Spectral properties of 441 radio pulsars

Abstract

1 INTRODUCTION

Although pulsars were discovered nearly 50 yr ago (Hewish et al. 1968), the exact mechanism by which they emit electro-magnetic radiation is far from being understood. The pulsar emission is often described using models that include a magnetosphere filled with an electron–positron plasma that corotates with the pulsar (Goldreich & Julian 1969). However, important details such as the location of the emission sites, including the polar cap close above the stellar surface (Sturrock 1971; Ruderman & Sutherland 1975; Baring 2004), the slot gap (Arons & Scharlemann 1979), or the outer gap (Cheng, Ho & Ruderman 1986), and the exact emission mechanism are still unclear. Potential mechanisms for the radio emission are coherent synchrotron-curvature radiation of particle bunches (Ruderman & Sutherland 1975; Gil, Lyubarsky & Melikidze 2004) and a form of plasma emission, apart from more exotic models (Melrose & Yuen 2016). Studying the radio spectra of pulsars could aid in developing an understanding of the emission mechanism, but accurate spectral data are sadly lacking for the majority of pulsars. The exception are a few pulsars that have been studied across a broader frequency range (e.g. Keith et al. 2011; Dai et al. 2015). A recent version of the Australia Telescope National Facility (ATNF) pulsar catalogue, version 1.54 (Manchester et al. 2005), shows that pulsar flux densities are relatively well known near 1.4 GHz and also 400 MHz, where most of the pulsars were discovered, but are poorly known at other radio frequencies. Out of the 2536 currently known pulsars, about 66  per cent have a recorded flux density measurement at 1.4 GHz; above 2 GHz, the fraction is only about 6  per cent; and, between 600 and 900 MHz, there are measurements for only 22  per cent of the pulsars. The low-frequency fluxes near 400 MHz are better known with about 30  per cent of the pulsars having measurements and the frequencies below that are subject of current study by low-frequency projects such as the Long Wavelength Array (LWA), Low Frequency Array (LOFAR) and Murchison Widefield Array (MWA) (Stovall et al. 2015; Bell et al. 2016; Bilous et al. 2016). The problem is not only that little data are available at intermediate and high (≥2 GHz) frequencies but also that the data in the catalogue were taken with different telescopes and different generations of receivers and backends, each with their own systematic errors. In addition, a large fraction of about 61  per cent at 1.4 GHz and nearly 80  per cent at 800 MHz of these flux density measurements are from the original discovery papers, which are typically not absolute flux density calibrated and simply estimates using the signal-to-noise ratio (S/N) of the detection converted into a flux density using the radiometer equation and supposedly known parameters of the observing system and sky temperature in that direction. That is done because observations of a calibrator source and the noise diode before each observation add a significant amount of overhead time to a survey. It was found that there can be significant differences between multiple measurements done this way (e.g. Levin et al. 2013). It is therefore crucial to obtain absolute flux density calibrated measurements.

Efforts to measure the radio spectra of a large number of pulsars began in earnest with Sieber (1973), followed by Malofeev & Malov (1980) and Izvekova et al. (1981) at low frequencies near 100 MHz and below. It was found that the majority of pulsars have steep spectra that can be described using a simple power law with spectral index α. There was also a realization that some pulsar spectra deviate from this at low frequencies and show a turn-over, which is usually attributed to either synchrotron self-absorption or thermal free–free absorption, while some show a high-frequency cut-off in the form of a spectral steepening or a break in the spectrum (Sieber 1973). For example, Lorimer et al. (1995) studied 280 pulsars and analysed their spectra, finding a mean spectral index of −1.6. Xilouris et al. (1996) summarized the frequency dependence of various parameters that characterize the pulsar emission, such as the shape of the spectrum, the modulation index, which describes the temporal variability in flux density, the separation of pulse components, the pulse width and the radius-to-frequency mapping, amongst others. They found a mean spectral index of −1.87 for simple power-law spectra and −1.55 before and −2.72 after a spectral break at 1 GHz for broken power-law spectra. Later Maron et al. (2000) extended the work of Lorimer et al. (1995) using mainly the same set of pulsars but broadened the frequency coverage. They derived a mean spectral index of −1.8 and realized that only 10 per cent of the pulsars studied show broken power-law spectra. Bates, Lorimer & Verbiest (2013) showed that the intrinsic spectral index distributions for three different pulsar surveys are Gaussian with a mean index of −1.4.

Spectral analysis also led to the recent discovery of a new class of pulsars that have gigahertz-peaked spectra (GPS): convex spectra where the flux density maximum occurs at a frequency around 1 GHz (Kijak, Gupta & Krzeszowski 2007; Kijak et al. 2011b; Dembska et al. 2015b; Kijak et al. 2017). It has been proposed that the GPS phenomenon occurs because of thermal free–free absorption in dense ionized material surrounding a pulsar, for example in a pulsar wind nebula (PWN), molecular cloud (MC) or supernova remnant (SNR; Lewandowski et al. 2015; Rajwade, Lorimer & Anderson 2016), or when the radio emission from a pulsar passes through the dense wind of a binary companion. In the case of the pulsar J1302–6350 (B1259–63), the only known radio pulsar with a main-sequence Be star companion, the pulsed flux density and its spectrum was found to vary with orbital phase (Johnston et al. 1996; Kijak et al. 2011a; Dembska et al. 2015a). There is considerable discussion about the spectral shape exhibited by GPS pulsars and their spectra have been modelled by broken power-law, log-parabolic (Bates et al. 2013; Dembska et al. 2014) and free–free absorption models by different authors. Throughout the paper, we define a GPS as any spectrum that peaks at a frequency near 1 GHz, regardless of spectral shape.

A major difficulty is that the observed flux densities of pulsars can vary significantly over time because of a combination of intrinsic fluctuations in the pulsar emission and strong diffractive interstellar scintillation (DISS) and refractive interstellar scintillation (RISS). DISS is the constructive and destructive interference of radio waves emitted by a pulsar at the observer's location, which have undergone scattering in the turbulent interstellar medium. The interference pattern changes over time because of the relative motion between the pulsar, scattering screen and observer and gives rise to strong fluctuations in observed flux density with a time-scale of minutes to hours (e.g. Armstrong, Rickett & Spangler 1995). RISS is responsible for weak long-term fluctuations in observed flux density when the effect of DISS is accounted for. It occurs because of focusing and defocussing of the pulsar emission by the scattering medium and has a time-scale of days to months (Sieber 1982; Romani, Narayan & Blandford 1986; Rickett 1990; Bhat, Rao & Gupta 1999). Experimentally, Stinebring et al. (2000) showed that high dispersion measure (DM) pulsars at large distances have nearly constant observed flux densities over years, indicating that the pulsar emission is stable when individual pulses are integrated for at least a few hours, and diffractive scintillation has been accounted for.

Another motivation to obtain accurate flux density measurements over a wide frequency range and corresponding spectral indices for a large fraction of the pulsar population is in order to make accurate predictions for pulsar surveys and observations with the Square Kilometre Array (SKA; Keane et al. 2015), SKA path finders and other radio telescopes, such as MeerKAT, the Five-hundred-meter Aperture Spherical radio Telescope (FAST) or the recently refurbished Molonglo Observatory Synthesis Telescope (Bailes et al. 2017). The flux density measurements contribute directly to the design of surveys via pulsar population synthesis (e.g. Bates et al. 2014), or to optimize observing strategies.

In this work, we concentrate on the stable radio spectrum as it appears after a minimum observation length of a few minutes. We use the total pulse profile to estimate the mean flux density, i.e. we do not consider the properties of individual pulses or profile components. Unless otherwise stated, we quote all uncertainties at the 1σ level. In Section 2, we describe the target selection and our observing programme at the Parkes telescope. In Section 3, we describe the data analysis methods including the absolute flux density calibration, flux density extraction and the modelling of the pulsar spectra. In Section 4, we present our results and show the objective spectral classification and correlation analysis, and in Section 5, we discuss these especially in relation to findings from the literature. Finally, in Section 6, we give our overall conclusions.

2 OBSERVATIONS

2.1 Target selection

We selected the target pulsars as follows. We extracted the available flux density data from version 1.54 of the ATNF pulsar catalogue and extrapolated to the centre frequencies 728 and 3100 MHz of the 10–50 cm receiver using a simple power law. For pulsars with a single flux density measurement, we used the median spectral index of −1.76 for the extrapolation. The sky temperature T_sky at each pulsar position was derived from the 408 MHz all-sky atlas of Haslam et al. (1982) and extrapolated to the centre frequencies using a power-law scaling with exponent of −2.6 (Lawson et al. 1987). We used the radiometer equation to determine the expected S/N of each pulsar (e.g. Lorimer & Kramer 2012). The parameters of the receivers used at Parkes were taken from the ATNF receiver fleet table.¹ Pulsars that were extensively studied as part of other Parkes projects such as P456 – Parkes Pulsar Timing Array (PPTA; Manchester et al. 2013) or P574 – Pulsar timing with the Parkes Radio Telescope for the Fermi mission (Weltevrede et al. 2010) were not considered, as they already had sufficient data. As one of the aims of the project was to inform the target selection at the refurbished Molonglo radio telescope, we chose only the pulsars from the above set which could be observed at Molonglo with an S/N of at least 20 in 10 min.

2.2 Observing set-up

We observed at Parkes simultaneously with two bands centred at 728 and 3100 MHz using the 10–50 cm receiver and used the CASPSR backend at 728 MHz and DFB4 or DFB3 at 3100 MHz. For the observations with the multibeam receiver centred at 1382 MHz, we also used CASPSR. One reason to use CASPSR was because of its advanced radio frequency interference (RFI) mitigation technique that uses spectral kurtosis to identify and eliminate RFI based on its deviation from a Gaussian distribution in the raw voltage stream (Nita & Gary 2010). This is particularly valuable for long period pulsars. Before each pulsar observation, we carried out a calibration observation 0.25° offset in right ascension from the pulsar position, which corresponds to about 1.1 full width at half-maximum (FWHM) central beam widths at 20 cm, lasting 50 s. In this observation, the noise diode inside the receiver was fired at a frequency of 11.123 Hz and the data folded. Afterwards, the pulsar observation was performed. Additionally, we observed an absolute flux density calibrator, the radio galaxy Hydra A, once every week of observations.

2.3 Observations

The P875 project was granted 76 h in total in semester 2014APR and 2014OCT. We combined the data with long-term pulsar observations from the Parkes project P574 (Weltevrede et al. 2010), especially in order to characterize the effect of strong RISS. The set comprises 8.5 yr of data beginning in 2007 July with a maximum number of 248 observing epochs. The raw data can be downloaded from the Parkes pulsar data archive (Hobbs et al. 2011).

3 ANALYSIS

3.1 Data analysis and calibration procedure

The polarization and absolute flux density of the observations were calibrated using the psrpl² calibration pipeline, which is based on PSRCHIVE tools (Hotan, van Straten & Manchester 2004). For the polarization calibration, we used the ideal feed assumption, and for the absolute flux density calibration, we used the radio galaxy Hydra A (3C218). While there are small discrepancies in published measurements, which could arise by different pointing positions along the two radio lobes, its flux density at 1.4 GHz and spectral index is well known (see Table 1). We adopted the value reported by Baars et al. (1977). In the data processing, we removed the band edges, where the gain falls off more than 25 per cent. The exact parameters of the data are shown in Table 2. We excised RFI in the frequency, time and pulsar rotational phase domains by searching for strong outliers that deviated at least 5σ from the smoothed mean values in each of these. In the worst case, we zero-weighted corrupted integrations.

3.2 Flux density extraction

The flux densities were extracted using a custom program that uses the python bindings to psrchive. We determined the peak and pulse averaged total flux density from the Stokes I component, for which we used the standard deviation of the baseline fluctuations as uncertainty. We extracted the flux densities for each observation of each pulsar individually and calculated the error-weighted mean and standard deviation after removing non-detections. The large bandwidth of the receivers and the high S/N of the observations allowed us to split them into frequency sub-bands and extract multiple flux density points. We determined the maximum number of sub-bands iteratively for each pulsar and all its observations for an S/N threshold of 10 per sub-band.

3.3 Flux density fluctuations due to scintillation

Interstellar scintillation can affect the measured flux densities of pulsars drastically. It is therefore crucial to take its effect into account in order to determine reliable estimates of the pulsar flux densities and their uncertainties. Therefore, we estimate the influence of scintillation on our flux density data in two ways: (1) using a theoretical simulation that incorporates the NE|$\scriptstyle{2001}$| Galactic free electron-density model (Cordes & Lazio 2002) and (2) directly from our flux density time series data. Here, we explain our methods, but show the results in Section 4.2.

3.3.1 Estimating the influence of scintillation using a theoretical simulation

We simulate the influence of strong diffractive and refractive, or weak scintillation on our observed pulsar flux density measurements theoretically using the NE|$\scriptstyle{2001}$| model and usual equations from the literature. We summarize the details of the simulation in Appendix A.

3.3.2 Flux density variability derived directly from our data

3.3.3 Combining the data

In the spectral plots in Figs 5, 7, 9 and 10, we show two error bars on our data: the inner and thicker one in lighter blue represents the statistical uncertainty due to scatter in the measurements |$\frac{\sigma _{{\rm r}, \nu }}{\sqrt{N}}$|⁠, whereas the outer error bar shows the total uncertainty (equation 3).

3.4 Robust regression and objective spectral classification

We combine our flux density data that we split into frequency sub-bands with spectral data from the literature (see Table 3). We compiled as much literature data as we could find, but do not claim that our database contains all pulsar flux density measurements ever obtained in the radio, which is simply infeasible. Nevertheless, we are confident that our combined data set represents the vast majority of pulsar spectral data available today. The literature data nicely extends our own measurements at frequencies below 400 MHz and above 3 GHz. To ensure uniqueness of data points, we gave preference to the data from individual publications rather than including them from summary publications such as the ATNF pulsar catalogue. For measurements where no uncertainty was given by the original authors we assumed a relative uncertainty of 50 per cent, which is a conservative value and was also adopted in earlier work (Sieber 1973). We show upper limits from this work and from the literature as such in the spectral plots, but exclude them from the spectral fit. That is because our fitting algorithm in its current form does not handle censored data.

3.5 Spectral models

We selected five spectral models from the literature that have distinctive spectral shapes, which is the key aspect that we can test using our measurements. They have been used successfully in the past to describe the spectra of pulsars, with the simple and broken power law being the most commonly applied ones. While these are morphological, the spectral index could potentially be related with other pulsar parameters (Section 4.12). Our model selection is sufficiently diverse to fully represent the spectral shapes seen in the data set. However, we cannot test every conceivable model and we limit ourselves to those five classes. In particular, we fit the following analytical models to the data, where ν is the centre frequency and ν₀ = 1.3 GHz a constant reference frequency:

• Simple power law of the form:

  \begin{eqnarray} S_\nu = b \, x^{\alpha }, \end{eqnarray}

  (12)

  where |$x = \frac{\nu }{\nu _0}$|⁠, α is the spectral index and b is a constant. The fit parameters are α and b.
• Broken power law of the form:

  \begin{eqnarray} S_\nu = b \, \big\lbrace \begin{array}{@{}l@{\quad }l@{}}x^{\alpha _1} & {\rm if} \, \, x \le x_{\rm b}\\ x^{\alpha _2} \, x_{\rm b}^{\alpha _1 - \alpha _2} & {\rm otherwise}\\ \end{array}\big., \end{eqnarray}

  (13)

  where |$x = \frac{\nu }{\nu _0}$|⁠, |$x_{\rm b} = \frac{\nu _{\rm b}}{\nu _0}$|⁠, ν_b is the frequency of the spectral break, and α₁ is the spectral index before and α₂ the one after the break. The fit parameters are b, α₁, α₂ and ν_b.
• Log-parabolic spectrum (LPS) of the form:

  \begin{eqnarray} \log _{10} S_\nu = a x^2 + b x + c, \end{eqnarray}

  (14)

  where |$x = \log _{10} ( \frac{\nu }{\nu _0} )$|⁠, a is the curvature parameter, b is the spectral index for a = 0 and c is a constant. This model has been used to describe the spectra of radio galaxies (e.g. Baars et al. 1977) and curved pulsar spectra (Bates et al. 2013; Dembska et al. 2014). The fit parameters are a, b and c.
• Power law with high-frequency cut-off of the form:

  \begin{eqnarray} S_\nu = b \, x^{-2} \, \left( 1 - \frac{x}{x_{\rm c}} \right), \qquad x < x_{\rm c} \end{eqnarray}

  (15)

  where |$x = \frac{\nu }{\nu _0}$|⁠, |$x_{\rm c} = \frac{\nu _{\rm c}}{\nu _0}$|⁠, b is a constant and ν_c is the cut-off frequency. The fit parameters are b and ν_c. The functional form is based on the work by Kontorovich & Flanchik (2013) and provides us with a direct test of their model for the coherent pulsar emission. Specifically, the radio emission should be created by electrons accelerated in the electric field of the pulsar and the cut-off frequency of the spectrum should depend on the maximum value of the electric field E_max as

  \begin{eqnarray} \nu _{\rm c} = \sqrt{\frac{ e E_{\rm max} }{ 2 m_e h }} = \sqrt{\frac{\pi e B}{m_e c P}}, \end{eqnarray}

  (16)

  where e is the charge and m_e is the mass of the electron, h is Planck's constant and c is the speed of light in vacuum. The second identity is true under an assumption about the maximum electric field in the centre of the polar cap, where P is the period and B is the magnetic field of the pulsar.
• Power law with low-frequency turn-over of the form:

  \begin{eqnarray} S_\nu = b \, x^{\alpha } \exp \left( \frac{ \alpha }{ \beta } \, x_{\rm c}^{-\beta } \right), \end{eqnarray}

  (17)

  where |$x = \frac{\nu }{\nu _0}$|⁠, |$x_{\rm c} = \frac{\nu }{\nu _{\rm c}}$|⁠, α is the spectral index, ν_c is the turn-over frequency, b is a constant and 0 < β ≤ 2.1 determines the smoothness of the turn-over. Fit parameters are α, ν_c, b and β. This functional form is motivated by a synchrotron self-absorption process proposed to be responsible for the low-frequency turn-over (Izvekova et al. 1981), but can describe the spectra expected due to both synchrotron self and thermal free–free absorption. For the special case β = 2.1, it is equivalent to a free–free absorption model (e.g. Rajwade et al. 2016; Kijak et al. 2017).

3.6 Quality of the spectral classification – classification categories

In the following sections, we discuss the spectral classifications of individual pulsars. They get classified by the best-fitting spectral model – the one with the lowest AIC. However, this classification has different qualities, for example the best-fitting spectral model might only be slightly preferred by the data over all the others tested. To quantify that in an objective way, we define the following categories (as opposed to classification) based on the value of p_best and the significance s in standard deviations σ of the spectral feature at hand (for a log-parabolic spectrum that is the curvature coefficient a for example):

• weak: p_best < 0.5 or s < 2.

• candidate: 0.5 ≤ p_best < 0.7 and s ≥ 2.

• strong: p_best ≥ 0.7 and s ≥ 2.

• clear: p_best ≥ 0.8 and s ≥ 3.

4 RESULTS

4.1 Parkes system temperature estimation

As part of the absolute flux density calibration procedure, the system equivalent flux density is estimated, which we denote as S_sys and is the sum over both polarizations of the receiver. We determined it from our Hydra A flux density calibrator observations and list it and the resulting system temperature T_sys for each receiver used in Table 2. The gain used to compute T_sys was taken from the ATNF receiver fleet table. Note that the measured system temperature of the 10 cm part of the 10–50 cm receiver and especially the one of the multibeam receiver are slightly lower than what is quoted in the receiver fleet table. They are lower by about 2 and 6 K, respectively. The value measured for the 50 cm part agrees well.

4.2 Flux density variability

4.2.1 Results from the theoretical simulation

We conducted a theoretical simulation using the NE|$\scriptstyle{2001}$| model to quantify the effect of scintillation on our data (see Section 3.3). We find that the vast majority of pulsars have a total modulation index close to zero. Specifically, about 83, 75 and 60  per cent of the pulsars have m_tot ≤ 0.3 at 728, 1382 and 3100 MHz. The modulation is less than that for the band-integrated flux densities.

4.2.2 Results from the data-driven approach

Figure 1.

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Measured robust modulation index versus DM of the pulsars with at least six observing epochs at a centre frequency of 3.1 GHz. The red solid line shows the best error-weighted fit of equation (18) to the data. The orange band is the uncertainty introduced by making the bootstrap errors symmetric in the fitting procedure.

We also compare the scaling of the robust modulation index with DM at the different observing frequencies (see Fig. 2). In order to reduce the scatter that is present in the data, we binned them in equal DM bins in logarithmic space and computed the weighted means for each. As uncertainty we estimated the standard error in each DM bin. We find that the modulation is the highest at 3.1 GHz, followed by 1.4 GHz and 728 MHz, except in the highest DM bin centred at 868 pc cm⁻³, where the value at 1.4 GHz exceeds the other two. Above a DM of 126 pc cm⁻³ the modulation index is nearly constant with a weighted mean value of around 0.14 at 1.4 GHz. At the other two frequencies, it is still decreasing with DM.

Figure 2.

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Comparison of the scaling with DM of the robust modulation index determined from our data for the three centre frequencies.

4.2.3 Comparison of the two approaches

We find that both approaches are largely consistent at 728 MHz and 3.1 GHz. Both modulation indices are in good agreement there, with the measured ones showing a larger scatter than the theoretical ones. However, at 1.4 GHz there is considerably more variability in the data than expected from the theoretical simulation. The variability at 1.4 GHz is presented in Fig. 3, where we show a comparison of the robust modulation indices measured from our flux density time series data with the theoretically expected modulation indices calculated using the NE|$\scriptstyle{2001}$| model. In the top panel, we directly compare the modulation indices, and in the bottom panel, we compare the modulation indices plotted against DM of the pulsars. Top panel: There is a large scatter around the identity line and the measured modulation indices reach a plateau with a median of about 0.22 for the majority of pulsars. This is most likely radiometer noise in combination with fluctuations in absolute flux density calibration and residual pulse-to-pulse fluctuations (see also Stinebring et al. 2000). In this noise-limited region, the theoretical simulation significantly underestimates the true scatter in the data. Bottom panel: The measured modulation indices exceed the theoretically expected ones in nearly all cases and the theoretical ones are effectively a lower limit for the flux density variability for a realistic data set that is limited in observation time and S/N. This is most obvious at 1.4 GHz.

Figure 3.

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Comparison between the modulation indices at 1.4 GHz measured from our flux density time series data and the theoretical prediction based on the NE|$\scriptstyle{2001}$| model. Top: direct comparison between the modulation indices, with the red solid line indicating the identity. Bottom: modulation index versus DM of the pulsars.

We conclude that our empirical scaling relationship (equation 18) based on the parameters determined from our data is therefore a more realistic and more conservative estimate for the flux density variability than the theoretically expected one. Therefore, we use it as an estimate for the uncertainty |$u_{{\rm scint}} ({\rm DM}, \nu ) = m_{{\rm r},\nu } \left( {\rm DM}, \nu \right) \, \bar{S}_\nu$| in flux density due to scintillation.

4.3 Calibrated flux densities

One of the main results of this paper are the band-integrated, calibrated flux densities at 728, 1382 and 3100 MHz for all the pulsars in our data set, which are listed in Table C1 in Appendix C. The table contains the data for 441 pulsars. In case a pulsar was never detected with an S/N of at least 6 at a certain frequency we quote a 3σ upper limit for its flux density. The table also contains the spectral classification, the range of frequencies over which the spectral classification was performed, the spectral index for pulsars that have simple power-law spectra and the robust modulation index in case we had at least six measurement epochs at that centre frequency.

4.4 Comparison with data from the literature

A comparison between the flux density measurements from this work and the ones from the ATNF pulsar catalogue is shown in Fig. 4. We show only the direct matches with the catalogue, i.e. measurements that are within 100 MHz of a particular centre frequency. While there are 219 matches at 1.4 GHz, there are only 10 and 20 at 728 and 3100 MHz, respectively. Effectively, all our measurements at those centre frequencies are new ones. Green rectangles show the flux density points that deviate by at least 5σ from the identity line. Our data and the pulsar catalogue data are generally in good agreement, with the majority of data points located close to the identity line. The agreement at 1.4 GHz is good, with only 33 measurements deviating significantly. At 3.1 GHz, only two measurements deviate significantly, but the uncertainties are generally larger. Unfortunately, sometimes no uncertainties are given in the catalogue. Of the 33 5σ outliers at 1.4 GHz, 30 are from publications of the Parkes Multibeam Pulsar Survey (PMPS). Twenty-three are from Hobbs et al. (2004), four from Lorimer et al. (2006), two from Kramer et al. (2003) and one from Manchester et al. (2001), and all are based on the same flux density estimation method. We had already noted that in multiple cases their flux densities are systematically about a factor of 2 lower than both our data and other data from the literature. The discrepancy is maybe not surprising, as PMPS flux densities were estimated using the radiometer equation and observations of high-DM pulsars used as standard candles (Manchester et al. 2001). We notice that three of their calibration pulsars have flux densities at 1.4 GHz that are roughly half of what we measure for them. The other outliers are each from different older publications. Apart from these outliers, the rms relative difference is 31 per cent. Nonetheless, more of our measurements at 1.4 GHz lay above the identity line than below, which indicates that our measurements are slightly biased high. That could be a result of a decrease in the flux density of Hydra A, or a slight offset in telescope pointing – the spectral index of Hydra A changes rapidly across its extent (Lane et al. 2004). It is not due to the exclusion of observations below an S/N of 6 in the flux density estimation; even if these are included we see the same slight bias.

Figure 4.

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Comparison between the measured flux densities from this work and directly matched flux densities within 100 MHz from the ATNF pulsar catalogue at a centre frequency of 1.4 GHz. The red solid line indicates the identity.

4.5 Modelling the pulsar spectra

We used our flux density measurements that are split into frequency sub-bands combined with data from the literature and fit different spectral models to the combined data set in a robust manner as described in Section 3.4. We decide to use the following requirements for the characterization of pulsar spectra: the spectra need to consist of at least four flux density measurements at four different centre frequencies and the data points must span at least a factor of 2 in frequency. This choice provides a balance between sufficient spectral coverage for the model fitting and the amount of pulsars that can be classified. In particular, it also ensures that we cannot characterize a spectrum using only observations at 10 cm that are split into frequency sub-bands. The pulsars for which the requirements are not fulfilled are excluded from the spectral analysis. We characterize the spectra based on the best-fitting model, that is the one with the lowest AIC. We can successfully characterize the spectra of 349 pulsars, which is about 79 per cent of the total data set after the removal of non-detections. The results are shown in Table 5 where we give an overview of the fraction of pulsar spectra that can be best characterized by a given spectral model.

We find that the majority of pulsar spectra, about 79 per cent, can best be characterized as a simple power law in the frequency range studied. The pulsars with log-parabolic spectra account for about 10 per cent and the broken power-law spectra for about 7 per cent. The spectral models with a low-frequency turn-over and hard high-frequency cut-off are rare cases accounting for about 3 and less than 1 per cent each. We list the pulsars that have spectra that deviate significantly from a simple power law in separate tables depending on the spectral classification: LPS pulsars in Table 7, pulsars with broken power-law spectra in Table 8, the ones with a hard high-frequency cut-off in Table 9 and the ones with power-law spectra that have a low-frequency turn-over in Table 10. For each spectral model, apart from a simple power law we show one example where it fits the data best in Fig. 5.

Figure 5.

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Example spectra where each spectral model fits best, from top left in clockwise direction: power law with low-frequency turn-over, broken power law, log-parabolic spectrum and power law with high-frequency hard cut-off. In this and all other spectral plots, we show two error bars on our data: the inner one in lighter blue represents the statistical uncertainty due to scatter in the measurements, whereas the outer error bar shows the total uncertainty, which additionally includes scintillation and the systematic uncertainty (equation 3).

4.5.1 How does the frequency coverage affect the spectral classification?

The classification depends naturally on the spectral coverage, i.e. whether spectral features can be determined from the data. Fortunately, the combination of our measurements with literature data provides reasonable to very good coverage in terms of the number of data points (median 13, maximum 90) and fractional frequency coverage (median 7.8, maximum 1600) for all pulsars that fulfilled our classification requirements. We examine the classification of pulsars for which we have good low- or high-frequency coverage separately. We have good low-frequency coverage, which we define as having at least two data points below 600 MHz for 119 pulsars and good high-frequency coverage with at least one data point above 4 GHz for 88 pulsars. For the ones with good low-frequency coverage, the simple power law is still the most common spectrum (56 per cent), followed by the broken power law with 16 per cent and the LPS with 14 per cent. Only 10 pulsars have a power-law spectrum with low-frequency turn-over. For the pulsars with good high-frequency coverage simple power-law spectra account for 56 per cent, LPS for 18 per cent and broken power-law spectra for 17 per cent. That means that with good low-frequency coverage, a spectral break is significantly favoured in comparison with the whole data set. At high frequencies, a spectral break is slightly favoured over spectral curvature with both showing an increase by a factor of 2 or more in fraction.

4.6 Simple power-law spectra and spectral indices

The majority of pulsars, 276 in total, have spectra that follow a simple power law. The best-fitting spectral indices are shown in Table C1 in Appendix C. A histogram of the resulting spectral indices is shown in Fig. 6. The mean spectral index is −1.57 ± 0.62, the one weighted by the uncertainty of each individual index is −1.60 ± 0.54 and the median spectral index is −1.65. The uncertainty given here is the (weighted) standard deviation. When quoting the standard error instead, the weighted mean spectral index is −1.60 ± 0.03. We also show a Gaussian and a shifted log-normal fit to the data, together with a kernel density estimation (KDE) using a Gaussian kernel. The KDE largely agrees with the log-normal fit. The spectral index distribution deviates from a Gaussian and has a tail that extends towards positive values. We used the Shapiro–Wilk (S–W) test for normality to check quantitatively whether the observed spectral indices are sampled from a Gaussian or a log-normal distribution, for which we transformed the data into logarithmic space according to z = log₁₀(α + 5) and applied the S–W test to the transformed data. We find that we have to reject the null hypothesis in both cases. However, for the log-normal distribution it is close with values 0.99 (0.02). A detailed investigation of the unbinned cumulative distribution function and quantile-quantile (Q–Q) plots for both distributions leads us to conclude that a log-normal distribution fits the data nearly perfectly, while a Gaussian shows deviations. When we constrain the spectral indices to the 257 that are well determined with uncertainties of less than 0.5, the S–W test indicates with values 0.99 (0.17), that we cannot reject the null hypothesis that the data was sampled from a log-normal distribution. The best-fitting values are 0.2, −4.6 and 3.0 for the shape, location and scale parameter, which corresponds to a skew of 0.6 and excess kurtosis of 0.7. We still have to reject the null hypothesis for a Gaussian distribution.

Figure 6.

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Histogram of the spectral indices α for all pulsars that were classified to have simple power-law spectra.

We investigate the pulsars at the extremes of the spectral index distribution individually and in particular the ones with flat, or positive spectral indices. We list these in Table 6. All pulsars except for PSR J1832–0644 have only a weak classification. Interestingly, PSR J1028–5819 is close to having a GPS with a peak at roughly 2 GHz. However, a simple power law is preferred because of the large uncertainty of the data at 3.1 GHz. In other aspects, the pulsar is special because it has an extremely narrow pulse profile with an FWHM of only 0.3 ms and a corresponding duty cycle of about 0.3 per cent at 1.4 GHz, which is amongst the narrowest profiles known. It was suggested that this is because we are grazing the emission beam (Keith et al. 2008). Moreover, profile components at the edges of beams seem to exhibit flatter spectral indices in general (Lyne & Manchester 1988; Kramer et al. 1994; Dai et al. 2015), and this is what we could see here. For PSR J1832–0644 the spectral index is determined from our data at 3.1 GHz and literature data at 1.4 GHz from PMPS (Morris et al. 2002) only. The positive spectral index is likely caused by a difference in absolute flux density scale (see Section 4.4). The pulsars with the steepest spectra are PSR J1059–5742 (−3.3 ± 0.4), which has a well-determined spectrum from 0.4 to 3.1 GHz, and PSR J1833–0338 (−2.8 ± 0.1) with an equally well-determined spectrum from 0.1 to 3.1 GHz and a hint of a spectral break below 400 MHz. Both have a candidate classification.

We also tested whether the Galactic plane affected the measured spectral indices and in particular if a hotspot in the Galactic plane led to an underestimate of the low-frequency flux densities, resulting in a shallow spectral index. We do not find any correlation of spectral index with Galactic latitude or longitude. Generally speaking, most of the pulsars are young (τ ≤ 3.2 × 10⁵ yr) or energetic (⁠|$\dot{E} \ge 10^{34} \, {\rm erg} \, {\rm s}^{-1}$|⁠), indicating that these have flatter spectral indices on average.

4.7 Log-parabolic spectra

The table contains three separate classes of LPS pulsars: (1) four pulsars with slightly concave spectra with positive curvature coefficients. Their classification categories are either weak or candidate and all of them have flux density measurements with relatively large uncertainties below 400 MHz. We expect that their spectral classifications will shift towards simple power laws once low-frequency data are available. (2) Twenty-one pulsars with spectral peaks at frequencies up to about 500 MHz and (3) 10 pulsars whose spectra peak between about 0.6 and 2 GHz, indicating that they belong to the class of GPS pulsars. The two pulsars with the highest DMs of close to 1000 pc cm⁻³, PSRs J1019–5749 and J1410–6132, show large amounts of scatter broadening of their profiles at 1.4 GHz, covering 50–70 per cent of pulse longitude, and large expected f_scat. Their flux densities at and below 1.4 GHz are most likely underestimated and the LPS classification a result of this. Interferometric techniques are needed to determine their flux densities accurately below 1.4 GHz (see e.g. Dembska et al. 2015b). We exclude them from further discussion. The remaining pulsars include four known, one newly identified and three potential GPS pulsars. We discuss the GPS pulsars separately in Section 4.11 and describe a small selection of LPS pulsars below.

PSR J0823+0159: This is the only pulsar in a binary system. We have good frequency coverage from 25 MHz to 4.8 GHz. A curved spectrum is slightly preferred with a peak at around 60 MHz. PSR J1024–0719: It is an MSP and studied as part of the PPTA with good spectral coverage from 100 MHz to 5 GHz. An LPS is preferred with a concave spectral shape and a candidate category. It seems that the spectrum curves up at low and high frequencies. PSR J1512–5759: The 50 cm data have a positive spectral index and an LPS is weakly preferred with a broken power law being second.

With the aim of understanding the physical origin of the LPS phenomenon, we list the available information about the environments in which the pulsars are located and search for associations with sources at other frequencies (see Table 7). The table contains two MSPs and only one pulsar that is known to be in a binary system. PSR J0823+0159 is the only case in which the LPS could be due to absorption in the wind of a companion. However, J0823+0159's companion is a DA white dwarf in a wide orbit (Koester & Reimers 2000; van Kerkwijk et al. 2005), which should not have a significant wind. In addition, we tested for correlations of the curvature parameter a and the parameter b with DM, spin frequency |$\tilde{\nu }$| and spin-down rate |$\dot{\tilde{\nu }}$| for all LPS pulsars. We do not find any significant dependence of a and b on DM. Apart from that we see an increase of b with increasing spin frequency, except for the MSPs, and with increasing absolute spin-down rate for all pulsars that have a measured |$\dot{\tilde{\nu }}$|⁠. The dependence is similar to what we find for the spectral indices of pulsars with simple power-law spectra (see Section 4.12). The curvature parameter a however appears to be uncorrelated with |$\tilde{\nu }$| and |$\dot{\tilde{\nu }}$|⁠.

4.8 Broken power-law spectra

The pulsars with broken power-law spectra are listed in Table 8. The most prominent examples are PSR J0437–4715, the brightest and closest MSP, whose spectrum breaks around 2 GHz and is well determined from 70 MHz to 17 GHz; the Vela pulsar (PSR J0835–4510), which is well studied from 70 MHz to 24.4 GHz and seems to have a flat spectrum below 900 MHz; and the bright pulsar J0742–2822, whose spectrum flattens slightly above 1.4 GHz. Other examples are PSR J1045–4509, an MSP studied as part of the PPTA, and the pulsar J1522–5829, whose spectrum seems to be flat above 1.4 GHz.

4.9 Power-law spectra with high-frequency cut-off and magnetic field in polar cap

The measured magnetic field strengths are between about 1 and 7 × 10¹¹ G and the inferred altitudes of the polar cap centres vary from 15 to 27 km (0.05 to 0.1 per cent R_LC). This suggests that the pulsar emission is produced extremely close to the surface. The derived values are much smaller than what is traditionally considered. Usually, the polar cap radio emission is thought to be created at an altitude lower than 10 per cent of the light cylinder radius and at a height of less than about 100–400 km, as determined by the most recent measurements that were conducted with focus on low frequencies (Hassall et al. 2012; Perera et al. 2016). However, radius-to-frequency mapping predicts that low frequency emission is created higher in the magnetosphere than that at higher frequency (Cordes 1978). As we are probing much higher frequencies of up to 8 GHz, the expected altitude should be lower than a few hundreds of kilometres. Interestingly, other authors have also derived very low emission altitudes of about 29 km (0.24 per cent R_LC) from pulse separation measurements at 42 and 74 MHz (Tsai et al. 2016), which is a completely different method than our spectral fitting. Nevertheless, the derived emission altitudes seem unreasonably low, which could point to problems in the spectral classification (the highest category is candidate), or to problems in the theoretical emission model of Kontorovich & Flanchik (2013), its underlying assumptions, or to deviations from a simple magnetic dipole scaling of the field strength near the neutron star surface. Further high-frequency data will help to better constrain the spectra and may resolve this issue.

4.10 Power-law spectra with low-frequency turn-over

The pulsars that have power-law spectra with a low-frequency turn-over are listed in Table 10. Notable examples are the pulsars J0630–2834, J0837+0610 and J0953+0755, whose spectra are very well determined from literature data and our own measurements from about 20 MHz up to 10, 4.9 and 14.8 GHz. They are amongst the pulsars with the most data points in our set. Their spectral classification is clear, and they have turn-over frequencies of 50–100 MHz. A further example is the pulsar J1803–2137, which has a turn-over near 1 GHz. None of these pulsars is in a binary system, but many are visible in X-rays, suggesting a connection of the turn-over with absorption in a PWN. Most of the pulsars have strong to clear classifications, or are very close to it. Half of them have smooth turn-overs with β near or below unity and half have sharper transitions with β ≥ 1.5.

4.10.1 Examining evidence for free–free absorption

The thermal free–free absorption model as implemented by Rajwade et al. (2016) or Kijak et al. (2017) is a special case of this model, where the smoothness parameter β is fixed to a constant value of 2.1. As such, it has three free parameters, one less than the more generic low-frequency turn-over model. For completeness and to enable comparison with previous work, we test this case separately from the rest of the analysis presented in this paper. That is, we compare the AIC of the free–free absorption model to those of the four other models (as described in Section 3.5) and present the parameters of the spectra that are best characterized by it in the bottom part of Table 10. However, the values of p_best and the classification categories in the bottom part of Table 10 should not be compared with p_best and the categories presented for other models (in Tables 7–9). The free–free absorption model is not included amongst the models compared in the rest of this work; therefore, its AIC is not included in the calculation of p_best (equation 11) as presented in other tables. Consequently, the bottom of Table 10 contains some pulsars that have higher values of p_best for other models (e.g. PSRs J1705–3950 and J1723–3659 in Table 7), although the free–free absorption model becomes the preferred model for these pulsars when it is included amongst those compared, the preference is less significant. On the other hand, the table also contains a few pulsars that have higher values of p_best for the free–free absorption model (e.g. PSRs J1743–3150 and J1835–1020 in Tables 8 and 7).

Generally, the table contains most of the pulsars that have low-frequency turn-over spectra, all newly identified GPS pulsars, one of the potential GPS pulsars and five, but not all, of the known GPS pulsars (see Section 4.11). In addition, it contains a small number of pulsars that otherwise have mainly weak log-parabolic, simple or broken power-law classifications. This suggests that the turn-over in these spectra could be due to free–free absorption, at least for the spectra with relatively high turn-over frequencies near 500 MHz or above. However, we do not classify them as free–free absorption for the following reason: there is considerable covariance between the β parameter and the other fit parameters, in the sense that for other fixed values of β between 0.5 and 2.1 a similar set of pulsars is objectively characterized as a power-law spectrum with low-frequency turn-over (and fixed β), but with shifted values of spectral index, turn-over frequency and normalization. This is because the spectral coverage at low frequencies is insufficient or too uncertain to unambiguously resolve the sharpness of the turn-over for many of these pulsars. For this reason, we prefer the more generic turn-over model, where β is a free fit parameter, and classify the spectra based on that.

4.11 Gigahertz-peaked spectra

Eleven pulsars in our data set show GPS, by which we mean spectra with peaks between about 0.6 and 2.0 GHz, regardless of best-fitting spectral model. Three others are potential GPS pulsars. The distribution of best-fitting models is dominated by the LPS, with eight pulsars having an LPS, four a broken power law and two a power law with low-frequency turn-over. In addition, we independently measured flux densities for all known GPS pulsars from the literature, except PSR J1740+1000. In particular, these are PSRs J1056–6258, J1743–3150, J1803–2137, J1809–1917, J1825–1446, J1826–1334 and J1852–0635 (Kijak et al. 2007, 2011b; Dembska et al. 2014, 2015b; Basu et al. 2016). Low-frequency measurements by Bilous et al. (2016) recently suggested that PSR J1740+1000, which was initially thought to show GPS, has a simple power-law spectrum with a large amount of flux density variability. During the refereeing process of this paper, Kijak et al. (2017) identified another five GPS pulsars (PSRs J1644–4559, J1723–3659, J1835–1020, J1841–0345 and J1901+0510), of which we have measurements for three. In the following, we discuss the most interesting spectra.

4.11.1 New identifications

We show the spectra of the newly identified GPS pulsars in Fig. 7. PSR J1705–3950: It is a newly identified GPS pulsar with a candidate LPS classification. Dembska et al. (2014) declared it as a potential GPS pulsar, but could not determine the spectral turn-over. PSR J1751–3323: Our and the data from Han et al. (2016) determine the spectral break around 1.3 GHz nicely. Its classification as a broken power law is only weak, but the second most-preferred model is an LPS with a peak around 1.1 GHz. The combined probability is 0.74 with respect to all other models tested. PSR J1806–1154: The pulsar has a broken power-law spectrum with a break around 900 MHz.

Figure 7.

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Spectra of the newly identified GPS pulsars.

4.11.2 Potential GPS pulsars

In addition to the new identifications, we find three further pulsars that seem to exhibit GPS. In fact, two of them have a clear classification, indicating that significant spectral curvature is present and that the spectral model is clearly preferred over all others tested. However, we consider them potential GPS pulsars until further low-frequency measurements are available that confirm their turn-overs beyond doubt. We present their spectra in Fig. 8. They are PSR J1055–6028, which has a clear LPS classification and a peak around 900 MHz. We resolve its turn-over partially. Its pulse profile at 1.4 GHz has a width of about 3.7 ms, occupies less than 10 per cent pulse longitude, and a single-sided exponential scattering tail is visible. The estimate of the scatter-broadening based on equation (19) is too high. PSR J1830–1059 is another potential GPS pulsar with a clear LPS classification. Kijak et al. (2011b) declared its spectrum as unclear and speculated that it might be a broken power law. Kijak et al. (2017) follows the same argument, but fits its spectrum using a free–free absorption model. PSR J1843–0211: The pulsar has an LPS spectrum with a peak around 600 MHz and a candidate classification.