Pulsars are rotating neutron stars that generate regular electromagnetic (EM) pulses at their spin rate. They were first discovered in the radio region of the EM spectrum by Jocelyn Bell and Anthony Hewish in 1967 [1]. Since this pioneering discovery, pulsars have also been detected in other regions of the EM spectrum such as the optical, x-ray and gamma ray regions.

This study will seek to detect pulsars from RXTE [2] x-ray observations, determine their intrinsic properties such as spin period, spin rate and period derivative¹. Using this information we will attempt to derive pulsar properties such as ‘characteristic age’, magnetic field strength, energy loss rate and contrast results with the ATNF [3] pulsar catalogue and published research in addition to performing a pulsar analysis. We’ll conclude with a summary of key points.

Back to Top | Back to High Mass Stellar Evolution

The following will provide background theory on supernovae, neutron stars and pulsars accompanied by information relating to the data source used for this study, namely the Rossi X-ray Timing Explorer (RXTE).

During neutronization many protons are converted to neutrons and vast amounts of neutrino energy is released. Neutron degeneracy pressure [6] in a similar way as electron degeneracy pressure halts further core collapse. A neutron star is created at this point; also described as a supernova by-product or stellar corpse. The supernova is powered by neutrino energy released from inverse beta decay and by explosive fission nucleosynthesis processes around the neutron star as part of the SNR². The newly created neutron star is extremely small, highly magnetised (magnetic field approximately 10^12 Gauss) and fast spinning (spin period is generally between 0.25 and 2 seconds) [7] compared to its pre-supernova stellar progenitor. The neutron star may also be ejected from the SNR due to explosion asymmetries resulting from the supernova.

There are three generic classes of pulsars arising from neutron stars, based on the energy source which powers them as follows:

Rotationally-powered pulsars emit electromagnetic radiation from their magnetic poles (shown in blue in figure 2) resulting from their inherent rotation energy. The electromagnetic dipole radiation emitted can be across a large portion of the EM spectrum, generally from the x-ray region down to the radio region. Typically the radiation is seen in the radio region of the EM spectrum.

Accretion-powered pulsars emit electromagnetic radiation via magnetic dipole radiation as well as by collecting material on their accretion disk typically from a binary companion in close proximity, or even by their closely associated SNR. Disk accretion creates x-ray ‘hot-spots’ which are responsible for periodic intensity variations (and non-periodic intensity variations) in these pulsars.

Magnetars are thought to be the sources of Soft Gamma ray Repeaters (SGR’s) and Anomalous X-ray Pulsars (AXP’s) [8]. Magnetars are highly magnetised neutron stars in fact much more than conventional neutron stars by a factor of up to 100 or more, with magnetic fields in the order of 10^14 Gauss and capable of emitting both x-rays and gamma rays by decay of their very strong magnetic field. The very strong magnetic field of a magnetar is thought to be inherited when the neutron star is first created during a supernovae [9].

Back to Top | Back to High Mass Stellar Evolution

The Ross X-ray Timing Explorer (RXTE) is a NASA mission which was launched in December of 1995. Originally designed as 2 year mission with a maximum lifespan of 5 years, RXTE is still in service today (November 2005) collecting x-ray data from galactic sources such as pulsars, galaxies and binary star systems.

Both PCA and HEXTE have been designed with microsecond time resolution capability i.e. the PCA instrument of our pulsar study is capable of detecting a range of pulsar periods down to 1 microsecond spin period accuracy. The EDS (Experiment Data System) onboard RXTE is responsible for capturing PCA pulsar data, processing it in ‘time binned mode’ [15], and inserting the results in the RXTE telemetry system for delivery to Earth based data collection systems. The PCA data used for our pulsar study relates to the detection of two young pulsars, namely PSR B0540-69 and PSR B1509-58. The former is associated with the LMC (Large Magellanic Cloud) at a distance from Earth of approximately 49.4 kpc and is associated with SNR 0540-693, whilst the latter is at a distance 4.4 kpc in the constellation Circinus as part of SNR G320.4-1.2 (shown on the cover page) [3].

Back to Top | Back to High Mass Stellar Evolution

The following describes technology and techniques required to analyse RXTE pulsar data. The method behind the chosen approach for data analysis will be described along with relevant signal processing theory required to place the subsequent sections of this study into context.

Back to Top | Back to High Mass Stellar Evolution

Fourier analysis is widely used in signal theory. It takes representations of a signal from a ‘real world’ analog system such as a pulsar and performs a Fourier Transform

1. Complex DFT’s can make use of complex numbers via substitution, thus
   making mathematical transformations⁶ easier to work with.
2. Complex DFT’s can handle negative spectral frequencies, generally required
   to deal with digital signal processing problems such as aliasing, circular
   convolution and amplitude modulation – whereas real DFT’s have difficulty
   handling these situations.
3. Complex DFT’s can satisfactorily handle corner cases, for instance special
   mathematical handling of the first and last points of a frequency spectrum –
   typically required when performing an inverse Fourier transform i.e. from the
   frequency domain to the time domain.

Real DFT implementations take a single N point time domain signal and create two N/2 + 1 point frequency domain signals containing only positive frequencies. The complex DFT takes two N point time domain signals and creates two N point frequency domain signals containing both positive and negative frequencies. The shaded boxes in figure 3 indicate pulsar data values common to both types of DFT transforms in both the time domain and frequency domain. One can then see how we might use a FFT to calculate a complex DFT to produce a real DFT transform of pulsar data (from the time domain to the frequency domain) – we simply compute the FFT (complex DFT) by:

1. Zeroing the imaginary input data of the complex DFT in the time domain.
2. Inserting the provided ‘binned’ RXTE pulsar data in the real part of the
   complex DFT in the time domain.
3. Perform the transform.
4. Discard the negative frequencies of both real and imaginary part in the frequency domain.
5. Compute the power spectrum in the frequency domain of the remaining positive frequencies.

The sampling rate also known as the sampling frequency is defined as the number of samples per second (Hz) taken from a continuous time domain signal to convert this into a ‘proper’ discrete signal. The inverse of the sampling rate also known as the sampling time (or sampling period) sets the length of the time bins for data folding so as to prepare the data in an evenly spaced manner required for the FFT. The sampling rate is related also to the Nyquist rate as per the Nyquist-Shannon sampling theorem [20].

The Nyquist-Shannon sampling theorem simply states that the sampling rate has to be greater than twice the Nyquist rate (or Nyquist frequency, which is representative of the bandwidth in the frequency domain of the pulsar signal) or equivalently at least twice the bandwidth of the time domain signal being sampled so as to avoid
alaising⁷. This consideration was taken into account as unwanted spectrum artifacts due to aliasing would have adversely altered the pulsar frequency domain representation. The relationship between the sampling rate and the Nyquist rate is given as:

We define Nf as the Nyquist rate (Hz), Ts as the sampling time (in seconds, also know as the sampling period) and Sf as the sampling rate (Hz). The sampling period serves two main purposes:

1. It ensures we avoid aliasing in the frequency domain equivalent of the time
   domain pulsar signal.
2. It ensures that we evenly space the data (fold data) as prerequisite for the FFT operation performed on the time domain data in our pulsar code.

We ensured selection of an appropriate sampling period to satisfy the Nyquist-Shannon sampling theorem to avoid aliasing, for each ‘test’ signal (sine & sawtooth trigonometric functions) and RXTE pulsar signals based on:

1. Knowing the value of the ‘test’ signal sine peaks in the frequency domain before the FFT. This assisted in understanding the likely Nyquist rate and sampling rate to use on time domain ‘test’ data.
2. Understanding the likely characteristics of pulsar spin periods at ‘worst case’ scenarios e.g. millisecond pulsars which would require a very high sampling rate, hence determining the Nyquist rate and sampling rates required for RXTE pulsar data to be sampled and subsequently transformed into the frequency domain spectrum.

One of the fundamental objectives of this study is to find the spin period and pulsar frequency from RXTE observations and use this result to derive other important pulsar properties such as period derivative, age, magnetic field and energy loss rate.

To calculate the pulsar frequency, firstly ‘test’ signals were sampled and transformed using FFTW to ensure that the approach of sampling, transformation and power spectrum generation in the frequency domain was correct and consistent for subsequent use on RXTE pulsar data. The ‘test’ time domain signals used were sine and sawtooth functions of an array of integers, which have known inherent fundamental frequencies and/or associated harmonics that transform to either/or a single/multiple frequency ‘spikes’ in the computed frequency spectrum. The approach common to both the test signals and the RXTE pulsar data used in our pulsar code was as follows:

The code implemented the algorithm in table 3, generating a ‘test’ sine signal in the time domain as per figure 4a, with a single frequency pre-set at 10 Hz (see Appendix). The results are shown in figure 4b:

Figure 4b shows a single peak with fundamental frequency 10Hz corresponding to the oscillations observed in the time domain. This provided adequate confidence and evidence that the algorithm to be applied to RXTE pulsar data was appropriate. As further confirmation before transforming RXTE pulsar data, a sawtooth function was transformed itself composed of a fundamental frequency accompanied by a number of overtones⁸ producing the FFT results in figure 5b. This further demonstrates the validity of the algorithm in table 3 as it clearly shows what is considered an appropriate FFT for a sawtooth function. The fundamental frequency occurs at 42Hz, the first overtone (2nd harmonic) at 84Hz, the second overtone (3rd harmonic) at 126 Hz.

Both the sine and sawtooth functions contained 512 data points which were sampled at a rate producing 256 ‘time bins’ (samples or intervals) containing time domain data. This sampling rate used was high enough so as to avoid any possible aliasing effects. As a further test for both test signals (not shown) decreasing the sampling rate shifted both spectrums in the frequency domain to the right, implying the detected fundamental frequency and associated harmonics were moved closer to the Nyquist rate set by the sampling frequency chosen.

The last 3 periods produced identical results and served as confirmation of appropriate choice of sampling rate(s) for RXTE data sets. Given the fastest spinning pulsar was found to have a spin rate below 20Hz (figure 7), a sampling period up to 0.05 seconds could have been employed and still satisfy the Nyquist-Shannon sampling criterion.

The FFT results produced for PSR B0540-69 and PSR B1509-58 are as follows:

The pulsar frequency and pulsar period results for all data sets analysed via FFTW (and compared to ATNF catalogue results and literature) is as follows:

Results from table 6 and 7 demonstrate an increase in spin period, accompanied by a decrease in pulsar frequency with increasing MET (mission elapsed time) as follows:

Back to Top | Back to High Mass Stellar Evolution

and is related to the pulsar age by:

where ‘dP’ is the change in pulsar period, ‘dt’ is the change in time related to dP (MET), and ‘P’ is the spin period of the pulsar in seconds. It should be noted that the ‘characteristic age’ calculation shown here assumes that pulsar spin period (and by consequence period derivative) is the key determinant for the ‘characteristic age’ calculation.

The period derivative for each pulsar was determined by subtracting the highest and lowest values of the spin period and MET respectively to determine ‘dP’ and ‘dt’ based on RXTE observations. The ratio of the two was taken to determine the period derivative for each pulsar. The resulting period derivative was used in the age calculation along with the pulsar period (from the last RXTE observation¹⁰ of each pulsar). ATNF catalogue and literature results are also shown:

Back to Top | Back to High Mass Stellar Evolution

where ‘I’ is the moment of inertia, ‘c’ is the speed of light, ‘R’ is the radius of the neutron star/pulsar, ‘P’ is the pulsar period and ‘dP/dt’ is the pulsar’s period derivative. Assuming the mass of each pulsar is 1.4 solar masses and the radius is 10^6 cm [7] an estimate for the moment of inertia ‘I’ can be obtained via: I = 2 / 5 * M * R^2, which gives I = 1.1e45 g.cm^2 [7]. In addition using a value of c equal to 2.99792458e10 cm s-1 [7] both magnetic field and energy generation rate can be calculated from previously calculated spin period and period derivative:

Back to Top | Back to High Mass Stellar Evolution

and energy generation (loss) rate:

there are a number of ‘fixed’ parameters and two ‘variable’ parameters namely ‘P’ being the spin period and ‘dP/dt’ the first period derivative of a pulsar. Changes in these two variable parameters will affect the magnetic field and energy generation rate of the pulsar. These relations shows that the product of spin period and period derivative for the magnetic field are both responsible for changes in a pulsar’s magnetic field and that the ratio of these same two (which is inversely proportional to pulsar age) given:

will in a similar way determine the energy generation rate at a given point in time for a pulsar. When the results of table 6,7,8,9 were analysed the following relative comparison was drawn:

Given the above relations for magnetic field and energy loss rate of a pulsar and table 10 analysis, the spin period appears to be the key determinant in both pulsar’s magnetic field and energy loss rate i.e. the spin period is directly proportional to the magnetic field and inversely proportional to the energy loss rate as shown in table 10. In addition one can see that faster spinning pulsars (small spin period) lose energy more quickly than their counterparts (slower spinning pulsars with larger spin periods). This is analogous to the behaviour observed by a ‘spinning top’ which loses a large amount of kinetic energy early in it’s life before gradually spinning down via surface friction. Consequently as a pulsar ages and spins down its ability to lose energy via magnetic dipole radiation is typically reduced.

Table 10 suggests also that if indeed PSR B1509-58 began life as a fast spinning pulsar (it has a larger spin period than it counterpart PSR B0540-69) in line with the accepted pulsar model whereby the spin period starts small and increases over time, it cannot be excluded that its magnetic field may have been in the high order of 10^13 Gauss or higher at ‘birth’. If this was the case in it’s early past, PSR B1509-58 may have been a very highly magnetised neutron star capable of emitting EM energy well into the x-ray region. We exclude the possibility that PSR B1509-58 may have been a magnetar based purely on characteristic age results (Magnetar models suggest ages of up to 10^4 years from birth) [8].

The results suggest also that small spin periods are typically associated with small period derivatives and vice versa, however in order to make this assertion with confidence one would need to determine both these characteristics across a larger population of pulsars. An argument in support of this assertion would be in the characteristics of millisecond pulsars and ordinary pulsars as shown in the following diagram:

Millisecond pulsars shown have small periods accompanied by small period derivatives whilst the same also appears to be true for ordinary pulsars. The characteristics described in table 10 and discussed earlier can be re-interpreted in figure 10 as follows:

• Both pulsars are young (order of 10^3 years) and belong to SNR’s
• Both pulsars have reasonably high magnetic fields (range 10^12 to 10^13 Gauss)
• Both pulsars have moderately high period derivatives and (with the exception of millisecond pulsars) and are reasonably fast spinning objects in line with their calculated ‘characteristic age’.

These observations place both PSR B0540-69 and PSR 1509-58 in the white circular region depicted in figure 10, in agreement with their SNR associations.

Back to Top | Back to High Mass Stellar Evolution

[20] Smith S. W., 1999, The Scientist’s & Engineers Guide To Digital Signal Processing, Chapter 3
http://www.dspguide.com

¹ ‘Period derivative’ is deemed the rate a which the period of a pulsar changes (‘first period derivative’ is also a term used) and is unit-less (sec sec-1).

³ Given the limitations imposed by the choice of operating system hence Cygwin choice, many tools listed were compiled directly from source code.

⁴ Discrete Fourier Transform hereafter is abbreviated to ‘DFT’.

⁵ Conventional DFT’s (DFT’s O(N^2)) can typically be implemented using simultaneous linear equations or correlation methods.

⁶ A transformation is defined as taking multiple inputs, performing an operation on these inputs based on a set of rules, and producing multiple outputs (many in, many out).

⁷ Aliasing causes reflection of high and low frequencies in the corresponding low and high frequency spectral regions computed in the FFT resulting in distortion of the sampled signal, hence causing a pulsar signal not be properly reconstructed from the sampled signal.

⁸ ‘Overtone’ is used in this instance, however the term implies additional sinusoidal frequencies which are not necessarily multiples of the fundamental frequency (unlike harmonics) – in the sawtooth example ‘harmonic’ is an appropriate description also, as well as for pulsar analysis.

⁹ Software limitations with Cygwin were encountered with small sampling periods (below 0.007 seconds) and large RXTE data sets. To mitigate this initially, some large pulsar data sets were reduced however the sampling periods were ultimately increased to avoid stack overflow issues caused by large arrays required for data folding eliminating the need to reduce pulsar data sets. The latter approach did not affect results as the corresponding Nyquist rates shown in table 5 (in bold italic) were still well above the Nyquist rate of each individual pulsar. A sampling period of 0.008 seconds was ultimately used in our code for both PSR B0540-69 and PSR B1509-58 to reduce computational time for all data sets.

¹⁰ The choice of observation was in the end arbitrary as the average values of all observations or other individual observations used in the age calculations did not adversely affect the age of the pulsar in calculated and shown in table 8.