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\begin{document}
\title[Reconstructing the velocity dispersion profiles]{Reconstructing the velocity dispersion profiles from the line-of-sight kinematic data in disc galaxies}\author[Aleksander~A.~Marchuk, Natalia~Ya.~Sotnikova]
{A.~A.~Marchuk$^{1}$\footnotemark[1], N.~Ya.~Sotnikova$^{1}$
\thanks{E-mail: [email protected] (AAM); [email protected] (NYS)}\\ $^{1}$ St. Petersburg State University, Sobolev Astronomical Institute, Universitetskij pr. 28, 198504 St. Petersburg, Stary Peterhof, Russia}
\date{Accepted ???? ??? 2014. Received ???? ??? 2014; in original form ??????? 2014 }
\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2002}
\maketitle
\label{firstpage}
\begin{abstract} We present \end{abstract}
\begin{keywords} galaxies: spiral --- galaxies: kinematics and dynamics --- methods: numerical \end{keywords}
\section{Introduction}
One of the most challenging problem of galactic
dynamics is reconstructing the full phase space model of a certain
galaxy from its limited observational data including surface photometry
and long-slit or two-dimensional field spectroscopy. Various techniques
to solve this problem have been invented last several years.
The most developed schemes are based on Schwarzschild linear programming
method (\cite{Schwarzschild1979}). They use a library of orbits in
a given potential to reproduce the structure and kinematics of a stellar
system (... -- examples). There is a particel-based extention of the
Schwarzscild method (``made-to-measure'' method) initially proposed
by \cite{Syer_Tremaine1996}. It works by adjusting individual particle
weights as the model evolves, until the $N$-particle system reproduces
a set of target constrains. Recently it has been substantially improved
by \cite{deLorenzi+2007} and applied to build a complete dynamical
model of the E4 galaxy NGC\textasciitilde{}4697 (\cite{deLorenzi+2008}).
Torus --- BLA-BLA-BLA (Binney). A new iterative method for constructing
equilibrium phase models of stellar systems (\cite{Rodionov_Sotnikova2006,Rodionov_Orlov2008})
has been tested on a modelled disc galaxy with imitated observational
data and showed its flexibility (\cite{Rodionov+2009}).
All these algorithms need a detailed mass model of a target galaxy,
so nearly all dynamical models have been applied for one-component
system, mainly for elliptical galaxies. There aren't still any reliable
complete phase models of observed spiral galaxies that are multicomponent
systems with a substantial mass contribution of invisible matter ---
dark haloes.
BLA-BLA-BLA about direct modelling \cite{Bottema_Geritsen1997}, Zasov
and Co).
In many cases we don't need a DF to judge about dynamical status of
a stellar disc. It is sufficient to know velocity dispersion profiles,
mainly $\sigma_{R}(R)$ and $\sigma_{z}(R)$. Radial velocity dispersion
profile $\sigma_{R}$ gives us $Q=\sigma_{R}/\sigma_{R}^{{\rm cr}}$,
where $\sigma_{R}^{{\rm cr}}=3.36G\Sigma/\kappa$, $\kappa$ is the
epicyclic frequency, $\Sigma$ is the disc surface density (\cite{Toomre1964}).
The $Q$-parameter shows us the level of disc heating and stability
against any perturbations in a disc plane. The ratio $\sigma_{z}/\sigma_{R}$
describes the shape of the velocity ellipsoid and tells us a great
deal about the relaxation in a disc and its dynamical history.
The $\sigma_{z}/\sigma_{R}$ ratio for stars in the solar neihborhood
is $0.53\pm0.07$ (\cite{Dehnen_Binney1998}). Until recently, there are no similar data available for external galaxies. Observations
has been concentrated on systems that are either close to face-on or to
edge-on (e.g., \cite{vanderKruit_Freeman1986,Bottema1993}). Face-on
systems provide direct information only about one component of random
velocities --- $\sigma_{z}(R)$. For edge-on systems one has to use
the gas and stellar velocity curves to derive $\sigma_{R}(R)$ via
the asymmetric drift equation that describes equilibrium in a rotating
stellar disc. This scheme needs some additional ad hoc assumptions
and a more or less robust parametrization of $\sigma_{R}(R)$ (..).
BLA-BLA-BLA.
The solution for the all three moments of random velocitiy distribution
in a stellar disc can be found for a galaxy at an intermediate inclination.
Long-slit data from a variety of position angles and large-area integral
field units provide information about line-of-sight velocity dispersion
$\sigma_{{\rm los}}$ at different points in a galaxy. The contributions
of $\sigma_{R},\sigma_{\varphi}$ and $\sigma_{z}$ to$\sigma_{{\rm los}}$
depend on azimuthal angle and can be picked out provided the epicicle
approximation that connects $\sigma_{R},\sigma_{\varphi}$ and rotational
velocity (\cite{Gerssen+1997,Gerssen+2000,Shapiro+2003,Verheijen+2004,Noordermeer2008}).
To derive the velocity dispersion profiles exponential distributions
for both the radial and the vertical velocity dispersion components
are usually assumed. The most conventional anzatz of a such parametric
approach is $\sigma_{z}^{2}(R)\propto\Sigma(R)$ and $\sigma_{z}/\sigma_{R}={\rm const}$
troughout a disc (\cite{Verheijen+2004}). For an exponential disc
with a scalelength $h$ the former assumption results in $\sigma_{z}(R)\propto\exp{-R/2h}$.
This agrees well with observational data (\cite{vanderKruit_Searle1981}).
It also has physical and numerical bases and means that the system
is vertically isothermal (\cite{Spitzer1942}). $N$-body simulatons
of stellar discs in equilibria show that the vertical equilibrium
is completly separated from the equilibrium in a disc plane and the
approximation of isothermal layers breaks down only if the system
posseses a very massive dark halo or a compact bulge (\cite{Rodionov_Sotnikova2006},
see Fig.\textasciitilde{}2 there). The assumption $\sigma_{z}/\sigma_{R}={\rm const}$
has no clear evidences but is commonly accepted.
\cite{Gerssen+1997,Gerssen+2000}, \cite{Shapiro+2003} used the less
robust parametrization of the velocity dispersion profiles. They let
the scale length for $\sigma_{R}(R)$ and $\sigma_{z}(R)$ be not
equal to $2h$ and obtained that it was a factor 3--5 larger than
the disc scale length $h$. The velocity dispersion profiles derived
from major- and minor-axis spectroscopic data by \cite{Noordermeer2008}
got also very shallow. These results contradict with the approximation
of isothermal layers and force to revise the procedure of extracting
the velocity dispersion profiles from the line-of-sight kinematics.
{[}We don't know the dynamical structure of any galaxy and can't check directly the reliability of the described reconstructing three dimensional kinematics from line-of-sight data. In this paper we address to an equilibrium stellar disc model with given velocity dispersion profiles, imitate observational data, restore galaxy kinematics and compare the restored kinematics with initial data.{]}
The outline of this paper is as follows.
{[}In Section 2, we describe the technique for constructing initial
equilibrium $N$-body model of a disc with a fixed radial velocity
dispersion profile by iterative method and demonstate a set of artificial
observational line-of-sigth data for different inclination of a disc.
In Section 3, we reconstruct the galaxy kinematics from imitated major-
and minor-axis velocity dispersions and via asymmetric drift equation.
In Section 4, we discuss the advantages and disadvantages of the algorithms
under test. In Section 5, we present our main results and recommendations.{]}
\section{Reconstruction}
\subsection{Main relations}
One of the main kinematic characteristics of disc galaxies is stellar
velocity dispersion. In cilindrical polar coordinates $(R,\,\varphi,\:z)$
the line-of-sight velocity dispersion relates to stellar velocity
ellipsoid components $\sigma_{R}$, $\sigma_{\varphi}$, $\sigma_{z}$
as
\begin{equation}
\sigma_{{\rm los,\phi}}^{2}=[\sigma_{R}^{2}\,\sin^{2}\phi+\sigma_{\varphi}^{2}\,\cos^{2}\phi]\,\sin^{2}i+\sigma_{z}^{2}\,\cos^{2}i\,,\label{eq:los_comm}
\end{equation}
where $\phi$ is intrinsic position angle and $i$ is galaxy inclination
angle. Following this relation for observed minor-axis and major-axis
line-of-sight velocity dispersions in intermedialely inclined led
to next system:
\begin{equation}
\begin{array}{rcl}
\sigma_{\text{los,min}}^{2}(R\,\cos i) & = & \sigma_{R}^{2}\,\sin^{2}i+\sigma_{z}^{2}\,\cos^{2}i\,,\\
\sigma_{\text{los,maj}}^{2}(R) & = & \sigma_{\varphi}^{2}\,\sin^{2}i+\sigma_{z}^{2}\,\cos^{2}i\,.
\end{array}\label{eq:slos}
\end{equation}
This system of equations contains three unknowns
and only two equations. It may be closed by some dynamical relation,
whish is valid if the whole system is in equilibrium. One such relation
connects radial and azimuthal dispersion components with mean azimutal
velocity of stars $\bar{v}_{\varphi}$ \cite{Binney_Tremaine2008}
\begin{equation}
\frac{\sigma_{\varphi}^{2}}{\sigma_{R}^{2}}=\frac{1}{2}\left(1+\frac{\partial\ln\bar{v}_{\varphi}}{\partial\ln R}\right)\,.\label{eq:vphi}
\end{equation}
If most orbits in
disc are well-described by epicycle approximation then we can use
local circular speed of gas $v_{{\rm c}}$ instead of $\bar{v}_{\varphi}$.
First attempt to recover velocity ellipsoid from three equations presented
above (\cite{Gerssen+1997}) was not effective because it includes
subtracton of very close values $\sigma_{\text{los,min}}^{2}$ and
$\sigma_{\text{los,maj}}^{2}$. A possible solution is to parametrize
the profiles and find best-fitting parameters. Such approach used
in many papers (\cite{Gerssen+2000,Shapiro+2003}), but strongly depend
on selected parametrization, usually exponential. Noordermeer in his
paper \cite{Noordermeer2008} introduce less-parametric approach using
equation for asymmetric drift (hereafter AD) and non-physical assumption
that $\sigma_{\varphi}=\sigma_{z}$. Equation for AD shows difference
between random stellar motions and the local circular speed \cite{Binney_Tremaine2008}:
\begin{equation}
v_{\mathrm{c}}^{2}-\bar{v}_{\varphi}^{2}=\sigma_{R}^{2}\left(\frac{\sigma_{\varphi}^{2}}{\sigma_{R}^{2}}-1-\frac{\partial\ln\Sigma_{\mathrm{s}}}{\partial\ln R}-\frac{\partial\ln\sigma_{R}^{2}}{\partial\ln R}\right)\,,\label{eq:AsDr}
\end{equation}
where $\Sigma$ is the stellar surface density. Notice that this form
of equation neglects the tilt term ${\rm d}(\overline{v_{R}v_{z}})/{\rm d}z$,
which is assumed to be small relative to other terms. Noordermeer's
method was adopted in \cite{Silchenko2011} for exponential dispersion
profiles and applied to NGC 7217 data. They suppose to use stellar
brightness for old population in $I$-band instead of surface density
in (\ref{eq:AsDr}) and then find $\sigma_{R}$ profile iteratively.
All mentioned methods interpret velocity dispersion profiles as exponential
with more or less valid typical scale. However, as was shown in (Sil'chenko??),
such simple parametrization not always correctly follow real random
stellar motion profiles. In this paper we present new parametrization
without exponential usage which allow us reconstruct velocity ellipsoid
more reasonably. We will combine equations \ref{eq:slos}-\ref{eq:vphi}
with addition assumption that ratio $\sigma_{z}/\sigma_{R}\equiv\alpha$
is constant on every radius $R$. Last statement is questionable and
obviously simpler than real profies. Hovewer it justified by disc
heating theory (\cite{Jenkins_Binney_1990}) and any observations
also estimated the ratio for the velosity anisotropy between 0.5 and
0.7 \cite{Gerssen+1997,Gerssen+2000,Shapiro+2003}.
Taking into account \ref{eq:slos} we recieve $\sigma_{{\rm los,min}}^{2}=\sigma_{R}^{2}[\sin^{2}i+\alpha^{2}\cos^{2}i]$
for minor axis. It means that radial velocity dispersion $\sigma_{R}$
proportional to $\sigma_{{\rm los,min}}$ and may be parametrized
by single value ---
$f=0.5(1+e^{-R/R_{0}}),$ where $R_{0}$ ---
typical scale
\setcounter{table}{0}
\begin{table}
\caption{Main parameters of data sample.}
\label{table:main_parameters}
\begin{tabular}{@{}lcccc}
\hline Galaxy & Type & Tilt angle $i$ & $r_{{\rm eff}}$ & Distance, Mpc
\tabularnewline \hline NGC 338 & Sab & 64$\degr$ & 15.0$\arcsec$ & 65
\tabularnewline \hline NGC 1167 & S0 & 36 & 6.7 & 67
\tabularnewline \hline NGC 2273 & SBa & 50 & 2.2 & 25.7
\tabularnewline \hline NGC 3245 & S0 & 62 & 15.1 & 18
\tabularnewline \hline NGC 4150 & S0 & 56 & 9.5 & 14
\tabularnewline \hline
\end{tabular}
\end{table}
\section{Observations}
Our dataset consists of three lenticular S0, one SBa and one Sab galaxies (see Table~\ref{table:main_parameters}). All galaxies were observed with the SCORPIO multimode instrument mounted at the primary focus of the 6-m telescope of the Special Astrophysical Observatory. Observations and data reduction described in detail in \citet{Zasov2008}. This paper explains only half of our galaxies dataset (NGC~1167, NGC~2273 and NGC~4150), but remained galaxies NGC~338 and NGC~3245 were obtained without considerable difference (spectral resolution was 2.6~$\text{\AA}$). Table~\ref{table:expositions} for last two galaxies shows dates, the exposures, the average seeing and the positional angle PA of the slit.
\setcounter{table}{1}
\begin{table}
\caption{Log of spectroscopic observations for NGC~338 and NGC~3245 (same as Table 2 in \citealt{Zasov2008}).}
\label{table:expositions}
\begin{tabular}{@{}l|l|l|l|l}
\hline Galaxy & Date & $T_{\rm{exp}},\:$s & Seeing & PA
\tabularnewline \hline NGC~338 & 1/2.10.2006 & 8400 & $2.7\arcsec$ & $289\degr$
\tabularnewline $ $ & 17/18.09.2007 & 10800 & 2.4 & 197
\tabularnewline \hline NGC~3245 & 13/14.04.2007 & 7200 & 4.0 & 355
\tabularnewline $ $ & 6/7.04.2008 & 7200 & 3.3 & 267
\tabularnewline \hline
\end{tabular}
\end{table}
The classical cross-correlation method, similar to employed in \citet{Moiseev2001} was used to calculate the radial velocities and stellar velocity dispersions. For spectral modelling PEGASE.HR model for SSP (Simple Stellar Population) applied to binarized object's spectra. After fitting by burst age $T$ and metallicity [Fe/H] parameters algorithm made convolution with line-of-sight velocity distribution (LOSVD). At every spectra bin fitting produce radial velocity $V_r$ and velocity dispersion $\sigma$. In addition NGC~2273 data was compared with profiles obtained by taking appropriate sections across the two-dimensional SAURON velocity and velocity-dispersion maps and led to good agreement.
In the next subsections we listed galaxies with their peculiarities.
${\bf NGC\,338}$
This is type Sa or Sab galaxy with weak spiral branches. On the SDSS and 2MASS images it looks like disc galaxy with bright core and extensive bulge. NGC~338 contains large amount of HI gas and can be consider as early type galaxy with slow starbirth rate. Bulge has substantial surface luminosity relatively to disc, which was associate with additional dust existence by some authors.
NGC~338 is highly rotated galaxy with maximum circular velocity near 280-320~km/s. Both gas and stellar rotation curves are measured up to $50\arcsec$ and have velocity uncertainties not greater than 10 km/. Radial velocity profile was obtained from absorption lines and emissions in $\rm{H}_{\beta}$, [NI] and [OIII]. Rotation curve gain saturation at $r \approx$~4~kpc ($12.5\arcsec$), but velocities at distances larger than 4~kpc become asymmetrical with gap up to 50 km/s along major axis. Such asymmetry in velocity field was noticed before and visible the most in HI line. Cause of this anomaly was supposed by some papers to be a small merging event and concerned both hot and cold rotation curves. Gas velocities exceed stellar curve noticeably, what explained by greater star velocity dispersion. Dispersions in line-of-sight profile along major axis (PA = $108\degr$) are high inside bulge-dominated region ($r\la 15\arcsec$) and decrease rapidly until become flat at $20\arcsec$. Dispersions along minor axis (PA = $17\degr$) harder to observe and have errors in measurement about 15-20 km/s.
${\bf NGC\,1167}$
This galaxy apparently does not belong to any group and doesn't have satellites with comparable luminosity. The bulge has a steep photometric profile (Sersic parameter $n \approx 3$ according to Zasov) and dominates over disc up to $15\arcsec-20\arcsec$ distance. The total disc luminosity is three times greater than bulge luminosity. Images of this galaxy display a remarkable system of ring-like arcs or spirals, which are thin and doesn't contain any irregularities inside. The nature of this formations stays unclear. It origin may possibly relate to star formation, but there are no any blue knots and the B-V color index in spirals is bluer very slightly ($0.05^{\rm{m}}-0.1^{\rm{m}}$). Also, according to Noordermeer et al. observations, the galaxy contains a significant amount of HI (total mass $\approx 1.7 \times 10^10 M_{\sun}$) that spread over a large area and the average HI surface density remains below the gravitational instability's critical value.
Observations of neutral hydrogen show a rotation curve extended over than 10 exponential disc scales. The maximum circular velocity is very large and is not far from 400 km/s (reached at $100\arcsec$). Stellar rotation velocities measured up to $r = 50\arcsec$ and have similar to gas values and uncertainties (10~km/s - 15~km/s). Line-of-sight dispersion profiles along major and minor axes have average errors around 20~km/s. They also have the same slope of profiles and nearest central dispersion values. All this factors do not allow separate them from each other with sufficient confidence.
${\bf NGC\,2273}$
This galaxy is like NGC~1167 in many cases. It is also isolated, contains large amount of gas with total hydrogen mass approximate to $2.4 \times 10^9 M_{\sun}$ and has badly separated dispersion profiles. The main difference is a bright bar in the central region with radius of $r_{\rm{bar}} \approx 25 \arcsec$. Moiseev et al. shows that in the bar region gas exhibits appreciable circular motions. Stellar rotation curve measured up to $100 \arcsec$ and demonstrate slight increase after bar region. exhibits appreciable circular motions. Velocities on this curve are very close to circular motions obtained by Noordermeer et al., which is also the same behaviour as previous galaxy show. Bulge dominated in photometric profile over $r < 40 \arcsec$ range.
${\bf NGC\,3245}$
This is the single isolated galaxy with two discs. The border between inner bright and outer main discs is very sharp and lies at $r \approx 15\arcsec-20\arcsec$. No spirals, HII areas and active star formation regions were found. Galaxy has the weak seyfert nuclei and compact bulge with effective radius approximately $5\arcsec$ \citep{Fabricius2012}.
Rotation curve and velocity dispersions are symmetrical, but galaxy has some dynamical peculiarities. First, it may be possible that gaseous disc incline relates to stellar disc, because gas velocities along minor axis have non-zero gradient. Second, stellar rotation curve has a maximum at $r \approx 2.5$~kpc (around $30\arcsec$) and slowly decreases after. Also galaxy has a huge central dispersion values (similar to NGC~1167, see Figure~\ref{figure:all_sig}). Line-of-sight dispersion profiles have good measure quality and small uncertainties, but divergence in center between them is large ($|\sigma_{\rm{los}}^{\rm{maj}} - \sigma_{\rm{los}}^{\rm{min}}| \approx 20$~km/s) and it is possibly testify that slit along major axis has some was shifted a little (see section \ref{subs:reconstr}).
${\bf NGC\,4150}$
This galaxy is low-luminosity and has the exponential photometric profile. There is no data obtained for gas motions in this galaxy. Stellar rotation curve stays flat for $r > 20\arcsec$ distances and has maximum velocity $\approx 100$~km/s. Dispersions data is very noisy and has greatest errors throw whole sample. It also contains a lot of zero data points across major axis. This makes hard to fit line-of-sight dispersion profiles without any additional data cleaning. It is also significant that distance estimates for this galaxy are uncertain and vary from 10 to 20 Mpc.
\section{Reconstruction methodology}
\section{Results}
\subsection{Direct method}
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\setcounter{figure}{0}
\begin{figure}
\includegraphics[width=84mm]{imgs/fig1_ngc338_svedata.eps}
\vspace{-1.0cm}
\caption{NGC~338.}
\label{figure:ngc338_sve}
\end{figure}
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\setcounter{figure}{1}
\begin{figure}
\includegraphics[width=84mm]{imgs/fig2_ngc338_maps.pdf}
\vspace{-2.0cm}
\caption{NGC~338 $\chi^2$ maps.}
\label{figure:n338_map}
\end{figure}
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\begin{figure}
\includegraphics[width=84mm]{imgs/fig3_ngc338_svedata.pdf}
\vspace{-1.0cm}
\caption{NGC~338 fitting data}
\label{figure:n338_fit}
\end{figure}
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\setcounter{figure}{3}
\begin{figure*}
\includegraphics[width=190mm]{imgs/fig4_ngc338_ad.pdf}
\vspace{-1.0cm}
\caption{NGC~338 asymmetric drift}
\label{figure:n338_ad}
\end{figure*}
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\setcounter{figure}{4}
\begin{figure*}
\includegraphics[width=190mm]{imgs/fig6_all_sig_los.eps}
\vspace{-1.0cm}
\caption{All $\sigma$ line-of-sight data.}
\label{figure:all_sig}
\end{figure*}
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\setcounter{figure}{5}
\begin{figure*}
\includegraphics[width=190mm]{imgs/fig6_all_rot_curv.eps}
\vspace{-1.0cm}
\caption{All rotation curves data.}
\label{figure:all_sig}
\end{figure*}
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\setcounter{figure}{6}
\begin{figure}
\includegraphics[width=84mm]{imgs/fig7_ngc3245_maps.pdf}
\vspace{-2.0cm}
\caption{NGC~3245 $\chi^2$ maps.}
\label{figure:n3245_map}
\end{figure}
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\subsection{Reconstruction metodology}
\label{subs:reconstr}
\subsection{Usage of asymmetric drift equation}
\section{Discussion}
\section{Summary}
\section*{Acknowledgments}
This work was partially supported by the Russian Foundation for Basic
Research (grant 06--02--16459 and 08--02--00361).\citep{Rodionov_Sotnikova2006}
\bibliographystyle{mn2e}
\bibliography{art}
\label{lastpage}
\end{document}
