pygwtf.response.transfer ======================== .. py:module:: pygwtf.response.transfer Functions --------- .. autoapisummary:: pygwtf.response.transfer.Ylms pygwtf.response.transfer.fill_P_lm pygwtf.response.transfer.fill_k pygwtf.response.transfer.dot_three_unpack_sum pygwtf.response.transfer.dot_R pygwtf.response.transfer.dot_three_unpack2 pygwtf.response.transfer.sinc pygwtf.response.transfer.threevector_diff_norm pygwtf.response.transfer.build_ysrl pygwtf.response.transfer.get_XYZ_TFs pygwtf.response.transfer.get_AET_TFs pygwtf.response.transfer.get_AET_TFs_equal_armlength Module Contents --------------- .. py:function:: Ylms(cosi, phi) ' Normalised spherical harmonics for Y_{2,2} and Y_{2,-2}. Parameters: ---------- cosi: float cosine of the inclination angle of the source (angle between the line of sight and the orbital angular momentum vector) phi: float Reference phase rotation of the source (angle between the line of sight and the major axis of the binary's orbit) NOTE: Usually set to zero within our code. Rotations are absorbed in the waveform phase definition itself. .. py:function:: fill_P_lm(P_lm, parameters) Polarization matrices from Eqn 16. in https://arxiv.org/pdf/2003.00357 Parameters: ---------- P_lm: array (3,3) array to be filled with the P_lm coefficients for the source, as defined in https://arxiv.org/pdf/2003.00357 parameters: array (4,) [0] cosi: cosine of the inclination angle of the source (angle between the line of sight and the orbital angular momentum vector) [1] pol: polarization angle of the source (angle between the line of sight and the major axis of the binary's orbit) [2] ecliptic_long: ecliptic longitude of the source [3] ecliptic_lat: ecliptic latitude of the source .. py:function:: fill_k(k, parameters) Wavevector pointing from the source to the centre of the SSB frame, in ecliptic coordinates. Parameters: ---------- k : array (3,) array to be filled with the components of the wavevector pointing from the source to the centre of the SSB frame. parameters: array (4,) [0] cosi: cosine of the inclination angle of the source (angle between the line of sight and the orbital angular momentum vector) [1] pol: polarization angle of the source (angle between the line of sight and the major axis of the binary's orbit) [2] ecliptic_long: ecliptic longitude of the source [3] ecliptic_lat: ecliptic latitude of the source .. py:function:: dot_three_unpack_sum(a, b, i, j) ' Computes the sum a[0] * (b[i, 0] + b[j, 0]) + a[1] * (b[i, 1] + b[j, 1]) + a[2] * (b[i, 2] + b[j, 2]) Parameters: ---------- a: array (3,) 3-vector b: array (3,3) 3x3 matrix i: int first index for b j: int second index for b Returns: ------- result: float the result of the sum a[0] * (b[i, 0] + b[j, 0]) + a[1] * (b[i, 1] + b[j, 1]) + a[2] * (b[i, 2] + b[j, 2]) .. py:function:: dot_R(k, p) Computes the dot product of the wavevector k with the sum of the position vectors p[i] for i=0,1,2. Parameters: ---------- k: array (3,) wavevector pointing from the source to the centre of the SSB frame, in ecliptic coordinates. p: array (3,3) positions of the spacecraft in the SSB frame (Ecliptic coordinates) Returns: ------- result: float the result of the dot product of the wavevector k with the sum of the position vectors .. py:function:: dot_three_unpack2(a, b, k) Computes the sum a[0] * b[k, 0] + a[1] * b[k, 1] + a[2] * b[k, 2] Parameters: ---------- a: array (3,) 3-vector b: array (3,3) 3x3 matrix k: int index for b Returns: ------- result: float the result of the sum a[0] * b[k, 0] + a[1] * b[k, 1] + a[2] * b[k, 2] .. py:function:: sinc(x) Sinc function, sin(x)/x Parameters: ---------- x: float input to the sinc function Returns: ------- result: float the result of the sinc function, sin(x)/x. Returns 1 if x=0 to avoid division by zero. .. py:function:: threevector_diff_norm(x, i, j) Computes the norm of the difference between two 3-vectors x[i] and x[j]. Parameters: ---------- x: array (3,3) array of 3-vectors i: int index of the first vector j: int index of the second vector Returns: ------- result: float the norm of the difference between x[i] and x[j], computed as sqrt((x[i, 0] - x[j, 0])^2 + (x[i, 1] - x[j, 1])^2 + (x[i, 2] - x[j, 2])^2) .. py:function:: build_ysrl(f, k, n, p, Ls, P_lm) Builds transfer function for *single* link responses. s: source index (0,1,2 for the 3 spacecraft) l: link index (0,1,2 for the 3 arms) r: response index (0,1 for the two directions of the link) NOTE: Usually the convention is 'slr' but here we use a slightly different convention 'srl' y_sr = [[1,2], [2,1], [1,3], [3,1], [2,3], [3,2]] Parameters: ---------- f: float frequency at which to evaluate the transfer function k: array (3,) Unit vector pointing from the source to the centre of the SSB frame n: array (3,3) (to be filled in within the function) unit vectors pointing along the arms of the interferometer. p: array (3,3) positions of the spacecraft in the SSB frame (Ecliptic coordinates) Ls: array (3,) arm lengths of the interferometer (metres) P_lm: array (3,3) P_lm coefficients for the source, as defined in https://arxiv.org/pdf/2003.00357 Returns: ------- ysrl_123, ysrl_213, ysrl_132, ysrl_312, ysrl_231, ysrl_321: complex the single link responses for the 6 possible combinations of s, r, l. The indices correspond to the following combinations of s, r, l: srl: 1->2, 2->1, 1->3, 3->1, 2->3, 3->2 respectively. Note that the convention here is 'srl' instead of 'slr'. .. py:function:: get_XYZ_TFs(f, P_lm, k, p, Ls, n, tdi2) Construct the single link response functions and then combine them appropriately to get the X,Y,Z transfer functions. Deals with slowly varying, unequal arm-lengths. In this case we are assuming - Arm lengths are unequal but slowly varying - Delays commute. - The delay factors effectively map to the (1-D^4) that is found in TDI 1.5 case to go from TDI-1 to TDI-2 (equal arm-length case) Under these approximations the TDI equations below reduce exactly to the equations for TDI-2. Checking the Rosetta stone equations for TDI-2, simplifying them, assuming commuting delays, one recovers the TDI-equations used here I think. X = (G31 + G13*z2 - G21 - G12*z3) - (G31*z3*z3 + G13*z2*z3*z3 - G21*z2*z2 - G12*z3*z2*z2) Y = (G12 + G21*z3 - G32 - G23*z1) - (G12*z1*z1 + G21*z3*z1*z1 - G32*z3*z3 - G23*z1*z3*z3) Z = (G23 + G32*z1 - G13 - G31*z2) - (G23*z2*z2 + G32*z1*z2*z2 - G13*z1*z1 - G31*z2*z1*z1) Parameters: ---------- f: float frequency at which to evaluate the transfer functions P_lm: array (3,3) P_lm coefficients for the source, as defined in https://arxiv.org/pdf/2003.00357 k: array (3,) Unit vector pointing from the source to the centre of the SSB frame p: array (3,3) positions of the spacecraft in the SSB frame (Ecliptic coordinates) Ls: array (3,) arm lengths of the interferometer (metres) n: array (3,3) (to be filled in) unit vectors pointing along the arms of the interferometer. tdi2: bool whether to apply the TDI 2.0 correction factor (1 - z1^2 z2^2 z3^2) to the transfer functions. .. py:function:: get_AET_TFs(f, P_lm, k, p, Ls, n, tdi2) ' 'Master function' - Called directly by fresnel to get the AET transfer functions. (Default transfer function for analysis within this library) Calls get_XYZ_TFs to get the X,Y,Z transfer functions, and then combines them appropriately to get A,E,T. Deals with slowly varying, unequal arm-lengths. Parameters: ---------- f: float frequency at which to evaluate the transfer functions P_lm: array (3,3) P_lm coefficients for the source, as defined in https://arxiv.org/pdf/2003.00357 k: array (3,) Unit vector pointing from the source to the centre of the SSB frame p: array (3,3) positions of the spacecraft in the SSB frame (Ecliptic coordinates) Ls: array (3,) arm lengths of the interferometer (metres) n: array (3,3) unit vectors pointing along the arms of the interferometer. tdi2: bool whether to apply the TDI 2.0 correction factor (1 - z1^2 z2^2 z3^2) to the transfer functions. Returns: ------- A, E, T: complex The A, E, T transfer functions for the source at frequency f. .. py:function:: get_AET_TFs_equal_armlength(f, P_lm, k, p, n, tdi2) ' DEPRECATED FUNCTION. Doing what is essentially the same thing as get_AET_TFs but assuming equal arm-lengths and doing some algebraic simplifications to get more compact expressions for the A,E,T transfer functions. Parameters: ---------- f: float frequency at which to evaluate the transfer functions P_lm: array (3,3) P_lm coefficients for the source, as defined in https://arxiv.org/pdf/ k: array (3,) Unit vector pointing from the source to the centre of the SSB frame p: array (3,3) positions of the spacecraft in the SSB frame (Ecliptic coordinates) n: array (3,3) unit vectors pointing along the arms of the interferometer. tdi2: bool whether to apply the TDI 2.0 correction factor (1 - D^4) to the transfer functions.