We have developed an IDL program based on the geometrical constraints developed in Clarke et al. (1998) to visualize the geometry of the NLR and the host galaxy disk with respect to the LOS. This geometry is shown in Figure 2.1. In the figure, the X-Y plane defines the plane of the galaxy. The
X′-Y′ plane defines the plane of the sky. The Z-axis is the normal to the
galaxy plane and Z′-axis is the LOS. The bicone is tilted from the normal
the galaxy plane by angle β, which we wish to estimate. The angle δ is the difference in position angle of the bicone axis as projected on the plane of the sky and the position angle of the photometric major axis of the galaxy as
measured on the plane of the sky. The angle between the LOS and the bicone axis is φ (inclination angle of the bicone). The angle φ defines the nature of the observed properties of the nucleus, in the framework of the unified model. The angle β is therefore the angle between the normal to accretion disk and the normal to the host galaxy disk. This is assuming that the NLR outflow is perpendicular to the accretion disk plane and the symmetry axis of the accretion disk matches the symmetry axis of the NLR bicone, which is presumably fixed by the interior torus. In the figure, the photometric major axis of a galaxy is along the X axis. There is a mirror symmetry about the Y-Z plane, which can be removed by observationally determining which side of the bicone is towards us. This can be done by fitting the kinematics of the NLR and then extracting the inclination from the biconical model fits. Apart from that, there is 180o ambiguity in determining the closer side of a
host galaxy disk. This is normally resolved by using a galactic rotation curve and the assumption that spiral arms trail the rotation (Kinney et al. 2000). The direction of the dusty spiral arms help to resolve this ambiguity.
The mathematical setup is as given in Clarke et al. (1998), which we reproduce here, the major difference being that we use the NLR axis instead of the axis of the radio emission. The coordinate system is setup as described above. In this coordinate system, the unit vector in the direction of the LOS is,
kLOS = (0,−sini,cosi) (2.1)
If we denote the unit vector along the NLR symmetry axis askN LR, then the
vector intersects a unit sphere centered on the origin and lies on a latitude defined by the angle β. The angle between kN LR and kLOS defines a great
circle on the unit sphere. The difference δ between the PA of the major axis of the galaxy (along X-axis) and the PA of the NLR axis, is constrained between 0 and 90o. Givenδ and inclinationi, the unit vectork
be written using direction cosines as,
kN LR = (kN LRx, kN LRy, kN LRz)
= (cosδsinφ,
sinδcosisinφ−sinicosφ,
sinδsinisinφ+ cosicosφ), (2.2) where φ is the angle between kLOS and kN LR and lies in the range −180o < φ < 180o. This leads to the mirror symmetry about the apparent minor
axis of the galaxy (Y’-axis). Thus the NLR axis may lie on one of the two great circles which are reflections of each other in the Y-Z plane. To remove this ambiguity, we need to know which side of the NLR is close to us. As mentioned above, this can be done by fitting a kinematical bicone model to the NLR kinematics (Das et al. 2005, 2006). The angle β between the kN LR
and Z-axis is then obtained from Eq. 2.2 as, cosβ = kN LRz
= sinδsinisinφ+ cosicosφ (2.3) The only relevant values of φ are those which give 0o < β <90o, thus, this
means that φ lies in the range φ1 < φ < φ1+ 180o, whereφ1 is given as,
φ1 = tan−1 µ −coti sinδ ¶ (2.4) which then lies in the range 90o > φ
1 > 0o. If φ < 0o, then the NLR is
projected against the half of the galaxy that is farther away from us. We can figure out which side of the host galaxy disk is close to us by tracing dusty spiral arms within optical images of the galaxy. The side of the galaxy with less pronounced dusty features will be farther away from us due to contamination from the light of the bulge. This allows us to remove the
ambiguity in φ. LOS Plane of Sky i Z Axis Y Axis X Axis i Y’ Axis Plane of Galaxy S Bicone Axis φ β X’ Axis N δ
Projected Bicone Axis
Figure 2.1: Geometrical model of NLR orientation with respect to the normal to the host galaxy disk.
The program is written in IDL and has a graphical user interface (GUI). The user inputs the position angle (P.A.gal) and inclination (igal) of the host
galaxy, the position angle (P.A.bicone) and inclination (φ) of the NLR bicone,
and the half-opening angle (H.A.bicone) of the NLR. Using these values and
the geometrical constraints described below, the software simulates a thin nuclear host disk, a torus and the NLR bicones, and then positions these 3-D surfaces in the coordinate system as shown in Figure 2.1. The program then orients the system such that North is up and East to the left in the view-port of the user, just as it would be on sky. After this point, the user can use the mouse pointer to move the entire model in 3-D by dragging it.
All components of the model can be transformed in three fundamental ways: translation, scaling and rotation about an arbitrary axis. The safest option among these is scaling, as the others will change the geometrical constraints we have put in the model. The entire system can be rotated around any axis to simulate different lines of sight. For example, one can view the galactic disk edge-on, to see how the NLR bicone is oriented with respect to the disk. The geometrical definitions and constraints on the system are given below.
• Inclination of the host galaxy disk, igal, is zero when face-on.
• Inclination of the bicone axis, φ, increases away from the line of sight.
• Position angle of the major axis of galaxy is such that P.A.gal + 90o
points along the minor axis of the host galaxy on the side of the disk that is closer to the observer.
• Position angle of the bicone axis is such that the cone along the given position angle points away from the observer.
• All angles used are positive and measured counter-clockwise.
The main parameters computed are |δ|, the difference between the posi- tion angle of the galactic major axis and the bicone axis, and β, the angle between the bicone axis and the normal to the galactic disk.
A detailed description of the usage of the program is given in Appendix A. An example of the visualization created by the program and the program GUI is shown in Figure 2.2.