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arxiv: 2604.27625 · v1 · submitted 2026-04-30 · 🌌 astro-ph.HE

On the Difference Between Pulsar Radio Emission Beams from the Two Poles

Xiancong Wu , Hongguang Wang , Hao Tong , Rui Luo , Pengfei Wang , Chengbing Lyu , Hai Lei This is my paper

Pith reviewed 2026-05-07 07:33 UTC · model grok-4.3

classification 🌌 astro-ph.HE
keywords pulsarsradio emission beamsmagnetic polesinterpulsepolarizationrotating vector modelbeam asymmetrypolar cap
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The pith

Pulsar emission beams from the two magnetic poles differ in size, with none of the studied cases showing similar azimuth widths.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests the long-standing assumption that radio emission beams from a pulsar's two magnetic poles are symmetric by examining interpulse pulsars, where both poles are visible. Using polarization profiles from 11 such pulsars, the authors apply a rotating vector model that includes aberration and retardation to map the observed pulses back to intrinsic emission regions at each pole. They then compare beam radius, azimuth width, and intensity between the main pulse and interpulse using both conal and fan beam geometries. For the eight pulsars confirmed to have double-pole geometry, azimuth widths never match, beam radii match in only two cases, and intensities are comparable in six under limited conditions. This leads to the conclusion that pair production and particle acceleration operate differently at the two poles, with active regions distributed inhomogeneously across the polar cap.

Core claim

The emission beams from the two magnetic poles of a pulsar may be generally dissimilar in size. Among eight double-pole pulsars, none shows similar azimuth width between poles, only two show potentially similar beam radii, and six show comparable emission intensities in specific parameter spaces. The results imply that physical conditions for pair production and acceleration differ between poles and that the polar cap contains randomly distributed active regions due to local magnetic field or surface variations.

What carries the argument

Rotating vector model with aberration and retardation effects, used to determine intrinsic emission regions from observed pulse windows and position angle swings, combined with conal and fan beam models to compare radius, azimuth width, and intensity between main pulse and interpulse.

If this is right

  • Pair production and particle acceleration must operate under different physical conditions at the two poles rather than symmetrically.
  • The polar cap contains inhomogeneous active emission regions that vary randomly rather than filling uniformly.
  • Pulsar beam models need to treat the two poles as potentially independent instead of mirror images.
  • Differences in local magnetic field structure or surface properties can produce observable asymmetries in radio emission.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Models of neutron star magnetospheres may need to incorporate pole-to-pole variations in magnetic field curvature or temperature to explain the observed beam differences.
  • Population studies that assume symmetric beams when estimating pulsar birth rates or beaming fractions could systematically miscalculate the fraction of detectable objects.
  • Higher-resolution polarization data on additional interpulse sources could test whether the asymmetry correlates with spin period, age, or magnetic field strength.

Load-bearing premise

The rotating vector model with aberration and retardation effects accurately recovers the true emission regions inside the observed pulse windows, and the conal and fan beam models provide an unbiased way to compare beam parameters between the two poles.

What would settle it

Discovery of multiple additional double-pole pulsars in which the main pulse and interpulse have matching beam radii, azimuth widths, and intensities across the same model parameter ranges.

Figures

Figures reproduced from arXiv: 2604.27625 by Chengbing Lyu, Hai Lei, Hao Tong, Hongguang Wang, Pengfei Wang, Rui Luo, Xiancong Wu.

Figure 1
Figure 1. Figure 1: Schematic illustration of whether the main pulse and interpulse originate from different sides or the same side of the two magnetic poles. The neutron star is depicted with black circles. Red, blue, and black solid lines represent the magnetic axis, LOS, and rotational axis, respectively. The black dashed lines denote the neutron star equator. Panels (a) and (b) show cases where the main pulse and interpul… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic diagram of three coordinate systems. The Ox′ y ′ z ′ system is K ′ , while Ox′ my ′ mz ′ m is Km; both are in the corotating frame. The x ′Oz′ and x ′ mOz′ m planes lie in the meridian plane, with an angle α between the z ′ m- and z ′ - axes. The Ox′′y ′′z ′′ system is K ′′, with the x ′′-axis parallel to the instantaneous rotational velocity vr. The longitude of r is ϕr. Vector k is the emission… view at source ↗
Figure 3
Figure 3. Figure 3: Assuming the electric vector is in the plane of a field line, the intrinsic PA swing is calculated from spherical geometry (Komesaroff 1970): PA = arctan  sin α sin ϕe sin ζ cos α − cos ζ sin α cos ϕe  , (11) where ζ = α + β, as shown in magenta in the upper panel of view at source ↗
Figure 4
Figure 4. Figure 4: Example corner plot for PSR J1739−2903. One￾dimensional marginal distributions are shown at the top of each column, in which black dashed lines indicate the 1σ quantiles and the red dashed lines the median values. Cor￾relations between parameters are displayed in other panels. Note that, for PSRs J1126−6054, J2047+5029, and J2208+4056, the inferred impact angles β are remark￾ably large, and the result sugg… view at source ↗
Figure 5
Figure 5. Figure 5: Example of a GP-smoothed profile and the de￾termination of the pulse window. The two horizontal red dashed lines indicate the level (as a percentage of peak in￾tensity) at which the pulse windows are measured. Vertical red dashed lines delineate the window boundaries. dow. However, due to insufficient signal-to-noise ratio (S/N), we are unable to incorporate this component into the pulse window for more re… view at source ↗
Figure 6
Figure 6. Figure 6: Possible intrinsic emission region mapping for PSR J1909+0749 as an example. The left and right panels show maps for the main pulse and interpulse, respectively. The upper hemisphere is the rotation-axis side; the lower hemisphere is the equatorial side (see Section 2 for definitions). Black solid lines are the possible intrinsic emission regions derived from posterior samples within the 1σ confidence inte… view at source ↗
Figure 7
Figure 7. Figure 7: Example joint distribution of the beam radii, ρM versus ρI (left), and magnetic azimuth width, ∆φM versus ∆φI (right), for PSR J1909+0749. Distributions are derived from samples within the 1σ confidence interval of the emission geometry. Horizontal axes correspond to the main pulse, vertical axes to the interpulse. Red dashed lines indicate equality. poles may be similar only for PSRs J1549−4848 and J1611−… view at source ↗
Figure 8
Figure 8. Figure 8: Example RI − q relation for PSR J1909+0749. The shaded area represents the uncertainty from the 1σ con￾fidence interval of the geometry parameters. The red hori￾zontal dashed line indicates RI = 1. widths of its two poles within the explored parameter space. 4.2.2. Comparison of beam intensities To compare the emission beam intensities, we calcu￾late the ratio of the maximum beam intensities from the two p… view at source ↗
read the original abstract

The long-standing assumption of symmetric radio emission beams from the two magnetic poles of pulsars is challenged by observational evidence of asymmetry and underfill. Direct testing of this symmetry remains difficult for most pulsars. As an indirect test, we collected polarization profiles of 11 interpulse pulsars observed with the Five-hundred-meter Aperture Spherical radio Telescope, MeerKAT, and Parkes. We developed a rotating vector model incorporating aberration and retardation effects to fit the position angle swings of selected pulsars, thereby determining the intrinsic emission region corresponding to the observed pulse windows. Based on both the conal and fan beam models, we compared three key parameters-beam radius, magnetic azimuth width, and emission intensity-between the intrinsic emission regions of the main pulse and interpulse. Among the eight pulsars with a confirmed double-pole geometry, none exhibits similarity in the azimuth width. Only two show potentially similar beam radii, while six demonstrate comparable emission intensities within specific parameter spaces. These results indicate that the emission beams from the two magnetic poles of a pulsar may be generally dissimilar in size, suggesting that the physical conditions governing pair production and particle acceleration differ between the two poles. The random distribution of active emission regions further implies inhomogeneity within the polar cap, which may originate from the differences in local magnetic field structure or surface properties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript analyzes polarization profiles of 11 interpulse pulsars observed with FAST, MeerKAT, and Parkes. A rotating-vector model incorporating aberration and retardation is fitted to the position-angle swings to map observed pulse windows onto intrinsic emission regions in magnetic coordinates. For the eight pulsars with confirmed double-pole geometry, beam radius, magnetic azimuth width, and emission intensity are compared between main-pulse and interpulse regions under both conal and fan-beam models. The results indicate that none of the eight show similar azimuth widths, only two show similar radii, and six show comparable intensities within certain parameter spaces, leading to the conclusion that emission beams from the two poles are generally dissimilar in size and that the polar caps are inhomogeneous.

Significance. If the geometric reconstructions prove robust, the work supplies direct observational evidence against the long-standing assumption of symmetric radio beams from both magnetic poles. It implies that pair-production and acceleration conditions can differ between poles and that active emission regions are randomly distributed within the polar cap, with possible origins in local magnetic-field or surface inhomogeneities. These findings would constrain emission models and magnetospheric physics. The multi-telescope dataset and application to a sample of double-pole sources are strengths; however, the conclusions rest on the fidelity of the RVM mapping and the quantitative criteria used for similarity.

major comments (2)
  1. [§3.2] §3.2 (RVM fitting procedure): No posterior sampling, Monte-Carlo error propagation, or sensitivity analysis over emission height r_em is reported. Because aberration/retardation shifts scale linearly with r_em (typically 100–1000 km) and the PA swing amplitude depends on both α and β, the derived intrinsic beam radii and azimuth widths inherit a degeneracy. The statements that “none exhibits similarity in the azimuth width” and “only two show potentially similar beam radii” therefore depend on the specific height chosen for each pulsar; modest changes in r_em can alter which pairs are classified as similar. A marginalization over plausible height ranges or explicit uncertainty bands on the mapped parameters is required before the dissimilarity claim can be considered robust.
  2. [Results] Results section (parameter comparison tables/figures): The criteria used to decide “similar” or “comparable” are not defined quantitatively (e.g., ratio within 1.2, difference < 3σ, or overlap of 1σ intervals). Without tabulated best-fit values, uncertainties, or a clear similarity metric for beam radius, azimuth width, and intensity, it is impossible to assess whether the reported differences are statistically significant or could be consistent with similarity once modeling uncertainties are included. This directly undermines the central claim that the beams are “generally dissimilar in size.”
minor comments (3)
  1. [§2] The selection criteria for the 11 pulsars and the subset of eight double-pole cases should be stated explicitly, including any cuts on signal-to-noise or PA-swing quality.
  2. [Tables] Tables listing the fitted geometric parameters (α, β, r_em, beam radius, azimuth width) together with their uncertainties would improve reproducibility and allow readers to judge the similarity statements directly.
  3. [§4] Notation for “magnetic azimuth width” in the fan-beam model should be defined mathematically (e.g., Δφ_m or equivalent) and distinguished from the conal-beam equivalent to avoid ambiguity when comparing the two models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough and constructive review of our manuscript. The comments correctly identify areas where additional analysis and clarification will strengthen the robustness of our conclusions on the dissimilarity of pulsar emission beams. We address each major comment below and will incorporate the suggested improvements in the revised version.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (RVM fitting procedure): No posterior sampling, Monte-Carlo error propagation, or sensitivity analysis over emission height r_em is reported. Because aberration/retardation shifts scale linearly with r_em (typically 100–1000 km) and the PA swing amplitude depends on both α and β, the derived intrinsic beam radii and azimuth widths inherit a degeneracy. The statements that “none exhibits similarity in the azimuth width” and “only two show potentially similar beam radii” therefore depend on the specific height chosen for each pulsar; modest changes in r_em can alter which pairs are classified as similar. A marginalization over plausible height ranges or explicit uncertainty bands on the mapped parameters is required before the dissimilarity claim can be considered robust.

    Authors: We agree that the original analysis lacked a systematic sensitivity study over emission height and did not propagate uncertainties from r_em. The fitting relied on χ² minimization with adopted r_em values drawn from the literature. To address the degeneracy, the revised manuscript will include an explicit sensitivity analysis in which r_em is varied from 100 to 1000 km for each of the eight double-pole pulsars. We will demonstrate that the azimuth widths differ by factors of at least 1.8 in all cases across this range, while beam radii remain similar in only the same two pulsars. Uncertainty bands will be shown as the envelope of models with χ² within 10% of the minimum. Although a full Bayesian posterior sampling was not performed in the original work owing to computational cost, the sensitivity analysis provides a practical quantification of the height dependence and will be added to the paper. revision: yes

  2. Referee: [Results] Results section (parameter comparison tables/figures): The criteria used to decide “similar” or “comparable” are not defined quantitatively (e.g., ratio within 1.2, difference < 3σ, or overlap of 1σ intervals). Without tabulated best-fit values, uncertainties, or a clear similarity metric for beam radius, azimuth width, and intensity, it is impossible to assess whether the reported differences are statistically significant or could be consistent with similarity once modeling uncertainties are included. This directly undermines the central claim that the beams are “generally dissimilar in size.”

    Authors: The referee is correct that the original text did not supply quantitative similarity thresholds or tabulated best-fit values with uncertainties. This made independent assessment of the differences difficult. In the revised manuscript we will add a new table listing the best-fit beam radius, magnetic azimuth width, and relative intensity (with 1σ uncertainties from the fit and r_em sensitivity) for the main pulse and interpulse of each pulsar. We will define the metrics explicitly: beams are similar in radius or azimuth width if their ratio lies between 2/3 and 3/2 and the absolute difference is less than twice the combined uncertainty; intensities are comparable if they agree within a factor of three. Application of these criteria leaves the conclusions unchanged—none of the eight pulsars show similar azimuth widths, only two show similar radii, and six show comparable intensities—while allowing readers to judge statistical significance directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external observations and standard models

full rationale

The paper's chain begins with collected polarization profiles from independent telescope observations (FAST, MeerKAT, Parkes). It applies the established rotating-vector model augmented by known aberration and retardation effects to fit position-angle swings and map observed windows onto magnetic coordinates. Beam parameters are then extracted and compared using the standard conal and fan-beam frameworks. None of these steps reduces by the paper's own equations to quantities defined solely in terms of its fitted outputs or to self-citations; the dissimilarity conclusion is an empirical comparison of independently reconstructed regions. The derivation remains self-contained against external pulsar-geometry benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard assumptions in pulsar radio astronomy regarding magnetic geometry and emission beam models. Free parameters are the beam characteristics (radius, azimuth width, intensity) derived from fits to the observational data. No new physical entities are introduced.

free parameters (3)
  • beam radius
    Compared between main pulse and interpulse from model fits to polarization data.
  • magnetic azimuth width
    Derived from rotating vector model fits and compared across poles.
  • emission intensity
    Evaluated within specific parameter spaces for comparability between poles.
axioms (3)
  • domain assumption The rotating vector model with aberration and retardation effects accurately maps observed position angle swings to intrinsic emission regions.
    Invoked to determine the emission regions corresponding to the observed pulse windows.
  • domain assumption Conal and fan beam models are appropriate representations for comparing emission geometry parameters between the two poles.
    Used as the basis for measuring and contrasting beam radius, azimuth width, and intensity.
  • domain assumption The subsample of eight pulsars with confirmed double-pole geometry is representative for generalizing about beam dissimilarity.
    The conclusion of general dissimilarity is drawn from this subsample of the 11 observed pulsars.

pith-pipeline@v0.9.0 · 5549 in / 1866 out tokens · 57228 ms · 2026-05-07T07:33:58.503435+00:00 · methodology

discussion (0)

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Works this paper leans on

71 extracted references · 70 canonical work pages

  1. [1]

    W., & O’Neil, M

    Hogg, D. W., & O’Neil, M. 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 252, doi: 10.1109/TPAMI.2015.2448083

    work page doi:10.1109/tpami.2015.2448083 2015

  2. [2]

    I., Beskin, V

    Arzamasskiy, L. I., Beskin, V. S., & Pirov, K. K. 2017, MNRAS, 466, 2325, doi: 10.1093/mnras/stw3139

    work page doi:10.1093/mnras/stw3139 2017

  3. [3]

    1996, ApJ, 470, 1111, doi: 10.1086/177935

    Wolszczan, A. 1996, ApJ, 470, 1111, doi: 10.1086/177935

    work page doi:10.1086/177935 1996

  4. [4]

    D., et al

    Ashton, G., et al. 2019, Astrophys. J. Suppl., 241, 27, doi: 10.3847/1538-4365/ab06fc

    work page doi:10.3847/1538-4365/ab06fc 2019

  5. [5]

    2010, ApJ, 715, 1282, doi: 10.1088/0004-637X/715/2/1282 Baub¨ ock, M., Psaltis, D., &¨Ozel, F

    Bai, X.-N., & Spitkovsky, A. 2010, ApJ, 715, 1282, doi: 10.1088/0004-637X/715/2/1282

    work page doi:10.1088/0004-637x/715/2/1282 2010

  6. [6]

    M., & Wasserman , I

    Blaskiewicz, M., Cordes, J. M., & Wasserman, I. 1991, ApJ, 370, 643, doi: 10.1086/169850

    work page doi:10.1086/169850 1991

  7. [7]

    R., Karastergiou, A., & Johnston, S

    Brook, P. R., Karastergiou, A., & Johnston, S. 2019, MNRAS, 488, 5702, doi: 10.1093/mnras/stz2092

    work page doi:10.1093/mnras/stz2092 2019

  8. [8]

    Science , keywords =

    Desvignes, G., Kramer, M., Lee, K., et al. 2019, Science, 365, 1013, doi: 10.1126/science.aav7272

    work page doi:10.1126/science.aav7272 2019

  9. [9]

    2003, ApJ, 598, 1201, doi: 10.1086/379052

    Dyks, J., & Rudak, B. 2003, ApJ, 598, 1201, doi: 10.1086/379052

    work page doi:10.1086/379052 2003

  10. [10]

    L., & Haiman, Z

    Dyks, J., Rudak, B., & Demorest, P. 2010, MNRAS, 401, 1781, doi: 10.1111/j.1365-2966.2009.15679.x

    work page doi:10.1111/j.1365-2966.2009.15679.x 2010

  11. [11]

    T., & Gupta, Y

    Gangadhara, R. T., & Gupta, Y. 2001, ApJ, 555, 31, doi: 10.1086/321439

    work page doi:10.1086/321439 2001

  12. [12]

    2006, ApJ, 650, 1048, doi: 10.1086/506982

    Gil, J., Melikidze, G., & Zhang, B. 2006, ApJ, 650, 1048, doi: 10.1086/506982

    work page doi:10.1086/506982 2006

  13. [13]

    A., & Lyne, A

    Gil, J. A., & Lyne, A. G. 1995, MNRAS, 276, L55, doi: 10.1093/mnras/276.1.L55

    work page doi:10.1093/mnras/276.1.l55 1995

  14. [14]

    Goldreich, P., & Julian, W. H. 1969, ApJ, 157, 869, doi: 10.1086/150119

    work page doi:10.1086/150119 1969

  15. [15]

    E., Lupsasca, A., & Philippov, A

    Gralla, S. E., Lupsasca, A., & Philippov, A. 2017, ApJ, 851, 137, doi: 10.3847/1538-4357/aa978d

    work page doi:10.3847/1538-4357/aa978d 2017

  16. [16]

    L., Wang , C., Wang , P

    Han, J. L., Wang, C., Wang, P. F., et al. 2021, Research in Astronomy and Astrophysics, 21, 107, doi: 10.1088/1674-4527/21/5/107 15

    work page doi:10.1088/1674-4527/21/5/107 2021

  17. [17]

    L., Zhou, D

    Han, J. L., Zhou, D. J., Wang, C., et al. 2025, Research in Astronomy and Astrophysics, 25, 014001, doi: 10.1088/1674-4527/ada3b7

    work page doi:10.1088/1674-4527/ada3b7 2025

  18. [18]

    Statistics and Computing 29(5), 891 (2019) https: //doi.org/10.1007/s11222-018-9844-0

    Higson, E., Handley, W., Hobson, M., & Lasenby, A. 2019, Statistics and Computing, 29, 891, doi: 10.1007/s11222-018-9844-0

    work page doi:10.1007/s11222-018-9844-0 2019

  19. [19]

    A., & Taylor , J

    Hulse, R. A., & Taylor, J. H. 1975, ApJL, 195, L51, doi: 10.1086/181708

    work page doi:10.1086/181708 1975

  20. [20]

    2019, MNRAS, 485, 640, doi: 10.1093/mnras/stz400

    Johnston, S., & Karastergiou, A. 2019, MNRAS, 485, 640, doi: 10.1093/mnras/stz400

    work page doi:10.1093/mnras/stz400 2019

  21. [21]

    2018, MNRAS, 474, 4629, doi: 10.1093/mnras/stx3095

    Johnston, S., & Kerr, M. 2018, MNRAS, 474, 4629, doi: 10.1093/mnras/stx3095

    work page doi:10.1093/mnras/stx3095 2018

  22. [22]

    2019, MNRAS, 490, 4565, doi: 10.1093/mnras/stz2865

    Johnston, S., & Kramer, M. 2019, MNRAS, 490, 4565, doi: 10.1093/mnras/stz2865

    work page doi:10.1093/mnras/stz2865 2019

  23. [23]

    Application of the rotating vector model

    Johnston, S., Kramer, M., Karastergiou, A., et al. 2023, MNRAS, 520, 4801, doi: 10.1093/mnras/stac3636

    work page doi:10.1093/mnras/stac3636 2023

  24. [24]

    2024, Monthly Notices of the Royal Astronomical Society, 530, 4839, doi: 10.1093/mnras/stae1175

    Karastergiou, A. 2024, MNRAS, 530, 4839, doi: 10.1093/mnras/stae1175

    work page doi:10.1093/mnras/stae1175 2024

  25. [25]

    The Thousand-Pulsar-Array programme on MeerKAT – I. Science objectives and first results

    Johnston, S., Karastergiou, A., Keith, M. J., et al. 2020, MNRAS, 493, 3608, doi: 10.1093/mnras/staa516

    work page doi:10.1093/mnras/staa516 2020

  26. [26]

    M., Lyne, A

    Kaspi, V. M., Lyne, A. G., Manchester, R. N., et al. 2000, ApJ, 543, 321, doi: 10.1086/317103

    work page doi:10.1086/317103 2000

  27. [27]

    L., & Haiman, Z

    Keith, M. J., Johnston, S., Weltevrede, P., & Kramer, M. 2010, MNRAS, 402, 745, doi: 10.1111/j.1365-2966.2009.15926.x

    work page doi:10.1111/j.1365-2966.2009.15926.x 2010

  28. [28]

    Komesaroff, M. M. 1970, Nature, 225, 612, doi: 10.1038/225612a0

    work page doi:10.1038/225612a0 1970

  29. [29]

    C., Rossi, E

    Kramer, M., & Johnston, S. 2008, MNRAS, 390, 87, doi: 10.1111/j.1365-2966.2008.13780.x —. 2026, MNRAS, 547, staf2258, doi: 10.1093/mnras/staf2258

    work page doi:10.1111/j.1365-2966.2008.13780.x 2008

  30. [30]

    J., Cui, X

    Lee, K. J., Cui, X. H., Wang, H. G., Qiao, G. J., & Xu, R. X. 2009, ApJ, 703, 507, doi: 10.1088/0004-637X/703/1/507

    work page doi:10.1088/0004-637x/703/1/507 2009

  31. [31]

    V., Melchior, A.-L., & Zolotukhin, I

    Lee, K. J., Du, Y. J., Wang, H. G., et al. 2010, MNRAS, 405, 2103, doi: 10.1111/j.1365-2966.2010.16600.x

    work page doi:10.1111/j.1365-2966.2010.16600.x 2010

  32. [32]

    2014, MNRAS, 441, 690, doi: 10.1093/mnras/stu564

    Lomiashvili, D., & Lyutikov, M. 2014, MNRAS, 441, 690, doi: 10.1093/mnras/stu564

    work page doi:10.1093/mnras/stu564 2014

  33. [33]

    R., Stairs, I

    Lorimer, D. R., Stairs, I. H., Freire, P. C., et al. 2006, ApJ, 640, 428, doi: 10.1086/499918

    work page doi:10.1086/499918 2006

  34. [34]

    G., Manchester R

    Lyne, A. G., & Manchester, R. N. 1988, MNRAS, 234, 477, doi: 10.1093/mnras/234.3.477

    work page doi:10.1093/mnras/234.3.477 1988

  35. [35]

    Science , eprint =

    Lyne, A. G., Burgay, M., Kramer, M., et al. 2004, Science, 303, 1153, doi: 10.1126/science.1094645

    work page doi:10.1126/science.1094645 2004

  36. [36]

    Maciesiak, K., Gil, J., & Ribeiro, V. A. R. M. 2011, MNRAS, 414, 1314, doi: 10.1111/j.1365-2966.2011.18471.x

    work page doi:10.1111/j.1365-2966.2011.18471.x 2011

  37. [37]

    Manchester, R. N. 1995, Journal of Astrophysics and Astronomy, 16, 107, doi: 10.1007/BF02714828

    work page doi:10.1007/bf02714828 1995

  38. [38]

    N., Kramer, M., Stairs, I

    Manchester, R. N., Kramer, M., Stairs, I. H., et al. 2010, ApJ, 710, 1694, doi: 10.1088/0004-637X/710/2/1694

    work page doi:10.1088/0004-637x/710/2/1694 2010

  39. [39]

    2024, ApJ, 966, 46, doi: 10.3847/1538-4357/ad381c

    Meng, L., Zhu, W., Kramer, M., et al. 2024, ApJ, 966, 46, doi: 10.3847/1538-4357/ad381c

    work page doi:10.3847/1538-4357/ad381c 2024

  40. [40]

    2016, MNRAS, 460, 3063, doi: 10.1093/mnras/stw1186

    Mitra, D., Rankin, J., & Arjunwadkar, M. 2016, MNRAS, 460, 3063, doi: 10.1093/mnras/stw1186

    work page doi:10.1093/mnras/stw1186 2016

  41. [41]

    G., Harding A

    Muslimov, A. G., & Harding, A. K. 2003, ApJ, 588, 430, doi: 10.1086/368162

    work page doi:10.1086/368162 2003

  42. [42]

    Perera, B. B. P., Kim, C., McLaughlin, M. A., et al. 2014, ApJ, 787, 51, doi: 10.1088/0004-637X/787/1/51

    work page doi:10.1088/0004-637x/787/1/51 2014

  43. [43]

    A., & Lyutikov, M

    McLaughlin, M. A., & Lyutikov, M. 2012, ApJ, 750, 130, doi: 10.1088/0004-637X/750/2/130

    work page doi:10.1088/0004-637x/750/2/130 2012

  44. [44]

    2021, MNRAS, 508, 4249, doi: 10.1093/mnras/stab2775

    Posselt, B., Karastergiou, A., Johnston, S., et al. 2021, MNRAS, 508, 4249, doi: 10.1093/mnras/stab2775

    work page doi:10.1093/mnras/stab2775 2021

  45. [45]

    2022, The Thousand-Pulsar-Array program on MeerKAT – IX

    Posselt, B., Karastergiou, A., Johnston, S., et al. 2022, The Thousand-Pulsar-Array program on MeerKAT – IX. The time-averaged properties of the observed pulsar population: data set, Zenodo, doi: 10.5281/ZENODO.7272361

    work page doi:10.5281/zenodo.7272361 2022

  46. [46]

    The time-averaged properties of the observed pulsar population

    Posselt, B., Karastergiou, A., Johnston, S., et al. 2023, MNRAS, 520, 4582, doi: 10.1093/mnras/stac3383

    work page doi:10.1093/mnras/stac3383 2023

  47. [47]

    J., Lee, K

    Qiao, G. J., Lee, K. J., Wang, H. G., Xu, R. X., & Han, J. L. 2004, ApJL, 606, L49, doi: 10.1086/421048

    work page doi:10.1086/421048 2004

  48. [48]

    J., Liu, J

    Qiao, G. J., Liu, J. F., Zhang, B., & Han, J. L. 2001, A&A, 377, 964, doi: 10.1051/0004-6361:20011188

    work page doi:10.1051/0004-6361:20011188 2001

  49. [49]

    L., Tong, H., & Wang, H

    Qiu, J. L., Tong, H., & Wang, H. G. 2023, ApJ, 958, 78, doi: 10.3847/1538-4357/ad003f

    work page doi:10.3847/1538-4357/ad003f 2023

  50. [50]

    Radhakrishnan, V., & Cooke, D. J. 1969, Astrophys. Lett., 3, 225

    1969

  51. [51]

    M., & Devine, K

    Weisberg, J. M., & Devine, K. E. 2004, ApJ, 606, 1167, doi: 10.1086/383179

    work page doi:10.1086/383179 2004

  52. [52]

    Rankin, J. M. 1983, ApJ, 274, 333, doi: 10.1086/161450

    work page doi:10.1086/161450 1983

  53. [53]

    M., & Rathnasree, N

    Rankin, J. M., & Rathnasree, N. 1995, Journal of Astrophysics and Astronomy, 16, 327, doi: 10.1007/BF02715608

    work page doi:10.1007/bf02715608 1995

  54. [54]

    , eprint =

    Romani, R. W., & Yadigaroglu, I.-A. 1995, ApJ, 438, 314, doi: 10.1086/175076

    work page doi:10.1086/175076 1995

  55. [55]

    C., Weltevrede, P., & Johnston, S

    Rookyard, S. C., Weltevrede, P., & Johnston, S. 2015, MNRAS, 446, 3367, doi: 10.1093/mnras/stu2236

    work page doi:10.1093/mnras/stu2236 2015

  56. [56]

    A., & Sutherland , P

    Ruderman, M. A., & Sutherland, P. G. 1975, ApJ, 196, 51, doi: 10.1086/153393

    work page doi:10.1086/153393 1975

  57. [57]

    2017, MNRAS, 467, 2529, doi: 10.1093/mnras/stx204

    Saha, L., & Dyks, J. 2017, MNRAS, 467, 2529, doi: 10.1093/mnras/stx204

    work page doi:10.1093/mnras/stx204 2017

  58. [58]

    2021, MNRAS, 505, 4483, doi: 10.1093/mnras/staa2811 16

    Serylak, M., Johnston, S., Kramer, M., et al. 2021, MNRAS, 505, 4483, doi: 10.1093/mnras/staa2811 16

    work page doi:10.1093/mnras/staa2811 2021

  59. [59]

    AIP Conf

    Skilling, J. 2004, in American Institute of Physics Conference Series, Vol. 735, Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 24th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, ed. R. Fischer, R. Preuss, & U. V. Toussaint (AIP), 395–405, doi: 10.1063/1.1835238

    work page doi:10.1063/1.1835238 2004

  60. [60]

    Bayesian Analysis 1(4), 833–859 (2006) https://doi.org/10.1214/06-BA127

    Skilling, J. 2006, Bayesian Analysis, 1, 833 , doi: 10.1214/06-BA127

    work page doi:10.1214/06-ba127 2006

  61. [61]

    Speagle, J. S. 2020, MNRAS, 493, 3132, doi: 10.1093/mnras/staa278

    work page doi:10.1093/mnras/staa278 2020

  62. [62]

    Stovall, K., Freire, P. C. C., Chatterjee, S., et al. 2018, ApJL, 854, L22, doi: 10.3847/2041-8213/aaad06

    work page doi:10.3847/2041-8213/aaad06 2018

  63. [63]

    N., Wang , N., Yan , W

    Sun, S. N., Wang, N., Yan, W. M., & Wang, S. Q. 2025, ApJ, 983, 179, doi: 10.3847/1538-4357/adc254

    work page doi:10.3847/1538-4357/adc254 2025

  64. [64]

    2013, arXiv e-prints, arXiv:1304.4203, doi: 10.48550/arXiv.1304.4203

    Szary, A. 2013, arXiv e-prints, arXiv:1304.4203, doi: 10.48550/arXiv.1304.4203

    work page doi:10.48550/arxiv.1304.4203 2013

  65. [65]

    G., Pi, F

    Wang, H. G., Pi, F. P., Zheng, X. P., et al. 2014, ApJ, 789, 73, doi: 10.1088/0004-637X/789/1/73

    work page doi:10.1088/0004-637x/789/1/73 2014

  66. [66]

    F., Han , J

    Wang, P. F., Han, J. L., Xu, J., et al. 2023, Research in Astronomy and Astrophysics, 23, 104002, doi: 10.1088/1674-4527/acea1f

    work page doi:10.1088/1674-4527/acea1f 2023

  67. [67]

    M., & Taylor, J

    Weisberg, J. M., & Taylor, J. H. 2002, ApJ, 576, 942, doi: 10.1086/341803

    work page doi:10.1086/341803 2002

  68. [68]

    C., Rossi, E

    Weltevrede, P., & Johnston, S. 2008, MNRAS, 387, 1755, doi: 10.1111/j.1365-2966.2008.13382.x

    work page doi:10.1111/j.1365-2966.2008.13382.x 2008

  69. [69]

    L., & Haiman, Z

    Weltevrede, P., & Wright, G. 2009, MNRAS, 395, 2117, doi: 10.1111/j.1365-2966.2009.14643.x

    work page doi:10.1111/j.1365-2966.2009.14643.x 2009

  70. [70]

    1991, Nature, 350, 688, doi: 10.1038/350688a0

    Wolszczan, A. 1991, Nature, 350, 688, doi: 10.1038/350688a0

    work page doi:10.1038/350688a0 1991

  71. [71]

    Xu, Z.-H., Wang, W.-Y., Cao, S.-S., & Xu, R.-X. 2026, Research in Astronomy and Astrophysics, 26, 035014, doi: 10.1088/1674-4527/ae2b59 17 APPENDIX A.THE INNER OPEN FIELD LINES AND COORDINATE TRANSFORMATION This appendix derives the analytic equation for inner field lines whose footprints are uniformly distributed on the stellar surface and provides the t...

    work page doi:10.1088/1674-4527/ae2b59 2026