The Virial Equations I We can obtain an important tensor equation relating global properties of the system, by multiplying the CBE by both vj and xk and then integrating over the entire phase-space. The first step of this has already been performed in our derivation of the Jeans equations, and yielded the momentum equations @(ρ vj ) @(ρ vivj ) @Φ h i + h i + ρ = 0 @t @xi @xj Multiplying all terms with xk and integrating over real space yields @ 3 @(ρ vivj ) 3 @Φ 3 ρxk vj d x = xk h i d ~x ρ xk d ~x @t h i − @xi − @xj Using integrationR by parts the firRst term on the r.h.s. becomesR @(ρ vivj ) 3 @(ρxk vivj ) 3 @xk 3 xk h i d ~x = h i d x~ ρ vivj d ~x @xi @xi − h i @xi = δ ρ v v d3x~ R −R ki h i ji R = ρ v v d3~x − R h k ji = 2 − RKkj where we have defined the kinetic energy tensor = 1 ρ v v d3x~ Kij 2 h i j i R The Virial Equations II It is customary to split the kinetic energy tensor into contributions from ordered and random motions: + 1 Π Kij ≡ Tij 2 ij where 1 ρ v v d3x~ Π ρ σ2 d3x~ Tij ≡ 2 h ii h j i ij ≡ ij In addition to the R we also define the potential energy Rtensor K @Φ 3 ij ρxi d ~x W ≡ − @xj Combining the above we obtain R @ ρ x v d3x = 2 + @t kh ji Kkj Wkj which allows us to writeR 1 d ρ [x v + x v ] = 2 + 2 dt k h ji j h ki Kjk Wjk where we have usedR that and are symmetric. K W The Virial Equations III Finally we also define the moment of inertia tensor ρ x x d3~x Iij ≡ i j Differentiating with respect to time,Rand using the continuity equation (i.e., the zeroth moment equation of the CBE) yields dIjk @ρ 3 dt = @t xj xk d ~x @ρ vi 3 = h i xj xk d ~x −R @xi @(ρ vi xj xk) 3 @(xj xk) 3 = h i d ~x + ρ vi d x~ − R @xi h i @xi = ρ v [x δ + x δ ] d3x~ Rh ii j ik k ij R = ρ [x v + x v ] d3x~ R j h ki kh j i so that R 2 1 d 1 d jk ρ [x v + x v ] = I2 2 dt k h j i j h ki 2 dt which allows us to writeR the Tensor Virial Theorem as 2 1 d jk I2 = 2 + Π + 2 dt Tjk jk Wjk which relates the gross kinematic and structural properties of gravitational systems. The Virial Equations IV If the system is in a steady-state the moment of inertia tensor is stationary, and the Tensor Virial Theorem reduces to 2 + = 0. Kij Wij Of particular interest is the trace of the Tensor Virial Theorem, which relates the total kinetic energy K = 1 M v2 to the total potential energy 2 h i 1 3 . W = 2 ρ(~x) Φ(~x) d ~x 3 R 1 2 2 2 3 tr( ) ii = 2 ρ(~x) v1 (~x) + v2 (~x) + v3 (~x) d ~x K ≡ i=1 K h i h i h i P = 1 R ρ(~x) v2 (~x)d3x~ 2 h i = 1 M v2 = K 2 R h i where we have used that v2 = 1 ρ(~x) v2 (~x)d3~x h i M h i Similarly, the trace of the potentialRenergy tensor is equal to the total potential energy (see next page for derivation): tr( ) = W = 1 ρ(~x) Φ(~x) d3~x W 2 We thus obtain the scalar virial theoremR 2K + W = 0 The Potential Energy Tensor I We have defined the potential energy tensor as @Φ 3 ij ρxi d ~x W ≡ − @xj ρ(~x) 3R Using that 0 we obtain Φ(~x) = G ~x ~x d x~ − j − j 0 x (x x ) R i j − j 3 3 ij = G ρ(~x) ρ(~x0) ~x0 ~x 3 d ~x0d x~ W j − j Using that ~x and ~x0 are dummR R y variables, we may relabel them, and write 0 0 x (x x ) j k− k 3 3 ij = G ρ(~x0) ρ(~x) ~x ~x0 3 d ~xd ~x0 W j − j Interchanging the order ofRintegrationR and summing the above two equations yields the manifestly symmetric expression 0 0 (x x )(x x ) G j − j k− k 3 3 ij = 2 ρ(~x) ρ(~x0) ~x0 ~x 3 d x~0d ~x W − j − j This expression allows usR Rto write 3 0 2 G ~x ~x 3 3 j − j tr( ) ii = 2 ρ(~x)ρ(~x0) ~x0 ~x 3 d ~x0d x~ W ≡ W − j − j i=1 0 PG ρ(R~x R) 3 3 1 3 = 2 ρ(~x) ~x0 ~x d ~x0d ~x = 2 ρ(~x)Φ(~x)d x~ = W − j − j R R R The Surface Pressure Term In our derivation on the previous pages we obtained @(ρ vivj ) 3 @(ρxk vivj ) 3 @xk 3 xk h i d ~x = h i d x~ ρ vivj d ~x @xi @xi − h i @xi = ρ v v d3~x = 2 R −R h k ji −R Kkj where we have used that R @(ρxk vivj ) 3 2 h i d x~ = ρxk vkvj d S = 0 @xi h i based on the assumptionR that ρ(r) =R0 when r . However, this is only true for an isolated system with ‘vacuum’ boundar! 1y conditions. In reality, a halo or galaxy is embedded in a cosmological density field, often with ongoing infall. This yields a non-zero surface pressure. In its most general form the scalar virial theorem therefore reads 2K + W + Sp = 0 with the surface pressure term S = v2 ~r ~nd2S~ p − h i · As long as Sp = 0 we thus expectRthat 2K= W = 1. See Shapiro et 6al. (astro-ph/0409173) for a detailedj j 6discussion. The Virial Equations V From a simple dimensional analysis one finds that W GM 2=R with j j / M the system’s mass and R a characteristic radius. A useful characteristic radius is the so-called gravitational radius defined by 2 GM rg W ≡ j j One can relate the gravitational radius to the half-mass radius rh, defined as radius enclosing half the total mass. As shown by Spitzer (1969), typical stellar systems have r 2:5r . g ' h Combining this with the scalar virial theorem we can write that 2 r v M 2:5 hh i ' G which is a useful equation to obtain a (rough) estimate of the virial mass from a measure of the half-mass radius and the rms motion The Virial Equations VI Using the scalar virial theorem we obtain E = K + W = K = 1 W − 2 Consider the formation of a virialized object. If the system forms by collecting material from large radii, the initial conditions are well approximated by Kinit = Winit = Einit = 0. Because of gravity the matter starts to collapse. Since W = GM 2=r − g this makes W more negative. At the same time K increases. Initially, during the early collapse, E = T + W = 0. After the first shell crossing, the system starts to virialize. When virialization is complete, 2T + W = 0 and E = W=2. Therefore, half the gravitational energy released by collapse is invested in kinetic form. The system somehow disposes of the other half in order to achieve a binding energy E = E. b − QUESTION Where does the other half of the energy go? Application: M/L of Spherical Systems As an application of the Virial Theorem, consider spherical, non-rotating systems (spherical galaxies or globulars) If the mass-to-light ratio Υ does not depend on radius then K = 1 ρ(~x) v2 d3x~ = Υ ν(~x) v2 d3~x xx 2 h xi 2 h xi where ν(~x) = ρ(~x)=RΥ is the 3D luminosity distribR ution, and Kxx is the kinetic energy associated with motion in the x-direction Since a spherical, non-rotating system is isotropic we have that K = Kxx + Kyy + Kzz = 3Kxx If one has observationally determined the surface brightness profile Σ(R) 2 and the line-of-sight velocity dispersion σp(R) then it is easy to see that 2π Υ 1 2 1 2 K = 3 2 dφ dRRΣ(R)σp(R) = 3πΥ dRRΣ(R)σp(R) ΥJ 0 0 0 ≡ R R R 2 where we defined the observationally accessible J = J(Σ; σp) Application: M/L of Spherical Systems 1 2 As seen in exersizes, for spherical system: G M (r) W = 2 r2 dr − 0 r R 2 Using that M(r) = 4π ρ(r0)r0 dr0 0 R where the density profile is related to Σ(R) according to ρ(r) = Υ 1 dΣ dR π dR pR2 r2 − r − we obtain that R r 2 2 1 dr 2 1 dΣ dR 2 ~ W = 8Υ r2 dr0 r0 dR 2 2 Υ J − 0 pR r ≡ 0 0 r − R R R where we have defined the observationally accessible integral J~ = J~(Σ) According to the virial theorem 2K + W = 0, and thus 2K=W = 1. − Substituting K = ΥJ and W = Υ2J~ we thus obtain that Υ = 2J − J~ Flattening of Oblate Spheroids I As another application of the virial theorem we relate the flattening of an oblate spheroid to its kinematics.