case D is the product of acollision between a 0.8 star and a 0.6 star. TheM

_M

_track shows the same sort of initial increase in luminosity ascase J, a signiÐcant but fairly short-lived main sequence,and the expected evolution through the subgiant branch tothe giant branch.

5. SUMMARY AND DISCUSSION

5.1. Comparison with PreÈMain-Sequence Stars& Livio proposed that the products ofLeonard (1995)

stellar collisions would be similar to preÈmain-sequencestars in their structure and evolution. They suggested thatthese stars gain a large amount of thermal energy in thecollision and so they expand. The swollen objects resemblepreÈmain-sequence stars in that they are not in thermalequilibrium, do not have sufficient temperature to starthydrogen burning, and will therefore release gravitationalenergy as they contract to thermal equilibrium and themain sequence on a thermal timescale.

While the collision products are indeed far from thermalequilibrium, they have some characteristics that signiÐ-cantly di†er from preÈmain-sequence stars. These di†er-ences all stem from the inhomogeneity of parent stars of thecollision products, unlike the parent gas of normal preÈmain-sequence stars. Since the parent stars are evolved, theyhave regions with nonzero chemical composition and spe-ciÐc entropy gradients. This is especially true for the starsclosest to the turno† and less true for lower mass stars,which have lived only a small fraction of their main-sequence lifetimes. Also, the speciÐc entropy of the Ñuiddi†ers between stars of di†erent masses. Therefore, if a colli-sion occurs between two evolved stars of high mass orbetween two stars of di†erent masses, the parent gas of thecollision product will not have a homogeneous entropycontent. This inhomogeneity of the entropy of the parentgas, plus the shock-heating that occurred during the colli-sions, results in an entropy gradient in the collision product.The condition for dynamical stability is that the entropygradient increases outward and so the collision product willcontinue to have Ñuid motions on a dynamic timescale untilthis condition is met. The calculations of show thatLRSthis condition is satisÐed quickly for most of the star,usually on the order of a day.

5.2. MixingIn this section, we will concentrate on an analysis of

mixing in case A, the collision between two turno† massstars. The mixing of this collision product is most inter-esting, since it has the largest semiconvective region of theseven cases presented in this paper. The other six cases havechemical composition proÐles that are less steep than thatof case A and smaller luminosities in the interior of the star.These two factors cause the stars to be more radiative thancase A.

shows the interior structure of case A as a func-Figure 7tion of time from 100 yr after the collision to the mainsequence. Very little of this star is convective at any timeprior to the main sequence. Never does any helium getmixed into the outer regions of the star, and only the inner

No. 1, 1997 STELLAR COLLISION PRODUCTS. I. 299

FIG. 7.ÈA plot of the convective state of case A as a function of time during the initial thermal relaxation stage of the star. The light gray regions aresemiconvective, the dark gray regions are convective, and the white regions are radiative. The crosshatched region is a transition region between thesemiconvective and radiative state. The majority of the star is always radiative. The inner one-third of the star develops small convective zones assemiconvection begins to level out the chemical proÐle. However, these convective zones never cover a signiÐcant fraction of the star.

m/M D0.1 even becomes largely convective. Small, short-lived convective regions do appear sporadically throughoutthe star, but these are not signiÐcant in mixing most of thestar. This can be seen most clearly in which showsFigure 8,helium proÐles of the star at six times during the initialthermal relaxation phase.

The core of the collision product is initially semi-convective because of the interplay between the luminosityat each shell in the star and the chemical compositionproÐle. The initial luminosity proÐle of case A has a““ hump ÏÏ in luminosity in the inner D25% of the star. Nor-mally, a large luminosity would cause the region to be con-vective, since convective energy transport is more efficientthan radiation. However, this star also has a strong gradientin mean molecular weight in the same region as the lumi-nosity excess ; thus, the region is only semiconvective, notconvective. As the star begins to settle down to thermalequilibrium, this luminosity moves outward in the star andthe luminosity proÐle begins to Ñatten out. The large lumi-nosity region leaves the innermost part of the star Ðrst, andso the center of the star becomes radiative before the rest ofthe star. The outer boundary of the semiconvective regionmoves outward as the increased luminosity moves outwardand causes regions of the star at higher mass fractions tobecome more semiconvective. The chemical proÐle does notbecome Ñat until a mass fraction of D0.5, so the meanmolecular weight gradient continues to stabilize the star.

If the luminosity excess were to move coherently out ofthe star, we would expect to see a region of the same widthmove from the center of the star to the edge in one thermaltime. This region would be semiconvective where the meanmolecular weight gradient is large enough to stabilize theregion against convection and convective where the gra-dient could not. However, the luminosity hump not onlymoves outward, but also dissipates and heats the star slight-ly. Therefore, the excess in luminosity mentioned above thatcan be carried by radiation gets smaller as the star evolves.This means that the semiconvective region shrinks as itmoves outward, and by t \ 105 yr, radiative energy trans-port alone is sufficient to carry away the remaining energy.The star becomes completely radiative at this time.

The gray crosshatched region near t \ 105 yr is a regionin which small regions of a few mass shells each alternatebetween being convective and semiconvective. This is anextreme version of the e†ect seen elsewhere in the star,where small convective regions are interspersed with semi-convective regions. Semiconvection is more efficient atmixing in regions of the star where the di†usion coefficient

is large. Therefore, the mean molecular weight gra-(eq. [5])dient is reduced more quickly in some regions than others.A reduced gradient causes more mixing, since a smaller +kmeans a smaller value of and therefore a larger di†usion+

Lcoefficient. Eventually, the gradient of mean molecularweight becomes zero and convection begins. Therefore,

300 SILLS ET AL. Vol. 487

FIG. 8.ÈSix proÐles of helium abundance as a function of mass fraction for six ages during the initial thermal relaxation stage for case A. The proÐles arefor ages of 0, 102, 103, 104, 105, and 106 yr after the collision. Note the appearance of the fully mixed zones, which correspond to the convective zones inFig. 7.Also notice that the overall shape of the proÐle does not change signiÐcantly during this phase.

regions that have more efficient semiconvection will becomeconvective sooner. This phenomenon is discussed in moredetail by et al. In the gray crosshatchedLanger (1985).region, the temperature gradient is very slightly larger thanthe adiabatic temperature gradient (so the region is onlybarely semiconvective) and the efficient semiconvectiveregions have reduced their mean molecular weight gradientvery close to zero. The distinction between radiative, con-vective, and semiconvective in this region is becomingunclear. Eventually, however, the luminosity excess Ðnallyleaves the star and the entire star becomes radiative. The

compositional e†ects of this merging of convective state canbe seen in Figure 7.

Toward the end of the thermal collapse phase, the stardevelops a small central convective zone. Stars with massesof more than about a solar mass have convective coreswhen on the main sequence, and this collision product isbehaving in accordance with standard stellar evolutiontheory.

None of the collision products modeled here, includingcase A, mix signiÐcant amounts of hydrogen into their coresor helium into their envelopes. The stars do not become

No. 1, 1997 STELLAR COLLISION PRODUCTS. I. 301

largely convective for two reasons. The Ðrst is the inÑuenceof the gradients in mean molecular weight, which stabilizethe Ñuid against convection but allow semiconvection. Thissemiconvection reduces the mean molecular weight gra-dient but is too slow to mix the region completely. Thesecond reason that these collision products do not becomeconvective is that the large luminosities that are present inthe outer regions of the star at the end of the collisionescape from the star in a few thousand years since thethermal timescale in these regions is so small. Therefore,there is little convection occurring in these stars, and theydo not mix.

While convection must be treated carefully, the generalconclusions presented here do not depend strongly on theinclusion and details of semiconvection. Since the outer50% or more of the case A product is radiative at any giventime during the initial thermal relaxation phase (see Fig. 7),this star will never completely mix. The details of the treat-ment of semiconvection will determine how much mixingoccurs in the core of this star, which in turn will determinethe main-sequence lifetime of this collision product. If semi-convection is neglected entirely, the inner one-third of thestar will be convective and mix immediately. Under thistreatment of convection, the collision product will have acentral helium abundance of Y \ 0.70. Since the star hasused up most of its core hydrogen, its main-sequence life-time will still be short compared to a normal star of thesame mass. Therefore, we can safely conclude that this colli-sion product will not mix fully during its thermal relaxationphase and that little hydrogen will be mixed into the core,resulting in a short main-sequence lifetime.

5.3. Comparison with Previous Evolutionary Calculations& Pinsonneault showed that the luminosityBailyn (1995)

function of blue stragglers in the globular cluster 47 Tucmatched the predicted luminosity function for single-starcollisions. However, their evolutionary tracks were basedon the assumption that a blue straggler created by a colli-sion between two stars was fully mixed. Both the SPH cal-culations and the evolutionary tracks presented here bringthis assumption into question. The SPH calculations showthat the collision itself does not induce a signiÐcant amountof mixing. The evolution calculations of head-on collisionproducts show that convection does not play a signiÐcantrole in mixing the star during the initial thermal relaxationphase. Therefore, the blue stragglers in 47 Tuc must havebeen mixed by some process we have not considered (forexample, rotational mixing). Alternatively, the total mass ofthe brightest blue stragglers could be greater than twice theturno† mass. This might be true if triple collisions (mediatedby binaryÈsingle star or binary-binary interactions) contrib-ute signiÐcantly to the blue straggler birth rate.

Two groups have recently used a portion of the SPHresults from stellar collision calculations to approximate theevolution of blue stragglers. et al. used the SPHSills (1995)chemical proÐle of a collision between two turno† massstars and imposed that proÐle on an otherwise normalstellar model. et al. also imposed a chemi-Sandquist (1997)cal proÐle on a normal stellar model. They then went onestep further than Sills and coworkers : following the sugges-tion of & Livio they arbitrarily addedLeonard (1995),energy until the star was on the Hayashi track and evolvedthe star from that point. The resulting evolutionary tracksof these two groups are generally similar, with one major

di†erence. The track calculated by Sandquist andcoworkers for the equivalent of case A has an obvious mainsequence, since the inner D0.2 of the star became con-M

_vective and mixed hydrogen into the core. et al.Sills (1995)began the evolution on the ““ main sequence, ÏÏ and sincetheir starting model had depleted its core hydrogen, itevolves immediately to the subgiant branch.

These two tracks can be compared to the equivalenttrack in this paper, that of case A. In general, all three tracksoccupy a similar place in the H-R diagram. However, thetrack presented in this paper shows that et al.Sills (1995)underestimated and et al. overestimatedSandquist (1997)the amount of mixing that occurs during the thermal relax-ation phase immediately after the collision. The amount ofhydrogen that is mixed into the core of the star a†ects boththe length of the main sequence in the H-R diagram and thetime spent as a blue straggler.

et al. did not evolve the end product ofSandquist (1997)their SPH collision calculations directly, as we have done.For this reason, they did not discover that, contrary to theassumption of & Livio the star never endsLeonard (1995),up on the Hayashi track. Sandquist and coworkers statethat they believe the evolution between the end of the SPHsimulation and the preÈmain-sequence is unimportant, andthey neglect all e†ects that occur on thermal relaxationscales or shorter. However, we have shown that the evolu-tion during the initial thermal relaxation phase is quiteimportant in determining the structure of the star on themain sequence. These blue stragglers never develop asurface convection zone and do not spend any time on theHayashi track. Therefore, they do not mix signiÐcantly.

5.4. Physical Reliability of the Collision ModelsThe evolutionary tracks we have presented here depend

on the assumptions made in the SPH simulations, in theevolution calculations, and in the transformation processbetween the two kinds of codes. Most of these assumptionsand their e†ect on the evolutionary tracks have been dis-cussed elsewhere in this paper. There are two points,however, which have not yet been addressed.

Given the importance of the entropy proÐles of themerger remnants, one might worry that the SPH artiÐcialviscosity might be signiÐcant for the subsequent stellar evol-ution. However, substantial changes in the artiÐcial vis-cosity parameters only weakly a†ect the resulting entropyand chemical composition proÐles (see ° 4.2 of LRS).Indeed, because of the typically low-velocity dispersions inglobular clusters, the stellar collisions we consider arerather weak and shocks are not the dominant e†ect. Never-theless, our SPH code has been well tested in the presence ofshocks (see, e.g., & Shapiro We have tested itsRasio 1991).treatment of one-dimensional shocks for Mach numbersMD a few and for the artiÐcial viscosity parameters used inLRS: we Ðnd that a shocked region typically has a Ðnalvalue of P/o5@3, which is only marginally (D2%) larger thanin accurate, high-resolution simulations done with a one-dimensional SPH code. Although artiÐcial viscosity canspuriously transport angular momentum and produceentropy in di†erentially rotating stars where there is shearbut no shocks, that is of no concern here since we treat onlyhead-on collisions.

One crucial assumption in the SPH simulation is the useof polytropes as parent stars in the collision. Polytropeswith indices n \ 1.5 or 3 (as used in are reasonableLRS)

302 SILLS ET AL. Vol. 487

approximations to zero-age main-sequence stars, but it isnot clear that they are sufficient to model evolved stars. Inparticular, the increase in central mean molecular weightdue to nuclear burning causes a signiÐcant decrease incentral entropy. Since the structure of the collision productis determined by its entropy proÐle, this change in entropyis important & Lombardi Therefore, the tracks(Sills 1997).presented here may not have initial conditions appropriatefor collisions between signiÐcantly evolved stars.

5.5. Future WorkHere we have evolved only nonrotating, head-on colli-

sion products. However, grazing incidence collisions aremore likely to occur in globular clusters. also calcu-LRSlated collisions with nonzero impact parameters, resultingin rapidly rotating collision products. Because of centrifugalsupport, rotating stars have lower central temperatures anddensities than their nonrotating counterparts. Consequent-ly, they will evolve more slowly and spend more time on themain sequence. Another crucial e†ect rapid rotation willhave on the evolution of a star is mixing, caused by merid-ional circulation. The global timescale for meridional circu-lation is the thermal timescaleq

c\ qtherm/s (Tassoul 1978),

divided by the ratio of the centrifugal to gravitational accel-eration. For the collision products calculated by LRS (seetheir Table 3), the values of s are typically on the order of20%È70% in the equatorial plane, which suggests that asigniÐcant fraction of the star could be mixed by rotationbefore it reaches the main sequence. Our fully mixed modelspresented in this paper represent the most extreme case ofmixing possible from rotation or other sources and soprovide a limiting case of the evolution of the rotating colli-sion products. The di†erences between a mixed star and anunmixed star in temperature, luminosity, and lifetime aresigniÐcant ; future evolutionary models of stellar collisionproducts will need to include the e†ects of rotation.

Although grazing incidence collision products are rapidlyrotating at the end of the SPH simulations, rapid rotationrates are not seen in blue stragglers in globular clusters andold, open clusters Therefore, if blue strag-(Mathys 1991).glers are indeed created in stellar collisions, some mecha-nism must spin the stars down between the collision andtheir appearance on the main sequence. Recent results haveshown that classical T Tauri stars, young stars with circum-stellar disks of masses down to about 0.01 are allM

_,

rotating with periods of about 8.5 days et al.(EdwardsOne current scenario suggests that1993). (Ko� nigl 1991)

magnetic coupling locks the disk to the star, which transfersangular momentum to the disk and forces these stars torotate with relatively long periods. If stellar collision pro-ducts have both magnetic Ðelds and disks created duringthe encounter, they could be slowed down by the samemechanism as preÈmain-sequence stars. Since low-massstars typically have magnetic Ðelds, their collision productsshould also. The collision also throws o† a few percent ofthe stellar material, which can amount to as much as 0.1

If even a tenth of this material were to remain in orbitM_

.around the new star, this disk could slow the starÏs rotationenough to match the observations. These speculations needto be conÐrmed with further calculations of the preÈmain-sequence evolution of blue stragglers.

Light elements, such as lithium, beryllium, and boron, areburned at low temperatures and are therefore excellenttracers of mixing and surface convection in stars. In particu-

lar, it is possible that if blue stragglers were created from thecollision of two stars that did not have signiÐcant surfaceconvection zones during their Ðrst foray on the mainsequence (such as two turno† mass stars), lithium could stillbe preserved on its surface. Since we predict that the colli-sion products presented in this paper do not mix and do notdevelop surface convection zones, we could also predict thatthese products should have observable lithium. Lithium willnot be destroyed in the collision since the collision timescaleis too short to burn any signiÐcant amount of lithium, evenif the temperature gets hot enough. Detailed models of thelight element abundances, including the e†ects of rotationalmixing would be needed to analyze observed abundancesquantitatively. A second popular creation mechanism forblue stragglers is the merger of the two stars of a binary.When a binary system merges, it is expected (Webbink

that the secondary star essentially pours itself on the1979)primary. This results in a star without observable lithium,since the hotter inner regions of the secondary, wherelithium has already been destroyed, presumably end up onthe surface of the blue straggler. If more sophisticatedmodels conÐrm that lithium is indeed present on the sur-faces of collision products, we may have a tracer to dis-tinguish between blue stragglers created by stellar collisionsand those created by the merger of binary stars.

5.6. SummaryWe have shown that it is possible to use the results for

SPH calculations directly as starting models for stellar evol-ution calculations with YREC. In order to deal correctlywith the problems arising from the di†erent physicalassumptions of each code, the evolution through the initialthermal relaxation must be dealt with carefully. Small timesteps must be taken, and semiconvection must be treatedexplicitly.

We have evolved seven representative cases of head-onstellar collisions in globular clusters and discussed three ofthose cases in detail. A collision between two turno† massstars forms the highest mass blue straggler possible in thiscreation mechanism. The other two cases represent typicalmass blue stragglers created in very di†erent kinds of colli-sions. A comparison of the latter two tracks demonstratethat collisional history is very important in determining thelifetimes and positions in the H-R diagram of these stars.From the seven cases presented here, we conclude that bluestragglers that are created by head-on collisions are notsigniÐcantly mixed either during the collision or subse-quently by convection.

The interesting scenario proposed by & LivioLeonardin which thermal energy deposited in the collision(1995),

products leads to thorough convective mixing, is not borneout by our calculations. This poses a problem for both thelow observed rotation rates of blue stragglers and for theblue stragglers in some clusters that have been identiÐed asbeing fully mixed stars on the basis of their location in theH-R diagram. Evolution studies including rotation may berequired to resolve this problem.

We would like to thank Constantine Deliyannis, MarcPinsonneault, and Ramesh Narayan for useful discussions.This work was supported by NASA grants NAGW-2469and NAG 5-2867 at Yale University and NSF grant AST91-19475 and NASA grant NAG 5-2809 at Cornell Uni-

No. 1, 1997 STELLAR COLLISION PRODUCTS. I. 303

versity. C. B. is partially supported by a National YoungInvestigator grant from NSF. F. A. R. is supported by anAlfred P. Sloan Foundation Fellowship. Hydrodynamicsimulations were performed at the Cornell Theory Center,

which receives major funding from the NSF and IBM Cor-poration, with additional support from the New York StateScience and Technology Foundation and members of theCorporate Research Institute.

APPENDIX

DEVIATIONS FROM SPHERICAL SYMMETRY

Head-on collision products have spherically symmetric density and entropy proÐles when the star has returned to hydro-static equilibrium. However, this does not necessarily mean that the chemical composition is also spherically symmetric (seeFig. 10 of Some of the SPH particles have been shock-heated during the collision, so their entropy has been increased.LRS).The particles will settle into their stable conÐguration, with entropy increasing outward. At the end of the SPH calculation,shock-heated particles will be at the same radius as other particles that were not shock heated, but rather simply started outwith higher entropy. The chemical composition of these two particles is most likely not the same, and we have a spherical shellof constant entropy made up of particles with di†erent chemical composition. Since YREC insists on spherical symmetry inall quantities, we assigned the chemical composition of each shell to be the average of the compositions of the particles in thatshell. This approximation is reasonable, as can be shown by considering the stability of these di†erent particles.

Consider a Ñuid element that is in mechanical equilibrium with its surroundings, but has a di†erent mean molecular weight.The Ñuid element must then have a di†erent temperature than its surroundings. In this case, the di†erential ideal gas equationof state can be written

dkk

\ dTT

since dP and do are equal to zero. Therefore, the element must transfer heat to or from its surroundings. Pressure balanceoccurs essentially instantaneously, and the mean molecular weight of the element is unchanged, so the density must change tocompensate for the elementÏs change in temperature. The element will now start to sink or rise, depending on the sign of thedi†erence in density between the element and the surroundings. The element will continue to sink (for example) until it reachesa layer of the star with the same density. There it would stop, except that the mean molecular weight of this new layer isprobably not that of the Ñuid element. If the chemical composition is di†erent, the temperature must be di†erent and theprocess continues. Eventually, the Ñuid element will end up in a region with the same mean molecular weight and the slowmovements will stop. The timescale over which this process operates is the local thermal timescale. The details of this processwill only be important during the initial thermal relaxation phase. The e†ect of these oscillations will be an averaging ofchemical composition, as elements with the same composition move toward each other. Therefore, we feel that averaging overspherical shells is a reasonable approximation and will not greatly a†ect the calculated amount of mixing or the evolution ofthe star.

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