Method for carrying out pulsar time measurement by using multi-component template and least square method

Abstract

The invention discloses a pulsar time measurement method using a multi-component template and a least square method, and the method comprises the following steps: S1, specifying a pulsar P, obtaining the average pulse contour of N times of radio observation of the pulsar P, each average pulse contour containing K data points; s2, assuming that M linearly independent pulse components exist in the pulse profile, each pulse component contains S peaks, and each peak has three parameters of position, intensity and width, and fitting to obtain a standard pulse profile of each pulse component; s3, fitting an average pulse profile obtained by observation of the pulsar by using the standard pulse profile of each pulse component to obtain a linear coefficient and a phase difference of the pulse components; and S4, calculating the pulse arrival time corresponding to the specific average pulse contour by using the phase difference. According to the invention, the effect of the least square method on the aspect of multi-parameter fitting is utilized, and the calculated amount and error of measuring the pulse arrival time by using the multi-component standard pulse profile are reduced.

Description

Technical Field

[0001] The present invention relates to the fields of astronomy and computer communications, and in particular to a method for measuring pulsar timing by utilizing a multi-component pulse profile template and a least squares method. Background Art

[0002] Pulsars are compact celestial bodies that emit periodic, stable pulse signals. By measuring and analyzing the arrival time of pulsar signals from Earth, we can develop a pattern for these arrival times, known as the pulsar ephemeris model. This model allows theoretical predictions of future pulsar pulse arrival times. By comparing these theoretically predicted pulse arrival times with those obtained from actual observations, we can make corrections to the time at the observation station using the pulsar ephemeris model or detect the effects of signal propagation during pulsar observations. This allows us to obtain more accurate pulsar ephemeris models, more precise pulsar time standards, and gravitational wave signals. Therefore, accurately measuring the arrival times of pulsar pulses, or in other words, timing pulsars, is extremely necessary.

[0003] Currently, the main method for timing pulsars is to superimpose average pulse profiles obtained from multiple observations or perform multi-peak fitting based on these to obtain a standard pulse profile. The standard pulse profile is then compared with the average pulse profile obtained from a specific observation using the phase gradient method to obtain the phase difference, thereby calculating the pulse arrival time corresponding to the specific observation. This method relies on the long-term stability of the average pulse profile, but the average pulse profile of some pulsars does not have long-term stability.

[0004] Some pulsars have multiple radiation modes, each corresponding to a mean pulse profile. A single observation may contain multiple radiation modes, so the mean pulse profile corresponding to each observation can be considered a linear combination of the mean pulse profiles corresponding to the different modes. Therefore, it is necessary to decompose the multiple mean pulse profiles to obtain a more stable standard pulse profile for analyzing the pulse arrival times corresponding to the mean pulse profiles.

[0005] Currently, a technique for timing pulsars is available that separates the simulated observed profiles of pulsars with multiple radiation modes into their components. However, this technique loosens the degrees of freedom of each point in each component during the component separation process. Therefore, fitting M components of an average pulse profile consisting of N data points requires fitting M(N+K)+N parameters. This large number of parameters leads to large fitting errors. Furthermore, when measuring the arrival times of pulsar pulses, this technique superimposes the different components in different linear combinations and uses the phase gradient method to calculate the phase difference between each linear combination and then select the one with the smallest error. However, this method is computationally intensive, and the resulting linear combination cannot be considered the optimal pulse shape, thus introducing additional timing errors. Summary of the Invention

[0006] In response to the problems existing in the prior art, the purpose of the present invention is to provide a method for pulsar timing using a multi-component template and the least squares method. On the one hand, a multi-peak profile template can be used to obtain the optimal shape of multiple components, thereby reducing the fitting error caused by an excessive number of parameters; on the other hand, the least squares method is used for timing, thereby reducing the computational complexity and error of measuring the pulse arrival time using a multi-component standard pulse profile.

[0007] To achieve the above object, the present invention provides a method for pulsar timing using a multi-component template and the least squares method, the method comprising the following steps:

[0008] S1. For a given pulsar P, obtain the average pulse profile of N radio observations, where each average pulse profile contains K data points.

[0009] S2. Assume that there are M linearly independent pulse components in the pulse profile, each pulse component contains S peaks, and each peak has three parameters: position, intensity, and width. Fit the standard pulse profile of each pulse component;

[0010] S3. fitting the average pulse profile obtained from the pulsar observation using the standard pulse profile of each pulse component to obtain the linear coefficient and phase difference of the pulse component;

[0011] S4. Calculate the pulse arrival time corresponding to the specific average pulse profile using the phase difference.

[0012] Furthermore, in step S1, the average pulse profiles corresponding to the N observations obtained need to have the same observation frequency band, and each average pulse profile contains the same number of data points.

[0013] Furthermore, the average pulse profile corresponding to N observations of the same pulsar with the same data frequency band is selected. If the observation frequency bands of the telescopes are different, the data frequency needs to be selected in the data preprocessing stage to ensure the consistency of the data frequency band.

[0014] Furthermore, it is required that each average pulse profile contains the same number of data points. If the obtained average pulse profile data have different numbers of data points, it is necessary to make the number of data points of each average pulse profile consistent through interpolation methods or merging adjacent data points.

[0015] Furthermore, in step S2, each of the N average pulse profiles is represented as a linear combination of M pulse components, which is periodically shifted and then added to the data error. The optimal solution of the M pulse components is obtained by least squares fitting as a standard profile template.

[0016] Furthermore, in step S2, the least squares method is used to fit the S peaks of the K pulse components, and it is necessary to ensure that N(KM-1)>3MS and the standard deviation of the fitting residual is greater than the error of the observation data itself.

[0017] Furthermore, in step S3, for any average pulse profile A of pulsar P, assuming that it is represented by the result of periodic translation of a linear combination of M pulse components, the least squares fitting method is used to obtain the linear coefficients of the M pulse components in the standard profile template and their periodic translation, and the periodic translation, i.e., the phase difference, is calculated.

[0018] Furthermore, in step S3, the least square method is used to fit the periodic translation. It is necessary to fit a total of M linear coefficients and periodic translations of the M components in the average pulse profile, and there are a total of M+1 parameters to be fitted.

[0019] Furthermore, in step S3, the periodic translation amount is fitted using the least squares method, and the absolute value of the phase difference in the fitting result is less than 0.5.

[0020] Furthermore, in step S4, the pulse arrival time is calculated using the phase difference, which requires the theoretical arrival time of the pulse corresponding to the center time of the observation period using the average pulse profile A.

[0021] The beneficial effects of the present invention are as follows: the method of the present invention for pulsar timing using a multi-component pulse profile template and the least squares method can, on the one hand, adopt a multi-peak profile template method to obtain the optimal shape of multiple components, thereby reducing the fitting error caused by an excessive number of parameters; on the other hand, the least squares method is used for timing, thereby reducing the amount of calculation and error in measuring the pulse arrival time using a multi-component standard pulse profile; the role of the least squares method in multi-parameter fitting is utilized, thereby reducing the inapplicability of the phase gradient method caused by profile deviation, and the proportion of the template in different components can be obtained in addition to obtaining the pulse arrival time. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1 In the examples, the average pulse profile (upper subfigure) and the residuals (lower subfigure) obtained using multi-peak, multi-component fitting of pulsar J0218+4232 obtained from 31 FAST observations in 2021 are shown. The horizontal axis represents the pulse phase, while the vertical axes of the upper and lower subfigures represent the relative radiation intensity and residual intensity at different phases, respectively.

[0023] Figure 2 31 average pulse profiles are fitted using multi-peak and multi-component fitting, and the standard pulse profiles of the three components are shown in FIG. The horizontal axis represents the pulse phase, and the vertical axis represents the relative intensity of different phases.

[0024] Figure 3 Figure 1 shows the average pulse profile obtained by FAST on January 10, 2022, and the fitting result for this pulsar. In the upper subplot, the blue dotted line represents the observed average pulse profile, the orange solid line represents the fitting result, and the lower subplot shows the fitting residual. The horizontal axis represents the pulse phase, while the vertical axes of the upper and lower subplots represent the relative intensity and residual intensity at different phases, respectively. DETAILED DESCRIPTION

[0025] The following will clearly and completely describe the technical solution of the present invention in conjunction with the accompanying drawings. Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0026] In the description of the present invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicating orientations or positional relationships, are based on the orientations or positional relationships shown in the accompanying drawings and are intended solely to facilitate and simplify the description of the present invention. They are not intended to indicate or imply that the devices or components referred to must have, be constructed, or operate in a specific orientation, and therefore should not be construed as limitations on the present invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0027] In the description of the present invention, it should be noted that, unless otherwise expressly specified or limited, the terms "mounted," "connected," and "connected" should be understood in a broad sense. For example, they may refer to fixed, detachable, or integral connections; mechanical or electrical connections; direct or indirect connections through an intermediate medium; and internal communication between two components. Those skilled in the art will understand the specific meanings of the above terms in the present invention based on specific circumstances.

[0028] The following combination Figure 1-Figure 3 The specific embodiments of the present invention are described in detail. It should be understood that the specific embodiments described herein are only used to illustrate and explain the present invention and are not used to limit the present invention.

[0029] like Figure 1 、 Figure 2 、 Figure 3 As shown, the present invention uses a multi-component template and the least squares method to perform pulsar timing, and the method includes the following steps:

[0030] S1. For a given pulsar P, obtain the average pulse profile of N radio observations, where each average pulse profile contains K data points.

[0031] In step S1 , the average pulse profiles corresponding to the N observations obtained need to have the same observation frequency band, and each average pulse profile contains the same number of data points.

[0032] It is required to select the average pulse profile corresponding to N observations of the same pulsar with the same data frequency band. If the observation frequency bands of the telescope are different, the data frequency needs to be selected in the data preprocessing stage to ensure the consistency of the data frequency band.

[0033] At the same time, it is required that each average pulse profile contains the same number of data points. If the average pulse profile data obtained have different numbers of data points, it is necessary to use interpolation methods or merge adjacent data points to make the number of data points of each average pulse profile consistent.

[0034] S2. Assume that the pulse profile contains M linearly independent pulse components, each with S peaks, each with three parameters: position, intensity, and width. Each of the N average pulse profiles can be represented as a linear combination of the M pulse components, periodically shifted, and then summed with the data error. Least squares fitting is used to obtain the optimal solution for the M pulse components, which serves as the standard profile template.

[0035] In step S2, the least squares method is used to fit the S peaks of the K components, and it is necessary to ensure that N(KM-1)>3MS and the standard deviation of the fitting residual is greater than the error of the observed data itself.

[0036] In step S2, the peak intensity is taken as von-Mises distribution, that is, the peak center is φ0, the peak height is I0, and the peak width is w. The intensity of the von-Mises peak at the pulse phase φ is defined as I(φ; φ0, I0, w), and the expression is as follows:

[0037]

[0038] In step S2, the least squares method is used to fit the S peaks of M components. Each peak has three parameters: position, intensity and width. During fitting, it is necessary to fit the 3S parameters of the S peaks, the MN linear coefficients of the M components in the N average pulse profiles, and the periodic translation of the N average pulse profiles. There are a total of MN+3MS+N parameters to be fitted.

[0039] In step S2, mark I nk is the kth data point of the nth average pulse profile (its corresponding phase is φ k ),φ ms is the peak center of the sth peak in the mth component, I ms is the peak height of the sth peak in the mth component, w ms is the peak width of the sth peak in the mth component, h mn is the intensity of the mth component in the nth average pulse profile, δφ n is the periodic translation of the multi-component template in the nth average pulse profile, and the function to be optimized used by the least squares method is:

[0040]

[0041] The parameter to be fitted is h mn ,φ ms , I ms , w ms and δφ n .

[0042] S3. fitting the average pulse profile obtained from the pulsar observation using the standard pulse profile of each pulse component to obtain the linear coefficient and phase difference of the pulse component;

[0043] In step S3, for any average pulse profile A of pulsar P, assuming that it can also be expressed as the result of periodic translation of the linear combination of M pulse components, the least squares fitting method is used to obtain the linear coefficients of the M pulse components in the standard profile template and their periodic translation, and the periodic translation, i.e., the phase difference, is calculated.

[0044] In step S3, the periodic shift is fitted using the least squares method. This requires fitting the M linear coefficients and periodic shifts of the M components in the average pulse profile, resulting in a total of M+1 parameters to be fitted. The periodic shifts in the fitting result are actually congruences modulo 1, and in practice, their absolute value should be less than 0.5.

[0045] In step S3, mark I k is the kth data point of the average pulse profile (its corresponding phase is φ k ), h m is the intensity of the mth component in the average pulse profile, δφ is the periodic translation of the multi-component template in the average pulse profile, and the function to be optimized used by the least squares method is:

[0046]

[0047] The parameter to be fitted is h m and δφ.

[0048] S4. Calculate the pulse arrival time corresponding to the specific average pulse profile using the phase difference.

[0049] In step S4, the phase difference is used to calculate the pulse arrival time. The product of the phase difference and the pulsar period is used, and the theoretical arrival time of the pulse at the center time of the observation period corresponding to the average pulse profile A is summed to obtain the measured pulse arrival time.

[0050] In step S4, the pulsar period is marked as P, and the theoretical arrival time of the pulse at the center time of the observation period corresponding to the average pulse profile A is T0, then the pulse arrival time corresponding to the average pulse profile A is T0+Pδφ.

[0051] Example:

[0052] In this embodiment, the method for pulsar timing using a multi-component pulse profile template and the least squares method includes the following steps:

[0053] S1. The standard pulse profile was extracted using the average pulse profile obtained from 31 FAST observations of pulsar J0218+4232 in 2021. The frequency band is 1050-1450 MHz, and each cycle contains 512 data points. The average pulse profile obtained from the 31 observations is as follows: Figure 1 As shown in the upper sub-figure, it can be seen that different average pulse profiles have different shapes.

[0054] S2. Assume that the average pulse profile of J0218+4232 has three pulse components, each of which contains 10 peaks, and the peak intensity is von-Mises distributed. Fit the standard pulse profiles of the three pulse components to obtain the shapes of the three pulse components. Figure 2 As shown, the fitting residuals are Figure 1 As shown in the lower sub-figure.

[0055] S3. The standard pulse profiles of the three pulse components are used to fit the average pulse profile obtained by FAST observation of this pulsar on January 10, 2022. The fitting results are shown in the figure below. Figure 3 The linear coefficients of the three pulse components obtained by fitting are 0.68615801, 0.4928097 and 0.05679789, respectively, and the phase difference is 0.13127129948162847.

[0056] S4. The pulsar observation on January 10, 2022, began at MJD date 59589.4930555825922056 and lasted 1200.8179641400131 seconds. This results in a center time of 59589.5000047606254233, corresponding to the 258430th pulse in the observation data, with a theoretical pulse arrival time of 59589.500002853616699. The period of this pulsar is 0.0023232899447627373 seconds, so the measured pulse arrival time is 0.0023232899447627373 / 86400×0.13127129948162847+59589.500002853616699=59589.5000028571465751.

[0057] The advantages of this technology are:

[0058] 1. The present invention can perform high-precision timing on pulsars with a variety of average pulse profiles;

[0059] 2. The present invention adopts a multi-peak contour template method to obtain the optimal shape of multiple components, with fewer parameters to be fitted and smaller fitting errors;

[0060] 3. The present invention uses the least square method to measure time, which has a small amount of calculation and a small timing error.

[0061] Throughout this specification, reference to terms such as "embodiment" and "example" indicates that a specific feature, structure, material, or characteristic described in conjunction with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the schematic representations of these terms do not necessarily refer to the same embodiment or example. Furthermore, those skilled in the art may combine or integrate different embodiments or examples described in this specification, as well as features therein, without creating any inconsistency.

[0062] Although the above content has shown and described the embodiments of the present invention, it can be understood that the above embodiments are exemplary and cannot be understood as limitations of the present invention. Ordinary technicians in this field can perform update operations such as changes, modifications, replacements and variations on the above embodiments within the scope of the present invention.

Claims

1. A method for pulsar timing using a multi-component template and the least squares method, characterized in that: The method comprises the following steps: S1. For a given pulsar P, obtain the average pulse profile of N radio observations, where each average pulse profile contains K data points. S2. Assume that there are M linearly independent pulse components in the pulse profile, each pulse component contains S peaks, and each peak has three parameters: position, intensity, and width. Fit the standard pulse profile of each pulse component; S3. fitting the average pulse profile obtained from the pulsar observation using the standard pulse profile of each pulse component to obtain the linear coefficient and phase difference of the pulse component; S4. Calculate the pulse arrival time corresponding to the specific average pulse profile using the phase difference.

2. The method for pulsar timing using a multi-component template and the least squares method according to claim 1, wherein: In step S1 , the average pulse profiles corresponding to the N observations obtained need to have the same observation frequency band, and each average pulse profile contains the same number of data points.

3. The method for pulsar timing using a multi-component template and the least squares method according to claim 2, wherein: The average pulse profile corresponding to N observations of the same pulsar with the same data frequency band is selected. If the observation frequency bands of the telescopes are different, the data frequency needs to be selected in the data preprocessing stage to ensure the consistency of the data frequency band.

4. The method for pulsar timing using a multi-component template and the least squares method according to claim 2, wherein: It is required that each average pulse profile contains the same number of data points. If the obtained average pulse profile data have different numbers of data points, it is necessary to use interpolation methods or merge adjacent data points to make the number of data points of each average pulse profile consistent.

5. The method for pulsar timing using a multi-component template and the least squares method according to claim 1, wherein: In step S2, each of the N average pulse profiles is represented as the result of a linear combination of M pulse components, which is periodically shifted and then added to the data error. The optimal solution of the M pulse components is obtained using the least squares fitting method as the standard profile template.

6. The method for pulsar timing using a multi-component template and the least squares method according to claim 5, wherein: In step S2, the least squares method is used to fit the S peaks of the K pulse components, and it is necessary to ensure that N(KM-1)>3MS and the standard deviation of the fitting residual is greater than the error of the observation data itself.

7. The method for pulsar timing using a multi-component template and the least squares method according to claim 1, wherein: In step S3, for any average pulse profile A of pulsar P, assuming that it is represented by the result of periodic translation of the linear combination of M pulse components, the least squares method is used to fit the linear coefficients of the M pulse components in the standard profile template and their periodic translation, and the periodic translation, i.e., the phase difference, is calculated.

8. The method for pulsar timing using a multi-component template and the least squares method according to claim 7, wherein: In step S3, the least square method is used to fit the periodic translation. It is necessary to fit a total of M linear coefficients and periodic translations of the M components in the average pulse profile, and there are a total of M+1 parameters to be fitted.

9. The method for pulsar timing using a multi-component template and the least squares method according to claim 7, wherein: In step S3, the periodic translation amount is fitted using the least squares method, and the absolute value of the phase difference in the fitting result is less than 0.

5.

10. The method for pulsar timing using a multi-component template and the least squares method according to claim 1, wherein: In step S4, the phase difference is used to calculate the pulse arrival time, which requires the theoretical arrival time of the pulse at the center time of the observation period corresponding to the average pulse profile A.

Images