A&A 446, L5L8 (2006)
DOI: 10.1051/00046361:200500225
A new method for determining masstolight ratios of nearly faceon spiral
galaxies
TaoHu
^{1} 
QiuHe Peng^{1,2,3} 
YingHe Zhao^{1}
1  Department of Astronomy, Nanjing University, Nanjing 210093, PR China
2  Joint Astrophysics Center of Chinese Academy of SciencePeking
University, Beijing 100871, PR China
3  The Open Laboratory of Cosmic Ray and High Energy
Astrophysics, Chinese Academy of Science
Received 12 August 2005 / Accepted 8 December 2005
Abstract
Aims. This letter gives a new method for determining masstolight ratios of nearly faceon spiral galaxies.
Methods. The method is based on the effective thickness of the galactic disk, the distribution of the vertical velocity dispersion, and the surface brightness of a spiral galaxy.
Results. As examples, the results of the determination of NGC 1566 and NGC 5247 in Bband are presented, and their masstolight ratios are
and
respective.
Key words: galaxy: disk  galaxies: fundamental parameters  galaxies: spiral 
galaxies: structure  galaxies: individual: NGC 1566, NGC 5247
1 Introduction
Van der Kruit & Searle (1981a,b) investigated surface
photometry of edgeon spiral galaxies, they fitted the z distribution of light at each r of the disk by a model of a
locally isothermal sheet. Let
denote the gravitational
force in the zdirection as a function of z,
the space
density of matter and
the
dispersion in the velocities in the zdirection. From the case
Poisson's and Liouville's equations, it could be reduced to

(1) 

(2) 
with
independent of zdirection,
and the solution is (Camm 1950; Spitzer 1942)

(3) 
with

(4) 
z_{0} is effective thickness of a galactic disk (H=z_{0}/2 is
scale height of a galactic disk).
By integrating Eq. (3), we have the total surface density of
disk

(5) 
so Eq. (4) can be reduced to

(6) 
Van der Kruit & Freeman (1986) pointed out that the
masstolight ratio (
)
of old disk is approximately
constant, in good approximation independent of position along the
radius. When a spiral galaxy is nearly faceon, the total surface
density can be taken as

(7) 
with I(r) the surface brightness and
the scale
length of galactic disk. Therefore, from Eqs. (6) and (7), we can find

(8) 
if we obtain the parameters z_{0}, V_{z}, I_{0}, and ,
masstolight ratio
could be calculated by Eq. (8).
Van der Kruit & Searle (1981a) proposed a method to determine
the scale heights of edgeon disk galaxies. It is based on
measuring surface brightness that is distributed with exponential
function of radius. A method for determining the scale heights of
spiral galaxies observed nonedgeon was proposed by Peng (1988)
on the basis of the asymptotic expression of the perturbed
gravitational potential. Zhao et al. (2004)
reinvestigate the method based on the rigorous expression of the
perturbed gravitational potential. We have already obtained
perturbed gravitational potential for such a logarithmic density
disturbance via the Poisson's equation for the galactic disk with
finite thickness (Peng et al. 1978, 1979), the perturbed
gravitational potential may be expressed
where
here,
is thickness factor of a spiral galactic disk,
and J_{m}(x) are the usual Gamma and Bessel functions,
respective. For an infinitely thin disk (
), the Eq. (9) has a simplified form as same as the
expression given by Kalnajs (1971)
The ratio ()
of the amplitude for the
perturbed gravitational potential for a disk galaxy with finite
thickness to that of an infinitely thin disk at the forbidden
radius r_{0} is (Zhao et al. 2004)

(12) 
it can be obtained the effective thickness (
)
through Eq. (12) by determining the winding parameter (), the number of spiral arms (m), and the forbidden
radius (A spiral arm can not extend to the forbidden region with
radius r_{0} to the galactic center.).
was assumed to be 0.5 based on the comparison between z_{0} and r_{0} which have
been observed and measured for Milky Way and Andromeda Nebula (M 31). In fact, we use the same values of the parameters of 71 spiral galaxies given by Ma et al. (1998) to calculate the factor
by Eq. (12), and obtain the average
.
In this paper, we take the average value of 0.5 and 0.486, i.e.,
.
It is well known that spiral arms can be represented by equiangular spirals
on the galactic disk. In Fig. 1, we determine the inclinations
and the winding parameters of the spiral galaxies NGC 1566 and
NGC 5247 (Bband) by fitting the curves of spiral arms in their
images, i.e., it is the optimum inclination of a spiral galaxy,
which is corresponding to the best fitted spiral curves.

Figure 1:
Images of NGC 1566 and NGC 5247 in Bband with superposed fitted curves of spiral arms. 
Open with DEXTER 
NGC 1566 and NGC 5247 were investigated based on studying the
surface photometric and spectroscopic by Bottema (1992) and Van der Kruit & Freeman (1986), respectively. The parameters V_{z},
I_{0}, and
we need are listed in Table 1 in this
paper. As shown in Table 1 Col. 5, the vertical velocity
dispersion V_{z} of NGC 1566 was obtained from Table 4 Col. 4 in
Bottema's paper (1992) and NGC 5247 from Table 1 Col. 4 in Van der Kruit & Freeman (1986), it only needs to get V_{z,r} at the
radius r in the galactic disk. However, we have obtained the
effective thickness z_{0}, the vertical velocity dispersion V_{z,r}, and the surface brightness I(r) at radius r, the
only unknown parameter is masstolight ratio
in Eq. (8).
On the basis of the method proposed in this paper, we obtain the
effective thicknesses (z_{0}) of NGC 1566 and NGC 5247 by Eq. (12) and their masstolight ratios ()
by Eq. (8).
The main results are listed in Table 2,
listed in Col. 8 and z_{0} in Col. 7.
Table 1:
V_{z} and parameters of surface brightness of NGC 1566 and NGC 5247 (Bband).
Table 2:
The effective thicknesses and the masstolight ratios of NGC 1566 and NGC 5247 (Bband).

Figure 2:
Panel a) shows
with different z_{0}
.
V_{V} is the
ratio for the amplitude of the gravitational potential
perturbation of a disk with finite thickness to that of an
infinitely thin disk. The curves from top down correspond to
,
,
and
,
respectively. Panels b) d) are three panels for NGC 1566 to
illustrate the changes of z_{0} with r_{0}, ,
and m,
respectively. 
Open with DEXTER 
 1.
 As shown in Fig. 2 with four panels
(Figs. 2ad), Fig. 2a is displayed the
radial dependence of
by Eqs. (9) and (11), it is noted that V_{V} is the
ratio for the amplitude of the gravitational potential
perturbation of a disk with finite thickness to that of an
infinitely thin disk. As an example, we choose a sample with two
spiral arms and winding parameter
.
The curves from
the top to bottom denote the galactic disk with different
effective thicknesses correspond to
,
,
and
,
respectively. In Fig. 2a, it is
shown that the amplitude of gravitational perturbation for a disk
with finite thickness is weaker than that of an infinitely thin
disk, and the thicker the galactic disk is, the weaker the
amplitude of the gravitational potential perturbation becomes.
When it is close to the center of the disk (
),
the decrease of the amplitude of gravitational potential
perturbation becomes even faster, the amplitude is too weak to
stir up the selfconsistent density waves due to the physical
concept of density waves suggested by Lindblad, spiral arms would
disappear in the forbidden region with radius r_{0} to the
galactic center. When
as shown in Fig. 2a, the
thicker the galactic disk is, the larger the forbidden radius
becomes (
r_{03}>r_{02}>r_{01}).
 2.
 Figures 2bd are three panels for NGC 1566. z_{0} is obtained
by calculating Eq. (12). Figure 2b shows the effective thickness
z_{0} changes with r_{0}, keeping the other two parameters m and
constant at their adopted values in Table 2. We could
note that z_{0} becomes thicker as the the forbidden radius r_{0}is increased, it is in agreement with the conclusion obtained from
Fig. 2a (when
,
the thicker the galactic disk is,
the larger the forbidden radius r_{0} becomes.).
Figure 2c is shown the change of z_{0} with ,
keeping mand r_{0} constant. We can learn from Peng (1988) and Ma et al.
(1998) that
,
for the spiral
galaxies with the same or similar value of z_{0}, the forbidden
radii r_{0} become larger as
be increased, it shows some
correlations with Hubble sequence. As displayed in Fig. 2c, r_{0}is kept constant at the value in Table 2, z_{0} becomes thinner as
becomes larger, it is in agreement with the method for
determining the thickness of a spiral galactic disk by Peng (1988)
and Ma et al. (1998).
Figure 2d shows the dependence of z_{0} on the parameter m, keeping
r_{0} and
constant. It is noted that the larger the
number of spiral arms (m) becomes, the narrower the gap of the
spiral arms is. It is similar to the result of increased, when
is increased, the gap of the spiral arms
becomes smaller. As shown in Fig. 2d, z_{0} becomes thinner as mbecomes larger.
 3.
 The masstolight ratios of NGC 1566 and NGC 5247 in
Bband given by Bottema (1992) and Van der Kruit & Freeman (1986)
respectively are both
.
As shown in Table 2 Col. 8 in this paper, we obtain the
masstolight ratios of NGC 1566 and NGC 5247 are
and
,
respective.
 4.
 Van de Kruit & Searle (1981a) and de Grijs & Van de Kruit (1996) found that, for scale heights
of edgeon spirals, H=z_{0}/2 is in good approximation
independent of position along the major axis, but Kent et al.
(1991), de Grijs & Peletier (1997) noted that the scale height
seems to increase with radius along the major axis, it seems to
occur more often in early type galaxies than in later type. Shaw
& Gilmore (1990) found that the radial variation of scale
heights is within ,
with no dependence on color or model
type. Here, we presume scale height to be approximately constant
along the axis of the galactic disk, and for Eq. (8), it is
obvious that masstolight ratio is approximately constant too, in
agreement with Van der Kruit & Freeman (1986).
Acknowledgements
We are very grateful to the referee for the careful and useful reviews of
this paper, the insightful comments and suggestions have improved
this paper very much. The images of NGC 1566 and NGC 5247 in Bband
are both got from the NASA/IPAC Extragalactic Database (NED),
which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with NASA. The image of
NGC 1566 is obtained with the 2.5 m telescope at LCO, with the image
size
arcmin, and the image of NGC 5247 is got
with the 0.9 m telescope at CTIO, with the image size
arcmin. This research is supported by Chinese National Science
Foundation No. 10573011, No.10273006, and the Doctoral Program
Foundation of State Education Commission of China.
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