Proffit User Guide Version 1deckert/proffit_guide.pdf · This chapter describes the main command to read images and save the output of a Proffit session. 3.1 Reading an image This

Proffit User Guide

Version 1.4

Dominique Eckert

August 19, 2016

Contents

1 Introduction 3

2 Installing and running Proffit 42.1 cmake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 autoconf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Running Proffit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Scripting Proffit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Reading Data 73.1 Reading an image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Reading an exposure map and correcting for vignetting . . . . . . . . . . . . . . . . 73.3 Reading a background map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Excluding areas in the image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Extracting Surface-brightness profiles 94.1 Azimuthally-averaged profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Elliptical annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 Profiles in sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 Growth curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.5 Grouping data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.6 Deprojection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.7 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 ProffitModels 125.1 Loading a model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 List of available models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3 Model Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4 Simulating an image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Plotting routines 166.1 Plotting surface brightness profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Plotting options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7 Fitting Data 187.1 Fitting tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2 Fitting options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3 Fitting statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.4 Error estimation and contour plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.5 PSF modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1

8 Image analysis tools 238.1 Profiles in sectors and azimuthal scatter . . . . . . . . . . . . . . . . . . . . . . . . 238.2 Median in sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3 Residual images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.4 Dynamical state indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9 Saving the results 27

A Appendices 30A.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2

Chapter 1

Introduction

Proffit is an interactive stand-alone package designed for the analysis of X-ray imaging data. It isa C++ code based on CERN’s Root data analysis framework1 and relies on the CFITSIO library forinput/output and uses the world coordinate system (WCS) library to convert between image and skycoordinates. In spite of its name, Proffit is free software and it is distributed under the GNU GPLv2licence. The software runs on any UNIX-based operating system (in particular GNU/Linux and MacOS X) and works with the data from any X-ray mission. The code was introduced for the first time inEckert et al. (2011) and has been used in more than 20 scientific papers thus far.

This guide provides an extensive documentation for the use of Proffit. The installation procedureis explained in detail in Chapter 2, and the following chapters provide a description of the mainfunctionalities of the code. Additionally, a short description of all commands and models is availableinteractively while running Proffit using the help command:

1 > help

List of commands:

allsectors error help plotcounts readback sector

angle2dist fakeit kpc plotgrowth readexp sectorellipse

backsub fit limits plotgrmod readimg showmod

centroidshift fitcounts logx plotmod region statistics

contour fixpar logy plotmodcounts save syst

csb flcomp ls psf savedeviations thawpar

density flobs mediansb profile savemodimg

deproject flux model quit savescat

ellipse group newpar r200 scatter

ellipticity growth plot r500 scripting

Command >

1http://root.cern.ch/

3

http://root.cern.ch/

Chapter 2

Installing and running Proffit

Starting from version 1.4 there are now two ways to install Proffit: the older autoconf method andthe new cmake method. The advantage of the autoconf version is that it allows dynamic linking oflibraries (set using the LD LIBRARY PATH environment variable) and it should be compatible withthe versions of WCS and CFITSIO provided with the HEASOFT package. This version is howeverincompatible with current versions of Mac OS X (El Capitan onwards), as these operating systemsdo not support dynamic linking. The cmake version allows static linking of the libraries, and thus itis compatible with newer Mac OS X versions. Using cmake is also recommended with non-standardinstallations of the core libraries.

Prior to installation, please verify that you have a working version of Root accessible within theterminal and that the ROOTSYS environment variable is properly set. This is usually done automaticallyby running the setup script1. Also make sure that the CFITSIO and WCS libraries are properly set upand accessible.

2.1 cmakeStarting from version 1.4, Proffit can now be installed using cmake. This is the recommended wayof installing Proffit and the only way for newer Mac OS X versions. I recommend to use the CFIT-SIO and WCS versions available within the standard package repositories (Linux) or within MacPorts(Mac OS X). This is done in the following way (for Ubuntu and derivatives):

sudo apt-get install build-essential wcslib-dev libcfitsio3-dev

For Mac OS X (using MacPorts):

sudo port install cfitsio wcslib

Then untar the package and build the software:

tar xvf proffit 1.4 cmake.tar.gz

cd proffit 1.4

make

A successful build should look like this:1$ROOTSYS/bin/thisroot.sh (Bourne shell versions) or $ROOTSYS/bin/thisroot.csh (C-shell versions)

4

mkdir -p /Users/deckert/devel/proffit_cmake/build

cd /Users/deckert/devel/proffit_cmake/build && cmake /Users/deckert/devel/proffit_cmake

-- The C compiler identification is AppleClang 7.0.2.7000181

-- The CXX compiler identification is AppleClang 7.0.2.7000181

-- Check for working C compiler: /usr/bin/cc

-- Check for working C compiler: /usr/bin/cc -- works

-- Detecting C compiler ABI info

-- Detecting C compiler ABI info - done

-- Detecting C compile features

-- Detecting C compile features - done

-- Check for working CXX compiler: /usr/bin/c++

-- Check for working CXX compiler: /usr/bin/c++ -- works

-- Detecting CXX compiler ABI info

-- Detecting CXX compiler ABI info - done

-- Detecting CXX compile features

-- Detecting CXX compile features - done

-- Found ROOT: /Users/deckert/root5-34/bin/root-config

-- Found wcs: /opt/local/lib/libwcs.a

-- Configuring done

-- Generating done

-- Build files have been written to: /Users/deckert/devel/proffit_cmake/build

/Applications/Xcode.app/Contents/Developer/usr/bin/make -C build all

Scanning dependencies of target proffit

[ 50%] Building CXX object CMakeFiles/proffit.dir/proffit.cpp.o

[100%] Linking CXX executable proffit

[100%] Built target proffit

If successful, the executable will be located in the build subdirectory. If the compilation fails,look carefully for the lines called Found ROOT and Found WCS to make sure that the required librariesare properly found.

2.2 autoconfTo build the software using the autoconf version, again set up ROOT as well as the CFITSIO and WCSlibraries. In the latter case, it is (usually) possible to use the libraries provided with the HEASOFTpackage2. In this case, you should first run the HEASOFT setup script,

setenv HEADAS /path/to/HEASOFT/

source $HEADAS/headas-init.csh

Then untar the bundle, run the configure script, and build the software:

tar xvf proffit_1.4_autoconf.tar.gz

cd proffit_1.4/

ac_stuff/configure

make

Note that when running the software you must set up the LD LIBRARY PATH environment variablesuch that it points to the appropriate libraries:

setenv LD_LIBRARY_PATH "/path/to/PROFFIT/lib:$LD_LIBRARY_PATH"

2http://heasarc.gsfc.nasa.gov/lheasoft/

5

http://heasarc.gsfc.nasa.gov/lheasoft/

2.3 Running Proffit

Once the software is built, launch the software by calling the executable:

$ ] proffit

*******************************

*** Welcome to proffit v1.4 ***

*******************************

Give me a command (for a list of commands, type help)

1 >

When launching the executable, the plotting window automatically opens (see Fig. 2.1). It willbecome callable only when a plotting or fitting command is used.

Figure 2.1: Launching the software in the terminal (left). The plotting window (right) automaticallyopens.

2.4 Scripting Proffit

Instead of the interactive mode, it is possible to run Proffit in scripting mode. In this case, a scriptcontaining a list of Proffit commands (one by line) can be fed to the executable, which will then exe-cute the script and exit the program. To this aim, the script should start with the following command:

scripting yes

This command switches off the interactive plotting window and allows the user to execute allcommands in one go. The script should always finish with the quit command to exit the programand return to the terminal.

To run the script myscript.txt, use:

$ ] proffit < myscript.txt

6

Chapter 3

Reading Data

This chapter describes the main command to read images and save the output of a Proffit session.

3.1 Reading an imageThis is achieved with the readimg command. This is the basic command to load an input image con-taining the data to be analyzed. Note that Proffit is designed to be used with raw images containingthe number of counts per pixel in order to get the proper statistics from the data. The uncertaintiescalculated by Proffitwill not be correct if a modified image (e.g. background subtracted or vignettingcorrected) is provided at this stage.

This is how it looks in a Proffit session:

1 > readimg

File name > mosaic_a1795.fits

The instrument is XMM/EMOS1

Image succesfully loaded

3.2 Reading an exposure map and correcting for vignettingTo correct for vignetting and calculate surface-brightness profiles in the correct units, Proffit alwaysrequires the user to supply an exposure map. When extracting a surface-brightness profile, the datawill be divided by the exposure map. The areas where the exposure is set to 0 are masked during theanalysis. The exposure map must have the same dimensions as the input image.

This is done using the readexp command:

3 > readexp

File name > mosaic_a1795_expo.fits

Exposure map succesfully loaded

For most missions (e.g. XMM, ROSAT), the exposure maps that can be extracted with the stan-dard analysis tools give the effective exposure time per pixel, taking vignetting into account. In thiscase, the surface-brightness profiles will be outputted in the unit of counts per second. For Chandra,the standard image extraction tool outputs an exposure map in units of cm2 s, such that the brightnessprofiles calculated by Proffit will be in unit of photon/cm2/s.

7

In case the input image is not vignetted, this step can be bypassed by using the command readexpnone. In this case, a uniform exposure of 1s is assumed for the extraction of the brightness profiles.

3.3 Reading a background mapA background map can be supplied by the user using the readback command. When extracting asurface-brightness profile, the user will have the possibility of calculating a background profile in thesame region as the image profile and subtracting it from the data. This step is however not requiredto run Proffit.

The command works similarly to readimg and readexp. The background map is expected to begiven in units of counts per pixel.

3.4 Excluding areas in the imageProffit supports SAO DS91-compatible region files to exclude regions of the image (e.g. includingpoint sources or artifacts). This can be achieved using the region command:

4 > region

Region file > s.reg

10 regions will be ignored

Excluding region: circle(952.14,350.791,5.58163)










The supplied region file must be in image coordinates. Circular and elliptical regions are sup-ported. The exposure in the corresponding regions is then set to 0. To revert to the original regions,the exposure map should be loaded again using readexp.

1http://ds9.si.edu/site/Home.html

8

http://ds9.si.edu/site/Home.html

Chapter 4

Extracting Surface-brightness profiles

This chapter describes the routines available within Proffit for the extraction of surface-brightnessprofiles.

4.1 Azimuthally-averaged profilesThe basic tool for extracting surface-brightness profiles is profile. This tool calculates vignetting-corrected profiles from the input image for any custom binning. The uncertainties are calculated byassuming that the probability distribution follows the Poisson distribution for each pixel. The centerof the surface-brightness profile can either be computed automatically or be supplied by the user. Theroutine can be invoked in the following way:

5 > profile

Center (1:centroid, 2:sb peak, 3:user input (image), 4:user input (FK5)) > 2

Bin size (arcsec) > 10

Maximal radius (arcmin) > 39

Logarithmic binning? (y/n) > n

Surface-brightness peak of the cluster (J2000.0): 207.219 26.5934

Exposure at the centre: 308252 sec

Do you want to subtract the background profile? (y/n) y

The various options for providing the center of the profile are shown in the example above. Forcase 1, the centroid of the image is calculated automatically within a given region. In case 2, the pixelwith the highest surface brightness in the image is used as center. In cases 3 and 4, any given centercan be supplied by the user. The coordinates of the center can be given either in image coordinates(option 3) or in WCS format (degrees, option 4).

The following options allow the user to specify the bin size (in arcsec) and the radius out to whichthe profile will be extracted (in arcmin). The user can then specify whether the output profile willlbe extracted in logarithmically-spaced or linearly-spaced bins. Finally, in case a background map hasbeen supplied, the user can choose to extract a background profile with the same binning and subtractit from the data.

4.2 Elliptical annuliProffit allows the user to calculate surface-brightness profiles in elliptical annuli instead of circularannuli. This is achieved with the command ellipse, which works very similarly to profile. In this

9

case, the mean surface brightness is extracted in regions which satisfy the condition

R2 = x2 + (a/b)2y2 = const , with x = x cos θ − y sin θ, y = y cos θ + x sin θ. (4.1)

The required parameters are the same as for profile, with the addition of the ratio between majorand minor axis (a/b) and the orientation angle of the ellipse (θ). The orientation angle is calculatedcounterclockwise from the Right Ascension axis, similarly to what is done in DS9.

4.3 Profiles in sectorsInstead of the whole azimuth, Proffit allows the user to calculate profiles in sectors, either circular orelliptical. This is done using the sector and sectorellipse commands. The use of these routinesis similar to profile and ellipse, with the exception of the position angles used for the extractionof the sectors. The position angles are defined counterclockwise from the Right Ascension axis,following the DS9 convention.

4.4 Growth curvesProffit allows the user to extract cumulative count rate profiles, or growth curves. This is especiallyuseful when looking for the maximum radius out to which a source is detected (see e.g. the case ofthe XXL survey, Pacaud et al. 2016). Calculating growth curves can be done easily in Proffit usingthe growth command, which works similarly to profile.

4.5 Grouping dataAfter a profile has been extracted, it is possible to rebin the data to increase the number of counts orthe signal-to-noise per bin. This is achieved with the group command. This routine can be called inthe following way:

6 > group

What do you want to group (minimum counts: 1, minimum S/N: 2) > 2

Minimum S/N > 3

If option 1 is used, the data will be grouped adaptively to reach a minimum number of counts perbin given by the user. In the second case, the data will be binned to reach a signal-to-noise thresholddefined by the user.

4.6 DeprojectionProffit includes a built-in routine to deproject surface- brightness profiles assuming spherical sym-metry. The routine follows the non-parametric “onion-peeling” algorithm of Kriss et al. (1983) tocalculate the deprojected profile recursively. Edge effects are taken into account using the correctionsintroduced by McLaughlin (1999). A light smoothing is applied to data to avoid the usual “rollercoaster” effect. The routine can be called in the following way:

7 > deproject

Redshift > 0.0622

Error estimation method (1: error propagation; 2: Monte Carlo) > 2

10

The output profile is given in units of volume emission density, usually counts/s/kpc3 (dependingon the unit of the exposure map). Two methods are provided to calculate the uncertainties: direct errorpropagation (1) and Monte Carlo (2). In the latter case, random realizations of the surface-brightnessprofiles are generated and deprojected. The mean and dispersion of the output values are then com-puted to estimate the deprojected profile and its uncertainty. This method is obviously more accurate,but requires a bit more time to calculate.

The output deprojected profile is then displayed in the plotting window. See Chapter 6 for detailson the plotting capabilities of Proffit.

4.7 Density profilesOnce the profiles have been deprojected, it is possible to estimate the 3-dimensional density profileassuming spherical symmetry. This is achieved using the density command. To evaluate the densityprofile, Proffit requires the user to provide the conversion between the measured count rate and theemission measure of the plasma. This conversion depends on the effective area of the telescope, thetemperature, metallicity and redshift of the plasma, the local Galactic absorption, and the energy band.It can be calculated for instance using the APEC model in Xspec (Smith et al. 2001). The conventionused for providing the count rate to emission measure conversion is that of APEC/MEKAL. Namely,the required number is the count rate corresponding to an APEC/MEKAL normalization of 1.0. Anexample Xspec script to calculate this conversion is given here.

The routine can then be invoked in the following way:

8 > density

Redshift > 0.0622

CR to EM conversion > 34.4825

The output gas density profile is given in units of electron number density, assuming ne = 1.21nH

in a fully ionized astrophysical plasma. For more details on the extraction of gas density profiles seeEckert et al. (2012).

The output gas density profile is then displayed in the plotting window. See Chapter 6 for detailson the plotting capabilities of Proffit.

Warning: This routine assumes that the conversion between count rate and emission measure isconstant. This assumption will fail for instance if the emissivity in the chosen energy band stronglydepends on temperature (e.g. above 2 keV) or metallicity (e.g. for low-temperature systems likegroups and ellipticals). In this case, temperature or metallicity gradients will induce significant varia-tions in the count rate at fixed emission measure, and the resulting density profile may be inaccurate.

11

http://www.isdc.unige.ch/~deckert/example_conversion.xcm

Chapter 5

ProffitModels

5.1 Loading a modelProffit includes a wide variety of built-in models for fitting surface-brightness profiles. Models canbe defined using the model command. This command allows the user to set the model that will beused for the fitting procedure and provide initial values for the model parameters. This is achieved inthe following way:

9 > model

model (type help for a list) > beta

beta > 0.7

rc > 2

norm > 0.1

const > 1e-4

Typing help provides the list of available models and their description (see Section 5.2). Thecurrent model and its values can always be displayed using the showmod command. The constparameter is usually interpreted as a constant background level (see backsub in Section 7.2).

5.2 List of available modelsThe following models are now available within Proffit:

• const: Well, just a constant:S (r) = const (5.1)

Free parameters: const

• beta: The standard β-profile (Cavaliere & Fusco-Femiano 1976) plus a constant:

S (r) = norm(1 + (r/rc)2

)(−3β+0.5)+ const (5.2)

Free parameters: β, rc, norm, const

• doublebeta: The sum of two β profiles with the same value of β (Mohr et al. 1999), plus aconstant:

S (r) = norm[(

1 + (r/rc1)2)(−3β+0.5)

+ ratio(1 + (r/rc2)2

)(−3β+0.5)]

+ const (5.3)

Free parameters: β, rc1, rc2, ratio, norm, const

12

• cuspbeta: A β profile with power-law (cuspy) central slope (Pratt & Arnaud 2002), plus aconstant:

S (r) = norm(r/rs)−α(1 + (r/rc)2

)(−3β+0.5+α/2)+ const (5.4)

Free parameters: β, rc, α, rs, norm, const

• gausbeta: A β profile plus a Gaussian plus a constant:

S (r) = norm(1 + (r/rc)2

)(−3β+0.5)+

normg√

2πσexp

(−

(r − µ)2

2σ2

)+ const (5.5)

Free parameters: β, rc, norm, µ, σ, normg, const

• gausdbeta: Double β profile plus a Gaussian plus a constant:

S (r) = norm[(

1 + (r/rc1)2)(−3β+0.5)

+ ratio(1 + (r/rc2)2

)(−3β+0.5)]

+normg√

2πσexp

(−

(r − µ)2

2σ2

)+ const (5.6)

Free parameters: β, rc1, rc2, ratio, norm, µ, σ, normg, const

• power: A projected single power law, plus a constant:

S (r) = norm(r/rs)−α + const (5.7)

Free parameters: α, rs, norm, const

• triplepl: Projected power law with two continuous breaks:

S (r) = norm

r−α1 , if r < rc1

r−α2 , if rc1 < r < rc2

r−α3 , otherwise+ const (5.8)

Free parameters: α1, α2, α3, rc1, rc2, norm, const

• bknpow: Broken power law with a density jump, numerically projected along the line of sight(Owers et al. 2009; Eckert et al. 2016b), plus a constant. This is the model normally used tostudy density discontinuities in X-ray images (shocks and cold fronts).

S (r) = norm∫

F(ω)2d` + const, with ω2 = r2 + `2, (5.9)

where the integration is performed over the line of sight, and F(ω) is the 3-dimensional densityprofile (in arbitrary units):

F(ω) =

ω−α1 , if ω < cutrad1

jumpω−α2 , otherwise (5.10)

Free parameters: α1, α2, cutrad, norm, jump, const

• bknbeta: Similar to bknpow, but the inner part of the profile is defined as a β profile (Rossettiet al. 2007):

F(ω) =

(1 + (ω/rc)2

)−3β/2, if ω < cutrad

1jumpω

−α2 , otherwise(5.11)

Free parameters: β, rc, α2, cutrad, norm, jump, const

13

• triplebkn: Similar to bknpow, but the density profile is a continuous triple power law (Ros-setti et al. 2013):

F(ω) =

ω−α1 , if ω < rc1

ω−α2 , if rc1 < ω < rc2

ω−α3 , otherwise(5.12)

Free parameters: α1, α2, α3, rc1, rc2, norm, const

• backfit: In cases where the subtraction of the background is uncertain, this model allows toadjust the background level included in the provided background map to the data in a source-free region,

S (r) = const + back ∗ backprofile, (5.13)

where backprofile is the background profile calculated from the user-given background map(see Section 3.3).Free parameters: const, back

5.3 Model FluxEasy functions are available to compute the total flux within any given radial range. To this aim,Proffit provides the three following routines:

• flux: Calculate the model flux in the radial range [R1,R2] by integrating the loaded model overthe corresponding spherical shell. The values of R1 and R2 should be given in arcmin. Theroutine can be invoked in the following way:

10 > flux 0 15

Model count rate: 3.18981 counts/sec

Note that this routine does not require the given radial range to lie within the range accessiblewith the data, i.e. extrapolation outside of the fitted range is possible.

• flobs: Calculate the observed flux in the radial range [R1,R2] by measuring the total vignetting-corrected photon flux within the corresponding area. If a background map is provided, thebackground can be optionally subtracted:

11 > flobs 0 15

Maximum detection radius (source intensity similar to bkg): 14.9167

Subtract the background? (y/n)y

Integrated count rate from 0 to 15: 3.18185 +/- 0.00473242 counts/sec

Note that in this case, extrapolation beyond the available radial range is not possible.

• flcomp: For models with multiple components only, calculate the model count rate for eachof the components separately. This feature works with the following models: doublebeta,gausbeta, gausdbeta. It can be invoked in the following way:

12 > flcomp 0 15

Model count rate for component 1: 1.22876 counts/sec

Model count rate for component 2: 1.83224 counts/sec

14

5.4 Simulating an imageGiven a model and an exposure map, Proffit allows the user to simulate a Poisson image for anycustom exposure time. This feature can be useful e.g. to simulate the expected improvement in modelparameters when increasing the exposure time. To simulate the data, the model surface brightnessper pixel is multiplied by the effective exposure for each pixel. The vignetting curve is assumed tofollow the curve encoded within the exposure map loaded with the readexp command. A Poissoniannumber is then generated for each pixel and a simulated image is constructed.

This can be achieved using the fakeit routine, which works in the following way:

13 > fakeit

Total exposure for simulation > 100000

Center (1: image coord, 2: FK5) > 2

RA center (J2000.0) > 207.219

Dec center (J2000.0) > 26.5934

Corresponding pixel coordinates : 942.236 730.97

Image succesfully simulated

As shown in the example above, the required parameters are the exposure time for the simulationand the center of the surface-brightness profile, which can be provided either in image or WCS coor-dinates. The simulated image is then loaded into Proffit, and it can be analyzed in the same way asan observed image.

15

Chapter 6

Plotting routines

This chapter defines the available plotting options provided by Proffit. The plotting interface comesfrom Root and thus it inherits all the options and functionalities of the Root plotting interface.

6.1 Plotting surface brightness profiles

Figure 6.1: Example of data displayed in the plotting window using the plot command. The bluecurve shows the background profile extracted in the same region as for the image. The plotting optionsand tools can be accessed using the interactive menus.

The basic plotting command is plot. After having extracted a surface-brightness profile, thiscommand allows the user to display the extracted data. The profile is then displayed in the plottinginterface, which is rendered clickable and editable (see Fig. 6.1). A number of plotting options arethen available using the Root interactive interface, in particular by clicking on View → Editor and

16

View → Toolbar options. The plots can be exported directly in image format using the File →Save As... menu.

Note that the command prompt is not accessible when the interactive plotting interface is active.To return to the command prompt, click File → Quit ROOT.

Proffit also includes the following alternative plotting routines, which work in the same way asplot:

• plotmod: Display the current model.

• plotcounts: Plot the number of counts per bin in the current brightness profile.

• plotmodcounts: Plot the number of counts per bin predicted by the current model.

• plotgrowth: Plot the current growth curve.

• plotgrmod: Plot the growth curve together with the growth curve predicted by the currentmodel.

6.2 Plotting optionsIn addition to the numerous Root plotting options, which can be accessed via the interactive plottingwindow, Proffit provides a few command-line tools to customize the appearance of the output figures.The following options are available:

• logx and logy: Turn on/off the log scale on the X and Y axes. This option can also bechanged interactively in the plotting window by right clicking on the plotting window andsetting SetLogx/SetLogy. By default, log scale is set to yes.

• kpc: Calculate the conversion between angular scale (arcmin) and physical scale (kpc) at agiven redshift (uses H0 = 70 km/s/Mpc, Ωm = 0.3, ΩΛ = 0.7). The user can then decidewhether the X axis will be plotted in angular or physical scale.

• r500 and r200: Compute the expected values of R500 and R200 using the mean cluster tem-perature and the mass-temperature relation of Arnaud et al. (2005). The user can then decidewhether the X axis will be rescaled by the calculated value of R500/R200.

17

Chapter 7

Fitting Data

This chapter describes the various fitting algorithms available within Proffit and various fitting func-tionalities.

7.1 Fitting toolsAfter having extracted a surface-brightness profile (see Chapter 4) and defined a model (see Chapter5), the model can be easily fitted to the data and the best-fitting parameters can be estimated. Forall cases, Proffit uses maximum-likelihood estimators and uses CERN’s Minuit1 library for multi-dimensional optimization (in particular the MIGRAD and MINOS algorithms). Two main fitting routinesare provided by Proffit: fit and fitcounts.

• fit: This routine adjusts the model directly to the surface-brightness data. This method hasthe advantage of being numerically stable and fast. However, this routine uses the Gaussianapproximation when propagating the uncertainty in the surface brightness from the data, whichfails when the number of counts per bin is too small (typically < 20). To alleviate this issue, thedata can be grouped together (see Section 4.5). Or, see fitcounts.

• fitcounts: In this case, the model count rate is folded through the vignetting curve and thearea of each annulus to predict the number of counts per annulus, which is then fitted directly tothe raw data. This method has the main advantage of preserving the statistical properties of thedata, thus providing accurate parameter estimation and error calculation, even in the low countrate regime. It is however slightly slower and less stable than fit.

The two routines can be called in exactly the same way in Proffit. After performing the optimiza-tion, the best-fit results are displayed within the command-line interface, and the data are displayed inthe plotting window together with the best-fit model. The bottom panel displays the model residuals(χ = (S i − f (ri))/σi). An example is provided in the following:

14 > fit

FCN=128.138 FROM MIGRAD STATUS=CONVERGED 455 CALLS 456 TOTAL

EDM=1.83239e-06 STRATEGY= 1 ERROR MATRIX ACCURATE

EXT PARAMETER STEP FIRST

NO. NAME VALUE ERROR SIZE DERIVATIVE

1 beta 6.16463e-01 6.92100e-03 1.63350e-06 -1.09296e+00

2 rc1 2.38101e+00 9.10056e-02 2.86688e-05 1.35922e-01

1http://www.fresco.org.uk/minuit/cern/minmain.html

18

http://www.fresco.org.uk/minuit/cern/minmain.html

3 rc2 6.64173e-01 1.12416e-02 4.94191e-06 -1.19336e-01

4 ratio 1.71134e+01 9.19129e-01 2.32547e-04 -4.22450e-03

5 norm 3.08541e-02 1.69829e-03 2.86953e-07 1.38161e+00

6 const -1.28195e-04 1.29638e-05 2.22491e-08 -1.38994e+02

Chi-squared = 128.138 for 70 d.o.f

Reduced chi-squared = 1.83054

Null-hypothesis probability = 2.78813e-05

Figure 7.1: Example of fit results displayed in the plotting window. The top panel shows the surface-brightness profile and the best-fit model (blue curve). The residuals are shown in the bottom panel.

7.2 Fitting optionsA number of options can be defined when fitting the data:

• Model parameters can either be fixed or released using the fixpar and thawpar commands.The value of a parameter can be manually changed using the newpar command. For instance,to set the const parameter (# 6) to 0 in the above example and fix the parameter while fitting,

15 > newpar 6 0.0

16 > fixpar 6

The fitting procedure can the be performed again with parameter 6 set to 0.

• The fitting range can be defined using the limits command. This routine sets a radial range[Rmin,Rmax] for which the fitting procedure will be performed, i.e. all the data points withr < Rmin or r > Rmax will be ignored. By default, the entire radial range is used. The commandworks in the following way:

19

17 > limits 1 15

• In case the data contain a source-free (or background-dominated) region, the const parameteris usually interpreted as a constant background brightness. In this case, Proffit provides theuseful backsub command, which subtracts the fitted value for the const parameter from thesurface-brightness profile. The uncertainties in the background are added in quadrature. Forinstance, if a source-free region is available (in this example, the range 33-39 arcmin), thesurface-brightness in this region can be fitted using the const model and subtracted from thedata:

18 > model

model (type help for a list) > const

const > 1e-4

19 > fit

FCN=37.3941 FROM MIGRAD STATUS=CONVERGED 14 CALLS 15 TOTAL

EDM=5.51538e-24 STRATEGY= 1 ERROR MATRIX ACCURATE

EXT PARAMETER STEP FIRST

NO. NAME VALUE ERROR SIZE DERIVATIVE

1 const 9.07378e-05 9.81065e-06 1.00000e-05 3.38536e-07

Chi-squared = 37.3941 for 35 d.o.f

Reduced chi-squared = 1.0684

Null-hypothesis probability = 0.359714

20 > backsub

• If the level of systematic uncertainties is known a priori, it can be set using the syst command:

21 > syst

Systematic error (%) > 2

A flat level of systematic uncertainty is then added in quadrature and taken into account in allfitting procedures and error estimations.

7.3 Fitting statisticsProffit provides two different likelihood functions: chi2 and cash. The fit statistics can be easilyset using the statistics command. If statistics is set to chi2 (which is the default option), theGaussian approximation is used. Let S i be the measured surface brightness in annulus i and σi thecorresponding Gaussian error. The likelihood function to be optimized is

−2 logL = χ2 =

N∑i=1

(S i − f (ri))2

σ2i

, (7.1)

where f (ri) is the model evaluated at radius ri.

If, on the other hand, statistics is set to cash, the Poisson likelihood derived by Cash (1979)is used. Let Ai, Ti be the area and the effective exposure time of annulus i. We set Fi = f (ri)AiTi thepredicted number of counts in the annulus. The Poisson likelihood is then given by

20

−2 logL = 2N∑

i=1

Fi −Ci log Fi −Ci + Ci log Ci, (7.2)

where Ci is the observed number of counts in annulus i. Note that this statistics only works whenusing the fitcounts command. This is the recommended method when dealing with a low numberof counts.

7.4 Error estimation and contour plotsProffit allows accurate error estimation of the model parameters using the MINOS algorithm. This isachieved using the error command, which works in the following way:

21 > error

Parameter > 1

Confidence level (%) > 68.3

Confidence interval (68.3%) : 9.0683e-05 ( -9.82416e-06 , 9.82416e-06)

To investigate correlations between model parameters, Proffit also includes the contour com-mand, which allows the user to plot 1, 2, and 3σ contours in the 2-dimensional parameter space of acouple of parameters:

22 > contour

Parameter 1 > 1

Parameter 2 > 2

An example of output is shown in Fig. 7.2.

7.5 PSF modelingWhen fitting small-scale features such as density jumps or high-redshift cool cores, it can be usefulto model the point-spread function (PSF) of the instrument. This can be achieved in Proffit bycalculating a matrix representing the instrumental PSF. During the fitting procedure, the model isconvolved with the PSF matrix, and thus the measured parameters are deconvolved from the PSF.Such a PSF can be reconstructed using the psf command,

23 > psf

Warning: This may take a while

PSF profile (gaus/king) > king

Number of ray-tracing photons per bin > 1e6

R0 (arcsec) > 5.5

Slope > 1.48

The implemented algorithm uses a “ray-tracing” approach. Namely, for each radial bin, a largenumber of ray-tracing photons (here 106) is simulated, and the position of each simulated photon israndomized according to the given properties of the PSF. The fraction of photons falling into eachradial bin is then computed and a convolution matrix is constructed. An example of convolutionmatrix is shown in Fig. 7.3. For more details on the algorithm, see Appendix C of Eckert et al.(2016a).

The radial profile of the PSF for the simulation can either be given as a Gaussian (in which casethe standard deviation should be provided) or as a King profile, as for the example given above. Therelevant parameters are usually provided within the calibration files of the relevant telescope/mission.

21

Figure 7.2: Contour plot showing the usual correlation between β and rc in the β model. The curvesshow 1, 2, and 3σ contours.

Figure 7.3: Example of PSF convolution matrix calculated using the psf routine (see Appendix C ofEckert et al. 2016a).

22

Chapter 8

Image analysis tools

In addition to the available tools to study surface-brightness profiles, Proffit also includes a numberof image analysis tools which are described in this chapter.

8.1 Profiles in sectors and azimuthal scatterA couple of tools are available to study the azimuthal variations of the surface brightness at fixedradius in the data and quantify the level of asymmetries: allsectors and scatter.

• allsectors: This tool allows to extract surface-brightness profiles in a user-given number ofsectors covering the entire azimuth and display the results to look for asymmetries in the data.The binning and center defined in the current surface-brightness profile are used (see Chapter4). For instance, in the following example we divide the azimuth into 6 sectors (correspondingto an opening angle of 60) and plot the resulting profiles on top of each other (Fig. 8.1):

24 > allsectors

Number of sectors > 6

Subtract the background? (y/n) > n

In this case, the routine highlights an excess surface brightness in the sector with position angle60-120 (red data points), which clearly shows an asymmetry in the North direction.

• scatter: To quantify the level of asymmetry in the provided image, Proffit allows the user tocompute the azimuthal scatter (Vazza et al. 2011; Eckert et al. 2012). For a given number ofsectors (N), the azimuthal scatter is defined as the standard deviation of the surface brightnessat fixed radius,

Σ2 =1N

N∑i=1

(S i − 〈S 〉)2

〈S 〉2, (8.1)

where S i is the measured surface brightness in sector i and 〈S 〉 is the azimuthally-averagedsurface brightness at the same radius. A large value of Σ indicates a very asymmetric image,while Σ = 0 means that no significant deviations from spherical symmetry are observed. Thestatistical scatter is evaluated analytically and subtracted from the data (see Appendix A ofEckert et al. 2012). The azimuthal scatter is constructed for each radius using the same binningas the current brightness profile. For example, the following command calls the routine tocalculate the azimuthal scatter with N = 12 sectors:

23

Radius [arcmin]

2−10 1−10 1 10

]-2

arc

min

-1S

B [

cou

nts

s

3−10

2−10

1−10

1

All sectors

Sector 0-60

Sector 60-120

Sector 120-180

Sector 180-240

Sector 240-300

Sector 300-360

Figure 8.1: Result of the allsectors command. Here, the profiles in 6 sectors of 60 opening arecompared.

25 > scatter


Subtract the background? (y/n) > n

8.2 Median in sectorsIn case the surface-brightness distribution is strongly asymmetric, the mean surface-brightness in anannulus is a biased tracer of the density (the “clumping bias”, Nagai & Lau 2011). One way to recovera surface brightness that is unaffected by this effect is to sample the surface-brightness distributionin an annulus and compute the median of the distribution (Eckert et al. 2015). This can be achievedin Proffit by calculating the surface brightness in sectors covering the entire azimuth and taking themedian value at each radius. The mediansb tool implements this method,

26 > mediansb


Subtract the background? (y/n) > y

In this example, the surface brightness is calculated in 16 sectors with the same center and binningas for the current profile. The resulting profile is then displayed in the plotting window.

8.3 Residual imagesIt can be useful to view an image of the deviations from the best-fit model to identify regions de-viating from spherical symmetry. Proffit provides two different tools to extract and save images of

24

1 5

-1.2

-0.9

-0.6

-0.3

0.0015

0.3

0.6

0.9

1.2

1.5

5

-4

-3

-2

-1

0.0049

1

2

3

4

5

Figure 8.2: Difference between the output of savedeviations (left) and savescat (right).savedeviations allows to detect small features at maximum resolution, while savescat is bet-ter to identify large-scale deviations from spherical symmetry.

deviations from spherical symmetry. These tools are called savedeviations and savescat. Es-sentially, savedeviations should be used to look for small-scale features, while savescat is moreefficient for identifying large-scale deviations from spherical symmetry. In Fig. 8.2 the output of thetwo routines for the same dataset is shown.

• savedeviations: This tool computes the pixel-by-pixel residuals (in units of σ) between thedata and the current model (spherical or elliptical). The resulting image is saved into an outputFITS image. The use of this tool is particularly simple:

27 > savedeviations

File name > deviations.fits

Image succesfully written

An example output image after obtaining the best fit with a doublebeta model is shown in theleft-hand panel of Fig. 8.2.

• savescat: In this case, the surface-brightness profile is extracted in a user-given number ofsectors (similar to scatter) and the deviations with respect to the azimuthal mean are calcu-lated for each sector and each radial bin, in a way similar to the use of scatter (see Section8.1). Thus, unlike savedeviations, this tool is model independent. The resulting image canthen be saved into an output FITS image. An example of output is shown in the right-handpanel of Fig. 8.2.

28 > savescat


Subtract the background? (y/n) > y

File name > scatimg.fits

Image succesfully written

25

8.4 Dynamical state indicatorsIt is now well known that the X-ray morphology of a cluster depends on its dynamical state (e.g. Rasiaet al. 2013). Proffit allows to compute easily some widely-used dynamical state indicators based onX-ray morphology.

• centroidshift: The centroid shift w (Mohr et al. 1993) evaluates the changes in the positionof the centroid of the X-ray image when changing the used aperture from a radius Rmax to thecenter of the cluster. It is defined as

w =1

Rmax

√∑(∆i − 〈∆〉)2

N − 1, (8.2)

where N is the number of apertures considered and ∆i designates the distance between thecentroid calculated at iteration i and the one obtained with the original aperture Rmax. The valueof the centroid shift thus depends on Rmax, which should be supplied by the user:

29 > centroidshift

Maximum radius (arcmin) > 15

Centroid shift: 0.00349674

Large values of w (> 0.03) indicate that the centroid strongly depends on the used aperture, i.e.the cluster is morphologically disturbed. Conversely, low values of w are found in clusters withregular morphology.

• csb: Surface-brightness concentration (cS B, Santos et al. 2008) is the ratio between the totalflux calculated within the core region to the flux in a larger region encompassing a signifi-cant fraction of the cluster volume. Proffit uses the current model to calculate cS B using thedefinition of Santos et al. (2008),

cS B =F(r < 40 kpc)

F(r < 400 kpc). (8.3)

For flux calculations, see Section 5.3. This tool requires the user to provide the cluster redshiftto compute the relation between angular and physical scales (using H0 = 70 km/s/Mpc, Ωm =

0.3, ΩΛ = 0.7):

30 > csb

Cluster redshift > 0.0622

Integrating the flux between 40 and 400 kpc (0.556186 and 5.56186 arcmin)

Surface-brightness concentration: 0.164741

High values of cS B (> 0.07) are found in centrally-peaked, cool-core cluster, which trace dy-namically relaxed systems. Conversely, merging systems typically show low values of cS B.

• ellipticity: Ellipticity of the surface-brightness distribution following the definition ofHashimoto et al. (2007),

ε = 1 −ba, (8.4)

with a, b the major, respectively minor axis of the ellipse. This tool can also be useful to searchfor major/minor axes of the surface-brightness distribution.

26

Chapter 9

Saving the results

The results of a Proffit session can be written in an output file using the save command, in ASCII,FITS, or Root format:

31 > save

Format (txt/root/fits) > fits

Output file name > a1795_test.fits

Results saved in file a1795_test.fits

It is recommended to save the results in FITS or Root format, as the ASCII option has limitedcapabilities.

• FITS: This is the most complete output format provided in Proffit. The output FITS file willcontain several extensions. The first extension contains the current extracted profile, number ofcounts per bin, deprojected profile, etc. In the header of the file, the parameters used to extractthe profile (center, position angles, ellipse parameters) are stored. The second extension con-tains the current model and its best-fit parameters. Finally, if a PSF matrix has been generated,it is saved in the third extension. An example of output FITS file is shown in Fig. 9.1.

Figure 9.1: Example of output file saved in FITS format, viewed with the fv utility.

• ROOT: In this case, the extracted products are saved as Root objects in TFile format, whichcan be viewed and modified within Root (or pyRoot). Profiles are stored as TH1F objects, asin the example shown below. Similarly to FITS, if a deprojected profile, PSF matrix, and so onhave been extracted, they are stored in the output TFile.

root [1] _file0->ls()

TFile** sb.root

TFile* sb.root

KEY: TTree t;1 t

KEY: TH1F chi;1 chi

27

KEY: TH1F back;1 back

KEY: TH1F profile;1 profile

KEY: TF1 model;1

• ASCII: In this case, limited information (surface-brightness profile, model, number of countsper bin) are stored in an output ASCII file.

28

Bibliography

Arnaud, M., Pointecouteau, E., & Pratt, G. W. 2005, A&A, 441, 893

Cash, W. 1979, ApJ, 228, 939

Cavaliere, A. & Fusco-Femiano, R. 1976, A&A, 49, 137

Eckert, D., Ettori, S., Coupon, J., et al. 2016a, A&A, 592, A12

Eckert, D., Jauzac, M., Vazza, F., et al. 2016b, MNRAS, 461, 1302

Eckert, D., Molendi, S., & Paltani, S. 2011, A&A, 526, A79+

Eckert, D., Roncarelli, M., Ettori, S., et al. 2015, MNRAS, 447, 2198

Eckert, D., Vazza, F., Ettori, S., et al. 2012, A&A, 541, A57

Hashimoto, Y., Henry, J. P., & Bohringer, H. 2007, MNRAS, 380, 835

Kriss, G. A., Cioffi, D. F., & Canizares, C. R. 1983, ApJ, 272, 439

McLaughlin, D. E. 1999, AJ, 117, 2398

Mohr, J. J., Fabricant, D. G., & Geller, M. J. 1993, ApJ, 413, 492

Mohr, J. J., Mathiesen, B., & Evrard, A. E. 1999, ApJ, 517, 627

Nagai, D. & Lau, E. T. 2011, ApJ L, 731, L10+

Owers, M. S., Nulsen, P. E. J., Couch, W. J., & Markevitch, M. 2009, ApJ, 704, 1349

Pacaud, F., Clerc, N., Giles, P. A., et al. 2016, A&A, 592, A2

Pratt, G. W. & Arnaud, M. 2002, A&A, 394, 375

Rasia, E., Meneghetti, M., & Ettori, S. 2013, The Astronomical Review, 8, 40

Rossetti, M., Eckert, D., De Grandi, S., et al. 2013, A&A, 556, A44

Rossetti, M., Ghizzardi, S., Molendi, S., & Finoguenov, A. 2007, A&A, 463, 839

Santos, J. S., Rosati, P., Tozzi, P., et al. 2008, A&A, 483, 35

Smith, R. K., Brickhouse, N. S., Liedahl, D. A., & Raymond, J. C. 2001, ApJ L, 556, L91

Vazza, F., Roncarelli, M., Ettori, S., & Dolag, K. 2011, MNRAS, 413, 2305

29

Appendix A

Appendices

A.1 AcknowledgementsThe development of Proffit is the result of several years of scientific development for my own benefit.However, the whole idea of writing an easy-to-use, self-contained code for the analysis of surface-brightness profiles was given to me by my good friend Fabio Gastaldello when we were sharing anoffice at IASF Milano. Big thanks to Fabio for pushing me to pursue this idea, which has now lead toa coherent and complete package.Thanks to Hubert Degaudenzi for his help with the C++ building tools, in particular cmake.I would also like to thank Geogiana Ogrean for using the code in an early stage and reporting bugsand bug fixes to me.Big thanks to Stephane Paltani for supporting me through all these years.Finally, I would like to thank all the users for their positive feedback and bug reports.

30

Documents

Proffit User Guide Version 1deckert/proffit_guide.pdf · This chapter describes the main command to read images and save the output of a Proffit session. 3.1 Reading an image This