Electron Density Variations in the Interstellar Medium and the Average Frequency Profile of a Scintle from Pulsar Scintillation Spectra

Scintillations of pulsars are related to electron density fluctuations in the ISM, which can be characterized by a spatial correlation function. Its Fourier transform is the spatial wavenumber spectrum, P(q). If the magnitude of the three-dimensional wavenumber, q, is within a limited range of wavenumbers with inner and outer boundaries, then P(q) can be described as a power law, P_P(q) = C_n q^−α , with α < 4 (Romani et al. 1986) including α = 11/3 for a Kolmogorov spectrum. For a Gaussian the spectrum is ${P}_{{\rm{G}}}{(q)={G}_{n}\exp (-(q/{q}_{0})}^{-2})$, corresponding to α = 4. The value of α can be obtained from the two-dimensional ACF of S(f, t) at zero time lag, ACF(Δf, Δt = 0). For simplicity, we refer to this frequency section after normalization as ACF(Δf) with

$\begin{eqnarray}&&\mathrm{ACF}({\rm{\Delta }}f)={\left\langle S(f,t)S(f+{\rm{\Delta }}f,t)\right\rangle }_{t}.\end{eqnarray} \tag{ 2 }$

The frequency section, ACF(Δf), is predicted to depend on α. For a thin-screen model of the ISM analytical solutions exist for the Gaussian and power-law models that can be fit to the ACFs. For an extended-medium model of the ISM (e.g., Lee & Jokipii 1975a) analytical solutions for the frequency section of the ACF exist for the Gaussian spectrum but only numerical solutions for the power-law spectrum (Lee & Jokipii 1975b).

Instead of computing the frequency section, ACF(Δf), for comparison with predictions, some authors have preferred to compare the Fourier transform of this ACF with predictions. Both approaches can be considered complementary. In this paper we use both approaches, by following Wolszczan (1983) concerning the computation of the predictions of the extended-medium scattering model and Armstrong & Rickett (1981) concerning the thin-screen model.

We studied 12 pulsars by analyzing the frequency section, ACF(Δf), of the dynamic spectra. From ACF(Δf) we determined the decorrelation bandwidth, Δf_1/2, as the half-width at half maximum (HWHM), and list the values for each pulsar in Table 2. The fractional statistical (rms) error can be estimated as the inverse square root of the number of scintles in each dynamic spectrum (Backer 1975). An approximate expression is given by ${\sigma }_{\mathrm{stat}}={\left({f}_{s}\tfrac{{{BT}}_{\mathrm{obs}}}{{\rm{\Delta }}{f}_{1/2}{t}_{\mathrm{scint}}}\right)}^{-1/2}$, where f_s is the filling factor, B the receiver bandwidth, T_obs the observing time, and t_scint the scintillation time. Assuming f _s = 0.5, and taking values for t_scint from Popov & Smirnova (2021), we list σ_stat in Table 2. For a consideration of possible systematic errors, see Popov & Smirnova (2021).

Table 2. Results of Data Reduction

PSR	Δf1/2	Δτ1/2	σstat	αACF	rmsACF	${\alpha }_{{ \mathcal F }{ \mathcal T }(\mathrm{ACF})}$	rmsscat. type	Scat. Type
	(kHz)	(μs)	(10−2)		(10−2)		(10−2)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
B0329+54	30	7.90	1.0	3.67 ± 0.02	0.05	3.80 ± 0.05	0.15	K
B0525+21	4000	0.04	8.6	3.73 ± 0.02	0.5	3.80 ± 0.05	3.1/4.2/4.2	K/E/L
B0823+26	180	0.83	1.2	3.56 ± 0.02	0.2	3.60 ± 0.05	1.8	K
B0834+06	400	0.55	5.8	3.74 ± 0.02	0.2	4.00 ± 0.10	1.8	K
B0919+06	170	1.15	1.9	3.59 ± 0.02	0.2	3.70 ± 0.10	1.7	K
B1133+16	96	2.50	0.7	3.80 ± 0.02	0.3	4.00 ± 0.10	2.3/3.7	K/L
B1237+25	2300	0.11	3.3	3.90 ± 0.04	0.7	3.80 ± 0.10	2.7/3.8	G/E
B1642−03	930	0.20	4.0	3.97 ± 0.05	1.4	3.80 ± 0.20	3.0/3.7	G/E
B1749−28	690	0.31	3.5	3.78 ± 0.02	0.2	3.90 ± 0.10	3.7/3.8/3.9	E/L/K
B1929+10	970	0.39	6.8	3.85 ± 0.02	0.4	3.90 ± 0.10	2.8	E
B1933+16	120	6.85	1.1	3.64 ± 0.04	0.7	3.80 ± 0.10	1.7	K
B2016+28	71	2.04	4.9	3.92 ± 0.02	0.2	3.90 ± 0.05	1.1/1.7	L/E

Note. Columns are as follows: (1) pulsar name; (2) HWHM of the ACFs of the dynamic spectra at zero time lag (the smaller value for for B1643-03 of ∼6 kHz is not considered in this paper); (3) HWHM of the ${ \mathcal F }{ \mathcal T }(\mathrm{ACF})$s; (4) fractional rms uncertainty of values in columns (2) and (3); (5) parameter α, from the fit of the extended-medium scattering model to the observed ACFs down to 0.45 of the maximum; the errors were calculated from the statistical errors derived from column (6) added in quadrature with systematic errors estimated from slight deviations of the ACF from the model (same values for extended-medium and thin-screen models); (6) the rms values in units of correlation coefficients for the ACFs with respect to the fit model; (7) parameter α, from the fit of the extended-medium scattering model to the Fourier transforms of the observed ACFs for the delay range from 0.2 to 20 Δτ/Δτ_1/2 or down to ∼10⁻³ of the maximum (same values for extended-medium and thin-screen models); the errors are largely systematic errors estimated from non-random deviations of ${ \mathcal F }{ \mathcal T }(\mathrm{ACF})$ from the models; (5), (7) values including errors larger than 4 are still valid but nonphysical within the context of the models and may indicate deviations from them; (8) the smallest rms values in units of (10⁻²) of the maximum of unity of the observed ACFs with respect to the analytical ACF(f) functions in column (9) from Table 4 up to 2 × Δf/Δf_1/2 (see Figure 7); (9) the best-matching functions from the four functions from Table 4, ACF(G) to ACF(K).

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We fit the functions with a curve corresponding to the predictions for the extended-medium model. The differences between the predictions for ACF(Δf) for the thin-screen and the extended-medium models are small for the inner part but larger for the tail of the functions. Using, for instance, the solutions for the Gaussian spectrum (α = 4.0; Lovelace 1970; Chashei & Shishov 1976; Lerche 1979) with a maximum at unity and scaling them so that the values for Δf_1/2 are identical, the functions differ by less than 0.002 down to the HWHM and by less than 0.007 down to 20% of the maximum. These differences are not large enough to be considered for our analysis of the inner part of ACF(Δf/Δf_1/2) but have to be considered for the tail of the ACFs, where they could perhaps become measurable. Although for some pulsars a good fit could be obtained even for the tail of ACF(Δf/Δf_1/2), a consistently good fit for all pulsars was only obtained up to frequency lags, Δf/Δf_1/2 ∼ 1.2, where ACF(Δf/Δf_1/2) =0.45. For this range, the fitted values of α for the two scattering models were the same within our sensitivity limits. We list the values of α together with their uncertainties in Table 2.

For a summary view of our 12 measured ACF(Δf/Δf_1/2) functions we plot their inner portions down to half of the maximum and compare them with the fit model predictions in Figure 2. It can be seen that the value of α is mostly determined by the shape of the inner part of ACF(Δf/Δf_1/2). All the functions are within a range of fit parameters, α, between 3.56 for the lower curve of PSR B0823+26 and 3.97 for the upper curve of PSR B1642-03. The mean with standard deviation is 〈α_ACF〉 = 3.76 ± 0.13 and is therefore close to α_K = 3.67 of the Kolmogorov spectrum. The quality of the fit varies. We list rms variations of the observed ACFs from the model for each pulsar in Table 2. The fits are for most pulsars quite good, with rms values smaller or equal to 0.005 on the scale of the ACF functions down to 0.45 of the maximum of unity. Larger deviations with rms values of 0.007–0.014 were found for PSRs B1237+25, B1642-03, and B1933+16. These are mostly not random variations but to a large part systematic deviations from the model.

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To inspect the fits in more detail down to lower correlation coefficients, we plot ACF(Δf/Δf_1/2) with the models from Figure 2 for each of the 12 pulsars separately for frequency lags up to 2 × Δf_1/2 in Figure 3. For about half the pulsars the data are still well fit by the model over the wider range of frequency lags. However, significant deviations are now apparent for the other half.

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Some of these deviations in the tail of ACF(Δf/Δf_1/2) could be due to technical effects such as (1) superposition of the contribution from neighboring scintles; (2) the limited band of the receiver filter, which distorts or cuts off wide diffraction structures; (3) inaccuracy of the determination of the off-pulse level for the computation of ACFs; and (4) relatively low signal-to-noise ratios. Some of these effects, if not all, could possibly increase the ACFs toward large lags. Such increase can be seen for PSRs B0834+06 and B1133+16. For the other pulsars with deviations, the tail of the ACFs is lower than the model predictions. In these cases the deviations are likely indications of deficiencies in the scattering models.

A complementary method to estimate α values is to fit the models to the Fourier transform of the ACFs, ${ \mathcal F }{ \mathcal T }(\mathrm{ACF})$ (e.g., Armstrong & Rickett 1981; Wolszczan 1983). We computed ${ \mathcal F }{ \mathcal T }[\mathrm{ACF}({\rm{\Delta }}f)]$ by using the full range of frequency delays of ±4 MHz. To suppress fluctuations, we applied to the observed ACF (Δf) functions a Gaussian filter with a half-width at the 1/e level of 4 × Δf_1/2. Only for PSR B0525+21, with its relatively large value of Δf_1/2, did we use a four times narrower Gaussian.

A comparison of the observed functions ${ \mathcal F }{ \mathcal T }[\mathrm{ACF}({\rm{\Delta }}f)]$ of all pulsars with the theoretical predictions for the extended-medium and for the thin-screen models is shown in Figure 4. In this subsection we focus on the fit of the former to the ${ \mathcal F }{ \mathcal T }(\mathrm{ACF})$s. All ${ \mathcal F }{ \mathcal T }(\mathrm{ACF})$ functions are normalized in amplitude and in width. Their HWHM values, Δτ_1/2, are listed in Table 2.

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The values of ${\alpha }_{{ \mathcal F }{ \mathcal T }(\mathrm{ACF})}$ and their estimated errors are listed in Table 2 (column (7)). The mean with standard deviation is $\langle {\alpha }_{{ \mathcal F }{ \mathcal T }(\mathrm{ACF})}\rangle =3.83\pm 0.12$, larger by 0.07 than 〈α_ACF〉. The mean of the magnitudes of the differences to those obtained from the fits to the ACFs with standard deviation is $\langle | {\alpha }_{\mathrm{ACF}}-{\alpha }_{{ \mathcal F }{ \mathcal T }(\mathrm{ACF})}| \rangle =0.11\pm 0.07$. In general the values are fairly consistent within the errors given that the fitting ranges are quite different. The largest discrepancy is 0.26 for PSR B0834+06. The difference could perhaps at least partly be due to the large deviations from the model in the ACF for lags outside the ACF fitting range.

The exponent, α, of the wavenumber power-law spectrum of electron density variations can be estimated by several methods. In Table 3 we compare our estimates from Table 2 in columns (5) and (7) with those from other authors derived from the Fourier transform of the frequency section of the ACFs, structure function, and frequency dependence of the decorrelation bandwidth.

Table 3. Comparison of Estimates of α

PSR	αACF(Δf)	${\alpha }_{{ \mathcal F }{ \mathcal T }(\mathrm{ACF})}$	${\alpha }_{{ \mathcal F }{ \mathcal T }(\mathrm{ACF})}$	${\alpha }_{{ \mathcal F }{ \mathcal T }(\mathrm{ACF})}$	αSF	${\alpha }_{{\rm{\Delta }}{f}_{1/2}(f)}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)
B0329+54	3.68 ± 0.02	3.80 ± 0.05	3.85	3.7-4.0	3.67 ± 0.01	${3.41}_{-0.25}^{+0.41}$
B0525+21	3.73 ± 0.02	3.80 ± 0.05
B0823+26	3.56 ± 0.02	3.60 ± 0.05			3.66 ± 0.01
B0834+06	3.74 ± 0.02	4.00 ± 0.10			3.528 ± 0.006
B0919+06	3.59 ± 0.02	3.70 ± 0.10			3.57 ± 0.01
B1133+16	3.80 ± 0.02	4.00 ± 0.10	3.90	3.4–3.8	3.86 ± 0.01
B1237+25	3.90 ± 0.04	3.80 ± 0.10			3.39 ± 0.01
B1642−03	3.97 ± 0.05	3.80 ± 0.20		3.2–3.9		${3.63}_{-0.21}^{+0.29}$
B1749−28	3.78 ± 0.02	3.90 ± 0.10			3.82 ± 0.01	${3.50}_{-0.14}^{+0.18}$
B1929+10	3.85 ± 0.02	3.90 ± 0.10			3.65 ± 0.02
B1933+16	3.64 ± 0.04	3.80 ± 0.10	3.90		3.18 ± 0.01	${3.80}_{-0.12}^{+0.14}$
B2016+28	3.92 ± 0.02	3.90 ± 0.05			3.36 ± 0.02*

Note. Columns are as follows: (1) pulsar name; (2) taken from Table 2, column (5); (3) taken from Table 2, column (7); (4) estimates of α of the extended-medium power-law model from the Fourier transform of the frequency ACFs, PSRs B0329+54 (327 and 480 MHz), B1133+16 (327 MHz), and B1933 (1416 MHz; Wolszczan 1983); (5) estimate of α of the thin-screen power-law model from the Fourier transform of the frequency ACFs, PSRs B0329+54 (340, 408 MHz), B1133+16 (340 MHz), and B1642-03 (340 MHz; Armstrong & Rickett 1981); (6) estimate of α of the power-law spectrum from the structure function of the time section of the ACFs, with "*" indicating a measurement biased toward lower values (Popov & Smirnova 2021); (7) estimate of α of the power-law model from the decorrelation bandwidth, Δf_1/2, as a function of observing frequency in the range 80–8100 MHz, (Cordes et al. 1985).

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The most similar analysis to ours was done by Wolszczan (1983, column (4)) and Armstrong & Rickett (1981, column (5)) with data taken some 30 years before ours, who fit an extended-medium and a thin-screen scattering model, respectively, to the FTs of the ACFs. Compared with the values of Wolszczan (1983), our corresponding values in column (3) are all consistent with theirs within our errors. Compared with the values of Armstrong & Rickett (1981), which are given with relatively large ranges, ours are within their ranges or slightly above but exclude the range below α = 3.5.

The values in the other columns were obtained with different analyses schemes. The most interesting for a comparison are those listed in column (6), which are based on the same data as ours in columns (2) and (3). They are derived from the structure function of the time section of the ACFs and are therefore best compared with our values in column (2). For five pulsars, the differences in α are ≤0.10. For the other four pulsars, ignoring B2016+28 (see caption of Table 3), the differences are as large as 0.51 as for B1237+25.

The last column lists estimates of α obtained from a fit of the decorrelation bandwidth as a function of observing frequency. They are consistent with our values in column (2) within 1.5 times their larger uncertainties. We comment on these comparisons in Section 4.

To search for any indications whether a thin-screen or an extended-medium scattering model is preferred by our observed ACF(Δf) functions, we extend the functions to frequency lags of 4.5 × Δf_1/2 and plot the observed ACFs from Figure 3 together with the predictions of the two scattering models in Figure 5. For pulsars with broad scintles relative to the bandwidth of 16 MHz (see, e.g., PSR B0525+21; Figure 1 and Table 2), we plot the functions only for a portion of the total range in frequency lags.

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Slight differences in the models can be seen at Δf ∼ 2.5 ×Δf_1/2 that grow larger with further frequency lags. For most pulsars the tail of the measured ACFs deviates substantially from either of the models and, as discussed in Section 3.2, can likely be interpreted in most cases as being due to deficiencies in the theoretical scattering models.

Three pulsars, B0329+54, B0823+26, and B0919+06, show fairly good consistency in the observed ACFs with either or both of the models for frequency lags at least up to 4.5 × Δf_1/2. For PSRs B0329+54 and B0919+06, deviations from either of the two models clearly grow for even larger lags. PSR B0823+26, however, is a candidate that warrants more scrutiny. In Figure 6 we show the observed ACF and the two model curves but now up to Δf ∼ 6 × Δf_1/2. The extended-medium model is drawn as in Figure 5 with α_em = 3.56. The thin-screen model curve lies for the same α value above the extended-medium model curve and is therefore a candidate for matching the tail of the ACF, but not that much that it would be a good fit to the observed ACF. To investigate this case further, we plot the thin-screen model for α_ts = 3.49, 3.5σ below the best-fit value in an attempt to deviate not too much from the best fit but also to match the tail of the ACF. However, even in this extreme case, deviations much larger than for the inner part of the ACF are still visible. We therefore think that also in this case the deviations are due to effects discussed already for the other pulsars. There is no convincing case that either of the two models is preferred.

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An alternative approach to search for differences is to analyze the Fourier transform of the ACFs, ${ \mathcal F }{ \mathcal T }[\mathrm{ACF}({\rm{\Delta }}f)]$ (e.g., Armstrong & Rickett 1981; Wolszczan 1983) and the respective thin-screen and extended-medium models as shown in Figure 4. Differences between the models are somewhat more pronounced than in Figure 5. However, also in this case the observed functions ${ \mathcal F }{ \mathcal T }[\mathrm{ACF}({\rm{\Delta }}f)]$ could not conclusively discriminate between either of the models.

The visibility function, V(τ, t), as a function of delay and time in our space VLBI observations is obtained from the inverse Fourier transform of the cross-spectrum between two different radio telescopes. In our case of observations of dynamic spectra with single telescopes, the cross-spectrum corresponds to the autospectrum at any particular time, t, in a dynamic spectrum, S(f, t). A Gaussian has been used to approximate the frequency section of the ACFs of dynamic spectra (e.g., Cordes 1986). Since for a Gaussian ACF, the underlying function and its Fourier transform are also Gaussians, the autospectrum would also be expected to be a Gaussian. Table 4 lists the function, y, as a Gaussian function, G(f), together with three other functions, L(f), E(f), and K(f), that we consider further in our analysis. In addition we list for each function, y, its Fourier transform, ${ \mathcal F }{ \mathcal T }(y)=\tilde{y}$, and its ACF(y).

Table 4. Fourier Transform and Self-convolution of Four Parametric Families of Functions

Name	y	$\tilde{y}$	ACF(y)
Gauss	$G(\sigma ,f)=\displaystyle \frac{\exp (-{f}^{2}/(2\,{\sigma }^{2}))}{\sqrt{2\pi }\,\sigma }$	$\displaystyle \frac{G\left(1/(2\pi \sigma ),t\right)}{\sqrt{2\pi }\,\sigma }$	$G(\sqrt{2}\,\sigma ,{\rm{\Delta }}f)$
Lorentz	$L(\lambda ,f)=\displaystyle \frac{\lambda }{\pi \left({\lambda }^{2}+{f}^{2}\right)}$	$\displaystyle \frac{E\left(1/(2\pi \lambda ),t\right)}{\pi \,\lambda }$	L(2λ, Δf)
Laplace	$E(\epsilon ,f)=\displaystyle \frac{\exp (-| f| /\epsilon )}{2\,\epsilon }$	$\displaystyle \frac{L{\left(1/(2\pi \epsilon ),t\right)}^{}}{2\,\epsilon }$	$\displaystyle \frac{(\epsilon +| {\rm{\Delta }}f| )E(\epsilon ,{\rm{\Delta }}f)}{2\,\epsilon }$
Bessel	$K(\phi ,f)=\displaystyle \frac{1}{\pi \phi }{K}_{0}\left(\displaystyle \frac{| f| }{\phi }\right)$	${\left(\displaystyle \frac{L\left(1/(2\pi \phi ),t\right)}{2\,\phi }\right)}^{1/2}$	E(ϕ, Δf)

Note. Name of the function, y, definition of y, Fourier transform of y with ${ \mathcal F }{ \mathcal T }(y)=\tilde{y}$. The Fourier transform is defined here as $\tilde{y}(p,t)={\int }_{-\infty }^{+\infty }y(p,f){e}^{-2\pi {ift}}\,{df}$, with p as a parameter. All the functions considered here are real and normalized so that $\tilde{y}(0)=1$. Therefore, the ACF is identical with self-convolution and $\widetilde{\mathrm{ACF}}={\tilde{y}}^{2}$. Because of the normalization chosen, the functions may be identified with probability densities of random variables. The functions named "Lorentz," "Gauss," and "Laplace" are related to the well-known statistical distributions whose mathematical properties are described by Feller (1971). For the last row, we used the term "Bessel" for the function since K₀(z) is the modified Bessel function of the second kind of order 0 and argument z which rises logarithmically to infinity at f → 0. The scale parameters σ and ϕ are equal to rms deviations of the corresponding distributions, and the parameters λ and are half-widths of the Lorentz and Laplace distributions at the level 1/2 and 1/e, respectively.

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Space VLBI with Radioastron revealed for some pulsars that the tail of the delay cross-section of V(τ, t) retained significant magnitudes at baseline projections where the pulsar scattering disk was completely resolved. The delay cross-section, ACF(Δτ, Δt = 0) of V(τ, t), was found to be well approximated by a Lorentz function (Popov et al. 2020). The Fourier transform of a Lorentz function is a two-sided exponential and therefore the autospectrum from our S(f, t) would also be expected to be a two-sided exponential (second row in Table 4). The ACF of such an exponential, ACF(E), is a function given in the third row in Table 4.

Independent of any prior knowledge from space VLBI observation, an inspection of our observed ACF functions in Figure 3 indicates that several of them have up to ∼0.5 × Δf_1/2 a concave profile, with a rounded top where the second derivative is negative. Such a profile is similar to the analytical functions ACF(G), ACF(L), and ACF(E) in Table 4. Some others have in contrast a convex inner profile where the second derivative is positive. This profile is similar to ACF(K).

In Figure 7 we plot as examples the observed ACFs of four pulsars that define the range in curvature for all 12 pulsars and compare them to the four analytical ACF(y) functions. PSR 1642-03 has the most concave inner ACF of all pulsars: it is between that of the ACF(G) and ACF(E) functions. For the inner part it resembles ACF(E) and for the outer part of the plot in lies between ACF(E) and ACF(G). For PSR 2016+28, the ACF is closest to ACF(L) and ACF(E) for the inner part and lies between the two functions for the outer part. The least ambiguous case is found for PSR 0329+54. The ACF is well matched by ACF(K). PSR B0823+26 has the most convex ACF, even slightly more so than ACF(K). In Table 2 we list the rms values of the observed ACFs to the closest analytical ACF(y) functions and indicate which functions they are. The residuals for all pulsars vary between rms values of 0.0015 for B0329+54 and 0.037 for B1749-28. They are all larger by a factor 2 to 9 than those for the scattering models, however they are also from a fit over a larger lag range and do not have a free parameter for adjustment. In general the ACFs of all 12 pulsars are approximately within the range of the four analytical functions.

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Does the ACF of any of these pulsars resemble a Gaussian? We fit a function of the form ${f}_{g}({\rm{\Delta }}f)=\exp (-| {\rm{\Delta }}f{| }^{\delta }/{b}_{f})$ to the ACF of PSR B1642-03, which is one of the two most likely candidates for its ACF to have a Gaussian shape. We obtained for the free exponent and its statistical error δ = 1.79 ± 0.03, still significantly smaller than δ = 2 for a Gaussian. However, in general it appears that for a few pulsars with a concave shape of the inner ACFs there can be a tendency in the shape of the ACF toward a Gaussian.

Listing the 12 pulsars according to the shape of their ACFs it is clear that the most concave inner ACFs are related to the highest values of α and the most convex to the lowest value. PSR B0823+26 has the most convex ACF, even slightly more so than ACF(K), and the lowest value of α of 3.56. At α = ∼3.75 the ACFs transition from concave to convex.

The average frequency profile of the scintles could be obtained analytically by finding the function y in Table 4 if our observed ACFs were exactly represented by ACF(y). However, only very few can be approximately associated with one particular ACF(y), and then only for a limited range of frequency lags. Most are between neighboring analytical functions or are closer to one function in their inner parts and to another function at the tail. Therefore it is best to use the comparison of the observed ACFs with the analytical functions as a guide. The inferences as to the scintle profile can only be approximate.

In Figure 8 we plot the functions, y, which are the underlying functions to ACF(y). The functions G and L are concave up to ∼HWHM with a bell-shaped profile and the other two functions, E and K, are convex with a profile of a cusp either with a moderate or an extreme peak. In the latter case, the function goes to infinity for f → 0.

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Comparing the ACF(y) with the y functions in Figures 7 and 8 and Table 4, it is clear that the observed ACFs matched best by ACF(G) and ACF(L) functions are associated with bell-shaped scintle profiles. The ACF(E) function is almost identical to ACF(L) for small lags and becomes clearly different for lags Δf > Δf_1/2. However, the function y assumes the shape of a cusp. Five of our pulsars have clear similarities to ACF(E) but also to any of the other two functions, G and L. The scintle profile is probably described best as a hybrid between a bell-shaped curve and a cusp.

For one pulsar, B1929+10, the ACF is best described only by ACF(E). None of the observed ACFs is best described only by the bell-shaped functions, G and L. For three pulsars, B0525+21, B1133+16, and B1749+28, there is a transition from concave to convex ACF shapes. Their ACFs are hybrids with ACF(K) as one element. For five other pulsars their ACFs are clearly convex and best described by only ACF(K). The scintle profile is less ambiguous. Although the function, K, goes to infinity for f → 0, because of deviations of the observed ACFs from ACF(K), the scintle profile would probably be best described as a pronounced cusp with a sharp peak.

What is the relation of the wavenumber power-law spectral index to the scintle profile? It appears that for pulsars with high α values, that is, steep wavenumber spectra, the scintle could have a profile intermediate between a cusp and a Gaussian or Lorentzian because of a tendency of the shape of their ACFs toward these functions. Below the transition from a concave shape of the inner ACFs to a convex shape at α ≃ 3.75 the cusp becomes more clearly defined with a sharp peak and its sides decrease even faster than exponentially in the case of pulsars with the flattest wavenumber spectra.