Exact deprojection of Sérsic surface brightness profiles.

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# -*- coding: utf-8 -*-

========

Sérsic

========

This is an implementation of the exact deprojection of Sérsic surface

brightness profiles as described in:

"Analytical expressions for the deprojected Sérsic model"

Baes and Gentile

Astronomy and Astrophysics, Volume 525:A136 (2011)

http://arxiv.org/abs/1009.4713

This code depends on the mpmath python library

(http://mpmath.googlecode.com) for an implementation of the Meijer G

function required by the Baes and Gentile (hereafter B+G) formulas for

rational values of the Sérsic index. B+G also give formulas for

irrational Sérsic indices, but I could not find an implementation of

the required Fox H function. Therefore this code requires rational

Sérsic indices, but any irrational number can be approximated

arbitrarily well by some rational number, so... no problem.

The code also depends on scipy (http://scipy.org), but the dependence

is mostly for testing: doing numerical integrals and such. If you

trust that the code passes the tests and don't want to run them

yourself (ha!), the only dependence on scipy is finding the root

(scipy.optimize.newton) in the function bg_constants(), which would be

pretty easy to replace.

This was implemented for use use in the paper:

"On Galaxies and Homology"

Novak et. al.

Monthly Notices of the Royal Astronomical Society, 424:635 (2012)

http://arxiv.org/abs/1205.2533

The B+G formulas are not simple but not terribly complex. However, I

wanted to be quite sure that I had correctly implemented the formulas

(and that the formulas themselves were correct), so I put significant

effort into comprehensive tests. That testing is the primary virtue

of this particular implementation of the B+G formulas, and is my

reason for releasing this code.

To test the implementation of the B+G formulas, I also implemented the

approximate formulas for deprojected Sérsic profiles described in:

"Dark matter in elliptical galaxies - I. Is the total mass density

profile of the NFW form or even steeper?"

Mamon and Lokas

Monthly Notices of the Royal Astronomical Society, Volume 362:95 (2005)

http://arxiv.org/abs/astro-ph/0405466

Additional tests use the parameterization of the Sérsic profile given

in:

Galactic Astronomy

Binney and Merrifield

Princeton University Press, 1998

For more information:

http://pypi.python.org/pypi/sersic/

For the source code:

http://launchpad.net/sersic

Installation

============

First install the mpmath python library (http://mpmath.googlecode.com).

To run the tests and avoid implementing your own root finding routine,

install numpy (http://www.numpy.org) and scipy (http://www.scipy.org).

But you've got those installed already, right?

The sersic module can be installed by any one of the following

standard incantations:

pip sersic

or

easy_install sersic

or

python setup.py install

If you want to install in your home directory, you can add the --user

flag to any of the above.

To run the tests:

python test.py

Usage

=====

Very likely the only function any user will care about is:

def luminosity(pp, qq, reff=1.0, lum=1.0) where pp and qq are the

numerator and denominator of the Sersic index (both integers) so that

nn=pp/qq, reff is the projected half-light radius, and luminosity is

the total luminosity. This returns a function that takes a radius and

returns a luminosity density.

>>> lum = luminosity(5,3)

>>> lum(1.1)

>>> lum([1.1, 2.2, 3.3])

All of the other functions in the module are just to help make sure

that the Baes + Gentile expressions are implemented correctly.

If you find that the calculations are taking a long time, you can try

reducing the precision of the calculation of the Meijer G function.

The mpmath library is designed to be able to work to arbitrary

precision, and is by default set to ~double precision. For

information on changing the precision, see:

http://mpmath.googlecode.com/svn/trunk/doc/build/basics.html#setting-the-precision

Keep in mind that small values of the numerator and denominator are

faster than large ones (e.g. luminosity(2,1) is fast,

luminosity(200,99) is slow.

The function returns arbitrary precision floats defined by the mpmath

library. They should behave as normal floats, but you may want to

cast them to Python floats to avoid invoking mpmath's more careful

arithmetic.

License

=======

The code is released under the MIT license, so you should be able to

do whatever you want with it.

If you use this software to produce a scientific publication or if you

incorporate this code into a larger project, I would appreciate it if

you send me a note at greg.novak@gmail.com

========

Sérsic

========

This is an implementation of the exact deprojection of Sérsic surface

brightness profiles as described in:

"Analytical expressions for the deprojected Sérsic model"

Baes and Gentile

Astronomy and Astrophysics, Volume 525:A136 (2011)

http://arxiv.org/abs/1009.4713

This code depends on the mpmath python library

(http://mpmath.googlecode.com) for an implementation of the Meijer G

function required by the Baes and Gentile (hereafter B+G) formulas for

rational values of the Sérsic index. B+G also give formulas for

irrational Sérsic indices, but I could not find an implementation of

the required Fox H function. Therefore this code requires rational

Sérsic indices, but any irrational number can be approximated

arbitrarily well by some rational number, so... no problem.

The code also depends on scipy (http://scipy.org), but the dependence

is mostly for testing: doing numerical integrals and such. If you

trust that the code passes the tests and don't want to run them

yourself (ha!), the only dependence on scipy is finding the root

(scipy.optimize.newton) in the function bg_constants(), which would be

pretty easy to replace.

This was implemented for use use in the paper:

"On Galaxies and Homology"

Novak et. al.

Monthly Notices of the Royal Astronomical Society, 424:635 (2012)

http://arxiv.org/abs/1205.2533

The B+G formulas are not simple but not terribly complex. However, I

wanted to be quite sure that I had correctly implemented the formulas

(and that the formulas themselves were correct), so I put significant

effort into comprehensive tests. That testing is the primary virtue

of this particular implementation of the B+G formulas, and is my

reason for releasing this code.

To test the implementation of the B+G formulas, I also implemented the

approximate formulas for deprojected Sérsic profiles described in:

"Dark matter in elliptical galaxies - I. Is the total mass density

profile of the NFW form or even steeper?"

Mamon and Lokas

Monthly Notices of the Royal Astronomical Society, Volume 362:95 (2005)

http://arxiv.org/abs/astro-ph/0405466

Additional tests use the parameterization of the Sérsic profile given

in:

Galactic Astronomy

Binney and Merrifield

Princeton University Press, 1998

For more information:

http://pypi.python.org/pypi/sersic/

For the source code:

http://launchpad.net/sersic

Installation

============

First install the mpmath python library (http://mpmath.googlecode.com).

To run the tests and avoid implementing your own root finding routine,

install numpy (http://www.numpy.org) and scipy (http://www.scipy.org).

But you've got those installed already, right?

The sersic module can be installed by any one of the following

standard incantations:

pip sersic

or

easy_install sersic

or

python setup.py install

If you want to install in your home directory, you can add the --user

flag to any of the above.

To run the tests:

python test.py

Usage

=====

Very likely the only function any user will care about is:

def luminosity(pp, qq, reff=1.0, lum=1.0) where pp and qq are the

numerator and denominator of the Sersic index (both integers) so that

nn=pp/qq, reff is the projected half-light radius, and luminosity is

the total luminosity. This returns a function that takes a radius and

returns a luminosity density.

>>> lum = luminosity(5,3)

>>> lum(1.1)

>>> lum([1.1, 2.2, 3.3])

All of the other functions in the module are just to help make sure

that the Baes + Gentile expressions are implemented correctly.

If you find that the calculations are taking a long time, you can try

reducing the precision of the calculation of the Meijer G function.

The mpmath library is designed to be able to work to arbitrary

precision, and is by default set to ~double precision. For

information on changing the precision, see:

http://mpmath.googlecode.com/svn/trunk/doc/build/basics.html#setting-the-precision

Keep in mind that small values of the numerator and denominator are

faster than large ones (e.g. luminosity(2,1) is fast,

luminosity(200,99) is slow.

The function returns arbitrary precision floats defined by the mpmath

library. They should behave as normal floats, but you may want to

cast them to Python floats to avoid invoking mpmath's more careful

arithmetic.

License

=======

The code is released under the MIT license, so you should be able to

do whatever you want with it.

If you use this software to produce a scientific publication or if you

incorporate this code into a larger project, I would appreciate it if

you send me a note at greg.novak@gmail.com

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