Dust within the old nuclear star cluster in the Milky Way

The mean absolute extinction towards the central parsec of the Milky Way is A_K~3 mag, including both foreground and Galactic center dust. Here we present a measurement of dust extinction within the Galactic old nuclear star cluster (NSC), based on combining differential extinctions of NSC stars with their u_l proper motions along Galactic longitude. Extinction within the NSC preferentially affects stars at its far side, and because the NSC rotates, this causes higher extinctions for NSC stars with negative u_l, as well as an asymmetry in the u_l-histograms. We model these effects using an axisymmetric dynamical model of the NSC in combination with simple models for the dust distribution. Comparing the predicted asymmetry to data for ~7100 stars in several NSC fields, we find that dust associated with the Galactic center mini-spiral with extinction A_K~=0.15-0.8 mag explains most of the data. The largest extinction A_K~=0.8 mag is found in the region of the Western arm of the mini-spiral. Comparing with total A_K determined from stellar colors, we determine the extinction in front of the NSC. Finally, we estimate that for a typical extinction of A_K~=0.4 the statistical parallax of the NSC changes by ~0.4%.


INTRODUCTION
Nuclear star clusters (NSC) are located at the centers of most spiral galaxies (Carollo et al. 1998;Böker et al. 2002). Their study became possible via high spatial resolution observations from HST in the 1990s. They have properties similar to those of globular clusters although they are more compact, more massive and on average 4 mag brighter than the old globular clusters of the Milky Way (Böker et al. 2004;Walcher et al. 2005). Many NSCs host an AGN (Seth et al. 2008) i.e. a supermassive black hole (SMBH) in their centers, have complex star formation histories (Rossa et al. 2006;Seth et al. 2006) and obey scaling-relations with host galaxy properties as do central SMBHs (Ferrarese et al. 2006;Wehner & Harris 2006).
The formation scenarios of NSCs can be split into two main categories: The merger scenario where several dense globular clusters migrate close to the center from the outskirts via dynamical friction and merge to form a compact stellar system (Tremaine & Ostriker 1975;Arca-Sedda & Capuzzo-Dolcetta 2014); and the 'in situ' episodic buildup scenario where stars form locally from infalling gas towards the center (Schinnerer et al. 2008). E-mail: sotiris@mpe.mpg.de, gerhard@mpe.mpg.de The study of NSCs is of great interest because several of the most extreme physical phenomena occur within them such as SMBHs, active galactic nuclei, star-bursts and extreme stellar densities. The Galactic NSC is particularly interesting because of its proximity. At a distance of about 8kpc from Earth it is the only NSC in which individual stars can be resolved.
The center of the galactic NSC harbors a SMBH (Genzel et al. 2010;Ghez et al. 2008). Joint statistical analysis based on orbits around Sgr A* (Gillessen et al. 2009), star counts and kinematic data gives M• = (4.23±0.14)×10 6 M and a statistical parallax R0 = 8.33±0.11 kpc (Chatzopoulos et al. 2015). Recent studies (Schödel et al. 2014;Chatzopoulos et al. 2015) have revealed that the NSC is flattened with an axial ratio q ≈ 0.73, which is consistent with the kinematic data (Chatzopoulos et al. 2015). One unsolved problem of the NSC is the absence of a stellar cusp near the center. If the nucleus containing the black hole is sufficiently old, a stellar cusp will form eventually (Bahcall & Wolf 1976;Preto & Amaro Seoane 2010). For the NSC we instead observe an almost flat core (Buchholz et al. 2009; Bartko et al. 2010).
Observations of the NSC in Optical-UV wavelengths are impossible because of the high extinction AV 30 mag due to interstellar dust (Scoville et al. 2003;Fritz et al. 2011). coordinate system, a + signifies an asymmetry in the υ l VHs (e.g. right peak higher) a -the opposite asymmetry and 0 no asymmetry at all. We notice that dust produces the same asymmetry in every quadrant.
Therefore we rely on the infrared, with average K-band extinction toward the central parsec close to AK ≈ 3 (Rieke et al. 1988;Schödel et al. 2010;Fritz et al. 2011) which is mostly foreground extinction. However it is very difficult observationally to measure the extinction variation along the line-of-sight within the NSC.
The area around Sgr A* contains ionized gas which can be well described by a system of ionized streamers or filaments orbiting Sgr A* (Ekers et al. 1983;Serabyn & Lacy 1985) that is presumably associated with some dust as well. This complex structure of ionized gas is called the 'minispiral' and consists of four main components: the northern arm, the eastern arm, the western arm and the bar (Zhao et al. 2009) surrounded by the circumnuclear disk of radius ∼ 1.6pc (Christopher et al. 2005;Jackson et al. 1993).
In our previous dynamical study of the NSC, an asymmetry in the υ l -proper motions was observed in the histograms which was attributed to dust causing stars on the far side of the NSC to fall out of the sample. The aim of this paper is to present estimates for the extinction within the NSC based on our dynamical model and see to what extent this is correlated with the mini-spiral, to try to understand the slight asymmetry in the υ l velocity histograms of the NSC and to check the impact of this on the fundamental parameters derived from the dynamical model such as the mass and the distance.
In section 2 we discuss briefly our current best dynamical model of the NSC and describe qualitatively the effects of dust on the dynamics of the NSC. In section 3 we show evidence based on dynamics for the presence of dust within the NSC. In section 4 we develop a method for making an analytical model for the dust extinction that can be used on top of an existing dynamical model. Finally in section 5 we present extinction values for the dust within the NSC based on the prediction of the model in conjunction with the mini-spiral observations.

EFFECTS OF DUST ON THE APPARENT DYNAMICS OF THE NSC
In this section we give a brief description of the current best dynamical model of the NSC based on Chatzopoulos et al. (2015) and we show initial evidence for dust extinction within the NSC.

Axisymmetric dynamical model of the NSC
For this work we use proper motions from ∼ 7100 stars obtained from AO assisted images and photometry data based on Fritz et al. (2014). The proper motion data are given in Galactic longitude l * and Galactic latitude b * angles centered on Sgr A*. In the following we always refer to the shifted coordinates but will omit the asterisks for simplicity. We assume that the rotation axis of the NSC is aligned with the rotation axis of the Milky Way disk. This is in accordance with the very symmetric Spitzer surface density distribution of Schödel et al. (2014). In Chatzopoulos et al. (2015) we used a two-component spheroidal γ-model (Dehnen 1993;Tremaine et al. 1994) which we fitted to the density data in the l and b directions provided by Fritz et al. (2014). The inner rounder component can be considered as the NSC and the outer, more flattened as the inner part of the nuclear stellar disk. Using the density we applied axisymmetric Jeans modeling in order to constrain the stellar mass M * , the black hole mass M • , and the distance R0 of the NSC which we found to be M * (r < 100 ) = (8.94 ± 0.31|stat ±0.9|syst) × 10 6 M M• = (3.86±0.14|stat ± 0.4|syst) × 10 6 M R0 = 8.27 ± 0.09|stat ± 0.1|systkpc (1) for the NSC only, not including the constraints from stellar orbits around Srg A*. Having this information we used the Qian et al. (1995) algorithm to calculate the even part of the 2-Integral distribution function (DF) f (E, Lz). This allowed us to calculate the velocity profiles (VP) of the model. We found that the even part of the DF can predict very well the characteristic 2-peak shape (Schödel et al. 2009;Fritz et al. 2014) of the velocity histograms (VH) for the υ l proper motion velocities. The addition of a suitable odd part in Lz to the even part of the DF represents the rotation of the cluster.

Asymmetry of the υ l proper motion histograms
Upon a closer look at the velocity histograms in l direction (VH l ) it is noticeable that the right peak is slightly higher than the left (e.g. Fig.2, the smooth blue line is a homogeneous dust model see Section 5). This means that seemingly there are more stars in the front of the cluster (positive velocities) than in the back. At least three effects could produce such an asymmetry. Figure 1 illustrates the effect of dust extinction without rotation, dust extinction with rotation, inclination and triaxiality on the VHs. Each small square represents a quadrant of the shifted Galactic coordinate system (l, b) where Sgr A* is at the center. A + signifies an asymmetry (e.g. right peak  Chatzopoulos et al. (2015). Each star is mapped to the first quadrant using (l, b) → (|l|, |b|). The blue smooth lines correspond to our best model of the NSC (based on Chatzopoulos et al. (2015)) plus a homogeneous dust model with A K = 0.4 that extends from −200 to +200 along the line-of-sight.
higher) a − the opposite asymmetry and 0 no asymmetry at all (both peaks same height). We will see in the following Sections that dust with rotation produces the same asymmetry in every quadrant. Because of the dust fewer stars will be visible at the back of the cluster, and because of the rotation the missing stars will be stars with negative velocities. Inclination of the NSC produces the opposite result for upper and lower quadrants because when the line-of-sight does not pass exactly through the center, it passes through areas of unequal density in front of and behind the NSC. On the other hand, triaxiality produces the opposite result for the right and left quadrants because of the geometry of the streamlines of the triaxial system. This plot shows that if we symmetrize the data to one quadrant only, the effect of the dust will remain while the effects of the inclination and triaxiality will cancel out. This is what we observe on the symmetrized data of the NSC (Fig. 2). Therefore we conclude that the observed asymmetry cannot be a result of inclination and triaxiality but might be a result of dust. This suggests that dust extinction in conjunction with rotation may produce the observed asymmetry. We note here that in order to observe this asymmetry (right peak higher than left) in the velocity profile in the l direction (VP l ), the dust should be inside the cluster (i.e. within a few parsecs of the Galactic center) where the density is maximum, otherwise only a change in scale of the VPs would take place.

DIFFERENTIAL EXTINCTION IN THE NSC
We use photometry data in the H and K bands for 7101 stars. We split the data into a central and an extended field. The central field is a square centered on Sgr A* with size of 40 and contains 5847 stars. The rest of the stars belong to the extended fields, as shown in Fig. 3.

Total extinction
We obtain the extinction towards each star from the H − K color (stars without H photometry and late-type stars are excluded). We obtain intrinsic color estimates by assuming that the stars are at the distance of the Galactic Center and that they are giants, as it is the case for most stars in the Galactic Center ). The intrinsic color varies between 0.065 and 0.34 but the majority of stars belong to a small magnitude range around the red clump. Therefore and also because the extinction is high, the influence of intrinsic color uncertainties on the extinction is small compared to other effects, like photometric uncertainties. We use the extinction law of Fritz et al. (2011) In this work we are mainly interested in the extinction variation AK within the Galactic center. We obtain an estimate for that by measuring for each star the extinction relative to its neighbors, more specifically relative to the median extinction of its 15 closest neighbors. Obvious foreground stars were already excluded in Fritz et al. (2014). By using 15 neighbors we obtain a robust median extinction estimate that is much less affected by extinction variations in the plane of the sky. To further reduce the influence of this extinction variation we exclude stars with too few close neighbors. Fig. 3 shows a map of extinction for the central and extended fields based on H − K colors. The area within the white frame is the central field which is consistent with Fig.  6 of Schödel et al. (2010). Most of the extinction of typically AK = 3 mag shown in this plot is foreground extinction but a fraction 0.4 mag or so is intrinsic to the NSC region as we show in the following. Fig. 4 shows a histogram of the extinction for the central field. The mean extinction inferred from this plot is AK = 2.94 mag with standard deviation 0.24 mag.  Figure 4. Histogram of the extinction A K based on H − K colors for all the stars. Mean extinction and standard deviation also given. Stars with small A K are excluded because they are foreground stars (Fritz et al. 2014).

Extinction in the NSC region
The average differential extinction of stars as a function of υ l and υ b velocities is an important photometric quantity that can also be modeled and gives us information about the dust within the NSC. Fig. 5 shows this for the central field. We notice that the average differential extinction for the υ l is negative for positive velocities (preferentially at the front of the cluster) and positive for negative velocities (back of the cluster) i.e., stars at the front of the cluster are observed with less extinction than their neighbors. This finding is consistent with the asymmetry of the VHs in l  Figure 5. Average differential extinction of nuclear cluster stars plotted as a function of υ l (blue) and υ b (red) proper motion. The differential extinction is inferred from the difference in the color of a star to the median of its 15 nearest neighbors using the extinction law of Fritz et al. (2011) and correcting also for the weak color variation with magnitude. direction ( Fig. 2) and implies AK 0.4 within the NSC, see below. In contrast the average differential extinction for υ b is relatively flat and consistent with the symmetric bell shape of the VH in b direction. However we still notice a scatter of the points which is indicative of the systematic variations we should expect in AK . Fig. 6 shows the extinction distribution from the observed ratio of Paα to H92α radio recombination-line emission (Roberts & Goss 1993) for the central field, from Scoville et al. (2003). The coloring signifies total extinction and the contours show the outline of the mini-spiral. The fact that the outline of the mini-spiral can be sheen also in the dust suggests that a fraction of the extinction is likely to be associated with the mini-spiral and therefore is located within the NSC. The dust associated with the nuclear spiral is likely to be concentrated in a small distance interval along the line-of-sight of order a fraction of the radius in the sky.

DUST MODELING
We saw in the previous section that the observed asymmetry of the VH l is likely to be associated with dust extinction. In this section we describe how one can make an analytical dust extinction model and use it with an already existing model of the NSC similar to that of section 2.
For the rest of this work, along with l and b we use a Cartesian coordinate system (x, y, z) where z is parallel to the axis of rotation as before, y is along the line of sight (smaller values closer to the earth) and x is along the direction of negative longitude, with the center of the NSC located at the origin.
First we need to model a luminosity function. We can do that by taking the product of two functions. The first represents a power law function in luminosity, corresponding to an exponential magnitude distribution, L(m) = 10 γ·m . The second is an error function that represents the completeness function, so that: Figure 6. Extinction derived from the observed ratio of Paα to H92α radio recombination-line emission (Roberts & Goss 1993) for the central field from on Scoville et al. (2003). The field is split into eight cells, according to the outline of the mini-spiral, the shape of the VH in υ l and the δA K curves.
In the previous function γ is the power law index of the luminosity function and m0 is the value where the completeness function C(m) has its half height. For the power law we set the index to γ = 0.27 ± 0.02 as in Schödel et al. (2010). For the completeness function we set m0 = 16.5 and σ = 1 because we found that these values represent well the K luminosity data as shown in Figure 7. The red curve of Figure  7 shows equation 2 with the chosen values.
Next we need the extinction variation over the line-ofsight which is just the derivative of the extinction over the line-of-sight i.e. daK /dy. The function daK /dy is general and could for example be represented as a sum of Gaussians but for simplicity we choose a square function, so that: in which y1 and y2 indicate the positions where the dust starts and ends respectively. The integral of eq.3 over all lineof-sight is the maximum extinction AK thus the constant c takes the value c = AK /∆y where ∆y = y2 − y1. Function 3 integrates to: The percentage reduction in observed stars as a function of line-of-sight distance is: With this we can calculate the percentage reduction in numbers of stars after the extinction with pmax = p(y2). The percentage of stars hidden by extinction for AK = 0.4 is about 25%. For the simple case where the luminosity function is a power law, the previous equation (5) takes the form p(y) = L(−aK (y)). The function for AK = 0.4 is shown in figure 8.
Having that we calculate the VP l after the effect of dust with: Where in the previous function ftot(E, Lz) = fe(E, Lz) + fo(E, Lz) is the total DF consisting of an even part in Lz (contributes to the density) and an odd part (contributes to rotation) as in Chatzopoulos et al. (2015). Figure 9 shows a typical VP l after adding dust with AK = 0.4. We observe that the right peak of the VP l is now higher than the left peak which is a combined effect of dust and rotation. The dust does not produce an asymmetry for the VP in b direction. One useful quantity is the average AK (υ l ) over the line-of-sight. This can be calculated with: AK (υ l ) = aK (y)p(y)ftot (E, Lz) dυzdυ los dy Σ × VPD (υ l ) Eq. 7 connects our model with the photometry. From this we calculate the average differential extinction variation along the line-of-sight which corresponds to the data of Fig. 5.
In order to understand the effects of dust extinction on δAK (υ l ), we use two simple models for the dust distribution. The first is a homogeneous dust model that extends a few parsecs along the line-of-sight. The second is a thin ∼ 10 screen of dust placed in several positions along the line-ofsight. We have verified that the width of the screen is not a sensitive parameter and the results are almost unchanged if we set it for example to ∼ 5 . Figures 9 and 10 shows these effects. In Fig. 9 we see the effect of three dust screen models and one homogeneous model on the VP l for AK = 0.4.
One important point to notice is that we can achieve the same effect (e.g. the same amount of asymmetry in the histograms) with several models by using different combinations of AK and the distances at which the dust is placed along the line-of-sight , therefore the dust extinction model is degenerate. In Fig. 10 we see plots of several screen and homogeneous models based on eq. 8. The top panels show the δAK (υ l ) curves of a screen dust model placed at several distances in front of (left) and at the back (right) of the cluster for AK = 0.4. The first thing to notice is that the further the screen of dust is placed from the center the smaller is the effect of dust. This makes sense since far from the center the density of stars is lower. We also note that if the screen of dust is placed in front of the cluster, the curves are close to constant for the stars behind the cluster since there is no dust there to affect the δAK (υ l ). The opposite happens when the screen of dust is behind the cluster. The bottom left panel shows the shape of δAK (υ l ) for different AK . The bottom right panel shows three homogeneous models that extend over different distance intervals along the line-of-sight. In this case the curves are symmetric relative to zero. The dust extinction model was implemented with Wolfram Mathematica.

PREDICTIONS OF THE DUST MODEL
In the last section we described how one can include the effects of dust extinction in the dynamical modeling of the NSC, and calculate differential reddening signatures for the NSC stars. Here we proceed to model predictions and com- . VPs in l direction for different models. The right peak is higher due to the combined effects of dust extinction and rotation. We can achieve the same effect (e.g. the same of asymmetry) for several combinations of A K s and the distances where the dust is placed along the line-of-sight.
pare with both photometric and kinematic evidence. We will see that the asymmetries seen in the VPs for υ l can be explained as due to dust within the NSC. Our goal is to see whether dust in the Galactic Center mini-spiral can explain our data, and also to provide a rough extinction map of the central field based on the model and the available data. Figure 6 shows the extinction derived from the observed ratio of Paα to H92α radio recombination-line emission (Roberts & Goss 1993) from Figure 5 of Scoville et al. (2003). This extinction map for the central 40 × 40 around Sgr A* includes both foreground and NSC extinction. It is limited by the signal-to-noise ratio (S/N) of the H92α recombination line flux, and the regions in which the extinction has an apparent value of AV = 15 is where the S/N is not sufficient to derive the extinction. However, these data show a better representation of the mini-spiral than the extinction map derived from the observed ratio of Paα to 6 cm radio continuum emission (Scoville et al. 2003). Fig. 6 is split into eight cells and sub-cells. The reasoning behind the choice of these cells is based on the outline of the mini-spiral, the statistics and shape of the velocity histograms in υ l and the mean differential extinction variation δAK along line-of-sight. The VH l s for all cells and the δAK as a function of υ l and υ b are shown in Fig. 11 for the following cells: • Cell A1: This cell's small δAK values are consistent with the lack of features in the extinction map in comparison with other cells. Also the δAK values are consistent with the VH l since both peaks look symmetric which is a sign of lack (or small amount) of dust within the NSC.
• Cell A2: The cell lacks strong mini-spiral features as cell A1. However both the VH l and δAK values show relatively strong effects of dust therefore this cell is separated from cell A1.
• Cell B: The δAK values and the VH l are consistent with the strong features of the mini-spiral in the extinction map since the δAK points are higher and lower for negative and positive velocities respectively than the other cells and the asymmetry of the VH l is intense. We also note that the dust effects are similar within the whole area B because after splitting it into 2 sub-areas (not shown) we observed the same signature.
• Cells C1 & C2: The cells C1 and C2 belong to the Northern Arm of the mini-spiral. The shape of the δAK data for both cells seems similar for negative velocities but the δAK for C2 is more symmetric and consistent with the extinction map hence we split the area into 2 halves. We notice also some asymmetry on the VH l s of both areas C1 and C2.
• Cells D: Cell D was separated from C1 & C2 because the δAK values look more symmetric than C1 & C2. We also note that the dust effects are similar within the whole area D because after splitting it into 2 sub-areas (not shown) we observed the same signature.
• Cells E & F: The δAK values of these two cells look similar but the VH l of the cell F lacks the asymmetry characteristic in contrast of cell E therefore we keep them separate.
The central field with these eight cells is surrounded by a a more extended area with observations for about 2000 stars. We split this area into three outer fields O1-O3 placed around the central field as shown in Fig. 3. The VH l for these cells and the δAK as a function of υ l are shown in Fig. 12.
Our goal is to give a model prediction of each of these cells (8+3 in total). For the model we use a thin screen of dust with width 10 because the dust associated with the nuclear spiral is likely to be concentrated in a small distance interval along the line-of-sight. The precise width of the dust screen is not important as the dust signatures are insensitive to this parameter. The two main parameters are the total extinction in the screen and its location along the line-of-sight. However, as explained in the last section, these two parameters are partially degenerate. The degeneracy is particularly strong for the VH l histograms. The δAK data in principle are sensitive to whether the extinction is in front or behind the Galactic center, as shown in Fig. 10. However, the AK data have large scatter between adjacent data points such that points with seemingly small error bars can even have the 'wrong' sign of δAK (Fig. 11). This large scatter is also seen in the δAK versus υ b plots (also shown in Fig. 11) where no dust signature is present. Therefore we decided to not try to fit the data using χ 2 .
Rather, we choose to place the dust screens along the line-of-sight according to other available information, and only deviate from this when this appears inconsistent with the shape of the δAK distribution. The total dust extinction of the dust screen is then chosen by eye mostly from the amplitude and shape of the δAK distribution, taking into account also the scatter of the δAK points, and to a lesser degree from the asymmetry of the VH l peaks.
Specifically for the central field we use the three orbitmodel of Zhao et al. (2009) for the three ionized gas structures in the central 3pc (the Northern Arm, Eastern Arm, and Western Arc). We then map the center of each of the cells to a point in the relevant orbit plane according to its R.A and Dec. position. The distance from the center along the line-of-sight is given from the coordinates of that point on the orbit plane. For each cell, we use one common mean distance. Table 1 shows to which orbital plane each cell is assigned, and the distance of the dust screen from Sgr A*. Fig. 11 shows the predictions of the screen dust model for the 8 cells of the central field. The reasoning behind the choice of AK and positions for the dust screen is based on the shape of data and the several examples of Fig. 10: We place the dust screen in front of the cluster according to the value of Table 1. These two cells are interesting because they both show a correlation between the outline of the mini-spiral and the photometry data. They also exhibit the maximum contrast between the amount of dust. Cell B needs ∼ 5 times more extinction than cell A.
• Cell A2: The dust screen is placed behind the center according to Table 1.
• Cell C1 & D: The shape of the data in conjunction with top right panel of Fig. 10 indicate that the dust screen should be behind of the cluster (also gives a much better χ 2 ) in contrast with the value of Table 1. The reason for this is probably because the single orbit description is not accurate for the centers of these cells.
• Cell C2 & E: For both cells the dust screen is placed in front of the cluster according to values of Table 1.
• Cell F: The shape of the data in conjunction with lower right panel of Fig. 10 indicate that the dust screen should be centered.
For the outer cells of the extended field the AK is selected according to the general characteristics of plot 10: • Cell O1 & O3: Both the VHs and the photometry have a consistent signature. For both cells the extinction is close to AK 0.35 mag.
• Cell O2: For this cell we note that the photometry is as expected but the peaks of the VH are almost even.
Having the prediction for the cells we can estimate the foreground extinction for each cell using the AK values of Fig. 3. The second row of table 2 shows the total extinction of each cell of the central field based on Fig. 3. The third row shows an estimate of the foreground extinction based on the AK (total) = AK (foreground) + x * AK (NSC) where we set x = 0.5. Figure 11. Predictions of the model with the VH and δA K data for each cell. The numbers in the brackets show where the screen of dust is placed relative to the center. Reduced χ 2 are also provided for the histograms and the photometry. For the cells A1, A2, B, C2, E, the screen dust distance is based on the orbit models of the mini-spiral (Zhao et al. 2009 Table 2. Extinction values per cell based on Fig. 3. The foreground extinction of each cell is estimated according to A K (total) = A K (foreground) + x * A K (NSC) where we set x = 0.5. In this section we try to answer how much the dust within the NSC will affect the derived (Chatzopoulos et al. 2015) statistical parallax, supermassive black hole and stellar mass of the NSC. The foreground dust will affect the VPs only by a scale factor which does not impact the derived values. In Chatzopoulos et al. (2015) we derived new constraints on the R0, M * and M• by fitting to the corresponding data the υ 2 1/2 l,b,los part of the 2 nd order Jeans moments, that are moments of the even part of the corresponding VPs of the 2-Integral distribution function. Therefore here the problem is reduced to how much the even part in Lz of the VPs changes after the addition of dust within the NSC.
Northern Arm E (-15") Eastern Arm A2 (5") C1 (-9") C2 (-25") D (-20") Western Arc B (-25") A1 (-10") F (-5") Table 1. Each one of the eight cells of Fig. 10 belongs to an orbital plane (Zhao et al. 2009) representing one of the three ionized gas (Northern Arm, Eastern Arm, and Western Arc) formations . The inferred line-of-sight distance of the dust screen from Sgr A* is given in the parentheses (negative points towards the earth.) Figure 13 shows the even parts of the VPs for υ l , υ b and υ los for a NSC dynamical model with no dust (best model from Chatzopoulos et al. (2015)) and a dynamical model that includes the screen dust prediction for a lineof-sight through cell B which has the largest amount of extinction (AK = 0.8) among the cells of the central and outer fields. We notice that the difference between the VPs for this amount of extinction is very small. Specifically the average difference of the 2 nd moments of the VPs between the two models is ∼ 1.5%. If instead we use AK = 0.4, close to the average extinction within the NSC inferred from this work, the difference is smaller than 0.5%. The relative differences of σ los /σ b and σ los /σ l between the model with no dust and the model with AK = 0.4 are similarly small, 0.2% and 0.6%, respectively. Therefore we conclude that the systematic effects on the statistical parallax due to dust are within the estimated errors of Chatzopoulos et al. (2015), causing the distance to the NSC to decrease by ∼ 0.4% 30pc.
That the changes in the even part are so small can be explained by the following formal argument for the VP l . We show that for small amounts (1 st assumption) of homogenized (2 nd assumption) dust around the center the even part of the renormalized VP in υ l is the same as that for no dust. The odd part is a direct indicator of dust at that point (l, b) and can be used to estimate AK .
When the luminosity function is a power law and the dust is homogeneously placed i.e. y1 = −y2 in equations 3 and 4, p(y) takes the form within the dust area: For small AK (γ ln(10)) −1 1.61 we have: where p(0) = 1 − γ ln(10) A K 2 and k = γ ln(10) A K ∆y . Therefore 1 the function g(y) = p(y) − p(0) is odd everywhere (including the area where there is no dust). Next for simplicity we use ftot(E, Lz) = f (E, xυ los − yυ l ) → f (υ l y) because y appears within f only with the form of y 2 and υ l y.
1 we showed this for the case where the luminosity function is a power law but the same holds in the general case L(m) where γ · ln(10) is replaced by The previous is 0 because g(y) is an odd function and (f (υ l y) + f (−υ l y)) dυzdυ los is an even function of y therefore: And thus: To find the constant p(0) we integrate once more over the velocity this time: Therefore p(0) is the normalization factor of the VPD l . The above means that to first order, dust does not affect the even part of the VPD in l direction significantly except for a scale factor. Fig 13 shows that the effects for VPD b and VPD los are similarly small. If extinction within a stellar system is small enough (AK 1.6 for the NSC) then fitting the even part of a model's VPs to the even part of the VHs is sufficient to get accurate estimates of the M•, M * and R0 parameters. In Chatzopoulos et al. (2015) we used the root mean square velocities that are moments of the even parts of the VHs to fit the M•, M * and R0 parameters of the axisymetric model. Therefore we expect that their values will not be affected from the dust more than 0.4% as we explained earlier.
In principle the odd part of the VPD l can be fitted to the odd part of the VHs and this part is scale free since the scaling factor is already known from the even part therefore one can fit the AK for some combination of cells.

DISCUSSION & CONCLUSIONS
The main goal of this work was to understand the slight asymmetries in the VP l s of the NSC and their influence on the dynamical modeling following the recent work of Chatzopoulos et al. (2015). Our interest was triggered by the observation that the right peak of the VH l is consistently slightly higher than the left. A plausible explanation was given based on the existence of dust within the NSC. Because of the dust, the apparent number of stars behind the NSC is smaller than that in front of the cluster. This in conjunction with the rotation can explain the observed characteristic.
In order to quantify the dust effects, we worked with proper motions and photometry for ∼ 7100 stars from Fritz et al. (2014). We applied an analytic dust extinction model together with our current best NSC dynamical model. The extinction model gave us reasonable results and was able to predict both the signature in the VPs and the photometry.
Observation of the NSC in the optical is almost impossible because of the 30 mag extinction. In the infrared the situation is much better since AK ∼ 3 mag. Most of this extinction belongs to the foreground and does not affect the shape of the VHs (except of a normalization factor). We find here that a small fraction of the total extinction value (∼ 15%) belongs within the NSC.
The area between ∼ 1 − 1.5pc radius consists of several streamers of dust, ionized and atomic gas with temperatures between 100K − 10 4 K and is called "ionized central cavity" (Ekers et al. 1983). The mini-spiral is a feature of the ionized cavity, and is formed from several streamers of gas and dust infalling from the inner part of the CND (Kunneriath et al. 2012). It consists of four main components: the Northern arm, the Eastern arm, the Western arm and the Bar (Zhao et al. 2009) that can be described well by streams of ionized gas or filaments orbiting Sgr A* (Serabyn & Lacy 1985).
We investigated how well the mini-spiral correlates with the extinction effects in the NSC data within the central field. To assess this we first investigated whether our extinction model puts the dust on the same side of the NSC as does the mini-spiral interpretation. This is true for six out of the eight cells (except cells C2 & F). We also found the largest extinction (AK = 0.8) in cell B where also the largest extinction is inferred from the extinction map of Scoville et al. (2003), and particularly low extinction in cell A where the extinction map is consistent with only foreground extinction.
We can estimate the mass that corresponds to a given amount of extinction using where ρ d is the dust density, κ λ = 1670cm 2 /g (Draine 2003a) is the mass extinction coefficient 2 for the K-Band, AK = 0.4 from the model prediction and ∆y = 10 is the width of the dust screen. We find that ρ d 1.8×10 −19 kg/m 3 and the dust mass within a parallelepiped with dimensions (10 , 40 , 40 ) centered on Sgr A* is M d ∼ 3M . Since the dust extinction model presented here is not precise we consider that this estimate is correct only within an order of magnitude. This value is within the range of 0.25 − 4M for the mini-spiral found from other works (Zylka et al. 1995;Kunneriath et al. 2012;Etxaluze 2011). Finally we showed that for small values of extinction the even parts of the VPs are not affected significantly. As a result, the measured M•, M * and R0 parameters of Chatzopoulos et al. (2015) do not change by more than ∼ 0.4% for extinction AK 0.4, which is less than the smallest systematic error (for the statistical parallax) inferred for these parameters in Chatzopoulos et al. (2015).
Our results can be summarized as follows: • We showed that extinction due to dust explains kinematic asymmetries and differential photometry of the NSC, and measured the amount of extinction within the NSC by combining a dynamical model with a dust extinction model.
• We presented an extinction table for the dust within the NSC in several cells.
• We found that the distribution of the dust is consistent with the extinction being associated with the mini-spiral for 6 out of 8 cells.
• Systematic effects due to dust with typical extinction AK 0.4 affect the M•, M * and R0 parameters deduced from previous dynamical modeling only by 0.4%, which is smaller than their estimated systematic errors.