\documentclass[journal]{rmaa}


\title{Kinematic ages of the central stars of planetary nebulae} 


\author{
W.~J. Maciel, 
T.~S. Rodrigues, 
and R.~D.~D. Costa\affil{Instituto de Astronomia, Geof\'\i sica e Ci\^encias Atmosf\'ericas\\ 
Universidade de S\~ao Paulo, Brazil}}


\shortauthor{Maciel, Rodrigues, \& Costa}

\shorttitle{Kinematic ages of CSPN}

\fulladdresses{
\item W. J. Maciel, T. S. Rodrigues and R. D. D. Costa: Instituto de Astronomia, Geof\'\i sica e Ci\^encias Atmosf\'ericas, 
Universidade de S\~ao Paulo, Rua do Mat\~ao 1226, CEP 05508-090, S\~ao Paulo SP, Brazil (maciel,~tsrodrigues,~roberto@astro.iag.usp.br).}


\listofauthors{W.~J.~Maciel, T.~S.~Rodrigues, \& R.~D.~D.~Costa}

\indexauthor{Maciel, W.~J.}
\indexauthor{Rodrigues, T.~S.}
\indexauthor{Costa, R.~D.~D.}


\SetYear{2011}
\ReceivedDate{2011 July 01}
\AcceptedDate{2011 August 11}



\abstract{The age distribution of the central stars of planetary nebulae (CSPN) is estimated using two methods based on their kinematic properties. 
First, the expected rotation velocities of the nebulae at their Galactocentric distances are compared with the predicted values from the rotation 
curve, and the differences are attributed to the different ages of the evolved stars. Adopting the relation between the ages and the velocity dispersions 
determined by the Geneva-Copenhagen survey, the age distribution can be derived. Second, the $U$, $V$, $W$, velocity components of the stars are 
determined, and the corresponding age-velocity dispersion relations are used to infer the age distribution. These methods have been applied to two 
samples of PN in the Galaxy. The results are similar for both samples, and show that the age distribution of the PN central stars concentrates at 
ages lower than 5~Gyr, peaking at about 1 to 3~Gyr.\medskip}%\vspace{0.176cm}}


\resumen{Se determina la distribuci\'on de edades de las estrellas centrales de nebulosas planetarias (CSPN) utilizando dos m\'etodos basados en 
sus propiedades cinem\'aticas.  En primer lugar, se comparan las velocidades de rotaci\'on de las nebulosas esperadas a partir de sus distancias 
galactoc\'entricas con los valores predichos por la curva de rotaci\'on, y las diferencias encontradas se atribuyen a las distintas edades de las 
estrellas evolucionadas. Se usa la relaci\'on entre las edades y la dispersi\'on de velocidades del relevamiento Geneva-Copenhagen para obtener 
la distribuci\'on de edades. En segundo lugar, se determinan las componentes $U$, $V$, $W$ de las velocidades de las estrellas, y se usan las 
relaciones edad-dispersi\'on de velocidades para inferir la distribuci\'on de edades. Hemos aplicado estos m\'etodos a dos muestras de nebulosas 
planetarias en nuestra galaxia. Los resultados son similares para ambas, y muestran que la distribuci\'on de edades de las CSPN est\'a concentrada 
en edades menores que 5 giga-a\~nos, y que tiene un pico entre 1 y 3 giga-a\~nos.\medskip}


\addkeyword{planetary nebulae: general}
\addkeyword{stars: AGB and Post-AGB}
\addkeyword{stars: fundamental parameters}
\addkeyword{stars: general}


\RescaleTitleLengths{1.064}


\begin{document}

\maketitle


\section{Introduction}
\label{section1}

Planetary nebulae (PN) are evolved objects ejected by stars with main sequence masses in the range of 0.8 and 8~$M_\odot$, so that the expected 
ages of their central stars are of the order of, or greater than, about 1~Gyr. However, the relatively large mass bracket of their progenitor 
stars implies that an age distribution is to be expected, which has some consequences for the interpretation of the PN data in the Galaxy and 
other stellar systems. The determination of ages of the central stars is a difficult problem, and most usual methods have large uncertainties 
when applied to intermediate and old age objects. We have recently developed three different methods to estimate the age distribution of the 
CSPN (Maciel, Costa, \& Idiart \citeyear{mci2010}, see also Maciel, Costa, \& Uchida 2003; Maciel, Lago, \& Costa 2005, 2006), and have applied 
these methods to a sample of PN in the disk of the Galaxy, most of which are located in the solar neighborhood, within 3~kpc of the Sun. These 
methods include the determination of the age distribution of CSPN using (i) an age-metallicity relation that also depends on the Galactocentric 
distance, (ii) an age-metallicity relation obtained for the disk, and (iii) the central star masses obtained from the observed nitrogen abundances. 
We concluded that most CSPN in our sample have ages under 6~Gyr, and that the age distribution is peaked around 2--4~Gyr. The average uncertainties 
were estimated as 1--2~Gyr, and the results were compared with the expected distribution based both on the observed mass distribution of white 
dwarfs and on the age distribution derived from available masses of CSPN. 

In the present work we develop two additional and more accurate methods to estimate the age distribution of the CSPN based on their kinematical 
properties, namely: (i) a method based on the expected rotation velocities of the nebulae at their Galactocentric distances, which are then 
compared with the predicted values for a given rotation curve, the differences being attributed to the different ages of the evolved stars; (ii) 
a method based on the derived $U$, $V$, $W$, velocity components of the stars and their corresponding dispersions. In both cases, the age-velocity 
dispersion relations from the Geneva-Copenhagen survey  are used to infer the age distribution. These methods are applied to two PN samples, (i) 
the previous sample of disk PN used by Maciel et al. (2010), for which a detailed data set is available, and (ii) a sample containing all PN 
for which accurate radial velocities are known. The methods are developed in \S~2, and the samples used are described in \S~3. The main results 
and discussion are given in \S~4.


\section{Determination of the age distribution of CSPN}
\label{section2}

\subsection{Method 1: The PN rotation velocity}

As objects of intermediate age, PN in the disk of the Galaxy describe a rotation curve similar to the one defined by younger objects, such as HII 
regions, although with a higher dispersion, as discussed in detail by Maciel \& Lago (\citeyear{ml2005}). Therefore, the discrepancies between 
the rotation velocities inferred from the PN  radial velocities and distances and the velocities expected from the known rotation curve may be 
at least partially ascribed to their evolved status. In other words, a given nebula located at a distance $d$, with galactic coordinates $\ell$ 
and $b$ and observed heliocentric radial velocity $V_r({\rm hel})$ can be associated with a rotation velocity $\theta(R)$, after obtaining its 
Galactocentric distance $R$ and its radial velocity relative to the Local Standard of Rest (LSR), $V_r(\mathrm{LSR})$. Assuming circular orbits, 
the rotation velocity $\theta(R)$  at the Galactocentric distance $R$ can be written as
\begin{equation}
\theta(R) = {R \over R_o} \ \biggl[{V_r(\mathrm{LSR}) \over \sin \ell \, \cos b} + \theta_0\biggr]\,,
\end{equation}
where $R_0$ and $\theta_0$ are the Galactocentric distance and rotation velocity at the solar position (see for example Maciel \& Lago \citeyear{ml2005}; 
Maciel \& Dutra \citeyear{md1992}). On the other hand, the expected rotation velocity at the given Galactocentric distance, $\theta_c(R)$, can be 
obtained from an adopted rotation curve. The difference $\Delta \theta=\vert{\theta(R)-\theta_c(R)}\vert$ can then be considered as proportional 
to the age difference between the PN and the objects defining the rotation curve. We have adopted the radial velocities from the catalogue by 
Durand, Acker, \& Zijlstra (1998), and two distance scales, those by Maciel (\citeyear{m1984}) and Stanghellini, Shaw, \& Villaver (2008). The 
first one was based on a relationship between the ionized mass and the radius of the nebulae, while the second is an update of the distance scale 
by Cahn, Kaler, \& Stanghellini (1992), using a modified Shklovksy method following Daub (\citeyear{daub}). Since the distances of planetary nebulae 
in the Galaxy may contain large individual uncertainties, the use of two different scales which are considered as ``short'' (Maciel \citeyear{m1984}) 
and ``long'' (Stanghellini et al. \citeyear{ssv2008}) ensures that these uncertainties will not affect the derived age distributions. We have adopted 
$R_0=8.0$~kpc for the distance of the Sun to the centre and $\theta_0=220$~km~s$^{-1}$ for the rotation velocity at $R_0$. Slightly different values 
can be found in the literature (see for example Perryman \citeyear{perryman}, and Reid \citeyear{reid}), but the values above are frequently adopted, 
so that a comparison with other work is made easier. For the ``theoretical'' rotation curve we have also adopted two possibilities, namely, the 
PN curve derived by Maciel \& Lago (\citeyear{ml2005}), and the HII region curve derived by Clemens (\citeyear{clemens}). In the first case, the 
rotation velocity can be written as
\begin{equation}
\theta_c(R) = a_0 + a_1 \, R + a_2 \, R^2\,,
\end{equation}
where the constants are $a_0 = 269.2549$, $a_1 = -14.7321$, and $a_2 = 0.7847$, the Galactocentric distance $R$ is given in kpc and $\theta_c(R)$ in km~s$^{-1}$. 
For the CO/HII region based Clemens (\citeyear{clemens}) curve, we have made an adjustment for $R_0 = 8.0$~kpc and $\theta_0(R)=220$~km~s$^{-1}$, in which 
case we have 
\begin{equation}
\theta_c(R) = \sum a_i\, R^i\,,
\end{equation}
where the constants are given in Table~1, with the same units as in equation~(2).


%%%%% TABLA 1

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{Coefficients of the polynomial given by Equation~(3)}
\label{table1}
\begin{tabular}{cr@{.}lr@{.}lr@{.}lr@{.}lc}
\toprule
$R$ (kpc) & \multicolumn{2}{c}{$0-0.765$} & \multicolumn{2}{c}{$0.765-2.9$} & \multicolumn{2}{c}{$2.9-3.825$} & \multicolumn{2}{c}{$3.825-13$} & $> 13$ \\
\midrule
$a_0$ &  0       & 0           & 325     & 0912    & 329     & 8       & $-$2346  & 0           & 230.6 \\
$a_1$ &  3069    & 81          & $-$248  & 1467    & $-$250  & 1       & 2507     & 60391       &  ...  \\
$a_2$ & $-$15809 & 8           & 231     & 87099   & 231     & 87099   & $-$1024  & 068760      &  ...  \\
$a_3$ &  43980   & 1           & $-$110  & 73531   & $-$110  & 73531   & 224      & 562732      &  ...  \\
$a_4$ & $-$68287 & 3           & 25      & 073006  & 25      & 073006  & $-$28    & 4080026     &  ...  \\
$a_5$ & 54904    & 0           & $-$2    & 110625  & $-$2    & 110625  & 2        & 0697271     &  ...  \\
$a_6$ & $-$17731 & 0           & .       & .       & .       & .       & $-$0     & 080508084   &  ...  \\
$a_7$ & .        & .           & .       & .       & .       & .       & 0        & 00129348    &  ...  \\
\bottomrule
\end{tabular}
\vspace{0.6cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% TABLA 2
 
\begin{table}[!t]
\vspace{0.1cm}
\centering
\setlength{\tabcolsep}{2\tabcolsep}
\caption{Coefficients of Equation~(4)}
\label{table2}
\begin{tabular}{ccc}
\toprule
      & $a$  & $b$ \\
\midrule
$U$   & 0.39 & 1.31 \\
$V$   & 0.40 & 1.10 \\
$W$   & 0.53 & 0.94 \\
Total & 0.40 & 1.40 \\
\bottomrule
\end{tabular}
\vspace{0.3cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The recent Geneva-Copenhagen Survey of the Solar Neighborhood (cf. Nordstr\"om et al. \citeyear{nordstrom}; Holmberg, Nordstr\"om, \& Andersen 2007, 
2009) has considerably improved the relations involving the ages, kinematics, and chemical composition of a large sample containing about 14000 F and 
G nearby stars. Using basically  the original {\it Hipparcos} parallaxes, $uvby-\beta$ photometry and the Padova stellar evolution models, several 
basic relations were investigated. In particular,  high correlations have been obtained between the velocity dispersions $\sigma_U$, $\sigma_V$, 
$\sigma_W$, and $\sigma_T$  and the ages of the stars, which clearly show a smooth increase of the velocity dispersions (of the $U$, $V$, $W$ components 
and the total velocity $T$) with time. From the calibration by Holmberg et al. (\citeyear{holmberg2009}) these correlations can be approximately written as
\begin{equation}
\log \sigma = a \, \log t + b\,,
\end{equation}
where the age $t$ is expressed in Gyr and the constants $a$, $b$ are given in Table~2. This approximation is valid in the age interval $0<t {\rm (Gyr)}<14$ with 
an estimated average uncertainty of about 25\%. Method~1 consists of assuming that the discrepancy in the rotation velocity $\Delta \theta$ is due 
to the evolved status of the CSPN, so that we should expect a correlation between $\Delta \theta$ and the velocity dispersion, as given by equation~(4). 
Since in this method we are using the rotation velocity, we have considered two possibilities, according to which the velocity discrepancy $\Delta \theta$ 
can be associated with (i) the $V$ component of the total velocity ($\sigma_V$), or (ii) the total velocity ($\sigma_T$). Moreover, since we are adopting 
two distance scales and two theoretical rotation curves, we have 8 different age distributions for Method~1, characterized by the timescales $t_1$ to 
$t_8$, as explained in Table~3.

 
%%%%% TABLA 3

\begin{table}[!t]
\vspace{0.1cm}
\centering
\caption{Parameters for Method 1}
\label{table3}
\small
\begin{tabular}{cccc}
\toprule
Distance & Rotation Curve & Dispersion & Age \\
\midrule
Maciel       & PN      & $\sigma_V$ & $t_1$ \\
Maciel       & PN      & $\sigma_T$ & $t_2$ \\
Maciel       & Clemens & $\sigma_V$ & $t_3$ \\
Maciel       & Clemens & $\sigma_T$ & $t_4$ \\
Stanghellini & PN      & $\sigma_V$ & $t_5$ \\
Stanghellini & PN      & $\sigma_T$ & $t_6$ \\
Stanghellini & Clemens & $\sigma_V$ & $t_7$ \\
Stanghellini & Clemens & $\sigma_T$ & $t_8$ \\
\bottomrule
\end{tabular}
\vspace{0.4cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\subsection{Method 2: The $U$, $V$, $W$ velocity components}

Method~2 is also a kinematic method, and in principle more accurate than Method~1, as discussed in more detail in \S~4. From the PN radial 
velocities and distances, we have estimated their proper motions both in right ascension and declination, $\mu_\alpha$ and $\mu_\delta$. We 
have assumed that, on average, the tangential velocities are similar to the radial velocities, namely $V_t \simeq V_r$. In view of the large 
distances of the nebulae, this hypothesis in practice does not introduce any major uncertainties in the results. Considering further the 
equatorial coordinates ($\alpha,\delta$) of the PN, we have used the equations by Boesgaard \& Tripicco (\citeyear{boesgaard}) to derive the 
$U$, $V$, $W$ velocity components of the nebulae, as well as the total velocity $T$ and the velocity dispersions $\sigma_U$, $\sigma_V$, 
$\sigma_W$, and $\sigma_T$. According to these equations we derive the following parameters: $C=f(d)$, $X=f(C, \mu_\alpha, \mu_\delta, \alpha, \delta, V_r)$, 
$Y=f(C, \mu_\alpha, \mu_\delta, \alpha, \delta, V_r)$, and $Z=f(C, \mu_\delta, \delta, V_r)$, from which the velocities can be written as 
$U=f(X, Y, Z)$, $V=f(X, Y, Z)$, $W=f(X, Y, Z)$, and $T=f(X, Y, Z)$, so that the dispersions are given by
\begin{equation}
\sigma_i = \sqrt{(V_i-\bar V_i)^2}\,,
\end{equation}
where $V_i$ stands for the velocities $U, V, W, T$. Then, we have again used the detailed correlations between the velocity dispersions and the 
ages as given by the Geneva-Copenhagen survey (Holmberg et al. \citeyear{holmberg2009}), adopting the same coefficients given in Table~2. We 
have used the same distance scales (Maciel \citeyear{m1984}; Stanghellini et al. \citeyear{ssv2008}), so that we have again 8 different age 
distributions, corresponding to the timescales $t_9$ to $t_{16}$, as described in Table~4.


%%%%% TABLA 4
 
\begin{table}[!t]
\vspace{0.1cm}
\centering
\setlength{\tabcolsep}{1.2\tabcolsep}
\caption{Parameters for Method 2}
\label{table4}
\begin{tabular}{ccc}
\toprule
Distance & Dispersion & Age \\
\toprule
Maciel & $\sigma_U$ & $t_9$ \\
Maciel & $\sigma_V$ & $t_{10}$ \\
Maciel & $\sigma_W$ & $t_{11}$ \\
Maciel & $\sigma_T$ & $t_{12}$ \\
Stanghellini & $\sigma_U$ & $t_{13}$ \\
Stanghellini & $\sigma_V$ & $t_{14}$ \\
Stanghellini & $\sigma_W$ & $t_{15}$ \\
Stanghellini & $\sigma_T$ & $t_{16}$ \\
\bottomrule
\end{tabular}
\vspace{0.4cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In practice, we have considered several additional cases, in order to better investigate the hypothesis of $V_t \simeq V_r$. Assuming that 
these velocities are of the same magnitude, but allowing for the possibility of different signs, we have as a result several possibilities 
for the proper motions $\mu_\alpha$ and $\mu_\delta$, all of which are consistent with either $V_t \simeq V_r$ or $\vert V_t\vert \simeq \vert V_r\vert$. 
It turns out that these possibilities produce very similar age distributions, which will be discussed in \S~4. Therefore, we will present 
only the distributions of the ages $t_9$ to $t_{16}$, as defined in Table~4, for the cases where $\mu_\alpha \simeq \mu_\delta \simeq 0$.

An interesting alternative to overcome the lack of proper motion and tangential velocity measurements would be to apply the singular value 
decomposition (SVD) technique, as used by Branham (\citeyear{branham}) to solve the inverse problem, that is, obtaining the space velocities 
from available proper motions. However, in view of the similarity of the results for different assumptions regarding the tangential velocities, 
it is unlikely that this technique would produce very different results than presented here. 


\section{The  samples}
\label{section3}

As mentioned in the Introduction, we have considered two samples of Milky Way PN. In order to make comparisons with our previous work, we have 
first considered the same sample used by Maciel et al. (\citeyear{mcu2003}, \citeyear{mlc2005}, \citeyear{mlc2006}), which we will call Sample~1. 
This sample contains 234 well-observed nebulae located in the solar neighborhood and in the disk, for which all data were obtained with the 
highest accuracy. Their Galactocentric distances are in the range $4<R {\rm (kpc)}<14$, and most (69\%) are located in the solar neighborhood, 
with distances $d<3$~kpc.

The second sample considered in this work, called Sample~2, includes all the nebulae for which accurate radial velocities are available in the 
catalogue by Durand et al. (\citeyear{durand}), comprising 867 objects. This is a more complete sample, so that it is expected that the derived 
results can be extended to the observed population of PN in the Galaxy. In both samples, the number of nebulae used depends on the availability 
of the statistical distances. The actual numbers of objects from the  Maciel (\citeyear{m1984}) and Stanghellini et al. (\citeyear{ssv2008}) 
distance scales are 195 and 170 for Sample 1 and 493 and 403 for Sample~2, respectively. We have then applied the approximation given by equation~(4) 
for both samples, with the coefficients shown in Table~2, considering only the objects for which ages in the interval $0<t ({\rm Gyr})<14$ could 
be obtained.


%%%%% FIGURA 1

\begin{figure}[!t]
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig1a.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig1b.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig1c.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig1d.eps}}
\caption{Age distribution of CSPN, Method 1, Sample 1.}
\label{fig1}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%% FIGURA 2

\begin{figure}[!t]
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig2a.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig2b.eps}}
\caption{Age distribution of CSPN, Method 2, Sample 1.}
\label{fig2}
\vspace{0.3cm} 
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 3

\begin{figure}[!t]
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig3a.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig3b.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig3c.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig3d.eps}}
\caption{Age distribution of CSPN, Method 1, Sample 2.}
\label{fig3}
\vspace{0.6cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 4

\begin{figure}[!t]
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig4a.eps}}
\vspace{0.4cm}
\centerline{\includegraphics[angle=-90,width=0.99\columnwidth]{wmacielfig4b.eps}}
\caption{Age distribution of CSPN, Method 2, Sample 2.}
\label{fig4}
\vspace{0.7cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Results and discussion}
\label{section4}

The main results for the age distribution of the CSPN are shown in Figures~1--4, where we have used the age parameter definitions given in Tables~3 
and 4 for Methods 1 and 2, respectively. Figures~1 and 2 refer to Sample~1, while Figures~3 and 4 refer to Sample~2. It can be seen that the age 
distributions obtained by both methods are similar, in the sense that most objects have ages under 5~Gyr, with a strong peak at ages typically between 
1 and 3~Gyr. The histograms of Figures~3--4 are summarized in Table~5, where the fraction of stars obtained by Method~1 (ages $t_1$ to $t_8$) and 
Method~2 (ages $t_9$ to $t_{16}$) are shown for three age bins, namely 0-3~Gyr, 3-6~Gyr, and $t>6$~Gyr.  


%%%%% TABLA 5

\begin{table}[!t]
\vspace{0.1cm}
\centering
\caption{Fraction of stars in\\ three age intervals}
\label{table5}
%\small
\begin{tabular}{ccccc}
\toprule
 & $\Delta t$ (Gyr) & 0--3 & 3--6  & $>6$ \\
\midrule
Method 1 & $t_1$    & $0.57$ & $0.13$ & $0.30$  \\
         & $t_2$    & $0.62$ & $0.18$ & $0.20$  \\
         & $t_3$    & $0.57$ & $0.19$ & $0.24$  \\
         & $t_4$    & $0.67$ & $0.18$ & $0.16$  \\
         & $t_5$    & $0.51$ & $0.13$ & $0.36$  \\
         & $t_6$    & $0.71$ & $0.17$ & $0.12$  \\
         & $t_7$    & $0.61$ & $0.15$ & $0.24$  \\
         & $t_8$    & $0.71$ & $0.11$ & $0.18$  \\
         &          &        &        &         \\
Method 2 & $t_9$    & $0.76$ & $0.12$ & $0.12$  \\
         & $t_{10}$  & $0.79$ & $0.10$ & $0.11$  \\
         & $t_{11}$  & $0.92$ & $0.04$ & $0.04$  \\
         & $t_{12}$  & $0.77$ & $0.18$ & $0.05$  \\
         & $t_{13}$  & $0.78$ & $0.10$ & $0.12$  \\
         & $t_{14}$  & $0.78$ & $0.11$ & $0.11$  \\
         & $t_{15}$  & $0.93$ & $0.03$ & $0.04$  \\
         & $t_{16}$  & $0.76$ & $0.18$ & $0.06$  \\
\bottomrule
\end{tabular}
\vspace{0.4cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The similarity of the results of both methods is remarkable, especially considering that Method~2 is probably more accurate than Method~1. 
Method~2 consists of straightforward calculations of the velocities and velocity dispersions followed by an application of relatively accurate 
correlations involving the kinematics and ages of the objects considered. On the other hand, Method~1 is based on the assumption that the 
differences between the observed and predicted rotation velocities are essentially due to age effects. However, other processes may be 
important, such as deviations from the circular rotation, which is particularly important for nearby objects.  According to Table~5, in all 
cases the vast majority of CSPN have ages under 3~Gyr. For Method~1 the total fraction of objects with $t \leq 3$~Gyr is 50--70\%, while for 
Method~2 this fraction is somewhat higher, 70--90\%. It is unlikely that this is a result from bias in the  samples, as the results for the larger 
Sample~2 are essentially the same as in the smaller Sample~1. It should be pointed out that the latter, albeit smaller, includes only well 
studied nebulae, for which all individual parameters (distances, velocities, abundances) are better determined. 

Also, there are no significant differences in the results using the different velocity components $U$, $V$, $W$, and $T$. For Method~1, the 
distributions using the $V$ velocity component are essentially the same as those using the total velocity, for both distance scales and samples. 
For Method~2, the distributions are slightly more concentrated in the first few age bins for the $W$ component, compared with the distributions 
for the $U$ and $V$ components and the  total velocity, again for both distance scales and samples. Since the $W$ component is more clearly 
associated with the disk heating, essentially caused by age effects, the corresponding distributions are probably more accurate. 

Similar remarks can be made regarding the adopted values for the proper motions. As mentioned at the end of \S~2, the results shown here 
assume that $\mu_\alpha \simeq \mu_\delta \simeq 0$. Adopting nonzero values for these quantities ($\mu_\alpha \simeq \mu_\delta \neq 0$), either 
the $V$ or $W$ component distributions become slightly less concentrated at the first few age bins, but most objects still have ages under 
about 4~Gyr. Again, the application of the SVD technique could be useful to confirm these results.

The uncertainties in the distances of the Milky Way PN are difficult to estimate, but the procedure adopted here ensures that the obtained age 
distributions are not particularly affected by the individual distances of the objects in the samples. As mentioned in \S~2, we have adopted 
two very different statistical scales, and the derived age distrbutions are essentially the same in both cases. The individual distances may 
depend on the particular scale, but the results shown in Figures~1--4 and in Table~5 do not depend on the choice of the distance scale. This 
can be seen by comparing the results for the timescales $t_1-t_4$ with those for $t_5-t_8$, or the results for $t_9-t_{12}$ with those for $t_{13}-t_{16}$.

The uncertainties in the radial velocities also do not seem to have an important effect on the age distributions. In the catalogue by Durand 
et al. (\citeyear{durand}), most objects ($\sim$90\%) have uncertainties smaller than 20~km~s$^{-1}$, and many objects have much lower uncertainties. 
Concerning Method~1, from Maciel \& Lago (\citeyear{ml2005}), the average rms deviation in the rotation velocity is about 50~km~s$^{-1}$ for PN, 
which can be compared with the values of about 20~km~s$^{-1}$ for HII regions (see also Clemens \citeyear{clemens}, and Maciel \& Dutra \citeyear{md1992}). 

Probably the main uncertainty of the age distributions is due to the calibration between the stellar ages and the velocity dispersions, given 
by equation~(4), which affects both Method~1 and 2. From the Geneva-Copenhagen Survey, this relation has a dispersion of about 20~km~s$^{-1}$ 
in average, which corresponds roughly to an age uncertainty of about 25\%, amounting to less than 1.2~Gyr for the objects of Figures~1--4. 
Therefore, the uncertainties of the present method are comparable and probably smaller than in the case of the methods based on age-metallicity 
relations considered by Maciel et al. (\citeyear{mci2010}).

The results for Sample~2 are not essentially different from those of Sample~1, so that a direct comparison can be made with the results by 
Maciel et al. (\citeyear{mci2010}). The results of both investigations are similar, even though the present methods are completely independent 
of the metallicity-based methods used by Maciel et al. (\citeyear{mci2010}). The main difference is that the kinematic methods used in the present 
investigation suggest somewhat lower ages for the CSPN in our samples. In this respect, these results fit nicely with the probability distribution 
for the progenitors of \,the \,CSPN according \,to \,Maciel \,et \,al. \,(\citeyear{mci2010}, \,cf. \adjustfinalcols Figure~7, dashed line). In this case the well known relation 
between the main sequence mass and the stellar ages  by Bahcall \& Piran (\citeyear{bahcall}) was adopted, taking $t=10$~Gyr for $1~M_\odot$ stars 
on the main sequence. Taking into account the uncertainties of the methods, which are typically in the range 1--2~Gyr, this case was considered 
as the most realistic, so that it is reassuring that the kinematic methods produce similar results.

\vspace{0.2cm}

\acknowledgements 

We thank Dr. R. Branham, Jr., for some interesting comments on an earlier version of this paper. 
This work was partly supported by FAPESP and CNPq.


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