Coordinate Transformations for VERITAS in OAWG - Stage 4

Transcription

1 Coordinate Transformations for VERITAS in OAWG - Stage 4 (11 June 2006) Tülün Ergin 1 Contents 1 COORDINATE TRANSFORMATIONS Rotation Matrices Rotations of the Coordinates The Physical Coordinates of the VERITAS Array Comparison of Coordinate systems in VEGAS and GrISU THE MAIN SHOWER RECONSTRUCTION ALGORITHM IN VEGAS PACKAGE 8 1 University of Massachusetts, Amherst/MA

2 1 COORDINATE TRANSFORMATIONS 1.1 Rotation Matrices There are two conventions to define rotations: rotation of the object with respect to a fixed coordinate system or rotation of the coordinate system itself, [1]. Here we will use the latter convention by defining all the positive rotations in clockwise direction. Here are the rotation matrices for rotations about x,y,z axes for clockwise rotations: R x,θ = cos θ sin θ 0 sin θ cos θ, (1) R y,ψ = cos ψ 0 sin ψ sin ψ 0 cos ψ, (2) R z,ϕ = cos ϕ sin ϕ 0 sin ϕ cos ϕ (3) 1.2 Rotations of the Coordinates On Figure 1 the x-, y- and z-axis in black represent the telescope s coordinate system, where the rotations around all axes are described in the clockwise direction. The unit vector of the tracking position, where we think the gamma-ray source is located in the sky, is shown as red arrow. To tip the telescope axis over to point at this position the telescope coordinate system is rotated. The first clockwise rotation is through an angle of 2 ϕ about the z-axis and it is followed by a second clockwise rotation through an angle of θ about x-axis. 2 The angles ϕ and θ selected to be positive, since the rotation matrices given in Section 1.1 have rotations defined in clockwise direction, and not counterclockwise. 1

3 Figure 1: Coordinate system in which the zenith- and azimuth-angles are described. The axes in black represent the telescope s coordinate system, where x-, y-,z-axis point toward the East, North, and Zenith. The red arrow gives the unit vector of the tracking position where we think the gamma-ray source position is in the sky. The direction coordinates of the proposed tracking position are given as dl, dm, dn and by looking at Figure 1 they can be formulated as follows: dl = sin θ sin ϕ, (4) dm = sin θ cos ϕ, (5) dn = 1 dl 2 dm 2 = cos θ. (6) where the angles θ, ϕ represent the zenith- and azimuth-angle of the tracking position. A parameter sv gives the unit length on the x-y-plane sv = dl 2 + dm 2, = sin 2 θ = sin θ. (7) 2

4 Multiplication of the matrices given in Equations (1) and (3) gives the following composite matrix: R = R x,θ R z,ϕ = cos ϕ sin ϕ 0 cos θ sin ϕ cos θ cos ϕ sin θ sin θ cos ϕ sin θ cos ϕ cos θ, (8) where ϕ and θ angles represent the azimuth- and zenith-angles. Since usually positions on the sky are specified with altitude- and azimuth-angles, θ (90 θ). After this replacement, the composite matrix R looks like the following: R = cos ϕ sin ϕ 0 sin θ sin ϕ sin θ cos ϕ cos θ cos θ cos ϕ cos θ cos ϕ sin θ. (9) In Figure 1 the upward pointing vector ˆn = (dl, dn, dm) that connects the telescope coordinate system to the coordinate system where altitudeand azimuth-angle are defined, is replaced by a downward pointing vector ˆn = ( dl, dn, dm) that gives the direction vector. The final form of the composite rotation matrix is as follows: R = cos ϕ sin ϕ 0 sin θ sin ϕ sin θ cos ϕ cos θ cos θ cos ϕ cos θ cos ϕ sin θ. (10) This matrix rotates the zenith into the direction of the proposed tracking position. The reconstructed source position on the camera plane might be shifted by a small amount from the center of the camera, where it is expected to be. Now the array has to be rotated so that the reconstructed source position must sit at the origin of the camera plane. To do this we use the same rotation matrices given in Section 1.1and apply small rotations. The first rotation is about the x-axis (negative/counterclockwise rotation) and then a positive rotation about the y-axis. This will align the telescope s pointing with the reconstructed shower direction. 3

5 The small rotation angles δx, δy are the RA and Dec angles of the reconstructed shower direction. The final form of the combined rotation matrix is as follows: R = 1 0 δx 0 1 δy δx δy 1. (11) 1.3 The Physical Coordinates of the VERITAS Array The coordinate systems used in the reconstruction of the shower direction and core position in VEGAS,[4], are the following: Ground (array) coordinate system, telescope coordinate system, camera coordinate system(s). Additionally there will be a shower (tilted) coordinate system. For all the explanations below it is assumed that the camera center overlaps with the center of the telescope dish. Also, when all the telescopes of the array are pointing at the same gamma-ray source position in the sky, the x-,y-, and z-axes of the telescopes are parallel to each other. Figure 2: The ground (array) coordinate system. 4

6 Ground (Array) Coordinate System: The ground system is a 3- dimensional Cartesian coordinate system (Figure 2). The center of this system is the center of the array. The positive x-axis is directed toward the geographical east, and the y-axis goes to geographical north, and the z-axis points toward the zenith, where the axes measure distances in units of meters. The precise position of a telescope in the ground system is measured from the center of the array to the center of the telescope dish. Here are the actual telescope positions for the Base Camp as given in VEGAS, [4], in the ground (array) coordinate system: Position of T1: X g = 40 m, Y g = 20 m, Z g = 0 m, Position of T2: X g = 41 m, Y g = 47 m, Z g = 0 m, Telescope Coordinate System: The telescope coordinate system is assumed to be a 3-dimensional Cartesian coordinate system. The center of this system is the center of telescope s frame(dish) as shown in Figure 3. The x-, y-, z-axis is directed toward the East, North, and Zenith, respectively. The z-axis of this system gives the pointing direction of the telescope. The units are in meters. Camera Coordinate System(s): There are two camera coordinate systems: Camera coordinate system-1 and camera coordinate system- 2. The center of both 2-dimensional camera coordinate systems are the center of the telescope s camera, and they are both parallel to the x-y-plane of the telescope coordinate system as shown in Figure 4. 5

7 Figure 3: The telescope coordinate system. Camera Coordinate System-1: This is the coordinate system for the actual camera. Every pixel in the camera has a specific coordinate in this system, which is given in meters. Camera Coordinate System-2: This represents the camera of a telescope having focal length of 1, i.e. the camera coordinate system- 1 is scaled down with the focal length, F. If the coordinates measured in the telescope coordinate system are projected onto the focal plane with F=1 and reflected such that every (x,y) (-x,-y), the camera coordinate system-2 can be found. 6

8 Figure 4: The camera coordinate system is a 2-dimensional system. Shower (Tilted) Coordinate System: The telescope coordinates can be rotated according to the gamma-ray source position or the tracking position in the sky and turn out to be tilted telescope coordinates. If all telescopes are rotated to the same position at the sky, then the array coordinates become the tilted array coordinates. 1.4 Comparison of Coordinate systems in VEGAS and GrISU The composite matrix given in GrISU and VEGAS match each other. The telescope park position in GrISU matches the telescope coordinate system defined here. The composite rotation matrix used in GrISU is derived in the same way as described in Section 1.2. Below is the rotation matrix used in GrISU as coded in the setup matrix program, [3]: 7

9 /* rotation about z axis to place y axis in the plane created by the vertical axis and the direction of the incoming primary followed by a rotation about the new x axis (still in the horizontal plane) until the new z axis points in the direction of the primary. */ matrix[0][0] =-dm / sv; matrix[0][1] = dl / sv; matrix[0][2] = 0; matrix[1][0] = dn * dl / sv ; matrix[1][1] = dn * dm / sv; matrix[1][2] = - sv; matrix[2][0] = -dl; matrix[2][1] = -dm; matrix[2][2] = -dn; The locations of T1 and T2 in the ground coordinate system in the simulations produced for 2-telescopes at the Base Camp, [6] are not exactly the same as for VEGAS. Position of T1: X g = 0 m, Y g = 0 m, Position of T2: X g = 81 m, Y g = 27 m. 2 THE MAIN SHOWER RECONSTRUC- TION ALGORITHM IN VEGAS PACK- AGE The shower reconstruction in VEGAS, [4], is an implementation of the method number zero, which is one of the (in total three) reconstruction algorithms developed by C.Duke as a part of the GrISU package, [3]. Here we give a short description of this algorithm. The algorithms consists of two main parts: The reconstruction of shower direction and the reconstruction of the shower core position. 8

10 In the first part the single image parameters from all telescopes that triggered for an event are carried into the camera plane. After all images are in the same plane, for each image a line that passes through the image s centroid having a slope,which is the tangent of the angle that is given between the image-axis and the x-axis of the camera plane, is calculated. Both the centroid location and the angle are on the list of Hillas parameters calculated per event per telescope. In ideal case the lines of all images available for an event should intersect at the source position in the camera. The point of intersection is the point minimizing the weighted sum of the perpendicular distances from a point to each line. No minimizing algorithm is used since there is an analytical expression to compute the intersection point. The intersection point is found in the camera plane and converted to RA and Dec J2000 coordinates using the event time. In the second part of the reconstruction the shower core location is found in the shower plane by assuming that the source point is at the center of this plane. To do this first the telescope locations are transformed into the shower (tilted) coordinate system by rotating them into the telescope s tracking direction. Then the telescope locations are rotated again by using the reconstructed source position, which are calculated in the camera plane. This second rotation corresponds to small rotations so that the second order terms appearing in the calculations are neglected. To find the shower core position at this shower plane, again the point that minimizes the sum of the perpendicular distances from a point to each line is calculated. The resulting value is the shower core position given in the shower (tilted) coordinate system. Acknowledgements I thank you for Charles Duke from Grinnell College and Glenn H. Sembroski from Purdue University for the valuable discussions and additional information. 9

11 References [1] MathWorld [2] Goldstein and Arfken Goldstein 1980, pp and 608; Arfken 1985, pp [3] GrISU [4] VEGAS (VERITAS Experiment Gamma-Analysis Suite) package [5] Slalib-Focal Plane Astrometry [6] 2-Telescope Simulations of VERITAS at the Base Camp Simulations 10