Axion-modified model-based hybrid star quark core structure prediction method and system

Abstract

The application discloses a hybrid star quark core structure prediction method and system based on a axial sub correction model, and the method comprises the following steps: S1, calculating the equation of state of the strong phase to generate the equation of state data table of the strong phase; S2, introducing the axial field correction NJL model to calculate the equation of state to generate the equation of state data table of the quark phase; S3, determining the strong-quark first-order phase transition point by using Maxwell construction and splicing the equation of state of each section to build the complete equation of state of the hybrid star; S4, solving the TOV equation to obtain the first data of the hybrid star; S5, in the process of solving the TOV equation, when the density of the radial position decreases to the phase transition density of the phase transition point, the quark core radius and the quark core mass are extracted as the second data, and the prediction result is output. The application combines the axial sub correction NJL model with the Maxwell construction, optimizes the calculation process, solves the physical inconsistency problem, and increases the quark core size.

Description

Technical Field

[0001] This invention relates to the technical field of astrophysical and nuclear physics computation, specifically to a method and system for predicting the core structure of hybrid star quarks based on an axion correction model. Background Technology

[0002] Accurate prediction of the internal structure of compact stars is crucial for understanding the equation of state (EOS) of matter under extreme conditions and for validating particle physics models. With the advancement of astronomical observations, theoretical models urgently need to explain whether large quark cores exist within massive stars.

[0003] In theoretical modeling, existing studies typically use nuclear physics models (ImMDI model, MIT model, etc.) to describe the low-density hadron phase and the Nambu-Jona-Lasinio (NJL) model to describe the high-density quark phase. The NJL model is widely used in quark phase calculations because it can effectively describe chiral dynamics.

[0004] Existing research indicates that some studies have introduced axion field corrections into the KMT determinant interaction terms of the NJL model to explore the influence of axions on the properties of quark matter; other studies have used Maxwell construction, Gibbs construction and other methods to handle hadron-quark phase transitions to satisfy charge conservation and describe strong first-order phase transitions.

[0005] However, existing research lacks the synergistic effect of axion fields and Maxwell structures, which makes it impossible for models to accurately describe the modulation effect of axions on the QCD (quantum dynamical) vacuum and the thermodynamic equilibrium conditions of the phase transition interface in high-density regions. Furthermore, existing models have failed to solve the problem of quark cores being too small due to late phase transitions in high-density regions through joint optimization of axion corrections and phase transition structures, resulting in a tension between theory and observation. Summary of the Invention

[0006] To address the aforementioned problems, the purpose of this invention is to propose a method and system for predicting the quark core structure of hybrid stars based on an axion correction model. This method solves the problems in existing technologies, which lack the synergistic effect of axion fields and Maxwell structures, cannot accurately describe the modulation effect of axions on the QCD vacuum and the thermodynamic equilibrium conditions of the phase transition interface in high-density regions, and cannot solve the problem of excessively small quark cores caused by late phase transitions in high-density regions through joint optimization of axion correction and phase transition structures. This invention provides a theoretical framework for the equation of state of hybrid stars that is both physically consistent and observationally reliable, and provides quantitative evidence for verifying the existence and properties of axions through astronomical observations.

[0007] This was achieved through the following technical solutions: First, a method for predicting the core structure of hybrid star quarks based on an axion correction model is proposed, including the following steps: S1: Input the fixed physical parameter I into the ImMDI model, calculate the equation of state of the hadron phase, and generate a hadron phase equation of state data table; S2: Introduce the axion field-corrected NJL model, input the control parameter II and the fixed physical parameter II into the corrected NJL model, calculate the equation of state of the axion-corrected quark phase, and generate a data table of the quark phase equation of state. S3: Based on the hadron phase equation of state data table and the quark phase equation of state data table, the first-order phase transition point of the hadron-quark transition is determined by Maxwell construction and the equations of state are spliced ​​together in sequence to construct the complete equations of state of the hybrid star. S4: Based on the complete equation of state of the hybrid star, the TOV equation is solved by numerical integration. By traversing different central densities, the mass-radius relationship and the maximum mass parameter of the hybrid star are obtained as the first data. S5: In the process of solving the TOV equation, when the density at the radial position drops to the phase transition density at the phase transition point, the quark core radius and quark core mass are extracted as the second data. The first and second data are used as the prediction results of the hybrid star quark core structure and output.

[0008] Optionally, in step S1, the fixed physical parameter I includes: nuclear material saturation parameters, saturation point binding energy, incompressibility coefficient, symmetry energy slope parameter, and interaction parameters of the ImMDI model.

[0009] Optionally, in step S1, the ImMDI model, for neutrons, protons, electrons, and muons, iteratively solves for isospin asymmetry based on each given baryon density, and satisfies β equilibrium and charge neutrality conditions; then it traverses the baryon density range to generate a hadron phase equation of state data table containing multiple physical quantities.

[0010] Optionally, in step S2, the control parameter II includes the quark phase vector coupling constant and the axion chiral rotation angle; the fixed physical parameter II includes the cutoff energy of the NJL model, the mass of the u / d quark flow, the mass of the s quark flow, the scalar coupling constant, and the KMT coupling constant.

[0011] Optionally, after obtaining the control parameter II and the fixed physical parameter II in step S2, the axion chiral rotation angle is first introduced into the KMT six fermion interaction term of the NJL model itself to construct the Lagrangian of the axion-corrected NJL model. Then, under the mean-field approximation, based on scalar condensed matter and pseudoscalar condensed matter, the effective mass of the u, d, and s quarks is obtained by solving the bandgap equation through self-consistent iteration. Finally, based on each given baryon density, the nonlinear equations are solved using the Newton-Raphson method. Under the three thermodynamic equilibrium conditions of baryon number conservation, charge neutrality, and β equilibrium, the data table of quark phase equations of state is generated by traversing the range of baryon densities.

[0012] Optionally, in step S3, the process of determining the first-order phase transition point of the hadron-quark phase using Maxwell's construction includes: reading the hadron phase equation of state data table and the quark phase equation of state data table respectively, plotting the curve of pressure versus baryon chemical potential, and the intersection of the two curves is the phase transition point.

[0013] Optionally, in step S3, the process of splicing the equations of state into segments includes: sequentially splicing the equations of state of the outer crust, the equations of state of the inner crust, the hadron phase equations of state, the phase transition plateau segment equations of state, and the quark phase equations of state to construct the complete equations of state of the hybrid star.

[0014] Optionally, when solving the TOV equation in step S4, the solution is based on the complete equation of state of the hybrid star; first, different stellar center densities are given, and the integral is performed from the center of the star outwards to the point where the surface pressure is zero to obtain the total mass and radius of the corresponding star; then, the entire mass-radius relationship curve is generated by traversing the range of center densities, and the maximum mass and corresponding radius of the hybrid star are solved.

[0015] Optionally, the process of extracting the second data in step S5 includes: recording the density value at each radial position in real time during the integration of the TOV equation; and recording the current radius and the integrated mass as the second data when the density drops to the phase transition density.

[0016] Secondly, a hybrid star quark core structure prediction system is proposed, which is run using the aforementioned hybrid star quark core structure prediction method based on the axion correction model. The system includes: The hadron phase equation of state calculation module is configured to: input a fixed physical parameter I, use the ImMDI model to calculate the hadron phase equation of state, and generate a hadron phase equation of state data table; The axion-corrected quark phase equation of state calculation module is configured to: input control parameter II and fixed physical parameter II, use the three-flavor NJL model with axion field correction to calculate the quark phase equation of state, and generate a quark phase equation of state data table; The whole star equation of state construction module is configured to: use Maxwell's construction to determine the hadron-quark first-order phase transition point, and then sequentially splice together the outer crust, inner crust, hadron phase, phase transition plateau segment and quark phase to construct the complete equation of state of the hybrid star; The stellar structure calculation module is configured to: solve the TOV equation based on the complete equation of state of the hybrid star, traverse different central densities, and obtain the mass-radius relationship and maximum mass parameter of the hybrid star; The quark core parameter extraction module is configured to extract the quark core mass and radius parameters of hybrid stars with different masses from the TOV equation solution results and output the final prediction results.

[0017] The beneficial effects of this invention compared to the prior art are: 1) This invention combines the axion-corrected three-flavor NJL model with Maxwell's construction, which can accurately describe the modulation effect of the axion field through the KMT interaction QCD vacuum, as well as the thermodynamic equilibrium conditions of the hadron-quark first-order phase transition interface, thus improving the physical inconsistency caused by the separation of the axion effect and the phase transition structure in the prior art.

[0018] 2) This invention, through joint optimization of axion correction and Maxwell construction, significantly reduces the occurrence density and pressure of hadron-quark phase transitions, effectively alleviating the problem of excessively small quark nuclei caused by late phase transitions in high-density regions. Under weak vector coupling conditions, when the axion angle parameter reaches its maximum value, the quark nucleus mass of the most massive hybrid star can be increased to 0.36 M. ⊙ Its radius could increase to 5.0 km, nearly half the total radius of a star.

[0019] 3) The mass-radius relationship of the hybrid star predicted by this invention is consistent with the high-precision mass-radius measurement results of two massive pulsars by NICER and the tidal deformation constraints of the GW170817 binary neutron star merger event, which can verify the observational consistency of the model.

[0020] 4) This invention can systematically study the structural characteristics of hybrid stars under different axion coupling strengths and quark interaction strengths by adjusting the axion angle parameter and vector coupling constant; at the same time, the modular design allows each part to be upgraded and replaced independently, which is convenient for expansion to the study of the influence of other particles beyond the Standard Model on the structure of compact stars. Attached Figure Description

[0021] Figure 1 The flowchart shows a method for predicting the core structure of hybrid star quarks based on the axion correction model. Figure 2 This is a block diagram of a prediction system. Detailed Implementation

[0022] The technical solutions in the embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0023] like Figure 1The diagram shows a flowchart of a method for predicting the quark core structure of hybrid stars based on an axion-corrected model. First, the ImMDI model is used to calculate the hadron phase equation of state. Then, an axion chiral rotation angle correction is introduced into the KMT interaction term of the NJL model to calculate the axion-corrected quark phase equation of state. Next, the Maxwell construction is used to determine the hadron-quark first-order phase transition point, and the complete equation of state for hybrid stars is constructed by piecing them together. Finally, the TOV equation is solved, and quark core parameters are extracted. This invention combines the axion-corrected NJL model with the Maxwell construction, optimizing the calculation process and solving the physical inconsistency problem caused by the separation of axion effects and phase transition structure. This significantly reduces the phase transition density and pressure, effectively increasing the quark core size. Example 1

[0024] Combination Figure 1 As shown, this scheme provides a method for predicting the core structure of hybrid star quarks based on the axion-corrected NJL model, specifically including the following steps S1-S5.

[0025] Step S1: Input the fixed physical parameter I into the ImMDI model, calculate the equation of state for the hadron phase, and generate a hadron phase equation of state data table. Specifically, step S1 includes the following sub-steps: Step 1.1: First, set the fixed physical parameters I of the ImMDI model, specifically including: nuclear material saturation density. saturation point binding energy Incompressibility coefficient Symmetric energy slope parameter The effective mass of s quark m s * ≡0.7m N Potential energy parameter U 0,∞ ≡75MeV, interaction parameters are taken as , , The selection of these parameters ensures that the equation of state for hadron phases is consistent with known nuclear physics constraints in the low-density region.

[0026] Step 1.2: Next, calculate the thermodynamic properties of the hadron phase matter. The calculation process simultaneously considers the contributions of four particles: neutrons, protons, electrons, and muons, for each given baryon density. r By iteratively solving for the isospin asymmetry (the iterative process is a well-known and conventional technique in the industry, and will not be elaborated here), the β equilibrium and charge neutrality conditions are satisfied: , , Among them, non-subscript m and r These are the chemical potential and number density of the particle, respectively, with subscripts. n、p、e、μ These refer to the neutron, proton, electron, and muon, respectively.

[0027] Step 1.3: Generate a table of hadron phase equations of state data. Traverse the baryon density range of 10. -10 r 0 ~20 r 0 This generates a data table containing multiple physical quantities such as baryon density, energy density, pressure, and baryon chemical potential. This data table covers the entire range from extremely low density (crust) to high density (hadron phase core region), providing basic data for subsequent phase transition structures.

[0028] It should be noted that the ImMDI (modified momentum-dependent interaction) model is a widely validated nuclear many-body effective model that can reproduce the properties of nuclear matter near saturation density and the behavior of isospin asymmetric nuclear matter well. Combined with rigorous β equilibrium and charge-neutral iteration, it ensures the thermodynamic self-consistency of the hadron phase equation of state in the low to medium density region, providing a reliable hadron phase boundary benchmark for subsequent phase transition calculations.

[0029] Step S2: Introduce the axion field modified NJL model, input the control parameters and fixed physical parameters into the modified NJL model, and calculate the equation of state of the axion modified quark phase.

[0030] Step 2.1: Set the fixed physical parameter II and control parameter II for the NJL model. The fixed physical parameter II is set to: cutoff energy. u / d quark flow mass s-quark flow mass scalar coupling constant KMT coupling constant (KMT stands for Kobayashi-Maskawa-'tHooft). Control parameter II includes: quark phase vector coupling constant. G V =0.0 G S Axle manual rotation angle In this context, u, d, and s refer to the up quark, down quark, and strange quark, respectively.

[0031] Step 2.2: Construct the axion-corrected Lagrangian of the NJL model. The axion chiral rotation angle is introduced into the KMT six-fermion interaction term to achieve the axion field correction (the KMT six-fermion interaction term is a key term widely used in the NJL model to describe the QCD anomalous effect; for a three-flavor quark system, it manifests as a six-fermion determinant interaction). The corrected Lagrangian is:

[0032] in For quark fields, superscript T For transpose, For the flow quark mass matrix, here i The imaginary unit is 'm', where 'm' represents mass. Represents an antiquark field. Represents the Dirac operator, l a Let be the Gell-Man matrix of the SU(3) group, and , c 5 represents the gamma matrix, and , c 0 、c 1 、c 2 、c 3 These are the time component and the three-dimensional spatial component, respectively, and SU(3) represents the third-order special unitary group.

[0033] Step 2.3: Self-consistently solve for quark condensate and thermodynamic equilibrium conditions. Under the mean-field approximation, simultaneously consider scalar condensate. and pseudoscalar condensed matter The subscript i is a positive integer. The effective mass of each flavor quark is obtained by solving the bandgap equation through self-consistent iteration (the self-consistent iteration method is a well-known solution method in this field and will not be elaborated here). The mean-field approximation can equate complex many-body nonlinear interactions to the motion of a single entity in a background field, enabling the solution of quark effective mass and thermodynamic potential even under high-density extreme conditions, thus ensuring the physical rigor and numerical stability of quark phase calculations.

[0034] For each given baryon density r b The nonlinear equations are solved using the Newton-Raphson method, specifically by determining the particle number density that satisfies the following thermodynamic equilibrium conditions: .

[0035] Step 2.4: Generate the quark phase equation of state data table. Traverse the baryon density range of 10. -10 p0 ~20 p0 This generates a data table containing multiple physical quantities such as baryon density, energy density, pressure, and baryon chemical potential.

[0036] Step S3: Construct the complete equation of state for the hybrid star using the Maxwell construction. The Maxwell construction ensures the thermodynamic self-consistency of the phase transition interface.

[0037] Step 3.1: Determine the first-order hadron-quark phase transition point. Read the hadron phase equation of state data tables and the quark phase equation of state data tables respectively to obtain the equation of state data for the hadron and quark phases. Plot the pressure versus baryon chemical potential curves in the same coordinate system. The intersection of the two curves is the phase transition point, which satisfies the conditions of equal pressure and equal baryon chemical potential. ,in, P Indicates pressure, H Indicates hadron phase, Q It represents the quark phase.

[0038] Step 3.2: Assemble the equations of state. Sequentially assemble the outer crust, inner crust, hadron phase, phase transition plateau, and quark phase segments to construct the complete equations of state for the hybrid star.

[0039] By modifying the quark phase with axions, the pressure-chemical potential curve of the quark phase shifts as a whole towards lower chemical potentials, thereby causing the phase transition point to appear at a lower baryon density, which can increase the size of the quark core.

[0040] Step S4: Solve the TOV equations to obtain the hybrid star structure parameters. Based on the complete equation of state, solve the Tolman-Oppenheimer-Volkoff (TOV) equations within the framework of general relativity: ,, Where G is the gravitational constant. e For energy density, M radius r Star masses within a certain range. A numerical integration method is used to traverse the central density range of 0.08. r 0 ~12 r 0 The mass-radius relationship and maximum mass parameter of the hybrid star were obtained as the first data.

[0041] Step S5: Extract quark core parameters and output prediction results. During the integration of the TOV equation, dynamic monitoring is performed. When the density at the radial position decreases to the phase transition density, the current radius and integrated mass are recorded, which are the radius and mass of the quark core, and used as the second data. Both the first and second data are output. The calculation results of this embodiment show that when At that time, the total mass of the most massive hybrid star was approximately 1.94. M ⊙ Its total radius is approximately 12.7 km, and the mass of its quark core is approximately 0.08. M ⊙ Its radius is approximately 3.2 km. M ⊙ This indicates the mass of the sun.

[0042] Further calculations with varying control parameters revealed that the mass and radius of the quark nucleus change with the axion chiral rotation angle. a / f a The value increases monotonically as it increases, when a / f a When the maximum value π is taken, the quark core mass of the most massive hybrid star can increase to 0.36. M ⊙ The radius can be increased to 5.0 km; the mass and radius of the quark core vary with the vector coupling constant. G V As it increases, it monotonically decreases, when G V More than 0.2 G S At that time, the quark core essentially disappeared. These results are consistent with NICER's high-precision mass radius measurements of two massive pulsars and the tidal deformation constraints of the GW170817 binary neutron star merger event. Example 2

[0043] Combination Figure 2 The system framework shown in this embodiment also discloses a hybrid star quark core structure prediction system based on the axion-corrected NJL model. The system is based on the hybrid star quark core structure prediction method based on the axion-corrected NJL model described in Embodiment 1. The system includes the following modules connected in series: hadron phase equation of state calculation module, axion-corrected quark phase equation of state calculation module, whole star equation of state construction module, stellar structure calculation module, and quark core parameter extraction module.

[0044] The hadron phase equation of state calculation module is configured to: input a fixed physical parameter I, use the ImMDI model to calculate the hadron phase equation of state, and generate a hadron phase equation of state data table.

[0045] The axion-corrected quark phase equation of state calculation module is configured to: input control parameter II and fixed physical parameter II, use the three-flavor NJL model with axion field correction to calculate the quark phase equation of state, and generate a quark phase equation of state data table.

[0046] The whole star equation of state construction module is configured to: use Maxwell's construction to determine the hadron-quark first-order phase transition point, and sequentially splice together the outer crust, inner crust, hadron phase, phase transition plateau segment and quark phase to construct the complete equation of state of the hybrid star.

[0047] The stellar structure calculation module is configured to: solve the TOV equation based on the complete stellar equation, traverse different central densities, and obtain the mass-radius relationship and maximum mass parameter of the hybrid star.

[0048] The quark core parameter extraction module is configured to extract the quark core mass and radius parameters of hybrid stars with different masses from the TOV equation solution results and output the final prediction results.

[0049] In summary, this invention first uses the ImMDI model to calculate the hadron phase equation of state, then introduces an axion chiral rotation angle correction into the KMT interaction term of the NJL model to calculate the axion-corrected quark phase equation of state. Next, it uses Maxwell's construction to determine the hadron-quark first-order phase transition point and assembles the complete equation of state for hybrid stars. Finally, it solves the TOV equation and extracts quark core parameters. This invention combines the axion-corrected NJL model with Maxwell's construction, optimizing the calculation process, solving the physical inconsistency problem caused by the separation of axion effects and phase transition structures, significantly reducing phase transition density and pressure, and effectively increasing quark core size, demonstrating significant progress.

[0050] The above embodiments are merely illustrative of the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made to the technical solutions based on the technical concept proposed in this invention shall fall within the scope of protection of this invention.

Claims

1. A method for predicting the core structure of hybrid star quarks based on an axion correction model, characterized in that, Includes the following steps: S1: Input the fixed physical parameter I into the ImMDI model, calculate the equation of state of the hadron phase, and generate the hadron phase equation of state data; S2: Introduce the axion field-corrected NJL model, input the control parameter II and the fixed physical parameter II into the corrected NJL model, calculate the equation of state of the axion-corrected quark phase, and generate a data table of the quark phase equation of state. S3: Based on the hadron phase equation of state data table and the quark phase equation of state data table, the first-order phase transition point of the hadron-quark transition is determined by Maxwell construction and the equations of state are spliced ​​together in sequence to construct the complete equations of state of the hybrid star. S4: Based on the complete equation of state of the hybrid star, the TOV equation is solved by numerical integration. By traversing different central densities, the mass-radius relationship and the maximum mass parameter of the hybrid star are obtained as the first data. S5: In the process of solving the TOV equation, when the density at the radial position drops to the phase transition density at the phase transition point, the quark core radius and quark core mass are extracted as the second data. The first and second data are used as the prediction results of the hybrid star quark core structure and output.

2. The method for predicting the hybrid star quark core structure based on the axion-corrected NJL model according to claim 1, characterized in that, In step S1, the fixed physical parameters I include: nuclear material saturation parameters, saturation point binding energy, incompressibility coefficient, symmetry energy slope parameters, and interaction parameters of the ImMDI model.

3. The method for predicting the hybrid star quark core structure based on the axion-corrected NJL model according to claim 2, characterized in that, In step S1, the ImMDI model, for neutrons, protons, electrons and muons, iterates to solve for isospin asymmetry based on each given baryon density, and satisfies the β equilibrium and charge neutrality conditions; then it traverses the baryon density range to generate a hadron phase equation of state data table containing multiple physical quantities.

4. The method for predicting the hybrid star quark core structure based on the axion-corrected NJL model according to claim 1, characterized in that, In step S2, control parameter II includes the quark phase vector coupling constant and the axion chiral rotation angle; fixed physical parameter II includes the cutoff energy of the NJL model, the mass of the u / d quark flow, the mass of the s quark flow, the scalar coupling constant, and the KMT coupling constant.

5. The method for predicting the hybrid star quark core structure based on the axion-corrected NJL model according to claim 4, characterized in that, In step S2, after the NJL model obtains the control parameter II and the fixed physical parameter II, the axion chiral rotation angle is first introduced into the KMT six fermion interaction term of the NJL model itself to construct the Lagrangian of the axion-corrected NJL model. Then, under the mean-field approximation, based on scalar condensed matter and pseudoscalar condensed matter, the effective mass of the u, d, and s quarks is obtained by solving the bandgap equation through self-consistent iteration. Finally, based on each given baryon density, the nonlinear equations are solved using the Newton-Raphson method. Under the three thermodynamic equilibrium conditions of baryon number conservation, charge neutrality, and β equilibrium, the data table of quark phase equations of state is generated by traversing the range of baryon densities.

6. The method for predicting the core structure of hybrid star quarks based on the axion-corrected NJL model according to claim 1, characterized in that, In step S3, the process of determining the first-order phase transition point of the hadron-quark transition using Maxwell's construction includes: reading the hadron phase equation of state data table and the quark phase equation of state data table respectively, plotting the curve of pressure versus baryon chemical potential, and the intersection of the two curves is the phase transition point.

7. The method for predicting the core structure of hybrid star quarks based on the axion-corrected NJL model according to claim 6, characterized in that, In step S3, the process of splicing the equations of state into segments includes: sequentially splicing the equations of state of the outer crust, the equations of state of the inner crust, the hadron phase equations of state, the phase transition plateau segment equations of state, and the quark phase equations of state to construct the complete equations of state of the hybrid star.

8. The method for predicting the hybrid star quark core structure based on the axion-corrected NJL model according to claim 1, characterized in that, In step S4, when solving the TOV equation, the solution is based on the complete equation of state of the hybrid star. First, different stellar center densities are given, and the integral is performed from the center of the star outwards until the surface pressure is zero to obtain the total mass and radius of the corresponding star. Then, the entire range of center densities is traversed to generate a complete mass-radius relationship curve, and the maximum mass and corresponding radius of the hybrid star are solved.

9. The method for predicting the hybrid star quark core structure based on the axion-corrected NJL model according to claim 1, characterized in that, The process of extracting the second data in step S5 includes the following steps: during the integration of the TOV equation, the density values ​​at each radial position are recorded in real time. When the density drops to the phase transition density, the current radius and the integrated mass are recorded and used as the second data.

10. A hybrid star quark core structure prediction system, operating using the hybrid star quark core structure prediction method based on the axion correction model as described in any one of claims 1 to 9, characterized in that, include: The hadron phase equation of state calculation module is configured to: input a fixed physical parameter I, use the ImMDI model to calculate the hadron phase equation of state, and generate a hadron phase equation of state data table; The axion-corrected quark phase equation of state calculation module is configured to: input control parameter II and fixed physical parameter II, use the three-flavor NJL model with axion field correction to calculate the quark phase equation of state, and generate a quark phase equation of state data table; The whole star equation of state construction module is configured to: use Maxwell's construction to determine the hadron-quark first-order phase transition point, and then sequentially splice together the outer crust, inner crust, hadron phase, phase transition plateau segment and quark phase to construct the complete equation of state of the hybrid star; The stellar structure calculation module is configured to: solve the TOV equation based on the complete equation of state of the hybrid star, traverse different central densities, and obtain the mass-radius relationship and maximum mass parameter of the hybrid star; The quark core parameter extraction module is configured to extract the quark core mass and radius parameters of hybrid stars with different masses from the TOV equation solution results and output the final prediction results.

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