Simulated Galactic methanol maser distribution to constrain Milky Way parameters
L.H. Quiroga-Nuñez, H.J. van Langevelde, M.J. Reid, J.A. Green
AAstronomy & Astrophysics manuscript no. quiroganunez˙v14 c (cid:13)
ESO 2018November 9, 2018
Simulated Galactic methanol maser distribution to constrain MilkyWay parameters
L. H. Quiroga-Nu˜nez , , Huib Jan van Langevelde , , M. J. Reid and J. A. Green , Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands.e-mail: [email protected] Joint Institute for VLBI ERIC (JIVE), Postbus 2, 7990 AA Dwingeloo, The Netherlands. Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA. CSIRO Astronomy and Space Science, Australia Telescope National Facility, PO Box 76, Epping, NSW 1710, Australia. SKA Organisation, Jodrell Bank Observatory, Lower Withington, Macclesfield, SK11 9DL, UK.Received 23 February 2017 / Accepted 7 June 2017
ABSTRACT
Context.
Using trigonometric parallaxes and proper motions of masers associated with massive young stars, the Bar and SpiralStructure Legacy (BeSSeL) survey has reported the most accurate values of the Galactic parameters so far. The determination of theseparameters with high accuracy has a widespread impact on Galactic and extragalactic measurements.
Aims.
This research is aimed at establishing the confidence with which such parameters can be determined. This is relevant for thedata published in the context of the BeSSeL survey collaboration, but also for future observations, in particular from the southernhemisphere. In addition, some astrophysical properties of the masers can be constrained, notably the luminosity function.
Methods.
We have simulated the population of maser-bearing young stars associated with Galactic spiral structure, generating severalsamples and comparing them with the observed samples used in the BeSSeL survey. Consequently, we checked the determination ofGalactic parameters for observational biases introduced by the sample selection.
Results.
Galactic parameters obtained by the BeSSeL survey do not seem to be biased by the sample selection used. In fact, thepublished error estimates appear to be conservative for most of the parameters. We show that future BeSSeL data and future observa-tions with southern arrays will improve the Galactic parameters estimates and smoothly reduce their mutual correlation. Moreover, bymodeling future parallax data with larger distance values and, thus, greater relative uncertainties for a larger numbers of sources, wefound that parallax-distance biasing is an important issue. Hence, using fractional parallax uncertainty in the weighting of the motiondata is imperative. Finally, the luminosity function for 6.7 GHz methanol masers was determined, allowing us to estimate the numberof Galactic methanol masers.
Key words.
Masers – Astrometry – Galaxy: fundamental parameters – Galaxy: kinematics and dynamics – Galaxy: structure.
1. Introduction
A lack of accurate distance measurements throughout the Galaxycombined with our location within the Milky Way have com-plicated the interpretation of astrometric measurements (Reid &Honma 2014). Consequently, the most fundamental Galactic pa-rameters, such as the distance to the Galactic center ( R ), therotation speed at the solar radius ( Θ ), and the rotation curve(e.g., d Θ / dR ) have not been established with high accuracy.At Galactic scales, distance estimates through radial velocities,mass and luminosity calculations of sources within the Galaxy,as well as the mass and luminosity estimates of the Milky Way,depend on the Galactic parameters. Additionally, extragalacticmeasurements are based on Galactic calibrations that are madeusing the Milky Way parameter values. Therefore, highly accu-rate estimates of the fundamental Galactic parameters are vitallyimportant.A step forward came with the Hipparcos satellite (Perrymanet al. 1997). It provided astrometric accuracies of the order of1 milliarcsecond (mas), which allows distance estimations in thesolar neighborhood ( ∼
100 pc) with 10% accuracy. However, thisis a tiny portion of the Milky Way. The ongoing European SpaceAgency mission, Gaia, aims to measure parallaxes and propermotions of 10 stars with accuracies up to 20 µ as at 15 mag with a distance horizon of 5 kpc with 10% accuracy and 10 kpcwith 20% accuracy (Perryman et al. 2001; Gaia Collaborationet al. 2016). Although Gaia will transform our knowledge of theMilky Way, the mission is restricted to optical wavelengths anddue to significant dust obscuration, it will not be able to probe theGalactic plane freely. In contrast, radio wavelengths are not af-fected by dust extinction and can be used throughout the Galaxy.Direct accurate distances and proper motions have been mea-sured for maser-bearing young stars (e.g. Sanna et al. 2014;Burns et al. 2017); this data was obtained employing Very LongBaseline Interferometry (VLBI). This astrometric informationhas provided us with a better understanding of the Milky Way’sspiral structure, insights into the formation and evolution of ourGalaxy, its 3D gravitational potential, and the Galactic baryonicand dark matter distribution (Efremov 2011).The most suitable radio beacons for astrometry are methanol(6.7 and 12.2 GHz) and water (22 GHz) masers (Brunthaler et al.2011). In addition to being bright, water masers can be associ-ated with high mass star forming regions (HMSFRs), while classII 6.7 and 12 GHz methanol masers are uniquely associated withHMSFRs (e.g. Breen et al. 2013; Surcis et al. 2013). By de-tecting 6.7 GHz methanol masers, we trace the Galactic spiralstructure because HMSFRs are expected to be born close to aspiral arm and evolve more quickly than low-mass stars (Yusof a r X i v : . [ a s t r o - ph . GA ] J u l .H. Quiroga-Nu˜nez et al.: Simulated maser distribution to constrain Milky Way parameters et al. 2013). Therefore, HMSFRs should follow the disk rota-tion with low dispersion (compared, for example, to masers inevolved stars).Given parallax, proper motion measurements, source co-ordinates, and line-of-sight velocities (from Doppler shifts ofspectral lines) to methanol and water masers, it is possible tosample complete phase-space information. This provides directand powerful constraints on the fundamental parameters of theGalaxy. The Bar and Spiral Structure Legacy (BeSSeL ) sur-vey has addressed this task using di ff erent arrays: the VeryLong Baseline Array (VLBA) in USA and the European VLBINetwork (EVN) in Europe, Asia and South Africa. Additionallysimilar parallax and proper motion data has come from the VLBIExploration of Radio Astrometry (VERA) in Japan. The mostrecent summary paper (Reid et al. 2014) lists astrometric datafor 103 parallax measurements with typical accuracies of 20 µ as.By fitting these sources to an axially symmetric Galactic model,they provide accurate values for the fundamental Galactic pa-rameters: R = . ± .
16 kpc, Θ = ± − , and d Θ / dR = − . ± . − kpc − between Galactocentric radiiof 5 and 16 kpc.Although the BeSSeL survey data is very accurate, the targetselection used was necessarily biased. It has targeted the bright-est known masers accessible to the (northern hemisphere) VLBIarrays used. Most of the published targets used by BeSSeL forastrometric measurements are 22 GHz water masers and 12 GHzmethanol masers that were originally selected based on 6.7 GHzsurveys. In the current study, a model used to simulate the 6.7GHz methanol maser distribution in the Milky Way is presented.The model was compared with systematic surveys, allowing usto determine the luminosity function. Also, it is used to gener-ate di ff erent artificial samples that can be used to test how ac-curately they can fit a Galactic model and how a given level ofincompleteness can bias the Galactic parameter values. This isparticularly important when more sources are being added to theBeSSeL sample.In Sect. 2 the components and assumptions of the model arepresented. Next, Sect. 3 describes the luminosity function fittedusing observational surveys, the Galactic parameter results, andthe correlation among parameters using several samples. Finally,the discussion and conclusions of the results compared to theBeSSeL findings are shown in Sects. 4 and 5, respectively.
2. Model for the 6.7 GHz methanol maserdistribution in the spiral structure
The main components of the Milky Way can be identified as ahalo, nuclear bulge (or bar), and two disk components: a thin anda thick disk (see e.g. Gilmore & Reid 1983; Rix & Bovy 2013).The current model is centered on the thin disk component, morespecifically on a spiral structure between 3 kpc and 15 kpc astraced by HMSFRs that contains methanol maser bearing stars.Following the analysis made by Reid et al. (2014), the modelis based on a galaxy with spiral structure. The analysis of therotation and scale of the galaxy does not seem directly dependenton this assumption.The aim of the model is to build a simulated database readyto be processed with the Galactic parameter fitting method usedby the BeSSeL survey. To do this, each simulated 6.7 GHzmethanol maser has spatial coordinates, velocity components,and an associated intrinsic luminosity (and their respective un-certainties). In the following subsections, we explain each of the http://bessel.vlbi-astrometry.org/ Model Distribution DistributionVariable Type ParametersGalactic Radial decay and h r = .
44 kpc (1)Plane (X,Y) Monte Carlo rejection σ d = µ z = σ z =
25 pcRadial Gaussian µ r = − Velocity (U) σ r = − Tangential Gaussian µ t = Θ =
240 km s − (1)Velocity (V) σ t = − Vertical Gaussian µ v = − Velocity (W) σ v = − Luminosity Power Cuto ff s: 10 − L (cid:12) , 10 − L (cid:12) Function (L) (3) Law and α = − . Table 1.
Spatial, velocity and luminosity distributions used inthe current model. We assumed the sun’s vertical position to bez = ffi ths (2011), and (3) Pestalozzi et al. (2007). Radial, tan-gential and vertical velocity dispersion values are discussed inSect. 2.3.distributions and the initial parameters adopted, as well as the fit-ting procedure used to obtain the Galactic parameters from theastrometric data. Table 1 presents a summary of the distributionsand values used. Reid et al. (2014) presented their best estimates of the Galacticparameter values (Model A5), which we adopt here (seeTable 2): – R , Θ , d Θ / dR : fundamental Galactic parameters. Wetook the current results of the BeSSeL survey, which as-sumes a Galactic model as a disk rotating at a speed of Θ ( R ) = Θ + d Θ dR ( R − R ); – ¯ U s , ¯ V s : average source peculiar motion. When velocitiesare measured, systematic extra velocity components can ap-pear as a result of two e ff ects: gas approaching a spiralarm with enhanced gravitational attraction and magneto-hydrodynamic shocks as the gas enters the arm; therefore,these extra velocity components, which are defined at the po-sition of each source, account for any average peculiar mo-tion of the masers; – U (cid:12) , V (cid:12) , W (cid:12) : solar motion. Because the model predicts thevelocities with respect to the local standard of rest (LSR) forall masers, the solar motion must be taken into account inorder to make the proper heliocentric corrections; – N : number of sources. The total number of 6.7 GHzmethanol masers in the Galaxy is a required parame-ter to populate the spiral arms. In Sect. 3, this parame-ter is fitted by comparing the model with the results ofMethanol Multibeam Survey (MMB, see: Green et al. 2009,2010, 2012; Caswell et al. 2010, 2011) results given theadopted spatial distribution (Sect. 2.2) and luminosity func-tion (Sect. 2.5). S o u r c e C o un t s Radial Distribution
Galactocentric Distance R (kpc) R o t a t i o n Sp ee d Θ ( R ) ( k m / s ) Masers d Θ /dR Sun
Fig. 1. Left:
Galactic plane distribution of 6.7 GHz methanol masers seen from the NGP overlaid on an artist impression of theMilky Way (R. Hurt: NASA / JPLCaltech / SSC). The spiral structure was constructed following Wainscoat et al. (1992) and thecentral molecular ring or 3 kpc arms (Green et al. 2011) was indicated, but it is not part of the model. The simulated spatial maserdistribution is presented in Sect. 2.2. The plot also includes the intrinsic peak luminosity for each source as the point size followingthe luminosity function described in Sect. 2.5. In this figure, the Galaxy rotates clockwise.
Bottom right:
Tangential velocitydistribution as a function of Galactocentric distance for the simulated 6.7 GHz methanol masers. It also displays the rotation curve, d Θ / dR = − . − kpc − . Top right:
Radial distribution of the 6.7 GHz methanol masers for our model is also shown. Table 1presents a summary of the distributions used.
Galactic Longitude (degrees) V L S R ( k m / s ) − L fl − L fl Fig. 2.
Velocity with respect to the LSR as a function of the Galactic longitude for the simulated 6.7 GHz methanol masersdistribution. The point size is a measure of the peak luminosity function (Sect. 2.5). Masers associated with di ff erent spiral armsare color-coded as in Fig. 1. The figure is overlaid on the CO emission ( J = −
0) plotted in grayscale and taken from Dame et al.(2001).
The spatial distribution along the spiral arms can be split into twocomponents, a Galactic plane distribution and a vertical compo-nent distribution (Z). The latter can be drawn using a randomgenerator from a Gaussian distribution with a mean of 0 pc and σ =
25 pc since massive young stars are found to be born close to the Galactic plane (see e.g. Green & McClure-Gri ffi ths 2011;Bobylev & Bajkova 2016).The Galactic plane distribution is drawn following two con-straints. First, the density of HMSFRs falls o ff exponentiallywith the Galactocentric distance ( R ) (Bovy & Rix 2013), andsecond, each source should be associated with a spiral arm (Reid Parameter Definition Value R Sun-Galactocentric distance 8.34 kpc Θ Solar rotation speed 240 km s − d Θ / dR Rotation curve -0.2 km s − kpc − U (cid:12) Inward radial solar velocity 10.7 km s − V (cid:12) Tangential solar velocity 15.6 km s − W (cid:12) Vertical solar velocity 8.9 km s − ¯ U s Inward radial average peculiar motion 2.9 km s − ¯ V s Tangential average peculiar motion -1.5 km s − Table 2.
Description of initial parameters values used in themodel which are based on the Model A5 results publishedin Reid et al. (2014).et al. 2014). For the first constraint, the maser radial distributionfollows Cheng et al. (2012) n ( R ) ∝ e − R / h R , (1)where n ( R ) is the number of sources and h R the exponential scalelength, which has been estimated from the maser parallax dataassuming a Persic Universal rotation curve formulation to be2 .
44 kpc (Reid et al. 2014) which we assumed valid for mas-sive young stars. The top right panel of Fig. 1 shows the radialdistribution of the simulated masers.For the second constraint, the spiral arm positions were setfollowing an analytic approximation made by Wainscoat et al.(1992). Each spiral arm (four main arms and the local arm) canbe located in the Galactic plane using a simple relation in polarcoordinates. The left plot of Fig. 1 depicts the position of thespiral arms as seen from the north galactic pole (NGP). In orderto populate the spiral arms with 6.7 GHz methanol masers, arejection sampling Monte Carlo method was implemented. Forthis, the model takes a source from the radial distribution (Eq. 1)and then the distance is calculated between the source and theclosest spiral arm. That distance d is evaluated in a probabilitydensity function of a Gaussian distribution P ( d ) ∝ exp − (d − µ ) σ , (2)where µ = σ d = .
35 kpc,which corresponds to the maximum spiral width arm observedfor HMSFRs (Reid et al. 2014) . The model evaluates P ( d ) foreach source and compares it with a random value k (0 < k < k > P ( d ), the source is rejected and the model takes anothersource from the radial distribution to calculate P ( d ) again andcompare it with a new k . However, if a source satisfies k < P ( d ),then the source is taken as a part of the model. The acceptanceprocess will continue until it reaches the total number of sources( N ). One example of a resulting spatial distribution can be seenin Figs. 1 and 2. For the velocity distribution, we used a cylindrical coordinatesystem ( U , V , W ) in a rotational frame with an angular velocityof Θ ( R ) in the direction of the Galaxy rotation, i.e., clockwiseseen from the NGP. In this system, U is the radial componentdefined positive towards the center of the Galaxy, V is the tan-gential velocity component defined positive in the direction ofthe Galactic rotation and W is the vertical velocity componentdefined positive towards the NGP. -8 -7 -6 -5 -4 -3 Peak luminosity L p (L fl ) S o u r c e C o un t s -1 Peak flux density S p (Jy)10 S o u r c e C o un t s Fig. 3. Top:
Peak luminosity function adopted in the model us-ing the fitted values for the total number of 6.7 GHz methanolmasers ( N = α = − . Bottom:
Peak flux densityfunction obtained without sensitivity limit.We drew Gaussian distributions for each velocity compo-nent independently using the values, distributions and disper-sions related in Table 1. For the tangential velocity, we adopteda Gaussian distribution with a mean value given by Θ ( R ) =Θ + d Θ / dR ( R − R ) and a dispersion of σ t = − (seeTable 1). The values for Θ , d Θ / dR and R are provided inTable 2. The lower right panel of Fig. 1 and Fig. 2 show the dis-tribution of the Galactic tangential velocities Θ ( R ) and the maservelocities with respect to LSR as a function of Galactocentricdistance and Galactic longitude respectively, assuming the val-ues listed in Table 2. The adopted dispersions for radial and ver-tical velocity components ( σ r , v = − ) are consistent withour estimates of virial motions of individual massive stars, basedon BeSSeL data, whereas σ t was set larger to allow for the pos-sible e ff ects of gravitational accelerations in the presence of ma-terial near spiral arms. The BeSSeL survey determined proper motions and parallaxesof water masers (at 22 GHz) and methanol masers (at 6.7 and 12GHz) and fit them to an axially symmetric Galactic model to es-timate the Galactic parameters. Compared with the BeSSeL sur- vey, we have made a simplification by assuming that all sourcesare selected from 6.7 GHz methanol masers surveys, but ob-served with VLBI at 12 GHz.
Notably, for our model it is important to estimate astrometricobservational errors based on maser detectability, which are di-rectly related to the peak flux density ( S p ) of each maser, i.e.,the flux density emitted in a specific line integrated over a singlechannel width. The peak flux density function can be estimatedif the peak luminosity function and the spatial distribution areknown, assuming isotropic emission. Although the individualmaser spots may not radiate isotropically, we assume that thisholds over the sample of randomly oriented masers.Pestalozzi et al. (2007) have suggested that the 6.7 GHzmethanol maser luminosity distribution takes the form of a sin-gle power law with sharp cuto ff s of 10 − L (cid:12) and 10 − L (cid:12) and aslope ( α ) between − . −
2. We assume the same dependencefor the peak luminosity function (see Fig. 3), but we refine it byvarying the parameters to match the results of the MMB survey.The results of this procedure are presented in Sect. 3.1.
In order to be able to use simulated data in tests to estimatethe Galactic parameters, it is necessary to assign observationalerror distributions. For our model, the errors in the parallaxand proper motions were estimated following a calculation forrelative motions of maser spots and statistical parallaxes, i.e., σ π ∝ Θ res / ( S / N ) and σ µ α,δ = σ π / (1yr), where Θ res is the VLBAresolution for 12 GHz methanol masers. The signal-to-noise ra-tio ( S / N ) depends on the peak flux density value ( S p ) and giventhat most of the current data of the BeSSeL survey are basedon VLBA observations, we adopted a channel width of 50 kHz(1 .
24 km s − ) at 12 GHz and an integration time of 2 hr. Thiswas used to estimate the S / N and thus the errors in parallax andproper motions. Reid et al. (2014) estimated an additional errorterm for σ V los (5 km s − ), which is associated with the uncertaintyon transferring the maser motions to the central star. This errordominates the BeSSeL observations of V los , and this uncertaintyis reflected in the value of σ V los .Parallax estimates in Reid et al. (2014) are often dominatedby residual, whereas troposphere-related errors dominate in theastrometry, and so we adopted a simple prescription for parallaxuncertainty (as shown above), which does not directly includesystematic e ff ects. However, when a large number of simulatedsources are used, many weak masers are included that would be S / N limited. Figure 4 shows a comparison between the two errordistributions for observational and simulated parallax measure-ments in which our S / N error estimate yields a similar distribu-tion to the uncertainties used in Reid et al. (2014).The fitting procedure described by Reid et al. (2014) usedto determine the Galactic parameters (combining BeSSeL andVERA data) requires high accuracy VLBI data as input. Thisdata consists of a 3D position vector ( α , δ , π ), a 3D velocity vec-tor ( µ α , µ δ , V los ), and the errors σ π , σ µ α , σ µ δ and σ V los . Althoughthe model gives exact values for position and velocities of eachmaser source seen from the Earth, we are interested in realis-tic values as input for the fitting procedure. Therefore, we adda noise component to each observable quantity ( π , µ α , µ δ , V los ),using random values following Gaussian distributions with stan-dard deviations equal to the estimated errors previously calcu- ∆ π (mas)0.00.10.20.30.40.50.60.7 N o r m a li z e d C o un t s Observational ErrorsSimulated Errors
Fig. 4.
Comparison between the error distribution for observa-tional and simulated parallax measurements. Observational er-rors are based on 103 astrometric sources published in Reid et al.(2014) as part of the BeSSeL survey.
MMB AreciboObservation Simulation Observation SimulationSensitivity (3 σ ) ≤ .
71 Jy ≤ .
27 JySky − ◦ ≤ l ≤ ◦ . ◦ ≤ l ≤ . ◦ coverage − ◦ ≤ b ≤ ◦ − . ◦ ≤ b ≤ . ◦ Sources 908 800 ±
20 76 95 ± β -0.60 ± ± ± ± Table 3.
Limits in sensitivity and source location, numbers ofmasers detected and the slope of the flux density functions ( β )for the 6.7 GHz methanol masers surveys: MMB and Arecibo.Limits of both surveys, numbers of masers ( N ) and slope of theluminosity function ( α ) fitted in Sect. 3.1, were applied to ourGalactic model; the results are displayed in the columns labeled”Simulation”. The simulated errors correspond to the standarddeviation after running 100 simulated galaxies.lated. By changing the error distribution, we can control thequality of the data entered in the fitting procedure. The fitting procedure used was adopted from the BeSSeL sur-vey (see Reid et al. 2009, 2014). The input data for the fit-ting procedure are 3D position and 3D velocity information ofthe masers, conservative priors for the solar motion, the av-erage source peculiar motion, and the Galactic scale and ro-tation. Convergence on the best Galactic parameters to matchthe spatial-kinematic model was made using a Bayesian fit-ting approach, where the velocities were used as known datato be fitted, and the sky positions and distances were used ascoordinates. The posterior probability density function (PDF)of the Galactic parameters were estimated with Markov chainMonte Carlo (MCMC) trials that were accepted or rejected bya Metropolis-Hastings algorithm (see Reid et al. 2009, 2014,for a detailed explanation). Finally, the procedure returns thebest Galactic parameter values that match the simulated data tothe spatial-kinematic model. The fitting procedure was improvedcompared to that used in Reid et al. (2009, 2014): first, the fittingprocedure now corrects for bias when inverting parallax to esti-mate distance, which becomes significant when fractional paral-
Fig. 5.
Grid of initial parameters displaying the ξ calculation foreach N , α pair. The dark blue region represents the best values of N and α that most closely match the MMB results. The projectedgray dashed lines show the profiles of the surface close to theminimum values of ξ . Peak Flux Density (Jy) S o u r c e C o un t s MMB -1 ARECIBO
Observation Simulation
Fig. 6.
In blue: flux density function obtained for the MMB(top) and the Arecibo survey (bottom). In green: simulated fluxdensity function obtained in the model (using N = α = − .
43) after the MMB and Arecibo limits were applied(Table 3).lax uncertainties exceet ≈
15% (note this is not a trivial inferenceproblem, see e.g. Bailer-Jones 2015); second, the fitting proce-dure was improved by adding a term to the motion uncertainties,which comes from parallax uncertainty. After these two modifi-cations, the fitting procedure yielded unbiased Galactic parame-ter values, even when weak and / or very distant masers with largefractional parallax uncertainties were simulated.
3. Results
A comparison of the systematic 6.7 GHz methanol maser ob-servational surveys and the simulated model peak flux density is shown in Sect. 3.1. In Sect. 3.2, di ff erent sample selectionsare used to compare the Galactic parameters obtained with re-spect to the initial values used (Table 2). Finally, in Sect. 3.3, thePearson correlation coe ffi cients are calculated to quantify corre-lations among the Galactic parameters. We compared the flux density distribution functions of the MMBsurvey and the Arecibo survey with the current model to fittwo parameters: the total number of sources ( N ) and the slopeof the peak luminosity function ( α ). The MMB survey is themost sensitive unbiased survey yet undertaken for 6.7 GHzmethanol masers. The Parkes Observatory was upgraded with aseven-beam receiver to carry out a full systematic survey of theGalactic Plane (Green et al. 2012, and the references within).The Arecibo Survey was a deep a 6.7 GHz methanol maser sur-vey over a limited portion of the Galactic plane (Pandian et al.2007).Table 3 summarizes the survey limits in sensitivity and skycoverage for the MMB and Arecibo surveys. The last two rowslist the number of sources detected and the slope of the flux den-sity function ( β ) for each survey. By using these data, we wereable to make a direct comparison between the simulated and ob-served flux density functions for each survey (green and bluehistograms in Fig. 6). For our comparison, we excluded MMBsources that reside inside a Galactocentric radius of 3 kpc as thisregion is not part of the model.In order to fit N and α to the results of the surveys, a grid ofinitial parameters (Fig. 5) was sampled using similar ranges tothose proposed by Pestalozzi et al. (2007) for N = [900 , α = [ − . , . N , α ) and for each pair, a set of simulated galaxies was gen-erated following the initial conditions described in Sect. 2. Next,the surveys limits (Table 3) were applied, and we compared theflux density function obtained for each N , α pair with the fluxdensity function of the MMB survey (blue histogram in the topof Fig. 6). Through a minimization procedure, we found valuesof N and α that best match the MMB results. This procedure wasimplemented only for the MMB data since it represents a largerand more complete sample than the Arecibo survey. The mini-mization procedure compares the MMB observed (blue) and thesimulated (green) flux density functions (see Fig. 6) and mini-mizes a quantity called ξ , where ξ = (cid:88) bins ( y − y obs ) y obs , (3)and y represents the number of sources per luminosity bin. Giventhat our Galactic model generates galaxies based on a stochas-tic method, the position, velocity and luminosity values for eachmaser vary each time the model is executed (even using the samepair of N and α ). By generating sets of ten independent galaxysimulations per N , α pair, we found that the fluctuations in thesimulations were smaller than the uncertainties in the binneddata, and hence this procedure was applied.Figure 5 shows the values obtained for ξ per N , α pair as a3D surface. The dark blue region in the projected contour plotrepresents the best set of parameters that mimic the MMB sur-vey results. We found that the surface near the minimum canbe approximated by a Gaussian in two dimensions (see pro-jections in Figure 5). Using the maximum likelihood estima-tion, which is well defined for multivariate Gaussian distribu- N u m b e r C o un t s R (kpc)
230 235 240 245 250 Θ (km / s) d Θ dR (km / s · kpc) U fl (km / s) V fl (km / s) W fl (km / s) Results Model A5 ¯ U s (km / s) Gaussian Fitting ¯ V s (km / s) Fig. 7.
Galactic parameters distributions found for 100 simulated galaxies mimicking the BeSSeL data sample selection (Sect. 3.2).The values listed in Table 4 correspond to the fitting made to the histograms and shown as black dashed lines. Bayesian fittingresults for the A5 model reported in Reid et al. (2014) are shown as gray regions.tions, we estimated the mean and its respective uncertainty. Thebest parameters were found to be N = ±
60 sources and α = − . ± .
18. Finally, Fig. 6 shows the flux density func-tion for the MMB (top), and Arecibo survey (bottom) in blue,and their respective simulated flux density function are shownin green for the best parameters of N and α found. Additionally,the number of sources detected and the slope of the flux densityfunction ( β ) for the simulated surveys are listed in Table 3. The model can reproduce the methanol maser distribution for theentire Galaxy including observational errors. In order to evaluatethe possible biases introduced by the observed BeSSeL sample(equivalent to the 103 brightest sources in the declination re-gion, − ◦ ≤ δ ≤ ◦ , which is equivalent to − ◦ ≤ l ≤ ◦ ),100 galaxies were simulated to mimic the BeSSeL sample. Then,they were fitted to test whether the adopted Galactic parameterswere returned. Figure 7 shows the distribution obtained on eachGalactic parameter for the simulated BeSSeL sample comparedwith the values reported in Reid et al. (2014). The histogramswere fitted to Gaussian distributions, and the results are shownin Table 4. Clearly, in all cases the distributions of fitted valuesare centered on the adopted value, and in most cases the widthsof the distributions are smaller than those reported in Reid et al.(2014).In addition to the 100 simulated galaxies that mimic theBeSSeL sample, we also simulated the first BeSSeL data sample, Galactic Simulated A5Parameter BeSSeL Sample Model R (kpc) 8 . ± .
07 8 . ± . Θ (km s − ) 240 . ± . . ± . Θ/ d R (km s − kpc − ) − . ± . − . ± . U (cid:12) (km s − ) 10 . ± . . ± . V (cid:12) (km s − ) 15 . ± . . ± . W (cid:12) (km s − ) 8 . ± . . ± . U s (km s − ) 3 . ± . . ± . V s (km s − ) − . ± . − . ± . Table 4.
Galactic parameter results for 100 simulated galaxiesmimicking the BeSSeL data sample. Additionally, the Bayesianfitting results for the A5 model reported in Reid et al. (2014),which were also the initial values adopted in the model (Table 2),are shown for comparison.where only 16 HMSFRs over the northern hemisphere were usedto estimate the same Galactic parameters but not the solar mo-tion (Reid et al. 2009). Moreover, we also started adding sourcesto form two additional sets of simulated data. Set A was made tostudy the impact of future viable observations with the VLBA,EVN, and VERA to obtain up to 500 sources in the northernhemisphere. Again, we selected the brightest sources first to fallin the same declination range that BeSSeL is targeting for this.We generated samples from 16 up to 500 sources, which were
Number of Sources S E T A R (kpc) Θ (km / s) d Θ dR (km / s · kpc) U fl (km / s) V fl (km / s) W fl (km / s) ¯ U s (km / s) Complete Simulation Model A5 Fit 3 Initial Values (-14.7) ¯ V s (km / s) Number of Sources S E T B R (kpc) Θ (km / s) d Θ dR (km / s · kpc) U fl (km / s) V fl (km / s) W fl (km / s) ¯ U s (km / s) (-14.7) ¯ V s (km / s) Fig. 8.
Galactic parameter values obtained for samples in sets A and B. In each sample, sources are added in the northern hemispheresimulating the future BeSSeL results (set A) and without location limit simulating samples when southern arrays can contribute withdata (set B). First and current BeSSeL results published in Reid et al. (2009, 2014), respectively labeled ”Fit 3” and ”Model A5”,are shown as stars for comparison. The initial values adopted in the model are represented as dashed lines. Gray regions correspondto values and uncertainties obtained for the complete sample ( N = R Θ d Θ/ d R U (cid:12) V (cid:12) W (cid:12) ¯ U s ¯ V s R Θ Θ/ d R U (cid:12) V (cid:12) W (cid:12) -0.01 (0.00) 0.02 (-0.01) -0.01 (0.03) -0.06 (-0.02) 0.01 (0.01) 1.00 (1.00)¯ U s V s Table 5.
Pearson product-moment correlation coe ffi cients calculated for 100 galaxies simulated to mimic the BeSSeL data sampleselection. The respective Pearson coe ffi cient reported in Reid et al. (2014) for the observed sample are listed in parentheses.
100 200 300 400 500Number of Sources1.00.50.00.51.0 P e a r s o n C o e ff i c i e n t SET A R - Θ Θ - V fl Θ - ¯ V s U fl - ¯ U s V fl - ¯ V s Model A5
200 400 600 800 1000Number of Sources1.00.50.00.51.0 P e a r s o n C o e ff i c i e n t SET B
Complete Simulation
Fig. 9.
Pearson product-moment correlation coe ffi cients calcu-lated for initially highly correlated values (see Table 5), whenmore sources are added in sets A (top panel) and B (bottompanel). Pearson coe ffi cients reported by Reid et al. (2014) areshown as stars and those for the complete sample ( N = N = ff ort when VLBI arrays in the Southern hemisphere can contribute to the astrometric sample. As wasdone in set A, we selected the brightest sources but now with-out declination limitation, generating samples from 16 up to thecomplete sample ( N = ± ∆ and ± ∆ for the obtained simulated BeSSeL values related inTable 4) and calculated a normalized di ff erence between the in-put parameters and the returned fits. We found that indeed thefitting procedure can properly recover the starting values. Using the Galactic parameter values obtained for 100 simulatedgalaxies mimicking the BeSSeL data sample selection, we cal-culated the Pearson product-moment correlation coe ffi cients be-tween all the parameters from the output distributions. The coef-ficients found are shown in Table 5; for comparison, the Pearsoncoe ffi cient estimates reported in Reid et al. (2014) from the fit-ting procedure are also listed. Pearson coe ffi cients in Reid et al.(2014) were calculated by MCMC trials, but in our case we havea large number of samples, which provides an independent wayto estimate the correlations. Our findings seems to be consistentwith the coe ffi cients published in Reid et al. (2014).We also estimated the Pearson coe ffi cients variation as moresources are added to the sample selection. In order to see whetherthe dependence between various parameters can be reduced,we focused on the more correlated parameters reported in Reidet al. (2014), i.e., r ( R , Θ ) , r ( Θ , V (cid:12) ) , r ( Θ , ¯ V s ) , r ( U (cid:12) , ¯ U s ) and r ( V (cid:12) − ¯ V s ) .Figure 9 shows the Pearson coe ffi cient evolution among theseparameters in sets A and B. Moreover, the Pearson coe ffi cients calculated for the complete sample and those published by Reidet al. (2014) are shown for comparison.
4. Discussion
We found that N = ±
60 and α = − . ± .
18 are the initialparameters that best match the MMB results. Using these values,the number of sources detected and the slope of the flux densityfunction ( β ) are slightly underestimated with respect to the ob-servational survey results (see Table 3). This di ff erence could berelated to the contamination from inner Galaxy sources includedin the MMB, which were not included in the simulation. This canaccount for approximately 100 sources in the N estimate, pro-ducing a value of N = + − . This estimate seems to be con-sistent with the initial calculation made by Green & McClure-Gri ffi ths (2011) of N = N = ffi ths (2011) reported α = − . ± . N since in our method the twoquantities were fitted simultaneously.There is no physical argument that predicts the luminosityfunction to be a single power law distribution. However, for thescope of this paper, we are only interested in deriving an em-pirical relation for the peak luminosity function for a popula-tion of 6.7 GHz methanol masers with the proper characteris-tics. Additionally, a single power law peak luminosity functionappears to be consistent with the results obtained for di ff erentsystematic surveys (including the Arecibo survey, see Figure 6)and, for bright sources, it has been previously suggested by sev-eral authors (e.g. Pandian et al. 2007; Green & McClure-Gri ffi ths2011). The di ff erent samples described in Sect. 3.2 were created totest how accurately the BeSSeL methodology can determine theGalactic parameters. When the sample testing was initially madeusing the same fitting procedure employed in Reid et al. (2009,2014), the resulting parameters start deviating from the initialparameters when more sources were added. When sources withlarge fractional errors in parallax are numerous, we found thatthis biases the determination towards larger distances, resultingin parameters that map to a bigger Galaxy. This observationale ff ect (see e.g. Bailer-Jones 2015) was corrected by allowing thefitting procedure to de-bias distance estimations based on mea-sured parallax. We note that the improvements to the fitting codedo not alter the results in Reid et al. (2014), which was based onthe brighter sources.Figures 7 and 8 summarize the Galactic parameters obtainedcompared with the initial values adopted (see Table 2), using thecurrent and possible future samples. The results in Table 4 andFig. 7 obtained for 100 simulated galaxies using the BeSSeLdata sample selection show that the Galactic parameter valuescan be determined very robustly. Figure 8 shows that the Galacticparameter results for the simulated samples of 100 sources (cur-rent BeSSeL data) in sets A and B are already very close to theinitial parameters, and as more sources are added the uncertain-ties become smaller. R , Θ , and d Θ / dR The di ff erences in R and Θ found when using 100 simu-lated galaxies mimicking the BeSSeL sample selection (Table 4and Fig. 7) are less than 0.2%, demonstrating that indeed wecan recover these parameters from the adopted model, evenwith samples that only contain northern hemisphere sources.Furthermore, the errors reported by Reid et al. (2014) for theseparameters (i.e. 0.16 kpc for R and 8 km s − for Θ ), which arerepresented in Fig. 7 as gray regions, are double compared toour findings. Consequently, we conclude that the errors assignedby Reid et al. (2014) to R and Θ are conservative, and there donot appear to be any bias, given the available maser samples sofar. For the rotation curve, the situation is somewhat di ff erent.Although the values found for d Θ / dR are very close to the ini-tial values adopted, the statistical spread is larger than expected.Reid et al. (2014) reported an error of 0.4 km s − kpc − in therotation curve which is optimistic compared with our findings.From our simulations, we would constrain the rotation curvevalue as -0.3 ± − kpc − given the BeSSeL data sampleselection. The larger error possibly indicates that our assumedvelocity distributions are too wide.When more sources are added to the sample selection (setsA and B), Figure 8 shows an initial improvement in the ac-curacy of the fundamental Galactic parameters. For set A, theerrors in R , Θ , and d Θ / dR can improve up to ± .
04 kpc, ± . − , and ± . − kpc − , respectively, when morenorthern hemisphere sources are added. In contrast when south-ern hemisphere sources are also added (set B), the errors candecrease to ± .
03 kpc, ± . − , and ± . − kpc − , re-spectively. Further improvements would require much better as-trometry for weak sources, requiring for example much moresensitive observations. Figure 7 shows the distribution obtained for solar velocity com-ponents ( U (cid:12) , V (cid:12) , and W (cid:12) ). The uncertainties derived werearound 2% for all solar velocity components with respect to theinitial parameters. For U (cid:12) , the standard deviation found was 1.2km s − , which compares well to the results published by Reidet al. (2014) (i.e., U (cid:12) = ± . − ). In addition, the spreadfor V (cid:12) and W (cid:12) are narrower with values of 2.4 km s − and0.5 km s − , respectively. Compared to the BeSSeL results (i.e., V (cid:12) = ± . − and W (cid:12) = ± . − ), we can a ffi rm thatthe solar motion results published by Reid et al. (2014) haveconservative estimates.For the radial and tangential average peculiar motions ( ¯ U s and ¯ V s ), Table 4 shows that indeed these peculiar velocities canbe fitted with high accuracy using the simulations; however, therelative spreads are high compared with other parameters (seeFigure 7). Even so, compared to the BeSSeL results (i.e., ¯ U s = ± . − and ¯ V s = ± . − ), the errors in the simulatedsample selection are still lower (i.e., ¯ U s = ± . − and ¯ V s = ± . − ).The reason for higher dispersions for ¯ U s and ¯ V s could be re-lated to the number of parameters that must fit independently.The parameters ¯ U s and U (cid:12) have largely the same e ff ect on theobservations for nearby sources and therefore are directly cor-related (see Table 5). For ¯ V s , the correlation is high with twocomponents ( V (cid:12) , Θ ), which a ff ects the fitting and hence the es-timated accuracy.
246 250 254 258 262 Θ + V fl (km/s)0.000.040.080.120.16 N o r m e d C o un t N u m b e r (Θ + V fl ) /R (km/s · kpc)0.00.40.81.21.6 Results Model A5 Gaussian Fitting V fl + | ¯ V s | (km/s)0.00.10.20.30.4 Fig. 10.
Marginalized posterior probability density distributions for correlated circular velocity parameters from 100 simulatedgalaxies mimicking the BeSSeL sample. Left panel: Circular orbital speed of the Sun. Middle panel: Orbital angular solar speed.Right panel: Di ff erence between the circular solar motion and average source peculiar motions. Pearson product-moment correlation coe ffi cients are shown inTable 5. All the parameters are in agreement with those foundby Reid et al. (2014), except for the Pearson coe ffi cient between d Θ / dR and ¯ V s (labeled ” r d Θ / dR , ¯ V s ”) where a low correlation wasfound in the observed sample instead of the moderate correlationfound in the simulated sample. However, we focus the discussionon the parameters that were reported to have considerable cor-relation in Reid et al. (2014), as it is interesting to see how it ispossible to disentangle these fundamental parameters.For the first BeSSeL summary paper (Reid et al. 2009),where only 16 HMSFRs were used, the estimated R and Θ were strongly correlated ( r R , Θ = . r R , Θ = . ffi cient by mimick-ing both samples using 100 simulated galaxies, finding similarresults for each sample, i.e, r R , Θ = .
77 and r R , Θ = . ffi cients to those found in the observations,even when the method used to calculate the Pearson coe ffi cientsare completely di ff erent (see Sect. 3.3). When more sources areadded to the sample selection, Figure 9 shows that the correlationbetween R and Θ is reduced. Furthermore, the Pearson coe ffi -cient will have a moderate value (0.3) when using the completedata sample, which demonstrates that the correlation betweenthese parameters can be unraveled smoothly as more sources areadded.For the tangential velocity component, we have three di ff er-ent Galactic parameters giving similar e ff ects: Θ , V (cid:12) and ¯ V s .Figure 9 and Table 5 show that the Pearson coe ffi cients amongthese parameters are always high, even when more sources areadded. This implies that di ff erent types of data are needed in or-der to better disentangle these Galactic parameters. Finally, inthe radial direction we have two Galactic parameters: U (cid:12) and¯ U s . The correlation between these parameters is around 0.5 fora low number of sources, as it shown in Figure 9 and listed inTable 5. When more sources are added, the correlation parame-ter seems to maintain the same value or slightly increase in bothsets of samples (see Figure 9). One could expect that the inclu- sion of southern hemisphere sources would help to disentanglesome of the dependences; however, comparing the top and bot-tom plots in Figure 9 we can see that using samples with a largercoverage of the Galaxy does not alter significantly the correla-tion values found for any of the Pearson coe ffi cients discussedhere.As some of the parameter correlations persist, even whenmore sources are added to the sample selection, we estimatedthe marginalized PDFs for di ff erent combined parameters thatwere also reported in the latest BeSSeL survey paper. Figure 10shows the PDFs for the circular orbital speed of the Sun, the or-bital angular solar speed, and the di ff erence between the circularsolar and average source peculiar motions. Additionally, thosePDFs reported by Reid et al. (2014) are shown as gray regions.We found 255 . ± . − , 30 . ± .
26 km s − kpc − , and16 . ± . − , respectively, for the correlated values of theparameters Θ + V (cid:12) , ( Θ + V (cid:12) ) / R and V (cid:12) + | ¯ V s | . Reid et al.(2014) found more conservative values for Θ + V (cid:12) = . ± . − and ( Θ + V (cid:12) ) / R = . ± .
43 km s − kpc − thanwe did. Although the mean value found for V (cid:12) + | ¯ V s | is inagreement with the BeSSeL results (i.e., 17 . ± . − ), weestimate a wider error of ± . − based on our simulations.
5. Conclusions
We constructed simulations of 6.7 GHz methanol maser distri-butions to test whether the Galactic parameter results obtainedby the BeSSeL survey (Reid et al. 2014) are biased in anyway and investigated the interdependencies between some pa-rameter estimates. We used our model to constrain the peakflux density function for the masers and obtained similar resultsto those of systematic unbiased surveys (MMB and Arecibo).This comparison allowed us to estimate the integral numberof sources ( N = + − ) and the slope of the luminos-ity function ( α = − . ± . ffi ths (2011);Urquhart et al. (2013).Assuming that the observations are predominantly of 12GHz methanol masers found through 6.7 GHz surveys, we sim-ulated the current database of the BeSSeL survey hundreds oftimes. We found that the fundamental Galactic parameters ( R , Θ , d Θ / dR ), the solar velocity components ( U (cid:12) , V (cid:12) , W (cid:12) ) andthe average peculiar motion ( ¯ U s , ¯ V s ) can be determined robustly.Furthermore, the results published by Reid et al. (2014) have aconservative error calculation given the current sample, exceptpossibly for the rotation curve error estimate. Also, correlationcoe ffi cients for the various Galactic parameters in our simula-tions and those reported by Reid et al. (2014) are similar.Additionally, the fitting procedures developed by Reid et al.(2009, 2014) for use with the BeSSeL data and improved in thisstudy estimate Galactic parameters correctly even when weakand / or distant sources with large fractional parallax uncertaintiesare included in the samples.For future BeSSeL observations, the simulations demon-strate that the Galactic parameter estimates can be improved andthe error bars reduced significantly. Moreover, using southernhemisphere data, the Galactic parameter estimates improve no-tably compared with samples limited to the northern sky.We find that the uncertainties in the values of certain com-bined velocity parameters that are highly correlated are similarto those published in Reid et al. (2014), except for the disper-sion in V (cid:12) + | ¯ V s | . However, when more sources are added tothe sample, the correlations among most Galactic parameters aresmoothly reduced; for the highly velocity parameters the corre-lation coe ffi cients do not decrease significantly.The framework proposed to test the results of the BeSSeLsurvey is useful for defining requirements for future astromet-ric campaigns that are similar or complementary to the BeSSeLdata. Southern arrays – like the Australian Long BaselineArray (see e.g. Krishnan et al. 2015, 2017) and, in the future,the African VLBI Network and the Square Kilometre Array inAustralia and South Africa – will supplement the lack of pre-cise astrometric data in quadrants III and IV of the Milky Wayplane, where only a few sources have been measured. Moreover,astrometric studies that include the inner Galactic region, suchas the Bulge Asymmetries and Dynamic Evolution (BAaDE )project, aim to resolve the dynamics of the bar by measuringproper motions and distances of SiO masers present in AGBstars (Sjouwerman et al. 2017). Out of the Galactic plane, Gaiawill soon provide astrometric results for a large number ofsources. All of these investigations will contribute to the deter-mination of Galactic parameters with even better accuracy withnew and improved astrometric data. Until then, Galactic simu-lations complement the current observations by demonstratingtheir robustness and potential. Acknowledgements. We sincerely thank the anonymous referee for makingvaluable suggestions that have improved the paper. L.H.Q.-N. acknowledgesthe comments and suggestions regarding the model implementation made byS. Solorzano-Rocha at ETH Z¨urich.
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