DEL SISTEMA DE SEGURIDAD SOCIAL CAPÍTULO

The fundamental datum of the pulsar timing technique is thetime of arrival (ToA)of the pulse, typically referred toSolar system barycentre (SSB). TheseToAsembed information about the telescopes, backends and algorithms used to measure them; the behaviour of the pulsars them- selves and the systems hosting them, if any; the effects of any large ob- jects near the line of sight, the Solar System and any other physical phe- nomena that would affect the propagation of the pulsar signals. Tim- ing models incorporating the spin and astrometric parameters, as well as parameters for theDMand when applicable, orbital parameters are fitted to theseToAsusing timing software like te m p o 2 (Hobbs et al.,

2006).

Often, however, the individual pulses are too weak to be distinguished from noise and it is preferable to integrate the archive along the time and/or frequency domain(s). This decreases the amplitude of the Gaus- sian noise and increases that of the pulse and any other non-Gaussian components of the noise. The scrunched or integrated profile then rep- resents an ensemble average of the individual pulses that are contained in the archive. Within the limits of the sensitivity of the receiver and the telescope, the ensemble average at one epoch can approximate the ensemble average at a different epoch exceedingly well. In fact, we can express the measured pulse profile P(t) as a function of a standard tem-

plate profile S(t), via the relation:

P(t) = a + b × S(t − Δ𝜏) + n(t) (2.12) where a is the offset between the baselines of the standard and meas- ured profiles, b is a scaling factor,Δ𝜏 is the phase offset between the profile and the template and, n(t) is the noise component.

If the discrete Fourier transforms of the profile and template are

Pke𝔦𝜃k = N−1 􏾜 j=0 pje𝔦2𝜋jk/N (2.13) and Ske𝔦𝜙k = N−1 􏾜 j=0 sje𝔦2𝜋jk/N (2.14)

50 t i m i n g & p r o p e r t ie s o f r e c ycl e d p u l s a r s

respectively, where k represents a frequency index, Pkand Skthe amp- litudes of the complex Fourier coefficients,𝜃 and 𝜙 are the respective phases and N is the number of frequency channels.

Since the transformation preserves linearity,eqn. (2.12)can be re- written as

Pke𝔦𝜃k = aN + bSke𝔦𝜙k+ Gk (2.15) where the index k runs from zero to N-1.

Measured Predicted Timing Pulses Pulses Residual

Figure 2.8: Schematic of the method of calculation ofToAs. In practice, the tem- plate matching or comparision between the predicted arrival time and the actual arrival time is carried out in the Fourier domain, as explained insection 2.5.1.

Once the individual transforms have been computed, the baseline offset can be measured from the zeroth components of the two amp- litudes,

a= (P0− bS0) /N. (2.16)

TheToAcan now be derived along with the scaling factor b by min- imising the goodness-of-fit statistic

𝜒2₌ N/2 􏾜 k=1 􏵶Pk− bSke 𝔦(𝜙k−𝜃k+k𝜏) 𝜎k 􏵶 2 (2.17)

where𝜎krepresents the root-mean-square intensity of the noise at frequency indexed by k. A more detailed derivation can be found in (Taylor,1992).

While the derivation above shows the simplest case, alternative meth- ods to recoverToAsfrom lowsignal-to-noise ratio (S/N)signals using Gaussian interpolation can be found inHotan et al.(2005) orvan Straten

(2006) which utilises the full Stokes information to produceToAs. An equivalent frequency domain formulation of the (Taylor,1992) method can also be found inDemorest(2007). Updated techniques forToAgen- eration for wideband backends, where frequency dependent evolution of the pulse profiles must be accounted for can be found inLiu et al.

(2014);Pennucci et al.(2014)

2.5.2

Pulsar timing with te m p o 2

First, theToAis time-stamped using a clock at the observatory. This local clock is referenced to a hydrogen maser, which itself is ultimately

p u l s a r a s t r o no my 51

not stable over extended periods of time. This time must then be trans- lated into theUniversal Coordinated Time (UTC)scale from which one can finally derive the corresponding value in theTemps Atomique In- ternational (TAI)timescale. However, the atomic clocks used to derive theTAItimescale do not measure the SI second exactly and offsets must be accounted for. The clocks used in theTAItimescale are used to define what is called theTerrestrial Time (TT)scale. TheBureau International

des Poids et Mesures (BIPM) publishes the transformation between “TT(BIPM) is a realization of Terrestrial Time

as defined by theInternational Astronomical

Union (IAU). It is computed annually by the

BIPMbased on a weighted average of the eval-

uations of the frequency ofTAIby the primary

and secondary frequency standards.”

–BIPM.org

The latest value for the correction is TT(BIPM15) =TAI+ 32.184 s + 27702.0 ns

pairs of timescales and these must be used to derive the correct refer- enceToAin SI seconds referred to theGeocentric Celestial Reference System (GCRS)based timescale, denoted byGeocentric Coordinate Time (TCG). The current standard for translating fromTTto any other times- cale is theTT(BIPM15), which applies to measurements extended bey- ondmodified Julian date (MJD)57379.

To extract meaningful information from the measuredToAsthey must be translated into a proper time of emission at the pulsar. The steps in- volved in this translation from aToAon the earth to the time of emis- sion at the pulsar are as follows.

t

t

Δ

Δ

Δ

Figure 2.9: Vector representation of the translation repsresented byeqn. (2.21), showing the change from the ToA at Earth,tearth_a to the time of emission at the pulsar,tpsr_e via the removal of the effect of the binary orbit􏸷BB, the effects due to

propagation through a non-neutral, tur- bulent interstellar medium􏸷ISand the

effects Solar System,􏸷⊙.

Having corrected the ToAto theGCRSwe then translate it to the

SSB. This involves, apart from translating the reference frame from the

GCRSto thebarycentric celestial reference system (BCRS), calculating the delays the photon must have encountered during its propagation through the Earth’s atmosphere (ΔAtm), the vacuum retardation due to the motion of the observatory (ΔR_⊙andΔp), that due to dispersion by

the ionised solar wind (ΔSW), that due to the relativistic frame trans- formations due to the co-movingSSBand observatory, also called the Einstein delay (i.e., the gravitational redshift,ΔE_⊙) and finally that due to the excess path it has to travel through the gravitational potential of the Solar System, called the Shapiro delay (ΔS_⊙).

Δ⊙= ΔAtm+ ΔR_⊙+ Δp+ ΔSW+ ΔE_⊙+ ΔS_⊙ (2.18)

Thebarycentred arrival time (BAT), or theToAtranslated to theSSB

must be corrected for the effects of propagation through theISMwhich include the vacuum propagation delay (ΔVP), the dispersion due to the

ISM(ΔISD) and other frequency dependent effects (ΔFDD) and finally the Einstein delay due to the relativistic motion of theSSBand the binary barycentre6(ΔE_SSB,BB).

6_{This is true for a pulsar in a binary,}

however for solitary pulsars this corres- ponds to centre of mass (effectively, the centre of the pulsar).

ΔIS= ΔVP+ ΔISD+ ΔFDD+ ΔE_SSB,BB (2.19)

If the pulsar is in a binary, we must now correct for the effects of bin- ary motion which include the Römer delay due to the binary compan- ion (ΔR_B), the aberration that is introduced due to the proper motion (ΔA_B), the Einstein delay due to the companion (ΔE_B) and the Shapiro delay due to the companion (ΔS_B).

ΔBB= ΔR_B+ ΔA_B+ ΔE_B+ ΔS_B (2.20)

52 t i m i n g & p r o p e r t ie s o f r e c ycl e d p u l s a r s

et al.(2006), which discusses these terms in context of the most com- monly used software package for pulsar timing nowadays, te m p o 2 .

Using theequations (2.18)to(2.20)we can now derive the time of emission at the pulsar as

tpsre = tEartha − Δ⊙− ΔIS− ΔBB, (2.21)

represented by the vector diagramfigure 2.9. Having obtained the time at which the photon was emitted at the pulsar, we can now model the rotational phase of the pulsar at this time as an integer number of cycles since the epoch, tP, at which the rate-change of the phase ̇𝜙, equals the

frequency of rotation𝜈 using the following expression (Taylor,1992) 𝜙(t) = 􏾜 n⩾1 𝜈n−1 n! 􏿴t psr e − tP􏿷 (2.22)

where𝜈n_{are the frequency derivatives.}

It is evident that many of the parameters listed earlier are not always known apriori. Instead, starting with a minimal set of parameters, a least squares minimisation must be carried out over the expression:

𝜒2= M 􏾜 i=1 􏿶𝜙(Ti) − ni 𝜎i/P 􏿹 2 (2.23) where niis the closest integer to the phase𝜙(Ti) and 𝜎iis the uncertainty of the ithToA.