The following software is freely available. However, if you use it for published research, you are requested to cite the paper where the method is described, highlighted in red. You are not allowed to redistribute any of these programs (modified or not) without explicit permission from the author.
NOTE: an outdated version of my programs in the language is also available HERE, however I strongly recommend the Python version.
My packages above are on the Python Package Index (PyPI).
See my review on galaxy structure and evolution:
Illustration of the steps involved in the MGE fit to the S0 galaxy NGC 4342 using the MgeFit package. The figures were produced by the mge_fit_example.py script included in the Python distribution of the software.

This software obtains an accurate MultiGaussian Expansion (MGE) parameterizations (Emsellem et al. 1994; Cappellari 2002) for a galaxy surface brightness with the fitting algorithm of Cappellari (2002, MNRAS, 333, 400). Given that Gaussians are not orthogonal functions, MGE fits are in general strongly degenerate, with difficult global convergence, but the MGE_FIT_SECTORS
method solves all problems, making MGE fitting an automated, reliable and robust process.
See Cappellari et al. (2013) for a large scale application of this software to the study of the M/L ratio and the Fundamental Plane of earlytype galaxies and Cappellari et al. (2012, Nature) for an application to the study of the stellar IMF. The MGE parameterization is useful in the construction of realistic dynamical models of galaxies (see JAM modelling below), for PSF deconvolution of images, for the correction and estimation of dust absorption effects, or for galaxy photometry.
MgeFit
package in The source code of the MgeFit
package, including four test galaxy images, is on the Python Package Index (PyPI). It was last updated on the 24 June 2021 and the changes are documented in the CHANGELOG.
How to install: Use "pip install mgefit
". Without write access to the global sitepackages
directory, use "pip install user mgefit
".
Usage examples: are in the directory "examples" inside the main package folder inside sitepackages.
MgeFit
requires the scientific core packages NumPy, SciPy and Matplotlib, and the examples use Astropy to read FITS images (it was tested with Python 3.10 using Numpy 1.22, SciPy 1.8 and Matplotlib 3.5).
Also included in the MgeFit
package is the mpfit
implementation by Markwardt (2009) of the MINPACK (Moré, Garbow & Hillstrom, 1980) LevenbergMarquardt nonlinear leastsquares optimization algorithm. The mpfit
routine was translated into Python by Mark River and updated here for NumPy by Sergey Koposov. I modified the latter version to support Python 3 and fixed some numerical instabilities. Note that MgeFit
uses the mpfit
implementation instead of the SciPy.optimize.least_squares
one, because of the ability of mpfit
to tie parameters or keep them fixed.
NOTE: I would appreciate if you drop me an email (address at the bottom) when you download the above procedures.
2. Jeans Anisotropic Modelling (JAM) 

Jeans Anisotropic Modelling of the dynamics, stellar kinematics or proper motions of axisymmetric or spherical galaxies 
Figure 1: Examples of JAM datamodel comparisons. The bisymmetrized SAURON stellar kinematics of 6 Elliptical (left) and 6 S0 (right) fastrotator earlytype galaxies is compared to the predictions of the anisotropic Jeans models with JAM. The kinematics varies widely for different galaxies, yet these twoparameters (!) models are able to correctly predict the shape of a pair of twodimensional functions (V and V_{rms}), once the observed surface brightness is given (this is fig.10 of Cappellari 2016, ARA&A). 
Figure 5: This is the first dynamical modelling (using JAM_{sph}) of the highquality Gaia DR2 kinematics of the Milky Way, with full 6D phasespace coordinates. For the first time, there are no dynamical degeneracies and the model has no freedom to fit the data, yet all features are reproduced. This is an important test of galaxy dynamics. (figure taken from Nitschai et al. 2020) 
Figure 4: Validating black hole (BH) recovery with JAM. A detailed JAM modelling accurately recovers the "benchmark" BH mass in NGC4258, as inferred from maser observations (figure taken from Drehmer et al. 2015). An equally good agreement was found with JAM for the "benchmark" BH in the Milky Way (Sec.4.1.2 of FeldmeierKrause et al. 2017). Other successful BH comparisons, between JAM and Schwarzschild's method, were presented e.g. in Cappellari et al. (2010), for 25 galaxies, and Krajnović et al. (2018), for 7 galaxies. 
Figure 3: The JAM package includes an implementation for both the cylindricallyaligned solution JAM_{cyl} of Cappellari (2008) and the sphericallyaligned solution JAM_{sph} of Cappellari (2020). Comparisons between these two extreme solutions allows for a robust assessment of possible systematic modeling uncertainties (see Cappellari 2020 for some examples). 
Figure 2: Including both proper motions and radial velocities. Top Panels: the components of the symmetric velocity second moment tensor measured from the particles of a realistic Nbody simulation. Bottom Panels: the corresponding JAM model predictions. Note the striking agreement! (Cappellari 2012) 
The JAM software in this section can be used for the dynamical modelling of galaxies or other gravitationallybound systems of particles. It was used e.g. to measure the mass of supermassive black holes in galaxies, to infer their darkmatter content or to measure galaxy masses and density profiles.
JAM implements a solution of the Jeans equations which allows for orbital anisotropy (threeintegrals distribution function) and also provides the full second moment tensor, including both proper motions and radial velocities, for both axisymmetric (Cappellari 2012) and spherical geometry (Cappellari 2015). The technique was introduced in Cappellari (2008), for the cylindricallyaligned case, and in Cappellari (2020), for the sphericallyaligned case (Figure 3), and I called it the Jeans Anisotropic Modelling (JAM) method. It relies on the MultiGaussian Expansion parameterization (Emsellem et al. 1994; Cappellari 2002) for the galaxy surface brightness.
With the addition of a single extra parameter β_{z}=1(σ_{z}/σ_{R})^{2} (in cylindrical alignment) or β=1(σ_{θ}/σ_{r})^{2} (in spherical alignment), the simple and userfriendly threeintegrals JAM method already provides a dramatic improvement over the classic but less general twointegrals f(E,L_{z}) Jeans (1922) models. However JAM also allows for tangential anisotropy γ=1(σ_{φ}/σ_{R})^{2} and/or for spatially varying anisotropy (different for every MGE Gaussian). The JAM models provide a remarkably good descriptions of stateoftheart integralfield stellar kinematics of real galaxies (Figure 1). This makes the technique well suited to measure the inclination, the dynamical M/L and angular momenta of earlytype fastrotators and spiral galaxies.
The JAM routines are designed for axisymmetric or spherical geometry, (i) they can provide the proper motion dispersion tensor (Cappellari 2012; Cappellari 2015; Figure 2), (ii) allow for the inclusion of dark matter, (iii) variable stellar M/L, (iv) spatially varying anisotropy and (v) multiple kinematic components and (vi) supermassive black holes (BH; Figure 4). The JAM package also includes a routine to compute the circular velocity from the MGE models. Some sample applications of the JAM method are given below:
To construct dynamical models with the JAM method one needs to describe the galaxies surface brightness via the MultiGaussian Expansion parametrization using my MGE_FIT_SECTORS package above.
Figure 7: JAM works better than Schwarzschild on simulated galaxies. A detailed comparison between the recovered total mass of Illustris simulations using the JAM and Schwarzschild's (SCH) methods, found that JAM recovers the known enclosed total masses more accurately than SCH (this is Fig.4 from Jin et al. 2019). Note the smaller scatter and error bars of the green symbols (JAM) with respect to the red ones (SCH). 
Figure 6: JAM works better than Schwarzschild on real galaxies. An extensive and detailed comparison between the circular velocity V_{CO} measured from CO gas and the corresponding circular velocity V_{JAM} and V_{SCH} inferred from dynamical modelling of integralfield stellar kinematics using the JAM and Schwarzschild's (SCH) methods, for 54 real disk galaxies with CALIFA data, found that JAM recovers the galaxy densities more accurately than SCH (this is Fig.8 from Leung et al. 2018). The smaller JAM scatter is particularly evident at large radii (inside the red boxes), where the gas V_{c} is better determined. 
The source code of the JamPy
package is on the Python Package Index (PyPI). It was last updated on the 7 July 2021 and the changes are documented in the CHANGELOG. The Python version of the mean velocity calculation is about two orders of magnitude faster than the IDL version, due to a redesign of the lineofsight integration approach and NumPy's broadcasting.
How to install: Use "pip install jampy
". Without write access to the global sitepackages
directory, use "pip install user jampy
".
Usage examples: are in the directory "examples" inside the main package folder in sitepackages.
Also required is my plotbin package (automatically installed by pip
).
The JamPy
package requires the scientific core packages NumPy, SciPy and Matplotlib (it was tested with Python 3.10 using Numpy 1.22, SciPy 1.8 and Matplotlib 3.5).
NOTE: I would appreciate if you drop me an email (address at the bottom) when you download the above procedures.
Left: Fourcoloring of Voronoi binning; Middle: Unbinned versus Voronoi binned stellar kinematics from IntegralField Spectroscopy (Cappellari & Copin 2003); Right: Voronoi binning of abundance from Xray data (Sanders et al. 2004). 
3. The Voronoi binning method (VorBin) 

Adaptively spatial bin twodimensional data to a constant signaltonoise ratio per bin 
The Voronoi Binning method of Cappellari & Copin (2003, MNRAS, 342, 345) optimally solves the problem of preserving the maximum spatial resolution of general twodimensional data (or higher dimensions), given a constraint on the minimum signaltonoise ratio.
The Voronoi binning method has been applied to a variety of types of data. A review of the concepts and applications to (i) Xray data, (ii) integralfield spectroscopy, (iii) FabryPerot interferometry, (iv) Nbody simulations, (v) standard images and (vi) other regularly or irregularly sampled data is given in Cappellari (2009).
The source code of the VorBin
package, with examples and instructions, is on the Python Package Index (PyPI). It was last updated on the 19 February 2020 and the changes are documented in the CHANGELOG.
How to install: Use "pip install vorbin
". Without write access to the global sitepackages
directory, use "pip install user vorbin
".
User Manual: Is available on THIS PAGE (most uptodate) or as PDF.
Usage example: is in the package folder inside sitepackages.
The VorBin
package requires the scientific core packages NumPy, SciPy and Matplotlib (it was tested with Python 3.10 using Numpy 1.22, SciPy 1.8 and Matplotlib 3.5).
My optional plotbin package contains routines to visualize Voronoi 2Dbinned or unbinned data like in the figures above.
NOTE: I would appreciate if you drop me an email (address at the bottom) when you download the above procedures.
4. Penalized PixelFitting (pPXF) 

Extract the stellar or gas kinematics and stellar population from galaxy spectra via full spectrum fitting 
Figure 1: Stellar and gas kinematics fit with pPXF. The black line (mostly hidden by the fit) is the relative flux of the observed spectrum. The red line is the pPXF fit for the stellar component, while the orange line is a fit to the gas emission lines. The green symbols at the bottom are the fit residuals, while the blue lines is the gasonly bestfitting spectrum. The main absorption and emission features are indicated at the top of the plot (taken from Cappellari 2017). 
This software implements the Penalized PixelFitting method (pPXF) to extract the stellar or gas kinematics and stellar population from absorptionline spectra of galaxies, using a maximum penalized likelihood approach. The method was originally described in Cappellari & Emsellem (2004) and was significantly upgraded in subsequent years and particularly in Cappellari (2017, MNRAS, 466, 798). The method is very general and robust. For this reason it was applied to a variety of situations. The following key features are implemented in the current pPXF program:
See Emsellem et al. (2004, SAURON), Cappellari et al. (2011, ATLAS^{3D}), FalconBarroso et al. (2017, CALIFA), van de Sande et al. (2017, SAMI) and Westfall et al. (2019, MaNGA) for some ever increasing largescale applications of the pPXF method to the measurement of the stellar or gas kinematics of galaxies. Cappellari et al. (2009) demonstrates the use of pPXF to measure the stellar velocity dispersion of a sample of passive galaxies at redshift z~2.
The ability of the pPXF method to fit a large set of stellar templates together with the kinematics allows the template mismatch problem to be virtually eliminated. This is particularly useful given the current availability of large stellar libraries spanning wide ranges of physical parameters and having good spectral resolution. Excellent results have been obtained by using a few hundred template stars with pPXF, from which generally about 1020 are selected by the program to provide detailed fits to high S/N galaxy spectra. An extensive account of the available stellar libraries is maintained by David Montes. An incomplete list of libraries that have been successfully used with pPXF for the kinematics extraction is given below:
Figure 2: Starformation history with pPXF. The plot shows the starformation history of ATLAS^{3D} galaxies, in bins of different galaxy stellar mass. This was recovered with pPXF using regularization (taken from McDermid et al. 2015).  Figure 3: Population and gas with pPXF. The plots show the pPXF fit to two MaNGA spectra for the spiral galaxy NGC2916 (from SDSS DR13). The Left Spectrum is located in the galaxy bulge, while the Right Spectrum is in the disk. The "Mass Fraction" maps, below each spectrum, show the corresponding pPXF solution in different age and metallicity intervals. The Left Maps have no regularization. These are the minimum χ^{2} solutions which all nonregularized fullspectrum fitting programs must return, if they properly converge. However these solutions are noisy and arbitrary as this inverse problem is illposed. Regularization is a standard approach to solve illposed problems (see Cappellari 2017, Sec.3.5). The Right Maps show the pPXF regularized solutions: here one can clearly recognize the change in the relative contribution of bulge (∼10Gyr) and disk (∼1Gyr) stars at the two spatial locations. The fits used as templates a subset of 24(Ages)×6([M/H])=144 MILES model spectra by Vazdekis et al. (2010), and 12 gas emission lines. 
pPXF is the most efficient, reliable and flexible public implementation of the so called "FullSpectrum Fitting" method to study stellar populations. This technique has replaced the traditional use of linestrength indices, due to the availability of highquality model spectra. pPXF was designed to be independent on any specific set of stellarpopulation models and has already been used with nearly every available one. An usage example, using the Vazdekis's models, is included with permission in the pPXF package. This can be easily adapted by users to employ pPXF with other stellarpopulation models. An incomplete list of the main stellar population models that were used with pPXF is the following:
pPXF allows one to extract multiple kinematic components, with different stellar populations, from a spectrum. Gas emission lines can be fitted simultaneously, avoiding the need for masking them. This is particularly useful when studying the stellar population of galaxies with prominent emission lines (e.g. the Balmer lines) filling important absorption features (see Figures 1 and 3).
Please also cite the source of the EMILES stellar population models (Vazdekis et al. 2016) if you use pPXF with the included library of spectral templates.
A number of similar, but less optimized and less general implementations of the Full Spectrum Fitting method exist (e.g. STARLIGHT by Cid Fernandes et al. 2005; STECMAP by Ocvirk, et al. 2006; VESPA by Tojeiro et al. 2007). A major limitation of most available codes is that they ignore the fact that the recovery of the star formation history and chemical composition distribution from the galaxy spectra is an illposed problem. A standard technique to solve illposed problems is regularization (see Cappellari 2017, Sec.3.5), which is invoked in pPXF via the keyword REGUL. Regularization is essential to make sense of the derived star formation histories and to explore model degeneracies (see Figure 2 and 3). The regularized solution has a simple Bayesian interpretation: it represents the most likely solution for the weights, given an adjustable prior on the amplitude of the fluctuations.
See Cappellari et al. (2012, Nature) for an early pPXF study of the M/L and IMF of stellar populations.
The source code of the ppxf
package, with examples and instructions, is on the Python Package Index (PyPI). It was last updated on the 10 February 2021 and the changes are documented in the CHANGELOG.
How to install: Use "pip install ppxf
". Without write access to the global sitepackages
directory, use "pip install user ppxf
".
User Manual: Detailed documentation is on THIS PAGE (always up to date) or from here as PDF ppxf_manual.pdf.
Usage examples: Notebooks ppxf
examples are available HERE. Python examples are in the directory "examples" inside the main ppxf
package folder inside sitepackages.
ppxf
requires the scientific core packages NumPy, SciPy and Matplotlib, and the examples use Astropy to read FITS data (it was tested with Python 3.10 using Numpy 1.22, SciPy 1.8 and Matplotlib 3.5).
NOTE: I would appreciate if you drop me an email (address at the bottom) when you download the above procedures.
Best fitting symmetrized kinematics of NGC 2974 
This software implements the method presented in Appendix C of Krajnovic et al. (2006, MNRAS, 366, 787) to measure the global kinematic positionangle (PA) from integral field observations of a galaxy stellar or gas kinematics.
See Cappellari et al. (2007) and Krajnovic et al. (2011) for two large scale application of this software to the study of the stellar kinematical misalignment of earlytype galaxies. See Davor's Krajnovic page for the related Kinemetry package.
PaFit
package in The PaFit
package is on the Python Package Index (PyPI). It was last updated on the 21 May 2018.
How to install: Use "pip install pafit
". Without write access to the global sitepackages
directory, use "pip install user pafit
".
Also required is my plotbin package (automatically installed by pip
).
PaFit
requires the scientific core packages NumPy and Matplotlib (it was tested with Python 3.10 using Numpy 1.22 and Matplotlib 3.5).
NOTE: I would appreciate if you drop me an email (address at the bottom) when you download FIT_KINEMATIC_PA.
Figure 1: Fitting of the (M/L)σ relation in the Virgo cluster using LTS_LINEFIT (taken from Cappellari et al. 2013a). The green outliers above the relation are automatically removed from the fit. They turn out to be background galaxies and for this reason they do not follow the cluster relation.

This software implements the method presented in Sec. 3.2 of Cappellari et al. (2013a, MNRAS, 432, 1709) to to perform extremely robust fit of lines or planes to data with errors in all variables, possible large outliers (bada data) and unknown intrinsic scatter.
The code combines the Least Trimmed Squares (LTS) robust technique, proposed by Rousseeuw (1984) and speeded up in Rousseeuw & Driessen (2006), into a leastsquares fitting algorithm which allows for intrinsic scatter and errors in all coordinates. This method makes the fit converge to the correct solution even in the presence of a large number of catastrophic outliers (like in Figure 1), where the much simpler σclipping approach can converge to the wrong solution.
LtsFit
package in The LtsFit
package is on the Python Package Index (PyPI). It was last updated on the 22 January 2021.
How to install: Use "pip install ltsfit
". Without write access to the global sitepackages
directory, use "pip install user ltsfit
".
Usage examples: are in the directory "examples" inside the main package folder inside sitepackages.
LtsFit
requires the scientific core packages NumPy, SciPy and Matplotlib (it was tested with Python 3.10 using Numpy 1.22, SciPy 1.8 and Matplotlib 3.5).
NOTE: I would appreciate if you drop me an email (address at the bottom) when you download LTS_LINEFIT
and LTS_PLANEFIT
.
Application of the CAP_LOESS_2D routine (with FRAC=0.6) to the recovery of two underlying functions from a noisy distribution of 200 scattered values. For comparison, the right panels show the distribution obtained via simple averaging on a grid, but using 100× more values than the ones used for the LOESS recovery. The LOESS approach achieves a comparable result while requiring a much smaller number of input values.

This software provides an improved implementation of the onedimensional (Cleveland 1979) and twodimensional (Cleveland & Devlin 1988) Locally Weighted Regression (LOESS) method to recover the mean trends of the population from noisy data in one or two dimensions. It includes a robust approach to deal with outliers (bad data).
These programs were implemented and used to produce the twodimensional maps and the onedimensional mean trends in Cappellari et al. (2013b, MNRAS, 432, 1862).
loess
package in The loess
package is on the Python Package Index (PyPI). It was last updated on the 31 Jan 2022 (it was tested with Python 3.10 using Numpy 1.22 and Matplotlib 3.5).
How to install: Use "pip install loess
". Without write access to the global sitepackages
directory, use "pip install user loess
".
Usage examples: are in the directory "examples" inside the main package folder inside sitepackages.
Also required to run the 2D example is my plotbin package (automatically installed by pip
).
NOTE: I would appreciate if you drop me an email (address at the bottom) when you download the code.
Figure 1: Corner plot for the posterior of the model parameters obtained when fitting the black hole mass in the galaxy NGC1277 using the jam modelling method and the AdaMet Bayesian code (taken from Krajnovic et al. 2018). The Python code to produce this figure is included in the JamPy package above.

This AdaMet
package is the implementation of Cappellari et al. (2013a, MNRAS, 432, 1709) of the Adaptive Metropolis algorithm of Haario (2001). In many realworld applications, it was found to be more efficient and robust than the emcee
approach, whose warmup phase scales linearly with the number of walkers. For this reason, and because of its didactic value, the AdaMet
code is provided here as an alternative.
AdaMet
package in The userfriendly AdaMet
package is on the Python Package Index (PyPI). It was last updated on the 6 October 2020 (it was tested with Python 3.10 using Numpy 1.22 and Matplotlib 3.5).
How to install: Use "pip install adamet
". Without write access to the global sitepackages
directory, use "pip install user adamet
".
Usage examples: are in the directory "examples" inside the main package folder inside sitepackages.
Also required for plotting is my plotbin package (automatically installed by pip
).
Comments and suggestions are welcome to the address michele.cappellari
@physics.ox.ac.ukLatest changes: 15/FEB/2022
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