Detailed Description

Extrapolate the Pulsar spin-paramters \( \{f^{(k)}\}\equiv\{f, \dot{f},\ddot{f},...\} \) , and "spin-ranges" \( \{ f^{(k)}, \Delta f^{(k)} \} \) from one SSB epoch to another.

Author: Reinhard Prix

The central function of this module is XLALExtrapolatePulsarSpinRange(), which extrapolates a complete "spin range" (defined as PulsarSpinRange) from one epoch to another. A "spin-range" contains an epoch, and two vectors, \( f^{(k)} \) and \( \Delta f^{(k)} \) (where "canonical" ordering refers to \( \Delta f^{(k)} >= 0 \) for all k.

The extrapolation is defined by the pulsar spindown-model:

\[ f(\tau_1) = f(\tau_0) + \frac{\dot{f}(\tau_0)}{1!} \,\Delta\tau + \frac{\ddot{f}(\tau_0)}{2!} \,\Delta\tau^2 + ... = \sum_{k=0}^s \frac{f^{(k)}(\tau_0)}{k!}\, \Delta\tau^k\,, \]

where

\[\Delta\tau \equiv \tau_1 - \tau_0\]

and therefore generally

\[ f^{(l)}(\tau_1) = \sum_{k=0}^{s - l} \frac{ f^{(k+l)}(\tau_0)}{k! }\, \Delta\tau^k\,. \]

This expression is used to extrapolate a whole "spin-range", namely at each spindown-order \( (l) \) the extrapolated range is given by

\[ \min\left[ f^{(l)}(\tau_1) \right] = \sum_{k=0}^{s - l} \frac{1}{k!} \min\left[ f^{(k+l)}(\tau_0) \, \Delta\tau^k \right]\,. \]

\[ \max\left[ f^{(l)}(\tau_1) \right] = \sum_{k=0}^{s - l} \frac{1}{k!} \max\left[ f^{(k+l)}(\tau_0) \, \Delta\tau^k \right]\,. \]

This ensures that the range will be correctly extrapolated even if \( \tau_1 < \tau_0 \) , i.e. \( \Delta\tau < 0 \) .

The initial-phase extrapolation in XLALExtrapolatePulsarPhase() proceeds in the other direction, extrapolating \( \phi(\tau_0) \) to \( \phi(\tau_1) \) , where the spins are given at \( \tau_1 \) , i.e. \( f^{(k)}(\tau_1) \) . By using the above equations, one can arrive at the following expression:

\[ \phi(\tau_1) = \phi(\tau_0) - \sum_{k=0}^s \frac{f^{(k)}(\tau_1)}{(k+1)!} \, (-\Delta\tau)^{k+1} \,. \]

This function is used in XLALEstimatePulsarAmplitudeParams() to propagate the estimated initial phase from the internal reference time back to the user-input reference time.

Prototypes
int	XLALInitPulsarSpinRangeFromSpins (PulsarSpinRange range, const LIGOTimeGPS refTime, const PulsarSpins fkdot1, const PulsarSpins fkdot2)
	Initialise a `PulsarSpinRange` struct from two PulsarSpins structs. More...

int	XLALExtrapolatePulsarSpinRange (PulsarSpinRange range1, const PulsarSpinRange range0, const REAL8 dtau)
	General pulsar-spin extrapolation function: given a "spin-range" (ie spins + spin-bands) `range0` at time \( \tau_0 \) , propagate the whole spin-range to time \( \tau_1 \) . More...

int	XLALExtrapolatePulsarSpins (PulsarSpins fkdot1, const PulsarSpins fkdot0, REAL8 dtau)
	Extrapolate the Pulsar spin-parameters \( \{f, \dot{f},\ddot{f},...\} \) (`fkdot0`) from the initial reference-epoch \( \tau_0 \) to the new reference-epoch \( \tau_1 \) . More...

int	XLALExtrapolatePulsarPhase (REAL8 *phi1, const PulsarSpins fkdot1, const REAL8 phi0, const REAL8 dtau)
	Extrapolate phase \( \phi_0 \) from \( \tau_0 \) to \( \tau_1 \) , given the spins `fkdot1` at \( \tau_1 \) . More...

int	XLALCWSignalCoveringBand (REAL8 minCoverFreq, REAL8 maxCoverFreq, const LIGOTimeGPS time1, const LIGOTimeGPS time2, const PulsarSpinRange *spinRange, const REAL8 binaryMaxAsini, const REAL8 binaryMinPeriod, const REAL8 binaryMaxEcc)
	Determines a frequency band which covers the frequency evolution of a band of CW signals between two GPS times. More...

int	XLALCWSignalBand (REAL8 minFreq, REAL8 maxFreq, const DetectorStateSeries detStates, const PulsarDopplerParams doppler)
	Determines the frequency band occupied by the frequency evolution of a given CW signal between two GPS times. More...

DetectorStateSeries *	XLALPrepareCWSignalBand (SkyPosition skypos_maxdoppler, const LIGOTimeGPS tStart, const REAL8 Tspan, const REAL8 dT, const LALDetector detector, const EphemerisData *edat)
	(Optional) Helper function for using XLALCWSignalBand(): compute DetectorStateSeries for given time-span and detector, and optionally also the sky-position with maximal Doppler band-width. More...

Function Documentation

◆ XLALInitPulsarSpinRangeFromSpins()

int XLALInitPulsarSpinRangeFromSpins	(	PulsarSpinRange *	range,
		const LIGOTimeGPS *	refTime,
		const PulsarSpins	fkdot1,
		const PulsarSpins	fkdot2
	)

Initialise a PulsarSpinRange struct from two PulsarSpins structs.

Parameters

[out]	range	output spin range
[in]	refTime	reference time
[in]	fkdot1	input spins
[in]	fkdot2	input spins

Definition at line 70 of file ExtrapolatePulsarSpins.c.

◆ XLALExtrapolatePulsarSpinRange()

int XLALExtrapolatePulsarSpinRange	(	PulsarSpinRange *	range1,
		const PulsarSpinRange *	range0,
		const REAL8	dtau
	)

General pulsar-spin extrapolation function: given a "spin-range" (ie spins + spin-bands) range0 at time \( \tau_0 \) , propagate the whole spin-range to time \( \tau_1 \) .

Note: *range1 is allowed to point to the same spin-range as *range0: the input will be overwritten with the output.; The output-range is in the 'canonical' order of \( [ f^{(k)}, f^{(k)} + \Delta f^{(k)}] \) , where \( \Delta f^{(k)} \ge 0 \) .

Parameters

[out]	range1	output spin range
[in]	range0	input spin range
[in]	dtau	time difference \( \tau_1 - \tau_0 \) to extrapolate `range0` to

Definition at line 99 of file ExtrapolatePulsarSpins.c.

◆ XLALExtrapolatePulsarSpins()

int XLALExtrapolatePulsarSpins	(	PulsarSpins	fkdot1,
		const PulsarSpins	fkdot0,
		REAL8	dtau
	)

Extrapolate the Pulsar spin-parameters \( \{f, \dot{f},\ddot{f},...\} \) (fkdot0) from the initial reference-epoch \( \tau_0 \) to the new reference-epoch \( \tau_1 \) .

This is equivalent to XLALExtrapolatePulsarSpins(), but uses the fixed-size array-type PulsarSpins instead, which is easier to handle and avoids any dynamic-memory hassles.

Note: This can be called with fkdot1 == fkdot0, in which case the input will be correctly replaced by the output.

Parameters

[out]	fkdot1	output spin-parameter array
[in]	fkdot0	input spin-parameter array
[in]	dtau	time difference \( \tau_1 - \tau_0 \) to extrapolate `fkdot0` to

Definition at line 159 of file ExtrapolatePulsarSpins.c.

◆ XLALExtrapolatePulsarPhase()

int XLALExtrapolatePulsarPhase	(	REAL8 *	phi1,
		const PulsarSpins	fkdot1,
		const REAL8	phi0,
		const REAL8	dtau
	)

Extrapolate phase \( \phi_0 \) from \( \tau_0 \) to \( \tau_1 \) , given the spins fkdot1 at \( \tau_1 \) .

Returns \( \phi_1 \) in the range \( [0, 2\pi] \) .

Parameters

[out]	phi1	output phase at \( \tau_1 \)
[in]	fkdot1	spin-params at reference \( \tau_1 \)
[in]	phi0	input phase at \( \tau_0 \)
[in]	dtau	time difference \( \tau_1 - \tau_0 \) to extrapolate `phi0` to

Definition at line 205 of file ExtrapolatePulsarSpins.c.

◆ XLALCWSignalCoveringBand()

int XLALCWSignalCoveringBand	(	REAL8 *	minCoverFreq,
		REAL8 *	maxCoverFreq,
		const LIGOTimeGPS *	time1,
		const LIGOTimeGPS *	time2,
		const PulsarSpinRange *	spinRange,
		const REAL8	binaryMaxAsini,
		const REAL8	binaryMinPeriod,
		const REAL8	binaryMaxEcc
	)

Determines a frequency band which covers the frequency evolution of a band of CW signals between two GPS times.

The calculation accounts for the spin evolution of the signals, and the maximum possible Dopper modulation due to detector motion, and (for binary signals) binary orbital motion.

Parameters

[out]	minCoverFreq	Minimum frequency of the covering band
[out]	maxCoverFreq	Maximum frequency of the covering band
[in]	time1	One end of the GPS time range
[in]	time2	The other end of the GPS time range
[in]	spinRange	Frequency and spindown range of the CW signals
[in]	binaryMaxAsini	Maximum projected semi-major axis a*sini/c (= 0 for isolated sources)
[in]	binaryMinPeriod	Minimum orbital period (s); must be 0 for isolated signals
[in]	binaryMaxEcc	Maximal binary eccentricity: must be 0 for isolated signals

Definition at line 249 of file ExtrapolatePulsarSpins.c.

◆ XLALCWSignalBand()

int XLALCWSignalBand	(	REAL8 *	minFreq,
		REAL8 *	maxFreq,
		const DetectorStateSeries *	detStates,
		const PulsarDopplerParams *	doppler
	)

Determines the frequency band occupied by the frequency evolution of a given CW signal between two GPS times.

The calculation accounts for the spin evolution of the signals, and the actual Dopper modulation due to detector motion, and (for binary signals) binary orbital motion.

Parameters

[out]	minFreq	Minimum frequency of the covering band
[out]	maxFreq	Maximum frequency of the covering band
[in]	detStates	detector state series, cf XLALGetDetectorStates()
[in]	doppler	Signal phase-evolution parameters

Definition at line 321 of file ExtrapolatePulsarSpins.c.

◆ XLALPrepareCWSignalBand()

DetectorStateSeries * XLALPrepareCWSignalBand	(	SkyPosition *	skypos_maxdoppler,
		const LIGOTimeGPS	tStart,
		const REAL8	Tspan,
		const REAL8	dT,
		const LALDetector *	detector,
		const EphemerisData *	edat
	)

(Optional) Helper function for using XLALCWSignalBand(): compute DetectorStateSeries for given time-span and detector, and optionally also the sky-position with maximal Doppler band-width.

The calculation accounts for the spin evolution of the signals, and the actual Dopper modulation due to detector motion, and (for binary signals) binary orbital motion.

Parameters

[out]	skypos_maxdoppler	[optional] sky-position of maximal Doppler band-width over the sky
[in]	tStart	start GPS time of observing interval
[in]	Tspan	total span of observing interval in seconds
[in]	dT	step-size (in seconds) to use to sample output detector-state series
[in]	detector	detector, cf XLALGetSiteInfo()
[in]	edat	ephemeris data, cf XLALInitBarycenter()

Definition at line 373 of file ExtrapolatePulsarSpins.c.

Detailed Description

Prototypes

Function Documentation

◆ XLALInitPulsarSpinRangeFromSpins()

◆ XLALExtrapolatePulsarSpinRange()

◆ XLALExtrapolatePulsarSpins()

◆ XLALExtrapolatePulsarPhase()

◆ XLALCWSignalCoveringBand()

◆ XLALCWSignalBand()

◆ XLALPrepareCWSignalBand()