diff options
| author | schnetter <schnetter@e296648e-0e4f-0410-bd07-d597d9acff87> | 2007-04-21 02:53:19 +0000 |
|---|---|---|
| committer | schnetter <schnetter@e296648e-0e4f-0410-bd07-d597d9acff87> | 2007-04-21 02:53:19 +0000 |
| commit | a29e3c43c4cd68052703cf6f37abdf4c6a5f921a (patch) | |
| tree | 5754134572da8f6191db4389b396604be1ac0018 | |
| parent | 0ab811fd67780bf002aca301e64dc34a9d0b65e0 (diff) | |
Add the capability to rotate initial data, similar to the existing
boost transformation.
Add spherical Kerr-Schild initial data, i.e., Kerr-Schild data in
which the horizon is a coordinate sphere.
Add the capability to smooth Kerr-Schild data with a parabolic term.
Re-indent param.ccl consistently.
git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/Exact/trunk@246 e296648e-0e4f-0410-bd07-d597d9acff87
247 files changed, 22001 insertions, 299 deletions
@@ -11,8 +11,8 @@ USES KEYWORD metric_type EXTENDS KEYWORD initial_data { - "exact" :: "Initial data from exact solution" - "slice" :: "Initial data from exact solution on arbitrary slice" + "exact" :: "Initial data from exact solution" + "slice" :: "Initial data from exact solution on arbitrary slice" } EXTENDS KEYWORD lapse_evolution_method @@ -26,11 +26,11 @@ EXTENDS KEYWORD shift_evolution_method EXTENDS KEYWORD initial_lapse { - "exact" :: "Initial lapse from exact solution" + "exact" :: "Initial lapse from exact solution" } EXTENDS KEYWORD initial_shift { - "exact" :: "Initial shift from exact solution" + "exact" :: "Initial shift from exact solution" } EXTENDS KEYWORD evolution_method @@ -79,18 +79,18 @@ private: REAL slice_Gauss_ampl "Amplitude of Gauss slice in exact" { -0.0: :: "Positive please" +0.0: :: "Positive please" } 0.0 REAL slice_Gauss_width "Width of Gauss slice in exact" { -0.0: :: "Positive please" +0.0: :: "Positive please" } 1.0 # this parameter is used directly in our schedule.ccl file KEYWORD overwrite_boundary "Overwrite g and K on the boundary" { - "no" :: "Do nothing" + "no" :: "Do nothing" "exact" :: "Use boundary data from an exact solution on a trivial slice" } "no" @@ -133,6 +133,25 @@ REAL shift_add_z "z component of added shift" } 0.0 ################################################################################ +##### parameters for rotating any model ######################################## +################################################################################ + +private: + +REAL rotation_euler_phi "Euler angle phi (first rotation, about z axis) (irrelevant for axisymmetric models)" +{ +*:* :: "any real number" +} 0.0 +REAL rotation_euler_theta "Euler angle theta (second rotation, about x axis)" +{ +*:* :: "any real number" +} 0.0 +REAL rotation_euler_psi "Euler angle psi (third rotation, about z axis)" +{ +*:* :: "any real number" +} 0.0 + +################################################################################ ##### general comments ######################################################### ################################################################################ @@ -209,38 +228,39 @@ KEYWORD exact_model "The exact solution/coordinates used in thorn exact" # # black hole spacetimes # -"Schwarzschild/EF" :: "Schwarzschild spacetime in Eddington-Finkelstein coordinates" -"Schwarzschild/PG" :: "Schwarzschild spacetime in Painleve-Gullstrand coordinates" -"Schwarzschild/BL" :: "Schwarzschild spacetime in Brill-Lindquist coordinates" -"Schwarzschild/Novikov":: "Schwarzschild spacetime in Novikov coordinates" -"Schwarzschild-Lemaitre":: "Schwarzschild metric in Schwarzschild coordinates, with cosmological constant" -"Kerr/Boyer-Lindquist" :: "Kerr spacetime in Boyer-Lindquist coordinates" -"Kerr/Kerr-Schild" :: "Kerr spacetime in Kerr-Schild coordinates" -"multi-BH" :: "Majumdar-Papapetrou or Kastor-Traschen maximally charged multi BH solutions" -"Alvi" :: "Alvi post-Newtonian 2BH spacetime (not fully implemented yet)" -"Thorne-fakebinary" :: "Thorne's fake-binary spacetime (non-Einstein)" +"Schwarzschild/EF" :: "Schwarzschild spacetime in Eddington-Finkelstein coordinates" +"Schwarzschild/PG" :: "Schwarzschild spacetime in Painleve-Gullstrand coordinates" +"Schwarzschild/BL" :: "Schwarzschild spacetime in Brill-Lindquist coordinates" +"Schwarzschild/Novikov" :: "Schwarzschild spacetime in Novikov coordinates" +"Schwarzschild-Lemaitre" :: "Schwarzschild metric in Schwarzschild coordinates, with cosmological constant" +"Kerr/Boyer-Lindquist" :: "Kerr spacetime in Boyer-Lindquist coordinates" +"Kerr/Kerr-Schild" :: "Kerr spacetime in Kerr-Schild coordinates" +"Kerr/Kerr-Schild/spherical" :: "Kerr spacetime in distorted Kerr-Schild coordinates such that the horizon is a coordinate sphere" +"multi-BH" :: "Majumdar-Papapetrou or Kastor-Traschen maximally charged multi BH solutions" +"Alvi" :: "Alvi post-Newtonian 2BH spacetime (not fully implemented yet)" +"Thorne-fakebinary" :: "Thorne's fake-binary spacetime (non-Einstein)" # # cosmological spacetimes # -"Lemaitre" :: "Lemaitre-type spacetime" -##"Robertson-Walker" :: "Robertson-Walker spacetime" -"de Sitter" :: "de Sitter spacetime (R-W cosmology, near t=0, p=0)" -"de Sitter+Lambda" :: "de Sitter spacetime with cosmological constant" -"anti-de Sitter+Lambda":: "anti-de Sitter spacetime with cosmological constant" -"Bianchi I" :: "approximate Bianchi type I spacetime" -"Goedel" :: "Goedel spacetime" -"Bertotti" :: "Bertotti spacetime" -"Kasner-like" :: "Kasner-like spacetime" -"Kasner-axisymmetric" :: "axisymmetric Kasner spacetime" -"Kasner-generalized" :: "generalized Kasner spacetime" -"Gowdy-wave" :: "Gowdy spacetime with polarized wave on a torus" -"Milne" :: "Milne spacetime for pre-big-bang cosmology" +"Lemaitre" :: "Lemaitre-type spacetime" +##"Robertson-Walker" :: "Robertson-Walker spacetime" +"de Sitter" :: "de Sitter spacetime (R-W cosmology, near t=0, p=0)" +"de Sitter+Lambda" :: "de Sitter spacetime with cosmological constant" +"anti-de Sitter+Lambda" :: "anti-de Sitter spacetime with cosmological constant" +"Bianchi I" :: "approximate Bianchi type I spacetime" +"Goedel" :: "Goedel spacetime" +"Bertotti" :: "Bertotti spacetime" +"Kasner-like" :: "Kasner-like spacetime" +"Kasner-axisymmetric" :: "axisymmetric Kasner spacetime" +"Kasner-generalized" :: "generalized Kasner spacetime" +"Gowdy-wave" :: "Gowdy spacetime with polarized wave on a torus" +"Milne" :: "Milne spacetime for pre-big-bang cosmology" # # miscelaneous spacetimes # -"boost-rotation symmetric":: "boost-rotation symmetric spacetime" -"bowl" :: "bowl (bag-of-gold) spacetime (non-Einstein)" -"constant density star":: "constant density (Schwarzschild) star" +"boost-rotation symmetric" :: "boost-rotation symmetric spacetime" +"bowl" :: "bowl (bag-of-gold) spacetime (non-Einstein)" +"constant density star" :: "constant density (Schwarzschild) star" } "Minkowski" ################################################################################ @@ -368,7 +388,7 @@ REAL Schwarzschild_EF__mass "Schwarzschild/EF: BH mass" REAL Schwarzschild_EF__epsilon "Schwarzschild/EF: numerical fudge" { -0.0:* :: "any real number >= 0.0" +0.0:* :: "any real number >= 0.0" } 1.e-16 ################################################################################ @@ -379,12 +399,12 @@ REAL Schwarzschild_EF__epsilon "Schwarzschild/EF: numerical fudge" REAL Schwarzschild_PG__mass "Schwarzschild/PG: BH mass" { -(0.0:* :: "any real number > 0.0" +(0.0:* :: "any real number > 0.0" } 1.0 REAL Schwarzschild_PG__epsilon "Schwarzschild/PG: numerical fudge" { -0.0:* :: "any real number >= 0.0" +0.0:* :: "any real number >= 0.0" } 1.e-16 ################################################################################ @@ -395,12 +415,12 @@ REAL Schwarzschild_PG__epsilon "Schwarzschild/PG: numerical fudge" REAL Schwarzschild_BL__mass "Schwarzschild/BL: BH mass" { -(0.0:* :: "any real number > 0.0" +(0.0:* :: "any real number > 0.0" } 1.0 REAL Schwarzschild_BL__epsilon "Schwarzschild/BL: numerical fudge" { -0.0:* :: "any real number >= 0.0" +0.0:* :: "any real number >= 0.0" } 1.e-16 ################################################################################ @@ -409,12 +429,12 @@ REAL Schwarzschild_BL__epsilon "Schwarzschild/BL: numerical fudge" REAL Schwarzschild_Novikov__mass "Schwarzschild/Novikov: BH mass" { -(0.0:* :: "any real number > 0.0" +(0.0:* :: "any real number > 0.0" } 1.0 REAL Schwarzschild_Novikov__epsilon "Schwarzschild/Novikov: numerical fudge" { -0.0:* :: "any real number >= 0.0" +0.0:* :: "any real number >= 0.0" } 1.e-16 ################################################################################ @@ -444,7 +464,7 @@ REAL Kerr_BoyerLindquist__spin \ REAL Kerr_BoyerLindquist__mass "Kerr/Boyer-Lindquist: BH mass" { -(0.0:* :: "any real number > 0" +(0.0:* :: "any real number > 0" } 1.0 ################################################################################ @@ -454,20 +474,29 @@ REAL Kerr_BoyerLindquist__mass "Kerr/Boyer-Lindquist: BH mass" # # the black hole is boosted in the z direction with this velocity, -# i.e. the black hole is located at x=0, y=0, and z=vt +# i.e., the black hole is located at x=0, y=0, and z=vt REAL Kerr_KerrSchild__boost_v "Kerr/Kerr-Schild: boost velocity of black hole in z direction" { -(-1:1) :: "any real number with absolute value < 1" +(-1:1) :: "any real number with absolute value < 1" } 0.0 REAL Kerr_KerrSchild__epsilon "Kerr/Kerr-Schild: numerical fudge" { -0.0:* :: "any real number >= 0.0" +0.0:* :: "any real number >= 0.0" } 1.e-16 +INT Kerr_KerrSchild__power "Kerr/Kerr-Schild: power (exponent) of numerical fudge" +{ +1:* :: "" +} 4 + +BOOLEAN Kerr_KerrSchild__parabolic "Kerr/Kerr-Schild: use a parabolic singularity-avoiding term" +{ +} "no" + REAL Kerr_KerrSchild__mass "Kerr/Kerr-Schild: BH mass" { -(0.0:* :: "any real number > 0" +(0.0:* :: "any real number > 0" } 1.0 REAL Kerr_KerrSchild__spin \ @@ -629,7 +658,7 @@ BOOLEAN Thorne_fakebinary__retarded "Thorne-fakebinary: use retarded time?" # FIXME: can this really be exactly 0.0? REAL Thorne_fakebinary__epsilon "Thorne-fakebinary: numerical fudge" { -0.0:* :: "any real number >= 0.0" +0.0:* :: "any real number >= 0.0" } 1.e-16 REAL Thorne_fakebinary__separation "Thorne-fakebinary: initial separation" @@ -719,7 +748,7 @@ REAL Lemaitre__R0 "Lemaitre: scale factor (radius) of the universe at time t=0" REAL de_Sitter__scale "de Sitter: multiplicative scale factor" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 0.1 ################################################################################ @@ -730,7 +759,7 @@ REAL de_Sitter__scale "de Sitter: multiplicative scale factor" REAL de_Sitter_Lambda__scale "de Sitter+Lambda: multiplicative scale factor" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 0.1 ################################################################################ @@ -742,7 +771,7 @@ REAL de_Sitter_Lambda__scale "de Sitter+Lambda: multiplicative scale factor" REAL anti_de_Sitter_Lambda__scale \ "anti-de Sitter+Lambda: multiplicative scale factor" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 0.1 ################################################################################ @@ -775,7 +804,7 @@ REAL Goedel__scale "Goedel: multiplicative scale factor" REAL Bertotti__Lambda "Bertotti: cosmological constant" { -*:* :: "any real number" +*:* :: "any real number" } -1.0 ################################################################################ @@ -843,19 +872,19 @@ REAL Gowdy_wave__amplitude "Gowdy-wave: amplitude parameter" REAL boost_rotation_symmetric__scale "boost-rotation symmetric: length scale" { -0.0: :: "Positive please" +0.0: :: "Positive please" } 1.0 REAL boost_rotation_symmetric__amp \ "boost-rotation symmetric: dimensionless amplitude" { -0.0: :: "Positive please" +0.0: :: "Positive please" } 0.1 REAL boost_rotation_symmetric__min_d \ "boost-rotation symmetric: dimensionless safety distance" { -(0.0: :: "any positive real number" +(0.0: :: "any positive real number" } 0.01 ################################################################################ @@ -878,44 +907,44 @@ BOOLEAN bowl__evolve "bowl: are we evolving the metric?" # FIXME: can this be negative? REAL bowl__strength "bowl: deformation strength" { -0.0:* :: "any real number >= 0" +0.0:* :: "any real number >= 0" } 0.5 REAL bowl__center "bowl: deformation center" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 2.5 # n.b. for bowl__shape = "Gaussian", this parameter is actually # sqrt(2) times the standard deviation of the Gaussian REAL bowl__sigma "bowl: width of deformation" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 1.0 REAL bowl__x_scale "bowl: scale for x coordinate" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 1.0 REAL bowl__y_scale "bowl: scale for y coordinate" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 1.0 REAL bowl__z_scale "bowl: scale for z coordinate" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 1.0 REAL bowl__t0 "bowl: center of Fermi step in time" { -*:* :: "any real number" +*:* :: "any real number" } 1.0 REAL bowl__sigma_t "bowl: width of Fermi step in time" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 1.0 ################################################################################ @@ -926,13 +955,13 @@ REAL bowl__sigma_t "bowl: width of Fermi step in time" REAL constant_density_star__mass "constant density star: mass of star" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 1.0 # FIXME: is this default value ok? (does it give a star or a BH?) -REAL constant_density_star__radius "constant density star: radius of star" +REAL constant_density_star__radius "constant density star: radius of star" { -(0.0:* :: "any positive real number" +(0.0:* :: "any positive real number" } 1.0 ################################################################################ diff --git a/schedule.ccl b/schedule.ccl index 09a7d32..59b4a50 100644 --- a/schedule.ccl +++ b/schedule.ccl @@ -66,7 +66,7 @@ if ( (CCTK_Equals(initial_data,"slice")) && ! (CCTK_Equals(evolution_method,"sli if ( (CCTK_Equals(initial_lapse,"exact")) || (CCTK_Equals(initial_shift,"exact")) ) { - schedule Exact__gauge in ADMBase_InitialData after Exact__decode_pars + schedule Exact__gauge in ADMBase_InitialGauge { LANG: Fortran } "Set initial lapse and/or shift from exact solution on a trivial slice" @@ -82,10 +82,17 @@ if ( (CCTK_Equals(lapse_evolution_method,"exact")) || (CCTK_Equals(shift_evoluti OPTIONS: global } "Register slicings" + # Necessary if MoL is not used schedule Exact__gauge at CCTK_PRESTEP { LANG: Fortran } "Set evolution lapse and/or shift from exact solution on a trivial slice" + + # Necessary if MoL is used + schedule Exact__gauge in MoL_PostStep + { + LANG: Fortran + } "Set evolution lapse and/or shift from exact solution on a trivial slice" } ####################################################################### diff --git a/src/boost.F77 b/src/boost.F77 index d81b100..dc26340 100644 --- a/src/boost.F77 +++ b/src/boost.F77 @@ -1,5 +1,7 @@ c This subroutine calculates the 4-metric and its inverse at an event, -c taking into account an optional Lorentz boost. +c taking into account an optional Lorentz boost and an optional rotation. +c The model is first rotated and then boosted, such that the boost is +c applied to the rotated model. c $Header$ c c The coordinates are @@ -9,6 +11,12 @@ c The 4-metrics are c Cgdd(a,b) = Cactus $g_{ab}$ Cguu(a,b) = Cactus $g^{ab}$ c Mgdd(a,b) = Model $g_{ab}$ Mguu(a,b) = Model $g^{ab}$ c +c For a definition of the Euler angles in the conventions used below, see +c http://mathworld.wolfram.com/EulerAngles.html +c Another useful resource may be +c http://en.wikipedia.org/wiki/Euler_angles +c although this uses (on 2006-11-29) different conventions. +c c This file is copyright (c) 2003 by Jonathan Thornburg <jthorn@aei.mpg.de>. c This file is covered by the GNU GPL license; see the files ../README c and ../COPYING for details. @@ -21,7 +29,7 @@ c #include "param_defs.inc" - subroutine Exact__metric( + subroutine Exact__metric( $ decoded_exact_model, $ x, y, z, t, $ gdtt, gdtx, gdty, gdtz, @@ -30,83 +38,94 @@ c $ guxx, guyy, guzz, guxy, guyz, guxz, $ psi) - implicit none - DECLARE_CCTK_FUNCTIONS - DECLARE_CCTK_PARAMETERS + implicit none + DECLARE_CCTK_FUNCTIONS + DECLARE_CCTK_PARAMETERS c input arguments - CCTK_INT decoded_exact_model - CCTK_REAL x, y, z, t + CCTK_INT decoded_exact_model + CCTK_REAL x, y, z, t c output arguments - CCTK_REAL gdtt, gdtx, gdty, gdtz, - $ gdxx, gdyy, gdzz, gdxy, gdyz, gdxz, - $ gutt, gutx, guty, gutz, - $ guxx, guyy, guzz, guxy, guyz, guxz, - $ psi + CCTK_REAL gdtt, gdtx, gdty, gdtz, + $ gdxx, gdyy, gdzz, gdxy, gdyz, gdxz, + $ gutt, gutx, guty, gutz, + $ guxx, guyy, guzz, guxy, guyz, guxz, + $ psi c intrinsic functions called - CCTK_REAL sqrt + CCTK_REAL sqrt c static local variables describing Lorentz transformation - logical firstcall - data firstcall /.true./ - CCTK_REAL gamma - CCTK_REAL vv(3), nn(3) - CCTK_REAL parallel(3,3), perp(3,3) - CCTK_REAL Cx_par(3), Cx_perp(3) - CCTK_REAL partial_Mx_wrt_Cx(0:3,0:3) - CCTK_REAL partial_Cx_wrt_Mx(0:3,0:3) - save firstcall - save gamma - save vv, nn - save parallel, perp - save Cx_par, Cx_perp - save partial_Mx_wrt_Cx - save partial_Cx_wrt_Mx + logical firstcall + data firstcall /.true./ + CCTK_REAL gamma + CCTK_REAL vv(3), nn(3) + CCTK_REAL parallel(3,3), perp(3,3) + CCTK_REAL Cx_par(3), Cx_perp(3) + CCTK_REAL partial_Mx_wrt_Cx(0:3,0:3) + CCTK_REAL partial_Cx_wrt_Mx(0:3,0:3) + CCTK_REAL R(0:3,0:3) + save firstcall + save gamma + save vv, nn + save parallel, perp + save Cx_par, Cx_perp + save partial_Mx_wrt_Cx + save partial_Cx_wrt_Mx + save R c coordinates and 4-metric - CCTK_REAL Ct, Cx(3) - CCTK_REAL Cgdd(0:3,0:3), Cguu(0:3,0:3) - CCTK_REAL Mt, Mx(3) - CCTK_REAL Mgdd(0:3,0:3), Mguu(0:3,0:3) + CCTK_REAL Cx(0:3) + CCTK_REAL Cgdd(0:3,0:3), Cguu(0:3,0:3) + CCTK_REAL Mx(0:3) + CCTK_REAL Mgdd(0:3,0:3), Mguu(0:3,0:3) + CCTK_REAL Nx(0:3) + CCTK_REAL Ngdd(0:3,0:3), Nguu(0:3,0:3) c misc temps - CCTK_REAL vnorm, vnormsq - CCTK_REAL delta_ij - CCTK_REAL Cx_par_i, Cx_perp_i - CCTK_REAL vdotCx - CCTK_REAL Cgdd_ab, Cguu_ab - character*100 warn_buffer + CCTK_REAL vnorm, vnormsq + CCTK_REAL delta_ij + CCTK_REAL Cx_par_i, Cx_perp_i + CCTK_REAL vdotCx + CCTK_REAL Cgdd_ab, Cguu_ab + CCTK_REAL cos_phi, sin_phi + CCTK_REAL cos_theta, sin_theta + CCTK_REAL cos_psi, sin_psi + CCTK_REAL R_phi(0:3,0:3), R_theta(0:3,0:3), R_psi(0:3,0:3) + character*1000 warn_buffer c flags, array indices, etc - logical Tmunu_flag - integer i, j - integer Ca, Cb, MA, MB + logical Tmunu_flag + integer i, j, k, l + integer Ca, Cb, MA, MB c constants - integer n - parameter (n = 3) + integer n + parameter (n = 3) cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c -c optimized fast-path if no Lorentz boost +c optimized fast-path if no Lorentz boost and no rotation c - if ( (boost_vx .eq. 0.0) - $ .and. (boost_vy .eq. 0.0) - $ .and. (boost_vz .eq. 0.0) ) then - call Exact__metric_for_model( - $ decoded_exact_model, - $ x, y, z, t, - $ gdtt, gdtx, gdty, gdtz, - $ gdxx, gdyy, gdzz, gdxy, gdyz, gdxz, - $ gutt, gutx, guty, gutz, - $ guxx, guyy, guzz, guxy, guyz, guxz, - $ psi, - $ Tmunu_flag) - return - endif + if ( (boost_vx .eq. 0.0) + $ .and. (boost_vy .eq. 0.0) + $ .and. (boost_vz .eq. 0.0) + $ .and. (rotation_euler_phi .eq. 0.0) + $ .and. (rotation_euler_theta .eq. 0.0) + $ .and. (rotation_euler_psi .eq. 0.0)) then + call Exact__metric_for_model( + $ decoded_exact_model, + $ x, y, z, t, + $ gdtt, gdtx, gdty, gdtz, + $ gdxx, gdyy, gdzz, gdxy, gdyz, gdxz, + $ gutt, gutx, guty, gutz, + $ guxx, guyy, guzz, guxy, guyz, guxz, + $ psi, + $ Tmunu_flag) + return + end if cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc @@ -125,203 +144,321 @@ cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c compute Lorentz transformation information on first call c - if (firstcall) then - firstcall = .false. + if (firstcall) then + firstcall = .false. c boost velocity - vv(1) = boost_vx - vv(2) = boost_vy - vv(3) = boost_vz + vv(1) = boost_vx + vv(2) = boost_vy + vv(3) = boost_vz c Lorentz gamma factor, unit vector in direction of boost velocity - vnormsq = 0.0d0 - do 100 i = 1,n - vnormsq = vnormsq + vv(i)*vv(i) -100 continue - gamma = 1.0 / sqrt(1.0 - vnormsq) - vnorm = sqrt(vnormsq) - do 110 i = 1,n - nn(i) = vv(i) / vnorm -110 continue + vnormsq = 0.0 + do i = 1,n + vnormsq = vnormsq + vv(i)*vv(i) + end do + gamma = 1.0 / sqrt(1.0 - vnormsq) + vnorm = sqrt(vnormsq) + if ( (boost_vx .eq. 0.0) + $ .and. (boost_vy .eq. 0.0) + $ .and. (boost_vz .eq. 0.0)) then + nn(1) = 1 + nn(2) = 0 + nn(3) = 0 + else + do i = 1,n + nn(i) = vv(i) / vnorm + end do + end if c projection operators parallel(*,*) and perp(*,*) - do 210 j = 1,n - do 200 i = 1,n - parallel(i,j) = nn(i) * nn(j) - if (i .eq. j) then - delta_ij = 1.0d0 - else - delta_ij = 0.0d0 - endif - perp(i,j) = delta_ij - parallel(i,j) -200 continue -210 continue + do j = 1,n + do i = 1,n + parallel(i,j) = nn(i) * nn(j) + if (i .eq. j) then + delta_ij = 1.0 + else + delta_ij = 0.0 + end if + perp(i,j) = delta_ij - parallel(i,j) + end do + end do c partial derivatives of Model coordinates with respect to Cactus coordinates - partial_Mx_wrt_Cx(0,0) = gamma - do 300 i = 1,n - partial_Mx_wrt_Cx(0,i) = -gamma*vv(i) -300 continue - do 320 i = 1,n - partial_Mx_wrt_Cx(i,0) = -gamma*vv(i) - do 310 j=1,n - partial_Mx_wrt_Cx(i,j) - $ = gamma*parallel(i,j) - $ + perp(i,j) -310 continue -320 continue + partial_Mx_wrt_Cx(0,0) = gamma + do i = 1,n + partial_Mx_wrt_Cx(0,i) = -gamma*vv(i) + end do + do i = 1,n + partial_Mx_wrt_Cx(i,0) = -gamma*vv(i) + do j=1,n + partial_Mx_wrt_Cx(i,j) = gamma*parallel(i,j) + perp(i,j) + end do + end do c partial derivatives of Cactus coordinates with respect to Model coordinates - partial_Cx_wrt_Mx(0,0) = gamma - do 400 i = 1,n - partial_Cx_wrt_Mx(0,i) = + gamma*vv(i) -400 continue - do 420 i = 1,n - partial_Cx_wrt_Mx(i,0) = + gamma*vv(i) - do 410 j=1,n - partial_Cx_wrt_Mx(i,j) - $ = gamma*parallel(i,j) - $ + perp(i,j) -410 continue -420 continue - - endif + partial_Cx_wrt_Mx(0,0) = gamma + do i = 1,n + partial_Cx_wrt_Mx(0,i) = + gamma*vv(i) + end do + do i = 1,n + partial_Cx_wrt_Mx(i,0) = + gamma*vv(i) + do j=1,n + partial_Cx_wrt_Mx(i,j) = gamma*parallel(i,j) + perp(i,j) + end do + end do + +c Sines and cosines of rotation angles + cos_phi = cos (rotation_euler_phi) + sin_phi = sin (rotation_euler_phi) + cos_theta = cos (rotation_euler_theta) + sin_theta = sin (rotation_euler_theta) + cos_psi = cos (rotation_euler_psi) + sin_psi = sin (rotation_euler_psi) + +c Set up individual rotation matrices + R_phi(0,0) = 1 + R_phi(0,1) = 0 + R_phi(0,2) = 0 + R_phi(0,3) = 0 + R_phi(1,0) = 0 + R_phi(1,1) = + cos_phi + R_phi(1,2) = + sin_phi + R_phi(1,3) = 0 + R_phi(2,0) = 0 + R_phi(2,1) = - sin_phi + R_phi(2,2) = + cos_phi + R_phi(2,3) = 0 + R_phi(3,0) = 0 + R_phi(3,1) = 0 + R_phi(3,2) = 0 + R_phi(3,3) = 1 + + R_theta(0,0) = 1 + R_theta(0,1) = 0 + R_theta(0,2) = 0 + R_theta(0,3) = 0 + R_theta(1,0) = 0 + R_theta(1,1) = 1 + R_theta(1,2) = 0 + R_theta(1,3) = 0 + R_theta(2,0) = 0 + R_theta(2,1) = 0 + R_theta(2,2) = + cos_theta + R_theta(2,3) = + sin_theta + R_theta(3,0) = 0 + R_theta(3,1) = 0 + R_theta(3,2) = - sin_theta + R_theta(3,3) = + cos_theta + + R_psi(0,0) = 1 + R_psi(0,1) = 0 + R_psi(0,2) = 0 + R_psi(0,3) = 0 + R_psi(1,0) = 0 + R_psi(1,1) = + cos_psi + R_psi(1,2) = + sin_psi + R_psi(1,3) = 0 + R_psi(2,0) = 0 + R_psi(2,1) = - sin_psi + R_psi(2,2) = + cos_psi + R_psi(2,3) = 0 + R_psi(3,0) = 0 + R_psi(3,1) = 0 + R_psi(3,2) = 0 + R_psi(3,3) = 1 + +c Combine individual rotation matrices + do i = 0,n + do j = 0,n + R(i,j) = 0 + do k = 0,n + do l = 0,n + R(i,j) = R(i,j) + R_psi(i,k) * R_theta(k,l) * R_phi(l,j) + end do + end do + end do + end do + +c Notes that help me (Erik Schnetter) think: +c This considers a rotation with phi=0, theta=pi/2, psi=0. +c Nx(i) = Nx(i) + R(j,i) * Mx(j) +c Nx(1) = - Mx(3) +c Nx(3) = Mx(1) +c Mgxx(1,1) = Ngxx(3,3) +c Mgxx(1,3) = - Ngxx(1,3) +c Mgxx(3,3) = Ngxx(1,1) +c Mgxx(i,j) = R(i,k) R(j,l) Ngxx(k,l) [correct] +c Mgxx(i,j) = R(k,i) R(l,j) Ngxx(k,l) [correct] +c Mbetax(1) = - Nbetax(3) +c Mbetax(3) = Nbetax(1) +c Mbetax(i) + R(i,j) Nbetax(j) [wrong] +c Mbetax(i) + R(j,i) Nbetax(j) [correct] + + end if cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c compute flat-space components of Cx(*) parallel and perpendicular to vv(*) c - Ct = t - Cx(1) = x - Cx(2) = y - Cx(3) = z - - do 530 i=1,n - Cx_par_i = 0.0d0 - Cx_perp_i = 0.0d0 - do 520 j=1,n - Cx_par_i = Cx_par_i - $ + parallel(i,j)*Cx(j) - Cx_perp_i = Cx_perp_i - $ + perp (i,j)*Cx(j) -520 continue - Cx_par (i) = Cx_par_i - Cx_perp(i) = Cx_perp_i -530 continue + Cx(0) = t + Cx(1) = x + Cx(2) = y + Cx(3) = z + + do i=1,n + Cx_par_i = 0.0 + Cx_perp_i = 0.0 + do j=1,n + Cx_par_i = Cx_par_i + parallel(i,j)*Cx(j) + Cx_perp_i = Cx_perp_i + perp (i,j)*Cx(j) + end do + Cx_par (i) = Cx_par_i + Cx_perp(i) = Cx_perp_i + end do c -c Lorentz-transform the Cactus coordinate to get the Model coordinates +c Lorentz-transform and rotate the Cactus coordinate +c to get the Model coordinates c - vdotCx = 0.0 - do 600 i = 1,n - vdotCx = vdotCx + vv(i)*Cx(i) -600 continue - Mt = gamma * (Ct - vdotCx) - do 610 i=1,n - Mx(i) = gamma * (Cx_par(i) - vv(i)*Ct) - $ + Cx_perp(i) -610 continue +c Boost + vdotCx = 0.0 + do i = 1,n + vdotCx = vdotCx + vv(i)*Cx(i) + end do + + Mx(0) = gamma * (Cx(0) - vdotCx) + do i=1,n + Mx(i) = gamma * (Cx_par(i) - vv(i)*Cx(0)) + Cx_perp(i) + end do + +c Rotation + do i=0,n + Nx(i) = 0 + do j = 0,n + Nx(i) = Nx(i) + R(i,j) * Mx(j) + end do + end do + c c compute the Model 4-metric and inverse 4-metric at the Model coordinates c - call Exact__metric_for_model( - $ decoded_exact_model, - $ Mx(1), Mx(2), Mx(3), Mt, - $ Mgdd(0,0), Mgdd(0,1), Mgdd(0,2), Mgdd(0,3), - $ Mgdd(1,1), Mgdd(2,2), Mgdd(3,3), - $ Mgdd(1,2), Mgdd(2,3), Mgdd(1,3), - $ Mguu(0,0), Mguu(0,1), Mguu(0,2), Mguu(0,3), - $ Mguu(1,1), Mguu(2,2), Mguu(3,3), - $ Mguu(1,2), Mguu(2,3), Mguu(1,3), - $ psi, Tmunu_flag) - - if (Tmunu_flag) then - write (warn_buffer, '(a,i8,a,a)') - $ 'exact_model = ', decoded_exact_model, - $ 'sets the stress-energy tensor', - $ ' ==> we cannot Lorentz-boost it! :(' - call CCTK_WARN(0, warn_buffer) - endif + call Exact__metric_for_model( + $ decoded_exact_model, + $ Nx(1), Nx(2), Nx(3), Nx(0), + $ Ngdd(0,0), Ngdd(0,1), Ngdd(0,2), Ngdd(0,3), + $ Ngdd(1,1), Ngdd(2,2), Ngdd(3,3), + $ Ngdd(1,2), Ngdd(2,3), Ngdd(1,3), + $ Nguu(0,0), Nguu(0,1), Nguu(0,2), Nguu(0,3), + $ Nguu(1,1), Nguu(2,2), Nguu(3,3), + $ Nguu(1,2), Nguu(2,3), Nguu(1,3), + $ psi, Tmunu_flag) + + if (Tmunu_flag) then + write (warn_buffer, '(a,i8,a,a)') + $ 'exact_model = ', decoded_exact_model, + $ 'sets the stress-energy tensor', + $ ' ==> we cannot Lorentz-boost or rotate it!' + call CCTK_WARN(0, warn_buffer) + end if c c symmetrize the Model 4-metric and inverse 4-metric arrays c (the Exact__metric_for_model() call only set the upper triangles) c - Mgdd(1,0) = Mgdd(0,1) - Mgdd(2,0) = Mgdd(0,2) - Mgdd(2,1) = Mgdd(1,2) - Mgdd(3,0) = Mgdd(0,3) - Mgdd(3,1) = Mgdd(1,3) - Mgdd(3,2) = Mgdd(2,3) - - Mguu(1,0) = Mguu(0,1) - Mguu(2,0) = Mguu(0,2) - Mguu(2,1) = Mguu(1,2) - Mguu(3,0) = Mguu(0,3) - Mguu(3,1) = Mguu(1,3) - Mguu(3,2) = Mguu(2,3) + Ngdd(1,0) = Ngdd(0,1) + Ngdd(2,0) = Ngdd(0,2) + Ngdd(2,1) = Ngdd(1,2) + Ngdd(3,0) = Ngdd(0,3) + Ngdd(3,1) = Ngdd(1,3) + Ngdd(3,2) = Ngdd(2,3) + + Nguu(1,0) = Nguu(0,1) + Nguu(2,0) = Nguu(0,2) + Nguu(2,1) = Nguu(1,2) + Nguu(3,0) = Nguu(0,3) + Nguu(3,1) = Nguu(1,3) + Nguu(3,2) = Nguu(2,3) c c tensor-transorm (the upper triangle of) the 4-metric and inverse 4-metric c - do 730 Ca = 0,n - do 720 Cb = Ca,n - Cgdd_ab = 0.0d0 - do 710 Ma = 0,n - do 700 Mb = 0,n - Cgdd_ab = Cgdd_ab - $ + Mgdd(Ma,Mb) - $ * partial_Mx_wrt_Cx(Ma,Ca) - $ * partial_Mx_wrt_Cx(Mb,Cb) -700 continue -710 continue - Cgdd(Ca,Cb) = Cgdd_ab -720 continue -730 continue - - do 830 Ca = 0,n - do 820 Cb = Ca,n - Cguu_ab = 0.0d0 - do 810 Ma = 0,n - do 800 Mb = 0,n - Cguu_ab = Cguu_ab - $ + Mguu(Ma,Mb) - $ * partial_Cx_wrt_Mx(Ca,Ma) - $ * partial_Cx_wrt_Mx(Cb,Mb) -800 continue -810 continue - Cguu(Ca,Cb) = Cguu_ab -820 continue -830 continue + +c Rotations + do i = 0,n + do j = 0,n + Mgdd(i,j) = 0 + Mguu(i,j) = 0 + do k = 0,n + do l = 0,n + Mgdd(i,j) = Mgdd(i,j) + R(k,i) * R(l,j) * Ngdd(k,l) +c The inverse of R is also its transpose. That means that the +c transpose of the inverse, which you would use for g^ij, is just R +c again. + Mguu(i,j) = Mguu(i,j) + R(k,i) * R(l,j) * Nguu(k,l) + end do + end do + end do + end do + +c Boost + do Ca = 0,n + do Cb = Ca,n + Cgdd_ab = 0.0 + do Ma = 0,n + do Mb = 0,n + Cgdd_ab = Cgdd_ab + $ + Mgdd(Ma,Mb) + $ * partial_Mx_wrt_Cx(Ma,Ca) + $ * partial_Mx_wrt_Cx(Mb,Cb) + end do + end do + Cgdd(Ca,Cb) = Cgdd_ab + end do + end do + + do Ca = 0,n + do Cb = Ca,n + Cguu_ab = 0.0 + do Ma = 0,n + do Mb = 0,n + Cguu_ab = Cguu_ab + $ + Mguu(Ma,Mb) + $ * partial_Cx_wrt_Mx(Ca,Ma) + $ * partial_Cx_wrt_Mx(Cb,Mb) + end do + end do + Cguu(Ca,Cb) = Cguu_ab + end do + end do c c unpack the Cactus-coordinates 4-metric and inverse 4-metric c into the corresponding output arguments c - gdtt = Cgdd(0,0) - gdtx = Cgdd(0,1) - gdty = Cgdd(0,2) - gdtz = Cgdd(0,3) - gdxx = Cgdd(1,1) - gdxy = Cgdd(1,2) - gdxz = Cgdd(1,3) - gdyy = Cgdd(2,2) - gdyz = Cgdd(2,3) - gdzz = Cgdd(3,3) - - gutt = Cguu(0,0) - gutx = Cguu(0,1) - guty = Cguu(0,2) - gutz = Cguu(0,3) - guxx = Cguu(1,1) - guxy = Cguu(1,2) - guxz = Cguu(1,3) - guyy = Cguu(2,2) - guyz = Cguu(2,3) - guzz = Cguu(3,3) - - return - end + gdtt = Cgdd(0,0) + gdtx = Cgdd(0,1) + gdty = Cgdd(0,2) + gdtz = Cgdd(0,3) + gdxx = Cgdd(1,1) + gdxy = Cgdd(1,2) + gdxz = Cgdd(1,3) + gdyy = Cgdd(2,2) + gdyz = Cgdd(2,3) + gdzz = Cgdd(3,3) + + gutt = Cguu(0,0) + gutx = Cguu(0,1) + guty = Cguu(0,2) + gutz = Cguu(0,3) + guxx = Cguu(1,1) + guxy = Cguu(1,2) + guxz = Cguu(1,3) + guyy = Cguu(2,2) + guyz = Cguu(2,3) + guzz = Cguu(3,3) + + end diff --git a/src/decode_pars.F77 b/src/decode_pars.F77 index 1b7da46..d51bb4a 100644 --- a/src/decode_pars.F77 +++ b/src/decode_pars.F77 @@ -76,6 +76,8 @@ c black hole spacetimes decoded_exact_model = EXACT__Kerr_BoyerLindquist elseif (CCTK_Equals(exact_model, "Kerr/Kerr-Schild") .ne. 0) then decoded_exact_model = EXACT__Kerr_KerrSchild + elseif (CCTK_Equals(exact_model, "Kerr/Kerr-Schild/spherical") .ne. 0) then + decoded_exact_model = EXACT__Kerr_KerrSchild_spherical elseif (CCTK_Equals(exact_model, "Schwarzschild-Lemaitre") .ne. 0) then decoded_exact_model = EXACT__Schwarzschild_Lemaitre elseif (CCTK_Equals(exact_model, "multi-BH") .ne. 0) then diff --git a/src/gauge.F77 b/src/gauge.F77 index 56088fb..b10bfa1 100644 --- a/src/gauge.F77 +++ b/src/gauge.F77 @@ -19,6 +19,7 @@ C $Header$ integer i,j,k integer nx,ny,nz + CCTK_REAL tt, xx, yy, zz CCTK_REAL gxxjunk, gyyjunk, gzzjunk, $ gxyjunk, gyzjunk, gxzjunk, $ hxxjunk, hyyjunk, hzzjunk, @@ -84,6 +85,11 @@ C do j=1,ny do i=1,nx + tt = cctk_time + xx = x(i,j,k) - cctk_time * shift_add_x + yy = y(i,j,k) - cctk_time * shift_add_y + zz = z(i,j,k) - cctk_time * shift_add_z + C Initialize the psi of exact C (also to tell the models about the conformal_state) if (conformal_state .ne. 0) then @@ -103,7 +109,7 @@ C (also to tell the models about the conformal_state) call Exact__Bona_Masso_data( $ decoded_exact_model, - $ x(i,j,k), y(i,j,k), z(i,j,k), cctk_time, + $ xx, yy, zz, tt, $ gxxjunk, gyyjunk, gzzjunk, $ gxyjunk, gyzjunk, gxzjunk, $ hxxjunk, hyyjunk, hzzjunk, @@ -147,6 +153,11 @@ C do j=1,ny do i=1,nx + tt = cctk_time + xx = x(i,j,k) - cctk_time * shift_add_x + yy = y(i,j,k) - cctk_time * shift_add_y + zz = z(i,j,k) - cctk_time * shift_add_z + C Initialize the psi of exact C (also to tell the models about the conformal_state) if (conformal_state .ne. 0) then @@ -166,7 +177,7 @@ C (also to tell the models about the conformal_state) call Exact__Bona_Masso_data( $ decoded_exact_model, - $ x(i,j,k), y(i,j,k), z(i,j,k), cctk_time, + $ xx, yy, zz, tt, $ gxxjunk, gyyjunk, gzzjunk, $ gxyjunk, gyzjunk, gxzjunk, $ hxxjunk, hyyjunk, hzzjunk, @@ -208,6 +219,11 @@ C do j=1,ny do i=1,nx + tt = cctk_time + xx = x(i,j,k) - cctk_time * shift_add_x + yy = y(i,j,k) - cctk_time * shift_add_y + zz = z(i,j,k) - cctk_time * shift_add_z + C Initialize the psi of exact C (also to tell the models about the conformal_state) if (conformal_state .ne. 0) then @@ -227,7 +243,7 @@ C (also to tell the models about the conformal_state) call Exact__Bona_Masso_data( $ decoded_exact_model, - $ x(i,j,k), y(i,j,k), z(i,j,k), cctk_time, + $ xx, yy, zz, tt, $ gxxjunk, gyyjunk, gzzjunk, $ gxyjunk, gyzjunk, gxzjunk, $ hxxjunk, hyyjunk, hzzjunk, diff --git a/src/include/Scalar_CalcTmunu.inc b/src/include/Scalar_CalcTmunu.inc index 346fe09..e4ae8da 100644 --- a/src/include/Scalar_CalcTmunu.inc +++ b/src/include/Scalar_CalcTmunu.inc @@ -61,6 +61,8 @@ c no stress-energy tensor in this model c no stress-energy tensor in this model elseif (decoded_exact_model .eq. EXACT__Kerr_KerrSchild) then c no stress-energy tensor in this model + elseif (decoded_exact_model .eq. EXACT__Kerr_KerrSchild_spherical) then +c no stress-energy tensor in this model cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc diff --git a/src/include/param_defs.inc b/src/include/param_defs.inc index 10bbefa..d2374cb 100644 --- a/src/include/param_defs.inc +++ b/src/include/param_defs.inc @@ -38,16 +38,17 @@ c Minkowski spacetime #define EXACT__Minkowski_conf_wave 6 c black hole spacetimes -#define EXACT__Schwarzschild_EF 10 -#define EXACT__Schwarzschild_PG 11 -#define EXACT__Schwarzschild_BL 12 -#define EXACT__Schwarzschild_Novikov 13 -#define EXACT__Kerr_BoyerLindquist 14 -#define EXACT__Kerr_KerrSchild 15 -#define EXACT__Schwarzschild_Lemaitre 16 -#define EXACT__multi_BH 17 -#define EXACT__Alvi 18 -#define EXACT__Thorne_fakebinary 19 +#define EXACT__Schwarzschild_EF 10 +#define EXACT__Schwarzschild_PG 11 +#define EXACT__Schwarzschild_BL 12 +#define EXACT__Schwarzschild_Novikov 13 +#define EXACT__Kerr_BoyerLindquist 14 +#define EXACT__Kerr_KerrSchild 15 +#define EXACT__Kerr_KerrSchild_spherical 16 +#define EXACT__Schwarzschild_Lemaitre 17 +#define EXACT__multi_BH 18 +#define EXACT__Alvi 19 +#define EXACT__Thorne_fakebinary 20 c cosmological spacetimes #define EXACT__Lemaitre 50 diff --git a/src/metric.F77 b/src/metric.F77 index a93ea12..f28c948 100644 --- a/src/metric.F77 +++ b/src/metric.F77 @@ -154,6 +154,15 @@ c $ guxx, guyy, guzz, guxy, guyz, guxz, $ psi, Tmunu_flag) + elseif (decoded_exact_model .eq. EXACT__Kerr_KerrSchild_spherical) then + call Exact__Kerr_KerrSchild_spherical( + $ x, y, z, t, + $ gdtt, gdtx, gdty, gdtz, + $ gdxx, gdyy, gdzz, gdxy, gdyz, gdxz, + $ gutt, gutx, guty, gutz, + $ guxx, guyy, guzz, guxy, guyz, guxz, + $ psi, Tmunu_flag) + elseif (decoded_exact_model .eq. EXACT__Schwarzschild_Lemaitre) then call Exact__Schwarzschild_Lemaitre( $ x, y, z, t, diff --git a/src/metrics/Kerr_KerrSchild.F77 b/src/metrics/Kerr_KerrSchild.F77 index f2fb45a..11b6ce1 100644 --- a/src/metrics/Kerr_KerrSchild.F77 +++ b/src/metrics/Kerr_KerrSchild.F77 @@ -38,6 +38,7 @@ c output arguments c local variables CCTK_REAL boostv, eps, m, a + integer power c local variables CCTK_REAL gamma, t0, z0, x0, y0, rho02, r02, r0, costheta0, @@ -54,6 +55,7 @@ C to the J/m definition used in the code here. boostv = Kerr_KerrSchild__boost_v eps = Kerr_KerrSchild__epsilon + power = Kerr_KerrSchild__power m = Kerr_KerrSchild__mass a = m*Kerr_KerrSchild__spin @@ -79,7 +81,18 @@ C Spherical auxiliary coordinate r and angle theta in BH rest frame. r02 = 0.5d0 * (rho02 - a**2) $ + sqrt(0.25d0 * (rho02 - a**2)**2 + a**2 * z0**2) - r0 = (r02**2 + eps**4)**0.25d0 + r0 = sqrt(r02) + if (Kerr_KerrSchild__parabolic .eq. 0) then +C Use a power law to avoid the singularity + r0 = (r0**power + eps**power)**(1.0d0/power) + else + if (r0 .lt. eps) then + r0 = (eps + r0**2 / eps) / 2 + end if + end if +C Another idea: +C r0 = r0 + eps * exp(-x/eps) + costheta0 = z0 / r0 C Coefficient H. Note this transforms as a scalar, so does not carry diff --git a/src/metrics/Kerr_KerrSchild_spherical.F77 b/src/metrics/Kerr_KerrSchild_spherical.F77 new file mode 100644 index 0000000..417b224 --- /dev/null +++ b/src/metrics/Kerr_KerrSchild_spherical.F77 @@ -0,0 +1,182 @@ +C Kerr-Schild form of boosted rotating black hole. +C Program g_ab = eta_ab + H l_a l_b, g^ab = eta^ab - H l^a l^b. +C Here eta_ab is Minkowski in Cartesian coordinates, H is a scalar, +C and l is a null vector. +C +C The coordinates are distorted, such that the event horizon is +C a coordinate sphere. +C +C Author: Erik Schnetter <schnetter@cct.lsu.edu> +C This formulation was invented by Nils Dorband <dorband@cct.lsu.edu> +C +C $Header$ + +#include "cctk.h" +#include "cctk_Parameters.h" +#include "cctk_Functions.h" + + subroutine Exact__Kerr_KerrSchild_spherical ( + $ x, y, z, t, + $ gdtt, gdtx, gdty, gdtz, + $ gdxx, gdyy, gdzz, gdxy, gdyz, gdzx, + $ gutt, gutx, guty, gutz, + $ guxx, guyy, guzz, guxy, guyz, guzx, + $ psi, Tmunu_flag) + + implicit none + + DECLARE_CCTK_PARAMETERS + DECLARE_CCTK_FUNCTIONS + +c input arguments + CCTK_REAL x, y, z, t + +c output arguments + CCTK_REAL gdtt, gdtx, gdty, gdtz, + $ gdxx, gdyy, gdzz, gdxy, gdyz, gdzx, + $ gutt, gutx, guty, gutz, + $ guxx, guyy, guzz, guxy, guyz, guzx + CCTK_REAL psi + LOGICAL Tmunu_flag + +c local variables + CCTK_REAL boostv, eps, m, a + +c local variables + CCTK_REAL t1, x1, y1, z1, rho1, + $ gamma, t0, z0, x0, y0, rho02, r02, r0, costheta0, + $ lt0, lx0, ly0, lz0, hh, lt, lx, ly, lz + + CCTK_REAL gd(3,3), gdt(3,3), det, jac(3,3) + + integer i, j, k, l + +cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc + +C This is a vacuum spacetime with no cosmological constant + Tmunu_flag = .false. + +C Get parameters of the exact solution, +C and convert from parameter file spin parameter J/m^2 +C to the J/m definition used in the code here. + + boostv = Kerr_KerrSchild__boost_v + eps = Kerr_KerrSchild__epsilon + m = Kerr_KerrSchild__mass + a = m*Kerr_KerrSchild__spin + +C Distort the coordinates such that the event horizon is a +C coordinate sphere + rho1 = sqrt (x**2 + y**2 + z**2) + rho1 = (rho1**4 + eps**4) ** 0.25d0 + t1 = t + x1 = x - a * y / rho1 + y1 = y + a * x / rho1 + z1 = z + +C Boost factor. + + gamma = 1 / sqrt(1 - boostv**2) + +C Lorentz transform t,x,y,z -> t0,x0,y0,z0. +C t0 is never used, but is here for illustration, and we introduce +C x0 and y0 also only for clarity. +C Note that z0 = 0 means z = vt for the BH. + + t0 = gamma * ((t1 - Kerr_KerrSchild__t) - boostv * (z1 - Kerr_KerrSchild__z)) + z0 = gamma * ((z1 - Kerr_KerrSchild__z) - boostv * (t1 - Kerr_KerrSchild__t)) + x0 = x1 - Kerr_KerrSchild__x + y0 = y1 - Kerr_KerrSchild__y + +C Spherical auxiliary coordinate r and angle theta in BH rest frame. + + r0 = rho1 + costheta0 = z0 / r0 + +C Coefficient H. Note this transforms as a scalar, so does not carry +C the suffix 0. + hh = m * r0 / (r0**2 + a**2 * costheta0**2) + +C Components of l_a in rest frame. Note indices down. + lt0 = 1.d0 + lx0 = (r0 * x0 + a * y0) / (r0**2 + a**2) + ly0 = (r0 * y0 - a * x0) / (r0**2 + a**2) + lz0 = z0 / r0 + +C Now boost it to coordinates x, y, z, t. +C This is the reverse Lorentz transformation, but applied +C to a one-form, so the sign of boostv is the same as the forward +C Lorentz transformation applied to the coordinates. + + lt = gamma * (lt0 - boostv * lz0) + lz = gamma * (lz0 - boostv * lt0) + lx = lx0 + ly = ly0 + +C Down metric. g_ab = flat_ab + H l_a l_b + + gdtt = - 1.d0 + 2.d0 * hh * lt * lt + gdtx = 2.d0 * hh * lt * lx + gdty = 2.d0 * hh * lt * ly + gdtz = 2.d0 * hh * lt * lz + + gd(1,1) = 1.d0 + 2.d0 * hh * lx * lx + gd(2,2) = 1.d0 + 2.d0 * hh * ly * ly + gd(3,3) = 1.d0 + 2.d0 * hh * lz * lz + gd(1,2) = 2.d0 * hh * lx * ly + gd(2,3) = 2.d0 * hh * ly * lz + gd(3,1) = 2.d0 * hh * lz * lx + gd(2,1) = gd(1,2) + gd(3,2) = gd(2,3) + gd(1,3) = gd(3,1) + +C Transform the tensor basis back + jac(1,1) = 1 + (a*x*y) / (rho1**3) + jac(1,2) = - a * (x**2+z**2) / (rho1**3) + jac(1,3) = a*y*z / (rho1**3) + + jac(2,1) = a * (y**2+z**2) / (rho1**3) + jac(2,2) = 1 - a*x*y / (rho1**3) + jac(2,3) = - a*x*z / (rho1**3) + + jac(3,1) = 0 + jac(3,2) = 0 + jac(3,3) = 1 + + do i = 1, 3 + do j = 1, 3 + gdt(i,j) = 0 + do k = 1, 3 + do l = 1, 3 + gdt(i,j) = gdt(i,j) + gd(k,l) * jac(k,i) * jac(l,j) + end do + end do + end do + end do + + gdxx = gdt(1,1) + gdyy = gdt(2,2) + gdzz = gdt(3,3) + gdxy = gdt(1,2) + gdyz = gdt(2,3) + gdzx = gdt(3,1) + +C Up metric. g^ab = flat^ab - H l^a l^b. +C Notice that g^ab = g_ab and l^i = l_i and l^0 = - l_0 in flat spacetime. + gutt = - 1.d0 - 2.d0 * hh * lt * lt + gutx = 2.d0 * hh * lt * lx + guty = 2.d0 * hh * lt * ly + gutz = 2.d0 * hh * lt * lz + + det = gdxx*gdyy*gdzz + 2*gdxy*gdzx*gdyz + . - gdxx*gdyz**2 - gdyy*gdzx**2 - gdzz*gdxy**2 + + guxx = (gdyy*gdzz - gdyz**2)/det + guyy = (gdxx*gdzz - gdzx**2)/det + guzz = (gdxx*gdyy - gdxy**2)/det + + guxy = (gdzx*gdyz - gdzz*gdxy)/det + guyz = (gdxy*gdzx - gdxx*gdyz)/det + guzx = (gdxy*gdyz - gdyy*gdzx)/det + + end diff --git a/src/metrics/make.code.defn b/src/metrics/make.code.defn index 12226e2..1e90de0 100644 --- a/src/metrics/make.code.defn +++ b/src/metrics/make.code.defn @@ -19,6 +19,7 @@ SRCS = Minkowski.F77 \ Schwarzschild_Novikov.F77 \ Kerr_BoyerLindquist.F77 \ Kerr_KerrSchild.F77 \ + Kerr_KerrSchild_spherical.F77 \ Schwarzschild_Lemaitre.F77 \ multi_BH.F77 \ Thorne_fakebinary.F77 \ diff --git a/test/KS-tilted.par b/test/KS-tilted.par new file mode 100644 index 0000000..445224d --- /dev/null +++ b/test/KS-tilted.par @@ -0,0 +1,96 @@ +Cactus::cctk_run_title = "Tilted Kerr/Kerr-Schild data" +Cactus::cctk_full_warnings = yes + +Cactus::cctk_itlast = 2 + +ActiveThorns = "IOUtil" +IO::out_dir = $parfile +IO::out_fileinfo = "axis labels" +IO::parfile_write = "no" + +ActiveThorns = "AEILocalInterp LocalInterp LocalReduce" + +ActiveThorns = "SphericalSurface" + +ActiveThorns = "PUGH PUGHInterp PUGHReduce PUGHSlab" +driver::global_nsize = 49 +driver::ghost_size = 2 + +ActiveThorns = "NaNChecker" + +ActiveThorns = "Boundary CartGrid3D CoordBase SymBase SymBase" +CartGrid3D::type = "byspacing" +CartGrid3D::domain = "full" +CartGrid3D::dxyz = 0.125 +CartGrid3D::avoid_origin = no + +ActiveThorns = "ADMBase ADMCoupling ADMMacros CoordGauge SpaceMask StaticConformal TmunuBase" +SpaceMask::use_mask = yes + +ActiveThorns = "Exact" +ADMBase::initial_data = "exact" +ADMBase::initial_lapse = "exact" +ADMBase::initial_shift = "exact" +Exact::exact_model = "Kerr/Kerr-Schild" +Exact::rotation_euler_theta = 0.4 # in radians, approx. 23 degrees + +ActiveThorns = "LegoExcision" +LegoExcision::fixed_excision = "sphere" +LegoExcision::fixed_size = 1.01 + +ActiveThorns = "Time" +Time::dtfac = 0.25 + +ActiveThorns = "MoL" +MoL::ODE_Method = "RK3" + +ActiveThorns = "BSSN_MoL" +BSSN_MoL::excise = yes +BSSN_MoL::excisionType = "lego" + +ADMBase::evolution_method = ADM_BSSN +ADM_BSSN::stencil_size = 2 +ADM_BSSN::advection = "upwind2" +ADM_BSSN::bound = "newrad" +ADM_BSSN::BSSN_MaxNumConstrainedVars = 16 + +ActiveThorns = "ADMConstraints" + +ActiveThorns = "AHFinderDirect" +AHFinderDirect::output_OpenDX_control_files = no +AHFinderDirect::geometry_interpolator_name = "Lagrange polynomial interpolation" +AHFinderDirect::geometry_interpolator_pars = "order=4" +AHFinderDirect::surface_interpolator_name = "Lagrange polynomial interpolation" +AHFinderDirect::surface_interpolator_pars = "order=4" + +ActiveThorns = "IOBasic" +IOBasic::outInfo_every = 1 +IOBasic::outInfo_reductions = "norm2" +IOBasic::outInfo_vars = " + ADMConstraints::ham + ADMBase::gxx +" + +IOBasic::outScalar_every = 1 +IOBasic::outScalar_reductions = "average count maximum minimum norm1 norm2 norm_inf" +IOBasic::outScalar_vars = " + ADMBase::metric + ADMBase::curv + ADMBase::lapse + ADMBase::shift + ADMConstraints::hamiltonian + ADMConstraints::momentum + SpaceMask::mask +" + +ActiveThorns = "IOASCII" +IOASCII::out1D_every = 1 +IOASCII::out1D_vars = " + ADMBase::metric + ADMBase::curv + ADMBase::lapse + ADMBase::shift + ADMConstraints::hamiltonian + ADMConstraints::momentum + SpaceMask::mask +" diff --git a/test/KS-tilted/BH_diagnostics.ah1.gp b/test/KS-tilted/BH_diagnostics.ah1.gp new file mode 100644 index 0000000..8114c5b --- /dev/null +++ b/test/KS-tilted/BH_diagnostics.ah1.gp @@ -0,0 +1,42 @@ +# apparent horizon 1/1 +# +# column 1 = cctk_iteration +# column 2 = cctk_time +# column 3 = centroid_x +# column 4 = centroid_y +# column 5 = centroid_z +# column 6 = min radius +# column 7 = max radius +# column 8 = mean radius +# column 9 = quadrupole_xx +# column 10 = quadrupole_xy +# column 11 = quadrupole_xz +# column 12 = quadrupole_yy +# column 13 = quadrupole_yz +# column 14 = quadrupole_zz +# column 15 = min x +# column 16 = max x +# column 17 = min y +# column 18 = max y +# column 19 = min z +# column 20 = max z +# column 21 = xy-plane circumference +# column 22 = xz-plane circumference +# column 23 = yz-plane circumference +# column 24 = ratio of xz/xy-plane circumferences +# column 25 = ratio of yz/xy-plane circumferences +# column 26 = area +# column 27 = m_irreducible +# column 28 = areal radius +# column 29 = expansion Theta_(l) +# column 30 = inner expansion Theta_(n) +# column 31 = product of the expansions +# column 32 = mean curvature +# column 33 = gradient of the areal radius +# column 34 = gradient of the expansion Theta_(l) +# column 35 = gradient of the inner expansion Theta_(n) +# column 36 = gradient of the product of the expansions +# column 37 = gradient of the mean curvature +0 0.000 -0.000000 0.000000 0.000000 1.800107386 1.897357126 1.865240773 1.199983840 -2.341261159e-08 -7.705129016e-08 1.181787313 0.04304125283 1.098188158 -1.897343169 1.897343169 -1.881558869 1.881558869 -1.813788206 1.813788206 12.40953204 11.74974772 11.61754522 0.9468324573 0.9361791549 45.23835967 0.9486772739 1.897354548 -1.081225865e-11 -1.466239844 1.657958671e-11 0.7331199219 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 +1 0.031 -0.000000 -0.000000 0.000000 1.799995950 1.897244949 1.865125428 1.199837619 -1.223698721e-07 -3.390703490e-07 1.181642415 0.04303875551 1.098048978 -1.897244949 1.897244949 -1.881459721 1.881459721 -1.813685028 1.813685028 12.40910569 11.74919833 11.61698218 0.9468207162 0.9361659475 45.23328718 0.9486240856 1.897248171 -3.293764153e-16 -1.466281593 4.966112274e-16 0.7331407966 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 +2 0.062 -0.000000 -0.000000 0.000000 1.799888324 1.897150818 1.865014207 1.199696765 -2.081014156e-07 -5.741339136e-07 1.181502773 0.04303642658 1.097914589 -1.897150818 1.897150818 -1.881364449 1.881364449 -1.813585228 1.813585228 12.40869291 11.74866887 11.61643983 0.9468095436 0.9361533823 45.22843886 0.9485732452 1.897146490 9.825935079e-17 -1.466319752 -1.506907866e-16 0.7331598758 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 diff --git a/test/KS-tilted/alp_3D_diagonal.xg b/test/KS-tilted/alp_3D_diagonal.xg new file mode 100644 index 0000000..6d025e2 --- /dev/null +++ b/test/KS-tilted/alp_3D_diagonal.xg @@ -0,0 +1,158 @@ +"x-label diagonal +"y-label ADMBASE::alp + + +"Time = 0.0000000000000 +-5.1961524227066 0.8491838137798 +-4.9796460717605 0.8440302314853 +-4.7631397208144 0.8385063860414 +-4.5466333698683 0.8325701064365 +-4.3301270189222 0.8261724128828 +-4.1136206679761 0.8192560464056 +-3.8971143170300 0.8117535880699 +-3.6806079660839 0.8035850231432 +-3.4641016151378 0.7946545420772 +-3.2475952641916 0.7848462727130 +-3.0310889132455 0.7740184844577 +-2.8145825622994 0.7619955561369 +-2.5980762113533 0.7485565831352 +-2.3815698604072 0.7334187813939 +-2.1650635094611 0.7162125660409 +-1.9485571585150 0.6964428409450 +-1.7320508075689 0.6734267500570 +-1.5155444566228 0.6461911061836 +-1.2990381056767 0.6133094408108 +-1.0825317547305 0.5727502701744 +-0.8660254037844 0.5231578523833 +-0.6495190528383 0.4890452737869 +-0.4330127018922 0.6319359177817 +-0.2165063509461 0.8340367852500 +0.0000000000000 0.0000000070711 +0.2165063509461 0.8340367852500 +0.4330127018922 0.6319359177817 +0.6495190528383 0.4890452737869 +0.8660254037844 0.5231578523833 +1.0825317547305 0.5727502701744 +1.2990381056767 0.6133094408108 +1.5155444566228 0.6461911061836 +1.7320508075689 0.6734267500570 +1.9485571585150 0.6964428409450 +2.1650635094611 0.7162125660409 +2.3815698604072 0.7334187813939 +2.5980762113533 0.7485565831352 +2.8145825622994 0.7619955561369 +3.0310889132455 0.7740184844577 +3.2475952641916 0.7848462727130 +3.4641016151378 0.7946545420772 +3.6806079660839 0.8035850231432 +3.8971143170300 0.8117535880699 +4.1136206679761 0.8192560464056 +4.3301270189222 0.8261724128828 +4.5466333698683 0.8325701064365 +4.7631397208144 0.8385063860414 +4.9796460717605 0.8440302314853 +5.1961524227066 0.8491838137798 + + +"Time = 0.0312500000000 +-5.1961524227066 0.8491838137798 +-4.9796460717605 0.8440302314853 +-4.7631397208144 0.8385063860414 +-4.5466333698683 0.8325701064365 +-4.3301270189222 0.8261724128828 +-4.1136206679761 0.8192560464056 +-3.8971143170300 0.8117535880699 +-3.6806079660839 0.8035850231432 +-3.4641016151378 0.7946545420772 +-3.2475952641916 0.7848462727130 +-3.0310889132455 0.7740184844577 +-2.8145825622994 0.7619955561369 +-2.5980762113533 0.7485565831352 +-2.3815698604072 0.7334187813939 +-2.1650635094611 0.7162125660409 +-1.9485571585150 0.6964428409450 +-1.7320508075689 0.6734267500570 +-1.5155444566228 0.6461911061836 +-1.2990381056767 0.6133094408108 +-1.0825317547305 0.5727502701744 +-0.8660254037844 0.5231578523833 +-0.6495190528383 0.4890452737869 +-0.4330127018922 0.6319359177817 +-0.2165063509461 0.8340367852500 +0.0000000000000 0.0000000070711 +0.2165063509461 0.8340367852500 +0.4330127018922 0.6319359177817 +0.6495190528383 0.4890452737869 +0.8660254037844 0.5231578523833 +1.0825317547305 0.5727502701744 +1.2990381056767 0.6133094408108 +1.5155444566228 0.6461911061836 +1.7320508075689 0.6734267500570 +1.9485571585150 0.6964428409450 +2.1650635094611 0.7162125660409 +2.3815698604072 0.7334187813939 +2.5980762113533 0.7485565831352 +2.8145825622994 0.7619955561369 +3.0310889132455 0.7740184844577 +3.2475952641916 0.7848462727130 +3.4641016151378 0.7946545420772 +3.6806079660839 0.8035850231432 +3.8971143170300 0.8117535880699 +4.1136206679761 0.8192560464056 +4.3301270189222 0.8261724128828 +4.5466333698683 0.8325701064365 +4.7631397208144 0.8385063860414 +4.9796460717605 0.8440302314853 +5.1961524227066 0.8491838137798 + + +"Time = 0.0625000000000 +-5.1961524227066 0.8491838137798 +-4.9796460717605 0.8440302314853 +-4.7631397208144 0.8385063860414 +-4.5466333698683 0.8325701064365 +-4.3301270189222 0.8261724128828 +-4.1136206679761 0.8192560464056 +-3.8971143170300 0.8117535880699 +-3.6806079660839 0.8035850231432 +-3.4641016151378 0.7946545420772 +-3.2475952641916 0.7848462727130 +-3.0310889132455 0.7740184844577 +-2.8145825622994 0.7619955561369 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-1.062183507 1.062183507 + + +### -z patch +# N_rho = 19 +# N_sigma = 19 +# ghosted_gfn=-6 +# dpx = dsigma = dnu +# dpy = 180 - drho = 180 - dmu +# +# dpx dpy gridfn global_x global_y global_z +135 45 1.83975591461287 1.062183573 1.062183573 -1.062183573 +140 45 1.83376544527668 0.9357221052 1.115150181 -1.115150181 +145 45 1.82859830236920 0.8113721186 1.158759474 -1.158759474 +150 45 1.82423606133886 0.6894964216 1.194242834 -1.194242834 +155 45 1.82064232687979 0.5701261575 1.222639490 -1.222639490 +160 45 1.81777442580895 0.4530685862 1.244795710 -1.244795710 +165 45 1.81559135102026 0.3379846540 1.261375901 -1.261375901 +170 45 1.81405826743836 0.2244425999 1.272877236 -1.272877236 +175 45 1.81314904171337 0.1119543117 1.279643638 -1.279643638 +180 45 1.81284758700941 -4.122830303e-16 1.281876822 -1.281876822 +185 45 1.81314849716269 -0.1119542780 1.279643253 -1.279643253 +190 45 1.81405722999529 -0.2244424715 1.272876508 -1.272876508 +195 45 1.81558991800743 -0.3379843873 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1.88751037172145 -1.089754621 -1.089754621 -1.089754621 + + diff --git a/test/KS-tilted/h.t1.ah1.gp b/test/KS-tilted/h.t1.ah1.gp new file mode 100644 index 0000000..5272109 --- /dev/null +++ b/test/KS-tilted/h.t1.ah1.gp @@ -0,0 +1,2337 @@ +# N_patches = 6 +# origin = 0 0 0 + +### +z patch +# N_rho = 19 +# N_sigma = 19 +# ghosted_gfn=-6 +# dpx = dsigma = dnu +# dpy = drho = dmu +# +# dpx dpy gridfn global_x global_y global_z +-45 -45 1.83963191932975 -1.062111984 -1.062111984 1.062111984 +-40 -45 1.83364194772965 -0.9356590876 -1.115075079 1.115075079 +-35 -45 1.82847565375891 -0.8113176978 -1.158681753 1.158681753 +-30 -45 1.82411442209835 -0.6894504463 -1.194163202 1.194163202 +-25 -45 1.82052172480133 -0.5700883915 -1.222558501 1.222558501 +-20 -45 1.81765479889428 -0.4530387699 -1.244713791 1.244713791 +-15 -45 1.81547257914480 -0.3379625438 -1.261293385 1.261293385 +-10 -45 1.81394019335102 -0.2244279913 -1.272794387 1.272794387 +-5 -45 1.81303148374545 -0.1119470530 -1.279560670 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drho = 180 - dmu +# +# dpx dpy gridfn global_x global_y global_z +135 45 1.83963191932975 1.062111984 1.062111984 -1.062111984 +140 45 1.83364194772964 0.9356590876 1.115075079 -1.115075079 +145 45 1.82847565375890 0.8113176978 1.158681753 -1.158681753 +150 45 1.82411442209834 0.6894504463 1.194163202 -1.194163202 +155 45 1.82052172480132 0.5700883915 1.222558501 -1.222558501 +160 45 1.81765479889428 0.4530387699 1.244713791 -1.244713791 +165 45 1.81547257914479 0.3379625438 1.261293385 -1.261293385 +170 45 1.81394019335102 0.2244279913 1.272794387 -1.272794387 +175 45 1.81303148374545 0.1119470530 1.279560670 -1.279560670 +180 45 1.81273034567639 -4.122563670e-16 1.281793920 -1.281793920 +185 45 1.81303135718677 -0.1119470451 1.279560581 -1.279560581 +190 45 1.81393995953738 -0.2244279623 1.272794223 -1.272794223 +195 45 1.81547227307698 -0.3379624869 1.261293172 -1.261293172 +200 45 1.81765446183886 -0.4530386859 1.244713560 -1.244713560 +205 45 1.82052140431247 -0.5700882912 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a/test/KS-tilted/h.t2.ah1.gp b/test/KS-tilted/h.t2.ah1.gp new file mode 100644 index 0000000..e90c5bd --- /dev/null +++ b/test/KS-tilted/h.t2.ah1.gp @@ -0,0 +1,2337 @@ +# N_patches = 6 +# origin = 0 0 0 + +### +z patch +# N_rho = 19 +# N_sigma = 19 +# ghosted_gfn=-6 +# dpx = dsigma = dnu +# dpy = drho = dmu +# +# dpx dpy gridfn global_x global_y global_z +-45 -45 1.83951245719592 -1.062043012 -1.062043012 1.062043012 +-40 -45 1.83352303858681 -0.9355984114 -1.115002768 1.115002768 +-35 -45 1.82835760060501 -0.8112653162 -1.158606944 1.158606944 +-30 -45 1.82399734918616 -0.6894061969 -1.194086560 1.194086560 +-25 -45 1.82040563611223 -0.5700520389 -1.222480542 1.222480542 +-20 -45 1.81753961965833 -0.4530100623 -1.244634917 1.244634917 +-15 -45 1.81535818556651 -0.3379412487 -1.261213910 1.261213910 +-10 -45 1.81382643169364 -0.2244139162 -1.272714563 1.272714563 +-5 -45 1.81291818137728 -0.1119400570 -1.279480706 1.279480706 +0 -45 1.81261731629151 0.000000000 -1.281713996 1.281713996 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drho = 180 - dmu +# +# dpx dpy gridfn global_x global_y global_z +135 45 1.83951245719591 1.062043012 1.062043012 -1.062043012 +140 45 1.83352303858681 0.9355984114 1.115002768 -1.115002768 +145 45 1.82835760060501 0.8112653162 1.158606944 -1.158606944 +150 45 1.82399734918616 0.6894061969 1.194086560 -1.194086560 +155 45 1.82040563611223 0.5700520389 1.222480542 -1.222480542 +160 45 1.81753961965832 0.4530100623 1.244634917 -1.244634917 +165 45 1.81535818556651 0.3379412487 1.261213910 -1.261213910 +170 45 1.81382643169363 0.2244139162 1.272714563 -1.272714563 +175 45 1.81291818137728 0.1119400570 1.279480706 -1.279480706 +180 45 1.81261731629150 -4.122306615e-16 1.281713996 -1.281713996 +185 45 1.81291840180820 -0.1119400706 1.279480862 -1.279480862 +190 45 1.81382686631158 -0.2244139700 1.272714868 -1.272714868 +195 45 1.81535882401949 -0.3379413676 1.261214354 -1.261214354 +200 45 1.81754042867544 -0.4530102639 1.244635471 -1.244635471 +205 45 1.82040657067291 -0.5700523315 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