Moon surface heat flow inversion method based on three-layer steady-state heat conduction model

Abstract

The invention discloses a lunar surface heat flow inversion method based on a three-layer steady-state heat conduction model, and the method comprises the steps: 1, calculating the total mass heat production rate through the specific heat production rate and mass heat production rate of radioactive elements in the moon; 2, constructing a three-layer steady-state heat conduction model comprising a giant weathered layer, a moon shell and a moon curtain, and setting boundary and heat flow continuity conditions; 3, solving a coefficient of a heat conduction equation of the three-layer steady-state heat conduction model to obtain a temperature profile and a temperature gradient; 4, calculating brittleness strength and plastic strength envelopes by using a temperature profile, and calculating a predicted bending moment by using an envelope integral; 5, calculating a target bending moment by using the effective elastic thickness of the lithosphere; 6, continuously adjusting the thickness of the giant weathered layer, reducing the deviation between the predicted bending moment and the target bending moment, and obtaining a convergent temperature structure; 7, solving a surface temperature gradient from the convergent temperature structure; and 8, calculating the lunar surface heat flow according to the surface temperature gradient. The estimation precision of the lunar surface heat flow is improved.

Description

Technical Field

[0001] This invention relates to planetary science and geophysics, specifically a method for retrieving lunar surface heat flow based on a three-layer steady-state heat conduction model. Background Technology

[0002] Currently, direct acquisition of lunar surface heat flow mainly relies on limited measurement points from the Apollo missions. These data are concentrated in the Oceanus Procellarum region on the near side of the Moon and cannot represent global characteristics. Moreover, existing heat flow estimates are mostly based on elastic thickness or radioactive element distribution models, ignoring the thermal insulation effect of the giant regolith and the significant uncertainty in the elastic thickness of the lithosphere. The giant regolith is characterized by low density, low thermal conductivity, and high porosity, reaching a thickness of 1-km in the lunar highlands. Its thermal resistance significantly affects surface heat flow conduction. However, existing heat flow estimation models do not consider the thermal insulation effect of the giant regolith and the uncertainty in the elastic thickness of the lithosphere, resulting in significant deviations in the heat flow estimation results.

[0003] Therefore, how to introduce the thickness of the giant regolith and the effective elastic thickness of the lithosphere to effectively constrain the heat flow on the lunar surface is a key challenge in the current study of planetary thermal evolution. Summary of the Invention

[0004] The purpose of this invention is to provide a lunar surface heat flow inversion method based on a three-layer steady-state heat conduction model. By introducing the thickness of the giant regolith and the effective elastic thickness of the lithosphere, the estimation accuracy of lunar surface heat flow is improved.

[0005] This invention is achieved through the following technical solution: A lunar surface heat flow inversion method based on a three-layer steady-state heat conduction model includes the following steps: Step 1: Using radioactive elements as the radioactive heat source, obtain the total mass heat production rate based on the specific heat production rate and mass heat production rate of radioactive elements Th, U, and K in the lunar crust and mantle. ; Step 2: Construct a three-layer steady-state heat conduction model and set boundary and heat flow continuity conditions. The specific process is as follows: Step 2.1: Divide the near-surface region of the moon into a giant regolith, lunar crust, and lunar mantle, and construct a three-layer steady-state heat conduction model. Its heat conduction equation is expressed as: In the formula: , , and These represent the temperatures of the giant regolith, lunar crust, lunar mantle, and lunar surface, respectively. It is 253 K; , and These represent the heat generated per unit volume of the giant regolith, lunar crust, and lunar mantle, respectively. , and The thermal conductivity density and total mass heat production rate are respectively for the giant regolith, lunar crust, and lunar mantle. The product; , and These represent the thermal conductivity of the giant regolith, lunar crust, and lunar mantle, respectively. , , , and All are coefficients; , and These represent the thicknesses of the giant regolith, the lunar crust, and the lunar mantle, respectively. Indicates the depth perpendicular to the surface of the moon; Step 2.2: Set the boundary and heat flow continuity conditions for the three-layer steady-state heat conduction model, including: the heat flux and temperature at the boundary between the giant regolith and the lunar crust, the heat flux and temperature at the boundary between the lunar crust and the lunar mantle, and the temperature at the boundary between the lunar mantle and the lunar core. Step 3: Using the temperature of the lunar surface, the boundary temperatures of the lunar mantle and the lunar core as boundary conditions for solving the heat conduction equation, solve for the coefficients. , , , and Substitute it into the heat conduction equation The temperature structure parameters of the giant regolith, lunar crust, and lunar mantle were solved, including temperature profiles. and temperature gradient ; Step 4: Calculate the envelopes of brittle strength and ductile strength using the temperature profile. The specific process is as follows: Step 4.1: Calculate the brittle strength of the lithosphere. and plastic strength They are represented as follows: In the formula: The density of the lithosphere; It is the acceleration due to gravity; It is the depth perpendicular to the surface of the moon; It is the strain rate; It refers to the grain size of the lithosphere; It is the gas constant; It is a temperature profile; , , and All are rheological parameters; Step 4.2, regarding brittle strength and plastic strength Envelope, represented as: Step 4.3: Using the intensity envelope integral, according to The lower limit value is used to calculate the predicted bending moment. , is represented as: In the formula: Indicates the depth of the neutral surface; Step 5: Target bending stiffness , is represented as: In the formula: and These represent the elastic modulus and Poisson's ratio, respectively. Indicates the effective elastic thickness of the lithosphere; Then the target bending moment Represented as: In the formula: Represents curvature, with a value of 1.0 × 10⁻⁶. -9 ; Step 6: Predict the bending moment With the target bending moment By comparing and continuously adjusting the thickness of the giant weathering layer Thus reducing and The deviation is used to obtain a convergent temperature structure; Step 7: Calculate the surface temperature gradient from the convergent temperature structure: First, calculate the convergent temperature profile... , is represented as: The temperature gradient is then expressed as: Step 8: Under the constraints of the boundary and heat flow continuity conditions of the three-layer steady-state heat conduction model, the heat flow on the lunar surface is... Represented as: In the formula: This indicates the thermal conductivity of the giant regolith.

[0006] Furthermore, the overall mass heat production rate in step 1 The calculation formula is: In the formula: , , and Representing radioactive elements 238 U、 235 U、 232 Th and 40 Specific heat production rate of K; , , and Representing radioactive elements 238 U、 235 U、 232 Th and 40 Mass heat production rate of K.

[0007] Furthermore, the specific process of step 2.2 is as follows: 1) At the boundary between the giant regolith and the lunar crust: 1.1) The condition that the heat flux needs to meet is expressed as follows: 1.2) The temperature requirement is expressed as follows: 2) At the boundary between the lunar crust and the lunar mantle: 2.1) The condition that the heat flux needs to meet is expressed as: 2.2) The temperature requirement is expressed as follows: 3) At the boundary between the lunar mantle and the lunar core, the temperature requirement is expressed as follows: In the formula: The boundary temperature between the lunar mantle and the lunar core is 1900 K.

[0008] Furthermore, the specific process of step 3 is as follows: Step 3.1, Apply the formula , , , and Written in matrix form, it can be represented as: In the formula: , , , and Represented as: Step 3.2, Elimination of coefficients and , is represented as: Step 3.3, Solve for coefficients , is represented as: Step 3.4: Solve for the coefficients and , is represented as: Step 3.5: Solve for the coefficients and , is represented as: By solving , , , and Substitute it into the heat conduction equation The temperature structure parameters of the giant regolith, lunar crust, and lunar mantle inside the moon were solved, including temperature profiles. and temperature gradient .

[0009] The present invention has the following beneficial technical effects: First, by introducing a giant regolith with low thermal conductivity, the inversion of lunar surface heat flow is corrected by considering its insulating effect, thus correcting the overestimation of lunar surface heat flow in traditional models and making the inversion results closer to reality. Second, based on the brittle-elastic-plastic transition exhibited by the lithosphere, the brittle and plastic strength envelopes are calculated, and then the predicted bending moment is calculated. Simultaneously, the target bending moment is calculated based on the effective elastic thickness of the lithosphere in existing studies. Then, by adjusting the thickness of the giant regolith, the deviation between the predicted and target bending moments is reduced, thereby iteratively correcting the temperature structure. This achieves joint constraints of the giant regolith and the effective thickness of the lithosphere, enabling accurate calculation of lunar surface heat flow and thus obtaining results that more realistically reflect the distribution of lunar surface heat flow. Third, the method of this invention has strong versatility and can be extended to the thermal structure inversion and crustal heat flow estimation of terrestrial planets such as Mars and Mercury, providing technical support for the study of planetary internal thermal history. Attached Figure Description

[0010] Figure 1 This is a schematic diagram of the structure of the invented three-layer steady-state heat conduction model. Detailed Implementation

[0011] The present invention will be further described in detail below with reference to specific embodiments. These descriptions are for explanation purposes only and are not intended to limit the scope of the invention.

[0012] This embodiment utilizes existing technologies and the latest research results on the effective elastic thickness distribution obtained by "inversion based on admittance spectrum using GRAIL gravity field and LRO topographic data" (e.g., about 20 km in highland areas) and the results of "research on giant regolith" (1-3 km) to determine the thickness of the giant regolith, the thickness of the lunar crust, the thickness of the lunar mantle, the lunar surface temperature, the boundary temperature of the lunar mantle-lunar core, the thermal conductivity density of each layer, the radioactive heat source, and the rheological parameters.

[0013] A method for inverting lunar surface heat flow based on a three-layer steady-state heat conduction model includes the following steps: Step 1: Using radioactive elements as the radioactive heat source, obtain the total mass heat production rate based on the specific heat production rate and mass heat production rate of radioactive elements Th, U, and K in the lunar crust and mantle. , is represented as: In the formula: , , and Representing radioactive elements 238 U、 235 U、 232 Th and 40 Specific heat production rate of K; , , and Representing radioactive elements 238 U、 235 U、 232 Th and 40 Mass heat production rate of K; Step 2: Construct a three-layer steady-state heat conduction model and set boundary and heat flow continuity conditions. The specific process is as follows: Step 2.1: Based on steady-state thermal conductivity, the near-surface region of the Moon is divided into three layers with different thermal conductivity and different radioactive heat sources along the vertical z-direction: the giant regolith, the lunar crust, and the lunar mantle, thus constructing a... Figure 1 The three-layer steady-state heat conduction model shown (the outermost giant regolith layer is very small relative to the lunar crust and mantle, so it is not shown in the figure) satisfies one-dimensional steady-state heat conduction in each layer. Therefore, the heat conduction equation of the three-layer steady-state heat conduction model can be expressed as: In the formula: , , and These represent the temperatures of the giant regolith, lunar crust, lunar mantle, and lunar surface, respectively. It is 253 K; , and These represent the heat generated per unit volume of the giant regolith, lunar crust, and lunar mantle, respectively. , and The thermal conductivity density and total mass heat production rate are respectively for the giant regolith, lunar crust, and lunar mantle. The product; , and These represent the thermal conductivity of the giant regolith, lunar crust, and lunar mantle, respectively. , , , and All are coefficients; , and These represent the thicknesses of the giant regolith, the lunar crust, and the lunar mantle, respectively. Indicates the depth perpendicular to the surface of the moon; Step 2.2: Define the boundary and heat flow continuity conditions for the three-layer steady-state heat conduction model, where: 1) At the boundary between the giant regolith and the lunar crust: 1.1) The heat flux satisfies the following conditions: Right now: 1.2) The temperature meets the following conditions: Right now: 2) At the boundary between the lunar crust and the lunar mantle: 2.1) The heat flux satisfies the following conditions: 2.2) The temperature meets the following conditions: 3) At the boundary between the lunar mantle and the lunar core, the temperature meets the following condition: In the formula: The boundary temperature between the lunar mantle and the lunar core is 1900 K; Step 3: Determine the temperature of the lunar surface. Boundary temperature of the lunar mantle and core As boundary conditions for solving the heat conduction equation, the coefficients are solved. , , , and The specific process is as follows: Step 3.1, Apply the formula , , , and Written in matrix form, it can be represented as: In the formula: , , , and Represented as: Step 3.2, Elimination of coefficients and , is represented as: Step 3.3, Solve for coefficients , is represented as: Step 3.4: Solve for the coefficients and , is represented as: Step 3.5: Solve for the coefficients and , is represented as: Will , , , and Substitute into the heat conduction equation The temperature structure parameters of the giant regolith, lunar crust, and lunar mantle inside the moon were determined, including temperature profiles. and temperature gradient Where: temperature gradient It is the key to obtaining surface heat flow; Step 4: Calculate the brittle and ductile strength envelopes using temperature profiles: The closer to the lunar surface, the lower the temperature and pressure, and brittle deformation dominates. Deformation at localized fracture points leads to brittle failure. The brittle strength is lowest at the lithosphere surface and increases with increasing pressure. In the region near the bottom of the lithosphere, ductile deformation occurs under high temperature and pressure, including dislocation creep and diffusion creep, where diffusion creep depends on grain size. Furthermore, ductile strength decreases exponentially with increasing temperature, with the lowest ductile strength at the lithosphere basement. That is, the lithosphere exhibits a brittle-elastic-ductile transition from top to bottom. In 1978, Byerlee proposed that when rocks undergo sliding compression, shear stress... and normal stress The relationship between them is: In the formula: It is the inherent shear strength; It is the coefficient of friction; Based on the relationship between normal stress, shear stress, and principal stresses, Brace and Kohlstedt expressed the Byerlee friction criterion as follows in 1980: In the formula: This is the maximum principal stress; It is the minimum principal stress; For a normal fault under tension structure, the maximum principal stress corresponds to the force in the vertical direction. ,Right now For thrust faults under compression structures, the forces in the vertical direction... The corresponding minimum principal stress, i.e. ; Under tension, the stress difference between the maximum stress of a normal fault and the minimum stress of a thrust fault. Represented as: Under compression conditions, the maximum stress difference between the normal fault and the thrust fault Represented as: The brittle strength of the upper lithosphere depends on the stress difference. Therefore, vertical stress This refers to the static stress in the rock, which is equal to the brittle strength of the lithosphere, and is expressed as: In the formula: The density of the lithosphere; It is the acceleration due to gravity; The depth perpendicular to the lunar surface is the key variable for solving the stress difference; As depth increases, the increase in pressure and temperature leads to plastic flow in the rock. The rate of plastic strain depends on the stress applied to it, and the plastic strength... It is strongly affected by temperature, and its expression is: In the formula: It is the strain rate; It refers to the grain size of the lithosphere; It is the gas constant; It is a temperature profile; , , and These are all rheological parameters, determined by the composition of the lithosphere, and obtained directly from existing studies on lithosphere structure. Plastic strength The brittle strength of the lithosphere decreases exponentially with increasing depth. and plastic strength Then, an envelope is applied to the brittle strength and ductile strength, which is represented as: envelope This is crucial for calculating and predicting bending moments, utilizing the strength envelope integral, based on... The lower limit value is used to calculate the predicted bending moment. , represented as: In the formula: This represents the depth of the neutral surface, with a value ranging from 0 to 50 km. Step 5: Utilize the effective elastic thickness of the lithosphere from existing research. Inversely calculate the target bending stiffness , represented as: In the formula: and These represent the elastic modulus and Poisson's ratio, respectively, both of which can be directly obtained from existing research. Then the target bending moment Represented as: In the formula: This represents curvature, typically taken as 1.0 × 10⁻⁶. -9 ; Step 6: Based on the external elastic thickness constraint, iteratively correct the temperature structure by adjusting the thickness of the giant weathering layer: predict the bending moment... The target bending moment obtained based on the external elastic thickness When comparing, and When deviations exist, the thickness of the giant weathered layer is continuously adjusted. This reduces the deviation, controls the temperature structure and intensity envelope, and yields a convergent temperature structure. Step 7: Calculate the surface temperature gradient from the convergent temperature structure: [This step involves] analyzing the convergent temperature profile... , is represented as: The temperature gradient is then expressed as: Step 8: Under the constraints of the boundary and heat flow continuity conditions of the three-layer steady-state heat conduction model, the heat flow on the lunar surface is... Represented as: Repeat steps 1 through 8 to obtain the heat flow of different geological units on the Moon, thereby obtaining the spatial distribution of heat flow across the entire lunar surface.

Claims

1. A method for inverting lunar surface heat flow based on a three-layer steady-state heat conduction model, characterized in that, Includes the following steps: Step 1: Using radioactive elements as the radioactive heat source, obtain the total mass heat production rate based on the specific heat production rate and mass heat production rate of radioactive elements Th, U, and K in the lunar crust and mantle. ; Step 2: Construct a three-layer steady-state heat conduction model and set boundary and heat flow continuity conditions. The specific process is as follows: Step 2.1: Divide the near-surface region of the moon into a giant regolith, lunar crust, and lunar mantle, and construct a three-layer steady-state heat conduction model. Its heat conduction equation is expressed as: In the formula: , , and These represent the temperatures of the giant regolith, lunar crust, lunar mantle, and lunar surface, respectively. It is 253 K; , and These represent the heat generated per unit volume of the giant regolith, lunar crust, and lunar mantle, respectively. , and The thermal conductivity density and total mass heat production rate are respectively for the giant regolith, lunar crust, and lunar mantle. The product; , and These represent the thermal conductivity of the giant regolith, lunar crust, and lunar mantle, respectively. , , , and All are coefficients; , and These represent the thicknesses of the giant regolith, the lunar crust, and the lunar mantle, respectively. Indicates the depth perpendicular to the surface of the moon; Step 2.2: Set the boundary and heat flow continuity conditions for the three-layer steady-state heat conduction model, including: the heat flux and temperature at the boundary between the giant regolith and the lunar crust, the heat flux and temperature at the boundary between the lunar crust and the lunar mantle, and the temperature at the boundary between the lunar mantle and the lunar core. Step 3: Using the temperature of the lunar surface, the boundary temperatures of the lunar mantle and the lunar core as boundary conditions for solving the heat conduction equation, solve for the coefficients. , , , and Substitute it into the heat conduction equation The temperature structure parameters of the giant regolith, lunar crust, and lunar mantle were solved, including temperature profiles. and temperature gradient ; Step 4: Calculate the envelopes of brittle strength and ductile strength using the temperature profile. The specific process is as follows: Step 4.1: Calculate the brittle strength of the lithosphere. and plastic strength They are represented as follows: In the formula: The density of the lithosphere; It is the acceleration due to gravity; It is the depth perpendicular to the surface of the moon; It is the strain rate; It refers to the grain size of the lithosphere; It is the gas constant; It is a temperature profile; , , and All are rheological parameters; Step 4.2, regarding brittle strength and plastic strength Envelope, represented as: Step 4.3: Using the intensity envelope integral, according to The lower limit value is used to calculate the predicted bending moment. , is represented as: In the formula: Indicates the depth of the neutral surface; Step 5: Target bending stiffness , is represented as: In the formula: and These represent the elastic modulus and Poisson's ratio, respectively. Indicates the effective elastic thickness of the lithosphere; Then the target bending moment Represented as: In the formula: Represents curvature, with a value of 1.0 × 10⁻⁶. -9 ; Step 6: Predict the bending moment With the target bending moment By comparing and continuously adjusting the thickness of the giant weathering layer Thus reducing and The deviation is used to obtain a convergent temperature structure; Step 7: Calculate the surface temperature gradient from the converged temperature structure: [This step involves] analyzing the converged temperature profile... , is represented as: The temperature gradient is then expressed as: Step 8: Under the constraints of the boundary and heat flow continuity conditions of the three-layer steady-state heat conduction model, the heat flow on the lunar surface is... Represented as: In the formula: This indicates the thermal conductivity of the giant regolith.

2. The lunar surface heat flow inversion method based on a three-layer steady-state heat conduction model according to claim 1, characterized in that, The overall mass heat production rate in step 1 The calculation formula is: In the formula: , , and Representing radioactive elements 238 U、 235 U、 232 Th and 40 Specific heat production rate of K; , , and Representing radioactive elements 238 U、 235 U、 232 Th and 40 Mass heat production rate of K.

3. The lunar surface heat flow inversion method based on a three-layer steady-state heat conduction model according to claim 1, characterized in that, The specific process of step 2.2 is as follows: 1) At the boundary between the giant regolith and the lunar crust: 1.1) The condition that the heat flux needs to meet is expressed as follows: 1.2) The temperature requirement is expressed as follows: 2) At the boundary between the lunar crust and the lunar mantle: 2.1) The condition that the heat flux needs to meet is expressed as: 2.2) The temperature requirement is expressed as follows: 3) At the boundary between the lunar mantle and the lunar core, the temperature requirement is expressed as follows: In the formula: The boundary temperature between the lunar mantle and the lunar core is 1900 K.

4. The lunar surface heat flow inversion method based on a three-layer steady-state heat conduction model according to claim 3, characterized in that, The specific process of step 3 is as follows: Step 3.1, Apply the formula , , , and Written in matrix form, it can be represented as: In the formula: , , , and Represented as: Step 3.2, Elimination of coefficients and , is represented as: Step 3.3, Solve for coefficients , is represented as: Step 3.4: Solve for the coefficients and , is represented as: Step 3.5: Solve for the coefficients and , is represented as: By solving , , , and Substitute it into the heat conduction equation The temperature structure parameters of the giant regolith, lunar crust, and lunar mantle inside the moon were solved, including temperature profiles. and temperature gradient .

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