Bayesian Analysis of RR Lyrae Distances and Kinematics Thomas R. - - PowerPoint PPT Presentation

Description of the Problem The Model and Sampling Program Results and Conclusions

Bayesian Analysis of RR Lyrae Distances and Kinematics

Thomas R. Jefferys1a Thomas G. Barnes III1b Andrei Dambis2 William H. Jefferys1b,3

• 1a. Department of Mathematics, b. Department of Astronomy

University of Texas at Austin

2Sternberg Astronomical Institute

Universitetskii pr. 13, Moscow, 119992 Russia

3Department of Mathematics

University of Vermont

MaxEnt 2007

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Abstract

We are using a hierarchical Bayes model to analyze the distances, luminosities, and kinematics of RR Lyrae stars. Our model relates these characteristics to the raw data of proper motions, radial velocities, apparent luminosities and metallicities of each star. A combination of Gibbs and Metropolis-Hastings sampling, using latent variables for the actual velocity and luminosity of each star, is used to draw a sample from the full posterior distribution of these variables, with consideration to admissibility and the properness of the hierarchical model, and draw inferences on the quantities of interest in the usual way. We have applied our model to the large HIPPARCOS database, and we have attempted to include metallicity and period in our model, which has not been done previously.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

RR Lyrae Stars: A Short Introduction

What are they? Why do we care? They are a class of pulsating variable stars. They are readily recognizable from their periods (0.75 ± 0.25 days) and characteristic light curves. Fairly bright (40 times as bright as the Sun) and so can be seen to fair distances in the galaxy. Their intrinsic mean visual-band luminosities are nearly constant.

This is known from studies of RR Lyrae stars in clusters, where all the stars are at the same distance.

Their consistent luminosities makes them useful as “standard candles” for estimating the distance of an object (like a star cluster)

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

RR Lyrae Stars: A Short Introduction

What are they? Why do we care? They are a class of pulsating variable stars. They are readily recognizable from their periods (0.75 ± 0.25 days) and characteristic light curves. Fairly bright (40 times as bright as the Sun) and so can be seen to fair distances in the galaxy. Their intrinsic mean visual-band luminosities are nearly constant.

This is known from studies of RR Lyrae stars in clusters, where all the stars are at the same distance.

Their consistent luminosities makes them useful as “standard candles” for estimating the distance of an object (like a star cluster)

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

RR Lyrae Stars: A Short Introduction

What are they? Why do we care? They are a class of pulsating variable stars. They are readily recognizable from their periods (0.75 ± 0.25 days) and characteristic light curves. Fairly bright (40 times as bright as the Sun) and so can be seen to fair distances in the galaxy. Their intrinsic mean visual-band luminosities are nearly constant.

This is known from studies of RR Lyrae stars in clusters, where all the stars are at the same distance.

Their consistent luminosities makes them useful as “standard candles” for estimating the distance of an object (like a star cluster)

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

RR Lyrae Stars: A Short Introduction

What are they? Why do we care? They are a class of pulsating variable stars. They are readily recognizable from their periods (0.75 ± 0.25 days) and characteristic light curves. Fairly bright (40 times as bright as the Sun) and so can be seen to fair distances in the galaxy. Their intrinsic mean visual-band luminosities are nearly constant.

This is known from studies of RR Lyrae stars in clusters, where all the stars are at the same distance.

Their consistent luminosities makes them useful as “standard candles” for estimating the distance of an object (like a star cluster)

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

RR Lyrae Stars: A Short Introduction

What are they? Why do we care? They are a class of pulsating variable stars. They are readily recognizable from their periods (0.75 ± 0.25 days) and characteristic light curves. Fairly bright (40 times as bright as the Sun) and so can be seen to fair distances in the galaxy. Their intrinsic mean visual-band luminosities are nearly constant.

This is known from studies of RR Lyrae stars in clusters, where all the stars are at the same distance.

Their consistent luminosities makes them useful as “standard candles” for estimating the distance of an object (like a star cluster)

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Goals of Our Investigation

Determine the mean absolute magnitude (log luminosity)

• f these stars

Investigate the kinematics of the stars as a group Investigate the “cosmic scatter” of the magnitude (i.e., the variation about the mean unexplained by the covariates) Investigate any variation of absolute magnitude with “metallicity” (i.e., content of elements heavier than helium) and period

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Goals of Our Investigation

Determine the mean absolute magnitude (log luminosity)

• f these stars

Investigate the kinematics of the stars as a group Investigate the “cosmic scatter” of the magnitude (i.e., the variation about the mean unexplained by the covariates) Investigate any variation of absolute magnitude with “metallicity” (i.e., content of elements heavier than helium) and period

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Goals of Our Investigation

Determine the mean absolute magnitude (log luminosity)

• f these stars

Investigate the kinematics of the stars as a group Investigate the “cosmic scatter” of the magnitude (i.e., the variation about the mean unexplained by the covariates) Investigate any variation of absolute magnitude with “metallicity” (i.e., content of elements heavier than helium) and period

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Goals of Our Investigation

Determine the mean absolute magnitude (log luminosity)

• f these stars

Investigate the kinematics of the stars as a group Investigate the “cosmic scatter” of the magnitude (i.e., the variation about the mean unexplained by the covariates) Investigate any variation of absolute magnitude with “metallicity” (i.e., content of elements heavier than helium) and period

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Overview of the Method

Our raw data: Proper motions µ (vector of angular motion across the sky per unit time) Radial velocities ρ (kilometers/second of the motions towards or away from the Sun, obtained via the Doppler shift) Apparent magnitudes m of the stars, assumed measured without error.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Overview of the Method

Our raw data: Proper motions µ (vector of angular motion across the sky per unit time) Radial velocities ρ (kilometers/second of the motions towards or away from the Sun, obtained via the Doppler shift) Apparent magnitudes m of the stars, assumed measured without error.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Overview of the Method

Our raw data: Proper motions µ (vector of angular motion across the sky per unit time) Radial velocities ρ (kilometers/second of the motions towards or away from the Sun, obtained via the Doppler shift) Apparent magnitudes m of the stars, assumed measured without error.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Overview of the Method

The proper motions are related to the cross-track velocities (in km/sec) by multiplying the former by the distance s to the star: sµ ∝ V ⊥ If we assume that the proper motions and radial velocities are characterized by the same kinematical parameters, we can (statistically) infer s and then the absolute magnitude M of the star through a defined relationship: s = 100.2(m−M+5) (1)

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Overview of the Method

The proper motions are related to the cross-track velocities (in km/sec) by multiplying the former by the distance s to the star: sµ ∝ V ⊥ If we assume that the proper motions and radial velocities are characterized by the same kinematical parameters, we can (statistically) infer s and then the absolute magnitude M of the star through a defined relationship: s = 100.2(m−M+5) (1)

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Overview of the Method

We measure the proper motions and radial velocities of the swarm of RR Lyrae stars as the Sun plunges through. These velocities are characterized by two vectors.

!!"# !!!! !"#

The velocity of the swarm as a group, relative to the Sun, V J The velocity of each star relative to the swarm, V − V J. The velocities V − V J are assumed characterized by a multivariate normal distribution with mean zero and covariance matrix W.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Overview of the Method

We measure the proper motions and radial velocities of the swarm of RR Lyrae stars as the Sun plunges through. These velocities are characterized by two vectors.

!!"# !!!! !"#

The velocity of the swarm as a group, relative to the Sun, V J The velocity of each star relative to the swarm, V − V J. The velocities V − V J are assumed characterized by a multivariate normal distribution with mean zero and covariance matrix W.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Overview of the Method

We measure the proper motions and radial velocities of the swarm of RR Lyrae stars as the Sun plunges through. These velocities are characterized by two vectors.

!!"# !!!! !"#

The velocity of the swarm as a group, relative to the Sun, V J The velocity of each star relative to the swarm, V − V J. The velocities V − V J are assumed characterized by a multivariate normal distribution with mean zero and covariance matrix W.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions RR Lyraes and Their Kinematics

Overview of the Method

We measure the proper motions and radial velocities of the swarm of RR Lyrae stars as the Sun plunges through. These velocities are characterized by two vectors.

!!"# !!!! !"#

The velocity of the swarm as a group, relative to the Sun, V J The velocity of each star relative to the swarm, V − V J. The velocities V − V J are assumed characterized by a multivariate normal distribution with mean zero and covariance matrix W.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

The Likelihood

If µo = (µo

α, µo δ) are the two components of the observed

proper motion vector (in the plane of the sky), and ρo is the

• bserved radial velocity, we assume

µo

α

∼ N(V ⊥

α /s, σ2 µα)

µo

δ

∼ N(V ⊥

δ /s, σ2 µδ)

(2) ρo ∼ N(V , σ2

ρ)

The variances are measured at the telescope and assumed known perfectly. The variables without superscripts are the “true” values. The likelihood is just the product of all these normal distributions over all stars in the sample

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

The Likelihood

If µo = (µo

α, µo δ) are the two components of the observed

proper motion vector (in the plane of the sky), and ρo is the

• bserved radial velocity, we assume

µo

α

∼ N(V ⊥

α /s, σ2 µα)

µo

δ

∼ N(V ⊥

δ /s, σ2 µδ)

(2) ρo ∼ N(V , σ2

ρ)

The variances are measured at the telescope and assumed known perfectly. The variables without superscripts are the “true” values. The likelihood is just the product of all these normal distributions over all stars in the sample

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

The Likelihood

If µo = (µo

α, µo δ) are the two components of the observed

proper motion vector (in the plane of the sky), and ρo is the

• bserved radial velocity, we assume

µo

α

∼ N(V ⊥

α /s, σ2 µα)

µo

δ

∼ N(V ⊥

δ /s, σ2 µδ)

(2) ρo ∼ N(V , σ2

ρ)

The variances are measured at the telescope and assumed known perfectly. The variables without superscripts are the “true” values. The likelihood is just the product of all these normal distributions over all stars in the sample

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: V i and V

The priors on the V i are assigned by assuming that the velocities of the individual stars are drawn from a (three- dimensional) multivariate normal distribution: V i|V J, W ∼ N(V J, W) (3) We choose an improper flat prior on V J, the mean velocity

• f the RR Lyrae stars relative to the velocity of the Sun

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: V i and V

The priors on the V i are assigned by assuming that the velocities of the individual stars are drawn from a (three- dimensional) multivariate normal distribution: V i|V J, W ∼ N(V J, W) (3) We choose an improper flat prior on V J, the mean velocity

• f the RR Lyrae stars relative to the velocity of the Sun

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: W

The prior on W is requires some thought.

One wants a proper posterior distribution Need estimators to be admissible under frequentist theory, e.g., under quadratic loss. Needs to be reasonably noncommittal.

To satisfy these criteria, we chose a “hierarchical independence Jeffreys prior” on W, which for a three- dimensional distribution implies π(W) ∝ |I + W|−2 Together with the prior on V i, this makes our model mathematically nice.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: W

The prior on W is requires some thought.

One wants a proper posterior distribution Need estimators to be admissible under frequentist theory, e.g., under quadratic loss. Needs to be reasonably noncommittal.

To satisfy these criteria, we chose a “hierarchical independence Jeffreys prior” on W, which for a three- dimensional distribution implies π(W) ∝ |I + W|−2 Together with the prior on V i, this makes our model mathematically nice.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: W

The prior on W is requires some thought.

One wants a proper posterior distribution Need estimators to be admissible under frequentist theory, e.g., under quadratic loss. Needs to be reasonably noncommittal.

To satisfy these criteria, we chose a “hierarchical independence Jeffreys prior” on W, which for a three- dimensional distribution implies π(W) ∝ |I + W|−2 Together with the prior on V i, this makes our model mathematically nice.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: W

The prior on W is requires some thought.

One wants a proper posterior distribution Need estimators to be admissible under frequentist theory, e.g., under quadratic loss. Needs to be reasonably noncommittal.

To satisfy these criteria, we chose a “hierarchical independence Jeffreys prior” on W, which for a three- dimensional distribution implies π(W) ∝ |I + W|−2 Together with the prior on V i, this makes our model mathematically nice.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: W

The prior on W is requires some thought.

One wants a proper posterior distribution Need estimators to be admissible under frequentist theory, e.g., under quadratic loss. Needs to be reasonably noncommittal.

To satisfy these criteria, we chose a “hierarchical independence Jeffreys prior” on W, which for a three- dimensional distribution implies π(W) ∝ |I + W|−2 Together with the prior on V i, this makes our model mathematically nice.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: W

The prior on W is requires some thought.

One wants a proper posterior distribution Need estimators to be admissible under frequentist theory, e.g., under quadratic loss. Needs to be reasonably noncommittal.

To satisfy these criteria, we chose a “hierarchical independence Jeffreys prior” on W, which for a three- dimensional distribution implies π(W) ∝ |I + W|−2 Together with the prior on V i, this makes our model mathematically nice.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: M and Ui

The distances si are related to the Mi and mi by the defined (exact) relationship referenced above, (1). We define for each star i, Mi = M + Ui. We choose a flat prior on M (a somewhat informative prior based on known data did not much change the results). Evidence from other sources (e.g., studies of RR Lyrae stars in clusters) indicates a “cosmic scatter” of about 0.15 magnitudes in Mi. Thus a prior on Ui of the form Ui ∼ N(0, (0.15)2) seems appropriate.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: M and Ui

The distances si are related to the Mi and mi by the defined (exact) relationship referenced above, (1). We define for each star i, Mi = M + Ui. We choose a flat prior on M (a somewhat informative prior based on known data did not much change the results). Evidence from other sources (e.g., studies of RR Lyrae stars in clusters) indicates a “cosmic scatter” of about 0.15 magnitudes in Mi. Thus a prior on Ui of the form Ui ∼ N(0, (0.15)2) seems appropriate.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: M and Ui

The distances si are related to the Mi and mi by the defined (exact) relationship referenced above, (1). We define for each star i, Mi = M + Ui. We choose a flat prior on M (a somewhat informative prior based on known data did not much change the results). Evidence from other sources (e.g., studies of RR Lyrae stars in clusters) indicates a “cosmic scatter” of about 0.15 magnitudes in Mi. Thus a prior on Ui of the form Ui ∼ N(0, (0.15)2) seems appropriate.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Prior Choice: M and Ui

The distances si are related to the Mi and mi by the defined (exact) relationship referenced above, (1). We define for each star i, Mi = M + Ui. We choose a flat prior on M (a somewhat informative prior based on known data did not much change the results). Evidence from other sources (e.g., studies of RR Lyrae stars in clusters) indicates a “cosmic scatter” of about 0.15 magnitudes in Mi. Thus a prior on Ui of the form Ui ∼ N(0, (0.15)2) seems appropriate.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

DAG of Model

The overall structure of the hierarchical model can be summarized by this directed acyclic graph (DAG)

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

The Kinematical Horn

We can use Gibbs sampling on the entire kinematical horn

• f the model (W, V i, V ⊙).

V i, V ⊙ may be sampled using appropriate normal distributions. Sampling on W, however, requires a rejection sampling

• scheme. We generate candidates from the distribution

W ∗|{V i}, V J ∼ InverseWishart(T, df = N) and accept with probability P = |W ∗|2/|I + W ∗|2, where T =

N

• i=1

(V i − V J)(V i − V J)′

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

The Kinematical Horn

We can use Gibbs sampling on the entire kinematical horn

• f the model (W, V i, V ⊙).

V i, V ⊙ may be sampled using appropriate normal distributions. Sampling on W, however, requires a rejection sampling

• scheme. We generate candidates from the distribution

W ∗|{V i}, V J ∼ InverseWishart(T, df = N) and accept with probability P = |W ∗|2/|I + W ∗|2, where T =

N

• i=1

(V i − V J)(V i − V J)′

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

The Kinematical Horn

We can use Gibbs sampling on the entire kinematical horn

• f the model (W, V i, V ⊙).

V i, V ⊙ may be sampled using appropriate normal distributions. Sampling on W, however, requires a rejection sampling

• scheme. We generate candidates from the distribution

W ∗|{V i}, V J ∼ InverseWishart(T, df = N) and accept with probability P = |W ∗|2/|I + W ∗|2, where T =

N

• i=1

(V i − V J)(V i − V J)′

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

The Luminosity Horn

The luminosity horn of the model (Ui, M) depends on conditional distributions of the form s2

i N(siµi + ˆ

siV

i − V J, W)

(4) and products thereof over all i, where si is defined as in (1).

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Sampling on Ui

Sampling each Ui is a Metropolis-Hastings step. Under the aforementioned conditional distribution (4), and

• ur informative prior on Ui, we use proposals U∗

i ∼ ctφ, tφ

is a t distribution with φ degrees of freedom, and c is a scaling of that distribution, each adjusted for good mixing.

We found that degrees of freedom φ = 10 and a scale factor of c = 0.11 mixed well. Due to the independence in the Ui’s, the M-H sampling on these variables can be done in parallel.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Sampling on Ui

Sampling each Ui is a Metropolis-Hastings step. Under the aforementioned conditional distribution (4), and

• ur informative prior on Ui, we use proposals U∗

i ∼ ctφ, tφ

is a t distribution with φ degrees of freedom, and c is a scaling of that distribution, each adjusted for good mixing.

We found that degrees of freedom φ = 10 and a scale factor of c = 0.11 mixed well. Due to the independence in the Ui’s, the M-H sampling on these variables can be done in parallel.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Sampling on Ui

Sampling each Ui is a Metropolis-Hastings step. Under the aforementioned conditional distribution (4), and

• ur informative prior on Ui, we use proposals U∗

i ∼ ctφ, tφ

is a t distribution with φ degrees of freedom, and c is a scaling of that distribution, each adjusted for good mixing.

We found that degrees of freedom φ = 10 and a scale factor of c = 0.11 mixed well. Due to the independence in the Ui’s, the M-H sampling on these variables can be done in parallel.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Sampling on Ui

Sampling each Ui is a Metropolis-Hastings step. Under the aforementioned conditional distribution (4), and

• ur informative prior on Ui, we use proposals U∗

i ∼ ctφ, tφ

is a t distribution with φ degrees of freedom, and c is a scaling of that distribution, each adjusted for good mixing.

We found that degrees of freedom φ = 10 and a scale factor of c = 0.11 mixed well. Due to the independence in the Ui’s, the M-H sampling on these variables can be done in parallel.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Sampling on M

We also sample on M using a Metropolis-Hastings step. Our proposal for M∗ ∼ N(M, w2) is a normal distribution with mean of the previous M and a standard deviation of w adjusted for good mixing.

We found that w = 0.15 mixed well.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Sampling on M

We also sample on M using a Metropolis-Hastings step. Our proposal for M∗ ∼ N(M, w2) is a normal distribution with mean of the previous M and a standard deviation of w adjusted for good mixing.

We found that w = 0.15 mixed well.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Results

Our results are based on the data of 347 stars in the HIPPARCOS data set. The mean magnitude M of RR Lyrae stars, as determined by this study, is 0.75 ± 0.07 The solar velocity V J of the RR Lyrae swarm is V̟ −22.67 ± 7.97 Vθ −141.38 ± 5.58 Vz −11.34 ± 3.96 The velocity ellipsoid of the swarm is σ̟ 145.60 ± 5.73 σθ 100.16 ± 4.04 σz 71.10 ± 2.96

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Results

Our results are based on the data of 347 stars in the HIPPARCOS data set. The mean magnitude M of RR Lyrae stars, as determined by this study, is 0.75 ± 0.07 The solar velocity V J of the RR Lyrae swarm is V̟ −22.67 ± 7.97 Vθ −141.38 ± 5.58 Vz −11.34 ± 3.96 The velocity ellipsoid of the swarm is σ̟ 145.60 ± 5.73 σθ 100.16 ± 4.04 σz 71.10 ± 2.96

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Results

Our results are based on the data of 347 stars in the HIPPARCOS data set. The mean magnitude M of RR Lyrae stars, as determined by this study, is 0.75 ± 0.07 The solar velocity V J of the RR Lyrae swarm is V̟ −22.67 ± 7.97 Vθ −141.38 ± 5.58 Vz −11.34 ± 3.96 The velocity ellipsoid of the swarm is σ̟ 145.60 ± 5.73 σθ 100.16 ± 4.04 σz 71.10 ± 2.96

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Results

Our results are based on the data of 347 stars in the HIPPARCOS data set. The mean magnitude M of RR Lyrae stars, as determined by this study, is 0.75 ± 0.07 The solar velocity V J of the RR Lyrae swarm is V̟ −22.67 ± 7.97 Vθ −141.38 ± 5.58 Vz −11.34 ± 3.96 The velocity ellipsoid of the swarm is σ̟ 145.60 ± 5.73 σθ 100.16 ± 4.04 σz 71.10 ± 2.96

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Sampling History of M

Every 3rd Point

• 5000

10000 15000 0.5 0.6 0.7 0.8 0.9 1.0

Sampling History of M (typ; every 3rd point)

Cycle M

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Histogram of M

Histogram of M

M Density 0.5 0.6 0.7 0.8 0.9 1.0 2 4 6 Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Unidentifiablity Problems

We believe that in RR Lyraes, metallicity is unidentified as a predictor in our model. Any change in the coefficient a in the model Mi = M + a[Fe/H]i + Ui is exactly offset by an appropriate change of the individual velocities of the stars. Since modeling individual velocities is at the core of our model, we suspect that this is a fundamental limitation of

• ur model.

This implies that modeling any effects on the individual magnitudes Mi will also result in an unidentified model, including dependance on log-periods. Ui would be unidentified but for the prior defined by the “cosmic scatter”. Investigating any models of the form Mi = M + cHi + Ui, where Hi is any predictor and c is its coefficient, would have to be done by binning the stars with respect to the predictor.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Unidentifiablity Problems

We believe that in RR Lyraes, metallicity is unidentified as a predictor in our model. Any change in the coefficient a in the model Mi = M + a[Fe/H]i + Ui is exactly offset by an appropriate change of the individual velocities of the stars. Since modeling individual velocities is at the core of our model, we suspect that this is a fundamental limitation of

• ur model.

This implies that modeling any effects on the individual magnitudes Mi will also result in an unidentified model, including dependance on log-periods. Ui would be unidentified but for the prior defined by the “cosmic scatter”. Investigating any models of the form Mi = M + cHi + Ui, where Hi is any predictor and c is its coefficient, would have to be done by binning the stars with respect to the predictor.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Unidentifiablity Problems

We believe that in RR Lyraes, metallicity is unidentified as a predictor in our model. Any change in the coefficient a in the model Mi = M + a[Fe/H]i + Ui is exactly offset by an appropriate change of the individual velocities of the stars. Since modeling individual velocities is at the core of our model, we suspect that this is a fundamental limitation of

• ur model.

This implies that modeling any effects on the individual magnitudes Mi will also result in an unidentified model, including dependance on log-periods. Ui would be unidentified but for the prior defined by the “cosmic scatter”. Investigating any models of the form Mi = M + cHi + Ui, where Hi is any predictor and c is its coefficient, would have to be done by binning the stars with respect to the predictor.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Unidentifiablity Problems

We believe that in RR Lyraes, metallicity is unidentified as a predictor in our model. Any change in the coefficient a in the model Mi = M + a[Fe/H]i + Ui is exactly offset by an appropriate change of the individual velocities of the stars. Since modeling individual velocities is at the core of our model, we suspect that this is a fundamental limitation of

• ur model.

This implies that modeling any effects on the individual magnitudes Mi will also result in an unidentified model, including dependance on log-periods. Ui would be unidentified but for the prior defined by the “cosmic scatter”. Investigating any models of the form Mi = M + cHi + Ui, where Hi is any predictor and c is its coefficient, would have to be done by binning the stars with respect to the predictor.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Unidentifiablity Problems

We believe that in RR Lyraes, metallicity is unidentified as a predictor in our model. Any change in the coefficient a in the model Mi = M + a[Fe/H]i + Ui is exactly offset by an appropriate change of the individual velocities of the stars. Since modeling individual velocities is at the core of our model, we suspect that this is a fundamental limitation of

• ur model.

This implies that modeling any effects on the individual magnitudes Mi will also result in an unidentified model, including dependance on log-periods. Ui would be unidentified but for the prior defined by the “cosmic scatter”. Investigating any models of the form Mi = M + cHi + Ui, where Hi is any predictor and c is its coefficient, would have to be done by binning the stars with respect to the predictor.

Jefferys, et al. RR Lyrae Distances and Kinematics

Description of the Problem The Model and Sampling Program Results and Conclusions

Acknowledgments

Thomas G. Barnes for his strong advice on the project, including background on RR Lyraes. James O. Berger for valuable input, especially on priors. Andrei Dambis for providing the radial velocity data. William H. Jefferys for his constant help on the project.

Jefferys, et al. RR Lyrae Distances and Kinematics