INNAME First RM grad image name INCLASS First RM grad image class INSEQ 0.0 9999.0 First RM grad image seq. no. INDISK 0.0 9.0 First RM grad image disk no. IN2NAME Second RM grad image name IN2CLASS Second RM grad image class IN2SEQ 0.0 9999.0 Second RM grad image seq. no. IN2DISK 0.0 9.0 Second RM grad image disk no. BLC BLC of input image(s) TRC TRC of input image(s) OUTNAME Output DPR image name OUTCLASS Output DPR image class OUTSEQ 0.0 9999.0 Output DPR image seq. # OUTDISK 0.0 9.0 Output DPR image disk drive CPARM User supplied parameters

BDEPO Type: Task Use: Depolarization can be produced if the Rotation Measure gradient is sufficiently large across the synthesized beam. BDEPO tries to predict the depolarization ratio (DPR) between two frequencies owing to a linear Rotation Measure (RM) gradient across the beam. The gradient image(s) may be calculated from your RM image (with NINER), or may be models of a foreground screen that you have generated in some other way. The output DPR image can be plotted (with IMVIM) against an observed DPR image in order to asses the importance of beam depolarization. The observed DPR should be m(F1)/m(F2) where F1 < F2 are the frequencies, and m is the degree of polarization. It assumed that the intrinsic degree of polarization is constant across the beam. A beam of arbitrary orientation must be present in the header and you must supply the two frequencies between which to calculate the DPR. If the RM structure is still possibly unresolved in your highest resolution RM image, you will run the risk of underpredicting the beam depolarization. This task runs in 2 modes. First, the more general case is treated if you input 2 RM gradient images. The first must be the x (RA) direction gradient, and the second the y (DEC) direction gradient. The second mode is invoked if you input only one RM gradient image. In this case it is assumed that the x and y components of the RM gradient are equal and given by the input image. The units of the gradient images should be reduceable to RAD/M/M/PIXEL by scale factors which you may supply via the CPARM(3) and CPARM(4) parameters. For example, consider BDEPO working in the second mode mentioned above. A generalised gradient image can be produced by NINER with opcode 'SOBL'. If DPARM(1) = 0, NINER makes a gradient of the form : G = 8 * (g_x**2 + g_y**2)**0.5 where g_x and g_y are x and y directional derivatives. Now, BDEPO assumes g = g_x = g_y so that g = G / (8 * sqrt(2)). Thus, a total scale factor of CPARM(3) = 1 / (8 * sqrt(2)) would be required. This factor is built in as the default when CPARM(3) = 0 and only one input image is specified. If two input images are specified, CPARM(3,4) = 0 => CPARM(3,4) = 1.0 Adverbs: INNAME........First (generalised or x-axis) RM gradient image name INCLASS.......First RM gradient image class INSEQ.........First RM gradient image seq. # INDISK........First RM gradient image disk drive # IN2NAME.......Second (y-axis) RM gradient image name IN2CLASS......Second RM gradient image class IN2SEQ........Second RM gradient image seq. # IN2DISK.......Second RM gradient image disk drive # BLC...........BLC of input image(s) TRC...........TRC of input image(s) OUTNAME.......Output predicted DPR image name OUTCLASS......Output predicted DPR image class OUTSEQ........Output predicted DPR image seq. # OUTDISK.......Output predicted DPR image disk drive # CPARM.........Frequencies (Hz) at which to calculate the DPR are stored in CPARM(1) and (2). The order is unimportant. CPARM(3) and CPARM(4) are scale factors by which the gradient images are multiplied. If there is no second image, CPARM(3) = 0 => invoke scale factor appropriate to a NINER 'SOBL' gradient image. If there is a second image, then both CPARM(3) and CPARM(4) = 0 give scale factors of unity.

TASK : BDEPO DOCUMENTOR : Neil Killeen, NRAO RELATED PROGRAMS : RM, COMB, NINER, IMVIM PURPOSE ----------- Depolarization can be produced by differential rotation across the beam owing to a Rotation Measure gradient. This rotation can occur anywhere along the line of sight. Thus, "beam" depolarization is distinguished from "internal" depolarization in which the rotating material is mixed in with the emitting material. BDEPO tries to predict the beam depolarization ratio (DPR) between two frequencies owing to a linear Rotation Measure (RM) gradient across the beam. The depolarization ratio is defined as DPR = m(F1) / m(F2) where m is the degree of polarization at the freqeuncy F. The convention used in this program is that F1 < F2 so that, in the absence of noise, DPR < 1.0. The output DPR image can be plotted (with IMVIM) against an observed DPR image in order to asses the importance of beam depolarization. The gradient image(s) may be calculated from your RM images (with NINER), or perhaps may be models of a foreground screen that you have generated in some other way. Note that because we generally make images with about 3 pixels per beam, you should set the XINC and YINC adverbs in IMVIM to 2 or 3 so that the plotted points are more or less independent of each other. Also. if you use NINER, remember that it operates on your image with a 3 x 3 matrix, so that there is a loss of independence between points here as well. BDEPO assumes that the intrinsic degree of polarization is constant across the beam. If there is unresolved and significant structure in your RM gradient image, then BDEPO will underpredict the beam depolarization. MODE 1:

Let us call the x (RA) and y (DEC) components of the RM gradient g_x and g_y. If we also assume that m is constant across the beam, then the DPR between wavelengths L_1 > L_2 is : -- -- ---- |-g_x**2 * (L_1**4 - L_2**4) | [A * g_y/g_x + C]**2| | DPR=exp|--------------------------- * |1+--------------------| | | 4 * ln2 * A | [A*B - C**2] | | -- -- ---- where a**2 B_min**2 + b**2 B_maj**2 A = ----------------------------- B_maj**2 B_min**2 a**2 B_maj**2 + b**2 B_min**2 B = ----------------------------- B_maj**2 B_min**2 ab (B_maj**2 - b**2 B_min**2) C = ----------------------------- B_maj**2 B_min**2 B_maj = FWHM of the beam major axis B_min = FWHM of the beam minor axis a = cos(theta) b = sin(theta) theta = angle between beam major axis and positive X-axis measured positive counter clockwise, and with 0-180 degree limits. This is BDEPO's most general mode. The x and y gradient images should be input as the first and second images, respectively. Before the DPR is calculated, g_x and g_y (as read directly from the input images) can be multipled by factors stored in CPARM(3) and CPARM(4), respectively. That is, g_x = CPARM(3) * First_RM_gradient_image_value g_y = CPARM(4) * Second_RM_gradient_image_value If these CPARMS are 0.0 on starting up BDEPO, they default to 1.0. They might be used to account for weighting in the input images, or conversion to radians from degrees. After application of the scale factors, the units of the images must be RAD/M/M/PIXEL. In the case that the beam is round, this expression simplifies to : -- -- | - B_maj**2 * (g_x**2 + g_y**2) * (L_1**4 - L_2**4) | DPR = exp | -------------------------------------------------- | | 4 * ln2 | -- -- MODE 2:

In the case that g = g_x = g_y and the beam is not round, we have : -- -- -- -- | - g**2 * (L_1**4 - L_2**4) | [A + C]**2 | | DPR= exp| ----------------------- * |1 + ------------ | | | 4 * ln2 * A | [A*B - C**2] | | -- -- -- -- To invoke this mode in BDEPO, supply only the first gradient image. The scale factor CPARM(3) must then reduce the units to RAD/M/M/PIXEL g = CPARM(3) * First_RM_gradient_image_value If the beam is round, then : -- -- | - (g * B_maj)**2 * (L_1**4 - L_2**4) | DPR = exp | ------------------------------------ | | 2 * ln2 | -- -- As an example, consider the output produced by NINER with opcode 'SOBL' and DPARM(1) = 0.0. This is a generalised gradient formed from a 3 x 3 matrix as follows : | A1 A2 A3 | | A8 A0 A4 | | A7 A6 A5 | NINER computes G = sqrt (X**2 + Y**2) where X = (A3 + 2A4 + A5) - (A1 + 2A8 + A7) Y = (A1 + 2A2 + A3) - (A7 + 2A6 + A5) Now, we can associate X with g_x and Y with g_y such that g_x ~ X / (4*2) and g_y ~ Y / (4*2) The factor of 4 for weighting, the factor of two because a gradient = DELTA Y / DELTA X and DELTA X = 2 pixels so that G ~ 8 sqrt (g_x**2 + g_y**2) Now in MODE 2, BDEPO assumes g = g_x = g_y so that G ~ 8 * sqrt (2) * g Thus, the scale factor in BDEPO input via CPARM(3) that is necessary for a NINER 'SOBL' (DPARM(1)=0)) gradient image is 1 / (8 sqrt(2)) This is the MODE 2 default scale factor applied when CPARM(3)=0 Other parameters: BLC, TRC : The usual windows for the input images. These default to the entire image CPARM(1 & 2) : Specify the frequencies (Hz) between which to calculate the DPR. The order is unimportant and there are no defaults.