GenerateHyperbolicSpinOrbitCW.c

Go to the documentation of this file.

/*
*  Copyright (C) 2007 Teviet Creighton
*
*  This program is free software; you can redistribute it and/or modify
*  it under the terms of the GNU General Public License as published by
*  the Free Software Foundation; either version 2 of the License, or
*  (at your option) any later version.
*
*  This program is distributed in the hope that it will be useful,
*  but WITHOUT ANY WARRANTY; without even the implied warranty of
*  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
*  GNU General Public License for more details.
*
*  You should have received a copy of the GNU General Public License
*  along with with program; see the file COPYING. If not, write to the
*  Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
*  MA  02110-1301  USA
*/
 
 
#include <lal/LALStdio.h>
#include <lal/LALStdlib.h>
#include <lal/LALConstants.h>
#include <lal/Units.h>
#include <lal/AVFactories.h>
#include <lal/SeqFactories.h>
#include <lal/PulsarSimulateCoherentGW.h>
#include <lal/GenerateSpinOrbitCW.h>
 
/**
 * \author Creighton, T. D.
 *
 * \brief Computes a continuous waveform with frequency drift and Doppler
 * modulation from a hyperbolic orbital trajectory.
 *
 * This function computes a quaiperiodic waveform using the spindown and
 * orbital parameters in <tt>*params</tt>, storing the result in
 * <tt>*output</tt>.
 *
 * In the <tt>*params</tt> structure, the routine uses all the "input"
 * fields specified in \ref GenerateSpinOrbitCW_h, and sets all of the
 * "output" fields.  If <tt>params->f</tt>=\c NULL, no spindown
 * modulation is performed.  If <tt>params->oneMinusEcc</tt> \f$ \not<0 \f$ (a
 * non-hyperbolic orbit), or if
 * <tt>params->rPeriNorm</tt> \f$ \times \f$ <tt>params->angularSpeed</tt> \f$ \geq1 \f$
 * (faster-than-light speed at periapsis), an error is returned.
 *
 * In the <tt>*output</tt> structure, the field <tt>output->h</tt> is
 * ignored, but all other pointer fields must be set to \c NULL.  The
 * function will create and allocate space for <tt>output->a</tt>,
 * <tt>output->f</tt>, and <tt>output->phi</tt> as necessary.  The
 * <tt>output->shift</tt> field will remain set to \c NULL.
 *
 * ### Algorithm ###
 *
 * For hyperbolic orbits, we combine $\eqref{eq_spinorbit-tr}$,
 * $\eqref{eq_spinorbit-t}$, and $\eqref{eq_spinorbit-upsilon}$ to get \f$ t_r \f$
 * directly as a function of \f$ E \f$ :
 * \f{eqnarray}{
 * \label{eq_tr-e3}
 * t_r = t_p & + & \left(\frac{r_p \sin i}{c}\right)\sin\omega\\
 * & + & \frac{1}{n} \left( -E +
 * \left[v_p(e-1)\cos\omega + e\right]\sinh E
 * - \left[v_p\sqrt{\frac{e-1}{e+1}}\sin\omega\right]
 * [\cosh E - 1]\right) \;,
 * \f}
 * where \f$ v_p=r_p\dot{\upsilon}_p\sin i/c \f$ is a normalized velocity at
 * periapsis and \f$ n=\dot{\upsilon}_p\sqrt{(1-e)^3/(1+e)} \f$ is a normalized
 * angular speed for the orbit (the hyperbolic analogue of the mean
 * angular speed for closed orbits).  For simplicity we write this as:
 * \f{equation}{
 * \label{eq_tr-e4}
 * t_r = T_p + \frac{1}{n}\left( E + A\sinh E + B[\cosh E - 1] \right) \;,
 * \f}
 *
 * \anchor inject_hanomaly
 * \image html inject_hanomaly.png "Function to be inverted to find eccentric anomaly"
 *
 * where \f$ T_p \f$ is the \e observed time of periapsis passage.  Thus
 * the key numerical procedure in this routine is to invert the
 * expression \f$ x=E+A\sinh E+B(\cosh E - 1) \f$ to get \f$ E(x) \f$ .  This function
 * is sketched to the right (solid line), along with an approximation
 * used for making an initial guess (dotted line), as described later.
 *
 * We note that \f$ A^2-B^2<1 \f$ , although it approaches 1 when
 * \f$ e\rightarrow1 \f$ , or when \f$ v_p\rightarrow1 \f$ and either \f$ e=0 \f$ or
 * \f$ \omega=\pi \f$ .  Except in this limit, Newton-Raphson methods will
 * converge rapidly for any initial guess.  In this limit, though, the
 * slope \f$ dx/dE \f$ approaches zero at \f$ E=0 \f$ , and an initial guess or
 * iteration landing near this point will send the next iteration off to
 * unacceptably large or small values.  A hybrid root-finding strategy is
 * used to deal with this, and with the exponential behaviour of \f$ x \f$ at
 * large \f$ E \f$ .
 *
 * First, we compute \f$ x=x_{\pm1} \f$ at \f$ E=\pm1 \f$ .  If the desired \f$ x \f$ lies
 * in this range, we use a straightforward Newton-Raphson root finder,
 * with the constraint that all guesses of \f$ E \f$ are restricted to the
 * domain \f$ [-1,1] \f$ .  This guarantees that the scheme will eventually find
 * itself on a uniformly-convergent trajectory.
 *
 * Second, for \f$ E \f$ outside of this range, \f$ x \f$ is dominated by the
 * exponential terms: \f$ x\approx\frac{1}{2}(A+B)\exp(E) \f$ for \f$ E\gg1 \f$ , and
 * \f$ x\approx-\frac{1}{2}(A-B)\exp(-E) \f$ for \f$ E\ll-1 \f$ .  We therefore do an
 * \e approximate Newton-Raphson iteration on the function \f$ \ln|x| \f$ ,
 * where the approximation is that we take \f$ d\ln|x|/d|E|\approx1 \f$ .  This
 * involves computing an extra logarithm inside the loop, but gives very
 * rapid convergence to high precision, since \f$ \ln|x| \f$ is very nearly
 * linear in these regions.
 *
 * At the start of the algorithm, we use an initial guess of
 * \f$ E=-\ln[-2(x-x_{-1})/(A-B)-\exp(1)] \f$ for \f$ x<x_{-1} \f$ , \f$ E=x/x_{-1} \f$ for
 * \f$ x_{-1}\leq x\leq0 \f$ , \f$ E=x/x_{+1} \f$ for \f$ 0\leq x\leq x_{+1} \f$ , or
 * \f$ E=\ln[2(x-x_{+1})/(A+B)-\exp(1)] \f$ for \f$ x>x_{+1} \f$ .  We refine this
 * guess until we get agreement to within 0.01 parts in part in
 * \f$ N_\mathrm{cyc} \f$ (where \f$ N_\mathrm{cyc} \f$ is the larger of the number
 * of wave cycles in a time \f$ 2\pi/n \f$ , or the number of wave cycles in the
 * entire waveform being generated), or one part in \f$ 10^{15} \f$ (an order
 * of magnitude off the best precision possible with \c REAL8
 * numbers).  The latter case indicates that \c REAL8 precision may
 * fail to give accurate phasing, and one should consider modeling the
 * orbit as a set of Taylor frequency coefficients \'{a} la
 * <tt>LALGenerateTaylorCW()</tt>.  On subsequent timesteps, we use the
 * previous timestep as an initial guess, which is good so long as the
 * timesteps are much smaller than \f$ 1/n \f$ .
 *
 * Once a value of \f$ E \f$ is found for a given timestep in the output
 * series, we compute the system time \f$ t \f$ via $\eqref{eq_spinorbit-t}$,
 * and use it to determine the wave phase and (non-Doppler-shifted)
 * frequency via $\eqref{eq_taylorcw-freq}$
 * and $\eqref{eq_taylorcw-phi}$.  The Doppler shift on the frequency is
 * then computed using $\eqref{eq_spinorbit-upsilon}$
 * and $\eqref{eq_orbit-rdot}$.  We use \f$ \upsilon \f$ as an intermediate in
 * the Doppler shift calculations, since expressing \f$ \dot{R} \f$ directly in
 * terms of \f$ E \f$ results in expression of the form \f$ (e-1)/(e\cosh E-1) \f$ ,
 * which are difficult to simplify and face precision losses when
 * \f$ E\sim0 \f$ and \f$ e\rightarrow1 \f$ .  By contrast, solving for \f$ \upsilon \f$ is
 * numerically stable provided that the system <tt>atan2()</tt> function is
 * well-designed.
 *
 * This routine does not account for relativistic timing variations, and
 * issues warnings or errors based on the criterea of
 * $\eqref{eq_relativistic-orbit}$ in \ref LALGenerateEllipticSpinOrbitCW().
 */
void
LALGenerateHyperbolicSpinOrbitCW( LALStatus             *stat,
                                  PulsarCoherentGW            *output,
                                  SpinOrbitCWParamStruc *params )
{
  UINT4 n, i;              /* number of and index over samples */
  UINT4 nSpin = 0, j;      /* number of and index over spindown terms */
  REAL8 t, dt, tPow;       /* time, interval, and t raised to a power */
  REAL8 phi0, f0, twopif0; /* initial phase, frequency, and 2*pi*f0 */
  REAL8 f, fPrev;          /* current and previous values of frequency */
  REAL4 df = 0.0;          /* maximum difference between f and fPrev */
  REAL8 phi;               /* current value of phase */
  REAL8 vDotAvg;           /* nomalized orbital angular speed */
  REAL8 vp;                /* projected speed at periapsis */
  REAL8 upsilon, argument; /* true anomaly, and argument of periapsis */
  REAL8 eCosOmega;         /* eccentricity * cosine of argument */
  REAL8 tPeriObs;          /* time of observed periapsis */
  REAL8 spinOff;           /* spin epoch - orbit epoch */
  REAL8 x;                 /* observed ``mean anomaly'' */
  REAL8 xPlus, xMinus;     /* limits where exponentials dominate */
  REAL8 dx, dxMax;         /* current and target errors in x */
  REAL8 a, b;              /* constants in equation for x */
  REAL8 ecc;               /* orbital eccentricity */
  REAL8 eccMinusOne, eccPlusOne; /* ecc - 1 and ecc + 1 */
  REAL8 e;                       /* hyperbolic anomaly */
  REAL8 sinhe, coshe;            /* sinh of e, and cosh of e minus 1 */
  REAL8 *fSpin = NULL;           /* pointer to Taylor coefficients */
  REAL4 *fData;                  /* pointer to frequency data */
  REAL8 *phiData;                /* pointer to phase data */
 
  INITSTATUS( stat );
  ATTATCHSTATUSPTR( stat );
 
  /* Make sure parameter and output structures exist. */
  ASSERT( params, stat, GENERATESPINORBITCWH_ENUL,
          GENERATESPINORBITCWH_MSGENUL );
  ASSERT( output, stat, GENERATESPINORBITCWH_ENUL,
          GENERATESPINORBITCWH_MSGENUL );
 
  /* Make sure output fields don't exist. */
  ASSERT( !( output->a ), stat, GENERATESPINORBITCWH_EOUT,
          GENERATESPINORBITCWH_MSGEOUT );
  ASSERT( !( output->f ), stat, GENERATESPINORBITCWH_EOUT,
          GENERATESPINORBITCWH_MSGEOUT );
  ASSERT( !( output->phi ), stat, GENERATESPINORBITCWH_EOUT,
          GENERATESPINORBITCWH_MSGEOUT );
  ASSERT( !( output->shift ), stat, GENERATESPINORBITCWH_EOUT,
          GENERATESPINORBITCWH_MSGEOUT );
 
  /* If Taylor coeficients are specified, make sure they exist. */
  if ( params->f ) {
    ASSERT( params->f->data, stat, GENERATESPINORBITCWH_ENUL,
            GENERATESPINORBITCWH_MSGENUL );
    nSpin = params->f->length;
    fSpin = params->f->data;
  }
 
  /* Set up some constants (to avoid repeated calculation or
     dereferencing), and make sure they have acceptable values. */
  eccMinusOne = -params->oneMinusEcc;
  ecc = 1.0 + eccMinusOne;
  eccPlusOne = 2.0 + eccMinusOne;
  if ( eccMinusOne <= 0.0 ) {
    ABORT( stat, GENERATESPINORBITCWH_EECC,
           GENERATESPINORBITCWH_MSGEECC );
  }
  vp = params->rPeriNorm * params->angularSpeed;
  vDotAvg = params->angularSpeed
            * sqrt( eccMinusOne * eccMinusOne * eccMinusOne / eccPlusOne );
  n = params->length;
  dt = params->deltaT;
  f0 = fPrev = params->f0;
  if ( vp >= 1.0 ) {
    ABORT( stat, GENERATESPINORBITCWH_EFTL,
           GENERATESPINORBITCWH_MSGEFTL );
  }
  if ( vp <= 0.0 || dt <= 0.0 || f0 <= 0.0 || vDotAvg <= 0.0 ||
       n == 0 ) {
    ABORT( stat, GENERATESPINORBITCWH_ESGN,
           GENERATESPINORBITCWH_MSGESGN );
  }
 
  /* Set up some other constants. */
  twopif0 = f0 * LAL_TWOPI;
  phi0 = params->phi0;
  argument = params->omega;
  a = vp * eccMinusOne * cos( argument ) + ecc;
  b = -vp * sqrt( eccMinusOne / eccPlusOne ) * sin( argument );
  eCosOmega = ecc * cos( argument );
  if ( n * dt * vDotAvg > LAL_TWOPI ) {
    dxMax = 0.01 / ( f0 * n * dt );
  } else {
    dxMax = 0.01 / ( f0 * LAL_TWOPI / vDotAvg );
  }
  if ( dxMax < 1.0e-15 ) {
    dxMax = 1.0e-15;
    LALWarning( stat, "REAL8 arithmetic may not have sufficient"
                " precision for this orbit" );
  }
  if ( lalDebugLevel & LALWARNING ) {
    REAL8 tau = n * dt;
    if ( tau > LAL_TWOPI / vDotAvg ) {
      tau = LAL_TWOPI / vDotAvg;
    }
    if ( f0 * tau * vp * vp * ecc / eccPlusOne > 0.25 )
      LALWarning( stat, "Orbit may have significant relativistic"
                  " effects that are not included" );
  }
 
  /* Compute offset between time series epoch and observed periapsis,
     and betweem true periapsis and spindown reference epoch. */
  tPeriObs = ( REAL8 )( params->orbitEpoch.gpsSeconds -
                        params->epoch.gpsSeconds );
  tPeriObs += 1.0e-9 * ( REAL8 )( params->orbitEpoch.gpsNanoSeconds -
                                  params->epoch.gpsNanoSeconds );
  tPeriObs += params->rPeriNorm * sin( params->omega );
  spinOff = ( REAL8 )( params->orbitEpoch.gpsSeconds -
                       params->spinEpoch.gpsSeconds );
  spinOff += 1.0e-9 * ( REAL8 )( params->orbitEpoch.gpsNanoSeconds -
                                 params->spinEpoch.gpsNanoSeconds );
 
  /* Determine bounds of hybrid root-finding algorithm, and initial
     guess for e. */
  xMinus = 1.0 + a * sinh( -1.0 ) + b * cosh( -1.0 ) - b;
  xPlus = -1.0 + a * sinh( 1.0 ) + b * cosh( 1.0 ) - b;
  x = -vDotAvg * tPeriObs;
  if ( x < xMinus ) {
    e = -log( -2.0 * ( x - xMinus ) / ( a - b ) - exp( 1.0 ) );
  } else if ( x <= 0 ) {
    e = x / xMinus;
  } else if ( x <= xPlus ) {
    e = x / xPlus;
  } else {
    e = log( 2.0 * ( x - xPlus ) / ( a + b ) - exp( 1.0 ) );
  }
  sinhe = sinh( e );
  coshe = cosh( e ) - 1.0;
 
  /* Allocate output structures. */
  if ( ( output->a = ( REAL4TimeVectorSeries * )
                     LALMalloc( sizeof( REAL4TimeVectorSeries ) ) ) == NULL ) {
    ABORT( stat, GENERATESPINORBITCWH_EMEM,
           GENERATESPINORBITCWH_MSGEMEM );
  }
  memset( output->a, 0, sizeof( REAL4TimeVectorSeries ) );
  if ( ( output->f = ( REAL4TimeSeries * )
                     LALMalloc( sizeof( REAL4TimeSeries ) ) ) == NULL ) {
    LALFree( output->a );
    output->a = NULL;
    ABORT( stat, GENERATESPINORBITCWH_EMEM,
           GENERATESPINORBITCWH_MSGEMEM );
  }
  memset( output->f, 0, sizeof( REAL4TimeSeries ) );
  if ( ( output->phi = ( REAL8TimeSeries * )
                       LALMalloc( sizeof( REAL8TimeSeries ) ) ) == NULL ) {
    LALFree( output->a );
    output->a = NULL;
    LALFree( output->f );
    output->f = NULL;
    ABORT( stat, GENERATESPINORBITCWH_EMEM,
           GENERATESPINORBITCWH_MSGEMEM );
  }
  memset( output->phi, 0, sizeof( REAL8TimeSeries ) );
 
  /* Set output structure metadata fields. */
  output->position = params->position;
  output->psi = params->psi;
  output->a->epoch = output->f->epoch = output->phi->epoch
                                        = params->epoch;
  output->a->deltaT = n * params->deltaT;
  output->f->deltaT = output->phi->deltaT = params->deltaT;
  output->a->sampleUnits = lalStrainUnit;
  output->f->sampleUnits = lalHertzUnit;
  output->phi->sampleUnits = lalDimensionlessUnit;
  snprintf( output->a->name, LALNameLength, "CW amplitudes" );
  snprintf( output->f->name, LALNameLength, "CW frequency" );
  snprintf( output->phi->name, LALNameLength, "CW phase" );
 
  /* Allocate phase and frequency arrays. */
  LALSCreateVector( stat->statusPtr, &( output->f->data ), n );
  BEGINFAIL( stat ) {
    LALFree( output->a );
    output->a = NULL;
    LALFree( output->f );
    output->f = NULL;
    LALFree( output->phi );
    output->phi = NULL;
  }
  ENDFAIL( stat );
  LALDCreateVector( stat->statusPtr, &( output->phi->data ), n );
  BEGINFAIL( stat ) {
    TRY( LALSDestroyVector( stat->statusPtr, &( output->f->data ) ),
         stat );
    LALFree( output->a );
    output->a = NULL;
    LALFree( output->f );
    output->f = NULL;
    LALFree( output->phi );
    output->phi = NULL;
  }
  ENDFAIL( stat );
 
  /* Allocate and fill amplitude array. */
  {
    CreateVectorSequenceIn in; /* input to create output->a */
    in.length = 2;
    in.vectorLength = 2;
    LALSCreateVectorSequence( stat->statusPtr, &( output->a->data ), &in );
    BEGINFAIL( stat ) {
      TRY( LALSDestroyVector( stat->statusPtr, &( output->f->data ) ),
           stat );
      TRY( LALDDestroyVector( stat->statusPtr, &( output->phi->data ) ),
           stat );
      LALFree( output->a );
      output->a = NULL;
      LALFree( output->f );
      output->f = NULL;
      LALFree( output->phi );
      output->phi = NULL;
    }
    ENDFAIL( stat );
    output->a->data->data[0] = output->a->data->data[2] = params->aPlus;
    output->a->data->data[1] = output->a->data->data[3] = params->aCross;
  }
 
  /* Fill frequency and phase arrays. */
  fData = output->f->data->data;
  phiData = output->phi->data->data;
  for ( i = 0; i < n; i++ ) {
 
    x = vDotAvg * ( i * dt - tPeriObs );
 
    /* Use approximate Newton-Raphson method on ln|x| if |x| > 1. */
    if ( x < xMinus ) {
      x = log( -x );
      while ( fabs( dx = log( e - a * sinhe - b * coshe ) - x ) > dxMax ) {
        e += dx;
        sinhe = sinh( e );
        coshe = cosh( e ) - 1.0;
      }
    } else if ( x > xPlus ) {
      x = log( x );
      while ( fabs( dx = log( -e + a * sinhe + b * coshe ) - x ) > dxMax ) {
        e -= dx;
        sinhe = sinh( e );
        coshe = cosh( e ) - 1.0;
      }
    }
 
    /* Use ordinary Newton-Raphson method on x if |x| <= 1. */
    else {
      while ( fabs( dx = -e + a * sinhe + b * coshe - x ) > dxMax ) {
        e -= dx / ( -1.0 + a * coshe + a + b * sinhe );
        if ( e < -1.0 ) {
          e = -1.0;
        } else if ( e > 1.0 ) {
          e = 1.0;
        }
        sinhe = sinh( e );
        coshe = cosh( e ) - 1.0;
      }
    }
 
    /* Compute source emission time, phase, and frequency. */
    phi = t = tPow =
                ( ecc * sinhe - e ) / vDotAvg + spinOff;
    f = 1.0;
    for ( j = 0; j < nSpin; j++ ) {
      f += fSpin[j] * tPow;
      phi += fSpin[j] * ( tPow *= t ) / ( j + 2.0 );
    }
 
    /* Appy frequency Doppler shift. */
    upsilon = 2.0 * atan2( sqrt( eccPlusOne * coshe ),
                           sqrt( eccMinusOne * ( coshe + 2.0 ) ) );
    f *= f0 / ( 1.0 + vp * ( cos( argument + upsilon ) + eCosOmega )
                / eccPlusOne );
    phi *= twopif0;
    if ( fabs( f - fPrev ) > df ) {
      df = fabs( f - fPrev );
    }
    *( fData++ ) = fPrev = f;
    *( phiData++ ) = phi + phi0;
  }
 
  /* Set output field and return. */
  params->dfdt = df * dt;
  DETATCHSTATUSPTR( stat );
  RETURN( stat );
}