Basic Stellar Dynamics James Binney University of Oxford Saas Fee, January 2019
Equilibria & orbits Actions Actions & Jeans theorem Computing actions DFs for spheroids Secular evolution Spiral structure
Basic principle of stellar dynamics We identify ◮ A steady state ◮ a high degree of symmetry (often axisymmetry)
Basic principle of stellar dynamics We identify ◮ A steady state ◮ a high degree of symmetry (often axisymmetry) ◮ Fluctuations drive the system through a succession of equilibria
Equilibrium models ◮ Most orbits in Φ( R, z ) prove quasiperiodic: x ( t ) = � n X n cos( n · Ω t )
Equilibrium models ◮ Most orbits in Φ( R, z ) prove quasiperiodic: x ( t ) = � n X n cos( n · Ω t ) ◮ ⇒ ∃ 3 constants of motion I ( x , v ) = const on orbit
Equilibrium models ◮ Most orbits in Φ( R, z ) prove quasiperiodic: x ( t ) = � n X n cos( n · Ω t ) ◮ ⇒ ∃ 3 constants of motion I ( x , v ) = const on orbit ◮ E, L z are 2 consts but there must exist a third (Arnold 1978)
Equilibrium models ◮ Most orbits in Φ( R, z ) prove quasiperiodic: x ( t ) = � n X n cos( n · Ω t ) ◮ ⇒ ∃ 3 constants of motion I ( x , v ) = const on orbit ◮ E, L z are 2 consts but there must exist a third (Arnold 1978) ◮ 3 constraints I i = const confine orbit to 3-surface in 6d phase space. ◮ surface proves to be a 3-torus (Arnold 1978) ◮ A 3-torus is a room with opposite walls identified and ceiling identified with the floor
Equilibrium models ◮ Most orbits in Φ( R, z ) prove quasiperiodic: x ( t ) = � n X n cos( n · Ω t ) ◮ ⇒ ∃ 3 constants of motion I ( x , v ) = const on orbit ◮ E, L z are 2 consts but there must exist a third (Arnold 1978) ◮ 3 constraints I i = const confine orbit to 3-surface in 6d phase space. ◮ surface proves to be a 3-torus (Arnold 1978) ◮ A 3-torus is a room with opposite walls identified and ceiling identified with the floor ◮ Since g ( I 1 , I 2 , I 3 ) = const we have choices
Actions ◮ Best to use actions J r , J z , and J φ = L z which are line integrals � J i = 1 p · d q 2 π γ i around closed paths γ i around the 3-torus ◮ In the room analogy, γ 1 runs from a point on the front wall to the corresponding point on the back wall, γ 2 runs between points on the left & right walls, γ 3 runs between floor and ceiling ◮ They have simple physical interpretations ◮ J r quantifies radial oscillations ◮ J z quantifies oscillations above & below the equatorial plane ◮ J φ = L z is cpt of angular momentum along symmetry axis
Properties of actions ◮ Adiabatic invariance : if Φ evolves slowly (accreting gas), orbit & its torus evolve but its J i = const
Properties of actions ◮ Adiabatic invariance : if Φ evolves slowly (accreting gas), orbit & its torus evolve but its J i = const ◮ J i can be complemented by canonically conjugate coordinate θ i so ( θ , J ) form set of canonical coordinates ◮ So d 3 x d 3 v = d 3 θ d 3 J ◮ Range θ i = (0 , 2 π ): x ( θ 1 + 2 π, θ 2 , θ 3 ) = x ( θ 1 , θ 2 , θ 3 ) � orbits d 3 x d 3 v = (2 π ) 3 � ◮ ⇒ d 3 J ◮ 3d action space a true map of 6d phase space
AA coords ◮ Hamilton’s eqs are trivial J = − ∂H 0 = ˙ ⇒ H ( J ) ∂ θ ◮ So θ = ∂H ˙ ∂ J = Ω ( J ) (consts) ◮ ⇒ θ ( t ) = θ (0) + Ω t (quasi-periodicity)
Jeans theorem ◮ Let f ( x , v ) d 3 x d 3 v be probability that a randomly chosen particle lies near ( x , v )
Jeans theorem ◮ Let f ( x , v ) d 3 x d 3 v be probability that a randomly chosen particle lies near ( x , v ) ◮ Liouville: d f d t = 0 ⇒ f is const of motion ◮ ⇒ f ( J )
Jeans theorem ◮ Let f ( x , v ) d 3 x d 3 v be probability that a randomly chosen particle lies near ( x , v ) ◮ Liouville: d f d t = 0 ⇒ f is const of motion ◮ ⇒ f ( J ) � d 3 x d 3 v f ( x , v ) = (2 π ) 3 � ◮ P orbits = d 3 J f ( J )
Jeans theorem ◮ Let f ( x , v ) d 3 x d 3 v be probability that a randomly chosen particle lies near ( x , v ) ◮ Liouville: d f d t = 0 ⇒ f is const of motion ◮ ⇒ f ( J ) � d 3 x d 3 v f ( x , v ) = (2 π ) 3 � ◮ P orbits = d 3 J f ( J ) ◮ Galaxy is an assembly of orbits ↔ points in 3d J-space
Jeans theorem ◮ Let f ( x , v ) d 3 x d 3 v be probability that a randomly chosen particle lies near ( x , v ) ◮ Liouville: d f d t = 0 ⇒ f is const of motion ◮ ⇒ f ( J ) � d 3 x d 3 v f ( x , v ) = (2 π ) 3 � ◮ P orbits = d 3 J f ( J ) ◮ Galaxy is an assembly of orbits ↔ points in 3d J-space ◮ (2 π ) 3 f ( J ) = density of stars in J-space � ◮ Mass = (2 π ) 3 m d 3 J f ( J ) follows immediately from f ( J ) ◮ Equilibrium Galaxy comprises f α ( J ) for each population α
Computing actions For decades the use of actions limited by difficulty of computing them ◮ Now θ ( x , v ) and J ( θ , v ) from St¨ ackel Fudge (Binney 2012) ◮ x ( θ , J ) and v ( θ , J ) from Torus Mapper (Binney & McMillan 2016) ◮ State-of-the-art Python/C++ implementations in AGAMA (Vasiliev 2018)
Computing actions For decades the use of actions limited by difficulty of computing them ◮ Now θ ( x , v ) and J ( θ , v ) from St¨ ackel Fudge (Binney 2012) ◮ x ( θ , J ) and v ( θ , J ) from Torus Mapper (Binney & McMillan 2016) ◮ State-of-the-art Python/C++ implementations in AGAMA (Vasiliev 2018) ◮ St¨ ackel Fudge related to older technology of fitting St¨ ackel Φ (Dejonghe & de Zeeuw 1988, Sanders 2012) but more accurate (Vasiliev 2018) ◮ St¨ ackel Fudge has been extended to triaxial Φ in absence of figure rotation (Sanders & Binney 2015) but not to case of figure rotation
Computing actions For decades the use of actions limited by difficulty of computing them ◮ Now θ ( x , v ) and J ( θ , v ) from St¨ ackel Fudge (Binney 2012) ◮ x ( θ , J ) and v ( θ , J ) from Torus Mapper (Binney & McMillan 2016) ◮ State-of-the-art Python/C++ implementations in AGAMA (Vasiliev 2018) ◮ St¨ ackel Fudge related to older technology of fitting St¨ ackel Φ (Dejonghe & de Zeeuw 1988, Sanders 2012) but more accurate (Vasiliev 2018) ◮ St¨ ackel Fudge has been extended to triaxial Φ in absence of figure rotation (Sanders & Binney 2015) but not to case of figure rotation ◮ TM extended to rotating bars by p-theory (Binney 2018) ◮ TM is a systematic approximation, but SF is not; unfortunately x ( θ , J ) less useful than J ( x , v )
Models from analytic DFs Role of models ◮ A model interprets observations in terms of what’s out there ◮ Discovering what’s there must preceed deducing how it got there ◮ So we need an apparatus that fits models to data independent of a creation myth
What must be included ◮ Bulge ◮ α -rich and α -poor discs ◮ stellar halo (impacts kinematics) ◮ dark halo (impacts dynamics) ◮ gas disc
From DF to observables ◮ Choose f α ( J ) for each component ◮ Guess Φ( x ) ackel Fudge to evaluate ρ = � � d 3 v f ( J ) on a grid ◮ Use St¨ α in x ◮ Solve for Φ( x ) and re-determine ρ ( x ) ◮ Converges in ≤ 5 cycles (5 min on my laptop) ◮ Hack your code from AGAMA example self consistent model.cpp
What you can do with the model ◮ MC sample initial conditions for N-body model ◮ Model starts uniquely close to equilibrium ◮ Ideal for study of perturbations: spirals, warps, . . . ◮ Predict v -distribution of DM particles for direct-detection expts ◮ Cross-section for detection expected to be speed-dependent ◮ MC sample with survey selection function to make mock catalogue ◮ Compute at x (i) star counts and (ii) stellar v -distributions for surveys ◮ Use f, Φ to understand dis-equilibria (streams, moving groups, Antoja spiral)
Principles of picking a DF ◮ Each component has its own DF ◮ DF has to arrange stars in 3d J-space J r ↔ σ R , σ φ f ( J r , J z , J φ ) J z ↔ σ z J φ ↔ Σ( R ) , v φ ( R ) ◮ Only 2 independent dispersions ◮ In case of disc ◮ σ z , z 0 and ρ DM ( R, 0) coupled ◮ f = f + + f − and only f + contributes to ρ
Spheroids: Posti DFs ◮ NFW, Hernquist, etc models generated by 2-power DFs (Posti + 2015) [1 + J 0 /h ( J )] a ρ 0 f ( J ) = M 0 ρ ( r ) = ↔ ( r/r 0 ) α (1 + r/r 0 ) β − α J 3 [1 + g ( J ) /J 0 ] b 0 with a = (6 − α ) / (4 − α ) b = 2 β − 3 and g, h ( J ) = J r + k φ | J φ | + k z J z 8 6 f ( J ) Hernquist f ( J ) NFW 6 NFW Hernquist 4 4 2 2 log 10 ρ log 10 ρ 0 0 −2 −4 −2 −6 −4 −8 −10 −6 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 log 10 r log 10 r
Discs: quasi-isothermal DF ◮ Introduced by Binney 2010 & Binney & McMillan 2011 ◮ Motivated by Schwarzschild DF (epicycle approx J = H/ Ω) E = E c + E R + E z J 2 φ E c = + Φ( R c ) , E R = κJ r , E z = νJ z 2 R 2 c f Schw ( J φ , E r , E z ) = f φ ( R c ) exp( − E r /σ 2 R ) exp( − E z /σ 2 z ) ◮ Quasi-isothermal DF is f ( J ) = f φ ( R c ) exp( − κJ r /σ 2 R ) exp( − νJ z /σ 2 z ) with κ ( R c ) and ν ( R c )
Discs: quasi-isothermal DF ◮ Quasi-isothermal DF does a superb job in extended solar nhd (e.g. Binney + 2014) ◮ But ◮ Behaves inappropriately when J φ → 0 through R c → 0 ◮ Undefined for some Φ( x ) because κ 2 can go -ve ◮ These flaws apparent when solving for self-consistent Φ
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