1. Inferring dynamics from kinematic snapshots Jo Bovy Distribution of Mass in the Milky Way Galaxy Leiden, 07/14/09 New York University The importance of the distribution function

2. Introduction Inference : p(data | model) p(model) p(model | data) = -------------------------------- p(data) Dynamics : 'model' = potential Kinematic snapshot: data = x, v ---> p(data | model) = p(data) ---> p(data | model) p(model) p(model | data) = -------------------------------- = p(model) !!! p(data) Distribution function: p(data | model) = f(x,v) ---> We need to make assumptions about the DF for p(data | model) ≠ p(data)

3. Assumptions about the DF Assumptions such as: - everything bound (e.g., local escape velocity, RAVE, Smith et al. 2007) - Angle-mixed, steady-state tracer population (e.g., satellite kinematicsmass constraints Little & Tremaine 1987, Galactic center 'non-orbits', Schwarzschild modeling) - Streams (nearly) trace out orbits (e.g., Eyre & Binney 2009, Koposov et al. 2009) + assumptions about the potential: axisymmetry, time-independence,... ---> large, often systematic, uncertainties

4. Outline 1) An example: the velocity distribution of nearby stars + how to infer distribution functions from noisy, heterogeneous, and incomplete data 2) The uncertainty in the Solar motion wrt the LSR due to the DF 3) When the DF is unknown: inferring both dynamics and the DF for Galactic masers 4) Going beyond the DF: marginalizing out the DF

5. The velocity distribution of stars near the Sun Toward the GC In the direction of Gal. rotation Toward the GC Toward the NGP Gal. rotation direction JB, Hogg, & Roweis (2009)

6. Inferring distributions functions: Extreme deconvolution JB, Hogg, & Roweis (2009) arXiv:0905.2979 Density estimation in the presence of noisy, heterogeneous, and incomplete data: -> Underlying distribution, not observed distribution -> Extreme deconvolution: each sample is drawn from a different distribution -> Ability to handle arbitrary uncertainties deals with incomplete data Underlying distribution= sum over K Gaussian distributions: -> Fit for amplitudes, means, and covariances -> Observations : Noisy projections of the true values -> L(model)= Π i p(data|model) -> p(data|model) = (Σ Gaussians) * (Noise) -> Optimize this scalar objective function

7. Model selection: how many components are necessary? More components will provide a better fit --> problem of overfitting We can use internal model selection tests: - leave-one-out cross validation -... or external model selection tests: - predict radial velocities of Geneva-Copenhagen Survey stars (Nordstrom et al. 2004) - The model that best predicts the radial velocities is the preferred model

8. Direction of Galactic rotation Toward the GC Radial velocity Predicted by model Observed 3 5 10 13 15 20 components # of components

9. One of the implications of this 'clumpy' velocity distribution The measurement of the Solar motion wrt the LSR assumes that the stars in the Solar neighborhood are in a fully angle-mixed, steady state (asymmetric drift) To say the least, it is not clear whether the clumps are due to unmixed phase- space structure, dynamical effects (resonances, non-axisymmetry, time- dependence), or caustics So how much does the Solar motion depend on what one assumes about the clump members?

10. Solar motion is established by looking at the motion of the Sun wrt to different subsamples selected by color (~ Dehnen & Binney 1998) Under the assumption of a axisymmetric Galaxy and an angle-mixed population of stars, the mean velocity of the different subsamples in the direction of Galactic rotation follows the asymmetric drift: mean velocity ∝ (velocity dispersion)^2 Extrapolating the asymmetric drift to zero velocity dispersion gives the Solar motion in the direction of Galactic rotation

11. All stars Stars with p(clump)>0.5 removed Toward the GC Direction of Gal. rotation Toward the NGP JB & Hogg, in prep. ∝ (velocity dispersion)^2

12. Another data set: parallaxes, proper motions and los velocities of high- mass star-forming regions (Reid et al. 2009) The full 6d phase-space information obtained for Galactic masers potentially holds much information about the dynamics and structure of the Galaxy, but mining it requires us to have some knowledge about the distribution function of masers. Non-trivial spatial selection function --> focus on the conditional velocity distribution: p(data | model) = p(v| x)= f_x (v) Since we do not know f_x(v) a priori, we need to infer it simultaneously with the dynamics

13. The conditional velocity distribution f_x (v) A Simple model for f_x (v): f_x (v) ≡ f( v – v_circ(x) φ^ ≡ v_peculiar) 1) f( v_peculiar) = δ ( v_peculiar – v') (Reid et al. 2009): Single offset from circular velocity 2) f( v_peculiar) = N ( v', V) : Gaussian with mean offset Thus, we need to infer the offset v' as well as the dispersion tensor V in addition to the dynamical parameters (R_0, Theta_0).

14. Reid et al. 2009 --> R_0= 8.5 kpc, v_circ= 220 km/s--> R_0= 8.4 kpc, v_circ= 254 km/s

15. 0.5 x Ghez et al. (2008) + 0.5 x Gillessen et al. (2009) Before the masers.... Prior information R₀R₀ + μ Sgr A* (Reid & Brunthaler 2005) + Solar motion (Hogg et al. 2005) Circular velocity at the Solar radius

16. After the masers.... fully marginalized posterior distributions JB, Rix, & Hogg, in prep. PRELIMINARY!! R₀R₀ Circular velocity at the Solar radius PRELIMINARY!! ---> After marginalizing over DF parameters v_circ is slightly reduced from its prior value

17. The inferred distribution function of the masers Toward the GCDirection of Gal. rotation Toward the NGP Trace(covariance) 0 (km/s)^21000 (km/s)^2 DispersionMean JB, Rix, & Hogg, in prep. PRELIMINARY!! Mean PRELIMINARY!! Toward the NGP Mean Toward the GC Dispersion 0 (km/s)^21000 (km/s)^2 Trace(covariance) Direction of Gal. rotation

18. Orbital phases of the masers Galactocentric distance (kpc) (r_ap, r, r_peri)/semi-major axis Most masers are near aphelion, moving inwards --> the masers are not at random orbital phases

19. Going beyond the DF: Marginalizing over the DF By marginalizing over the parameters of the DF we can integrate over the uncertainty in the DF: p(dynamics | data) = ∫ d(DF params) p(dynamics,DF params | data) Example: The gravitational force law in the Solar System from a snapshot of the kinematics of the 8 planets: -> We assume that the system is angle-mixed and non-resonant -> Jeans's theorem: DF ≡ DF( Integrals of the motion) -> Spherical symmetry: DF ≡ DF( Energy, eccentricity) Now assume a tophat in ln E and a tophat in e, and marginalize over the boundaries of the tophat JB, Murray, & Hogg (2009)

20. power-law exponent alpha Acceleration at the Earth's radius power-law exponent Model: a(r) = - A (r/1 AU)^{-alpha} r^ JB, Murray, & Hogg (2009) power-law exponent alpha

21. Dependence on the priors/parametrization of the DF Using bins in eccentricity^2 instead of the tophat in eccentricity: 3 bins256 bins131072 bins power-law exponent alpha Acceleration at the Earth's radius ---> structure in DF helps (i.e., e^2 ≃ 0 for many planets)

22. Conclusion Detailed inference of the DF is important in order to get the best, most assumption-free knowledge of the dynamics of the Galaxy from a kinematic snapshot We can obtain accurate and precise constraints on the potential by integrating over the uncertainties in the DF Discovering structure in the DF is key in order to get the best constraints Dfs can be messy and lead to significant systematic uncertainties: e.g., Solar Motion is uncertain at the few km/s level because of clumps in the velocity distribution With more general accounting for the orbital DF of Galactic masers, the Reid et al. maser measurements seem to slightly lower the posterior estimate of v_circ (to 240 km/s) from its prior value

23. power-law exponent alpha Acceleration at the Earth's radius Virial considerations