Ay 20Fall 2004Star Clusters andStellar Dynamics(This file has a bunch of picturesdeleted, in order to save space)Stellar Dynamics• Gravity is generally the only important force inastrophysical systems (and almost always a Newtonianapprox. is OK)• Consider astrophysical systems which can beapproximated as a self-gravitating “gas” of stars (~ pointmasses, since in most cases R << r.m.s.): open andglobular star clusters, galaxies, clusters of galaxies• If 2-body interactions of stars are important in drivingthe dynamical evolution, the system is called collisional(star clusters); if stars are mainly moving in thecollective gravitational field, it is called collisionless(galaxies)– Sometimes stars actually collide, but that happens only in thedensest stellar systems, and rarely at thatBlue Stragglers: Stellar Merger Products?M3 CMDCommonly seen in globular clusters, as an extension of the mainsequence, and with masses up to twice the turnoff mass• Isolated stellar systems conserve energy and angularmomentum (integrals of motion), which balance theself-gravity• The form of the kinetic energy is important: 1. Ordered (e.g., rotation): disk (spiral) galaxies, >90% of Ekin is in rotation 2. Random (pressure supported): ellipticals, bulges, and star clusters, >90% of Ekin is in random motions• We will focus on the pressure supported systemsStar Clusters• Great “laboratories” for stellar dynamics• Dynamical and evolutionary time scales < or << Galaxy’sage, and a broad range of evolutionary states is presentOpen (or Disk):N ~ 102 - 103Ages ~ 107 - 109 yrGlobular:N ~ 104 - 107Ages ~ 10 - 13 GyrBasic Properties of Typical, Pressure-SupportedStellar SystemsDynamical Modeling of Stellar Systems• A stellar system is fully described by an evolving phase-spacedensity distribution function, f(r,v,t)– Unfortunately, in most cases we observe only 3 out of 6 variables:2 positional + radial velocity; sometimes the other 2 velocity comp’s(from proper motions); rarely the 3rd spatial dimension– … And always at a given moment of t. Thus we seek families ofstellar systems seen at different evolutionary states• Not all of the phase space is allowed; must conserve integrals ofmotion, energy and angular momentum:• The system is finite and f(r,v,t) ≥ 0. The boundary conditions:Dynamical Modeling of Stellar Systems• The evolution of f(r,v,t) is described by the Bolzmann eqn., butusually some approximation is used, e.g., Vlasov (= collisionlessBoltzmann) or Fokker-Planck eqn.– Their derivation is beyond the scope of this class …• Typically start by assuming f(v,t), e.g., a Maxwellian• Density distribution is obtained by integrating ρ(r) = ∫ f(r,v) dv• From density distribution, use Poisson’s eqn. to derive thegravitational potential, and thus the forces acting on the stars:• The resulting velocities must be consistent with the assumeddistribution f(r,v)• The system can evolve, i.e., f(r,v,t), but it can be usuallydescribed as a sequence of quasi-stationary statesDynamics of Stellar Systems• The basic processes are acceleration (deflection) of starsdue to encounters with other stars, or due to the collectivegravitational field of the system at large• Stellar encounters lead to dynamical relaxation, whebythe system is in a thermal equilibrium. The time to reachthis can be estimated as the typical star to change itsenergy by an amount equal to the mean energy; or thetime to change its velocity vector by ~ 90 deg.• There will be a few strong encounters, and lots of weakones. Their effects can be estimated through Coulom-like scattering(From P. Armitage)Strong encountersIn a large stellar system, gravitational force at any point dueto all the other stars is almost constant. Star traces out an orbitin the smooth potential of the whole cluster.Orbit will be changed if the star passes very close to anotherstar - define a strong encounter as one that leads to Δv ~ v.Consider two stars, of mass m. Suppose thattypically they have average speed V.Kinetic energy:€ 12mV2When separated by distance r, gravitationalpotential energy:€ Gm2rBy conservation of energy, we expect a large change in the(direction of) the final velocity if the change in potential energyat closest approach is as large as the initial kinetic energy:Strong encounter€ Gm2r≈12mV2⇒ rs≡2GmV2Strong encounter radiusNear the Sun, stars have random velocities V ~ 30 km s-1, which for a typical star of mass 0.5 Msun yields rs ~ 1 au.Good thing for the Solar System that strong encounters arevery rare…(From P. Armitage)Time scale for strong encounters:In time t, a strong encounter will occur if any other star intrudeson a cylinder of radius rs being swept out along the orbit.rsVtVVolume of cylinder:€ πrs2VtFor a stellar density n, mean number of encounters:€ πrs2VtnTypical time scale between encounters:€ ts=1πrs2Vn=V34πG2m2n(substituting for the strong encounterradius rs)Note: more important for small velocities.(From P. Armitage)Plug in numbers (being careful to note that n in the previousexpression is stars per cubic cm, not cubic parsec!)€ ts≈ 4 ×1012V10 km s-1      3mMsun      −2n1 pc-3      −1 yrConclude:• stars in the disks of galaxies (V ~ 30 km s-1, n ~ 0.1 pc-3near the Sun), never physically collide, and are extremely unlikely to pass close enough to deflecttheir orbits substantially.• in a globular cluster (V ~ 10 km s-1, n ~ 1000 pc-3 or more),strong encounters will be common (i.e. one or moreper star in the lifetime of the cluster).(From P. Armitage)Weak encountersStars with impact parameter b >> rs will also perturb the orbit. Path of the star will be deflected by a very small angleby any one encounter, but cumulative effect can be large.Because the angle of deflection is small, can approximatesituation by assuming that the star follows unperturbedtrajectory:Velocity VStar mass MStar mass mImpactparameter bDistance VtDefine distance of closest approach to be b; define this moment to be t = 0.θDistance d(From P. Armitage)Force on star M due to gravitational attraction of star m is:€ F =GMmd2=GMmb2+ V2t2(along line joining two stars)Component of the force perpendicular to the direction of motion of star M is:€ F⊥= F sinθ= F ×bd=GMmbb2+ V2t2( )3 2Using F = mass x acceleration:€ F⊥= MdV⊥dtVelocity componentperpendicular tothe original