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th 11 • Phase Space Lec Collisionless Systems • We showed collisions or deflections are rare • Collisionless: stellar motions under influence of mean gravitational potential! • Rational: • Gravity is a long-distance force, decreases as r-2 – as opposed to the statistical mechanics of molecules in a box Collisionless Systems • stars move under influence of a smooth gravitational potential – determined by overall structure of system • Statistical treatment of motions – collisionless Boltzman equation – Jeans equations • provide link between theoretical models (potentials) and observable quantities. • instead of following individual orbits • study motions as a function of position in system • Use CBE, Jeans eqs. to determine mass distributions and total masses Fluid approach:Phase Space Density PHASE SPACE DENSITY:Number of stars per unit volume per unit velocity volume f(x,v) (all called Distribution Function DF). number of stars m Nm f(x, v) 3 space volume velocity volume pc (kms 1 )3 The total number of particles per unit volume is given by: m n( x ) f ( x , v )dvx dv y dvz • E.g., air particles with Gaussian velocity (rms velocity = σ in x,y,z directions): vx2 v y2 vx2 m n o exp 2 2 f(x, v) ( 2 )3 • The distribution function f(x,v) is defined by: mdN=f(x,v)d3xd3v where dN is the number of particles per unit volume with a given range of velocities. 3 3 mdN f ( x , v )d xd v • The total mass is then given by the integral of the mass distribution function over space and velocity volume: 3 3 3 M total ( x)d x f ( x , v )d v d x • Note:in spherical symmetry d3x=4πr2dr, • for isotropic systems d3v=4πv2dv • The total momentum is given by: 3 3 Ptotal v mdN f ( x , v )v d xd v • Example:mean speed of air molecules in a box of dx3 : v2 v2 v2 y m n o exp x 2 2 f(x, v) ( 2 )3 x v2 exp 3 3 2 vdN vfd xd v 2 0 2 3 3 dN fd xd v exp v 2 2 0 These are gamma functions 3 4 v dv 2 4 v dv • Gamma Functions: (n) e x dx x n 1 0 (n) (n 1)(n 1) 1 2 How to calculate 2 0 0 d sin d 3 dx and 3 dv 2 2 2 d 2 2 , r x x y z d 3 x dxdydz r 2 drdΩ 4πr 2 dr (if spherical) d 3 v dvx dv y dvz v 2 dvdΩ 4πv 2 dv (if isotropic) V v vx2 v y2 vz2 2[ E ( x)] DF and its moments d 3 x A(x) dM A d 3 x Af ( x, v )d 3 v d 3 x dM d 3 x f ( x, v)d 3 v 1 For : A(x,v ) Vx , VxVy, (Vx 2 Vy 2 Vz 2 ) x , x v 2 AdM A A dM , mass-weighted average, Additive: subcomponents add up to the total gravitational mass A B A B f f1 f 2 1 2 Full Notes online • http://www-star.st-and.ac.uk/~hz4/gravdyn/ GraviDynFinal3.ppt GraviDynFinal3.pdf Liouvilles Theorem We previously introduced the concept of phase space density. The concept of phase space density is useful because it has the nice property that it is incompessible for collisionless systems. A COLLISIONLESS SYSTEM is one where there are no collisions. All the constituent particles move under the influence of the mean potential generated by all the other particles. INCOMPRESSIBLE means that the phase-space density doesn’t change with time. Consider Nstar identical particles moving in a small bundle through spacetime on neighbouring paths. If you measure the bundles volume in phase space (Vol=Δx Δ p) as a function of a parameter λ (e.g., time t) along the central path of the bundle. It can be shown that: dVol dNstar 0, 0, d d px ' LIOUVILLES THEOREM' px x x It can be seen that the region of phase space occupied by the particle deforms but maintains its area. The same is true for y-py and z-pz. This is equivalent to saying that the phase space density f=Nstars/Vol is constant. df/dt=0! motions in phase-space • Flow of points in phase space corresponding to stars moving along their orbits. • phase space coords: ( x, v) w (w1 , w2 ,..., w6 ) w ( x, v) (v,) • and the velocity of the flow is then: – where wdot is the 6-D vector related to w as the 3-D velocity vector v relates to x • stars are conserved in this flow, with no encounters, stars do not jump from one point to another in phase space. • they drift slowly through phase space • In the COMBINED potential of stars and dark matter fluid analogy • regard stars as making up a fluid in phase space with a phase space density f (x, v,t) f (w,t) • • assume that f is a smooth function, continuous and differentiable – good for N >105 • as in a fluid, we have a continuity equation • fluid in box of volume V, density , and velocity v, the change in mass is then: dM dt V d 3 x v d 2 S t S 3 2 F d x F d S V S v d 3 x 0 t V – Used the divergence theorem continuity equation • must hold for any volume V, hence: t v 0 • in same manner, density of stars in phase space obeys a continuity equation: 6 f fw 0 t 1 w If we integrate over a volume of phase space V, then 1st term is the rate of change of the stars in V, while 2nd term is the rate of outflow/inflow of stars from/into V. 0 3 3 vi vi w vi i 1 vi 1 w i 1 xi 6 x 0 i Collisionless Boltzmann Equation • Hence, we can simplify the continuity equation to the CBE: 6 f f w 0 t 1 w 3 f f f vi 0 t i 1 xi xi vi • Vector form f f v f 0 t v • in the event of stellar encounters, no longer collisionless • require additional terms to rhs of equation CBE cont. • can define a Lagrangian derivative • Lagrangian flows are where the coordinates travel along with the motions (flow) – hence x= x0 = constant for a given star • then we have: • and d w 6 dt t 6 df f f w 0 dt t 1 w incompressible flow • example of incompressible flow • idealised marathon race: each runner runs at constant speed • At start: the number density of runners is large, but they travel at wide variety of speeds • At finish: the number density is low, but at any given time the runners going past have nearly the same speed DF & Integrals of motion • If some quantity I(x,v) is conserved i.e. dI ( x, v ) 0 dt • Assume f(x,v) depends on (x,v) through the function I(x,v), so f=f(I(x,v)). • Such phase space density is incompressible, i.e df 0 dt Jeans theorem • For most stellar systems the DF depends on (x,v) through generally three integrals of motion (conserved quantities), Ii(x,v),i=1..3 f(x,v) = f(I1(x,v), I2(x,v), I3(x,v)) • E.g., in Spherical Equilibrium, f is a function of energy E(x,v) and ang. mom. vector L(x,v).’s amplitude and z-component f ( x, v) f ( E, || L ||, L zˆ) 3D Analogy of 6D Phase space • If DF(x,v) is analogous to density(x,y,z), • Then DF(E,L,Lz) is ~ density(r,theta,phi), • Integrals analogous to spherical coordinates – E(x,v) analogous to r(x,y,z) • Isotropic DF(E) ~ spherical density(r) – Normalization dM=f(E)dx3dv3 ~ dM=density(r)dr3 – Have non-self-gravitating subcomponents: DF1+DF2, like rho1+rho2 to make up total gravity. th 12 • Phase Space Lec Tensor Virial Theorem • Equation of motion: dv dt T T 1 dv 1 dt r dtr . T 0 dt T 0 d (rv ) dr v dt dt T T T (rv ) 1 1 v vdt dtr . T 0 T 0 T 0 v v r This is Tensor Virial Theorem • E.g. vx vx x x vx v y x y etc v 2 vx2 v y2 vz2 r . d 2 r vcir dr 2 v 2 vcir ( spherical ) • So the time averaged value of v2 is equal to the time averaged value of the circular velocity squared. Scalar Virial Theorem • the kinetic energy of a system with mass M is just where <v2> is the mean-squared speed of the system’s stars. 1 K M v2 2 • Hence the virial theorem states that W GM v M rg v 2 r . 2 2K W 0 Virial Stress Tensor n ij 2 • describes a pressure which is anisotropic – not the same in all directions • and we can refer to a “pressure supported” system n P x • the tensor is symmetric. • can chose a set of orthogonal axes such that the tensor is diagonal ij 2 ii 2 ij • Velocity ellipsoid with semi-major axes 11 , 22 , 33 given by 2 ij i Subcomponents in Spherical Equilibrium Potential • Described by spherical potential φ(r) • SPHERICAL subcomponent density ρ(r) depends on modulus of r. r , r r , 0, 0 (0, r , 0) (0, 0, r ) x 0 xy 0 • EQUILIBRIUM:Properties do not evolve with time. f 0 0 0 t t t • In a spherical potential r 2 x2 y2 z 2 d (r 2 ) d ( y 2 ) rdr ydy dr y dy r (r ) r (r ) x x y y r y d (r ) x r dr xy r r So <xy>=0 since the average value of xy will be zero. <vxvy>=0 Spherical Isotropic f(E) Equilibrium Systems • ISOTROPIC:The distribution function f(E) only depends on the modulus of the velocity rather than the direction. f E , E v / 2 (r ) 2 2 2 x vx v y 0 Note:the tangential direction has and components 2 y 2 z 1 2 tangential 2 2 r Anisotropic DF f(E,L) in spherical potential. • Energy E is conserved as: 0 t • Angular Momentum Vector L is conserved as: 0 • DF depends on Velocity Direction through L=r X v • Hence anisotropic 1 2 2 r tangential 2 e.g., f(E,L) is an incompressible fluid • The total energy of an orbit is given by: 1 2 E v (r , t ) 2 df ( E , L) f ( E , L) dE f ( E , L) dL 00 dt E dt L dt 0 for static potential, 0 for spherical potential So f(E,L) constant along orbit or flow • spherical Jeans eq. of a tracer density rho(r) • d ( r2 ) / dr (2 r2 t2 ) / r d / dr • Proof : vr 2 E 2 L2 / r 2 2 vr dvr / dr d / dr L2 / r 3 dv 3 d (vt2 )dvr d (L2 )dE / vr r 2 fdv3 f ( E , L)d (L2 )dE / vr r 2 d / dr L / r r 2 3 2 fdv3 f ( E , L)d (L2 )dE (dvr / dr ) Jeans eq. Proof cont. fdv3 f ( E , L)d L2 dE /( vr r 2 ) t2 f ( E , L)d L2 dE * L2 /( r 4 vr ) r2 r 2 d L f ( E , L)dEv 2 L 0 r vr2 0 f ( E , L)d L dE (d / rdr ) / v f ( E , L)d L dE L / r /v d ( r2 r 2 ) / rdr f ( E , L)d L2 dE *(dvr / dr ) / r 2 2 2 4 r rd ( r2 ) / dr 2 r2 rd / dr t2 r • SELF GRAVITATING:The masses are kept together by their mutual gravity. • In non-self gravitating systems the density that creates the potential is not equal to the density of stars. e.g a black hole with stars orbiting about it is NOT self gravitating. th 13 • Phase Space Lec Velocity dispersions of a subcomponent in spherical potential • For a spherically symmetric system we have d n 2 2 2 2 n vr 2vr v v dr r • a non-rotating galaxy has n ddr – and the velocity ellipsoids are spheroids with their symmetry axes pointing towards the galactic centre v 2 v 2 2 2 r 1 v / v r • Define anisotropy Spherical mass profile from velocity dispersions. • Get M(r) or Vcir from: 1 d 2 vr d GM r 2 n vr n dr r dr r2 vcirc 2 2 2 d GM r d ln vr 2 d ln n r vr 2 dr r d ln r d ln r • RHS: observations of dispersion and as a function of radius r for a stellar population. • Isotropic Spherical system, β=0 d ( 2 ) d (r ) dr dr Note: 2=P • This is the isotropic JEANS EQUATION, relating the pressure gradient to the gravitational force. d P dr | g |dr dr r r 2 Above Solution to Isotropic Jeans Eq: negative sign has gone since we reversed the limits. Hydrostatic equilibrium Isotropic spherical Jeans equation dP d ( 2 ) d (r ) dr dr dr • Conservation of momentum gives: 0 P g 1 g P Tutorial g M 2 vesc (r) (E) (r) Tutorial Question 3 • Question: Show dispersion sigma is constant in potential Phi=V02ln(r). What might be the reason that this model is called Singular Isothermal Sphere? • 1 d 2 r dr vc2 r r r ro r 2 c r 2 2 v v v 1 2 dr c 2 c dr r 4G r r At r ro , P 2 0 2 2 v v c 2 c 4G 2r 2 vc2 vc2 4Gr 2 2 • 2 v 2 c 2 2 v 2 c 2 vc 2 • Since the circular velocity is independent of radius then so is the velocity dispersionIsothermal. Flattened Disks • Here the potential is of the form (R,z). • No longer spherically symmetric. • Now it is Axisymmetric 1 2 ( R, z ) ( R, z ) R 2 R 4G R R z gr R gz z Question 4: Oblate Log. potential • oblate galaxy with Vcirc ~ V0 =100km/s ( R, z ) 12 v0 2 ln R 2 2 z 2 C0 • Draw contours of the corresponding Selfgravitating Density to show it is unphysical. • Let Lz=1kpc*V0 , E=0.55*V02 +C0, Plot effective potential contours in RZ plane to show it is an epicycle orbit. • Taylor expand the potential near (R,z)=(1,0) to find epicycle frequencies and the approximate z-height and peri-apo range. Orbits in Axisymmetric Potentials (disk galaxies) z y x R2=x2+y2 R • cylindrical (R,,z) symmetry z-axis • stars in equatorial plane: same motions as in spherically symmetric potential – non-closed rosette orbits • stars moving out of plane – can be reduced to 2-D problem in (R,z) – conservation of z-angular momentum, L • Angular momentum about the z-axis is conserved, toque(rF=0) if no dependence on . d 2 2 2 LZ R ( R ) 0 dt 2 • Energy is also conserved (no time-dependence) 1 2 R R22 z 2 (R, z) const 2 Specific energy density in 3D • Eliminating in the energy equation using conservation of angular momentum gives: 2 1 2 J ( R z 2 ) ( R, z ) z 2 E 2 2R eff Total Angular momentum almost conserved • These orbits can be thought of as being planar with more or less fixed eccentricity. • The approximate orbital planes have a fixed inclination to the z axis but they process about this axis. • star picks up angular momentum as it goes towards the plane and returns it as it leaves. Orbital energy • Energy of orbit is (per unit mass) E 1 2 1 2 1 2 2 R R R R 2 z 2 2 z 2 2 z 2 2 Lz 2 2R eff • effective potential is the gravitational potential energy plus the specific kinetic energy associated with motion in direction • orbit bound within E eff • The angular momentum barrier for an orbit of energy E is given by eff ( R, z ) E • The effective potential cannot be greater than the energy of the orbit. R 2 z 2 2 E 2 ( R, z ) eff 0 • The equations of motion in the 2D meridional (RZ)plane then become: . eff R R eff z z R 2 J z • Thus, the 3D motion of a star in an axisymmetric potential (R,z) can be reduced to the motion of a star in a plane (Rz). • This (non uniformly) rotating plane with cartesian coordinates (R,z) is often called the MERIDIONAL PLANE. • eff(R,z) is called the EFFECTIVE POTENTIAL. • The orbits are bound between two radii (where the effective potential equals the total energy) and oscillates in the z direction. • The minimum in eff occurs at the radius at which a circular orbit has angular momentum Lz. • The value of eff at the minimum is the energy of this circular orbit. eff J z2 2R 2 R E Rcir • The effective potential is the sum of the gravitational potential energy of the orbiting star and the kinetic energy associated with its motion in the direction (rotation). • Any difference between eff and E is simply kinetic energy of the motion in the (R,z) plane. • Since the kinetic energy is non negative, the orbit is restricted to the area of the meridional plane satisfying E - eff . (R,z)>= 0 • The curve bounding this area is called the ZERO VELOCITY CURVE since the orbit can only reach this curve if its velocity is instantaneously zero. Nearly circular orbits: epicycles • In disk galaxies, many stars (disk stars) are on nearly-circular orbits eff R eff ; • EoM: z R • x=R-Rg z eff eff 0 R z at R Rg , z 0 – expand in Taylor series about (x,z)=(0,0) 2 2 eff 2 eff 1 1 eff 2 x 2 2 2 R z ( R ,0) – then 2 x2 / 2 2 z 2 / 2 g z2 ( Rg , 0 ) • When the star is close to z=0 the effective potential can be expanded to give eff 1 2 2 eff ( R, z ) eff ( R,0) z z 2 z 2 z Zero, changes sign above/below z=0 equatorial plane. 1 2 2 eff ( R, z ) eff ( R,0) z ....... 2 z 2 z So, the orbit is oscillating in the z direction. 2 epicyclic approximation • ignore all higher / cross terms: • EoM: harmonic oscillators – epiclyclic frequency : R 2 R , – vertical frequency : eff ( R, z ) – with 2 2 3 L z 2 ; 2 4 Rg R ( Rg ,0) and Lz 2 2R2 z 2 z epicycles cont. • using the circular frequency , given by 1 Lz 2 ( R) 4 R R ( R ,0 ) R 2 2 3 L – so that 2 z R 2 3 2 R R R 4 R R 2 4 2 R R disk galaxy: ~ constant near centre g – so ~ 2 ~ declines with R, Vrot » slower than Keplerian R-3/2 » lower limit is ~ in general < 2 R Example:Oort’s constants near Sun A R ; R R0 1 2 1 B 2 R R R0 – where R0 is the galacto-centric distance • then 2 = -4A(A-B) + 4(A-B)2 = -4B(A-B) = -4B0 • Obs. A = 14.5 km/s /kpc and B=-12 km/s /kpc B 0 0 2 A B 1.3 0.2 the sun makes 1.3 oscillations in the radial direction per azimuthal (2) orbit – epicyclic approximation not valid for z-motions when |z|>300 pc General Jeans Equations • CBE of the phase space density f is eq. of 7 variables and hence generally difficult to 3 f f f solve vi 0 t i 1 xi xi vi • Gain insights by taking moments of the CBE f 3 f f 3 3 t Ud v vi xi Ud v xi vi Ud v 0 • where integrate over all possible velocities – U=1, vj, vjvk 1st Jeans (continuity) equation • define spatial density of stars n(x) n fd 3 v • and the mean stellar velocity v(x) 1 v i fvi d 3 v n • then our zeroth moment equation becomes n nv i 0 x i t 3rd Jeans Equation 2 vj n vj ij n nvi n t xi xi xi v 1 v v P t similar to the Euler equation for a fluid flow: – last term of RHS represents pressure force JEANS EQUATION for oblate rotator : a steady-state axisymmetrical system in which ASSUME ij2 is isotropic and the only streaming motion is azimuthal rotation: 1 ( ) z z 2 1 ( ) v rot R R R 2 2 • The velocity dispersions in this case are given by: 2 2 ( R, z ) vr2 vz2 v2 vrot 2 since (v vrot ) v 2v vrot v 2 2 2 2 rot but v v rot since apart from v rot it' s isotropic 2 2 v2 vrot • If we know the forms of (R,z) and (R,z) then at any radius R we may integrate the Jeans equation in the z direction to obtain 2. Obtaining 2 ( R, z ) dz z z 2 1 Inserting this into the jeans equation in the R direction gives: v 2 rot R R dz R R z z th 20 • orbits Lec Applications of the Jeans Equations • I. The mass density in the solar neighbourhood • Using velocity and density distribution perpendicular to the Galactic disc – cylindrical coordinates. – Ignore R dependence E.g.: Total Mass of spherical Milky WAY • Motions of globular clusters and satellite galaxies around 100kpc of MW – Need n(r), vr2, to find M(r), including dark halo • Several attempts all suffer from problem of small numbers N ~ 15 • For the isotropic case, Little and Tremaine TOTAL mass of 2.4 (+1.3, -0,7) 1011 Msol • 3 times the disc need DM Power-law model of Milky Way 0, v 2 vr 2 • Isotropic orbits: 1, v 2 0 • Radial orbits • If we assume a power law for the density distribution n r , M (r ) r – E.g. Flat rotation a=1, Self-grav gamma=2, Radial anis. 0. – E.g., Point mass a=0, Tracer gamma=3.5, Isotro 2 2 M 4 . 5 ( v v r )r / G 0 Mass of the Milky Way: point-mass potential model We find GM d r vcirc 2 vr 2 p r dr vr 2 1/ r d ln n d ln vr 2 p 2 1 2 d ln r d ln r For =3.5, and isotropic tracer =0, we have p 4.5 M pvr 2 r / G pv2 r / G Vertical Jeans equation • Small z/R in the solar neighbourhood, R~8.5 kpc, |z|< 1kpc, R-dependence neglected. • Hence, reduces to vertical hydrostatic eq.: 2 nv z n z z mass density in solar neighbourhood • Drop R, theta in Poisson’s equation in cylindrical coordinates: 1 1 2 2 4G R R R R R 2 z 2 2 2 4G 2 z local mass density = 0 Finally - 1 2 n v z / 4G z n z • all quantities on the LHS are, in principle, determinable from observations. RHS Known as the Oort limit. • Uncertain due to double differentiation! local mass density • Don’t need to calculate for all stars – just a well defined population (ie G stars, BDs etc) – test particles (don’t need all the mass to test potential) • Procedure – determine the number density n, and the mean square vertical velocity, vz2, the variance of the square of the velocity dispersion in the solar neighbourhood. local mass density • > 1000 stars required • Oort : 0 = 0.15 Msol pc-3 • K dwarf stars (Kuijken and Gilmore 1989) – MNRAS 239, 651 • Dynamical mass density of 0 = 0.11 Msol pc-3 • also done with F stars (Knude 1994) • Observed mass density of stars plus interstellar gas within a 20 pc radius is 0 = 0.10 Msol pc-3 • can get better estimate of surface density • out to 700 pc S ~ 90 Msol pc-2 • from rotation curve Srot ~ 200 Msol pc-2 • Question 5: In potential ( R, z ) 0.5v02 ln( R 2 2 z 2 ) v02 (1 ( R 2 z 2 ) / 1kpc2 ) 1/ 2 , due to dark halo (1st term) and stars (2nd term), assume V0 100km / s. calculate total mass of stars and star density s ( R, z ). What is the dark halo density on equator (R, z) (1kpc,0)? 0 Calculate stellar s (1kpc,0) 2 s (1kpc, z ) z 2 Show isotropic rotator have unphysical vrot (1kpc,0) dz Helpful Math/Approximations (To be shown at AS4021 exam) • Convenient Units • Gravitational Constant • Laplacian operator in various coordinates • Phase Space Density f(x,v) relation with the mass in a small position cube and velocity cube 1km/s 1kpc 1pc 1Myr 1Gyr G 4 10 3 pc (km/s) 2 M - 1 sun G 4 10 6 kpc (km/s) 2 M - 1 sun 2 2 2 (rectangul ar) z y x R - 1 ( R ) 2 R - 2 2 (cylindric al) z R R 2 (r 2 ) (sin ) r (spherical ) r 2 2 2 2 r sin r r sin dM f ( x, v)dx 3dv3 th 21 • orbits Lec: MOND