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From: Sabine Philipp sphilipp@mpifr-bonn.mpg.de
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\thesaurus{4(08.12.3; 09.04.1; 10.03.1; 10.05.1; 10.19.2; 13.09.6)}
\authorrunning{P.G.\ Mezger, R.\ Zylka, S.\ Philipp, R.\ Launhardt}
\titlerunning{The Nuclear Bulge II: The K band Luminosity Function}
\title{The Nuclear Bulge of the Galaxy}
\subtitle{II. The K band Luminosity Function of the Central 30\,pc}
\author{P.G.\ Mezger\inst{1},
R.\ Zylka\inst{2,1},
S.\ Philipp\inst{1},
R.\ Launhardt\inst{1}}
\offprints{P.G.\ Mezger, Bonn}
\mail{sphilipp@mpifr-bonn.mpg.de}
\institute{Max-Planck-Institut f\"ur Radioastronomie, Auf dem H\"ugel 69,
D-53121 Bonn, Germany
\and
Institut f\"ur Theoretische Astrophysik, Tiergartenstra{\ss}e 15,
D-69121 Heidelberg, Germany
}
\date{Received 19.01.1999 ; accepted \dots}
\maketitle
\begin{abstract}
Philipp et al. (1999, Paper I) investigated the $K$ band emission from a mosaic
of size $\Delta\alpha\times\Delta\delta \sim 650''\times710''$ centered
approximately on Sgr~A* ($R_{\rm equiv} \sim 15.8$\,pc for $R_0 = 8.5$\,kpc).
For the $\sim 6\times10^4$ stars above the detection limit
($S_{\rm K}^\prime \sim 100\,\mu$Jy)
an {\it observed} $K$-band luminosity function (KLF) has been obtained.
Below the completeness limit ($S_{\rm K}^\prime \sim 2\,000\,\mu$Jy),
an ever increasing fraction of stars merges into the background continuum.
In this paper we combine the {\it observed} with {\it model} KLFs
and thus obtain a {\it complete} KLF for the flux density range $3\times10^{-3}
\ga S_{\rm K}^\prime / \mu{\rm Jy} \ga
2\times10^6$. The overall KLF consists of four sectors obeying power laws
of the form $dN(S_{\rm K}^\prime) / d{\rm log} S_{\rm K}^\prime \propto
S_{\rm K}^{\prime \gamma_1 + 1}$, where $\gamma_1 + 1$ decreases from --0.6 to
--1.75. Sector\,{\tt I} corresponds to a Salpeter Initial Mass Function
(IMF) and represents Main Sequence (MS) stars with $M_* \la 1\,{\rm M}_\odot$,
which account for $\sim 90\,\%$ of the dynamical mass but only $\sim 6\,\%$
of the K band flux density. Sector\,{\tt II} represents MS stars with $M_*
\ga 1\,{\rm M}_\odot$ and red giants. These stars account for only $\sim
6\,\%$ of the dynamical mass and a similar percentage of the integrated
$K$-band surface brightness but represent $\sim 80\,\%$ of the bolometric
stellar luminosity in the mosaic. The Mass Function (MF) of MS stars is
$dN / dM_*$ $\propto M_*^{-2.35}$ (i.e., the Salpeter IMF) for $M_*
\la 1\,{\rm M}_\odot$ and $dN / dM_*$ $\propto M_*^{-4.5}$ for more
massive stars, which is similar to the Present Day MF in the solar vicinity.
Part of sector\,{\tt II} of the KLF, as well as sectors\,{\tt III} and
{\tt IV}, represent giants and supergiants which, though they account for
only a small fraction of the mass, dominate the integrated $K$-band surface
brightness.
... shortened ..
Paper I shows, in agreement with earlier observations, that massive stars are
preferentially formed in the central parsec. A preliminary discussion of star
formation rates suggests that bimodal star formation (introduced by
G\"usten \& Mezger [1983] for the spiral arm region of the Galactic Disk)
may also apply to the central 30\,pc. Preferential formation of stars
with masses $M_* \ga M_c \sim 1 - 5\,{\rm M}_\odot$ would
make conversion of matter into radiation by star formation much more
efficient and could be the process which powers star burst galaxies. There is
an overabundance of evolved stars which can be explained by a strongly
increased star formation rate $\sim 10^8 - 10^9$\,yrs ago.
\end{abstract}
\keywords{Stars: luminosity function, mass function -- ({\it ISM:\/}) dust,
extinction -- Galaxy: center -- evolution -- stellar content --
Infrared: stars}
\section{Introduction}
In a recent review (Mezger et al., 1996; hereafter MDZ96) we have
compared the central region of our galaxy with active nuclei in
external galaxies and have found that it belongs to the class of mildly
active Seyfert nuclei, which fuel their luminosity by starbursts and
account for $\sim 10\,\%$ of all spiral galaxies.
The conclusion is nearly inevitable that there is a black hole of mass
$\sim 2.6 \times 10^6\,{\rm M}_\odot$ at the center of our galaxy, coinciding
with the compact radio source Sgr A\,* (Genzel et al., 1997, and references
therein). However, all observations agree that accretion onto this black
hole does not at present contribute more than a few hundred ${\rm L}_\odot$
to the dust luminosity of $\sim 4\times10^6\,{\rm L}_\odot$ of the central
parsec, which is mainly powered by the absorption of radiation from massive
young stars.
Are there indications of greater activity in the past? Apart from the
enigmatic source Sgr A East, there are few observations supporting recent
violent energy releases which could be connected to the black hole. It
rather appears that the ISM is swept by a bar from the Galactic Disk (GD)
towards the Center, where it assembles in a disk-like structure of stars
and ISM referred to as the Nuclear Bulge (NB). The bar, on the other hand,
is identified with the Galactic Bulge (GB) which extends in the galactic
plane as far as $R \sim 3.5$\,kpc. Episodic star formation, i.e. the more
or less periodic transformation of the inward flowing ISM into stars, appears
to account for most of the more recent activity in the Galactic Center
(see also Serabyn \& Morris, 1996). Within the inner Lindblad resonance
of the GD, i.e. $R \la 3.5$\,kpc, star formation occurs nearly exclusively
in the NB which -- as seen both in the starlight dominated near infrared
($\lambda \la 5 - 7\,\mu$m) and in the dust emission dominated
mid- to far-infrared -- extends $\sim 200 - 300$\,pc in the galactic plane
and $\sim 30 - 50$\,pc perpendicular to it and may be adequately described
as a massive ($\sim 4\times10^9\,{\rm M}_\odot$) disk or torus consisting
of stars and interstellar matter. The physical conditions in the NB --
specifically the matter and radiation densities -- are extreme: In a volume
$V_{\rm NB} \sim 10^{-4} V_{\rm GD}$ the NB contains $\sim 5\,\%$ of the total
mass of stars and ISM in the GD. Similar percentages hold for the stellar and
dust luminosities. To further investigate its evolution we have begun
a program to investigate the physical state of the NB. In paper I (Philipp
et al., 1999), we derived the $K$-band luminosity function (KLF) of the
central $\sim 30$\,pc, which we here analyze and interpret it in terms of
star formation history. The star formation history of the NB will be dealt
with in a later paper.
To keep this paper concise we have used a number of abbreviations whose
meanings are explained in Table \ref{tababbrev}.
\begin{table}
\caption{{\bf Abbreviations used in this paper}}
\label{tababbrev}
\begin{tabular}{cc}
\hline
abbreviations & Meaning\\
\hline
BW & Baade's Window, an area of low extinction\\
& centered approximately at $l \sim 1^\circ$, $b \sim -3.9^\circ$\\
GB & Galactic Bulge\\
GC & Galactic Center\\
GD & Galactic Disk\\
IMF & Initial Mass Function\\
KLF & K band Luminosity Function\\
LF & Luminosity Function\\
Lyc & Lyman Continuum\\
MF & Mass Function\\
Mosaic & Area of size $\Delta \alpha\times \Delta \delta \sim
650''\times710''$\\
& centered approximately on Sgr A\,*\\
MS & Main Sequence\\
NB & Nuclear Bulge\\
PDMF & Present Day Mass Function\\
SFR & Star Formation Rate\\
\hline
\end{tabular}
\end{table}
\section{Observations}
A visual extinction of typically $\sim 31$\,mag caused by dust in the GD
and NB (the GB to a first approximation is free of dust) prevents
observations of stars in the NB at wavelengths $\lambda < 1.6\,\mu$m.
We have constructed mosaic images in the $H$- and $K$-bands ($\lambda 1.65$
and $2.2\,\mu$m) of an area $\Delta \alpha \times \Delta \delta \sim 650''
\times 710''$ (equivalent radius $R_{eq} \sim 383''$ or 15.8\,pc for
$R_0 = 8.5$\,kpc),\footnote{Here we use ``mosaic'' and ``central 30\,pc''
as synonymous expressions.} centered approximately on Sgr A\,*. In paper
I (Philipp et al., 1999) we reported on results obtained from the $K$-band
mosaic. With a detection limit $S_{\rm K}^\prime
\sim 100\,\mu$Jy\,\footnote{In the following we refer to $S_{\rm K}^\prime$
as observed and $S_{\rm K}$ as dereddened $K$-band flux densities.} and
a completeness limit of $S_{\rm K}^\prime \ga 2\,000\,\mu$Jy,
we separated $\sim 6\times10^4$ sources from an extended unresolved
background and derived a logarithmic $K$-band luminosity function (KLF)
$dN(S_{\rm K}^\prime) / d{\rm log}S_{\rm K}^\prime \propto
S_{\rm K}^{\prime \gamma_1 + 1}$ which increases with $\gamma_1 +1 \sim -1.75$
for $2\times10^6 \ga S_{\rm K}^\prime / \mu{\rm Jy} \ga 4.5\times10^4$,
flattens to $\gamma_1 + 1 \sim -1.0$ for $4.5\times10^4 \ga S_{\rm K}^\prime /
\mu{\rm Jy} \ga 3.6\times10^3$ and attains a maximum at $S_{\rm K}^\prime
\sim 4\times10^3\,\mu$Jy.
Fig.\,9 of paper I shows computed and reddened $K$-band flux densities of Main
Sequence (MS), giant and supergiant stars. It is clear from this diagram that
only giants, supergiants and MS stars earlier than B5 are observable above
the detection limit. MS stars later than B5 and a decreasing number of stars
with $S_{\rm K}^\prime$ between the detection and completeness limits are
not detected as individual sources but merge into the background continuum.
In paper I we find that, of a flux density of $S_{\rm K,integr}^\prime \sim
752$\,Jy integrated over the mosaic, about half is contributed by stars
above the detection limit and the other half is contributed by the background
continuum. We also estimate that a flux density $S_{\rm GB+GD}^\prime \sim
131$\,Jy is contributed to $S_{\rm K,integr}^\prime$ by stars in Galactic
Bulge and Disk.
\section{Interrelations between Mass and Luminosity Functions}
The logarithmic $K$-band luminosity function (KLF) $dN(S_{\rm K} ^\prime) /$
$d{\rm log}S_{\rm K}^\prime$ counts the number of sources within a given flux
density range in logarithmic bins. This allows a convenient graphical
presentation of data taken over as many as 8 decades. For numerical
evaluations a linear KLF $dN(S^\prime) / dS_{\rm K}^\prime$ (i.e. with
sources counted in linear bins) is, however, more convenient. There exists
the relation
\begin{equation}
\frac{dN}{dS_{\rm K}^\prime} = 0.43 S_{\rm K}^{\prime -1} \frac{dN(S_{\rm K}
^\prime)}{d{\rm log}S_{\rm K}^\prime}.
\end{equation}
\noindent For ease in making comparisons one usually normalizes observed
KLFs to an area of $1^{\widehat{\prime\prime}}$. Here we discuss luminosity
and mass functions relating to the mosaic and thus have to multiply the above
equation by $\Omega_{\rm mosaic} \sim 4.1\times10^{5\widehat{\prime\prime}}$,
the solid angle that it subtends. We thus have
\begin{eqnarray}
\frac{dN({\rm mosaic})}{dS_{\rm K}^\prime} & = & 0.43 \Omega_{\rm mosaic}
S_{\rm K}^{\prime -1} \frac{dN(S_{\rm K}^\prime)}{d{\rm log}S_{\rm K}^\prime}
\nonumber \\
& = & C_1 S_{\rm K}^{\prime \gamma_1}.
\end{eqnarray}
\noindent In paper I we showed that a mass function (MF) -- in this case
the Salpeter Initial Mass Function (IMF)
\begin{equation}
\frac{dN}{dM_*} = A M_*^{\alpha},
\end{equation}
\noindent where $\alpha = -2.35$, can be transformed into a model KLF given
by the above relation (2) with $\gamma_1 = -1.6$, using the observed
relation between stellar mass and reddened $K$-band flux density
\begin{equation}
\left(\frac{S_{\rm K}^\prime}{\rm \mu Jy} \right) \sim 1.6\left( \frac{M_*}{
{\rm M}_\odot} \right)^{2.3} = B_1 M_*^{\beta_1}.
\end{equation}
\noindent With the stellar parameters given in Appendix A this relation
holds approximately for all Main Sequence (MS) stars.
In a similar way the luminosity function (LF), i.e. the distribution function
of the total stellar luminosity
\begin{equation}
\frac{dN}{dL_*} = C_2 L_*^{\gamma_2}
\end{equation}
\noindent can be constructed with
\begin{equation}
\frac{L_*}{{\rm L}_\odot} = B_2 M_*^{\beta_2} = \left \{
\begin{array}{cc}
M_*^{2.8} & for M_* \la 1 {\rm M}_\odot \\
M_*^{4} & for M_* \ga 1 {\rm M}_\odot
\end{array}
\right \}
\end{equation}
\noindent
which is valid for MS stars (see e.g. Lang, 1992). Relations showing how to
compute the numerical values $A$ or $C$ and the exponents $\alpha$ or
$\gamma$ with $B_1 = 1.6$, $B_2 = 1$ and $\beta_2 = 2.8$, and 4, respectively,
can be found in Appendix B. They can be used to convert $K$-band luminosity
functions into mass and bolometric luminosity functions.
\section{A Complete Mosaic KLF}
A complete mosaic KLF is shown in Fig.\,\ref{figmosaicklf} and given in Table
\ref{tabklf} in analytical form. It consists of
four sectors described by power-law approximations of the form $dN(S_{\rm K}
^\prime) / d{\rm log}S_{\rm K}^\prime \propto S_{\rm K}^{\prime \gamma_1 + 1}$.
Here $\gamma_1 + 1$ increases from --1.75 to --0.6 while $S_{\rm K}^\prime$
decreases from $2\times10^6$ to $3\times10^{-3}\,\mu$Jy.
The four sectors are labeled {\tt I} through {\tt IV} and are listed in
column\,1 of Table \ref{tabklf}. Column\,2 gives their ranges of validity.
Power law approximations $dN(S_{\rm K}^\prime) / d{\rm log}S_{\rm K}^\prime$,
normalized to an area of $1^{\widehat{\prime\prime}}$, are given in column\,3
and shown in Fig.\,1. The corresponding linear KLFs $dN(S_{\rm K}^\prime)
/ dS_{\rm K}^\prime$, but now referred to the mosaic with $\Omega_{\rm mosaic}
\sim
4.1\times10^{5\widehat{\prime\prime}}$, are given in column\,4. Stellar
luminosity (LF) and mass functions (MF) related to the mosaic are given in
columns\,8 and 10. Column\,7 delineates the range of MS stellar
masses to which the MFs apply. Cumulative numbers of stars and their reddened
K band flux densities are given in columns\,5 and 6, while cumulative stellar
luminosities and masses are given in columns\,9 and 11.
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(55,64)
%\special{psfile=lumf_add_mod_bwfit_pgm_bw.ps
\special{psfile=h1354f1.ps
angle=-90 hoffset=0 voffset=192 hscale=32 vscale=32}
\end{picture}
\caption{The complete mosaic KLF is shown together with the observed
KLF and datapoints of the luminous part of the adjusted BW KLF (Tiede et al.,
1995, and references therein). See also sect.\,4.3.}
\label{figmosaicklf}
\end{figure}
\begin{table*}
\caption{\bf Mass and Luminosity Functions of the central 30\,pc}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Sector &\multicolumn{2}{|c|}{$S_{\rm K,min}^\prime$, $S_{\rm K,max}^\prime$}&
$dN(S_{\rm K}^\prime) / d{\rm log}S_{\rm K}^\prime$ & $dN(S_{\rm K}^\prime) /
dS_{\rm K}^\prime$ & $N_{\rm cum}$ & $S_{\rm cum}^\prime$ \\
&\multicolumn{2}{|c|}{$\mu$Jy} & $S_{\rm K}^\prime$ in $\mu$Jy& $S_{\rm
K}^\prime$ in $\mu$Jy& & Jy\\
(1) &\multicolumn{2}{|c|}{(2)} & (3) & (4) & (5) & (6)\\
\hline
{\tt I} & 3\,10$^{-3}$ & 1.6 & \qquad 89.6\,$S_{\rm K}^{\prime -0.6}$ &
1.58\,$10^7$\,$S_{\rm K}^{\prime -1.6}$ & 8.4\,$10^8$ & 43.8 \\
{\tt II} & 1.6 & 5.3\,$10^3$ & \qquad 98.4\,$S_{\rm K}^{\prime -0.8}$ &
1.73\,$10^7$\,$S_{\rm K}^{\prime -1.8}$ & 1.5\,$10^7$ & 385.7 \\
{\tt IIa} & 1.6 & 5.3\,$10^3$ & \quad\, 136.1\,$S_{\rm K}^{\prime -1.5}$ &
2.40\,$10^7$\,$S_{\rm K}^{\prime -2.5}$ & (7.9\,$10^6$) & (37) \\
{\tt III} & 5.3\,$10^3$ & 4.5\,$10^4$ & \qquad 547\,$S_{\rm K}^{\prime -1.0}$ &
9.64\,$10^7$\,$S_{\rm K}^{\prime -2.0}$ & 1.6\,$10^4$ & 206.2\\
{\tt IV} & 4.5\,$10^4$ & 2\,$10^6$ & \,1.69\,$10^6$\,$S_{\rm K}^{\prime -1.75}$
& 2.98\,$10^{11}$\,$S_{\rm K}^{\prime -2.75}$ & 1.2\,$10^3$ & 121.1\\
$\Sigma^{\tt IV}_{\tt I}$ & & & & & 8.6\,$10^8$ & 756.8\\
\hline
\end{tabular}
\end{center}
\vspace*{3mm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Sector & \multicolumn{2}{|c|}{$M_{\rm *,min}$, $M_{\rm *,max}$} & $dN / dL_*$ &
$L_{*,{\rm cum}}$ & $dN / dM_*$ & $M_{*,{\rm cum}}$\\
&\multicolumn{2}{|c|}{${\rm M}_\odot$} & $L_*$ in ${\rm L}_\odot$ &
${\rm L}_\odot$ & $M_*$ in ${\rm M}_\odot$ & ${\rm M}_\odot$\\
(1)&\multicolumn{2}{|c|}{(7)} & (8) & (9) $[a],[b]$ & (10) & (11) $[a]$\\
\hline
{\tt I} & 0.06 & 1 & 9.75\,$10^6$\,$L_*^{-1.50}$ & 1.5\,$10^7$ &
2.73\,$10^7$\,$M_*^{-2.35}$ & 1.3\,$10^8$\\
{\tt II} & 1 & 34 & 6.83\,$10^6$\,$L_*^{-1.45}$ &(2.9\,$10^{10}$) &
2.73\,$10^7$\,$M_*^{-2.85}$ & (3.1\,$10^7$)\\
{\tt IIa} & 1 & 34 & 6.83\,$10^6$\,$L_*^{-1.9~}$ & 2.8\,$10^8$ &
2.73\,$10^7$\,$M_*^{-4.5~~}$ & 8.2\,$10^6$\\
{\tt III} & & & & $\ga 4.2\,10^7$ & & \\
{\tt IV} & & & & $\ga 2.1\,10^7$ & & \\
$\Sigma^{\tt IV}_{\tt I}$ & & & & $\ga 3.6\,10^8$ & & 1.4\,$10^8$ \\
\hline
\end{tabular}
\end{center}
\vspace*{1mm}
\hspace*{30mm}Footnotes:\\
\hspace*{30mm}$[a]$ Values in brackets have not been used to compute
$\Sigma^{\tt IV}_{\tt I}$.\\
\hspace*{30mm}$[b]$ Cumulative luminosities {\tt I, II}, and {\tt IIa}
have been computed by integration of the LF, \\
\hspace*{33mm} column\,8, within the limits $3.8\times10^{-4} -
1\,{\rm L}_\odot$ for sector\,{\tt I} and $1 - 1.3\times10^6\,{\rm L}_\odot$
for\\
\hspace*{33mm} sector\,{\tt II}; luminosities related to sectors\,{\tt III} and
{\tt IV} have been computed by substituting \\
\hspace*{33mm} $T_{\rm eff} \sim 3000$\,K and $S_{\rm cum}^\prime$ from
column\,6 in Eq.\,(11), the contribution of evolved stars to \\
\hspace*{33mm} the stellar luminosity of sector\,{\tt II} has been neglected.
\label{tabklf}
\end {table*}
\subsection{Conditions a complete mosaic KLF must comply with}
As mentioned in sect.\ 2, the $\sim 6\times10^4$ sources with reddened flux
densities above the detection limit $S_{\rm K}^\prime \sim 100\,\mu$Jy account
for only about half of the flux density integrated over the mosaic. The
weaker sources form a continuum for which one has to find an appropriate
model KLF. This model KLF should comply with the following conditions:
\begin{enumerate}
\item It should smoothly join with the observed KLF above the completeness
limit.
\item It should be physically plausible.
\item The cumulative flux density must agree with
\begin{eqnarray}
S^\prime_{\rm cum} & = &\int\limits_{S_{\rm K}^\prime = 3\times10^{-3}\,\mu{\rm
Jy}}^{2\times10^6\,\mu{\rm Jy}} S_{\rm K}^\prime \frac{dN}{S_{\rm K}^\prime}
dS_{\rm K}^\prime \nonumber\\
& = & S_{\rm K,integr}^\prime \sim 752\,{\rm Jy},
\end{eqnarray}
\end{enumerate}
\noindent the $K$-band flux density integrated over the mosaic. The sum
at the bottom of column\,6 Table \ref{tabklf} shows that the cumulative
flux density of the complete KLF actually agrees within $< 1\,\%$ with the
integrated flux density.
\subsection{The model KLF $dN_{\tt I}(S_{\rm K}^\prime) / d{\rm log}
S_{\rm K}^\prime$ based on the
\newline
Salpeter IMF}
In paper I we adopted the Salpeter IMF (see Table
\ref{tabklf}, column\,10, sector\,{\tt I}) and converted it into the KLF
$\propto S_{\rm K}^{\prime -0.6}$ (column\,3). The IMF describes the mass
distribution of a newly formed generation of stars; hence, the use of an IMF
is physically plausible only for stars with masses $M_* \la 1\,{\rm M}_\odot$
(or $S_{\rm K}^\prime \la 1.6\,\mu$Jy) whose MS lifetime $\tau_{\rm MS} \ga
10^{10}$\,yrs is the assumed age of the NB. This determines the range of
validity as given in Table \ref{tabklf}, column\,2 for the KLF in
sector\,{\tt I}.
\subsection{The model KLF $dN_{\tt II}(S_{\rm K}^\prime) / d{\rm log}
S_{\rm K}^\prime$ based on
\newline
observations in Baade's Window}
\label{s43}
The NGC\,6522 Baade's Window (BW) located at $l \sim 1^\circ$, $b \sim
-3.9^\circ$ is distinguished by its low extinction ($A_{\rm K}
\sim 0.13$\,mag compared to $A_{\rm K} \sim 3.5$\, mag) in the
direction of the Galactic Center. The BW KLF has been the subject of
a number of investigations by Davidge et al. (1997), DePoy et
al. (1993) and Tiede et al. (1995). We use the latters' KLF, which appears to
be complete for $K_0 \la 16.25$\,mag; it is given as number of stars per
0.5$m_{\rm K}$ in an area of 2.58 arcmin$^2$. After conversion of $K$
magnitudes into reddened (by $A_{\rm K} \sim 0.13$\,mag) flux densities
$S_{\rm K,BW}^\prime$ and normalization to an area of
$1^{\widehat{\prime\prime}}$ we obtain the following observed
logarithmic BW KLF:
\begin{equation}
\frac{dN(BW)}{d{\rm log}S_{\rm K,BW}^\prime} = 21.15 S_{\rm K,BW}
^{\prime -0.80}
\end{equation}
\noindent valid within the limits $m_{\rm K} \sim$ 9.75 and 16.25\,mag or
$8\times10^4 \ga S_{\rm K,BW}^\prime / \mu{\rm Jy} \ga 2\times10^2$. For a
comparison with the mosaic KLF one has to compensate the BW KLF for the
extra reddening towards the GC, using the relation
$S_{\rm K,mosaic}^\prime = e^{-\tau_{\rm K}} S_{\rm K,BW}^\prime$. For
$A_{\rm K,mosaic} \sim 3.5$\,mag and $A_{\rm K,BW} \sim 0.13$\,mag one obtains
$\tau_{\rm K} \sim 3.1$ and $S_{\rm K,mosaic}^\prime \sim 4.5\times10^{-2}
S_{\rm K,BW}^\prime$, which, when substituted into eqn (8), yields
\begin{equation}
KLF({\rm BW,reddened}) \sim 1.77 S_{\rm K,mosaic}^{\prime -0.80}
\end{equation}
\noindent
Since we have fixed the slope of the KLF at $\gamma_1 + 1 = -0.80$\, the only
free parameter available to adjust the modified BW KLF to the KLF in
sector\,{\tt I} (see previous section) is the surface density $N_*$ of MS stars
at $M_* \sim 1\,{\rm M}_\odot$. We obtain the best fit approximation, as
given in Table \ref{tabklf}, column\,3, sector\,{\tt II}, by multiplying the
above
relation by
\begin{equation}
\label{eq10}
\frac{N_*({\rm mosaic})}{N_*({\rm BW})} = \frac{98.4}{1.77} = 55.6
\end{equation}
\noindent
The corresponding space-averaged volume density is $n_*(NB)$ $\sim$
$10^3\,n_*(GB)$. The observed KLF, shifted to the mosaic, is valid in the
range $9 \la S_{\rm K,mosaic}^\prime / \mu{\rm Jy} \la 3.6\times10^3$.
We enlarged
this range of validity slightly to $1.6 \la S_{\rm K}^\prime / \mu{\rm Jy} \la
5.3\times10^3$ so that sector\,{\tt II} now spans the range of stellar masses
from 1 to 34\,${\rm M}_\odot$. A star of $M_* \sim 8\,{\rm M}_\odot$ has a
reddened $K$-band flux density of $S_{\rm K}^\prime \sim 2\times10^2\,\mu$Jy.
It is of interest to note that in the region $5.3\times10^3 \la S_{\rm
K,mosaic}^\prime / \mu{\rm Jy} \la 10^5$, where the shifted BW KLF deviates
from the power law approximation $\propto S_{\rm K}^{\prime -0.8}$, the
observations by DePoy et al. (1993), reduced as above and shown in
Fig.\,\ref{figmosaicklf}, agree well with the observed mosaic KLF.
\section{Physical Plausibility of the Mosaic KLF}
The complete mosaic KLF, i.e. the combination of the model KLF sectors\,{\tt I
+ II} with the observed KLF sectors\,{\tt III + IV}, complies with conditions
1) and 3) stated in sect.\,4.1. Here we discuss condition 2), i.e. the physical
plausibility of the model KLFs, by comparing observed and predicted stellar
luminosities. The dust luminosity of the mosaic of $L_{\rm IR} \sim
1.2\times10^8\,{\rm L}_\odot$ (Table\,C1) is clearly a lower limit to the
stellar luminosity. The $K$-band surface brightness integrated over the
mosaic yields a comparable lower limit if we assume that all stars in the
NB are old and evolved with an effective temperature of $\sim 3000$\,K
(see Eq.\,12 below). The high Lyman continuum photon production rate
derived from free-free radio emission (Table\,C1) indicates, however,
that the stellar population of the NB contains a substantial fraction
of young, massive and hot stars which can ionize the surrounding gas.
The predicted stellar luminosity depends on the number of massive MS stars
which are mixed with evolved stars. For a separation of evolved and MS stars
we can concentrate our efforts on sector\,{\tt II} of the KLF which contains
MS stars in the mass range $M_* \sim 1 - 34\,{\rm M}_\odot$. Sector\,{\tt I}
(which contain mainly MS stars with $M_* < 1\,{\rm M}_\odot$) as well as
sectors\,{\tt III} and {\tt IV} (which contain mainly giants and supergiants)
each contribute only $\sim 1/10$-th of the estimated total stellar luminosity
of $\sim 3.6\times10^8\,{\rm L}_\odot$ (see Table\,\ref{tabklf}, bottom line).
In the following we estimate the number of MS stars contained in sector\,{\tt
II}, labelling it as subgroup {\tt IIa} and referring to the result as the
Present Day Mass Function (PDMF).
\subsection{Stellar Luminosity of the Mosaic}
One can obtain a lower limit to the integrated luminosity of the ensemble of
mosaic stars by assuming that all of these stars have effective temperatures
as low as $T_{\rm eff} \sim 3000$\,K. Combination of Eqs.\,(5.4) and (5.5) of
MDZ96 yields for the stellar luminosity (with $x = 6.55\times10^3({\rm K} /
T_{\rm eff})$)
\begin{equation}
\left(\frac{L}{{\rm L}_\odot} \right) = 7.14\times10^{-8} \left(\frac{T_{\rm
eff}}{{\rm K}} \right)^3 \frac{e^x - 1}{x} \left(\frac{S_{\rm K}}{{\rm Jy}}
\right)
\end{equation}
\noindent and, specifically for the mosaic, we take $\tau_{\rm K} \sim 3.2$,
$S_{\rm K}^\prime \sim 752$\,Jy and $S_{\rm K} = S_{\rm K}^\prime
e^{\tau_{\rm K}} \sim 1.9\times10^4$\,Jy, obtaining
\begin{equation}
L_{\rm *,mosaic} \ga 1.3\times10^8 {\rm L}_\odot
\end{equation}
\noindent Likewise one obtains an upper limit for the luminosity by assuming
that all stars in the KLF sector\,{\tt II} are MS stars; the corresponding
luminosity function $dN_{\tt II} / dL_*$ (column\,8 of Table \ref{tabklf})
yields a cumulative luminosity of (see column\,9)
\begin{equation}
L_{\rm *,mosaic} \la 2.9\times10^{10} {\rm L}_\odot
\end{equation}
\noindent which we consider as a strict upper limit (actually this luminosity
equals that of the whole Galactic Disk (see e.g. MDZ96, Table 5).
These lower and upper limits of the stellar luminosity in the central $\sim
30$\,pc have to be compared to an estimated dust luminosity of $L_{\rm IR}
\sim 1.2\times10^8\,{\rm L}_\odot$ (Table \ref{tabirlumlyc}, column\,2), which
coincides with the lower limit given by Eq.\,(12). To conclude, however,
that dust in the mosaic is mainly heated by cool stars would be wrong: since
the dust absorption cross section in the NIR decreases $\propto
\lambda^{-1.7}$ only a small fraction of the photons emitted by stars with
$T_{\rm eff} \sim 3000$\,K will be intercepted by dust. The absorption by dust
of radiation from hot stars is much more efficient. Furthermore, the
presence of radio free-free emission (quantitatively represented by
$N_{\rm Lyc}^\prime$, Table \ref{tabirlumlyc}, column\,4; see also MDZ96)
reveals convincingly the presence of hot and luminous stars in the mosaic
outside the central parsec.
The upper limit of the integrated stellar luminosity of
$L_{\rm IR} \la 3\times10^{10}\,{\rm
L}_\odot$ is $\sim 250$ times higher than the value of $\sim 1.2\times
10^8\,{\rm L}_\odot$ consistent with both the observed dust luminosity (Table
\ref{tabirlumlyc}, column\,2) and the lower limit of the stellar luminosity
(Eq.\,(12)). The reason for this high luminosity, contributed by medium and
high mass MS stars, is the (physically implausible) assumption that all
stars contained in sector\,{\tt II} of the model KLF are actually still
on the MS. In a more realistic approach one has to consider and account
for the short MS lifetimes of the medium and high mass stars and
therefore has to use the Present Day Mass Function (PDMF) rather than
the IMF which describes the mass distribution of a newly born star generation.
HST observations (Holtzmann et al., 1993) indicate for the central parsec a MS
turnoff at stellar masses of $\sim 1\,{\rm M}_\odot$ which we adopt in the
following discussion of the mosaic.
\begin{table*}
\caption{\bf Numbers of Main Sequence Stars ($\mathbf 1 \mathbf - \mathbf
3\mathbf4\,{\rm{\bf M}}_{\bf \odot}$),
Giants and Supergiants contained in the Mosaic}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$M_{\rm *,min}$ & $\tau_{\rm MS}$ & $M_{\rm *,max}$ & $\tau_{\rm MS}$ &
stars on MS & stars off MS & Red Giants & Supergiants & Stars not accounted
for \\
${\rm M}_\odot$ & yrs & ${\rm M}_\odot$ & yrs & $[a]$ & $[b]$ & $[c]$ & $[d]$
& [e] \\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9)\\
\hline
1& 9.5\,10$^9$ & 8 & 1.3\,10$^7$ & 7.8\,10$^6$ & 1.1\,10$^7$ & 4.8\,10$^5$ &
& 6.6\,10$^6$\\
8& 1.3\,10$^7$ & 34 & 4.2\,10$^6$ & 5.4\,10$^3$ & 1.0\,10$^6$ & & 4.4\,10$^4$
& \\
\hline
\end{tabular}
\end{center}
\vspace*{1mm}
\hspace*{22mm}Footnotes:\\
\hspace*{22mm}$\tau_{\rm MS}$ for $Z = 0.04$ from Schaerer et al. (1993).
Accumulated numbers of stars have been computed from: \\
\hspace*{22mm}$[a]$ $dN_{\tt IIa} / dM_*$\\
\hspace*{22mm}$[b]$ $dN_{\tt I} / dM_* - dN_{\tt IIa} / dM_*$; $dN_{\tt I} /
dM_* \propto M_*^{-2.35}$ is supposed to be the birthrate function (IMF).\\
\hspace*{22mm}$[c]$ $dN_{\tt II} / dS_{\rm K}^\prime - dN_{\tt IIa} /
dS_{\rm K}^\prime$ ($10^2 \la S_{\rm K}^\prime / \mu{\rm Jy} \la 2\times10^3$)\\
\hspace*{22mm}$[d]$ $dN_{\tt II} / dS_{\rm K}^\prime - dN_{\tt IIa} /
dS_{\rm K}^\prime$ ($2\times10^3 \la S_{\rm K}^\prime / \mu{\rm Jy} \la
5.3\times10^3$) $+ N_{\rm cum\,III} + N_{\rm cum\,IV}$.\\
\hspace*{22mm}$[e]$ $dN_{\tt II} / dS_{\rm K}^\prime - dN_{\tt IIa} /
dS_{\rm K}^\prime$ ($1.6 \la S_{\rm K}^\prime / \mu{\rm Jy} \la 100$).
\label{tabstars}
\end {table*}
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(55,66)
%\special{psfile=lumf_add_mod_bwfit_pgm_3_2.ps
\special{psfile=h1354f2.ps
angle=-90 hoffset=-5 voffset=187 hscale=32 vscale=32}
\end{picture}
\caption{Mass functions for MS stars in the mosaic discussed in this
paper. The combination of the Salpeter IMF (sector\,{\tt I}) with the PDMF
(sector\,{\tt IIa}) has been adopted (see Table \ref{tabklf}, column\,10).}
\label{figpdmfklf}
\end{figure}
\subsection{The Present Day Mass Function (PDMF)}
In this section we search for a PDMF whose Lyc- and bolometric luminosity
are consistent with the observations. In Appendix A, Eqs.\,(A.1) and (A.2)
we give some integral properties of young O star clusters. Substitution of
$N_{\rm Lyc} \sim 3.6\times10^{51}$\,s$^{-1}$ (Table \ref{tabirlumlyc},
column\,6) for the actual Lyc photon production rate in Eq.\,(A.1) yields the
mass $<$$M_*$$>$ $\sim 1.1\times10^4\,{\rm M}_\odot$ for the O star cluster
responsible for the ionization of gas in the mosaic. This mass can be used to
find the slope (for $M_* > 1\,{\rm M}_\odot$) of an appropriate PDMF. With
$dN / dM_* = 2.73\times10^7 M_*^{\alpha}$ (see Table \ref{tabklf},
column\,9 and 10) the cumulative mass is
\begin{equation}
\frac{M_{\rm cum}}{{\rm M}_\odot} = 2.73\times10^7 \int \limits_{M_* = 15
{\rm M}_\odot}^{34 {\rm M}_\odot} M_*^{\alpha +1}dM_*
\end{equation}
\noindent The integration limits are determined by the fact that only O stars
with masses $M_* \sim 15 - 34\,{\rm M}_\odot$ have high
enough effective temperatures to contribute substantially to the ionization.
Evaluation of the above Eq.\,(14) shows that for $\alpha \sim -4.5$ the
cumulative mass is $\sim 1.36\times10^4\,{\rm M}_\odot$ sufficiently close to
the above value $<$$M_*$$>$. The corresponding cumulative bolometric luminosity
of all O stars of $L_* \sim 2.8\times10^8\,{\rm L}_\odot$ is found in Table
\ref{tabklf}, column\,9, sector\,{\tt IIa} together with the luminousity of
all stars in the mosaic of $\Sigma_{\tt I}^{\tt IV} L_* \sim 3.6\times
10^8\,{\rm L}_\odot$ which is $\sim 3 L_{\rm IR}$, the dust luminosity given in
Table \ref{tabirlumlyc}.
Fig.\,\ref{figpdmfklf} shows the three different MFs which entered into the
above discussion. $dN_{\tt I} / d{\rm log}M_* \propto M_*^{-1.35}$ is the
Salpeter IMF, which is the Present Day Mass Function (PDMF) of MS stars
with $M_* \la 1\,{\rm M}_\odot$. Similarly, $dN_{\tt II} / d{\rm log}M_*
\propto M_*^{-1.85}$ is the MF derived for the BW KLF adjusted
for the stellar surface density of the central 30\,pc. The mass and luminosity
functions $dN_{\tt II} / dM_*$ and $dN_{\tt II} / dL_*$, given in columns\,10
and 8 of Table\,\ref{tabklf}, would hold if all stars in sector\,{\tt II} of
the KLF were MS stars. The much too high cumulative stellar luminosity of
$\sim 2.9\times10^{10}\,{\rm L}_\odot$ given in column\,9 shows that this
assumption is unrealistic. To get agreement with observed luminosities we
had to make the assumption that the stellar population in sector\,{\tt II}
consists of MS stars with a Present Day Mass Function (PDMF) $dN_{\tt IIa}
/ dM_* \propto M_*^{-4.5}$ and evolved stars with the KLF {\tt II - IIa}
shown in Fig.\,\ref{figpdmf} which account for the bulk of the observed
$K$-band luminosity in sector {\tt II}. This PDMF yields good agreement
with the observed values $L_*$ and $N_{\rm Lyc}$. Its slope is also in
agreement with the PDMF in the solar vicinity (see e.g. von Hoerner,
1968; Kroupa et al., 1993). We adopt for the following discussion
$dN_{\tt I} / dM_* + dN_{\tt IIa} / dM_*$ as the Main Sequence PDMF of
the central $\sim 30$\,pc in the sectors\,{\tt I} and {\tt II}. This PDMF
is shown as heavy solid line in Fig.\,\ref{figpdmfklf}.
The stellar mass contained in the mosaic had been assumed to be $\Sigma M_*
\sim 1.2\times10^8\,{\rm M}_\odot$. The cumulative mass of MS stars (Table
\ref{tabklf}, column\,11), bottom) is slightly higher; however, a somewhat
flatter IMF as suggested by Scalo (1986) will decrease the stellar mass
contributed by sector\,{\tt I}. The contribution of $N_* \sim 1.7\times10^4$
evolved stars (column\,5 sector\,{\tt III + IV}) to the total mass should be
negligible. The main contributors to the stellar mass ($> 90\,\%$) are MS stars
($M_* \la 1\,{\rm M}_\odot$) which account, however, for only $\sim 6\,\%$ of
the integrated K band flux density of the mosaic. MS stars ($M_* >
1\,{\rm M}_\odot$) account for $\sim 7\,\%$ of the total mass, $\sim 6\,\%$ of
the K band flux density but $\sim 80\,\%$ of the stellar luminosity of the
mosaic.
\subsection{The KLF of MS and Evolved stars} \label{s53}
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(55,69)
%\special{psfile=lumf_add_mod_bwfit_pgm_3_3.ps
\special{psfile=h1354f3.ps
angle=-90 hoffset=-5 voffset=183 hscale=32 vscale=32}
\end{picture}
\caption{The complete mosaic KLF (sectors\,{\tt I - IV} from
Fig.\,\ref{figmosaicklf} together with sector\,{\tt IIa} from Table
\ref{tabklf}, column\,3). Also shown is the difference KLF $dN_{\tt II} /
d{\rm log}S_{\rm K}^\prime - dN_{\tt IIa} / d{\rm log}S_{\rm K}^\prime$ (see
text) which represents stars which have moved off the MS. The light dotted
curve indicates the KLF of MS stars, the heavy solid curve indicates the KLF of
evolved stars.}
\label{figpdmf}
\end{figure}
Figure \ref{figpdmf} shows once more the complete mosaic KLF together with
the KLF corresponding to the PDMF (Table \ref{tabklf}, column\,3,
sector\,{\tt IIa}) which represents MS stars with $M_* > 1\,{\rm M}_\odot$.
Also shown is the difference KLF $dN_{\tt II} / d{\rm log}S_{\rm K}^\prime -
dN_{\tt IIa} / d{\rm log}S_{\rm K}^\prime$ which represents stars which have
already moved off the MS. These stars clearly dominate the KLF in sector\,{\tt
II}. Only in the mass range around $M_* \sim 1 - 4\,{\rm M}_\odot$
do we find more MS stars than evolved ones.
In Table \ref{tabstars} we give the numbers of MS ($M_* > 1\,{\rm M}_\odot$)
and evolved stars contained in the mosaic. Columns\,1 through 4 give the
range of stellar masses of MS stars together with the corresponding MS life
times. The letters in brackets in the headings of columns\,5 through 8 refer to
footnotes which indicate how the numbers of stars have been computed.
Column\,5 shows the number of stars still on the MS and column\,6 the
number of stars which have moved off the MS since stars have formed in the NB.
Remember that $dN_{\tt I} / dM_*$ is the Salpeter IMF which we adopt as
an appropriate birth rate function in the NB and hence in sector\,{\tt II} of
the KLF.
MS stars with masses $\sim 1 - 8\,{\rm M}_\odot$ are supposed to evolve into
red giants whose reddened flux densities range from $S_{\rm K}^\prime \sim
10^2$ to $2\times10^3\,\mu$Jy (paper I, Fig.\,9). Column\,7 of Table
\ref{tabstars} gives $4.8\times10^5$ as the number of stars which fall in this
flux density range. This corresponds to $\sim 6\,\%$ of the MS stars in this
mass range. Schaerer et al (1993) have computed that the lifetimes of MS stars
in the He-burning phases are about 10 -- 20\,\% of those in
the H-burning phase. Massive MS stars in the range $M_* \sim 8 -
34\,{\rm M}_\odot$ are supposed to evolve through a short supergiant phase
and to end as supernovae. Column\,8 of Table \ref{tabstars} gives the number
of $4.4\times10^4$ stars which could be evolved objects with progenitor
masses $M_* \ga 8\,{\rm M}_\odot$. There are, however, only $\sim 12\,\%$
of this number counted as MS stars. A much higher rate of formation of
massive stars $\sim 10^7 - 10^8$\,yrs ago (see below) would be a possible
explanation of these numbers. There are also $\sim 6.6\times10^6$
additional stars with flux densities $1.6 \la S_{\rm K,BW}^\prime /
\mu{\rm Jy} \la 10^2$ which appear to be neither single MS stars
nor to fit into the flux density range expected to be occupied by Red
Giants. All these numbers should, however, be considered with caution
since the KLF of sector\,{\tt II} is not based on star counts in the mosaic
but rather has been inferred from Baade's Window.
\section{Star formation and stellar population in the central 30\,pc}
\subsection{The overabundance of luminous stars at $K$}
Several papers compare the KLF of the central parsecs with the KLF of Baade's
Window (see sect. \ref{s43} and references therein, Narayan et al. (1996),
Blum et al. (1996)). Most authors agree that the KLF of the central parsecs
has an overabundance of $K$-luminous stars, which are interpreted as remnants of
high star formation activity some $10^7$\ to $10^8$\,yrs ago (Genzel et al.
1994; Tamblyn et al. 1996). We find for $S_{\rm K}^\prime \ge 5\times10^3$\,Jy
good agreement between the KLF derived here for the central 30\,pc and the
shifted BW KLF (see Fig. \ref{figmosaicklf}). In BW there are no stars observed
with flux densities corresponding to $S_{\rm K, mosaic}^\prime \ge
10^5\,\mu$Jy. This can be explained, however, as a purely statistical effect.
The number of stars in the mosaic with $S_{\rm K}^\prime \ge 10^5\,\mu$Jy is
$\sim$\,358. The KLF in BW is based on observations in a field of
$2.58^{\widehat{\prime}}$\ while the mosaic KLF covers a field of
$128^{\widehat{\prime}}$. In Sect. \ref{s43}, eq. (\ref{eq10}), we show that
the stellar surface density in the NB is $\sim$\,55.6 times higher than in BW.
Hence, one expects in BW a surface density of stars with $S_{\rm K,
mosaic}^\prime \ge 10^5\,\mu$Jy of only $3.6\,10^{-4}\,\times\,358 \sim
0.13$\ stars.
The difference KLFs shown in Paper\,I, Fig.\,4b compare the NB KLF
(Sgr\,A East minus M-0.13-0.08) with the GD KLF (M--0.13--0.08 minus Dark
Cloud).
Here the overabundance of $K$-luminous stars in the central 30\,pc is clearly
visible.
\subsection{Bimodal star formation in the NB}
In Appendix D we estimated star formation rates (SFRs) based on the Lyc photon
production rate $N_{\rm Lyc}$\ using procedures developed by G\"usten \&
Mezger (1983, hereafter GM83). We obtain for a standard or continual Saltpeter
IMF an observed SFR of $\psi(t_0)$\,$\sim$\,0.9\,M$_{\odot}$\,yr$^{-1}$\ for
the NB and 0.2\,M$_{\odot}$\,yr$^{-1}$\ for the mosaic. The observed SFR is
even higher than the time-averaged SFR $\langle \psi
\rangle$\,$=$\,$M_{\ast}\,/\,(1$\,$-$\,$r)\,\tau_{\rm NB} \sim
0.6$\,M$_{\odot}$\,yr$^{-1}$\ of the NB with a total stellar mass
$M_{\ast}$\,$\sim$\,$4\times10^9$\,M$_{\odot}$, an instantaneous return rate of
$r \sim 0.3$\ and an assumed age of the NB of $\tau_{\rm NB} \sim 10^{10}$\,yr.
This result appears to suggest a constant SFR over the lifetime of the NB -
a rather unlikely situation. GM83 had obtained similar contradictory results
for the GD. These authors had introduced bimodal star formation to explain
the abundance gradient observed in the ISM of the GD; as a by-product the
concept of bimodal star formation also solved the apparent contradiction
between time-averaged and present-day SFR. Bimodal star formation discriminates
between spontaneous and induced star formation. In the GD spontaneous (i.e.,
normal) star
formation occurs in dense clouds in the interarm region with a continual IMF
including stars of all masses. Induced star formation occurs when ISM flows
into spiral arms and gets compressed by density waves. In this case only stars
above a critical mass $M_{\rm c}$\,$\sim$\,1--5\,M$_{\odot}$\ are formed. It is
clear that in the case of bimodal star formation for a given Lyc photon
production rate $N_{\rm Lyc}$\ less ISM is converted into stars and an even
lower fraction of ISM remains permanently locked-up in low-mass MS stars and
stellar remnants. Table \ref{tabd1} shows that for a specific bimodal model
combining spontaneous and induced star formation, i.e. $M_{\rm c} \sim
3$\,M$_{\odot}$, $\psi_{\rm ind}/(\psi_{\rm ind}+\psi_{\rm spont})
\sim 0.7$\ for a given $N_{\rm Lyc}$\ the SFR is reduced by
a factor of 2.25 and the lock-up rate by a factor of 6. Alternatively we can
state that in the case of bimodal star formation the luminosity per
${\rm M}_\odot$ converted into stars is between $\sim 10$ and 40 times higher
(depending on $M_c$) than for star formation according to a continual IMF (see
Mezger et al. 1974).
Our observations (Paper\,I Fig.\,2b) show that for $R \le 15$\arcsec\ the
contribution of $K$-band luminous (i.e. resolved) stars compared with that
of low-luminosity (i.e. unresolved) stars is more than 80\%\ but decreases
to 50\%\ at $R \sim 300$\arcsec. In the context of bimodal star formation
this would mean that the ratio $\psi_{\rm ind}/(\psi_{\rm ind}+\psi_{\rm
spont})$\ decreases with increasing distance from the GC. This suggestion
is supported by Genzel et al. (1996) who find that in the central 24\arcsec\
early-type stars are concentrated in the central 12\arcsec\ while the
brightest late-type stars, which are red supergiants and AGB stars, seem
to avoid the central 4\arcsec\ and form a ring which peaks at
$R$\,$\sim$\,7\arcsec. This ring extends as far as 10\arcsec\ and appears
to delineate the region of recent high-mass star formation. For the
intermediate- and low-brightness stars Genzel et al. (1994) find a central
depression. It is interesting to note that a bimodal star burst
with $<\Psi> \sim 1\,{\rm M}_\odot$yr$^{-1}$ lasting for $5\times10^6$\,yrs
and producing mainly O stars would produce a luminosity of $\sim
2.5\times10^{10}\,{\rm L}_\odot$, equaling the stellar luminosity of
the GD. As already suggested by GM83, bimodal star formation could explain
the extremely high central luminosities of starburst galaxies.
\vspace*{5mm}
\subsection{$K$-band luminosity as tracer of stellar mass}
An evaluation of the KLF (sect.\,5.2) shows that low-mass stars ($M_{\ast} <
1$\,M$_{\odot}$) contribute $>$\,90\%\ of the stellar mass but only
$\sim$\,6\%\ of the integrated $K$-band flux density. If the populations of
low-mass stars and red giants are not well mixed, the $K$-band surface
brightness will not be proportional to the surface density of stellar
mass. Fig. 6 of Paper I actually shows that the surface brightness of
stars of medium and high luminosity stars at $K$ is much more concentrated
than that of the low-luminosity stars.
\appendix
\setcounter{section}{0}
\setcounter{table}{0}
\setcounter{section}{1}
\setcounter{table}{0}
\begin{table*}
\caption{\bf Parameters of Main Sequence Stars}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Spectral & $M_*$ & $T_{\rm eff}$ & \multicolumn{2}{|c|}{$L_*$} &
$N_{\rm Lyc}$& $S_{\rm K,*}$ & \multicolumn{2}{|c|}{$S_{\rm K,*}^\prime$}\\
Type & ${\rm M}_\odot$ & K & \multicolumn{2}{|c|}{${\rm L}_\odot$} & s$^{-1}$&
$\mu$Jy & \multicolumn{2}{|c|}{$\mu$Jy}\\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9)\\
\hline
O4 & 34 & 5.2\,$10^4$ & 1.05\,$10^6$ & 1.33\,$10^6$ & 1.02\,$10^{50}$ &
1.43\,$10^5$ & 5.8\,$10^3$ & 5.3\,$10^3$\\
O6 & 24 & 4.8\,$10^4$ & 3.94\,$10^5$ & 3.32\,$10^5$ & 2.38\,$10^{49}$ &
4.65\,$10^4$ & 1.9\,$10^3$ & 2.4\,$10^3$\\
O8 & 17 & 4.16\,$10^4$ & 1.03\,$10^5$ & 8.35\,$10^4$ & 4.90\,$10^{48}$ &
1.83\,$10^4$ & 7.5\,$10^2$ & 1.1\,$10^3$\\
O9 & 15 & 3.72\,$10^4$ & 6.38\,$10^4$ & 5.06\,$10^4$ & 2.12\,$10^{48}$ &
1.59\,$10^4$ & 6.5\,$10^2$ & 8.1\,$10^2$\\
B0 & 13 & 3.22\,$10^4$ & 3.45\,$10^4$ & 2.86\,$10^4$ & 4.17\,$10^{47}$ &
1.24\,$10^4$ & 5.1\,$10^2$ & 5.8\,$10^2$\\
B5 & 5.0 & 1.64\,$10^4$ & 7.14\,$10^2$ & 6.25\,$10^2$ & 1.48\,$10^{43}$ &
1.82\,$10^3$ & 7.4\,$10^1$ & 6.5\,$10^1$\\
A0 & 2.8 & 1.08\,$10^4$ & 7.19\,$10^1$ & 6.15\,$10^1$ & &
5.77\,$10^2$ & 2.4\,$10^1$ & 1.7\,$10^1$\\
A5 & 2.0 & 8.62\,$10^3$ & 1.73\,$10^1$ & 1.60\,$10^1$ & &
2.49\,$10^2$ & 1.0\,$10^1$ & 7.9\\
G0 & 1.1 & 5.92\,$10^3$ & 1.45 & 1.46 & &
5.30\,$10^1$ & 2.2 & 2.0\\
K5 & 0.6 & 3.97\,$10^3$ & 2.0$10^{-1}$ & 2.39\,$10^{-1}$ & &
1.75\,$10^1$ & 7.1\,$10^{-1}$ & 4.9\,$10^{-1}$\\
M8 & 0.06& 2.6\,$10^3$ & 1.0\,$10^{-3}$ & 3.79\,$10^{-4}$ & &
7.68\,$10^{-2}$ & 3.1\,$10^{-3}$ & 2.5\,$10^{-3}$\\
\hline
\end{tabular}
\end{center}
\label{tabmasslum}
\end{table*}
In Sect \ref{s53} we assigned sectors\,{\tt IV, III}, and part of sector\,{\tt
II} (viz. that for $S_{\rm K}^\prime \ge 100\,\mu$Jy) to red giants and
supergiants but were left with $\sim$\,6.6\,$10^6$\ stars with $1.6 \le
S_{\rm K}^\prime/\mu{\rm Jy} \le 100$\ which could not be accounted for in
terms of such stars (column\,9 of Table \ref {tabstars}). These stars
contribute $\sim$\,122\,Jy or $\sim$\,16\%\ to the integrated mosaic flux
density of $\sim$\,752\,Jy. A similar discrepancy between observed $K$-band
surface brightness and kinematic mass has already been noted by Maihara et
al.\ (1978) and Hayakawa et al.\ (1977), who scaled the $\lambda 2.4\,\mu$m
volume emissivity in the solar vicinity with Innanen's (1973) mass model
(see e.g. Fig. III.7 in GM83). Serra et al.\ (1980) suggested that the
excess NIR emission comes from M giants with progenitor stars of mass
$M_{\ast} \sim 1.5 - 3$\,M$_{\odot}$ together with an increased star
formation rate some $10^8$ to $10^9$\,yrs ago. As discussed above (see
sect.\,5.3) such an explanation may also apply to the NB.
\vspace*{6mm}
\noindent {\bf Acknowledgements}\\
\\
The data reductions and analyses were carried out on workstations
provided by the {\it Alfried Krupp von Bohlen und Halbach
Stiftung\/} through a joint grant to the Max-Planck-Institut f\"ur
Radioastronomie and the Institut f\"ur Theoretische Astrophysik, University
of Heidelberg.
This support is gratefully acknowledged. We benefitted much from discussions
with P.L.\ Biermann, W.J.\ Duschl and R.\ Kudritzki; our special thanks go to
I.S.\ Glass whose suggestions substantially improved this paper\\
\\
\setcounter{section}{0}
\setcounter{table}{0}
\noindent{\bf Appendix}
\section {Mass-Luminosity Relations:}
Parameters of a number of stars of different spectral type are compiled
in Table \ref{tabmasslum}. For stars O4 through K5 we use the stellar
parameters given by Mezger et al. (1974); for M8 we use the stellar
parameters given by Lang (1992).
The K band luminosities $S_{\rm K,*}$ in column\,7 of Table \ref{tabmasslum}
have been computed with Eqs.\,5.4 from MDZ96 using a distance $R_0 \sim
8.5$\,kpc to the Galactic Center. The reddened flux density $S_{\rm K}^\prime
= e^{-\tau_{\rm K}}S_{\rm K}$ in column\,8 of Table \ref{tabmasslum} has
been computed for $\tau_{\rm K} \sim 3.2$, which corresponds to a visual
extinction $A_{\rm V} \sim 31$\,mag. In column\,9 we give the approximate
flux densities computed with Eq.\,(4).
The stellar luminosities of Mezger et al. (1974) in column\,4 should be
compared with the approximation given in column\,5 computed with Eq.\,(6).
In the following we also make use of integral properties of young OB star
clusters. Mezger et al. (1974) give for stars earlier then B0\\
\\
\\
\begin{equation}
\frac{}{} = 3.16\times10^{47} \qquad{\rm
photons~per~s~and~M}_\odot
\end{equation}
\noindent and
\begin{equation}
\frac{}{} = 5.38\times10^{3} {\rm L}_\odot {\rm per~M}_\odot
\end{equation}
\section {How to convert Mass into Luminosity Functions and vice versa}
\begin{table}
\caption{\bf Combinations of Exponents}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Eq.(3) & Eq.(4) & Eq.(6) & Eq.(2) & Eq.(2) & Eq.(5) \\
$\alpha$ & $\beta_1$ & $\beta_2$ & $\gamma_1$ & $\gamma_1+1$ & $\gamma_2$ \\
\hline
-2.35 & 2.3 & & -1.6 & -0.6 & \\
& & 2.8 & & & -1.5 \\
& & 4.0 & & & -1.35 \\
\hline
-2.85 & 2.3 & & -1.8 & -0.8 & \\
& & 2.8 & & & -1.65 \\
& & 4.0 & & & -1.45 \\
\hline
-4.50 & 2.3 & & -2.50& -1.50 & \\
& & 2.8 & & & -2.25 \\
& & 4.0 & & & -1.85 \\
\hline
\end{tabular}
\end{center}
\label{tabparam}
\end {table}
In sect.\,3 we gave each of the MF, LF and KLF in a general algebraic form.
The exponents are related by
\begin{equation}
\gamma_1 = \frac{\alpha - \beta_1 + 1}{\beta_1}
\end{equation}
\begin{equation}
\gamma_2 = \frac{\alpha - \beta_2 + 1}{\beta_2}
\end{equation}
\noindent The constants are related by
\begin{equation}
C_1 = \frac{A}{\beta_1 B_1^{\gamma+1}}
\end{equation}
\begin{equation}
C_2 = \frac{A}{\beta_2 B_2^{\gamma+1}}
\end{equation}
\noindent with $\beta_1 = 2.3$ , $B_1 = 1.6$ and $\beta_2 = 2.8$ or 4.0 (see
Eq.\,(6)) and $B_2 = 1$. Some combinations of $\alpha$, $\beta$ and $\gamma$
used to compute MFs, LFs and KLFs in Table \ref{tabklf} are given in Table
\ref{tabparam}. Remember that the exponent of the logarithmic KLF is
$\gamma+1$.
\section {Dust Luminosity and Lyc-Photon Production Rate in the Mosaic}
\begin{table}
\caption{\bf IR-luminosity and Lyc-photon production rate related to central pc,
the mosaic, and the Nuclear Bulge}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$R$ & $L_{\rm IR}$ & Ref. & $N_{\rm Lyc}^\prime$ & Ref. & $N_{\rm Lyc}$ & Ref.\\
arcsec& ${\rm L}_\odot$ & & s$^{-1}$ & & s$^{-1}$ &\\
(1) & (2) & (3) & (4) & (5) & (6) & (7)\\
\hline
1.5\,$10^1$ & 3.7\,$10^6$ & $[a]$ & 6.3\,$10^{49}$ & $[b]$ & 1.9\,$10^{50}$
& $[e]$ \\
3.83\,$10^2$ & 1.2\,$10^8$ & $[c]$ & 1.2\,$10^{51}$ & $[d]$ & 3.6\,$10^{51}$
& $[e]$ \\
3.04\,$10^3$ & 1.3\,$10^9$ & $[b]$ & 6.4\,$10^{51}$ & $[b]$ & 1.9\,$10^{52}$
& $[e]$ \\
\hline
\end{tabular}
\end{center}
\vspace*{1mm}
References:\\
$[a]$ Paper I, Table 1b (Note that in paper I Table 1c the IR luminosity
(corr.) of $(7.5\pm3.5)\,10^7 {\rm L}_\odot$) relates to the stellar
luminosity.\\
$[b]$ see MDZ96, Table 6 and references therein.\\
$[c]$ Interpolated with $L_{\rm IR} \propto R^{1.1}$.\\
$[d]$ Interpolated with $N_{\rm Lyc}^\prime \propto R^{0.9}$.\\
$[e]$ Correction for Lyc-Photons directly absorbed by dust $N_{\rm Lyc} \sim
3 N_{\rm Lyc}^\prime$ (see Mezger \& Pauls, 1978).
\label{tabirlumlyc}
\end {table}
To check the stellar luminosities ($L_*$) computed in sect.\,5.1 it is
important to estimate the dust luminosity $L_{\rm IR}$ and the Lyc-photon
production rate $N_{\rm Lyc}$ integrated over the mosaic. There is an
unpublished dust luminosity $L_{\rm IR} \sim 4.4\times10^7\,{\rm L}_\odot$
derived by Launhardt from COBE-DIRBE and IRAS observations but there are no
observations of $N_{\rm Lyc}^\prime$ (i.e. the number
of Lyc-photons directly absorbed by the interstellar gas which contribute to
the radio free-free emission) and $L_{\rm IR}$ available which would relate
directly to the mosaic; the corresponding values in Table \ref{tabirlumlyc}
have therefore been interpolated between measurements related to the central
$R \la 15''$ and to the NB ($R \la 3\times10^3$$''$; see MDZ96 and paper I).
Column\,1 gives the radii adopted for the central 30\,pc, mosaic and Nuclear
Bulge (in~$l$).
\section{Star Formation and lock-up Rates}
The star formation rate (SFR) $\psi(t)$\ is the rate at which ISM is
transformed into stars. Stars with masses $M_{\ast} > 1$\,M$_{\odot}$\
have MS lifetimes shorter than the age of the GD (assumed to be $\tau_{\rm GD}
\sim 10^{10}$\,yr). They evolve to giants and supergiants, loose part of their
outer shell to the ISM and end as white dwarfs, neutron stars, or black holes.
With $r \sim 0.3$, the fraction of matter instantaneously returned to the
ISM by a new generation of stars, the lock-up rate $(1-r) \psi(t)$\ is the
fraction of matter transformed into stars which is permanently locked up in
low-mass stars ($M_{\ast} < 1$\,M$_{\odot}$) and in the remnants of massive
stars. The average star formation rate in the NB is then
\begin{equation} \label{eqd1}
\langle \psi(t)\rangle = \frac{M_{\ast}}{(1-r)\,\tau_{\rm NB}}
\sim 0.6\,{\rm M}_{\odot}\,{\rm yr}^{-1}
\end{equation}
\noindent with $M_{\ast} \sim 4\,10^9$\,M$_{\odot}$\ the total stellar mass and
$\tau_{\rm NB} \sim \tau_{\rm GD} \sim 10^{10}$\,yrs the age of the NB.
For a closed system and a SFR proportional to the mass of ISM available ,
$\psi \propto M_{\rm ISM}$, G\"usten \& Mezger (1983; in the following
referred to as GM83) give a relation between present-day and time-averaged SFR:
\begin{equation}
\frac{\psi(t_0)}{\langle \psi \rangle } = \frac{\mu\,{\rm ln}\mu^{-1}}{(1-\mu)}
= 9\,10^{-2}
\end{equation}
\noindent which - with $M_{\rm ISM} \sim 10^8$\,M$_{\odot}$\ and
$\mu = M_{\rm ISM}/(M_{\ast} + M_{\rm ISM}) \sim 0.024$\ - predicts a
present-day SFR $\psi(t_0)_{\rm pred} \sim 6\,10^{-2}$\,M$_{\odot}$\,yr$^{-1}$.
On the other hand, if one estimates the SFR from the Lyc photon production
rate, one obtains a present-day SFR (see GM83 and Table \ref{tabd1}) of
$\psi(t_0)_{\rm obs} \sim 0.9$\,M$_{\odot}$\,yr$^{-1}$, which is $\sim$\,15
times higher than the predicted SFR and close to the time-averaged SFR given
above (Eq. \ref{eqd1}).
\begin{table}
\caption{\bf Star formation and lock-up rates in the GD, GB,
and in the central 30\,pc}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
&
\multicolumn{2}{|c|}{Galactic Disk} &
\multicolumn{2}{|c|}{Nuclear Bulge} &
\multicolumn{2}{|c|}{Mosaic} \\
&
\multicolumn{2}{|c|}{M$_{\odot}$\,yr$^{-1}$} &
\multicolumn{2}{|c|}{M$_{\odot}$\,yr$^{-1}$} &
\multicolumn{2}{|c|}{M$_{\odot}$\,yr$^{-1}$} \\
& $\psi$ & $\psi$\,(1-$r$) & $\psi$ & $\psi$\,(1-$r$) & $\psi$ &
$\psi$\,(1-$r$) \\
& (1) & (2) & (3) & (4) & (5) & (6) \\ \hline
contin. IMF & 9.4 & 6.5 & 0.9 & 0.6 & 0.2 & 0.1 \\ \hline
bimod. IMF & 4.3 & 1.1 & 0.4 & 0.1 & 0.08 & 0.02 \\ \hline
\end{tabular}
\end{center}
\vspace*{1mm}
Remarks:\\
$[a]$\ GM83. {\em Continual IMF} refers to Salpeter IMF,
{\em bimodal IMF} has the following parameters:
$M_{\rm c} \sim 3$\,M$_{\odot}$;
$\psi_{\rm (ind)}/\psi_{\rm (spont+ind)} \sim 0.7$.\\
$[b]$\ Scaled with $N$(NB)/$N$(GD) and $N$(mos)/$N$(GD) with $N$(NB)
and $N$(mos) given in Table \ref{tabirlumlyc} and
$N$(GD)\,$\sim$\,2.0\,10$^{53}$\ given above.
\label{tabd1}
\end {table}
As discussed above (see sect.\,6.2) GM83 found that bimodal star formation
could solve this problem. SFR and lock-up rates for the GD as derived by GM83
are given in Table \ref{tabd1}, columns\,1 and 2 for $M_{\rm c} \sim
3$\,M$_{\odot}$. The corresponding rates for the NB and the Mosaic, scaled
according to their Lyc-Photon production rate $N_{\rm Lyc}$, are given in
columns\,3 through 6.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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