Time modeling method for pulsar photons to reach different centroids

Abstract

The invention relates to the technical field of cosmic navigation, in particular to a modeling method for time when pulsar photons reach different centroids, which comprises the following steps of: 1, calculating Julian day (JD) values according to a formula, and looking up a DE series calendar to obtain the position of a geocentric celestial sphere in a solar system centroid coordinate system (BCRS); 2, converting the position coordinates of the satellite in the GCRS into a BCRS position vector; 3, deducing a unit vector pointing to the pulsar direction from the satellite; 4, according to the distance from the crab-shaped nebula to the earth, determining a position vector of the pulsar based on a solar system centroid coordinate system (BCRS) through a unit vector; and 5, substituting the position vector of the pulsar relative to the BCRS and the position vector of the satellite relative to the BCRS into an Euclidean distance formula, and calculating the vacuum geometric propagation path time difference by combining the difference and the light velocity. The method can more accurately calculate the vacuum geometric propagation time delay when the pulsar X-ray photons arrive between the satellite and the center of mass of the solar system.

Description

Technical Field

[0001] The present invention relates to the field of space navigation technology, and in particular to a method for modeling the time when pulsar photons arrive at different mass centers. Background Art

[0002] In the field of deep space exploration, the propagation time modeling of pulsar X-ray signals is a key technology. It provides theoretical support for deep space exploration, autonomous navigation of spacecraft, and maintenance of high-precision time bases. As extremely stable celestial bodies, the X-ray signals emitted by pulsars have extremely high time accuracy, making them an important tool for studying high-precision navigation and time bases.

[0003] Existing technologies mainly rely on modeling the propagation characteristics of pulsar X-ray signals to provide navigation data support for deep space exploration missions. However, as exploration missions deepen, existing methods have certain limitations when calculating the arrival time of pulsar photons, such as the inability to accurately consider the impact of the motion of celestial bodies in the solar system, as well as the divergence and bending effects of light at great distances. Therefore, it is necessary to make improvements based on existing technologies. Summary of the invention

[0004] The purpose of the present invention is to solve the shortcomings existing in the prior art and to propose a time modeling method for pulsar photons arriving at different centers of mass. This method can more accurately calculate the vacuum geometric propagation delay of pulsar X-ray photons arriving at the satellite and the center of mass of the solar system.

[0005] In order to achieve the above object, the present invention adopts the following technical solutions:

[0006] A method for modeling the time at which pulsar photons arrive at different centers of mass comprises the following steps:

[0007] Step 1: Calculate the Julian day (JD) value according to the formula, and obtain the position of the geocentric celestial sphere in the solar system barycentric coordinate system (BCRS) according to the DE series ephemeris table;

[0008] Step 2: By adding the position coordinates of the Earth relative to the solar system barycenter and the position coordinates of the satellite relative to the geocentric celestial reference system (GCRS), the satellite's geocentric celestial reference system (GCRS) is converted to the coordinates of the solar system barycenter reference system (BCRS) to obtain the position vector;

[0009] Step 3: Use the pulsar's right ascension α and declination δ to represent the directional unit vector of the pulsar's location;

[0010] Step 4: According to the research, the distance from the Crab Nebula to the Earth is about 6.14965×10 16Kilometers, the direction vector of the satellite pointing to the pulsar Crab is obtained by the unit vector formula, and the position vector of the pulsar based on the solar system barycenter coordinate system (BCRS) is determined;

[0011] Step 5: Substitute the position vector of the pulsar relative to the solar system barycenter coordinate system (BCRS) and the position vector of the satellite relative to the solar system barycenter coordinate system (BCRS) into the Euclidean distance formula, then subtract the two and combine them with the speed of light and the vacuum geometry propagation formula to obtain the delayed time.

[0012] Preferably, in step 1, the geocentric celestial reference system (GCRS) is relative to the mass center of the earth, and the solar system barycentric coordinate system (BCRS) coordinate system is relative to the mass center of the solar system. The JD value can be obtained by the reduced Julian day (MJD) and Julian day (JD) conversion formula, and then the position of the geocentric celestial sphere in the solar system barycentric coordinate system (BCRS) can be obtained by looking up the DE series ephemeris table; the satellite's geocentric celestial reference system (GCRS) coordinates are converted to coordinates in the solar system barycentric coordinate system (BCRS), using the reduced Julian day (MJD) and Julian day (JD) conversion formula as shown in (1):

[0013] MJD=JD-2400000.5 (1).

[0014] Preferably, in step 2, the position of the Earth relative to the mass center of the solar system is added to the position of the satellite relative to the geocentric celestial reference system (GCRS) to obtain the position of the satellite relative to the mass center of the solar system (BCRS), as shown in formula (2):

[0015]

[0016] in, is the satellite position vector relative to the Geocentric Celestial Reference System (GCRS), is the position vector of the geocentric celestial sphere relative to the solar system's barycentric reference system, is the obtained satellite position vector relative to the Solar System Barycenter Reference System (GCRS).

[0017] Preferably, in step three, the position of the pulsar is expressed as a direction unit vector n by the following formula:

[0018] n=(cos(α)·cos(δ),sin(α)·cos(δ),sin(δ)) (3).

[0019] Preferably, in step 4, by multiplying the distance between the Crab Nebula where the Crab pulsar is located and the Earth by the unit direction vector from the satellite to the pulsar obtained by formula (3), it can be obtained:

[0020] n satellite,crab=d1·n (4)

[0021] Among them, n satellite,crab is the direction vector of the satellite pointing to the pulsar Crab, and d1 is the distance from the Crab Nebula to the Earth;

[0022] Then the satellite position coordinates obtained in formula (2) relative to the geocentric celestial reference system GCRS are Adding the position coordinates of the Earth in the solar system barycenter coordinate system obtained by formula (4), we can obtain the position coordinates of the pulsar in the solar system coordinate system (BCRS):

[0023]

[0024] Preferably, in step 5, the distances of the pulsar from the center of mass of the solar system and the satellite are determined, respectively, in combination with the Euclidean distance formula:

[0025]

[0026] The position coordinates of the pulsar equivalent to the barycenter coordinate system of the solar system (BCRS), the position coordinates of the satellite equivalent to the barycenter coordinate system of the solar system (BCRS) and the barycenter of the solar system, with the barycenter of the sun as the origin of the coordinate system;

[0027] set up for:

[0028]

[0029] set up for:

[0030]

[0031] Substituting into the Euclidean distance formula, we get:

[0032]

[0033] In formula (10), d crabpath and d satellitepath are the distances from the pulsar to the mass center of the solar system and from the pulsar to the satellite, respectively;

[0034] From the above, the vacuum geometric delay formula can be obtained as follows:

[0035]

[0036] In formula (11), Δt is the geometric propagation delay in vacuum, in seconds; ΔL is the geometric length difference of the propagation path, in kilometers; c is the speed of light in vacuum, which is approximately 299792.458, in kilometers per second.

[0037] Compared with the prior art, the present invention has the following beneficial effects:

[0038] The present invention uses geometric principles to select a better time delay model, establishes a model hypothesis for the huge star measurement problem, and uses Python to evaluate the effect of pulsar signal propagation delay, thereby minimizing the problem of too large data values ​​and the inability of the current scientific level to accurately measure the accurate values ​​of pulsars in distant galaxies. It can achieve the effect of using existing condition parameters to maximize the accurate calculation of delay. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 is a flow chart of the present invention;

[0040] Figure 2 Schematic diagram of the conversion of the distance between a pulsar and a satellite and the distance between a pulsar and the sun from the GCRS coordinate system to the BCRS coordinate system in an embodiment of the present invention;

[0041] Figure 3 A flowchart of using Monte Carlo simulation to evaluate the propagation delay of pulsar signals and calculating the standard deviation and confidence interval in an embodiment of the present invention;

[0042] Figure 4 is a geometric propagation delay distribution diagram obtained by Monte Carlo simulation in an embodiment of the present invention;

[0043] Figure 5 Graph showing the cumulative distribution function of geometric propagation delay obtained by Monte Carlo simulation in an embodiment of the present invention. DETAILED DESCRIPTION

[0044] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the accompanying drawings, so that those skilled in the art can better understand the advantages and features of the present invention, thereby making a clearer definition of the protection scope of the present invention. The embodiments described in the present invention are only a part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in the field without making creative work are within the scope of protection of the present invention.

[0045] Example:

[0046] A method for modeling the time at which pulsar photons arrive at different centers of mass comprises the following steps:

[0047] Step 1: Calculate the Julian day (JD) value according to the formula, and obtain the position of the geocentric celestial sphere in the solar system barycentric coordinate system (BCRS) according to the DE series ephemeris table;

[0048] Step 2: By adding the position coordinates of the Earth relative to the solar system barycenter and the position coordinates of the satellite relative to the geocentric celestial reference system (GCRS), the satellite's geocentric celestial reference system (GCRS) is converted to the coordinates of the solar system barycenter reference system (BCRS) to obtain the position vector;

[0049] Step 3: Use the pulsar's right ascension α and declination δ to represent the directional unit vector of the pulsar's location;

[0050] Step 4: According to the research, the distance from the Crab Nebula to the Earth is about 6.14965×10 16 Kilometers, the direction vector of the satellite pointing to the pulsar Crab is obtained by the unit vector formula, and the position vector of the pulsar based on the solar system barycenter coordinate system (BCRS) is determined;

[0051] Step 5: Substitute the position vector of the pulsar relative to the solar system barycenter coordinate system (BCRS) and the position vector of the satellite relative to the solar system barycenter coordinate system (BCRS) into the Euclidean distance formula, then subtract the two and combine them with the speed of light and the vacuum geometry propagation formula to obtain the delayed time.

[0052] like Figure 1 As shown in the figure, the idea flow of establishing the vacuum geometry propagation delay model of pulsar photons reaching different mass centers is as follows. First, the distance from the pulsar to the mass center of the solar system is explored, and then the distance from the pulsar to the sun and satellite is calculated respectively. Then, the propagation time difference of pulsar photons reaching different mass centers is obtained from the distance difference and the speed of light. Finally, the standard deviation of right ascension α and declination δ in a small range is established through the Monte Carlo algorithm, and the stability and accuracy of the model are tested under the conditions of randomly generated right ascension α and declination δ of photons within the standard deviation range 10,000 times.

[0053] Specifically, in step 1, the geocentric celestial reference system (GCRS) is relative to the Earth's barycenter, while the solar system barycenter coordinate system (BCRS) coordinate system is relative to the solar system barycenter. The JD value can be obtained by using the reduced Julian day (MJD) and Julian day (JD) conversion formula, and then the position of the geocentric celestial sphere in the solar system barycenter (BCRS) can be obtained by looking up the DE series ephemeris table. The satellite's geocentric celestial reference system (GCRS) coordinates are converted to coordinates in the solar system barycenter coordinate system (BCRS), using the reduced Julian day (MJD) and Julian day (JD) conversion formula as shown in (1):

[0054] MJD=JD-2400000.5 (1).

[0055] Specifically, in step 2, if Figure 2As shown, the distance between the pulsar and the satellite and the distance between the pulsar and the sun are converted from the GCRS coordinate system to the BCRS coordinate system. The JD value can be obtained by formula (1). The position of the geocentric celestial sphere in the solar system mass center (BCRS) is obtained by looking up the DE series ephemeris. The position of the earth relative to the solar system mass center is added to the position of the satellite relative to the geocentric celestial reference system (GCRS) to obtain the position of the satellite relative to the solar system mass center (BCRS), as shown in formula (2):

[0056]

[0057] in, is the satellite position vector relative to the Geocentric Celestial Reference System (GCRS), is the position vector of the geocentric celestial sphere relative to the solar system's barycentric reference system, is the obtained satellite position vector relative to the Solar System Barycenter Reference System (GCRS).

[0058] Specifically, in step 3, the unit vector pointing from the satellite to the pulsar is derived. The position of the pulsar in the sky is given by right ascension α and declination δ. The position of the pulsar is expressed as a directional unit vector n by the following formula:

[0059] n=(cos(α)·cos(δ),sin(α)·cos(δ),sin(δ)) (3).

[0060] Specifically, in step 4, the position coordinates of the pulsar based on the solar system barycenter coordinate system (BCRS) are determined. According to the above-mentioned known position of the geocentric sphere relative to the solar system barycenter coordinate system. By consulting the literature, it is known that the Crab Nebula where the Crab pulsar is located is about 6500 light years away from the Earth. By conversion, it can be obtained that it is about 6.14965×10 16 By multiplying the distance between the Crab Nebula where the Crab pulsar is located and the Earth by formula (3), we can get the unit direction vector from the satellite to the pulsar:

[0061] n satellite,crab =d1·n (4)

[0062] Among them, n satellite,crab is the direction vector of the satellite pointing to the pulsar Crab, and d1 is the distance from the Crab Nebula to the Earth;

[0063] Then the satellite position coordinates obtained in formula (2) relative to the geocentric celestial reference system GCRS are Adding the position coordinates of the Earth in the solar system barycenter coordinate system obtained by formula (4), we can obtain the position coordinates of the pulsar in the solar system coordinate system (BCRS):

[0064]

[0065] Specifically, in step 5, the distances of the pulsar from the center of mass of the solar system and the satellite are determined, combined with the Euclidean distance formula:

[0066]

[0067] The position coordinates of the pulsar equivalent to the barycenter coordinate system of the solar system (BCRS), the position coordinates of the satellite equivalent to the barycenter coordinate system of the solar system (BCRS) and the barycenter of the solar system, with the barycenter of the sun as the origin of the coordinate system;

[0068] set up for:

[0069]

[0070] set up for:

[0071]

[0072] Substituting into the Euclidean distance formula, we get:

[0073]

[0074] In formula (10), d crabpath and d satellitepath are the distances from the pulsar to the mass center of the solar system and from the pulsar to the satellite, respectively;

[0075] From the above, the vacuum geometric delay formula can be obtained as follows:

[0076]

[0077] In formula (11), Δt is the geometric propagation delay in vacuum, in seconds; ΔL is the geometric length difference of the propagation path, in kilometers; c is the speed of light in vacuum, which is approximately 299792.458, in kilometers per second.

[0078] The mathematical model of the time it takes for pulsar photons to reach different centers of mass is:

[0079]

[0080] Substituting the above pulsar position vector relative to the solar system barycenter coordinate system (BCRS) and the satellite position vector relative to the solar system barycenter coordinate system (BCRS) into equation (12), the solution can be obtained, as shown in Table 1:

[0081] Table 1 Solution of the mathematical model for the arrival time of pulsar photons at different centers of mass

[0082]

[0083] Monte Carlo simulation is used to evaluate the pulsar signal propagation delay model. The Monte Carlo algorithm can simulate the system through a large number of random samples and evaluate the comparative model under different input conditions.

[0084] like Figure 3 , the figure shows the calculation process of Monte Carlo standard deviation and confidence interval.

[0085] First, the speed of light, pulsar, Earth, and the center of mass of the solar system are set as constants and in fixed positions. Monte Carlo simulation is used to set a small range of standard deviations for right ascension and declination. Within the set standard deviation range, values ​​are randomly drawn from a normal distribution to provide different pulsar position vectors and pulsar unit direction vectors. For each sample, the geometric propagation delay of the pulsar signal is calculated.

[0086] like Figure 4 , the horizontal axis represents the propagation delay in seconds, while the vertical axis represents the density, that is, the relative density of each delay value. The area is the largest at about 277.8 seconds to 278 seconds, indicating that under the conditions of right ascension and declination randomly generated within the standard deviation of the simulation, the geometric propagation delay of most photons is between 277.8 seconds and 278 seconds.

[0087] like Figure 5 , the horizontal axis still represents the propagation delay, which is in seconds, while the vertical axis represents the probability density, which ranges from 0 to 1. The accumulated probability density gradually increases, and when the delay reaches about 278.1 seconds, the corresponding probability density reaches its peak.

[0088] The standard deviation is a statistic that measures the degree of dispersion of a set of numerical distributions. It indicates the deviation of the numerical values ​​in the data set from the mean value. Its calculation formula is as follows:

[0089]

[0090] Among them, σ represents the standard deviation we require, N is the number of data points, and x i is each data point, μ represents the average value of the data, and its calculation formula is:

[0091]

[0092] In the calculated results, the smaller the standard deviation, the more concentrated the data is, which means the better the stability of the model prediction is. The larger the standard deviation, the more discrete the data is, which means the worse the stability of the model prediction is.

[0093] Substituting the specific numerical values ​​into the formula, we can obtain the standard deviation of the model propagation delay as 0.0646150934, and the model prediction has better stability.

[0094] The confidence interval (95%) represents a range used to estimate an unknown population parameter (such as the population mean, population proportion, etc.). The calculation formula is as follows:

[0095]

[0096] Wherein, SE represents the confidence interval (95%) required by the present invention, x i It refers to each time delay, and n refers to the number of time delays. In the calculated results, the narrower the confidence interval, the more accurate and stable the prediction effect is, and the wider the confidence interval, the inaccurate and fluctuating the prediction effect is.

[0097] Substituting the obtained specific numerical value into the formula for calculation, the 95% confidence interval of the propagation delay is [277.7977903971 seconds, 278.0510815635 seconds], and the 95% confidence interval of the propagation delay is narrow.

[0098] Table 2 Summary of numerical evaluation of mathematical models for the arrival time of pulsar photons at different centers of mass

[0099]

[0100] According to the Monte Carlo algorithm simulation, the predicted values ​​of the mathematical model of the time it takes for pulsar photons to arrive at different centers of mass are stable and accurate, whether in terms of standard deviation or 95% confidence interval.

[0101] Description of the basic symbols of the present invention:

[0102]

[0103] In summary, the present invention uses geometric principles to select a better time delay model, establishes a model hypothesis for the huge star measurement problem, and uses Python to evaluate the effect of pulsar signal propagation delay, thereby minimizing the problem of too large data values ​​and the inability of existing scientific level to accurately measure the accurate values ​​of pulsars in distant galaxies, and can achieve the effect of using existing condition parameters to maximize the accurate calculation of delay.

[0104] The description and practice disclosed in the present invention are easy to think and understand for ordinary technicians in the technical field, and several improvements and modifications can be made without departing from the principles of the present invention. Therefore, modifications or improvements made without departing from the spirit of the present invention should also be regarded as the protection scope of the present invention.

Claims

1. A method for modeling the time at which pulsar photons arrive at different centers of mass, characterized in that: The steps include: Step 1: Calculate the Julian day JD value according to the formula, and obtain the position of the geocentric celestial sphere in the solar system barycentric coordinate system BCRS according to the DE series ephemeris table; Step 2: By adding the position coordinates of the Earth relative to the solar system barycenter and the position coordinates of the satellite relative to the geocentric celestial reference system GCRS, the satellite's geocentric celestial reference system GCRS is converted to the coordinates of the solar system barycenter reference system BCRS to obtain the position vector; Step 3: Use the pulsar's right ascension α and declination δ to represent the directional unit vector of the pulsar's location; Step 4: According to the research, the distance from the Crab Nebula to the Earth is 6.14965×10 16 Kilometers, the direction vector of the satellite pointing to the pulsar Crab is obtained by the unit vector formula, and the position vector of the pulsar based on the solar system barycenter coordinate system BCRS is determined; Step 5: Substitute the position vector of the pulsar relative to the solar system barycenter coordinate system BCRS and the position vector of the satellite relative to the solar system barycenter coordinate system BCRS into the Euclidean distance formula, then subtract the two and combine them with the speed of light and the vacuum geometry propagation formula to obtain the delayed time.

2. A method for modeling the time at which pulsar photons arrive at different centers of mass according to claim 1, characterized in that: In step 1, the geocentric celestial reference system GCRS is relative to the Earth's barycenter, while the solar system barycenter coordinate system BCRS is relative to the solar system barycenter. The JD value can be obtained by the conversion formula between the reduced Julian day MJD and the Julian day JD. Then, the position of the geocentric celestial sphere in the solar system barycenter BCRS can be obtained by looking up the DE series ephemeris. The satellite's geocentric celestial reference system GCRS coordinates are converted to coordinates in the solar system barycenter coordinate system BCRS, using the reduced Julian day MJD and Julian day JD conversion formula as shown in (1): MJD=JD-2400000.5 (1).

3. A method for modeling the time at which pulsar photons arrive at different centers of mass according to claim 1, characterized in that: In step 2, the position of the Earth relative to the mass center of the solar system is added to the position of the satellite relative to the geocentric celestial reference system GCRS to obtain the position of the satellite relative to the mass center of the solar system BCRS, as shown in formula (2): in, is the satellite position vector relative to the geocentric celestial reference system GCRS, is the position vector of the geocentric celestial sphere relative to the solar system's barycentric reference system, is the obtained satellite position vector relative to the solar system barycenter reference system GCRS.

4. A method for modeling the time at which pulsar photons arrive at different centers of mass according to claim 1, characterized in that: In step 3, the position of the pulsar is expressed as a directional unit vector n using the following formula: n=(cos(α)·cos(δ),sin(α)·cos(δ),sin(δ)) (3).

5. A method for modeling the time at which pulsar photons arrive at different centers of mass according to claim 4, characterized in that: In step 4, by multiplying the distance between the Crab Nebula where the Crab pulsar is located and the Earth by formula (3), the unit direction vector from the satellite to the pulsar can be obtained: n satellite,crab =d1·n (4) Among them, n satellite,crab is the direction vector of the satellite pointing to the pulsar Crab, and d1 is the distance from the Crab Nebula to the Earth; Then the satellite position coordinates obtained in formula (2) relative to the geocentric celestial reference system GCRS are Adding the position coordinates of the Earth in the solar system mass center coordinate system obtained by formula (4), we can get the position coordinates of the pulsar in the solar system coordinate system BCRS:

6. A method for modeling the time at which pulsar photons arrive at different centers of mass according to claim 1, characterized in that: In step 5, determine the distances of the pulsar from the center of mass of the solar system and the satellite, respectively, using the Euclidean distance formula: The pulsar is equivalent to the position coordinates of the solar system barycenter coordinate system BCRS, the satellite is equivalent to the position coordinates of the solar system barycenter coordinate system BCRS and the solar system barycenter, and the sun's barycenter is the origin of the coordinate system; set up for: set up for: Substituting into the Euclidean distance formula, we get: In formula (10), d crabpath and d satellitepath are the distances from the pulsar to the mass center of the solar system and from the pulsar to the satellite, respectively; From the above, the vacuum geometric delay formula can be obtained as follows: In formula (11), Δt is the vacuum geometric propagation delay, in seconds; ΔL is the geometric length difference of the propagation path, measured in kilometers; c is the speed of light in a vacuum, which is 299792.458, measured in kilometers per second.

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