1. Masers as Probes of Galactic Structure
Mark J. Reid
Harvard-Smithsonian Center for Astrophysics
Collaborators:
K. Menten, A. Brunthaler, K. Immer ,
Y. Choi, A. Sanna, B. Zhang (MPIfR)
X-W Zheng, Y. Xu, Y. Wu (Nanjing)
L. Moscadelli (Arcetri)
G. Moellenbrock (NRAO)
M. Honma, T. Hirota, M. Sato (NAOJ)
T. Dame (CfA)
A. Bartkiewicz (Torun)
K. Rygl (INAF, Rome)
K. Hachisuka (Shanghai)
Here to present preliminary results from a major effort, world-wide, to measure distances to SFRs using gold-standard method: trigonometric parallax
Goals to give solid distances to MSFRs….allowing accurate measures of mass, size and luminosity of constituents of SFRs
and to study the structure and dynamics of the Galaxy

2. What does the Milky Way look like?
Hipparcos range
Distances to stars long solved…even distances to galaxies solved using “standard candles” eg, Cepheids
But still we don’t even know what the Milky Way looks like…perhaps a mix of these two spirals.
(Spirals traced by newly formed, massive stars that are very hot 10’s of thousands K, ionize surrounding clouds of Hydrogen which glow blue)
Why? Distances are very great (Hipparcos surveyed just a very small volume: Solar Neighborhood)…cannot give Galactic structure
…Significant progress optically (even with future $1B space mission like GAIA) will be limited by dust extinction in the plane of the Galaxy.
GAIA range (± 10 to 20 mas); but cannot see through dust in Galactic plane
VLBI range (± 5 to 20 mas): can “see” through plane to massive star forming regions that trace spiral structure

3. Very Long Baseline Interferometry: VLBA, VERA & EVN
Fringe spacing (eg, VLBA):
qf~l/D ~ 1 cm / 8000 km = 250 mas
Centroid Precision:
0.5 qf / SNR ~ 10 mas
Systematics:
path length errors ~ 2 cm (~2 l)
shift position by ~ 2qf ~ 500 mas
Relative positions (to QSOs):
DQ ~ 1 deg (0.02 rad)
cancel systematics: DQ*2qf ~ 10 mas
Key is not optical light (absorbed by dust throughout M.W.), but other wave bands, eg, Radio
Explains why we can achieve ~0.01 mas accuracy
Radio waves “see” through galaxy
Can “synthesize” telescope the size of the Earth

4. Parallax Signatures What do parallax data look like?
Earth moves in near circular orbit; viewed from source, projects to ellipse on sky
Relative motion of Sun & Target gives proper motion
Combination of Pi and P.M. gives curliques on sky
Can separate E and N vs time;
Can plot with remove p.m.
If well designed expt; pi and p.m. uncorrelated
For Galactic sources, EW amplitude > NS amplitude…often pays to only go after the EW signature.

5. Orion Nebular Cluster Parallax
Orion is archetypal MSFR…extensively studied from radio – x-rays
T Tauri –like stars exhibit compact, non-thermal, radio continuum (from magnetic field activity from inner accretion disk)
3 stars vs 1 QSO overplotted…one star is distant companion of Theta-1A Orionis (Trapezium)
2% parallax measured; slightly better than Bowers measure of 1 star, and recently equalled by VERA…all agree well within joint errors
Orion 10% closer than had been assumed…changes young stellar luminosities by 20%, and hence YSO ages and masses
VLBA: P = 2.42 ± 0.04 mas
D = 414 ± 7 pc
VERA: D = 419 ± 6 pc
Menten, Reid, Forbrich & Brunthaler (2007)

6. Mapping the Milky Way 6.7/12.2 GHz CH3OH masers 22 GHz H2O masers
Key Project to map spiral structure of Milky Way
6.7/12.2 GHz CH3OH masers
22 GHz H2O masers
VLBA Key Science Project: hours over 5 years to measure hundreds of parallaxes/proper motions
Observations for ~70 masers started 2010/2011 recently completed

7. Parallax for Sgr B2(Middle) H2O masers
P = 129 ± 12 mas (D=7.8 ± 0.8 kpc)

8. Parallax for W 49N H2O masers
P = 82 ± 6 mas (D=12.2 ± 0.9 kpc)

9. Mapping Spiral Structure
Preliminary results of parallaxes
from VLBA, EVN & VERA:
Arms assigned by CO l-v plot
Tracing most spiral arms
Inner, bar-region is complicated
Background: artist conception by Robert Hurt (NASA: SSC)

10. Spiral Arm Pitch Angles
For a log-periodic spiral:
log( R / Rref ) = -( b – bref) tan y
Outer spiral arms: ~13˚ pitch angles
Inner arms may have smaller pitch
angels (need more observations)
Sun
Sources with uncertainty in log(R/kpc) < 0.1 and beta < 10 deg
Dashed lines correspond to perfect log-spirals with 16 degree pitch angles (no attempt to adjust pitch angle to optimize)
Undoubtedly more complicated than single, global pitch angle, but fairly clear that MW has 4 arms, not 2

11. Convert observations from Heliocentric to Galactocentric coordinates
Galactic Dynamics
Qo ~ 220 km/s
Vsun ~ 20 km/s
Vsun
Convert observations from Heliocentric to Galactocentric coordinates Qo l d Ro
VGC R
VHelio
Qo+Vsun
Why? …
Measure in Heliocentric frame Vhelio (ie, Vsource – Sun’s full motion)
To obtain VGC (in Galactocentric frame) , must add back To+Vsun:
Vsun directly measurable in Solar Neighborhood…but has held surprises recently
Theta_o more difficult but of great astrophysical interest

12. The Milky Way’s Rotation Curve
Q0 = 245 km/s
Blue points moved up 25 km/s
Q0 = 220 km/s
Theta_o not easy to measure since all observables are Heliocentric and have removed Theta_o from measurements
When one adds back a Theta_o one “basically gets one one puts back”…see two examples of RC for two Theta_o values.
In right panel, move the blue points up by the difference in Theta_o used (25 km/s). .. Mostly line up!
But, there are differences which give a handle on To (also, for parallax/p.m.’s can compare D vs To to help fit for To);
It takes the entire data set to solve for Theta_o!
Remember we have all phase space info…imagine how tricky if only have radial velocity and less reliable distances!
Controversial!

13. Modeling Parallax & Proper Motion Data
Data: have complete 3-D position and velocity information for each source: Independent variables: a, d Data to fit: p, ma, md, V Data uncertainties include: measurement errors source “noise” of 7 km/s per component (Virial motions in MSFR) Model: Galaxy with axially symmetric rotation: R0 Distance of Sun from G. C. Q0 Rotation speed of Galaxy at R0 ¶Q/¶R Derivative of Q with R: Q(R) º Q0 + ¶Q/¶R ( R – R0 ) Usun Solar motion toward G. C. Vsun “ “ in direction of Galactic rotation Wsun “ “ toward N. G. P. <Usrc> Average source peculiar motion toward G. C. <Vsrc> “ “ “ “ in direction of Galactic rotation
How we fit the data to a model of the Galaxy:
Every sources gives us 4 measurements to fit: parallax, 2 pm’s, Vlsr
In interest of keeping simple now, only use two parameters for Rotation Curve: Theta_o and dTheta/dR or other 2-param formulations

14. “Outlier-tolerant” Bayesian fitting
Prob(Di|M,si) µ exp(- Ri2 /2)
Ri = (Di – Mi) / si
Prob(Di|M,si) µ (1 – exp(- Ri2 /2) ) / Ri2
Since we expect departures from simple model
(eg, near bar or from supernova induced shells),
using outlier-tolerant Bayesian fitting…not sensitive to outliers
(but will return larger uncertainties…very conservative)
Sivia “A Bayesian Tutorial”

15. Model Fitting Results for 93 Sources
Method / R0 Q0 dQ/dR <Vsrc> <Usrc> Q0/R0 Rotation Curve used (kpc) (km/s) (km/s/kpc) (km/s) (km/s) (km/s/kpc) “Outlier-tolerant” Bayesian fitting Flat Rotation Curve 8.39 ± ± 7 [0.0] -8 ± 2 5 ± 3 (28.2) Sloped “ “ 8.38 ± ± ± ± 2 6 ± 2 (29.0) Least-Squares fitting: removing 13 outliers (>3s): Sloped “ “ 8.30 ± ± ± ± 2 5 ± 2 (29.4) Notes: Assuming Solar Motion V-component = 12 km/s (Schœnrich et al 2010) <Vsrc> = average deviation from circular rotation of maser stars <Usrc> = average motion toward Galactic Center Q0/R0 = 28.8 ± 0.2 km/s/kpc from proper motion of Sgr A* (Reid & Brunthaler 2004)
Using 62 sources with parallax/pm measurements from VLBA, VERA and EVN.
Expect simple model will not characterize all data (eg, doesn’t account for local effects of Bar or super-bubbles).
Therefore, adopt an “error tolerant” Bayesian approach…maximize the Prob(D|M) with function that falls as 1/R^2
instead of exponentially for a Gaussian prob
This handles outliers very well…doesn’t let them dominate the fit as it would for least squares.
Get Ro~8.5 kpc and Theta_o~245 km/s…close to Paper VI results with only 18 sources.
Can remove 11 outliers (>3sigma residuals) based on Bayesian fit and perform a (more accurate) Least Squares fit.
Get Ro=8.6 kpc and Theta_o=248…almost same but with reduced uncertainties (one loses accuracy with the error-tolerant Bayesian fit as the price for not being sensitive to outliers)
Other info: average counter-rotation (C.R.) 6 km/s;
average toward G. C. 5 km/s
Angular rotation Theta_o/Ro = /- 0.8 km/s/kpc (matches value inferred from Sgr A*’s proper motion of /- 0.2 km/s when using Vsun=12 km/s)

16. The Milky Way’s Rotation Curve
For R0 = 8.4 kpc, Q0 = 243 km/s
Assumes Schoenrich Solar Motion
Corrected for maser counter-rotation
New and direct result based on
3-D motions
“gold standard” distances
Examples of Rotation Curves over plotted on our data (for Theta_o=245)
Blue points have >3sigma residuals, probably owing to 1) near bar, 2) near super-bubbles

17. Conclusions
VLBA, VERA & EVN parallaxes tracing spiral structure of Milky Way
Milky Way has 4 major gas arms (and minor ones near the bar)
Outer arm spiral pitch angles ~13o
Star forming regions “counter-rotate” by ~8 km/s (for Vsun=12 km/s)
Parallax/proper motions: Ro ~ 8.38 ± 0.18 kpc; Qo ~ 243 ± 7 km/s/kpc
VLBI can achieve <10 uas parallax uncertainty in 1 year…adding ~60 measures per year
Spiral arms are being traced and it appears that a “global” average pitch angle is ~13 deg (~ for Sbc galaxy)
3) On average, MSFRs counter rotate (ie, are near apocenter in slightly elliptical orbits)
4) Ro, To being tightly constrained using full 3D info; compares well to only other truly DIRECT measurements

19. Conclusions
VLBA, VERA & EVN parallaxes to massive young stars (via masers)
tracing spiral structure of Milky Way
Milky Way has 4 major gas arms (and minor ones near the bar)
Outer arm spiral pitch angles ~13o
Star forming regions “counter-rotate” by ~8 km/s (for Vsun=12 km/s)
Parallax/proper motions: Ro ~ 8.38 ± 0.18 kpc; Qo ~ 243 ± 7 km/s/kpc
G.C. stellar orbits + Sgr A* p.m.: Ro ~ 8.2 ± 0.3 kpc; Qo ~ 236 ± 10 km/s/kpc
VLBI can achieve <10 uas parallax uncertainty in 1 year…adding ~60 measures per year
Spiral arms are being traced and it appears that a “global” average pitch angle is ~13 deg (~ for Sbc galaxy)
3) On average, MSFRs counter rotate (ie, are near apocenter in slightly elliptical orbits)
4) Ro, To being tightly constrained using full 3D info; compares well to only other truly DIRECT measurements

20. Is Q0 really >220km/s ?
Parallax/Proper Motions of Star Forming Regions
R0 = 8.4 ± 0.2 kpc & Q0 = 243 ± 7 km/s
Q0 / R0 = 29.0 ± 0.9 km/s/kpc
(assuming Schoenrich, Binney & Dehnen 2010 Solar Motion)
Sgr A*’s proper motion (caused by Sun’s Galactic orbit)
Q0 / R0 = ± 0.15 km/s/kpc
(Reid & Brunthaler 2004)
IR stellar orbits
R0 = 8.3 ± 0.3 kpc
(Ghez et al 2008; Gillessen et al 2009)
Hence, Q0 = 238 ± 9 km/s
Now back to our question…can we confirm large To value?
Yes…totally independent and direct measurements consistent result.
Combined result:
Q0 = 241 ± 6 km/s

21. Carbon Monoxide (CO) Longitude-Velocity Plot
Doppler Velocity
Dame, Hartmann & Thaddeus (2001)
Galactic Longitude

22. Counter-Rotation of Star Forming Regions
Compute Galacto-centric V
Transform to frame rotating at
Qo = 250 km/s (yellow)
See peculiar (non-circular) motions …clear counter-rotation
Transform to frame rotating at Qo = 235 km/s (red)
Still counter-rotating
Now on to dynamics…remember not only have parallax, but also proper motion and Vlsr (3-D velocity vector in km/s)
Transform to frame rotating with the Galaxy (flat rotation curve) and plot peculiar (non-circular) velocities
Clearly see an average counter rotation.
First thought might be that this indicates To too large…but value of To largely cancels.
Dropping To by 15 km/s makes very little difference

23. Sensitivity to Rotation Curve
Method / R Q dQ/dR C.R G.C Q0/R0
Rotation Curve used (kpc) (km/s) (km/s/kpc) (km/s) (km/s) (km/s/kpc)
“Error-tolerant” Bayesian fitting: Prob(Di|M) µ (1 – exp(- Ri2 /2) ) / Ri2 where Ri = (Di – Mi) / si
Flat Rotation Curve ± ± [0] ± ± (28.6)
Sloped “ “ ± ± ± ± ± (28.9)
R.C. params
a a2
Brand-Blitz formulation ± ± ± [0] ± ± (29.0)
Polynomial formulation ± ± ±1 -1.5± ± ± (28.8)
“Universal” formulation ± ± ± ± ± ± (28.4)
Brand-Blitz Q = Q0 ra1 + a where r = R/R0
Polynomial Q = Q0 + a1 (r - 1) + a2 (r - 1)2
Universal Q = f( Q0, Ropt = a1 R0, L = a2 L* )
Checking sensitivity to various rotation curves (RC)
Not much change in Ro, Theta_o ; if anything favors slightly larger Ro, To values (keeping ratio constant near 28.8 km/s/kpc)

24. Sgr A*’s Proper Motion m = ( Q0 + V ) / R0 220 km/s 8.4 kpc
Apparent motion of Sgr A* (Galactic Center compact radio source) traces Galactic orbit directly gives (To+Vsun)/Ro
Here’s how we measure it…

25. Reid & Brunthaler (2004) + new data
Proper Motion of Sgr A*
Parallel to Galactic Plane:
6.379 ± mas/yr 
Qo/Ro = ± 0.15 km/s/kpc
(after removing V=12 km/s)
Remove Qo/Ro = 29.4 ± 0.9 km/s/kpc
Sgr A*’s motion || to Gal. Plane
-7.2 ± 8.5 km/s (Ro/8 kpc)
Perpendicular to Gal. Plane:
-7.6 ± 0.7 km/s
Remove 7.2 km/s motion of Sun
Sgr A*’s motion ^ to Gal. Plane
-0.4 ± 0.9 km/s !
Galactic Plane
What do we see…
Plotted is small piece of sky; major tics at 10 mas; position of Sgr A* plotted from 1995 (top) to 2003 (bottom)
Value is 6 mas/yr measure to 0.3% accuracy as expected from Sun’s orbit
Implies Rot Speed/Distance = (for a flat rotation curve -> A=14.75)
Fit is red dashes; Gal Plane is blue line…difference is Vsun out of G.P.
Could use to help redefine G.P.
Sgr A* intrinsic motion extremely small…less than military aircraft! …must be extremely massive
fit to data
Reid & Brunthaler (2004) + new data

26. Effects of Increasing Q0
Reduces kinematic distances: Dk by 15%, hence…
Molecular cloud sizes (R µ jD) by 15%
Young star luminosities: L µ R2 by 30% (increasing YSO ages)
Cloud masses (from column density & size): M µ R2 by 30%
Milky Way’s dark matter halo mass:
M µ (Vmax) 2 RVir
Vmax µ Q0 & RVir µ Q0
M µ Q or up by 50%
Increasing Q0, increases expected dark matter annihilation signals
Largest uncertainty for modeling Hulse-Taylor binary pulsar timing is accounting for the acceleration of the Sun in its Galactic Orbit: Q2/R0

27. Effects of Increasing Q0
1) Increases mass and overall size of Galaxy
2) Decreases velocity of LMC with respect to M.W.
Both help bind LMC to M.W. (Shattow & Loeb 2009)
Increases likelihood of an Andromeda-Milky Way collision MW
LMC
Measurement so accurate, need to consider some pretty exotic possibilities for apparent motion of Sgr A*
This effect comparable to our measurement accuracy