{VERSION 3 0 "SGI MIPS UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "terminal" 1 11 255 0 0 1 0 2 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "fixed" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 0 0 128 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Sans" -1 256 "helvetica" 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 174 243 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 174 243 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 256 3 1 {CSTYLE "" -1 -1 " " 1 18 0 0 128 1 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple O utput" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 256 18 1 {CSTYLE "" -1 -1 "" 1 24 0 0 0 0 0 1 1 0 0 0 0 0 0 } 3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 256 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "Sans" -1 256 1 {CSTYLE "" -1 -1 "helvetica" 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT 258 83 "Beam Pattern Response Functions and Times of Arrival\nfor Earthbound Interferometers" }}{PARA 19 "" 0 "" {TEXT 259 215 "Warren G. Anderson and John T. Whelan (The Univers ity of Texas at Brownsville), \nPatrick R. Brady and Jolien D. E. Crei ghton (University of Wisconsin - Milwaukee),\nDavid Chin and Keith Ril es (University of Michigan)" }}{PARA 19 "" 0 "" {TEXT 260 57 "First Ve rsion March 30, 2000; Last Revised July 18, 2002 " }}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 13 "Preliminaries" }}{SECT 1 {PARA 4 "" 0 "results sup ressed" {TEXT -1 14 "Notes on usage" }}{PARA 0 "" 0 "" {TEXT -1 560 "H opefully everything one needs in order to calculate response functions and times of arrival is contained in this worksheet, along with inter mediate results which one can use to check calculations. There are a \+ few results in this worksheet that are so long and unweildy that we ch ose to supress the output by using a colon (:) to end the input statem ent rather than the usual semicolon (;). We always warn you that this \+ is the case by putting (results supressed) into the preceding text. If you need to see these results, just change the colon to a semicolon. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Technical asides" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 41 "LIGO technical document LIGO-T980044-08-E " }}{PARA 0 "" 0 "" {TEXT -1 60 "Much of this document is based upon L IGO technical document " }{HYPERLNK 17 "LIGO-T980044-08-E" 1 "" "T9800 44-08-E" }{TEXT -1 188 ". There are, however, several definitions that are left to the reader to determine, presumably because their is only one clear choice. We have made those definitions obvious here, to wit : " }}{PARA 0 "" 0 "" {TEXT -1 362 "- the LIGO document refers to a su face, a point, and a line through the point prependiular to the surfac e. There can in principle be an arbitrary number of such lines. Clearl y, they mean the line passing through the surface whose intersection w ith the surface is closest to the point. It can be shown that this lin e always intersects the surface perpendicularly." }}{PARA 0 "" 0 "" {TEXT -1 200 "- the term local horizontal at a point is undefined. We \+ have taken this to mean the plane through that point which is parallel to the tangent to the surface at the intersection point mentioned abo ve. " }}{PARA 0 "" 0 "" {TEXT -1 174 "- they give values for angles of vectors \"above the local horizontal\". Again, there are an infinite \+ number of such angles. Clearly, they mean the angle of smallest magnit ude." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 " We have also corrected one typographical error in the document - the d enominator of the equation for " }{XPPEDIT 18 0 "R;" "6#%\"RG" } {XPPEDIT 18 0 "[phi];" "6#7#%$phiG" }{TEXT -1 47 " should be the squar e root of the value quoted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Finally, the document also refers to an eccentr icity, " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 70 ". Howe ver, it is a quantity derived from two other quoted quantities, " } {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b;" " 6#%\"bG" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "[1-epsilon^2] = b^2/(a^2);" "6#/7#,&\"\"\"\"\"\"*$%(epsilonG\"\"#!\"\"*&%\"bG\"\"#*$%\"aG\"\"#F+" }{TEXT -1 56 "), and is therefore unnecessary, so we have not used it. " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 15 "Note about GMST" }}{PARA 0 " " 0 "" {TEXT -1 7 "In the " }{HYPERLNK 17 "Times of Arrival" 1 "" "Tim e of Arrival" }{TEXT -1 910 " section, one encounters the following si tuation: the vector describing the location of the detector depends on the GMST at which the wave arrives at the detector. However, the whol e purpose of this section is to calculate the difference between the G MST at which the wave arrives at the detector and the GMST at which it arrives at the center of the earth. This could be dealt with by inver ting the functions of GMST, at the cost of computational complexity. H owever, the dependence of the time of arrival on this difference betwe en the two GMST's is clearly a higher order effect, and we will ignore it. Note however that this may not be the case for the response funct ions themselves. The earth can rotate by approximately 0.3 arcseconds \+ during the time it takes light to travel from the center of the earth \+ to it's limb. This might have a significant effect on the response fun ctions for some applications." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Housekeeping duties" }}{PARA 0 "" 0 "" {TEXT -1 17 "Getting starte d. " }{HYPERLNK 17 "restart" 2 "restart" "" }{TEXT -1 29 " for a clean slate. Load the " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 5 " \+ and " }{HYPERLNK 17 "tensor" 2 "tensor" "" }{TEXT -1 62 " packages to \+ give us matrix and tensor handling capabilities. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart; with(linalg):with(tensor):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}}}{SECT 1 {PARA 3 "" 0 "WGS-84" {TEXT -1 18 "Earth Model WGS-84" }}{PARA 0 "" 0 "" {TEXT -1 248 "It is convenient to first specify an earth model with which to relate the detector position and orientation to the gravitat ional wave propagation direction and polarization. The earth model use d here is the same one used in LIGO technical document " }{HYPERLNK 17 "LIGO-T980044-08-E" 1 "" "T980044-08-E" }{TEXT -1 76 "; earth model WGS-84. The reference figure of this model is the surface of (" } {XPPEDIT 18 0 "Sigma;" "6#%&SigmaG" }{TEXT -1 43 ") an oblate ellipsoi d with semi-major axis " }{XPPEDIT 18 0 "a = 6378137*m;" "6#/%\"aG*&\" (P\"yj\"\"\"%\"mGF'" }{TEXT -1 18 ", semi-minor axis " }{XPPEDIT 18 0 "b = 6356752.314*m;" "6#/%\"bG*&$\"+9Bvcj!\"$\"\"\"%\"mGF)" }{TEXT -1 48 ". The document also refers to the eccentricity, " }{XPPEDIT 18 0 " epsilon;" "6#%(epsilonG" }{TEXT -1 41 ". However, it is a quantity der ived from " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 7 " (i.e. " }{XPPEDIT 18 0 "[1- epsilon^2] = b^2/(a^2);" "6#/7#,&\"\"\"\"\"\"*$%(epsilonG\"\"#!\"\"*&% \"bG\"\"#*$%\"aG\"\"#F+" }{TEXT -1 102 " ), and it is therefore unnece ssary. The set of relevant parameters is therefore (in units of meters ):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "WGS84_params:=\{a = 63 78137,b = 6356752.314\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-WGS84_p aramsG<$/%\"aG\"(P\"yj/%\"bG$\"+9Bvcj!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "In WGS-84, one specifies the po sition of any point in space " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 23 " in terms of the point " }{XPPEDIT 18 0 "sigma(x);" "6#-%&sigma G6#%\"xG" }{TEXT -1 58 ", which is defined to be the closest point on \+ the surface " }{XPPEDIT 18 0 "Sigma;" "6#%&SigmaG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 21 ". The coordinates of " } {XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 23 " are then specified by " } {XPPEDIT 18 0 "l;" "6#%\"lG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "lambd a;" "6#%'lambdaG" }{TEXT -1 43 ", the North latitude and East longitud e of " }{XPPEDIT 18 0 "sigma(x);" "6#-%&sigmaG6#%\"xG" }{TEXT -1 33 " \+ respectively, and by the height " }{XPPEDIT 18 0 "h;" "6#%\"hG" } {TEXT -1 29 ", which is the distance from " }{XPPEDIT 18 0 "sigma(x); " "6#-%&sigmaG6#%\"xG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x;" "6#%\"xG " }{TEXT -1 29 " along the outward normal to " }{XPPEDIT 18 0 "Sigma; " "6#%&SigmaG" }{TEXT -1 2 ". " }{TEXT 263 132 "N.B. the line segment \+ connecting any point in space to the nearest point on any smooth surfa ce intersects the surface orthogonally. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Coordinate Frames" }}{PARA 0 "" 0 "" {TEXT -1 110 "There \+ are at least four coordinate frames which are useful in discussing the detection of gravitational waves:" }}{PARA 0 "" 0 "" {TEXT -1 78 "- \+ source frame, which is chosen to provide a simple description of the s ource" }}{PARA 0 "" 0 "" {TEXT -1 6 "- the " }{HYPERLNK 17 "wave propa gation frame" 1 "" "wave prop frame" }{TEXT -1 91 ", which provides a \+ simple description of the waves propagating from the source to the ear th" }}{PARA 0 "" 0 "" {TEXT -1 6 "- the " }{HYPERLNK 17 "earth fixed f rame" 1 "" "earth fixed frame" }{TEXT -1 86 ", which provides a standa rd earthbound coordinate frame in which to describe the waves" }} {PARA 0 "" 0 "" {TEXT -1 6 "- the " }{HYPERLNK 17 "detector frame" 1 " " "detector frame" }{TEXT -1 90 ", which is adapted to the particular \+ geometry of the detector which is to detect the waves" }}{PARA 0 "" 0 "" {TEXT -1 300 "This worksheet is primarily concerned with the latter three frames. Each frame is an orthogonal Cartesian frame. It is assu med that the spatial curvature of space-time near the earth is suffici ently small that Euclidean translations may be used to relate frames a t the earth's center and its surface." }}{SECT 1 {PARA 4 "" 0 "wave pr op frame" {TEXT -1 22 "Wave Propagation Frame" }}{PARA 0 "" 0 "" {TEXT -1 77 "The wave propagation frame is chosen to be compatible wit h the definition of " }{HYPERLNK 17 "Will and Wiseman" 1 "" "Will and \+ Wiseman" }{TEXT -1 73 ". The origin of this frame is taken to be the c entroid of the earth. The " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 78 "-axis points lies along the line joining the origin to the source, and points " }{TEXT 257 4 "away" }{TEXT -1 22 " from the source. The \+ " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 157 "-axis will typically be chosen to make the transition from the source frame to the wave propa gation frame convenient. For the purposes of this document, the " } {XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 89 "-axis can be considered to \+ be an arbitrary vector chosen in the plane orthogonal to the " } {XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 11 "-axis. The " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 129 "-axis is then chosen to complete a rig ht-handed coordinate system. The coordinates in the wave propagation f rame are denoted by (" }{XPPEDIT 18 0 "x[w],y[w],z[w];" "6%&%\"xG6#%\" wG&%\"yG6#F&&%\"zG6#F&" }{TEXT -1 24 "). The unit vectors are:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "e_wx:=create([-1],array(1.. 3,[1,0,0]));e_wy:=create([-1],array(1..3,[0,1,0]));e_wz:=create([-1],a rray(1..3,[0,0,1]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_wxG-%&TAB LEG6#7$/%+index_charG7#!\"\"/%'comptsG-%'vectorG6#7%\"\"\"\"\"!F4" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_wyG-%&TABLEG6#7$/%+index_charG7#! \"\"/%'comptsG-%'vectorG6#7%\"\"!\"\"\"F3" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_wzG-%&TABLEG6#7$/%+index_charG7#!\"\"/%'comptsG-%' vectorG6#7%\"\"!F3\"\"\"" }}}}{SECT 1 {PARA 4 "" 0 "earth fixed frame " {TEXT -1 17 "Earth Fixed Frame" }}{PARA 0 "" 0 "" {TEXT -1 60 "The e arth fixed frame is defined in LIGO technical document " }{HYPERLNK 17 "LIGO-T980044-08-E" 1 "" "T980044-08-E" }{TEXT -1 61 ". The origin \+ of this frame is the centroid of the earth. The " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 47 "-axis points from the origin to the North pole " }{XPPEDIT 18 0 "\{l = 90*N, lambda = 0\};" "6#<$/%\"lG*&\"#!*\"\"\" %\"NGF(/%'lambdaG\"\"!" }{TEXT -1 6 ". The " }{XPPEDIT 18 0 "x;" "6#% \"xG" }{TEXT -1 91 "-axis points from the origin to the intersection o f the earth's equator and prime meridian " }{XPPEDIT 18 0 "\{0, 0\};" "6#<$\"\"!F$" }{TEXT -1 6 ". The " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT -1 61 "-axis is chosen to complete a right-handed coordinate sys tem " }{XPPEDIT 18 0 "\{0, 90*E\};" "6#<$\"\"!*&\"#!*\"\"\"%\"EGF'" } {TEXT -1 290 ". Note that this coordinate system rotates with earth (w ith respect to the fixed background stars). Its relationship to the wa ve propagation frame therefore changes as a periodic function of time, with a period of one sidereal day. The coordinates in the earth fixed frame are denoted by \n(" }{XPPEDIT 18 0 "x[e],y[e],z[e];" "6%&%\"xG6 #%\"eG&%\"yG6#F&&%\"zG6#F&" }{TEXT -1 24 ").The unit vectors are: " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "e_ex:=create([-1],array(1.. 3,[1,0,0]));e_ey:=create([-1],array(1..3,[0,1,0]));e_ez:=create([-1],a rray(1..3,[0,0,1]));\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_exG-%&T ABLEG6#7$/%+index_charG7#!\"\"/%'comptsG-%'vectorG6#7%\"\"\"\"\"!F4" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_eyG-%&TABLEG6#7$/%+index_charG7# !\"\"/%'comptsG-%'vectorG6#7%\"\"!\"\"\"F3" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_ezG-%&TABLEG6#7$/%+index_charG7#!\"\"/%'comptsG-%' vectorG6#7%\"\"!F3\"\"\"" }}}}{SECT 1 {PARA 4 "" 0 "detector frame" {TEXT -1 14 "Detector Frame" }}{PARA 0 "" 0 "" {TEXT -1 189 "We define the detector frame in terms of cardinal compass points at the positio n of the detector on the earth's surface. The definition of the detect or frame assumes the oblate ellipsoidal " }{HYPERLNK 17 "earth model W GS-84" 1 "" "WGS-84" }{TEXT -1 50 ". We begin with some preliminary de finitions. Let " }{XPPEDIT 18 0 "sigma(x);" "6#-%&sigmaG6#%\"xG" } {TEXT -1 46 " be the on the reference ellipsoid's surface (" } {XPPEDIT 18 0 "Sigma;" "6#%&SigmaG" }{TEXT -1 23 ") nearest to the poi nt " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 6 ". Let " }{XPPEDIT 18 0 "Lambda(x);" "6#-%'LambdaG6#%\"xG" }{TEXT -1 25 " be the tangent pla ne to " }{XPPEDIT 18 0 "Sigma;" "6#%&SigmaG" }{TEXT -1 4 " at " } {XPPEDIT 18 0 "sigma(x);" "6#-%&sigmaG6#%\"xG" }{TEXT -1 13 ". Define \+ the " }{TEXT 271 16 "local horizontal" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 28 " to be the plane containing " } {XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 22 " which is parallel to " } {XPPEDIT 18 0 "Lambda(x);" "6#-%'LambdaG6#%\"xG" }{TEXT -1 102 ". With in the local horizontal, the directions North and East are inherited f rom their definitions in " }{XPPEDIT 18 0 "Lambda(x);" "6#-%'LambdaG6 #%\"xG" }{TEXT -1 84 ". Note that there is no ambiguity in this since \+ the local horizontal is parallel to " }{XPPEDIT 18 0 "Lambda(x);" "6#- %'LambdaG6#%\"xG" }{TEXT -1 2 ". " }{XPPEDIT 18 0 "sigma(x);" "6#-%&si gmaG6#%\"xG" }{TEXT -1 241 " is the point on the surface of the ellips oid that is closest to the corner of the interferometer (this will be \+ unique if the corner is near enough the surface). The origin of the de tector frame is then the corner of the interferometer. The " } {XPPEDIT 18 0 "x-y;" "6#,&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 50 " plane \+ is defined to be the local horizontal. The " }{XPPEDIT 18 0 "x;" "6#% \"xG" }{TEXT -1 57 "-axis is chosen to point due East in this plane, a nd the " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 34 "-axis to point du e North, so that " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 21 " are orthogonal. The " } {XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 117 "-axis points along the out ward normal to the earth's surface. The coordinates in the detector f rame are denoted by (" }{XPPEDIT 18 0 "x[d],y[d],z[d];" "6%&%\"xG6#%\" dG&%\"yG6#F&&%\"zG6#F&" }{TEXT -1 23 ").The unit vectors are:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "e_dx:=create([-1],array(1.. 3,[1,0,0]));e_dy:=create([-1],array(1..3,[0,1,0]));e_dz:=create([-1],a rray(1..3,[0,0,1]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_dxG-%&TAB LEG6#7$/%+index_charG7#!\"\"/%'comptsG-%'vectorG6#7%\"\"\"\"\"!F4" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_dyG-%&TABLEG6#7$/%+index_charG7#! \"\"/%'comptsG-%'vectorG6#7%\"\"!\"\"\"F3" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_dzG-%&TABLEG6#7$/%+index_charG7#!\"\"/%'comptsG-%' vectorG6#7%\"\"!F3\"\"\"" }}}}}{SECT 1 {PARA 3 "" 0 "Grav Wave Tens" {TEXT -1 36 "Gravitational Wave Tensor Components" }}{PARA 0 "" 0 "" {TEXT -1 54 "The components of the wave are naturally given in the " } {HYPERLNK 17 "wave propagation frame" 1 "" "wave prop frame" }{TEXT -1 51 ". As a first step, define the polarization tensors " }{XPPEDIT 18 0 "e_plus;" "6#%'e_plusG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "e_cro ss;" "6#%(e_crossG" }{TEXT -1 7 " to be:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "e_plus:=lin_com(1,prod(e_wx,e_wx),-1,prod(e_wy,e_wy)) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'e_plusG-%&TABLEG6#7$/%+index_c harG7$!\"\"F,/%'comptsG-%'matrixG6#7%7%\"\"\"\"\"!F57%F5F,F57%F5F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "e_cross:=lin_com(1,prod(e _wx,e_wy),1,prod(e_wy,e_wx));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(e_ crossG-%&TABLEG6#7$/%+index_charG7$!\"\"F,/%'comptsG-%'matrixG6#7%7%\" \"!\"\"\"F47%F5F4F47%F4F4F4" }}}{PARA 0 "" 0 "" {TEXT -1 57 "The gravi tational wave tensor is a linear combination of " }{XPPEDIT 18 0 "e[pl us];" "6#&%\"eG6#%%plusG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "e[cross] ;" "6#&%\"eG6#%&crossG" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "h_ tensor:=lin_com(h_plus,e_plus,h_cross,e_cross);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)h_tensorG-%&TABLEG6#7$/%+index_charG7$!\"\"F,/%'comp tsG-%'matrixG6#7%7%%'h_plusG%(h_crossG\"\"!7%F5,$F4F,F67%F6F6F6" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 62 "The Transformation from Wave Prop agation to Earth Fixed Frames" }}{PARA 0 "" 0 "" {TEXT -1 49 "We will \+ need to write an arbitrary vector in the " }{HYPERLNK 17 "wave propaga tion frame" 1 "" "wave prop frame" }{TEXT -1 13 " in terms of " } {HYPERLNK 17 "earth fixed frame" 1 "" "earth fixed frame" }{TEXT -1 160 " unit vectors. The transformation between the wave propagation an d earth fixed frames that will allow us to do this is carried out usin g Euler angle rotations. " }}{SECT 1 {PARA 4 "" 0 "Euler angles" {TEXT -1 30 "Definition of the Euler Angles" }}{PARA 0 "" 0 "" {TEXT -1 221 "First, some initial definitions. \"Counterclockwise about an a xis\" means in a counterclockwise direction as viewed from that axis w hile facing the origin. The line of nodes is defined to be the line of intersection of the " }{XPPEDIT 18 0 "x[e]-y[e];" "6#,&&%\"xG6#%\"eG \"\"\"&%\"yG6#F'!\"\"" }{TEXT -1 15 " plane and the " }{XPPEDIT 18 0 " x[w]-y[w];" "6#,&&%\"xG6#%\"wG\"\"\"&%\"yG6#F'!\"\"" }{TEXT -1 54 " pl ane. If one draws a circle about the origin on the " }{XPPEDIT 18 0 "x [e]-y[e];" "6#,&&%\"xG6#%\"eG\"\"\"&%\"yG6#F'!\"\"" }{TEXT -1 140 " pl ane, their will be two points of intersection between the circle and t he line of nodes, and these are called nodes. Draw a circle in the " } {XPPEDIT 18 0 "x[w]-y[w];" "6#,&&%\"xG6#%\"wG\"\"\"&%\"yG6#F'!\"\"" } {TEXT -1 126 " plane about the origin and passing through the nodes. D raw tangent vectors to the circle pointing counterclockwise about the \+ " }{XPPEDIT 18 0 "z[w];" "6#&%\"zG6#%\"wG" }{TEXT -1 71 "-axis. At one node, the tangent vector thus drawn will have a positive " }{XPPEDIT 18 0 "z[e];" "6#&%\"zG6#%\"eG" }{TEXT -1 65 " component, at the other \+ the tangent vector will have a negative " }{XPPEDIT 18 0 "z[e];" "6#&% \"zG6#%\"eG" }{TEXT -1 269 " component. The former is called the ascen ding node, and the half line from the origin through the ascending nod e is called the line of ascending nodes. The Euler angles are defined \+ with respect to the axes of the two frames and the line of ascending n odes. They are:\n " }{XPPEDIT 18 0 "Psi;" "6#%$PsiG" }{TEXT -1 62 " - \+ the angle from the line of ascending nodes to the positive " } {XPPEDIT 18 0 "x[w];" "6#&%\"xG6#%\"wG" }{TEXT -1 42 "-axis counterclo ckwise about the positive " }{XPPEDIT 18 0 "z[w];" "6#&%\"zG6#%\"wG" } {TEXT -1 22 "-axis. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 43 " - the smallest ang le between the positive " }{XPPEDIT 18 0 "z[e];" "6#&%\"zG6#%\"eG" } {TEXT -1 23 "-axis and the positive " }{XPPEDIT 18 0 "z[w];" "6#&%\"zG 6#%\"wG" }{TEXT -1 31 "-axis. It is always positive.\n " }{XPPEDIT 18 0 "Phi;" "6#%$PhiG" }{TEXT -1 31 " - the angle from the positive " } {XPPEDIT 18 0 "x[e];" "6#&%\"xG6#%\"eG" }{TEXT -1 64 "-axis to the lin e of ascending nodes counterclockwise about the " }{XPPEDIT 18 0 "z[e] ;" "6#&%\"zG6#%\"eG" }{TEXT -1 6 "-axis." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "The Rotation Matrices" }}{PARA 0 "" 0 "" {TEXT -1 32 "The three rotation matrices are:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "R[Psi]:=array(1..3,1..3,[[cos(Psi),sin(Psi),0],[-sin(Psi),cos(Ps i),0],[0,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"RG6#%$PsiG-%' matrixG6#7%7%-%$cosGF&-%$sinGF&\"\"!7%,$F/!\"\"F-F17%F1F1\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "R[Theta]:=array(1..3,1..3,[[ 1,0,0],[0,cos(Theta),sin(Theta)],[0,-sin(Theta),cos(Theta)]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"RG6#%&ThetaG-%'matrixG6#7%7%\"\" \"\"\"!F.7%F.-%$cosGF&-%$sinGF&7%F.,$F2!\"\"F0" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 80 "R[Phi]:=array(1..3,1..3,[[cos(Phi),sin(Phi),0] ,[-sin(Phi),cos(Phi),0],[0,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"RG6#%$PhiG-%'matrixG6#7%7%-%$cosGF&-%$sinGF&\"\"!7%,$F/!\"\"F-F17 %F1F1\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 101 "The full rotation matrix which transforms wave propagation frame vectors in the fixed earth fr ame is:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "R[Euler]:=multipl y(R[Psi],multiply(R[Theta],R[Phi]));" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>&%\"RG6#%&EulerG-%'matrixG6#7%7%,&*&-%$cosG6#%$PsiG\"\"\"-F06#%$Phi GF3F3*(-%$sinGF1F3-F06#%&ThetaGF3-F9F5F3!\"\",&*&F/\"\"\"F=FAF3*(F8FAF :FAF4FAF3*&F8FA-F9F;F37%,&*&F8FAF4FAF>*(F/FAF:FAF=FAF>,&*&F8FAF=FAF>*( F/FAF:FAF4FAF3*&F/FAFDFA7%*&FDFAF=FA,$*&FDFAF4FAF>F:" }}}}}{SECT 1 {PARA 3 "" 0 "sky angles" {TEXT -1 11 "Sky Angles " }}{PARA 0 "" 0 "" {TEXT -1 243 "While the Euler angles are useful for defining the rotat ion matrix, the gravitational wave community has historically used ang les specifying the sky position rather than the Euler angles above. We will recalculate in terms of these angles now." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Definition of Sky Angles" }}{PARA 0 "" 0 "" {TEXT -1 98 "The three angles in terms of which the response functions are u sually calculated are the altitude " }{XPPEDIT 18 0 "theta;" "6#%&thet aG" }{TEXT -1 14 ", the azimuth " }{XPPEDIT 18 0 "phi;" "6#%$phiG" } {TEXT -1 29 ", and the polarization angle " }{XPPEDIT 18 0 "psi;" "6#% $psiG" }{TEXT -1 108 ". Altitude and azimuth specify the position of t he gravitational wave source on the sky with respect to the " } {HYPERLNK 17 "earth fixed coordinates" 1 "" "earth fixed frame" } {TEXT -1 107 ", i.e. they define a vector in the earth fixed frame tha t points toward the source. The polarization angle " }{XPPEDIT 18 0 "p si;" "6#%$psiG" }{TEXT -1 148 " contains information about the polariz ation of the wave. Altitude and azimuth might best be understood by th eir simple relationship to declination " }{XPPEDIT 18 0 "delta;" "6#%& deltaG" }{TEXT -1 21 " and right ascension " }{XPPEDIT 18 0 "alpha;" " 6#%&alphaG" }{TEXT -1 21 ". The relationship is" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "sky_to_celestial:=\{phi=alpha-GMST,theta=Pi/2- delta\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1sky_to_celestialG<$/%$p hiG,&%&alphaG\"\"\"%%GMSTG!\"\"/%&thetaG,&%#PiG#F*\"\"#%&deltaGF," }}} {PARA 0 "" 0 "" {TEXT -1 170 "where GMST, the Greenwich Mean Sidereal \+ Time of the observation, allows us to relate a frame fixed with respec t to the celestial sphere to the rotating earth fixed frame." }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Relationship Between Sky Angles a nd Euler Angles" }}{PARA 0 "" 0 "" {TEXT -1 48 "The relationship betwe en the sky angles and the " }{HYPERLNK 17 "Euler angles" 1 "" "Euler a ngles" }{TEXT -1 42 " is also particularly simple. Recall that " } {XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 32 " measures the angle \+ between the " }{XPPEDIT 18 0 "z[e];" "6#&%\"zG6#%\"eG" }{TEXT -1 14 "- axis and the " }{XPPEDIT 18 0 "z[w];" "6#&%\"zG6#%\"wG" }{TEXT -1 20 " -axis, which points " }{TEXT 261 4 "away" }{TEXT -1 34 " from the sour ce. Thus, the angle " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 13 " between the " }{XPPEDIT 18 0 "z[e];" "6#&%\"zG6#%\"eG" }{TEXT -1 19 " and the direction " }{TEXT 262 2 "to" }{TEXT -1 29 " the source m ust be given by " }{XPPEDIT 18 0 "theta = Pi-Theta;" "6#/%&thetaG,&%#P iG\"\"\"%&ThetaG!\"\"" }{TEXT -1 24 ". Likewise, recall that " } {XPPEDIT 18 0 "Phi;" "6#%$PhiG" }{TEXT -1 23 " is the angle from the \+ " }{XPPEDIT 18 0 "x[e];" "6#&%\"xG6#%\"eG" }{TEXT -1 86 "-axis to the \+ line of ascending nodes. Furthermore, observe that the projection of t he " }{XPPEDIT 18 0 "z[w];" "6#&%\"zG6#%\"wG" }{TEXT -1 15 "-axis onto the " }{XPPEDIT 18 0 "x[e]-y[e];" "6#,&&%\"xG6#%\"eG\"\"\"&%\"yG6#F'! \"\"" }{TEXT -1 10 " plane is " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\" \"\"\"#!\"\"" }{TEXT -1 63 " clockwise from the line of ascending node s about the positive " }{XPPEDIT 18 0 "z[e];" "6#&%\"zG6#%\"eG" } {TEXT -1 13 "-axis. Thus, " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 21 ", the angle from the " }{XPPEDIT 18 0 "x[e];" "6#&%\"xG6#%\"eG " }{TEXT -1 56 "-axis to the line of ascending nodes about the positiv e " }{XPPEDIT 18 0 "z[e];" "6#&%\"zG6#%\"eG" }{TEXT -1 19 "-axis, is g iven by " }{XPPEDIT 18 0 "phi = Phi+Pi/2;" "6#/%$phiG,&%$PhiG\"\"\"*&% #PiGF'\"\"#!\"\"F'" }{TEXT -1 34 ". Finally, since the Euler angle " }{XPPEDIT 18 0 "Psi;" "6#%$PsiG" }{TEXT -1 107 " encodes all the neces sary polarization information already, we take it to be the polarizati on angle, i.e. " }{XPPEDIT 18 0 "psi = Psi;" "6#/%$psiG%$PsiG" }{TEXT -1 53 ". The complete coordinate transformation is therefore" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Euler_to_sky:=\{Theta=Pi-the ta, Phi=phi-Pi/2, Psi=psi\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-Eul er_to_skyG<%/%&ThetaG,&%#PiG\"\"\"%&thetaG!\"\"/%$PhiG,&%$phiGF*F)#F, \"\"#/%$PsiG%$psiG" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "The Rotat ion Matrices in Terms of Sky Angles" }}{PARA 0 "" 0 "" {TEXT -1 105 "I t is now a straightforward matter to express the rotation matrices in \+ terms of the sky angles. They are:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "R[psi]:=map2(subs,Euler_to_sky,R[Psi]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"RG6#%$psiG-%'matrixG6#7%7%-%$cosGF&-%$sinGF &\"\"!7%,$F/!\"\"F-F17%F1F1\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "R[theta]:=map(expand,map2(subs,Euler_to_sky,R[Theta]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"RG6#%&thetaG-%'matrixG6#7%7 %\"\"\"\"\"!F.7%F.,$-%$cosGF&!\"\"-%$sinGF&7%F.,$F4F3F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "R[phi]:=map(expand,map2(subs,Euler_ to_sky,R[Phi]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"RG6#%$phiG-%' matrixG6#7%7%-%$sinGF&,$-%$cosGF&!\"\"\"\"!7%F0F-F37%F3F3\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "R[sky]:=map(expand,map2(subs ,Euler_to_sky,R[Euler]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"RG6# %$skyG-%'matrixG6#7%7%,&*&-%$cosG6#%$psiG\"\"\"-%$sinG6#%$phiGF3F3*(-F 5F1F3-F06#%&thetaGF3-F0F6F3!\"\",&*&F/\"\"\"F=FAF>*(F9FAF4FAF:FAF>*&F9 FA-F5F;F37%,&*&F9FAF4FAF>*(F/FAF:FAF=FAF>,&*&F9FAF=FAF3*(F/FAF4FAF:FAF >*&F/FAFDFA7%,$*&FDFAF=FAF>,$*&FDFAF4FAF>,$F:F>" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 54 "The Gravitational Wave Tensor in the Earth Fixed \+ Frame" }}{PARA 0 "" 0 "" {TEXT -1 78 "In order to find detector respon se functions, it is convenient to express the " }{HYPERLNK 17 "gravita tional wave vector" 1 "" "Grav Wave Tens" }{TEXT -1 17 " in terms of t he " }{HYPERLNK 17 "earth fixed frame" 1 "" "earth fixed frame" } {TEXT -1 92 " coordinate basis. To do this, we first create a triad to hold the earth fixed frame vectors" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "triad[e]:=vector([e_e_x,e_e_y,e_e_z]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%&triadG6#%\"eG-%'vectorG6#7%%&e_e_xG%&e_e_yG% &e_e_zG" }}}{PARA 0 "" 0 "" {TEXT -1 55 "We then transform then multip ly by the rotation matrix " }{XPPEDIT 18 0 "R[Euler];" "6#&%\"RG6#%&Eu lerG" }{TEXT -1 25 " to get the triad in the " }{HYPERLNK 17 "wave pro pagation frame" 1 "" "wave prop frame" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "triad[w]:=multiply(R[sky],triad[e]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>&%&triadG6#%\"wG-%'vectorG6#7%,(*&,&* &-%$cosG6#%$psiG\"\"\"-%$sinG6#%$phiGF4F4*(-F6F2F4-F16#%&thetaGF4-F1F7 F4!\"\"F4%&e_e_xGF4F4*&,&*&F0\"\"\"F>FDF?*(F:FDF5FDF;FDF?F4%&e_e_yGF4F 4*(F:FD-F6FFDF?F4F@FDF4*& ,&*&F:FDF>FDF4*(F0FDF5FDF;FDF?F4FFFDF4*(F0FDFHFDFIFDF4,(*(FHFDF>FDF@FD F?*(FHFDF5FDFFFDF?*&F;FDFIFDF?" }}}{PARA 0 "" 0 "" {TEXT -1 99 "The in dividual wave propagation frame vectors can now be written in the eart h fixed frame. They are" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 " ind:=[x,y,z]:\nfor c1 from 1 to 3 do e_w.(ind[c1]):=lin_com(c oeff(triad[w][c1],e_e_x),e_ex,coeff(triad[w][c1],e_e_y),e_ey,coeff(tri ad[w][c1],e_e_z),e_ez);\nod; c1:='c1':" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_wxG-%&TABLEG6#7$/%+index_charG7#!\"\"/%'comptsG-%'vectorG6# 7%,&*&-%$cosG6#%$psiG\"\"\"-%$sinG6#%$phiGF9F9*(-F;F7F9-F66#%&thetaGF9 -F6F%%e_wyG-%&TABLEG6#7$/%+index_charG7#!\"\"/%'com ptsG-%'vectorG6#7%,&*&-%$sinG6#%$psiG\"\"\"-F66#%$phiGF9F,*(-%$cosGF7F 9-F?6#%&thetaGF9-F?F;F9F,,&*&F5\"\"\"FCFFF9*(F>FFF:FFF@FFF,*&F>FF-F6FA F9" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%e_wzG-%&TABLEG6#7$/%+index_ch arG7#!\"\"/%'comptsG-%'vectorG6#7%,$*&-%$sinG6#%&thetaG\"\"\"-%$cosG6# %$phiGF9F,,$*&F5\"\"\"-F6F " 0 "" {MPLTEXT 1 0 164 "`tensor/lin_com/simp`:= proc(x) simplify(x, trig) en d:\ne_plus:=lin_com(1,prod(e_wx,e_wx),-1,prod(e_wy,e_wy)):\ne_cross:=l in_com(1,prod(e_wx,e_wy),1,prod(e_wy,e_wx)):" }}}{PARA 0 "" 0 "h on ea rth fixed" {TEXT -1 69 "And finally, the gravitational wave tensor in \+ the earth fixed frame (" }{HYPERLNK 17 "results supressed" 1 "" "resul ts supressed" }{TEXT -1 2 ")." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "`tensor/lin_com/simp`:= proc(x) collect(simplify(x, trig),\{h_p lus,h_cross\}) end:\nh_tensor:=lin_com(h_plus,e_plus,h_cross,e_cross): " }}}{PARA 0 "" 0 "" {TEXT -1 106 "To verify, calculate the response f unctions for a detector at the center of the earth with arms along the " }{XPPEDIT 18 0 "x[e];" "6#&%\"xG6#%\"eG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y[e];" "6#&%\"yG6#%\"eG" }{TEXT -1 13 " axes. Since " } {XPPEDIT 18 0 "h_tensor;" "6#%)h_tensorG" }{TEXT -1 66 " has already b een written in the detector frame above, the strain " }{XPPEDIT 18 0 " h;" "6#%\"hG" }{TEXT -1 18 " is just given by " }{XPPEDIT 18 0 "h_tens or[1,1]-h_tensor[2,2];" "6#,&&%)h_tensorG6$\"\"\"\"\"\"\"\"\"&F%6$\"\" #\"\"#!\"\"" }{TEXT -1 2 ". " }{XPPEDIT 18 0 "F_plus;" "6#%'F_plusG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "F_cross;" "6#%(F_crossG" }{TEXT -1 37 " are then simply the coefficients of " }{XPPEDIT 18 0 "h[plus];" " 6#&%\"hG6#%%plusG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h[cross];" "6#& %\"hG6#%&crossG" }{TEXT -1 14 " respectively." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "F_plus:=coeff(get_compts(h_tensor)[1,1]/2-get_c ompts(h_tensor)[2,2]/2,h_plus);\nF_cross:=coeff(get_compts(h_tensor)[1 ,1]/2-get_compts(h_tensor)[2,2]/2,h_cross);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'F_plusG,4*$)-%$cosG6#%$psiG\"\"#\"\"\"\"\"\"*&F'F-)- F)6#%$phiGF,F-!\"#*,F(F.-%$sinGF2F.-F7F*F.-F)6#%&thetaGF.F1F.!\"%*&)F9 F,F-F0F-F.*(F>F-F0F-F'F-F4#!\"\"F,F.*$F0F-F.*$F>F-F@*&F>F-F'F-F." }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%(F_crossG,.*&-%$cosG6#%$psiG\"\"\"-% $sinGF)F+!\"\"*(F'\"\"\"F,F0)-F(6#%$phiG\"\"#F0F5**)F'F5F0-F-F3F+-F(6# %&thetaGF+F2F+!\"%*(F9F0F2F0F8F0F5**F,F0)F9F5F0F1F0F'F0F5*(F,F0F?F0F'F 0F." }}}{PARA 0 "" 0 "" {TEXT -1 38 "This is a bit messy, so let's sim plify" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "tmp1:=expand(cos(2 *phi))=cos(2*phi):\ntmp2:=expand(cos(2*psi))=cos(2*psi):\ntmp3:=expand (sin(2*phi))=sin(2*phi):\ntmp4:=expand(sin(2*psi))=sin(2*psi):\nF_plus :=algsubs(tmp1,algsubs(tmp2,algsubs(tmp3,algsubs(tmp4,F_plus))));\nF_c ross:=algsubs(tmp1,algsubs(tmp2,algsubs(tmp3,algsubs(tmp4,F_cross)))); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'F_plusG,(*(-%$cosG6#%&thetaG \"\"\"-%$sinG6#,$%$psiG\"\"#F+-F-6#,$%$phiGF1F+!\"\"*()F'F1\"\"\"-F(F. F+-F(F3F+#F6F1*&F:F9F;F9F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(F_cro ssG,(*()-%$cosG6#%&thetaG\"\"#\"\"\"-%$sinG6#,$%$psiGF,\"\"\"-F)6#,$%$ phiGF,F3#F3F,*&F.F-F4F-F8*(F(F3-F/F5F3-F)F0F3!\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 21 "This is exactly what " }{HYPERLNK 17 "Anderson, Brady, \+ Creighton and Flanagan" 1 "" "ABCF" }{TEXT -1 5 " get!" }}}{SECT 1 {PARA 3 "" 0 "Location and Orientation vectors" {TEXT -1 50 "Location \+ and Arm Orientation Vectors for Detectors" }}{PARA 0 "" 0 "" {TEXT -1 29 "The primary advantage of the " }{HYPERLNK 17 "earth fixed frame" 1 "" "" }{TEXT -1 214 " is that one can conveniently express all the v ectors describing the detector arms in it, and these vectors, once det ermined, are fixed for all time. Our goal, therefore, is to find for e ach detector three vectors:" }}{PARA 0 "" 0 "" {TEXT -1 4 "- a " } {TEXT 268 15 "location vector" }{TEXT -1 51 " describing the position \+ of the corner station and " }}{PARA 0 "" 0 "" {TEXT -1 6 "- two " } {TEXT 269 23 "arm orientation vectors" }{TEXT -1 41 " describing the o rientation of the arms. " }}{PARA 0 "" 0 "" {TEXT -1 32 "These can the n be used with the " }{HYPERLNK 17 "gravitational wave tensor in the e arth fixed frame" 1 "" "h on earth fixed" }{TEXT -1 422 " to obtain th e strain produced by the gravitational wave anywhere. Information whic h can be used to construct these vectors for each detector is (mostly) contained within the data frames (not to be confused with coordinate \+ frames) produced by the detector. However, this information is not spe cified in terms of the earth fixed coordinates. The information used t o reconstruct the location vector is given in terms of the " } {HYPERLNK 17 "earth model WGS-84" 1 "" "WGS-84" }{TEXT -1 115 " coordi nates. The information needed for the arm orientation vectors is encod ed in a set of arm orientation angles." }}{SECT 1 {PARA 4 "" 0 "orient ation angles" {TEXT -1 33 "Arm orientation angle definitions" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are four arm orientation angles, " } {XPPEDIT 18 0 "psi[1];" "6#&%$psiG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "psi[2];" "6#&%$psiG6#\"\"#" }{TEXT -1 37 " which are fo und in data frames, and " }{XPPEDIT 18 0 "omega[1];" "6#&%&omegaG6#\" \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "omega[2];" "6#&%&omegaG6#\" \"#" }{TEXT -1 202 " which are not in the data frames at the present t ime. We note in passing that while three angles are sufficient to spec ify the orientation of any frame (and hence of the detector arms), we \+ will follow " }{HYPERLNK 17 "LIGO-T980044-08-E" 1 "" "T980044-08-E" } {TEXT 270 1 " " }{TEXT -1 85 "in using a four-angle system. The four o rientation angles most easily defined in the " }{HYPERLNK 17 "detector frame" 1 "" "detector frame" }{TEXT -1 25 ". They are defined to be: " }}{PARA 0 "" 0 "" {TEXT -1 2 "- " }{XPPEDIT 18 0 "psi[1];" "6#&%$psi G6#\"\"\"" }{TEXT -1 89 ": the angle North of East of the projection o f one arm (arm 1) onto the local horizontal." }}{PARA 0 "" 0 "" {TEXT -1 2 "- " }{XPPEDIT 18 0 "psi[2];" "6#&%$psiG6#\"\"#" }{TEXT -1 95 ": \+ the angle North of East of the projection of the other arm (arm 2) ont o the local horizontal." }}{PARA 0 "" 0 "" {TEXT -1 2 "- " }{XPPEDIT 18 0 "omega[1];" "6#&%&omegaG6#\"\"\"" }{TEXT -1 124 ": the angle of s mallest magnitude between arm 1 and the local horizontal. Angles above (below) the horizontal are +ve (-ve)." }}{PARA 0 "" 0 "" {TEXT -1 2 " - " }{XPPEDIT 18 0 "omega[2];" "6#&%&omegaG6#\"\"#" }{TEXT -1 124 ": t he angle of smallest magnitude between arm 1 and the local horizontal. Angles above (below) the horizontal are +ve (-ve)." }}}{PARA 0 "" 0 " " {TEXT -1 47 "We furthermore require the transformation from " } {HYPERLNK 17 "earth model WGS-84" 1 "" "WGS-84" }{TEXT -1 40 " coordin ates to earth fixed coordinates," }}{SECT 1 {PARA 4 "" 0 "WGS-84 to ea rth fixed" {TEXT -1 78 "Transformation from earth model WGS-84 coordin ates to earth fixed coordinates." }}{PARA 0 "" 0 "" {TEXT -1 26 "This \+ transformation from (" }{XPPEDIT 18 0 "l,lambda,h;" "6%%\"lG%'lambdaG% \"hG" }{TEXT -1 7 "), the " }{HYPERLNK 17 "earth model WGS-84" 1 "" "W GS-84" }{TEXT -1 20 " coordinates to the " }{HYPERLNK 17 "fixed earth \+ frame" 1 "" "earth fixed frame" }{TEXT -1 14 " coordinates (" } {XPPEDIT 18 0 "x[e],y[e],z[e];" "6%&%\"xG6#%\"eG&%\"yG6#F&&%\"zG6#F&" }{TEXT -1 18 ") is described in " }{HYPERLNK 17 "LIGO-T980044-08-E" 1 "" "T980044-08-E" }{TEXT -1 23 ". The transformation is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "M2E_x:=x[e]=(R+h)*cos(l)*cos(lambd a);\nM2E_y:=y[e]=(R+h)*cos(l)*sin(lambda);\nM2E_z:=z[e]=(b^2*R/a^2+h)* sin(l);\nearthModel_to_earthFixed:=[M2E_x,M2E_y,M2E_z];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&M2E_xG/&%\"xG6#%\"eG*(,&%\"RG\"\"\"%\"hGF-F-- %$cosG6#%\"lGF--F06#%'lambdaGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%& M2E_yG/&%\"yG6#%\"eG*(,&%\"RG\"\"\"%\"hGF-F--%$cosG6#%\"lGF--%$sinG6#% 'lambdaGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&M2E_zG/&%\"zG6#%\"eG* &,&*&*&)%\"bG\"\"#\"\"\"%\"RG\"\"\"F1*$)%\"aG\"\"#F1!\"\"F3%\"hGF3F3-% $sinG6#%\"lGF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%9earthModel_to_ear thFixedG7%/&%\"xG6#%\"eG*(,&%\"RG\"\"\"%\"hGF.F.-%$cosG6#%\"lGF.-F16#% 'lambdaGF./&%\"yGF)*(F,\"\"\"F0F;-%$sinGF5F./&%\"zGF)*&,&*&*&)%\"bG\" \"#F;F-F.F;*$)%\"aG\"\"#F;!\"\"F.F/F.F.-F=F2F." }}}{PARA 0 "" 0 "" {TEXT -1 52 "where R is the local radius of curvature, defined as" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Rdef:=R=a^2/sqrt(a^2*cos(l)^ 2+b^2*sin(l)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RdefG/%\"RG*&*$ )%\"aG\"\"#\"\"\"F,*$-%%sqrtG6#,&*&F)F,)-%$cosG6#%\"lGF+F,\"\"\"*&)%\" bGF+F,)-%$sinGF6F+F,F8F,!\"\"" }}}}{PARA 0 "" 0 "" {TEXT -1 8 "and the " }{HYPERLNK 17 "detector frame" 1 "" "detector frame" }{TEXT -1 41 " basis vectors expressed in terms of the " }{HYPERLNK 17 "earth fixed \+ frame" 1 "" "earth fixed frame" }{TEXT -1 7 " basis." }}{SECT 1 {PARA 4 "" 0 "detector frame to earth fixed frame" {TEXT -1 53 "Detector fra me basis vectors in the earth fixed frame" }}{PARA 0 "" 0 "" {TEXT -1 22 "The components of the " }{HYPERLNK 17 "detector frame" 1 "" "detec tor frame" }{TEXT -1 22 " basis vectors in the " }{HYPERLNK 17 "earth \+ fixed frame" 1 "" "earth fixed frame" }{TEXT -1 193 " basis are can be calculated in terms of the derivatives of the transformation from ear th model WGS-84 coordinates to earth fixed coordinates (i.e. the Jacob ian of the transformation). For the " }{XPPEDIT 18 0 "lambda;" "6#%'la mbdaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 46 " derivatives, which give us the components of " }{XPPEDIT 18 0 "e_d_x ;" "6#%&e_d_xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "e_d_z;" "6#%&e_d_z G" }{TEXT -1 61 " respectively, this is relatively trivial. The compon ents of " }{XPPEDIT 18 0 "e_d_y;" "6#%&e_d_yG" }{TEXT -1 72 " are cons iderably more difficult. First, we calculate the components of " } {XPPEDIT 18 0 "e_d_x;" "6#%&e_d_xG" }{TEXT -1 26 " in the earth fixed \+ frame." }}{SECT 1 {PARA 5 "" 0 "" {XPPEDIT 18 0 "lambda;" "6#%'lambdaG " }{TEXT -1 17 " derivatives and " }{XPPEDIT 18 0 "e_d_x;" "6#%&e_d_xG " }}{PARA 0 "" 0 "" {TEXT -1 38 "First the derivatives with respect to " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "M2E_derivs[lambda] := [ diff(rhs(M 2E_x),lambda), diff(rhs(M2E_y),lambda), diff(rhs(M2E_z),lambda) ];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%+M2E_derivsG6#%'lambdaG7%,$*(,&%\"R G\"\"\"%\"hGF-F--%$cosG6#%\"lGF--%$sinGF&F-!\"\"*(F+\"\"\"F/F7-F0F&F- \"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 64 "This gives an un-normalized vec tor. The normalization factor is:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "e_dx_norm:=simplify(sqrt(algsubs(sin(lambda)^2+cos(l ambda)^2=1,factor(M2E_derivs[lambda][1]^2+ M2E_derivs[lambda][2]^2+M2E _derivs[lambda][3]^2))),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%*e_dx_normG*&-%$cosG6#%\"lG\"\"\",&%\"RGF*%\"hGF*F*" }}}{PARA 0 "" 0 "" {TEXT -1 37 "The components of the detector frame " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 53 " basis vector in the earth fixed basis \+ are therefore:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "e_d_x:=arr ay([M2E_derivs[lambda][i]/e_dx_norm$i=1..3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&e_d_xG-%'vectorG6#7%,$-%$sinG6#%'lambdaG!\"\"-%$cosG F,\"\"!" }}}}{PARA 0 "" 0 "" {TEXT -1 22 "Now the components of " } {XPPEDIT 18 0 "e_d_z;" "6#%&e_d_zG" }{TEXT -1 1 "." }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 17 " de rivatives and " }{XPPEDIT 18 0 "e_d_z;" "6#%&e_d_zG" }}{PARA 0 "" 0 " " {TEXT -1 28 "Derivatives with respect to " }{XPPEDIT 18 0 "h;" "6#% \"hG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "M2E_ derivs[h] := [ diff(rhs(M2E_x),h), diff(rhs(M2E_y),h), diff(rhs(M2E_z) ,h) ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%+M2E_derivsG6#%\"hG7%*&-% $cosG6#%\"lG\"\"\"-F+6#%'lambdaGF.*&F*\"\"\"-%$sinGF0F.-F5F," }}} {PARA 0 "" 0 "" {TEXT -1 28 "This is, in fact normalized:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "e_dz_norm:=sqrt(simplify(M2E_derivs [h][1]^2+ M2E_derivs[h][2]^2+M2E_derivs[h][3]^2,trig));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%*e_dz_normG\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 92 "The components of the detector frame x; basis vector in the ear th fixed basis are therefore:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "e_d_z:=array([M2E_derivs[h][i]/e_dz_norm$i=1..3]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&e_d_zG-%'vectorG6#7%*&-%$cosG6#%\"lG\"\"\"-F+ 6#%'lambdaGF.*&F*\"\"\"-%$sinGF0F.-F5F," }}}}{PARA 0 "" 0 "" {TEXT -1 32 "We could find the components of " }{XPPEDIT 18 0 "e_d_y;" "6#%&e_d _yG" }{TEXT -1 165 " by taking a cross product of the other two basis \+ vectors, to complete a right handed frame. However, for pedagogy's sak e, we calculate it using derivatives as well." }}{SECT 1 {PARA 5 "" 0 "" {XPPEDIT 18 0 "l;" "6#%\"lG" }{TEXT -1 17 " derivatives and " } {XPPEDIT 18 0 "e_d_y;" "6#%&e_d_yG" }}{PARA 0 "" 0 "" {TEXT -1 38 "Fir st, let's compute the derivatives (" }{HYPERLNK 17 "results supressed " 1 "" "results supressed" }{TEXT -1 2 ")." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 113 "M2E_derivs[l] := [ diff(rhs(subs(Rdef,M2E_x)),l), \+ diff(rhs(subs(Rdef,M2E_y)),l), diff(rhs(subs(Rdef,M2E_z)),l) ]:" }}} {PARA 0 "" 0 "" {TEXT -1 269 "Since we will be normalizing anyway, we \+ can always rescale. When algebraic functions are involved, one will te nd to get repeated multiplicative factors. A simple way to remove and \+ identify these is to consider ratios of components. We do so to get si mple forms for the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 15 " components of " } {XPPEDIT 18 0 "e_d_y;" "6#%&e_d_yG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 106 "Lx:=numer(normal(M2E_derivs[l][1]/M2E_deriv s[l][2]));Ly:=denom(normal(M2E_derivs[l][1]/M2E_derivs[l][2]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#LxG-%$cosG6#%'lambdaG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#LyG-%$sinG6#%'lambdaG" }}}{PARA 0 "" 0 "" {TEXT -1 36 "Now remove the same factor from the " }{XPPEDIT 18 0 "z; " "6#%\"zG" }{TEXT -1 11 " component." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Lz:=algsubs(sin(l)^2+cos(l)^2=1,normal(M2E_derivs[l][ 3]/M2E_derivs[l][1])*Lx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#LzG,$* &-%$cosG6#%\"lG\"\"\"-%$sinGF)!\"\"!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 27 "The normalization factor is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "e_dy_norm:=simplify(sqrt(algsubs(sin(l)^2+cos(l)^2=1,algsubs( sin(lambda)^2+cos(lambda)^2=1,normal(Lx^2+Ly^2+Lz^2)))),symbolic);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*e_dy_normG*&\"\"\"F&-%$sinG6#%\"lG! \"\"" }}}{PARA 0 "" 0 "" {TEXT -1 116 "Up to a sign (which we may have lost by rescaling by removing a negative multiplicative factor above) , the vector is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "e_d_y_tmp :=array([Lx/e_dy_norm,Ly/e_dy_norm,Lz/e_dy_norm]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*e_d_y_tmpG-%'vectorG6#7%*&-%$cosG6#%'lambdaG\"\"\" -%$sinG6#%\"lGF.*&-F0F,F.F/\"\"\",$-F+F1!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 36 "We can check the sign by computing (" }{XPPEDIT 18 0 "e_d _x*X*e_d_y;" "6#*(%&e_d_xG\"\"\"%\"XGF%%&e_d_yGF%" }{TEXT -1 28 ") and seeing whether we get " }{XPPEDIT 18 0 "e_d_z;" "6#%&e_d_zG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "-e_d_z;" "6#,$%&e_d_zG!\"\"" }{TEXT -1 1 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "crossprod(e_d_x,e_d_y_t mp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$*&-%$cosG6#%\" lG\"\"\"-F*6#%'lambdaGF-!\"\",$*&F)\"\"\"-%$sinGF/F-F1,&*&)F5\"\"#F4-F 6F+F-F1*&)F.F:F4F;F4F1" }}}{PARA 0 "" 0 "" {TEXT -1 8 "This is " } {XPPEDIT 18 0 "-e_d_z;" "6#,$%&e_d_zG!\"\"" }{TEXT -1 5 ", so " } {XPPEDIT 18 0 "e_d_y;" "6#%&e_d_yG" }{TEXT -1 21 " is actually given b y" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "e_d_y:=array([-e_d_y_tm p[1],-e_d_y_tmp[2],-e_d_y_tmp[3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&e_d_yG-%'vectorG6#7%,$*&-%$cosG6#%'lambdaG\"\"\"-%$sinG6#%\"lGF/! \"\",$*&-F1F-F/F0\"\"\"F4-F,F2" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Location vector" }}{PARA 0 "" 0 "" {TEXT -1 20 "The location vecto r " }{XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" }{TEXT -1 39 " for a detect or is given simply by the " }{HYPERLNK 17 "WGS-84 to earth fixed coord inate transformation" 1 "" "WGS-84 to earth fixed" }{TEXT -1 1 "," }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Lambda:=[rhs(M2E_x),rhs(M2E_ y),rhs(M2E_z)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'LambdaG7%*(,&%\" RG\"\"\"%\"hGF)F)-%$cosG6#%\"lGF)-F,6#%'lambdaGF)*(F'\"\"\"F+F3-%$sinG F0F)*&,&*&*&)%\"bG\"\"#F3F(F)F3*$)%\"aG\"\"#F3!\"\"F)F*F)F)-F5F-F)" }} }{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "R;" "6#%\"RG" } {TEXT -1 10 " is given " }{HYPERLNK 17 "above" 1 "" "WGS-84 to earth f ixed" }{TEXT -1 21 ", and the parameters " }{XPPEDIT 18 0 "a;" "6#%\"a G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 16 " ar e also given " }{HYPERLNK 17 "above" 1 "" "WGS-84" }{TEXT -1 1 "." }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Orientation vectors" }}{PARA 0 " " 0 "" {TEXT -1 133 "For each of the two arms, we want to have unit or ientation vectors, which will determine the orientation of the arms in terms of the " }{HYPERLNK 17 "orientation angles" 1 "" "orientation a ngles" }{TEXT -1 48 ". In the detector coordinate frame, the angles \+ " }{XPPEDIT 18 0 "psi[1, 2];" "6#&%$psiG6$\"\"\"\"\"#" }{TEXT -1 72 " \+ play the role of azimuthal angles for arms 1 and 2 respectively, while " }{XPPEDIT 18 0 "omega[1,2];" "6#&%&omegaG6$\"\"\"\"\"#" }{TEXT -1 96 " are the polar complementary angles to the polar angle. Thus, it i s straightforward to see that:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "Omega_d[1]:=vector([cos(psi1)*cos(omega1),sin(psi1)*cos(omega1) ,sin(omega1)]);\nOmega_d[2]:=vector([cos(psi2)*cos(omega2),sin(psi2)*c os(omega2),sin(omega2)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(Omega _dG6#\"\"\"-%'vectorG6#7%*&-%$cosG6#%%psi1GF'-F.6#%'omega1GF'*&-%$sinG F/F'F1\"\"\"-F6F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(Omega_dG6#\" \"#-%'vectorG6#7%*&-%$cosG6#%%psi2G\"\"\"-F.6#%'omega2GF1*&-%$sinGF/F1 F2\"\"\"-F7F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "transpose( evalm(Omega_d[1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*transposeG6# -%'vectorG6#7%*&-%$cosG6#%%psi1G\"\"\"-F,6#%'omega1GF/*&-%$sinGF-F/F0 \"\"\"-F5F1" }}}{PARA 0 "" 0 "" {TEXT -1 66 "To transform these into t he earth fixed frame, we need to use the " }{HYPERLNK 17 "detector fra me to earth fixed frame transformation" 1 "" "detector frame to earth \+ fixed frame" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "for C1 from 1 to 2 do\n Omega_.(C1) := evalm(Omega_d[C1][1]*e_d _x+Omega_d[C1][2]*e_d_y+Omega_d[C1][3]*e_d_z)\nod;C1:='C1':" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Omega_1G-%'vectorG6#7%,(*(-%$cosG6#%%psi1 G\"\"\"-F,6#%'omega1GF/-%$sinG6#%'lambdaGF/!\"\"**-F4F-F/F0\"\"\"-F,F5 F/-F46#%\"lGF/F7*(-F4F1F/-F,F=F/F;F:F/,(*(F+F:F0F:F;F:F/**F9F:F0F:F3F: F%(Omega_2G-%'vectorG6#7%,(*(-%$cosG6#%%psi2G\"\"\"- F,6#%'omega2GF/-%$sinG6#%'lambdaGF/!\"\"**-F4F-F/F0\"\"\"-F,F5F/-F46#% \"lGF/F7*(-F4F1F/-F,F=F/F;F:F/,(*(F+F:F0F:F;F:F/**F9F:F0F:F3F:F " 0 "" {MPLTEXT 1 0 202 "Omega_1m:= convert(Omega_1,matrix): Omega_2m:=convert(Omega_2,matrix):\nOmega_1M: =transpose(Omega_1m): Omega_2M:=transpose(Omega_2m):\nresponse_tensor: =evalm((Omega_1m &* Omega_1M-Omega_2m &* Omega_2M)/2):" }}}}{SECT 1 {PARA 4 "" 0 "known detectors" {TEXT -1 52 "Location and Orientation v ectors for known detectors" }}{PARA 0 "" 0 "" {TEXT -1 35 "First, put \+ all vectors into a list." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "Det_vecs:=[Lambda,convert(Omega_1,list),convert(Omega_2,list),eval(re sponse_tensor)];\nDet_vecs:=subs(WGS84_params, subs(Rdef,Det_vecs)):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)Det_vecsG7&7%*(,&%\"RG\"\"\"%\"hG F*F*-%$cosG6#%\"lGF*-F-6#%'lambdaGF**(F(\"\"\"F,F4-%$sinGF1F**&,&*&*&) %\"bG\"\"#F4F)F*F4*$)%\"aG\"\"#F4!\"\"F*F+F*F*-F6F.F*7%,(*(-F-6#%%psi1 GF*-F-6#%'omega1GF*F5F4!\"\"**-F6FHF*FJF4F0F4FCF4FM*(-F6FKF*F,F4F0F4F* ,(*(FGF4FJF4F0F4F***FOF4FJF4F5F4FCF4FM*(FQF4F,F4F5F4F*,&*(FOF4FJF4F,F4 F**&FQF4FCF4F*7%,(*(-F-6#%%psi2GF*-F-6#%'omega2GF*F5F4FM**-F6FgnF*FinF 4F0F4FCF4FM*(-F6FjnF*F,F4F0F4F*,(*(FfnF4FinF4F0F4F***F]oF4FinF4F5F4FCF 4FM*(F_oF4F,F4F5F4F*,&*(F]oF4FinF4F,F4F**&F_oF4FCF4F*-%'matrixG6#7%7%, &*$)FEF=F4#F*F=*$)FZF=F4#FMF=,&*&FEF*FRF*F_p*&FZF*F`oF*Fbp,&*&FEF4FVF* F_p*&FZF4FdoF*Fbp7%Fcp,&*$)FRF=F4F_p*$)F`oF=F4Fbp,&*&FRF4FVF4F_p*&F`oF 4FdoF4Fbp7%FfpF_q,&*$)FVF=F4F_p*$)FdoF=F4Fbp" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Now the detectors:" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 51 "Reference to make sure we're getting it right (REF)" }}{PARA 0 "" 0 "" {TEXT -1 114 "Let's try a detector with arms due east and due sou th at 0 degrees longitude and 0 degrees latitude ... should be " } {XPPEDIT 18 0 "[[a, 0, 0], [0, 1, 0], [0, 0, 1]];" "6#7%7%%\"aG\"\"!F& 7%F&\"\"\"F&7%F&F&\"\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "params_REF:=[\n l = 0,\n lambda = 0, \n h = 0,\n psi1 = 0,\n psi2 = Pi/2,\n omega1 \+ = 0,\n omega2 = 0\n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+params _REFG7)/%\"lG\"\"!/%'lambdaGF(/%\"hGF(/%%psi1GF(/%%psi2G,$%#PiG#\"\"\" \"\"#/%'omega1GF(/%'omega2GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "DV_REF:=evalf(subs(params_REF,Det_vecs),12);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%'DV_REFG7&7%$\"--+q8yj!\"&\"\"!F*7%F*$\"\"\"F*F*7%F *$!-oo2Q.^!#BF,-%'matrixG6#7%7%F*F*F*7%F*$\"-+++++]!#7$\"-M%Q!p^DF17%F *F;$!-+++++]F:" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 30 "LIGO-Hanford \+ Observatory (LHO)" }}{PARA 0 "" 0 "" {TEXT -1 154 "Reference:\nWilliam Althouse, Larry Jones, Albert Lazzarini (1999)\n\"Determination of Gl obal and Local Coordinate Axes for the LIGO Sites\"\nLIGO-T980044-08-E " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "params_LHO:=[\n h \+ = 142.554,\n l = (46 + 27/60 + 18.528/3600)*Pi/180,\n \+ lambda = (240 + 35/60 + 32.4343/3600)*Pi/180,\n psi1 = 125.99 94*Pi/180,\n psi2 = 215.9994*Pi/180,\n omega1 = -6.195e-4,\n \+ omega2 = 1.25e-5 \n];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+param s_LHOG7)/%\"hG$\"'aD9!\"$/%\"lG,$%#PiG$\"+#[T3e#!#5/%'lambdaG,$F.$\"+F TiO8!\"*/%%psi1G,$F.$\"+nm'***pF1/%%psi2G,$F.$\"+nm***>\"F7/%'omega1G$ !%&>'!\"(/%'omega2G$\"$D\"FF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "DV_LHO:=evalf(subs(params_LHO,Det_vecs),12);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%'DV_LHOG7&7%$!-6E\\Th@!\"&$!-By^pMQF)$\"-AF-N+YF)7% $!-t:m#*QA!#7$\"-[viI)*zF1$\"-r\"y[!pbF17%$!-Pb=yR\"*F1$\"-K,RS4E!#8$! -k$F1$\"-yG$y*zAF17%FGFL$\"-&>H.!4tF;" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 33 "LIGO-Livingston Observatory (LLO)" }}{PARA 0 "" 0 "" {TEXT -1 154 "Reference:\nWilliam Althouse, Larry Jones, Albert Lazzar ini (1999)\n\"Determination of Global and Local Coordinate Axes for th e LIGO Sites\"\nLIGO-T980044-08-E" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "params_LLO:=[\n h = -6.574,\n l \+ = (30 + 33/60 + 46.4196/3600)*Pi/180,\n lambda = (269 + 13/60 + 3 2.7346/3600)*Pi/180,\n psi1 = 197.7165*Pi/180,\n psi2 = 287. 7165*Pi/180,\n omega1 = -3.121e-4,\n omega2 = -6.107e-4 \n];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+params_LLOG7)/%\"hG$!%ul!\"$/%\"l G,$%#PiG$\"+u&Qzp\"!#5/%'lambdaG,$F.$\"+k')p&\\\"!\"*/%%psi1G,$F.$\"++ ]U)4\"F7/%%psi2G,$F.$\"++]U)f\"F7/%'omega1G$!%@J!\"(/%'omega2G$!%2hFF " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "DV_LLO:=evalf(subs(para ms_LLO,Det_vecs),12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'DV_LLOG7&7 %$!-hW`gFu!\"($!-s>PG'\\&!\"&$\"-A " 0 "" {MPLTEXT 1 0 229 "params_VIRGO:=[ \n h = 51.884,\n l = (43 + 37 /60 + 53.0921/3600)*Pi/180,\n lambda = (10 + 30/60 + 16.1878/3600) *Pi/180,\n psi1 = 70.5674*Pi/180,\n psi2 = 160.5674*Pi/180,\n \+ omega1 = 0,\n omega2 = 0\n];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% -params_VIRGOG7)/%\"hG$\"&%)=&!\"$/%\"lG,$%#PiG$\"+ru'RU#!#5/%'lambdaG ,$F.$\"+]9$e$e!#6/%%psi1G,$F.$\"+66T?RF1/%%psi2G,$F.$\"+66T?*)F1/%'ome ga1G\"\"!/%'omega2GFD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DV _VIRGO:=evalf(subs(params_VIRGO,Det_vecs),12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)DV_VIRGOG7&7%$\"-j)4uja%!\"&$\"-nup*)H%)!\"'$\"-vipd yVF)7%$!-^[@e/q!#7$\"-?i[*[3#F2$\"-'pi;c#oF27%$!,^Pb#z`F2$!-#[0=3p*F2$ \"-zq^/3CF2-%'matrixG6#7%7%$\"-cf.uQCF2$!-=AzP3**!#8$!-Dp@wDBF27%FE$!- E*Re#yWF2$\"-vK5Ly=F27%FHFM$\"-oR!=&R?F2" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 31 "GEO-600 Interferometer (GEO600)" }}{PARA 0 "" 0 "" {TEXT -1 73 "Reference:\nhttp://www.geo600.uni-hannover.de/geo600/project/lo cation.html" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "params_GEO60 0:=[\n h = 114.425,\n l = (52 + 14/60 + 42.528/3600) *Pi/180,\n lambda = (9 + 48/60 + 25.894/3600)*Pi/180,\n psi1 = 21.6117*Pi/180,\n psi2 = 115.9431*Pi/180,\n omega1 = 0,\n ome ga2 = 0\n];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%.params_GEO600G7)/%\" hG$\"'DW6!\"$/%\"lG,$%#PiG$\"+[\"3D!H!#5/%'lambdaG,$F.$\"+K/W[a!#6/%%p si1G,$F.$\"+++l+7F1/%%psi2G,$F.$\"+LLGTkF1/%'omega1G\"\"!/%'omega2GFD " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "DV_GEO600:=evalf(subs(p arams_GEO600,Det_vecs),12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*DV_G EO600G7&7%$\"-`\\*4j&Q!\"&$\"-_j&*)fm'!\"'$\"-#pTT'>]F)7%$!-M!pnIX%!#7 $\"-,8a8l')F2$\"-IJ68bAF27%$!-)ynv0E'F2$!-X^4'=_&F2$\"-=\\s$e]&F2-%'ma trixG6#7%7%$!-!>)z\\#o*!#8$!-[XJ#yl$F2$\"-_fHP@7F27%FF$\"-$z;\"oHAF2$ \"-\"p@ur\\#F27%FHFM$!-wp8Vh7F2" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 33 "TAMA-300 Interferometer (TAMA300)" }}{PARA 0 "" 0 "" {TEXT -1 91 " Reference:\nMasa-Katsu Fujimoto (1995) (unpublished), E-mail: fujimoto @gravity.mtk.nao.ac.jp" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "p arams_TAMA300:=[\n h = 90,\n l = (35 + 40/60 + 35.6/ 3600)*Pi/180,\n lambda = (139 + 32/60 + 9.8/3600)*Pi/180,\n psi1 = 180*Pi/180,\n psi2 = 270*Pi/180,\n omega1 = 0,\n omega2 = 0\n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/params_TAMA300G7)/%\"hG\" #!*/%\"lG,$%#PiG$\"+k3.#)>!#5/%'lambdaG,$F,$\"+$3.?v(F//%%psi1GF,/%%ps i2G,$F,#\"\"$\"\"#/%'omega1G\"\"!/%'omega2GF?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DV_TAMA300:=evalf(subs(params_TAMA300,Det_vecs), 12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+DV_TAMA300G7&7%$!-r()*3k%R! \"&$\"-UK!fiO$F)$\"-*=p]\"*p$F)7%$\"-:hSp*['!#7$\"-$G/X\"3wF1$!-?R*p&z ;!#C7%$!-n(oPrV%F1$\"-J_r%[y$F1$!-LRBAB\")F1-%'matrixG6#7%7%$\"-T2pR@6 F1$\"-7#3@%3LF1$!-O*z#>-=F17%FE$\"-L^,%z<#F1$\"-$puds`\"F17%FGFL$!-req L*H$F1" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 33 "Caltech-40 Interferom eter (CIT40)" }}{PARA 0 "" 0 "" {TEXT -1 79 "Reference:\nB. Allen, \"G ravitational Wave Detector Sites,\" gr-qc/9607075 (1996)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "params_CIT40:=[ \n h = \+ 0,\n l = 34.17*Pi/180,\n lambda = -118.13*Pi/180,\n psi 1 = 270*Pi/180,\n psi2 = 0*Pi/180,\n omega1 = 0,\n omega2 = \+ 0\n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-params_CIT40G7)/%\"hG\"\"! /%\"lG,$%#PiG$\"+LLL)*=!#5/%'lambdaG,$F,$!+yxxilF//%%psi1G,$F,#\"\"$\" \"#/%%psi2GF(/%'omega1GF(/%'omega2GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DV_CIT40:=evalf(subs(params_CIT40,Det_vecs),12);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)DV_CIT40G7&7%$!-*Re\\1\\#!\"&$!-H#o *peYF)$\"-P8T1iNF)7%$!- " 0 "" {MPLTEXT 1 0 169 "para ms_MPQ30:=[\n h = 0,\n l = 35.57*Pi/180,\n lambda \+ = 139.47*Pi/180,\n psi1 = 132*Pi/180,\n psi2 = 225*Pi/180,\n omega1 = 0,\n omega2 = 0\n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %-params_MPQ30G7)/%\"hG\"\"!/%\"lG,$%#PiG$\"+666w>!#5/%'lambdaG,$F,$\" +LLL[xF//%%psi1G,$F,#\"#6\"#:/%%psi2G,$F,#\"\"&\"\"%/%'omega1GF(/%'ome ga2GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DV_MPQ30:=evalf(s ubs(params_MPQ30,Det_vecs),12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%) DV_MPQ30G7&7%$!-=jPqZR!\"&$\"-uRRBvLF)$\"-is!)[*o$F)7%$\"-JB9(Rj(!#7$ \"-.VUkwAF1$\"-Y+0yWgF17%$\"-`*)zyo9F1$\"-9+pVZ!)F1$!-o%3X;v&F1-%'matr ixG6#7%7%$\"-pw!4g!GF1$\"-X8#H*zF!#8$\"-1T9oHFF17%FD$!-i#e1*yHF1$\"-*3 y!R-IF17%FGFL$\",$f](*G " 0 "" {MPLTEXT 1 0 169 "params_TAMA20:=[\n h \+ = 0,\n l = 35.68*Pi/180,\n lambda = 139.54*Pi/180,\n \+ psi2 = 45*Pi/180,\n psi1 = 135*Pi/180,\n omega1 = 0,\n omeg a2 = 0\n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.params_TAMA20G7)/%\"h G\"\"!/%\"lG,$%#PiG$\"+AAA#)>!#5/%'lambdaG,$F,$\"+AAA_xF//%%psi2G,$F,# \"\"\"\"\"%/%%psi1G,$F,#\"\"$F:/%'omega1GF(/%'omega2GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "DV_TAMA20:=evalf(subs(params_TAMA20 ,Det_vecs),12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*DV_TAMA20G7&7%$! -U4bTYR!\"&$\"-f`_zlLF)$\"-'Gk3%*p$F)7%$\"-M:M^Ex!#7$\"-O3xy.FF1$\"-Du wtVdF17%$!-6%=f0X\"F1$!-_7gQc!)F1$\"-CuwtVdF1-%'matrixG6#7%7%$\"-a!QW( zGF1$\"-!*ROH-Y!#8$\"-X\")[`NEF17%FD$!-b!QW(zGF1$\"-6f2=!4$F17%FGFL\" \"!" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 25 "Glasgow-10 (1995) (G1095 )" }}{PARA 0 "" 0 "" {TEXT -1 79 "Reference:\nB. Allen, \"Gravitationa l Wave Detector Sites,\" gr-qc/9607075 (1996)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "params_G1095:=[\n h = 0,\n l \+ = 55.87*Pi/180,\n lambda = -4.28*Pi/180,\n psi1 = 152*Pi/180, \n psi2 = 242*Pi/180,\n omega1 = 0,\n omega2 = 0\n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-params_G1095G7)/%\"hG\"\"!/%\"lG,$%#PiG$ \"+*))))Q5$!#5/%'lambdaG,$F,$!+yxxxB!#6/%%psi1G,$F,#\"#Q\"#X/%%psi2G,$ F,#\"$@\"\"#!*/%'omega1GF(/%'omega2GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DV_G1095:=evalf(subs(params_G1095,Det_vecs),12);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)DV_G1095G7&7%$\"-&R@Iod$!\"&$!-8'Rw on#!\"'$\"-w7ZLc_F)7%$!-S(4TU`%!#7$!-#fQG[^)F2$\"-n#yvSj#F27%$\"-ZrR*z $pF2$!-J%fzqA&F2$!-9Og(R&\\F2-%'matrixG6#7%7%$!-%4!4#)y8F2$\"-$fd'oVPF 2$\"-JggN@6F27%FE$\"-#y1(**eAF2$!-r9l<;CF27%FGFL$!-()o;w,))!#8" }}}}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Response Functions and a Simple E xample" }}{PARA 0 "" 0 "" {TEXT -1 226 "We now have the information ne cessary in to calculate the beam pattern functions for a gravitational wave that arrives from a source at known declination and right ascens ion at a given GMST and with a given polarization angle " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 83 ". We begin by defining a procedur e that will return the required results as a list." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Definition of the \"Response\" procedure" }} {PARA 0 "" 0 "" {TEXT -1 101 "MAPLE Response() procedure. '#' denotes \+ the beginning of a comment. Comments end at carriage returns." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1863 "Response:= proc(\n dec, \+ # declination\n ra, # right ascension\n time, \+ # Greenwich mean sidereal time\n pol_angle, # polarization angle\n lat, # North latitude\n long, # East longitude\n h eight, # height above reference ellipsoid\n psi_arm1, # arm 1 \+ polarization angle\n psi_arm2, # arm 2 polarization angle\n omeg a_arm1, # arm 1 angle to horizontal\n omega_arm2) # arm 2 angle to h orizontal\n\n# specify the local and global variables. Note that delta , alpha etc. MUST be global.\n local parms,arm_vec1,arm_vec2,strain, F_p,F_c;\n global delta,alpha,GMST,psi,l,lambda,h,\n WGS84_ params,Rdef,sky_to_celestial,Omega,h_tensor,R,F_plus,F_cross;\n\n# cre ate a set of parameter values to with which to evaluate.\n parms:=\{ \n delta=dec,\n alpha=ra,\n GMST=time,\n psi=pol_a ngle,\n l=lat,\n lambda=long,\n h=height,\n psi1=p si_arm1,\n psi2=psi_arm2,\n omega1=omega_arm1,\n omega2 =omega_arm2\n \};\n\n# create the two arm vectors\n arm_vec1:=crea te([1], subs(WGS84_params, subs(Rdef, subs(parms, op(Omega_1)))));\n \+ arm_vec2:=create([1], subs(WGS84_params, subs(Rdef, subs(parms, op(Om ega_2)))));\n \n# dot the strain vectors into h_tensor to get the st rain in the detector\n strain:= get_compts( prod(h_tensor, \n \+ lin_com(1,prod(arm_vec1,arm_vec1),-1,prod(arm_vec2,arm_vec2)), [1,1] , [2,2]));\n \n# extract response functions, i.e. the coefficients o f the plus and cross polarizations \n F_p:=subs(parms,subs(sky_to_ce lestial,coeff(strain,h_plus)))/2;\n F_c:=subs(parms,subs(sky_to_cele stial,coeff(strain,h_cross)))/2;\n \n# make sure the labels we are u sing for the response functions are not assigned.\n F_plus:= 'F_plus ';\n F_cross:='F_cross';\n\n# return a list with the response functi ons \n RETURN([F_plus=simplify(F_p),F_cross=simplify(F_c)]);\nend: " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Response for reference dete ctor" }}{PARA 0 "" 0 "" {TEXT -1 118 "As a check, calculate the known \+ response functions for a refernce detector and source. The detector is at the equator " }{XPPEDIT 18 0 "l = 0;" "6#/%\"lG\"\"!" }{TEXT -1 20 " and prime meridian " }{XPPEDIT 18 0 "lambda = 0;" "6#/%'lambdaG\" \"!" }{TEXT -1 30 ", with arms running due North " }{XPPEDIT 18 0 "psi 1 = 0;" "6#/%%psi1G\"\"!" }{TEXT -1 10 " and East " }{XPPEDIT 18 0 "ps i2 = Pi/2;" "6#/%%psi2G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 29 ", on the reference ellipsoid " }{XPPEDIT 18 0 "h = 0;" "6#/%\"hG\"\"!" }{TEXT -1 30 ", and in the local horizontal " }{XPPEDIT 18 0 "[omega1 = 0, om ega2 = 0];" "6#7$/%'omega1G\"\"!/%'omega2GF&" }{TEXT -1 31 ". The sour ce is at declination " }{XPPEDIT 18 0 "delta = 0;" "6#/%&deltaG\"\"!" }{TEXT -1 21 " and right ascension " }{XPPEDIT 18 0 "alpha = 0;" "6#/% &alphaG\"\"!" }{TEXT -1 52 ", and the wave arrives at the reference de tector at " }{XPPEDIT 18 0 "GMST = 0;" "6#/%%GMSTG\"\"!" }{TEXT -1 29 " and with polarization angle " }{XPPEDIT 18 0 "psi = 0;" "6#/%$psiG\" \"!" }{TEXT -1 35 ". The response functions should be " }{XPPEDIT 18 0 "[F_plus = 1, F_cross = 0];" "6#7$/%'F_plusG\"\"\"/%(F_crossG\"\"!" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Response_R EF:=eval(subs(params_REF,Response(0,0,0,0,l,lambda,h,psi1,psi2,omega1, omega2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-Response_REFG7$/%'F_p lusG\"\"\"/%(F_crossG\"\"!" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 77 " Response Functions for LIGO Hanford (LHO) and LIGO Livingston (LLO) De tectors" }}{PARA 0 "" 0 "" {TEXT -1 78 "Response functions for Hanford and Livingston detectors for an generic source." }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 16 "Response for LHO" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Response_LHO:=simplify(subs(params_LHO,Response(delt a,alpha,GMST,psi,l,lambda,h,psi1,psi2,omega1,omega2)));" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%-Response_LHOG7$/%'F_plusG,L*(-%$cosG6#%$psiG \"\"\"-%$sinGF,F.-F06#%&deltaGF.$\"+jl`/J!#5*(-F06#,&%&alphaG$!\"\"\" \"!%%GMSTGF.F.-F+F9F.)-F+F2\"\"#\"\"\"$!+p#oAb\"F6*()F*FCFDF8FDF@FD$!+ 1J24iF6*(FBF.F1FDF@FD$\"+34yZ\\F6**FBFDF1FDF@FDFHFD$!+1=c&*)*F6*(FBFDF 8FDF1FD$\"+Zm&*fXF6*(FHFD)F@FCFDFAFD$!+LjFC9!\"**$FHFD$!+NjFC9FX*&FHFD FAFD$\"+q1oG\\F6**F/FDFBFDF*FDF8FD$\"+2=c&*)*F6**F/FDF1FDFUFDF*FD$!+-J 24iF6*&F8FDF@FD$\"+ol`/JF6*&FHFDFUFD$\"+kEb[GFX*,F*FDF8FDF/FDF1FDF@FD$ !+kEb[GFX*&FUFDFAFD$\"+\\;Q@rF6**FHFDF8FDF@FDFAFD$\"+ll`/JF6$\"+i;Q@rF 6F.**F/FDFBFDF*FDF@FD$!+4L\"*>\"*F6*$FUFDFV*$FAFD$!+[.MkCF6**FBFDF8FDF 1FDFHFDFap/%(F_crossG,F*&FBFDF8FD$!+34yZ\\F6*,F*FDF8FDF/FDF@FDFAFD$!+p l`/JF6*&F*FDF/FD$\"+MjFC9FX*&FBFDF@FD$\"+]m&*fXF6*(F*FDF/FDFAFD$!+y1oG \\F6F1FE*&FHFDF1FD$\"+kl`/JF6**FHFDF8FDF1FDF@FDFfo**F*FDF8FDF/FDF@FD$ \"+#4t!4iF6**F*FDF/FDFUFDFAFDFbq*(F1FDF@FDF8FDFbq*,F*FDFBFDF/FDF1FDF@F D$\"+1=c&*)*F6*(FHFDF1FDFUFD$!+(4t!4iF6*,F*FDFBFDF/FDF8FDF1FD$\"++L\"* >\"*F6*&F1FDFUFDF[r*(F*FDF/FDFUFDFfo*(FHFDFBFDF8FDFjn*(FHFDFBFDF@FD$!+ 5L\"*>\"*F6" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Response for LLO " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Response_LLO:=simplify( subs(params_LLO,Response(delta,alpha,GMST,psi,l,lambda,h,psi1,psi2,ome ga1,omega2)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-Response_LLOG7$/% 'F_plusG,L*()-%$cosG6#%$psiG\"\"#\"\"\")-F,6#,&%&alphaG$!\"\"\"\"!%%GM STG\"\"\"F/F0)-F,6#%&deltaGF/F0$\"+&4t0/\"!\"**(F+F:-%$sinGF-F:-FDF=F: $!+61T3c!#5*(FFH**F*F0FJF0F2F0F;F0FF*,F+F0FJF0FCF0FEF0F2F0$\"+&=Y63#FA**FCF0F_iksFH**FCF0F " 0 "" {MPLTEXT 1 0 68 "e_sky:=vector([cos(ph i)*sin(theta),sin(phi)*sin(theta),cos(theta)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&e_skyG-%'vectorG6#7%*&-%$sinG6#%&thetaG\"\"\"-%$cosG 6#%$phiGF.*&F*\"\"\"-F+F1F.-F0F," }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "The time delay from the origin to the detector" }}{PARA 0 "" 0 "" {TEXT -1 43 "First, calculate the required inner product" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "det_orig_dist:=simplify(subs(sky_t o_celestial,subs(WGS84_params,subs(Rdef,dotprod(e_sky,convert(Lambda,a rray),orthogonal)))));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%.det_orig_ distG*&,.**-%$cosG6#%&deltaG\"\"\"-F)6#,&%&alphaG$!\"\"\"\"!%%GMSTGF,F ,-F)6#%\"lGF,-F)6#%'lambdaGF,$\"+fJ1oS\"\"#*.F(\"\"\"F-F?F5F?F8F?%\"hG F,-%%sqrtG6#,&*$)F5F=F?$\")hJBFF3$\"+)**H3/%F3F,F?F,**F(F?-%$sinGF.F,F 5F?-FMF9F,$!+fJ1oSF=*.F(F?FLF?F5F?FNF?F@F?FAF?F1*&-FMF*F,-FMF6F,$FJF=* *FSF?FTF?F@F?FAF?F,F?*$-FB6#FDF?!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 360 "The GMST appearing in the above formula is actually the GMST at w hich the wave arrives at the detector. However, we lose only higher or der corrections by taking it to be the GMST at which the wave arrives \+ at the origin of the earth fixed coordinates. We therefore have that t he time difference between the arrival of a wave at the origin and at \+ the detector is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Delta_GMS T:= -simplify(evalf(det_orig_dist/(3*10^8)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+Delta_GMSTG,$*&,.**-%$cosG6#%&deltaG\"\"\"-F*6#,&%&a lphaG$!\"\"\"\"!%%GMSTGF-F--F*6#%\"lGF--F*6#%'lambdaGF-$\"+`5-c8\"#7*. F)\"\"\"F.F@F6F@F9F@%\"hGF--%%sqrtG6#,&*$)F6\"\"#F@$\")hJBFF4$\"+)**H3 /%F4F-F@$\"+LLLLLF4**F)F@-%$sinGF/F-F6F@-FQF:F-$!+`5-c8F>*.F)F@FPF@F6F @FRF@FAF@FBF@$!+LLLLLF4*&-FQF+F--FQF7F-$\"+LL%pM\"F>**FYF@FZF@FAF@FBF@ FMF@*$-FC6#FEF@!\"\"$!+++++5!#F" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 90 "An example: Time difference between LIGO Hanford (LHO) and LIGO Li vingston (LLO) detectors" }}{PARA 0 "" 0 "" {TEXT -1 143 "Using this r esult, we find that the time difference between a wave arriving at the Hanford and Livingston observatories for a generic source is" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "GMST[LHO-LLO]:=evalf(subs(pa rams_LHO,Delta_GMST)-subs(params_LLO,Delta_GMST));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%%GMSTG6#,&%$LHOG\"\"\"%$LLOG!\"\",(*&-%$cosG6#%&de ltaGF)-F/6#,&%&alphaG$F+\"\"!F%F)F)$\"+o&Hr&p!#7*&F.\"\"\"-%$sinGF3F)$ \"*YG'Qb!#6-F>F0$!*Qxpe%FA" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 " References" }}{PARA 0 "" 0 "Will and Wiseman" {TEXT -1 38 "Clifford M. Will and Alan G. Wiseman, " }{TEXT 264 15 "Physical Review" }{TEXT -1 1 " " }{TEXT 265 4 "D54," }{TEXT -1 18 " page 4813 (1996)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "T980044-08-E" {TEXT -1 49 "William Althouse, Larry Jones, Albert Lazzarini, " }{TEXT 266 68 " Determination of Global and Local Coordinate Axes for the LIGO Sites" }{TEXT -1 39 ", \nLIGO-T980044-08-E (1999)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "ABCF" {TEXT -1 85 "Warren G. Anders on, Patrick R. Brady, Jolien D. E. Creighton, and \311anna \311. Flana gan, " }{TEXT 267 85 "An excess power statistic for detection of burst sources of gravitational radiation, " }{TEXT -1 13 "gr-qc/0008066" }} }}{MARK "2 0" 56 }{VIEWOPTS 1 1 0 1 1 1803 }