1. Made with OpenOffice.org 1 Inferring the dynamics of the Milky Way Jo Bovy New York University MPE, Garching, 2009/08/07

2. Made with OpenOffice.org 2 Introduction >>Dynamics of the MW : - M halo - M BH - Σ disk (R) - ρ DM (r) -... >>Why? - MW in a cosmological context - M BH : PPN, DM, M-σ relation,... >> Why think hard? - x,v are initial conditions for any potential --> no constraint without assumptions about the DF - Current data set are at the level in which detailed MW inference becomes important - Gaia: positions & proper motions for 10 9 stars

3. Made with OpenOffice.org 3 >> Most models assume a random distribution of orbital phases: - Jeans modeling: e.g., central parsec: Eckart & Genzel97, Schoedel+09 - f(x,v) == f(I): e.g., Σ(R 0 ): Kuijken & Gilmore89 - Oort constants: Feast & Whitelock97 - Escape velocity: Smith+07 >> data binning, maximum-likelihood, fixed or best fit DF

4. Made with OpenOffice.org 4 >>Going to larger data sets and more precise modeling of MW dynamics, we need to: - Model the DF in more general, assumption-free way - lock into informative features either of the full DF (e.g., Koposov+09, Eyre&Binney09) or in the space of actions (e.g., Beloborodov+06) - Marginalize over uncertainties on the inferred DF - Combine different measurements and use prior information

5. Made with OpenOffice.org 5 >>Going to larger data sets and more precise modeling of MW dynamics, we need to: - Model the DF in more general, assumption-free way - lock into informative features either of the full DF (e.g., Koposov+09, Eyre&Binney09) or in the space of actions (e.g., Beloborodov+06) - Marginalize over uncertainties on the inferred DF - Combine different measurements and use prior information >>Luckily, MW dynamics is mature enough to be fully ''Bayesian'': - We know that the general model is good (Φ and DFs) - DF is generative model that specifies the likelihood (and we're interested in it!)

6. Made with OpenOffice.org 6 Outline >>Warm-up problem: Inferring the force law in the Solar System from a small, precise snapshot of its kinematics: We infer the amplitude and radial dependence of the gravitational force law in the Solar System to a reasonable accuracy by inferring and marginalizing over the DF But we lose a lot of accuracy in inferring the DF >>Inferring the circular velocity of the MW from the kinematics of masers occurring in high-mass star-forming regions: As the DF of the masers is unknown, we propose a general model, infer, and marginalize over it; we find that the masers are not very constraining and are consistent with other recent measurements of V c

7. Made with OpenOffice.org 7 The force law in the Solar System Given the positions and velocities of the 8 planets, can we infer the force law (amplitude A and radial dependence r -α ): a(r) = -A (r/1 AU)^(-α) [Given the positions/proper motions/los velocities of a set of stars, can we infer the dynamics of the MW/central parsec/...] JB, Hogg, & Murray09

8. Made with OpenOffice.org 8 Starting point? E.g., virial relation for each planet: = (1-α)/2 Data: JPL ephemerides (x,v) for the 8 planets on 2009 April 1.0 Accurate to one part in 10 -6 -10 -8

9. Made with OpenOffice.org 9 Starting point? E.g., virial relation for each planet: = (1-α)/2 Lines cross near a single point --> system seems virialized Data: JPL ephemerides (x,v) for the 8 planets on 2009 April 1.0 Accurate to one part in 10 -6 -10 -8

10. Made with OpenOffice.org 10 Orbital roulette >>Basic idea: distribution of angles should be flat when we use the right dynamical parameters---if the force is too large/small each planet will be at aphelion/perihelion (Beloborodov & Levin 04) >>Thus, calculate all of the radial angles, and use a statistic to test whether the distribution is consistent with being flat: - E.g., mean angle = π/2; - KS - Kuiper (test on the circle) >> Also test that there is no correlation between the angles and the energies (Beloborodov & Levin04)

11. Made with OpenOffice.org 11 Angles (JB, Hogg, & Murray09)

12. Made with OpenOffice.org 12 Angles

13. Made with OpenOffice.org 13 Different frequentist tests

14. Made with OpenOffice.org 14 These are tricky to combine:

15. Made with OpenOffice.org 15 Orbital roulette works fairly well, but >>It's hard to marginalize over nuisance parameters; >>It does not teach you much about the distribution function; >>It cannot handle large uncertainties and/or missing data well; >>It only uses a very crude model of the data, which does not allow it to find structure in the DF ---> A more Bayesian approach is called for...

16. Made with OpenOffice.org 16 Our approach and model >>The likelihood of the model parameters (A,α) given the data is given by the DF: f(x,v) >>We assume that the system is angle-mixed and non-resonant ----> f(x,v) == f(I) >>The system is spherically symmetric: ----> f(I) ∝ f(E,e) >> We model f(E,e) as a tophat in ln E, and a tophat in e ----> 4 extra parameters in our model: ln E a, ln E b, e a, e b (bounds of the tophat) >>We marginalize over these last 4 parameters (analytically)

17. Made with OpenOffice.org 17 f(I) = J(ln E,e; x,v) f(ln E,e) alpha=2:

18. Made with OpenOffice.org 18 f(I) = J(ln E,e; x,v) f(ln E,e) alpha=2:

19. Made with OpenOffice.org 19 Results Marginalized over the DF+ marginalized over A

20. Made with OpenOffice.org 20 >>We infer the force law to good accuracy for precise measurements of only 8 planets >>Much of the precision of the measurements (10 -6 -10 -8 ) goes to inferring the DF >>This approach can be generalized to missing data problem (M BH ) and (part of) the full Gaia problem Solar System: conclusions

21. Made with OpenOffice.org 21 The circular velocity of the MW >>V c important for 1) correcting observed velocities of stars and galaxies for the motion of the Sun around the GC 2) MW dynamics >>Recently, Reid+09 measured parallaxes, proper motions, and los velocities for 18 masers occuring in high-mass star-forming regions in the Galactic disk Reid+09 inferred a circular velocity of 254 +/- 16 km/s!! JB, Hogg, & Rix09, ApJ submitted

22. Made with OpenOffice.org 22 The data R 0 = 8.5 kpc, V c = 220 km/sR 0 = 8.4 kpc, V c = 254 km/s Reid+09

23. Made with OpenOffice.org 23 The distribution function of the masers >>The distribution function of masers is unknown, thus, we need to infer it >>The masers do not constitute a spatially uniform sample, therefore, we focus on the distribution of the velocities conditioned on the position of the masers: Likelihood = f(v|x) >>We assume that f(v|x) = f(v peculiar == v-v lsr (R) φ), and that this distribution is Gaussian: f(v pec ) = N(m,σ) (in cylindrical coordinates) (The Reid+09 model assumed, basically, f(v pec = δ(v pec - m)

24. Made with OpenOffice.org 24 Further model specification >>Our model contains 14 model parameters: -The distance to the Galactic Center; - The proper motion of Sgr A*; - The three components of the Solar motion; - The mean offset and the velocity dispersion tensor of the masers >>The likelihood given a single data point is obtained by convolving the intrinsic Gaussian with the observational uncertainties: >>Combining all of the likelihoods and adding in prior information, we find the posterior distribution for all 14 of the model parameters:

25. Made with OpenOffice.org 25 Prior information >>On the distance to the Galactic Center: ½ * Gillessen+09 + ½ * Ghez+08 >>On the proper motion of Sgr A*: Reid&Brunthaler04 measurement 30.24 +/- 0.12 km/s/kpc >>On the Solar motion: Hipparcos measurement (Hogg+05) U,V,W = (10.1,4.0,6.7)+/-(0.5,0.8,0.2) km/s >>Flat priors on the mean m and velocity dispersion σ of the masers

26. Made with OpenOffice.org 26 MCMC exploration of the posterior distribution >>14 model parameters is a lot: MCMC algorithm returns samples from the posterior probability distribution >> We use the Metropolis-Hastings algorithm: 1) Propose a step in the space of model parameters from a proposal distribution Q(x';x_current) 2) Compute 3) Draw u uniform in [0,1] 4) Accept the new parameters if a > u, otherwise add the old parameters again to the sampling >>The list of model parameter values thus obtained is a sampling from the posterior

27. Made with OpenOffice.org 27 Results Thick gray: prior Black: posterior distribution Thin gray: posterior distribution when dropping the prior on the proper motion of Sgr A* Marginalized over: - DF parameters - R 0 - Sgr A* - Solar motion

28. Made with OpenOffice.org 28 Results Thick gray: prior Black: posterior distribution Marginalized over: - DF parameters - Sgr A* - Solar motion

29. Made with OpenOffice.org 29 Results: the DF of the masers -

30. Made with OpenOffice.org 30 Solar motion? >>It has been claimed recently that instead of the masers having a mean offset from Galactic rotation, the Solar motion in the direction of Galactic rotation could be much larger than currently thought (McMillan & Binney09) V solar =~ 5 km/s (Hipparcos) ----> V solar =~ 20 km/s >>If we set the Solar motion free in our analysis, the best fit V c does not change (its uncertainty does increase to the uncertainty in the Solar motion) >>But the velocity distribution of nearby stars does seem to break the assumptions going into the measurement of the Solar motion (axisymmetry and phase-mixing)

31. Made with OpenOffice.org 31 Velocity distribution of nearby stars JB, Hogg, & Roweis09: a:ApJ 700, 1794 b:arXiv:0905:2979

32. Made with OpenOffice.org 32 Excluding stars in clumps Asymmetric drift: Excluding stars with probability of being in a clump p MG Uncertainty in the Solar motion is larger than thought, but no indication of V solar =~ 20 km/s JB&Hogg, in prep.

33. Made with OpenOffice.org 33 Conclusion >>Inferring the DF simultaneously with the dynamics: (1) is possible; (2) is necessary to avoid biases (3) leads to precise and robust inferences of the dynamics (marginalization over uncertainties is key) >>V c from MW masers: V c = 246 +/- 30 km/s as opposed to 254 +/-16 km/s (from the same data set!); >>V c is consistent with other recent measurements: combined estimate: V c = 236 +/- 11 km/s >>But masers are bad because all the information in them goes to inferring their distribution function

34. Made with OpenOffice.org 34 Conclusion >>Inferring the DF simultaneously with the dynamics: (1) is possible; (2) is necessary to avoid biases (3) leads to precise and robust inferences of the dynamics (marginalization over uncertainties is key) >>V_c from MW masers: V_c = 246 +/- 30 km/s as opposed to 254 +/- 16 km/s (from the same data set!); >>V_c is consistent with other recent measurements: combined estimate: V_c = 236 +/- 11 km/s >>But masers are bad because all the information in them goes to inferring their distribution function >>The force law in the Solar System ∝ 1/r^2!