Stella elaxation Time [Chandasekha 1960, Pinciples of Stella Dynamics, Chap II] [Ostike & Davidson 1968, Ap.J., 151, 679] Do stas eve collide? Ae inteactions between stas (as opposed to the geneal system potential) impotant? We can answe this question by calculating the time it takes fo a sta s obit to be significantly petubed by individual encountes with othe stas. To calculate this elaxation time, let s fist define the wod significant. One way of doing this is though total enegy: when does the kinetic enegy exchanged duing stella encountes equal the sta s oiginal kinetic enegy, i.e., T E = ( E) = E (30.01) But fo simplicity, we ll define significant as the time it takes a sta to lose all memoy of its oiginal tajectoy, i.e., T D = sin φ = 1 (30.0) We then assume that a) all deflections ae two-body encountes, b) each encounte is statistically independent, and c) close encountes ae insignificant compaed to long-ange encountes, so that duing each encounte, E E. Unde these assumptions, all the deflections ae small (sin φ 1), and we can use the Bon appoximation, whee (v init v final v). θ s b M v v ϕ 1
Fo a single encounte, the deflection angle, φ is elated to the initial impact paamete, b, by F = m dv dt = v = dv = 1 m F dt (30.03) Fom the geomety of the encounte F = F sin θ = F ( ) b = ( GMm ) ( ) b (30.04) Also, fom the Bon appoximation v dt = vdt = ds = dt = ds v (30.05) So v = 1 m F dt = m 0 ( GMm ) ( b ) ( ) 1 ds (30.06) v o, since = (s + b ) 1/ v = GM v Letting x = s/b 0 v = GM vb = GM vb b GM ds = (s + b 3/ ) v 0 dx GM = (1 + x 3/ ) vb 0 ds/b (1 + (s/b) ) 3/ x (1 + x ) 1/ (30.07) 0 (30.08)
Since fo small deflections, tan φ φ = v /v φ = GM v b (30.09) Now, let s sum this ove all possible collisions. The numbe of collisions that take place in time dt depends on the impact paamete, the distance a sta tavels in dt, and the density of stas in the stella system, N, i.e., So, to deflect the sta by 90, sin φ φ = 1 = TD N coll = (πb db) (vdt) N (30.10) 0 bmax b min (πb db)(vdt) N φ = TD bmax 0 b min (πb db)(vdt)n ( GM v b ) = 8πG M N v 3 T D bmax b min db b (30.11) As fo the limits on the log quantity, we can use the obvious fact that no deflection angle can be geate than π. Thus φ = GM v b min = π = b min = GM πv (30.1) Similaly, it is clea that the maximum impact angle must be less than the mean distance between stas, so N = 1 (4/3)π b 3 max = b max = ( ) 1/3 3 (30.13) 4πN 3
So giving ( 8πG M ) { N bmax πv } v 3 T D ln = 1 (30.14) GM T D = v 3 { 8πG M N ln bmax v } π GM (30.15) Obviously, the above deivation involves a numbe of appoximations. A moe igoous deivation by Chandasekha gives and T D = T E = v 3 { 8πG M NH(χ) ln bmax v } GM v 3 { 3πG M NG(χ) ln bmax v } GM (30.16) (30.17) whee H(χ) and G(χ) ae factos of the ode unity that depend on the stella distibution function. Finally, Ostike & Davidson (1968) give an impoved, ecusive expession fo the elaxation time T P = v 3 8πG M N ln { v 3 T P GM } (30.17) In the sola neighbohood, this timescale is much lage than a Hubble time. Thus, the motions of stas ae collisionless, and contolled only by the oveall Galactic potential. 4
The Collisionless Boltzmann Equation [Binney & Temaine 1987, Galactic Dynamics, 1987] The basis of undestanding galactic dynamics lies with the collisionless Boltzmann equation. Imagine a closed volume, V, bounded by a suface S, and containing a mass, M(t). The net amount of mass flowing though a suface is equal to the change of mass in the volume, i.e., S ρ v ds = dm dt = V ρ But the divegence theoem in mathematics says S Q ds = V dv (30.18) Q dv (30.19) so o V ρ dv + V ρ (ρv) dv = 0 (30.0) + (ρv) = 0 (30.1) 5
Now expand this concept to stas flowing in a 6-dimensional phase space (x, y, z, v x, v y, v z ). Assume that the flow of stas is smooth, and is contolled by a potential pe unit mass Φ. Now let f = the stella density in the 6-D space ω = the 6-D position coodinate x = the 3-D space coodinate v = the 3-D velocity coodinate With these definitions, the continuity equation fo a fluid of stas is + (f ω) = 0 (30.) Now let s expand this out to + 6 ( α=1 ω α + f ω ) α ω α ω α = 0 (30.3) and explicitly wite out the space and velocity pats of the second pat of the equation. + 6 α=1 ω α ω α + f ẋ i + f v i = 0 (30.4) Since x and v ae independent dimensions, ẋ i = = 0 (30.5) Also, fo consevative foces with no collisions v = Φ x (30.6) 6
so + 6 α=1 ω α ω α + f { Φ } = 0 (30.7) Also, since the potential a paticle feels depends only on its location, and not on its velocity ( ) Φ = 0 (30.8) which leaves only 6 + ω α = 0 (30.9) ω α α=1 Finally, to make the equation moe tanspaent, we can again explicitly wite out the position and velocity pats of the sum + ẋ i + v i = 0 (30.30) o, if we substitute ẋ = v and v = Φ + which, in vecto notation is x, { v i Φ + ( v f) } ( Φ ) v = 0 (30.31) = 0 (30.3) This is sometimes called the collisionless Boltzmann equation, the Vlasov equation, o the equation of continuity. Note that this equation, when expessed in tems of a Lagangian deivative (i.e., fom the point of view of an obseve moving though space with the fluid) is ( ) ( ) v v dv = dt + dx = df x dt = 0 (30.33) 7
Jeans Equations As it stands, the collisionless Boltzmann equation is athe useless, as it is not only a function of 7 vaiables (many of which ae not easily obsevable), but thei deivatives. Even if f( ω) wee measuable, uncetainties due to Poisson noise would play havoc with the deivatives. Fotunately, moe tactable equations can be found by finding the moments (aveages) of the equation. Such moments poduce the Jeans Equations. Fist, note that the spatial aveage of any quantity associated with the 6-D stella fluid is simply found by integating ove velocity, i.e., Q = Qfd v/ 3 fd 3 v (30.34) Moeove, since the density of stas in the fluid, ν, is simply ν( x) = f d 3 v (30.35) the spatial mean of any quantity is Q = 1 Q fd 3 v (30.36) ν So, fo the fist Jeans equation, stat with the Boltzmann equation { + v i Φ } = 0 (30.31) and integate ove the 3 velocity dimensions d3 v + v i d 3 v Φ d 3 v = 0 (30.37) 8
Since time, position, and velocity ae independent quantities, we can extact those dependencies fom the integals fd 3 v + v i fd 3 v Φ d 3 v = 0 (30.38) The fist integal in the equation is simply the density, and the second is just the density times mean velocity, so (30.38) becomes ν + (ν v i ) Φ d 3 v = 0 (30.39) But if you expand that last integal d 3 v = = = dv i dv j dv k dv i dv k dv i dv k f + = 0 (30.40) (because obviously, the phase-space density of objects at infinite velocity is zeo). This leaves us with the fist Jean s equation: the 3-D equation of continuity ν + (ν v i ) = 0 (30.41) 9
Fo the second Jeans equation, we again stat with the Boltzmann equation + { v i Φ } = 0 (30.31) This time, howeve, we multiply though by v j befoe integating v j d3 v + v i v j d 3 v v j Φ d 3 v = 0 (30.4) Once again, the time and space deivatives can be taken outside the integal, leaving mean quantities v j fd 3 v + (ν v j ) + v i v j fd 3 v (ν v i v j ) Φ Φ v j d 3 v = 0 v j d 3 v = 0 (30.43) And, once again, we can expand and evaluate the individual integals of the last tem v j d 3 v = If j i, the esult is the same as befoe v j dv i dv j dv k = v j dv i dv j dv k (30.44) v j dv j dv k df = 0 (30.45) 10
But if j = i, v i dv i dv j dv k = dv j dv k v i df (30.46) We can evaluate this last tem by integating by pats v i df = v i f fdv i (30.47) The fist tem is zeo, and the second is just the space density, ν, so the second Jean s equation becomes (ν v j ) + (ν v i v j ) + ν Φ x j = 0 (30.48) The final Jeans equation comes fom noting that the anisotopic pessue tem (i.e., the stess tenso), σ ij is σ i,j = (v i v i )(v j v j ) = v i v j v i v j (30.49) So, if we substitute this into the nd Jeans second, (ν v j ) + { ( ν σ,j + v i v j )} + ν Φ = 0 (30.50) i x j If we expand this out ν v j + v j ν + { (νσij ) + ν j (ν v i ) + ν v i v } j = ν Φ x j (30.51) 11
multiply the 3-D continuity equation by v j v j ν + v j (ν v i ) = 0 (30.5) and subtact the two equations, we get ν v j + ν v i v j = ν Φ x j (νσ ij ) (30.53) Hee, νσ ij is the anisotopic pessue (stess) tenso. But since σ ij is symmetic, its matix can be diagonalized. This poduces the pinciple axes of the velocity ellipsoid. This is the equivalent of Eule s equation fo fluid flows ρ v + ρ (v ) v = ρ Φ p (30.54) whee ρ is the density and p the pessue. 1
Jeans Equations in Cylindical Coodinates The Jeans and Boltzmann equation ae almost neve used in thei Catesian coodinate fom. Fo most applications in spial galaxies, cylindical coodinates ae used; in elliptical galaxies, the equations ae given in spheical coodinates. The deivation of these equations is staightfowad, but tedious. Fo the cylindical coodinate equations, the Boltzmann equation becomes + v 1 + v θ ( v v θ + Φ v θ θ + v z ) Φ v θ z ( v z + θ Φ ) v v z = 0 (30.54) and if we assume azimuthal symmety, the Jeans equations ae ν + 1 ν v d + ν v z z = 0 (30.55) ν v + ν v + ν v v z z ( v + ν vθ + Φ ) = 0 (30.56) ν v θ + ν v v θ + ν v θv z z + ν v θv = 0 (30.57) ν v z + ν v v z + ν v z z + v v z + ν Φ z = 0 (30.58) 13
This last equation is often used in its simplified fom. Fist, assume the galaxy is in a steady state, so that the time deivative is zeo. Next, conside v v z. Fom symmety, this tem should be zeo in the plane of the galaxy. Above and below the plane, this tem might be zeo, but wost case is that the pincipal axes of the stella velocity ellipsoid ae otated to align with that of the spheoidal component. To estimate the effect this would have on v v z, we must fist tansfom the spheical coodinate system (v, v θ, v ϕ ) to the cylindical system (v, v θ, v z ). This is tedious, but staightfowad. If the pinciple axis of the system is aligned with the disk (i.e., if v v θ = 0), the the esult is v v z = { v v θ } z + z = { v v θ } { v v θ } z z 1 (1 + z / ) (30.59) Since most of the stas ae nea the plane of the galaxy, v v and v θ v z, so v v z { v v z } z (30.60) Now, when we evaluate the thid tem of the Jeans equation v v z z { z ( v v z )} { v v z v v z } (30.61) Since most of the stas in a disk galaxy ae nea the galactic plane, i.e., z, this tem is small. So except fo the egion 14
nea the galactic cente v v z 0 (30.6) Similaly, the deivative ν v v z { νz ( v vz )} z { ν v v z + ( v vz ) ( ν ν )} 0 (30.63) (And emembe this is the wost case scenaio. The close to the plane you ae, the moe likely that v v z = 0 by symmety.) So, to fist ode, the Jeans equation elates the z velocity dispesion to the galactic potential by Φ z = 1 ν ν v z z (30.64) To see the impotance of this, we can take the deivative of this equation, Φ z = { 1 ( ν v z ν z z )} (30.65) and then note that nea the galactic plane, the gadient of the potential is almost entiely in the z diection. Hence Φ z Φ = z { 1 ν z ( ν v z )} = 4πGρ (30.66) via Poisson s equation. Thus the mass in the disk can be measued via the z-motions of stas nea the galactic plane. 15
Jeans Equations in Spheical Coodinates The Boltzmann equation in spheical coodinates is + v + v θ ( v θ + v ϕ 1 ) Φ θ + v ϕ sin θ v + 1 ϕ + { v ϕ (v + v θ cot θ) + 1 sin θ ( vϕ cot θ v v θ Φ ) θ v θ } Φ = 0 (30.67) ϕ v ϕ and, fo steady-state systems with v = v θ = 0, the Jeans equation is ν v + ν { v ( v θ + v ϕ)} = ν Φ (30.68) Anothe way to look at this equation is to think of it in tems of hydostatic equilibium. ecall that the definition of pessue is P = 1 3 ρ v = ρ v i (30.69) whee v i epesents one component of the motion. So let P be the adial pessue impated by the stella motions, and Q be the tangential pessue tem, i.e., P = ν v Q = ν ( v ϕ + v θ ) (30.70) In this case, the Jeans equation looks is dp d + P Q = ν dφ d (30.71) 16
If the stella obits ae andomly distibuted (i.e., isotopic, so that ( v = vθ = v ϕ ), then P = Q, and the equation educes to the simple hydostatic equilibium. To descibe the degee of obital anisotopy in a spheical system, one often uses the paamete β, β = 1 v θ v (30.7) Fo isotopic obits, β = 0; fo puely adial obits, β = 1, and fo puely cicula obits, β =. Note that β need not be a constant thoughout the system; ealistic models of spheical systems often have β() as thei most impotant fee paamete. 17