%
\documentclass[11pt,twoside]{article} % Leave intact
\usepackage{adassconf}
\usepackage{graphicx}
%\documentstyle[11pt,twoside,adassconf]{article}
\begin{document} % Leave intact
\paperID{P7-8}
%%%% ID=P7-8
\title{Determination of Initial Conditions of M81 Triplet Using Evolution Strategies}
\titlemark{Evolution Strategies in Galaxy Interaction}
\author{Juan Carlos Gomez and Olac Fuentes}
\affil{I.N.O.E., Luis Enrique Erro No. 1, Tonantzintla, Puebla,
Mexico Email: jcgc@inaoep.mx}
\author{Lia Athanassoula and Albert Bosma}
\affil{Observatoire de Marseille, 2 Place le Verrier, 13248, Cedex
4, Marseille, France }
\contact{Juan Carlos Gomez} \email{jcgc@inaoep.mx}
\paindex{Gomez, J. C.}
\aindex{Athanassoula, L.}
\aindex{Fuentes, O.}
\aindex{Bosma, A.}
\authormark{Gomez, Athanassoula, Fuentes, \& Bosma}
\keywords{galaxies: interactions, kinematics, dynamics, methods: evolution strategies}
%-----------------------------------------------------------------------
% Abstract
%-----------------------------------------------------------------------
\begin{abstract} % Leave intact
In this work we present Evolution Strategies (ES) as an efficient
method to approximate the initial conditions of the main
interacting group of three galaxies in M81.
The M81 group is one of the nearest groups of galaxies. Its
biggest galaxy, M81, sits at the core of the group together with
its two companions M82 and NGC3077. The interaction between these
three galaxies is very well defined on an image taken in HI. In
this first attempt we use non-self-gravitating simulations to
approximate the initial conditions; even with that restriction our
method reproduces the density distribution of the three galaxies
with great precision.
Results presented here show that ES is an ideally suited method to
work in optimization problems in Astrophysics, where the solution
is hard to find by common methods. In particular we argue that ES
is a good method to find initial conditions of groups of
interacting galaxies, where a large number of parameters need to
be determined.
\end{abstract}
%-----------------------------------------------------------------------
% Main Body
%-----------------------------------------------------------------------
\section{Introduction}
It now established that galaxies are not "island universes", but
rather interact with each other in pairs or in small or big
groups. Interactions can form spirals, bars, warps, rings and
bridges. Thus, observing the morphological and kinematics results
of an interaction can give us crucial information about the
interaction scenario.
Yet very few detailed studies of individual interacting systems
exist. The reason is that the number of free parameters in such
simulations is very large and one needs an excessively large
number of simulations to cover the corresponding parameter space.
The problem of finding the right parameters for modelling the
interaction of a given system of galaxies can be posed as an
optimization problem. Indeed any simulation will give a projected
surface density map and line-of-sight velocities. These can be
compared to the corresponding observed quantities, and then the
best model is the one that minimizes the difference.
We will here use a method called Evolution Strategies (ES), which,
while having much in common with Genetic Algorithms (GA)
(Charbonneau 1995), is better suited for working with continuous
spaces, i.e. real parameters. Since most of the parameters of
interacting systems are continuous, this constitutes a clear
incentive for trying out ES. We have chosen the M81 triplet as the
interacting system to be studied.
The M81 group is one of the nearest groups of galaxies. Its
biggest galaxy, M81, sits in the core of the group together with
its two nearby companion M82 (in the upper part of the image) and
NGC3077 (in the lower part of the image).
\section{Evolution Strategies}
% We reset the footnote counter for the hyperlink since it does not
% appear to recognize the previous 3 footnotes generated from the
% altaffilmarks. The command \htmladdnormallinkfoot puts the link as a
% footnote in the printed paper. The command \htmladdnormallink with
% the same arguments will ignore the link in the printed copy.
%\setcounter{footnote}{3}
Evolution Strategies (ES) (Rechenberg 1975) is a technique for
finding the minimum of a function with a large number of variables
using ideas based on biological evolutionary process. We start by
choosing $K$ individuals, each characterized by an object
parameter vector $\bf O$ and a corresponding strategy parameter
vector $\bf S$:
\begin{equation}
\mathbf{O}=o_{i}=\langle q_{1,i}, q_{2,i},\cdots,q_{L,i} \rangle
i=1,\cdots,K
\end{equation}
\begin{equation}
\mathbf{S}=s_{i}=\langle
\sigma_{1,i},\sigma_{2,i},\cdots,\sigma_{L,i} \rangle i=1,\cdots,K
\end{equation}
The elements $q$ are the parameters we need to find, and $\bf S$
contains standard deviations of the $L$ variables
$q_{l,i},l=1,\cdots,L$
In the initialization of the process, the first generation, the
elements of the $\bf O$ and $\bf S$ vectors, can be chosen either
totally at random, or with some help from previous knowledge about
the system.
Each of the $K$ individuals, i.e. each set of initial conditions,
is used as input for the simulation program. The result of each
simulation has to be evaluated with the help of a fitness
function.
The next sept is to produce a new population with the help of
genetic operators: cross-over and mutation. For cross-over two
individuals are chosen at random and in a such way that each
individual is used once and only once. Mutation is applied to the
individuals resulting from the cross-over operation. Each element
of the new individual is calculated from the old individual using
the simple equation:
\begin{equation}
q_{\mathrm{mut}}=q_{j}+N(0,\sigma_{j})
\end{equation}
where $N(0,\sigma_{j})$ is a random number obtained from a normal
distribution with a zero mean an a standard deviation
$\sigma_{j}$, which is given from the strategy parameter vector.
The process is repeated until the population converges.
\section{Aplication to Interacting Systems of Galaxies}
We use 30 individuals per iteration ($K=30$). In constituting the
children population we apply first cross-over and then mutation to
the parent population, then we merge both populations, select the
$K$ best individuals from this merged population, and use the
result as input for the next iteration.
In order to obtain the fitness fuction we used mainly the HI
density information, summarized in a $48\times48$ grid. We also
grided the simulation results on a similar grid and then obtained
the fitness as the Kullback-Leibler distance (Kullback \& Leibler
1951).
For the simulation we use the test particle approach. In this
approximation the mass of each galaxy is assumed to be
concentrated in a single point in its center, while the disc,
which responds most to the interaction, is represented by test
particles, initially on co-planar circular orbits around the
center of the galaxy. This approach is very fast and thus allows
us to run the very large number of simulations necessary for
tackling this problem. Furthermore, in our case the galaxies are
not inter-penetrating and thus they are perturbed only in their
outer parts, making the test particle approach fairly adequate.
\section{Results}
We obtained the density matrix of the original image simply by
scanning the HI density map given in (Yun 1997). By using only
this information we obtained a good fit to the density matrix
after 100 generations, i.e. a total of $100\times30$ simulations.
The images in Figure 1 show the best simulation reached and the
original HI image. Table 1 shows the corresponding parameters for
the best simulation.
\begin{figure}[t]
\epsscale{.8} \plotone{P7-8_f1.eps} \centering \caption{Simulated
and HI images for M81 group}
\end{figure}
Although the density was fairly well reproduced in this way, the
velocities were innacurate. So, in future work we are planning to
introduce some velocity information to improve the velocity
distribution.
\section{Conclusions}
In this work we presented an efficient method, based on ES, to
approximate the initial conditions of the M81 triplet.
Even with the several simplifying assumptions done in simulations,
searching with ES has demonstrated to be an excellent method for
optimization problems where an exploration of continuous
parameters spaces is needed; ES could find a good simulation that
match very well the HI density distribution in this problem. We
are planning to extend the application of ES to the study of other
interacting systems.
\subsection{Future Work}
In order to improve the method, the possibility of implementing a
parallelization of the ES could be considered with the purpose of
reducing the computing time required. Also, methods based on
self-gravitating N-body simulations can be used to improve the
match between simulations and the HI density distribution.
\begin{table}[t]
\resizebox{\linewidth}{!}{%
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$z_{2}$ &$z_{3}$ &$V_{x1}$ &$V_{y1}$ &$V_{x2}$ &$V_{y2}$ &$V_{x3}$ &
$V_{y3}$ &$i_{1}$ &$PA_{1}$ &$i_{2}$ &$PA_{2}$ &$i_{3}$ &
$PA_{3}$ &$m_{1}$ &$m_{2}$ &$m_{3}$ &$t$ \\
\hline
62.05 &11.14 &3.17 &1.59 &-61.91 &41.78 &-168.75 &-0.90 &44.90 &113.37 &38.62 &32.58
&53.83 &232.87 &19.47 &1.04 &1.06 &812\\
\hline
\end{tabular}%
}\caption{Parameters to produce the simulation in Figure 1}
\tablenotetext{}{Index 1 is for M81, 2 for M82 and 3 for NGC3077}
\end{table}
%\begin{deluxetable}{crrrrrrrrrrr}
%\scriptsize
%\tablecaption{Terribly relevant tabular information. \label{P7-8:T1.10-tbl-1}}
%\tablehead{
%\colhead{Star} & \colhead{Height} & \colhead{$d_{x}$} & \colhead{$d_{y}$} &
%\colhead{$n$} & \colhead{$\chi^2$} & \colhead{$R_{min}$} &
%\multicolumn{1}{c}{$P$\tablenotemark{a}} & \colhead{$P R_{maj}$} &
%\colhead{$P R_{min}$} & \multicolumn{1}{c}{$\Theta$\tablenotemark{b}}}
%\startdata
%1 &33472.5 &$-$0.1 &0.4 &53 &27.4 &1.940 &3.900 &68.3 &116.2 &$-$27.639 \nl
%2 &27802.4 &$-$0.3 &$-$0.2 &60 &3.7 &1.510 &2.156 &6.8 &7.5 &$-$26.764 \nl
%3 &29210.6 &0.9 &0.3 &60 &3.4 &1.551 &2.159 &6.7 &7.3 &$-$40.272 \nl
%4 &32733.8 &$-$1.2 &$-$0.5 &41 &54.8 &2.156 &4.313 &117.4 &78.2 &$-$35.847 \nl
%5 & 9607.4 &$-$0.4 &$-$0.4 &60 &1.4 &1.574 &2.343 &8.0 &8.9 &$-$33.417 \nl
%6 &31638.6 &1.6 &0.1 &39 &315.2 &3.075 &7.488 &92.1 &25.3 &$-$12.052 \nl
%\enddata
% Text for table footnotes must follow the tabular environment but must
% be inside the table environment. Note that it is OK to put \ref's
% in \tablenotetext's.
%\tablenotetext{a}{Sample footnote for Table~\ref{P7-8:T1.10-tbl-1}}
%\tablenotetext{b}{Another sample footnote for
%Table~\ref{P7-8:T1.10-tbl-1}} \tablenotetext{c}{Footnote with no call
%out} \tablenotetext{d}{Another footnote with no call out}
%\tablenotetext{e}{A further additional footnote with no call out}
%\end{deluxetable}
% In the figure environment the \caption command should contain only the
% caption text. The "Figure N." identification is generated by the
% \caption command on its own.
%
% In this example we insert the \epscale command before the \plotone
% command to override the default scaling of the figure.
%
% You cannot use footnotes within figures.
%\begin{figure}
%\epsscale{.80}
%\plotone{figure.eps}
%\caption{A particularly ghostly figure.} \label{P7-8:T1.10-fig-1}
%\end{figure}
% The same tabular data presented above in the deluxetable environment is
% aligned below within the "tabular" environment. Observe
% that our tabular environment is embedded within a "center" environment,
% which is in turn inside a "table" environment.
%
% We need the table environment for autonumbering and caption generation,
% which is why it is not enough to have a centered tabular.
%
% Within the tabular environment, please note that we use no vertical
% rules, and the horizontal rules are inserted with \tableline (*not* \hline).
% Note that a couple of the column headings require special annotation, i.e.,
% footnotes for tables. They are marked and tagged with \tablenotemark.
% \tablenotemarks could be placed on individual data entries as well,
% but try not to go berserk doing this.
%
% It is necessary to \label tables and figures *after* the \caption has been
% specified because the table/figure number is generated by \caption, not
% by \begin{whatever}.
% Text for table footnotes must follow the tabular environment but must
% be inside the table environment. Note that it is OK to put \ref's
% in \tablenotetext's.
% Finally, we have a little acknowledgments section.
\acknowledgments
We are grateful to CONACYT for partially supports this work.
% That's the end of the main body of the paper. Now we will have some
% back matter.
%
%-----------------------------------------------------------------------
% References
%-----------------------------------------------------------------------
\begin{references}
\reference Charbonneau, P.\ 1995, \apjs, 101, 309 \reference
Kullback, S., \& Leibler, R.\ A.\ 1951, Ann. of Math. St., 22, 76
\reference Rechenberg, I.\ 1973, Evolutionsstrategie: Optimierung
technischer Systeme nach Prinzipien der biologischen Evolution,
Stuttgart: Fromman-Holzboog \reference Wahde, M., \& Donner, K.\
J.\ 2001, \aap, 115
\reference Wahde, M.\ 1998, \aaps, 132, 417 \\
\reference Yun, M.\ S.\ 1997, IAU Symposium, 186
\end{references}
% Do not place any material after the references section
\end{document} % Leave intact