The Chocolate Chip Cookie Model: Dust Geometry of Milky-Way like Disk Galaxies

Our galaxy sample is drawn from the Main Galaxy Sample (MGS) of the Sloan Digital Sky Survey Data Release 7 (SDSS DR7)¹¹1http://skyserver.sdss.org/dr7/. We take the measurements of total stellar mass, fiber magnitude, H$\alpha$, H$\beta$, [OIII]$\lambda$5007 and [NII]$\lambda$6584 emission line fluxes from the MPA-JHU data release ²²2https://wwwmpa.mpa-garching.mpg.de/SDSS/DR7/ (Kauffmann et al., 2003a; Tremonti et al., 2004; Brinchmann et al., 2004). To study the dust attenuation properties of both the stellar continua and nebular emission lines, we select star-forming galaxies (SFGs) with emission line features and then use the classical BPT diagram (Baldwin, Phillips, & Terlevich, 1981) to remove galaxies with signs of active galactic nuclei (AGNs). We adopt the criteria of Kauffmann et al. (2003b) and require signal-to-noise ratio (S/N) larger than 3 for the four emission lines used. This selection results in 260,856 SFGs.

We take the stellar mass estimation from MPA-JHU and obtain the inclination angle $\theta$ of each SFG from Simard et al. (2011). According to Simard et al. (2011), $\theta$ is estimated from morphological fitting by applying a disk+bulge model and therefore gives a better approximation of the real inclination of disk galaxies than the simple axis ratio ($b/a$) parameter, especially for highly inclined disks. To avoid possible observational biases in the physical properties of our sample galaxies(Driver et al., 2007) , we further require sample galaxies at $z<0.1$, where the PSF effects on the modelling of disk inclination could be minimized. To illustrate this effect, we show the $\cos\theta$ distributions of all the SFGs in MGS and the selected SFGs at $z<0.1$ as the dotted and solid histograms in the top panel of Figure 1 respectively . As can be seen, compared with all SFGs, SFGs at $z<0.1$ show a much flatter cos$\theta$ distribution, indicating a better and unbiased measurement of disk inclination. In the bottom panel of Figure 1, we show the number density of the SFGs at $z<0.1$ as function of their stellar mass and disk inclination. We see little systematic bias in the stellar mass of these SFGs at different inclinations.

In this study, our main goal is to construct a modeling framework for the geometry of different dust components in SFGs using a statistical approach. The Milky-Way like (MW-like) galaxies have stable and well-formed disks and the largest sample size in MGS, making them most suitable for this study. Therefore, we further select the SFGs with the stellar mass in the range $10^{10.2}-10^{10.6}M_{\odot}$. These selection criterics result our final MW-like galaxy sample of 33,273 galaxies. We leave the exploration of the geometry of disk galaxies with other stellar masses to an upcoming study.

We show the joint distribution of $E_{\rm{g}}$ and $E_{\rm{s}}$ for our sample galaxies in the top panel of Figure 2, where we have removed 23 outliers with $E_{\rm{g}}$ or $E_{\rm{s}}$ values deviate more than 5$\sigma$ from their median values. The final sample contains 33,250 galaxies. As can be seen, $E_{\rm{g}}$ spans a wide range from 0 to $\sim 1$ mag with a median value of $\sim 0.45$ mag, while $E_{\rm{s}}$ is mostly distributed in the range 0-0.4 mag and with a median value 0.17 mag.

In general, $E_{\rm{g}}$ and $E_{\rm{s}}$ show a monotonic correlation and with a mean ratio of $E_{\rm{s}}/E_{\rm{g}}\sim 0.4$. Considering the conversion factor from $E_{\rm{g}}$ to $E_{\rm{g}}(B-V)$ ($\sim 0.9$ for the Calzetti attenuation curve), this mean ratio is consistent with the conical value $f\sim 0.44$ of Calzetti et al. (2000).

We show the inclination dependence of $E_{\rm{s}}$ and $E_{\rm{g}}$ for our MW-like galaxies by color-coding each galaxy with their disk inclination $\theta$ in the top panel of Figure 2. As expected, there is a general trend that the heavily attenuated galaxies are mostly these galaxies with large inclination angle $\theta$. However, at given $\theta$, there are large scatters on both $E_{\rm{g}}$ and $E_{\rm{s}}$, which reflects the variation of the intrinsic dust attenuation among different galaxies.

To better characterize the inclination dependence of these two types of dust reddening, we divide the sample galaxies into 90 $\theta$ bins and require that each $\theta$ bin includes the same number of galaxies. We show the median values (square dots connected with a solid line) of $E_{\rm{s}}$ and $E_{\rm{g}}$ for each $\theta$ bin in the bottom left and bottom right panels of Figure 2, where the $16$ and $84$ percentiles of the $E_{\rm{g}}$ and $E_{\rm{s}}$ distribution are indicated by the gray solid lines. Because of the large number of the galaxies in each $\theta$ bin ($n\sim 300$), the uncertainty of the median values of $E_{\rm{g}}$ and $E_{\rm{s}}$ are both smaller than 0.01.

As shown in Figure 2, at inclination $\theta<75^{\circ}$, both $E_{\rm{g}}$ and $E_{\rm{s}}$ increase monotonically with the disk inclination $\theta$, while $E_{\rm{g}}$ shows a sharper increase trend. For highly inclined disk galaxies ($\theta>75^{\circ}$), the trend of $E_{\rm{s}}$ becomes flat, whereas the trend of $E_{\rm{g}}$ is reversed, i.e. $E_{\rm{g}}$ decreases with increasing $\theta$.

The monotonic increase trend of $E_{\rm{s}}$ as a function of $\theta$ has been reported by many studies, and can be reasonably explained by either simple parametric models or dedicated RT models (e.g. Shao et al., 2007; Maller et al., 2009; Masters et al., 2010; Yip et al., 2010). Different trends of $E_{\rm{g}}$ and $E_{\rm{s}}$ as functions of disk inclination also have been found in observations (e.g. Yip et al., 2010; Chevallard et al., 2013; Battisti et al., 2017). In our study, because of the equal-size binning of $\theta$ and the large sample size, for the first time, we reveal a complicate inclination dependence of $E_{\rm{g}}$, especially at high $\theta$ values.

On galactic scale, the dust attenuation of diffuse ISM dust on stellar emission can be simplified and parameterized by a uniform mixture model. We show a schematic map for such a simple mixture model in the top panel of Figure 3. In specific of a disk galaxy, we assume that the density profiles of stellar population and diffuse ISM dust follow the same exponential disk model (Equation 5),

where $R_{\rm{s}}$ and $h_{\rm{s}}$ are the scale-length and scale-height of the stellar disk, $\rho_{\rm{s,0}}$ is the central effective density of dust particles or stellar population, depending on the usage of this Equation.

With this exponential disk modelling, for a disk galaxy with inclination $\theta$, the total line of sight optical depth along the galactic center direction is

where $\kappa_{\lambda}$ is the dust extinction coefficient (absorption cross-section) at given wavelength $\lambda$, and $F_{\rm{s}}(\theta)$ is the inclination dependence term shown in Equation 6. We parameterize the wavelength dependence of dust extinction coefficient, namely the dust extinction curve $\kappa(\lambda)$ with a simple power-law,

which is normalized by $V$ band extinction coefficient $\kappa_{V}$ at its effective wavelength $5500\text{\AA}$. We set the power-law index $\beta$ to be $1.32$, so that it has $R_{V}=\frac{A_{V}}{E(B-V)}=3.1$, a typical value for the diffuse ISM dust in the Milky-Way (Weingartner & Draine, 2001; Fitzpatrick et al., 2019; Li et al., 2017).

Moreover, during the calculation of $\tau_{\lambda,\rm{s,A}}(\theta)$, $\kappa_{V}$ is coupled with the central dust density parameter $\rho_{\rm{s,0}}$. Therefore, for simplicity, we define a new parameter $\alpha_{\rm{s,0}}\equiv\rho_{\rm{s,0}}\kappa_{V}$, which directly represents the central dust absorption density. With this new parameter, we have

For the uniform mixture model, the total emitted intensity $I$ is linked to the total unextincted intensity $I_{*}$ through

where $\tau_{\lambda,\rm{s,A}}$ is the total optical depth along the given line of sight. The dust reddening then follows

As shown by Equations 11 and 12, for the uniform mixture model, when the optical depth is thick ($\tau>>1$), there is a simple correlation of emitted intensity with $\tau$ ($I\approx I_{*}/\tau$) and a nearly constant color excess $E(\lambda_{1}-\lambda_{2})\approx$ $2.5\rm{log}(\tau_{\lambda_{1}}/\tau_{\lambda_{2}})$.

The optical depth $\tau$ of the uniform mixture model (Equation 10) is composed of two terms. One is that describing the global amount of dust attenuation, which could be parameterized by the $V$ band dust optical depth to the galactic center in edge-on case, $\tau_{\rm{s,A}}(90^{\circ})\equiv 2\alpha_{\rm{s,0}}R_{\rm{s}}$. The other is the inclination dependence term $F_{\rm{s}}(\theta)$ (Equation 6), which has the only parameter $h_{\rm{s}}/R_{\rm{s}}$, i.e., the disk scale-height to scale-length ratio. Since the disk scale-length $R_{\rm{s}}$ is included in both terms, for simplicity, we set $R_{\rm{s}}=1$. Then, the two fitting model parameters become $\alpha_{\rm{s,0}}$ and $h_{\rm{s}}$ , which are in units of $R_{\rm{s}}^{-1}$ and $R_{\rm{s}}$, respectively. We fit the observed inclination dependence of $E_{\rm{s}}$ with these two free model parameters. The best fitting of the $E_{\rm{s}}-\theta$ relation is shown as the red solid line in the top panel of Figure 3 and the two best model parameters are listed in Table 1.

Figure 3 (top panel) shows that this uniform mixture model fits the observed $E_{\rm{s}}-\theta$ relation quite well. In our best fitting, the disk component has a height-to-radius ratio $h_{\rm{s}}/R_{\rm{s}}=0.27$. The best estimate of $\alpha_{\rm{s,0}}$ is $1.9$, which means that the global $V$ band optical depth along the galaxy center direction is varying from $\sim 1$ (face-on) to $\sim 4$ (edge-on), corresponding to the effective dust attenuation from $A_{V}\sim 0.50$ mag (face-on) to $1.41$ mag (edge-on), which is broadly consistent with the observational values ($0.5-1.5$ mag, Bianchi, 2007; De Geyter et al., 2014; Casasola et al., 2017).

The nebular emission of star-forming (HII) regions is known to be distributed on a much thinner disk than the stellar component (e.g. Anderson et al., 2019). Considering that the SDSS fiber spectra only target the central regions of galaxies, these HII regions might be viewed as a point-like source in galaxy center if they do not overlap each other along the line of sight. In this simplified model, the nebular emission lines emitted from the central star-forming regions are then only subject to the extinction of outer dust layers along the line of sight, which is consistent with a screen model.

For the dust screen model, the observed intensity of an object with intrinsic intensity $I_{0}$ is determined by the optical depth $\tau$ of the dust layer along the line of sight, $I=I_{0}e^{-\tau}$. We show a schematic map for this simple dust screen model in the bottom panel of Figure 3. The line of sight dust consists of two parts. One is the dust component in diffuse ISM and the other is the dust shell of nebular region itself. For the former component, we take the best model estimate of Section 3.1, i.e. an exponential diffuse ISM dust disk layer with geometric configuration $h_{\rm{s}}/R_{\rm{s}}=0.27$ and the central dust absorption density $\alpha_{\rm{s,0}}=1.9$. Since the nebular emission region is assumed to be located at galaxy center, the inclination dependence of its line of sight optical depth from diffuse ISM dust is therefore half of the $\tau_{\lambda,s,A}(\theta)$ that parameterized by Equation 10. For the dust component of the nebular emission region itself, we assume that the dust layer is distributed in a spherical shell. In this case, the dust optical depth of the shell has no inclination dependence and therefore can be parameterized by a constant $\tau_{\lambda,\rm{cl}}$. Putting these two dust components together, for disk galaxies with inclination $\theta$, the total line of sight optical depth of the modelled nebular emission line region in galaxy center is

We assume that the extinction curve of the dust particles in nebular regions is the same as that of diffuse ISM dust (Equation 9) and then parameterize $\tau_{\lambda,\rm{cl}}$ with the $V$ band optical depth $\tau_{\rm{cl}}$. In this simple dust screen model, the observed color excess of emission lines is

With above model, we fit the observed inclination dependence of $E_{\rm{g}}$ shown in the bottom right panel of Figure 2 with the only parameter $\tau_{\rm{cl}}$. We obtain the best fit with $\tau_{\rm{cl}}=0.54$, which is shown as the solid line in the bottom panel of Figure 3 and listed in Table 1. As can be seen, by including the dust component in nebular regions, the dust screen model of Equation 13 generally reproduces the observed $E_{\rm{g}}-\theta$ relation for low inclination disks ($\theta<70^{\circ}$). However, for these high inclined disks ($\theta>70^{\circ}$), the observed $E_{\rm{g}}-\theta$ relation shows a saturation effect, which can not be accounted by our simple dust screen model. This saturation effect resembles the behavior of a mixture model in optically thick case. That means, at very high inclinations, these nebular emission regions along the line of sight can no more be approximated by a point source in galaxy center, which suggests a more complex geometric configuration of these nebular line regions.

We assume that each clumpy region is self-extincted by its dust shell with optical depth $\tau_{\rm{cl}}$. When a clumpy region is in front of a emission source (e.g. a star or another clumpy region) along the line of sight, its mean (effective) optical depth is

where $R_{\rm{cl}}$ is the radius of a single clumpy region. We note that the optical depth $\tau$ is always a function of wavelength $\lambda$ (Equation 9). Therefore, we henceforth omit the subscript $\lambda$ for $\tau_{\lambda}$ to avoid complexity of notation.

In our sample, the flux of a galaxy is collected from the SDSS fiber aperture with a radius of 1.5 arcsec⁴⁴4For SDSS I and II, the fibers of SDSS spectrograph have a diameter of 3 arcsec, while for SDSS III $\&$ IV, the BOSS spectrograph has a fiber diameter of 2 arcsec., which corresponds to a radius of $R_{\rm{a}}\sim 2.2$ kpc at redshift $z\sim 0.07$ (the median redshift of our MW-like galaxy sample). We parameterize the number of clumpy regions in each SDSS aperture as $N\propto n_{\rm{g}}{\pi}R_{\rm{a}}^{2}$ , where $n_{\rm{g}}$ is the column number density of clumpy regions that equals to the integrate of $\rho_{\rm{g}}$ along the line of sight. We note $R_{\rm{cl}}$ is far smaller than $R_{\rm{a}}$ and define a covering factor $p$ as the ratio of the cross-sectional area of each clump to the aperture area,

which also indicates the probability of a random emission source being covered by a foreground clumpy region along the line of sight. Because $R_{\rm{cl}}<<R_{\rm{a}}$, the covering factor $p<<1$.

We assume that there are $N$ discrete clumpy regions randomly distributed within the aperture. Then, the number of foreground clumpy regions along a particular line of sight obeys a binomial distribution,

Therefore, for an emitting source with intrinsic flux density $I_{e}$, the observed flux density after extinction by these front clumpy regions is

When $\bar{\tau}_{\rm{cl}}<1$ and $p<<1$, the discrete binomial distribution can be approximated by a continuous Gaussian distribution ($\mathcal{N}$) with mean $\mu=Np$ and variance $\sigma^{2}=Np(1-p)$ . Then, we obtain an approximation

That is to say, the equivalent optical depth of foreground clumpy regions can be approximated by:

Also, as $p<<1$ and if $\bar{\tau}_{\rm{cl}}<1$, the term $p\bar{\tau}_{\rm{cl}}$ in Equation 21 can be neglected and $Np$ can be written as $n_{\rm{g}}{\pi}R^{2}_{\rm{cl}}$. Thus, we finally get

This final approximation shows that $\hat{\tau}_{\rm{cl}}$ is determined by the size $R_{\rm{cl}}$, the column number density $n_{\rm{g}}$ and the mean optical depth $\bar{\tau}_{\rm{cl}}$ of an individual foreground clump.

During the derivation of Equation 19, we have assumed $p<<1,N>>1$ and $\bar{\tau}_{\rm{cl}}<1$. In Appendix B, we provide a detailed comparison of the approximation of Equation 21 to the numerical solutions of Equation 19 for different parameter sets, where we show that Equation 22 only has a difference from the numerical solutions at the level of $\sim 10\%$ for $\bar{\tau}_{\rm{cl}}<1$. As we will show in Section 4.6.3, our best model estimation indeed have $\bar{\tau}_{\rm{cl}}<1$.

We remind that the continuous approximation of Equation 19 is only for solving Equation 20, not that the distribution of these star forming regions is itself continuous. Actually, the dust extinction effect of these clumpy regions is different from that of the continuously distributed dust. We show such a comparison in Appendix C.

With the geometric models of the clumpy nebular disk (Equation 15 and 22) and continuous ISM distribution (Equation 7), we model the overall dust extinction effect on a specific nebular region. For a given nebular cloud in a model galaxy along the line of sight to galaxy center, we parameterize its position with central distance $l$ and disk inclination $\theta$ and get $r=l\sin\theta,h=l\cos\theta$, where $l$ takes positive and negative values at the proximal and distal ends respectively.

For a given nebular region, its nebular emission is extincted by the dust of itself, foreground diffuse ISM dust, and other foreground clumpy regions. The total line of sight optical depth then is

where $\tau_{\rm{cl}}$, $\tau_{\rm{s}}$ and $\hat{\tau}_{\rm{cl}}$ are the optical depths of the clumpy region itself, the foreground diffuse ISM dust, and other foreground clumpy regions, respectively.

For the double exponential diffuse ISM dust disk parameterized by $h_{\rm{s}}$ and $R_{\rm{s}}$ (Equation 7), the optical depth of the foreground ISM dust along the line of sight at $(l,\theta)$ is

where $\tau_{\rm{s,A}}$ is the total optical depth along the central line of sight which follows Equation 10.

For $\hat{\tau}_{\rm{cl}}$, according to Equation 22, $n_{\rm{g}}$ is the only parameter being function of $(l,\theta)$. With the global exponential disk distribution model of the clumpy regions (parameterized by $h_{\rm{g}},R_{\rm{g}}$), we have

where $n_{\rm{g,A}}(\theta)$ is the total column number density of clumpy regions along the central line of sight:

Similar to Equation 23, the line of sight optical depth to a stellar emission region at ($l,\theta$) is contributed by the foreground diffuse ISM dust and clumpy dust,

Obviously, $\tau_{\rm{s}}(l,\theta)$ and $\hat{\tau}_{\rm{cl}}(l,\theta)$ follow Equations 24 and 22, respectively.

In Sections 4.2 and 4.3, we have parameterized the foreground dust extinction on the stellar and nebular emissions at a specific region $(l,\theta)$ along the galaxy central line of sight. However, the SDSS fiber has an aperture of $1.5$ arcsec instead of being an area-free point toward the center of each galaxy. Therefore, there is a certain fraction of photons not along the line of sight to galaxy center, which also enter the fiber. Because the SDSS fiber aperture is smaller than the radius of a typical disk galaxy in our sample, the on-axis (line of sight to galaxy center) assumption is a good approximation when galaxies are viewed in face-on cases. However, when galaxies are viewed edge-on, the scale-height of our sample galaxies becomes comparable or even smaller than the SDSS fiber aperture. In this case, the off-axis effect (line of sight not along the galaxy center), as we show next, will cause significant biases.

We assume that the scale-height of the nebular emission regions $h_{\rm{g}}$ is smaller than that of the stellar disk $h_{\rm{s}}$ and both scale-heights are smaller than the SDSS aperture size. In this case, as shown by the schema figure (top panel of Figure 4), along the line of sight to the galaxy center, the stellar emissions suffer from the dust extinction effects of both of the clumpy nebular dust and diffuse ISM dust. While for the stellar emissions off the disk plane, the dust extinction from nebular regions becomes negligible.

To approximate and correct for this off-axis effect, we assume that there are a fraction of photons $(f_{\rm{off}})$ emitted from the stellar disk being off-axis, which is defined as:

These off-axis stellar emissions are not extincted by the nebular dust. On the other hand, the emissions from the nebular regions obviously do not have this off-axis effect.

We consider a case where the disk inclination $\theta$ is relatively large but not completely edge-on. In this case, the effective disk height in the line of sight is expressed as

As we will show later, the nebular disk has larger scale-length than that of the stellar disk. As a result, with the decreasing of disk inclination $\theta$, the effective height of nebular disk ($h^{\prime}_{\rm{g}}$) starts to approach that of the stellar disk ($h^{\prime}_{\rm{s}}$). We write the specific inclination when $h^{\prime}_{\rm{g}}=h^{\prime}_{\rm{s}}$ as $\theta_{\rm{crit}}$. We assume that there is no off-axis effect on the stellar disk when $\theta<\theta_{\rm{crit}}$. Actually, when disk galaxy becomes face-on, both $h^{\prime}_{\rm{s}}$ and $h^{\prime}_{\rm{g}}$ will be larger than the SDSS fiber aperture, and thus there is no off-axis effect on both disks.

In summary, because of the off-axis effect for the highly inclined disk galaxies, inside the SDSS fiber, a fraction of off-axis photons emitted from the thicker disk have not been extincted by the thinner component. In specific, this fraction is parameters by

with above modeling, the dust extincted stellar intensity at given position $(l,\theta)$ is written as

where $\tau_{\rm{s}}$ and $\tau^{\prime}_{\rm{s}}$ are given by Equations 24 and 27 respectively and $I_{\rm{s}}$ represents the stellar emissivity in arbitrary unit. For nebular emission, there is no off-axis effect. Therefore, the dust extincted intensity is

where $\tau^{\prime}_{\rm{g}}$ is given by Equation 23.

In the above sections, we have modelled the dust extinction effect on both nebular emission and stellar emission for a given point $(l,\theta)$ in model galaxy along the line of sight. Next, we model the global dust attenuation effect by integrating the dust extincted radiation along the line of sight while neglecting the dust scattering effect,

For a given inclination $\theta$ of a disk galaxy, the unreddened intensity along the line of sight to galaxy center is

In Equation 33 and Equation 34, the subscript $`\rm{comp}^{\prime}$ refers to $`\rm{s}^{\prime}$ for stellar disk and $`\rm{g}^{\prime}$ for nebular disk. With the unattenuated and attenuated nebular emission line intensity, the dust attenuation then is easily defined as

and so that the reddening is $E_{\rm{comp}}(\lambda_{1}-\lambda_{2})=A_{\lambda_{1},\rm{comp}}-A_{\lambda_{2},\rm{comp}}$. In above equations, we have assumed the dust extinction curves of both clumpy nebular and diffuse ISM dust follow the same simple power-law (Equation 9).

We first compare the geometry parameter of our model galaxy with observational or modelling results of the Milky-Way and other nearby disk galaxies.

For the overall structural parameters of the dust component of the Milky-Way, many studies have reached consistent conclusions that dust is thinner and more extend than stars (e.g. Drimmel & Spergel, 2001; Misiriotis et al., 2006; Li et al., 2018). For the extra-galactic galaxies, most of the studies on the dust geometry in optical wavelengths also have assumed only one global dust component and obtained similar conclusions. For example, Xilouris et al. (1999)(hereafter X99) investigated the surface brightness profiles of five nearby edge-on galaxies in B, V, and I bands, Bianchi (2007) analyzed another seven nearby edge-on galaxies in V and K bands. By applying radiative transfer analysis on these galaxies, they find that the radial scale-length of dust is about 1.4 times larger than that of stars, while its vertical scale-height is about half of the stellar disk. More recently, De Geyter et al. (2014) studied the SDSS $g,r,i,$ and $z$ band images of 12 edge-on spiral galaxies selected from the CALIFA survey and found similar conclusions. Besides edge-on galaxies, Casasola et al. (2017) investigated 18 face-on spiral galaxies in DustPedia from UV to sub-millimeter bands and found that the dust scale-length is about 1.6 times of the stellar one. For our model galaxy, the clumpy nebular disk has a larger scale-length and smaller scale-height than the stellar disk, while the diffuse ISM dust component has been assumed to follow the same geometry as the stellar disk. Qualitatively, merging the dust in the clumpy regions with that in the diffuse ISM will give a global dust component that is larger in scale-length and smaller in scale-height than the stellar component, which is consistent with other studies. However, considering the non-linearity of the dust attenuation effect, the geometric parameters of different dust models are not quantitatively comparable.

To quantify the global dust attenuation effect of the two dust components in our CCC model, we reconstruct the projected image of our model galaxy from the edge-on view using the best model parameters. More specifically, the image projection process is an integral of the dust attenuated stellar emissions along the line of sight, which is given by

where $\rho_{s}$ and $\tau$ denote the density of stellar emission and dust optical depth at given position $(X,Y,l_{z})$ respectively, and $(X,Y,l_{z})$ is a Cartesian coordinate system with the centre of the model galaxy as the origin. $(X,Y)$ forms the projection plane and $X,Y$ is the major and minor axis along the disk plane,separately, while $l_{z}$ is the axis perpendicular to the projection plane. This Cartesian coordinate system can be easily mapped to the cylindrical coordinate system $(r,h)$ that has been defined in Equation 5 through

As can be seen from the Equation 38, the resulted image is not only a function of geometric parameters (e.g. scale-length and scale-height, discussed in detail below), but also is a function of the normalization parameter of dust optical depth (or density). In addition, depending on the model assumptions, $\rho_{s}(X,Y,l_{z})$ and $\tau(X,Y,l_{z})$ can both be combined by multiple components. For example, $\rho_{s}$ of our CCC model is only an exponential disk characterized by $R_{s},h_{s}$, while $\tau$ is combined of two disks, the diffuse dust (c.f. Equation 8) and the clumpy dust (c.f. Equation 22).

With above equations, the resulted $V$-band edge-on model galaxy image of our best fit CCC modelling is shown in the top panel of Figure 7. To do the projection, we have set the apparent magnitude of our model galaxy without any dust attenuation to be 17.5 mag and its stellar disk scale-length $R_{s}=1.5$ arcsec (typical values of our sample galaxy). In this reconstructed image, a dust lane structure along the middle plane of the model galaxy is clearly seen, which is originated from the extra obscuring effects of the clumpy dust component. To further quantify the global properties of the projected image, we plot the effective optical depth $\tau_{\rm eff}$⁵⁵5The effective dust optical depth is defined through $I=I_{0}*{\rm exp}(-\tau_{\rm eff})$, where $I$ and $I_{0}$ are the observed surface brightness along the line of sight with and without dust attenuation respectively. and the surface brightness profiles along the galactic mid-plane in the bottom two panels of Figure 7 as dotted lines. With these two quantitative profiles, we make more detailed discussions on the dust geometry of our best fit CCC model and further compare it with other dust attenuation models of disk galaxies.