Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mrao.cam.ac.uk/~bn204/publications/2009/specbasic.pdf Äàòà èçìåíåíèÿ: Fri Aug 10 20:18:02 2012 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 11:18:13 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: arp 220

Fitting and Comparison of Models of Radio Spectra
Bojan Nikolic
Astrophysics Group, Cavendish Laborator y/Kavli Institute for Cosmology University of Cambridge

December 2009 @ NRAO Green Bank
R60

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Introduction

Outline
1 2

Introduction Method Bayesian analysis Implementation Visualisation Simple examples NGC 628 NGC 3627 NGC 7331 Free-free component in a supernova remnant Summar y/Fur ther Directions/References

3

4 5

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Introduction

Introduction

This work was done in preparation for analysing for thcoming data from GBT+MUSTANG It is a little bit of a "spare-par ts" project in which I reused various software components I developed for other purposes (e.g., phase calibration for ALMA) The approach is described in full detail in Nikolic (2009) All of the source code available for download (GPL license) from website: http://www.mrao.cam.ac.uk/~bn204/galevol/speca/ index.html

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Introduction

Schematic radio spectrum of a star-forming galaxy
1000 500 200 100 50 20 10 1 10 ( GHz )
Schematic & hypothetical (continuum-only) spectrum of NGC 3627: the dashed line is the synchrotron the component; the dotted line is the free-free component; the dash-dot-dash line is the dust component; the solid line is the total emission. B. Nikolic (Cambridge) Fitting of radio spectra 4 / 37

F (mJy)

100

1000


Introduction

Why analyse radio spectra
Energetics
Reconstruct the total energy balance from few/sparse measurements of the spectrum

Inference of proper ties of the source:
Geometr y (Filling factor from low-frequency turnover) Dynamics (e.g., through electron ageing)

Redshift determination radio "photometric" redshifts
Currently mostly used for sub-millimetre selected ("SCUBA") sources

Physics:
Free-free emission Slope of the dust continuum ­ physics of interstellar dust

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Introduction

Analysis strategy

Model fitting In radio, sub-mm and far-IR, the physics is fairly well understood and candidate models are computational easy. So analysis often consists of "fitting" a set of models to the observations. Synchrotron radiation (analytic or 1-D integral) Thermal free-free (analytic) Modified black-body emission from dust (analytic or 1-D integral) Spinning dust models (analytic)

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Introduction

Requirements for model fitting

Objective measure of how well the model fits observed data For all model parameters:
Unbiased estimates Error on these estimates Correlations between the errors Full probability distributions if significantly non-Gaussian

Objective way of comparing how well different models fit the data A mechanism to incorporate already known constraints on model parameters Visualisation of the fit in comparison to observations

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Method

Outline
1 2

Introduction Method Bayesian analysis Implementation Visualisation Simple examples NGC 628 NGC 3627 NGC 7331 Free-free component in a supernova remnant Summar y/Fur ther Directions/References

3

4 5

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Method

Bayesian analysis

Bayesian analysis in a nutshell

1 2 3 4

Can handle "nuisance" parameters Fully describes non-Gaussian distributions Unbiased Objective model (or hypothesis) selection

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Method

Bayesian analysis

Bayes equation & the evidence

p(|D , H ) =

p(D |, H )p(|H ) p(D |H )

D : Obser ved data flux density at several frequencies H : Hypothesis model for emission & priors for parameters p(D |, H ): Likelihood given a model and its parameters, how likely are the obser ved data? p(|D , H ): Posterior given a model, what we know about it's parameters p(D |H ): "Evidence", objective measure of how good the model is
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Method

Bayesian analysis

Bayes equation & the evidence
p(D |, H )p(|H ) p(D |H )

p(|D , H ) =

D : Obser ved data flux density at several frequencies H : Hypothesis model for emission & priors for parameters p(D |, H ): Likelihood given a model and its parameters, how likely are the obser ved data? p(|D , H ): Posterior given a model, what we know about it's parameters p(D |H ): "Evidence", objective measure of how good the model is
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Method

Bayesian analysis

Calculating the evidence

Evidence is an integral over the likelihood surface p(D |H ) = dp(D |, H )p(|H )

Evidence is not available from standard Markov Chain Monte Carlo calculations I use a new implementation of the nested sampling algorithm by Skilling (2006). Compared to MCMC, this algorithm is:
Efficient (fewer likelihood function evaluations) Reliable (less chance of getting stuck in local maxima) The output is both the evidence and the posterior distribution

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Method

Implementation

Inputs/outputs
Inputs
1

The model: a C++ class. Possible to compose different models in the Python layer Priors: only flat, independent priors suppor ted. Supplied as a dictionar y in the Python layer Observed data: supplied as simple list in Python layer

2

3

Outputs
1 2 3 4

The evidence value Histograms of marginalised distributions of each model parameter Fan-diagram of flux vs frequency Maximum likelihood plot of flux vs frequency
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Method

Implementation

Inputs/outputs
Inputs
1

The model: a C++ class. Possible to compose different models in the Python layer Priors: only flat, independent priors suppor ted. Supplied as a dictionar y in the Python layer Observed data: supplied as simple list in Python layer

2

3

Outputs
1 2 3 4

The evidence value Histograms of marginalised distributions of each model parameter Fan-diagram of flux vs frequency Maximum likelihood plot of flux vs frequency
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Method

Visualisation

NGC 628 observations
10

1 F (Jy) 0.1 0.01 101 102 (MHz)
Obser vations at five frequencies of the near-by galaxy NGC 628 collected by Paladino et al. (2009)
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103

10

4


Method

Visualisation

NGC 628 ­ max. likelihood line fit
10

1 F (Jy) 0.1 0.01 101 102 (MHz)
Obser vations at five frequencies of the near-by galaxy NGC 628 collected by Paladino et al. (2009)
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103

10

4


Method

Visualisation

NGC 628 ­ doubled errors & max. likelihood line fit
10

1 F (Jy) 0.1 0.01 101 102 (MHz)
I have scaled up the error estimates by a factor of two
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103

10

4


Method

Visualisation

NGC 628 ­ original errors & fan-diagram
40 2

1 0.5 F (Jy)

30

20 0.2 0.1 0.05 0 101 102 (MHz)
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10

103

10

4


Method

Visualisation

NGC 628 ­ doubled errors & fan-diagram
2 6 1 0.5 F (Jy) 4 0.2 0.1 0.05 0 101 102 (MHz)
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2

103

10

4


Simple examples

Outline
1 2

Introduction Method Bayesian analysis Implementation Visualisation Simple examples NGC 628 NGC 3627 NGC 7331 Free-free component in a supernova remnant Summar y/Fur ther Directions/References

3

4 5

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Simple examples

Four simple models for synchrotron emission
Underlying synchrotron spectrum Power law 0 F ( ) = F ·
1 GHz

Continuous injection of electrons ver y approximately broken power law
1 GHz " " ,, « 0 F · 1 GHz br 0 F ·

Low-frequency optical depth effects None



Synchrotron self-absorption x = /pk As = x
-+5/2 -5/2

"

"

br
-1/2

> br .

â 1 - exp 1 - x

(1)
For the pur poses of these examples, I've taken the models from Paladino et al. (2009) to go with their data ­ both more complex and more physical models could be used
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Simple examples

NGC 628

Model fits for NGC 628 I

Power-law (Z = 185.874)
40 2 2 1 0.5 F (Jy) 20 0.2 0.1 0.05 0 101 102 (MHz) 103 10
4

Power-law + SSA (Z = 76.3034)
15 12.5 10 7.5 5 2.5 0 101 102 (MHz) 103 10
4

30

1 0.5 F (Jy) 0.2 0.1 0.05

10

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Simple examples

NGC 628

Model fits for NGC 628 II
CI (Z = 83.4132)
2 15 12.5 10 7.5 5 2.5 0 101 102 (MHz) 103 10
4

CI + SSA (Z = 27.65)
2 5 1 0.5 F (Jy) 3 0.2 0.1 1 0.05 0 101 102 (MHz) 103 10
4

1 0.5 F (Jy)

4

0.2 0.1 0.05

2

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Simple examples

NGC 3627

Model fits for NGC 3627 I

Power-law (Z = 9.7 â 10-15 )
2 4 · 10 1 0.5 F (Jy) 3 · 10
- 15 - 15

Power-law + SSA (Z = 3.9 â 10-15 )
2 2 · 10-
15

1 0.5 F (Jy)

1.5 · 10-

15

1 · 10- 0.2 0.1 0.05 5 · 10-

15

0.2 0.1 0.05

2 · 10-

15

1 · 10-

16

15

0 101 102 (MHz) 103 10
4

0 101 102 (MHz) 103 10
4

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Simple examples

NGC 3627

Model fits for NGC 3627 II
CI (Z = 1.3 â 10
2 5 · 10- 1 4 · 10- 0.5 F (Jy) F (Jy) 3 · 10- 0.2 2 · 10- 0.1 1 · 10- 0.05 0 101 102 (MHz) 103 10
4 10 10 10 10 10

-09

)
6 · 10-
10

CI + SSA (Z = 2.6 â 10
2 1

-10

)
1 · 10- 8 · 10-
10

11

0.5 6 · 10- 0.2 0.1 2 · 10- 0.05 0 101 102 (MHz) 103 10
4 11 11

4 · 10-

11

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Simple examples

NGC 3627

Probability distributions of parameters

Slope
3 · 10-
10

Break-frequency
2 · 10- 1.5 · 10-
10 10

2 · 10- f

10

1 · 10- 5 · 10-

1 · 10-

10
11

f

10

0 -0.8 -0.6 -0.4 -0.2

0 8.6 8.8 9 log10 (br /Hz) 9.2 9.4

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Simple examples

NGC 7331

Model fits for NGC 7331 I

Power-law (Z = 2.9 â 10-17 )
2 2.5 · 10-
17

Power-law + SSA (Z = 0.5)
2 0.4
17

1 0.5 F (Jy)

2 · 10-

1 0.5 0.3

17

F (Jy)

1.5 · 10- 0.2 0.1 5 · 10- 0.05

1 · 10-

17

0.2 0.1

0.2

18

0.1

0.05 0 101 102 (MHz) 103 10
4

0 101 102 (MHz) 103 10
4

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Simple examples

NGC 7331

Model fits for NGC 7331 II
CI (Z = 70)
2 60 2 4 · 104 1 0.5 F (Jy) 50 1 0.5 F (Jy) 3 · 104

CI + SSA (Z = 5.0 â 105 )

40

30 0.2 20 0.1 10 0.05 0 101 102 (MHz) 103 10
4

0.2 0.1 0.05

2 · 104

1 · 104

0 101 102 (MHz) 103 10
4

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Free-free component in a supernova remnant

Outline
1 2

Introduction Method Bayesian analysis Implementation Visualisation Simple examples NGC 628 NGC 3627 NGC 7331 Free-free component in a supernova remnant Summar y/Fur ther Directions/References

3

4 5

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Free-free component in a supernova remnant

Introduction
Data cour tesy of D. A. Green in Cambridge

Analysis of spectrum of supernova remnant HB3 Is there evidence for flattening of the spectrum?
Could be inter preted as a thermal free-free component due to interaction of shock with the molecular cloud

´ See Urosevic et al. (2007), Green (2007), Onic & Urosevic (2008)

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Free-free component in a supernova remnant

Condon model

Single power-law synchrotron with slope () as a free parameter Free-free emission component (H is the thermal fraction at 1 GHz) Thermal free-free absorption at low frequencies ( is the optical depth at 1 GHz) 1 GHz
2

A( ; ) = 1 - exp -10 F ( ; H , ) =



-2.1

A( ) A(1 GHz) 1 GHz

H + (1 - H )

1 GHz

0.1+

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Free-free component in a supernova remnant

Model fits for HB3

Power-law Z = 9 â 10-
1000 500

25
1000 1.25 · 10-
25

Condon Z = 1.9 â 10
500 1 · 10-
25

-24
2.5 · 10-
25

2 · 10- 200 F (Jy)

25

200 F (Jy) 7.5 · 10- 100 5 · 10-
26 26

100

1.5 · 10-

25

50

50

1 · 10-

25

20

2.5 · 10-

26

20

5 · 10-

26

10 101 102 (MHz) 103 10
4

0

10 101 102 (MHz) 103 10
4

0

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Free-free component in a supernova remnant

Marginalised distribution of

Power-law Z = 9 â 10-
8 · 10- 6 · 10
26 - 26

25
1 · 10
- 25

Condon Z = 1.9 â 10
7.5 · 10- 5 · 10- 2.5 · 10-
26

-24

4 · 10- 2 · 10-

f

26

26

f

26

26

0 -0.8 -0.6 -0.4

0 -0.8 -0.6 -0.4

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Free-free component in a supernova remnant

Marginalised distribution of H
6 · 10-
26

4 · 10- f

26

2 · 10-

26

0 0.1 0.2 0.3 H 0.4 0.5

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Free-free component in a supernova remnant

Error estimates reduced by half

Power-law Z = 8 â 10-
1000 500

24
2.5 · 10
- 24

Condon Z = 9 â 10-
1000 500 2 · 10-
24

23

2 · 10-

23

200 F (Jy) F (Jy) 1.5 · 10- 100 1 · 10-
24 24

200

1.5 · 10-

23

100 1 · 10- 50
23

50

20

5 · 10-

25

5 · 10- 20

24

10 101 102 (MHz) 103 10
4

0

10 101 102 (MHz) 103 10
4

0

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Free-free component in a supernova remnant

Marginalised distribution of H
4 · 10- 3 · 10- 2 · 10- 1 · 10-
24

24

f

24

24

0 0.1 0.2 0.3 H 0.4 0.5

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Summary/Fur ther Directions/References

Outline
1 2

Introduction Method Bayesian analysis Implementation Visualisation Simple examples NGC 628 NGC 3627 NGC 7331 Free-free component in a supernova remnant Summar y/Fur ther Directions/References

3

4 5

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Summary/Fur ther Directions/References

Future direction
Analysis of Luminous Infrared Galaxies (with Marcel Clemens from U. Padua)
2 3000 1 0.5 F (Jy) 2000 0.2 0.1 0.05 0 101 102 103 (MHz)
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1000

104

10

5


Summary/Fur ther Directions/References

References

Green D. A., 2007, Bulletin of the Astronomical Society of India, 35, 77. arXiv:0705.2642 Nikolic B., 2009, ArXiv e-prints. arXiv:0912.2317 Onic D., Urosevic D., 2008, Serbian Astronomical Journal, 177, 67. arXiv:0809.2693 ~ Paladino R., Murgia M., Orra E., 2009, A&A, 503, 747. arXiv:0905.3643 Skilling J., 2006, in ISBA 8th World Meeting on Bayesian Statistics ´ Urosevic D., Pannuti T. G., Leahy D., 2007, ApJ, 655, L41. arXiv:arXiv:astro- ph/0612691

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