Stellar Dynamics Ay 124 – Lecture 4 – Supplementary Slides •! Gravity is generally the only important force in Stellar Dynamics astrophysical systems (and almost always a Newtonian approx. is OK) •! Consider astrophysical systems which can be approximated as a self-gravitating “gas” of stars (~ point masses, since in most cases R! << r.m.s.): open and globular star clusters, galaxies, clusters of galaxies •! If 2-body interactions of stars are important in driving the dynamical evolution, the system is called collisional (star clusters); if stars are mainly moving in the collective gravitational field, it is called collisionless (galaxies) –! Sometimes stars actually collide, but that happens only in the densest stellar systems, and rarely at that Dynamical Modeling of Stellar Systems Dynamical Modeling of Stellar Systems •! A stellar system is fully described by an evolving phase-space •! The evolution of f(r,v,t) is described by the Bolzmann eqn., but density distribution function, f(r,v,t) usually some approximation is used, e.g., Vlasov (= collisionless –! Unfortunately, in most cases we observe only 3 out of 6 variables: Boltzmann) or Fokker-Planck eqn. 2 positional + radial velocity; sometimes the other 2 velocity comp’s –! Their derivation is beyond the scope of this class … (from proper motions); rarely the 3rd spatial dimension •! Typically start by assuming f(v,t), e.g., a Maxwellian –! … And always at a given moment of t. Thus we seek families of stellar systems seen at different evolutionary states •! Density distribution is obtained by integrating !(r) = ! f(r,v) dv •! Not all of the phase space is allowed; must conserve integrals of •! From density distribution, use Poisson’s eqn. to derive the motion, energy and angular momentum: gravitational potential, and thus the forces acting on the stars: •! The resulting velocities must be consistent with the assumed •! The system is finite and f(r,v,t) ! 0. The boundary conditions: distribution f(r,v) •! The system can evolve, i.e., f(r,v,t), but it can be usually described as a sequence of quasi-stationary states Dynamics of Stellar Systems Strong encounters! •! The basic processes are acceleration (deflection) of stars due In a large stellar system, gravitational force at any point due" to all the other stars is almost constant. Star traces out an orbit" to encounters with other stars, or due to the collective in the smooth potential of the whole cluster." gravitational field of the system at large •! Stellar encounters lead to dynamical relaxation, wheby the Orbit will be changed if the star passes very close to another" system is in a thermal equilibrium. The time to reach this can star - define a strong encounter as one that leads to "v ~ v." be estimated as the typical star to change its energy by an Consider two stars, of mass m. Suppose that" amount equal to the mean energy; or the time to change its typically they have average speed V." velocity vector by ~ 90 deg. 1 Kinetic energy:" mV 2 •! There will be a few strong encounters, and lots of weak ones. 2 Their effects can be estimated through Coulom-like scattering When separated by distance r, gravitational" potential energy:" Gm2 ! r (From P. Armitage)! ! By conservation of energy, we expect a large change in the" Time scale for strong encounters:" (direction of) the final velocity if the change in potential energy" at closest approach is as large as the initial kinetic energy:" In time t, a strong encounter will occur if any other star intrudes" on a cylinder of radius rs being swept out along the orbit." 2 Strong encounter" Gm 1 2 2Gm " mV # rs $ r " r 2 V 2 s V" Strong encounter radius" Vt" 2 ! -1 Volume of cylinder:" Near the Sun, stars have random velocities V ~ 30 km s , " "rs Vt which for a typical star of mass 0.5 M yields r ~ 1 au." sun s 2 For a stellar density n, mean number of encounters:" "rs Vtn Good thing for the Solar System that strong encounters are" Typical time scale between encounters:" very rare…" 3 1 V (substituting for the strong encounter" ts != 2 = 2 2 radius rs)" "rs Vn 4"G m n Note: more important for small velocities." ! (From P. Armitage)! (From P. Armitage)! ! Plug in numbers (being careful to note that n in the previous" Weak encounters! expression is stars per cubic cm, not cubic parsec!)" Stars with impact parameter b >> rs will also perturb the " orbit. Path of the star will be deflected by a very small angle" 3 *2 *1 by any one encounter, but cumulative effect can be large." 12$ V ' $ m ' $ n ' ts " 4 #10 & -1 ) & ) & -3 ) yr % 10 km s ( % Msun ( % 1 pc ( Because the angle of deflection is small, can approximate" Conclude:" situation by assuming that the star follows unperturbed" trajectory:" •! stars in the disks of galaxies (V ~ 30 km s-1, n ~ 0.1 pc-3" ! "near the Sun), never physically collide, and are " Distance Vt! "extremely unlikely to pass close enough to deflect" "their orbits substantially." Velocity V! #$ Impact! Star mass M! parameter b! Distance d! •! in a globular cluster (V ~ 10 km s-1, n ~ 1000 pc-3 or more)," "strong encounters will be common (i.e. one or more "" Star mass m! "per star in the lifetime of the cluster)." Define distance of closest approach to be b; define this " moment to be t = 0." (From P. Armitage)! (From P. Armitage)! Force on star M due to gravitational attraction of star m is:" Final perpendicular velocity is:" GMm GMm % (along line joining two stars)" dV# F = 2 = 2 2 2 "V = dt d b + V t # & $% dt Component of the force perpendicular to the direction of " 1 % motion of star M is:" = & F# (t)dt M $% ! b GMmb 1 % GMmb 2Gm As before, small V leads" to larger deflection" F = F sin# = F $ = = 3 2 dt = " 2 2 2 3 2 M & 2 2 2 bV d b + V t $% b + V t during the flyby" ( ) Velocity component" ( ) perpendicular to" Using F = mass x acceleration:" dV" the original direction" Deflection angle is:" F" = M dt of motion" ! V" ! Integrate this equation with respect to time to get final velocity" $V 2Gm %$ "V " # tan" = % = in the perpendicular direction." # V bV 2 Note: in this approximation, consistent! to assume that " V|| Setting V = c, see that this is exactly half the correct " remains unchanged. Whole calculation is OK provided that" relativistic value for massless particles (e.g. photons)." the perpendicular velocity gain is small." ! ! (From P. Armitage)! (From P. Armitage)! ! "V2 Using identical reasoning as for the strong encounter case:" dN = n "Vt " 2#bdb "V "V! 3 Area of the annulus" 1 between impact" Distance star travels" If the star receives many independent deflections, each with " Number density " parameter b and" in time t" a random direction, expected value of the perpendicular " of perturbing stars" b + db" velocity after time t is obtained by !summing the squares of the" ! This gives for the expected velocity:" individual velocity kicks:" ! 2 bmax 2 2 2 2 $ 2Gm' "V = "V + "V + "V + ... "V 2 = nVt& ) 2+bdb # 1 2 3 # * % bV ( bmin b Writing this as an integral (i.e. assuming that there are very" 8+G2m2nt max db many kicks):" = Logarithm means in a uniform " V * b bmin density stellar system, `encounters#" bmax 2 Where dN is the expected number" 2 $ 2Gm' with stars at distances (b -> 10b)" ! "V = & ) dN of encounters that occur in time t" 8+G2m2nt , b / # * % bV ( between impact parameter b and" = ln max and (10b -> 100b) etc contribute" bmin . 1 equally to the deflection." b + db." V - bmin 0 (From P. Armitage)! (From P. Armitage)! ! ! Relaxation time! Relaxation time for a stellar cluster! After a long enough time, the star#s perpendicular speed will" The factor ln[bmax / bmin] depends upon the limits of integration." (on average) grow to equal its original speed. Define this as" Usually take:" the relaxation time - time required for the star to lose all memory" of its initial orbit." •! bmin to be the strong encounter radius rs (~1 au for " "the Sun). Approximations made in deriving the" 8 G2m2nt % b ( Set:" 2 2 $ relax max "relaxation time are definitely invalid for r < rs." V = "V# = ln' * V & bmin ) 3 V •! bmax to be the characteristic size of the whole stellar" …and solve for t :" relax trelax = 2 2 "system - for the Sun would be reasonable to " 8"G m nln[bmax bmin ] "adopt either the thickness of the disk (300 pc)" ! V 3 "or the size of the galaxy (30 kpc)." Recall that the strong encounter time scale was:" t = s 4"G2m2n t ! t = s relax ln[bmax bmin ] =18 " 23 2ln[bmax bmin ] Frequent distant interactions are more effective at !changing ! Because the dependence is only logarithmic, getting the limits" the orbit than rare close encounters…! exactly right isn#t critical." (From P. Armitage)! (From P. Armitage)! ! ! 12 3$ '* 2 *1 Can use the virial theorem to write this result in an alternate" 2 #10 $ V ' m $ n ' form:" trelax " & -1 ) & ) & -3 ) yr ln[bmax bmin ]% 10 km s ( % Msun ( % 1 pc ( 2 KE + PE = 0 Evaluate the relaxation time for different conditions:" ! Sun! Globular cluster! Open cluster! Average value of" Average value of gravitational" the kinetic energy" potential energy" For a cluster of N stars, each of mass m, moving at average" V / km s-1! 30" 10" 1" ! velocity V in a system of size R:" n / pc-3! 0.1" 104" 10" •! total mass M = Nm" 1 size / pc! 1000" 5" 5" •! total kinetic energy:" NmV 2 2 2 2 13 GM G(Nm) 3 x 10 yr" ~100 Myr" ~100 Myr" •! gravitational potential energy:" ~ = R R Predict that clusters ought to evolve due to star-star" Applying the virial theorem:" GNm interactions during the lifetime of the Galaxy." V = ! R (From P.