Inferring the mass of the Dark Matter Halo from Globular Cluster 3D Kinematics

Transcription

1 Inferring the mass of the Dark Matter Halo from Globular Cluster 3D Kinematics Gwendolyn Eadie, PhD Candidate Supervisor: William Harris Aaron Springford (Queen s University) SCMA VI - Carnegie Mellon University, Pittsburgh, PA - June 10, 2016

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5 The Bayesian Paradigm Thomas Bayes ( )

6 Bayes Theorem Posterior Likelihood Prior Distribution model parameters data

7 Bayes Theorem Posterior Likelihood Prior Distribution Probability distribution function of model parameters, given the data, the model, and prior assumptions.

8 Bayes Theorem Posterior Likelihood Prior Distribution data model parameters Assuming data y are independent: The astrophysics part

9 Probability Distribution Function (Binney & Tremaine, Galactic Dynamics, 2008, Cuddeford 1991, Little & Tremaine 1987) vt r HST The astrophysics part vr

10 The astrophysics part Probability Distribution Function (Binney & Tremaine, Galactic Dynamics, 2008, Cuddeford 1991, Little & Tremaine 1987) Specific Energy: Specific Angular Momentum: vr vt r

11 Bayes Theorem Posterior Likelihood Prior Distribution The astrophysics part Probability Distribution Function (DF) (Binney & Tremaine, Galactic Dynamics, 2008, Cuddeford 1991, Little & Tremaine 1987) data model parameters Assuming data y are independent: Assuming tracers are independent:

12 The astrophysics part Probability Distribution Function (Binney & Tremaine, Galactic Dynamics, 2008, Cuddeford 1991, Little & Tremaine 1987) Specific Energy: Specific Angular Momentum: vr vt r c i r t n e e c to nc c la ere! a f me G e R ra F

13 vlos The astrophysics part Probability Distribution Function (Binney & Tremaine, Galactic Dynamics, 2008, Cuddeford 1991, Little & Tremaine 1987) Specific Energy: Specific Angular Momentum: μ vr vt r c i r t n e e c o nc o i l ere! e H ef e R ram F

14 Not really a problem vlos vr μ Heliocentric solar motion vt D rotation of disk RA & Declination distance Galactocentric Good guide: Johnson & Soderblom (1987), Astronomical Journal, 23:4.

15 except that Globular Cluster 3D Kinematics are Incomplete GC r μ vlos Eadie & Harris (2016), submitted to ApJ

16 Our solution to include incomplete data: 1. Make approximation vr vlos when appropriate ɣ vlos vr

17 Our solution to include incomplete data: 1. Make approximation vr vlos when appropriate 2. Sample missing vt s as nuisance parameters in the Markov chain BONUS: Helps to break mass- velocity anisotropy degeneracy Simple Model! Past work with this method: Eadie, Harris, & Widrow (2015), ApJ 806, 54 Eadie, G. 2014, JSM Proceedings, ASA, Section 175. Eadie, G. MSc Thesis, Queen s University (2013).

18 Gravitational Potential of Dark Matter Halo Recent Work Model where the Dark Matter and Visible Matter have different radial profiles (Evans et al 1997, Deason et al 2011, 2012) Density Profile of Tracers

19 Cumulative Mass Profile Recent Work Model where the Dark Matter and Visible Matter have different radial profiles (Evans et al 1997, Deason et al 2011, 2012) Isothermal Sphere: 0 Keplerian Case: 1

20 Distribution Function (DF) (Evans et al 1997, Deason et al 2011, 2012)

21 Distribution Function (DF) Velocity Anisotropy Parameter

22 What do we want to know?

23 Group 1 Eadie & Harris (2016), submitted to ApJ

24 Group 1

25 Group 2

26 Group 3

27 Scenario X Eadie & Harris (2016), submitted to ApJ

28 Scenario X s cumulative mass profile Virial Mass: 7.04 x 1011 Msun 50% cred. (6.23, 7.76) Eadie & Harris (2016), submitted to ApJ

29 Latest Results Hierarchical Bayes --- including measurement uncertainties

30 Setting up the Hierarchical Bayesian Model Measurements are inherently uncertain True values of positions and velocities are unknown Uncertainties are well understood

31 Data drawn from a distribution Parameter (true value, unknown) r ~ N ( r, r ) Data Uncertainty (fixed)

32 Probability distribution Parameter (true value, unknown) p (r r, r ) Data Uncertainty (fixed)

33 Hierarchical Bayesian Model Data Uncertainty (fixed) Parameter

34 Hierarchical Bayesian Model Data Parameter Uncertainty (fixed)

35 Hierarchical Bayesian Model Data Parameter Uncertainty (fixed)

36 Hierarchical Bayesian Model distance line-of-sight velocity proper motion (DEC) proper motion (RA)

37 Hierarchical Bayesian Model

38 Hierarchical Bayesian Model

39 Hierarchical Bayesian Model

40 Hierarchical Bayesian Model Hyperpriors on model parameters Eadie, Harris, and Springford (2016) in prep Eadie, Harris, Widrow, and Springford (2015) IAU Proceedings Eadie, Harris, & Springford (2015) JSM Proceedings

41 Scenario X (new) same data (89 GCs) but including uncertainties Virial Mass: 5.98 x 1011 Msun 50% cred. (5.39, 6.52) Eadie, Harris, and Springford (2016), in prep

42 Conclusions Mass profile credible regions easy to compare to other studies Includes incomplete and complete data simultaneously Mass of the Milky Way is light under power-law model Mvir= ( ) x 1011 Msun (95% credible region) Future Work & Challenges Other galaxies where 3D motions are never known Implement more realistic, physical models First GAIA data release coming soon!

43 Thank #figureskatingastrophysicist Eadie, Harris, & Springford, in prep. Eadie & Harris (2016), submitted to ApJ. Eadie, Harris, Widrow (2015), ApJ 806, 54. Eadie, Harris, Widrow & Springford (2015), IAU Proceedings, Symposium 317. Eadie, Harris, Springford (2015), JSM Proceedings, Section on Physical & Engineering Sciences.