A method for calculating escape velocity of asteroid surface considering rotation and terrain influence

Abstract

The application relates to a kind of asteroid surface escape velocity calculation method considering rotation and terrain influence, comprising: obtaining the three-dimensional shape model of target asteroid and carrying out discrete, reading the vertex set and triangular facet set of polyhedron, carrying out gravity field modeling, establishing standard polyhedral gravity model;Define the logarithmic kernel function of each edge of each triangular facet, to determine the gravitational potential of target asteroid on any field point;Establish star-fixed coordinate system in standard polyhedral gravity model, determine the rotation speed of any point on the surface of target asteroid, thereby define the effective potential upper bound and the inertial system speed of the release instant of the particle on the surface of target asteroid;Based on the minimum escape condition that the kinetic energy of the particle on the surface of target asteroid is equal to the corresponding effective potential upper bound, determine the minimum normal escape velocity of the particle. Compared with the prior art, the application can quickly obtain accurate minimum escape velocity and corresponding escape direction at any surface point.

Description

Technical Field

[0001] This invention relates to the field of asteroid surface escape velocity calculation technology, and in particular to a method for calculating asteroid surface escape velocity that takes into account the effects of rotation and topography. Background Technology

[0002] Asteroids are irregular small celestial bodies that are widespread in the solar system. They are characterized by their small mass, weak gravity, complex shapes, and often significant surface topographic features such as craters, ridges, boulder deposits, and depressions. With the rapid development of deep space exploration missions, close-range flybys, landings, and sample return missions of asteroids have gradually become a key research direction. Among these, accurately analyzing the dynamic characteristics of probes on the asteroid surface, especially escape conditions and escape velocities, is of great significance for mission planning, landing constraint design, and safety assessment.

[0003] Traditional escape velocity calculation methods are typically based on ideal spherical celestial bodies, deriving the classical expression of the escape formula using the law of universal gravitation. This method assumes a uniform mass distribution and a gravitational field that is centrally symmetric with respect to distance, making it only applicable to large-scale, approximately spherical celestial bodies such as planets or satellites. However, for asteroids with highly irregular shapes and significantly uneven gravitational distribution, the simplified model deviates from reality, resulting in calculated escape velocities that do not fully meet engineering requirements. In asteroid contact operations, even minute velocity errors can cause probe instability, ejection from the surface, or even mission failure; therefore, more accurate escape velocity calculation methods are needed.

[0004] Asteroids typically exhibit rapid rotation, and the effective potential energy at their surface points is determined by both gravitational potential and centrifugal potential caused by rotation. When a probe is located at the equator, low latitudes, or bulges, the centrifugal force generated by rotation further weakens the local effective gravity, thus altering the minimum escape conditions and resulting in significant differences in escape velocities at different locations. Existing research indicates that on some rapidly rotating asteroids, rotation can even create localized negative gravity regions, allowing the probe to detach from the surface with minimal disturbance. Therefore, considering the effects of rotation in escape velocity calculations has become a necessary prerequisite.

[0005] The complex topography of an asteroid's surface also has a significant impact on escape velocity. Irregular terrain causes fluctuations in local gravitational potential energy; for example, crater walls and boulder structures on an asteroid's surface may require the probe to overcome additional kinetic energy demands due to geometric resistance when leaving the surface. Therefore, relying solely on the effective gravity of the asteroid's surface is insufficient to accurately reflect the escape dynamics under real surface topography.

[0006] In recent years, with the development of the polyhedral gravity model (PGM) and asteroid surface dynamics theory, researchers have begun to attempt to calculate position-dependent local escape velocities. Existing methods mainly include gravity models based on the asteroid surface, using energy integration on the asteroid surface to analyze escape conditions. However, most of these methods still rely on the global minimum potential point or maximum potential barrier, making them unsuitable for local escape analysis under complex terrain. To calculate escape velocities that accurately reflect the actual conditions of asteroid surfaces, it is necessary to develop analysis methods that take into account terrain orientation, thus better meeting the requirements of engineering mission design.

[0007] Therefore, it is necessary to propose a minimum escape velocity calculation method that can simultaneously consider the irregular gravitational field, rotational centrifugal potential, and complex terrain effects of asteroids, so that the accurate minimum escape velocity and corresponding escape direction can be obtained quickly at any surface point, thereby providing reliable data support for key tasks such as asteroid lander attachment strategy planning and probe return. Summary of the Invention

[0008] The purpose of this invention is to overcome the shortcomings of the existing technology, such as insufficient accuracy of the near-surface gravitational field of asteroids and insufficient consideration of the coupling effect of irregular terrain and rotation in the escape velocity calculation. This invention provides a method for calculating the escape velocity of asteroid surfaces that takes into account the effects of rotation and terrain, and realizes the analytical solution of the minimum escape velocity along the outward normal at any point on the surface of a real irregular asteroid and the analysis of the distribution of the escape velocity across the entire surface.

[0009] The objective of this invention can be achieved through the following technical solutions: A method for calculating the surface escape velocity of an asteroid that takes into account the effects of rotation and terrain includes the following steps: Obtain the three-dimensional shape model of the target asteroid, discretize the three-dimensional shape model using a three-dimensional polyhedron model, read the vertex set and triangular face set of the polyhedron, and model the gravitational field of the target asteroid as a rigid body to establish a standard polyhedron gravity model. In the standard polyhedral gravity model, a logarithmic kernel function is defined for each edge of each triangular element to determine the gravitational potential of the target asteroid at any field point. A star-fixed coordinate system is established in the standard polyhedral gravity model to determine the rotational velocity of any point on the surface of the target asteroid in the standard polyhedral gravity model, thereby defining the upper bound of the effective potential energy and the inertial frame velocity at the instant of particle release from the surface of the target asteroid. Based on the minimum escape condition that the kinetic energy of the particles on the surface of the target asteroid is equal to the upper bound of the corresponding effective potential energy, the minimum normal escape velocity of the particles on the surface of the target asteroid is determined.

[0010] Furthermore, the expression for the logarithmic kernel function of each edge of each triangular element is as follows: In the formula, Let p be the logarithmic kernel function of the field point p with respect to edge e; and These are the distances from field point p to the two endpoints of edge e, respectively. Let e ​​be the length of the edge.

[0011] Furthermore, the expression for the gravitational potential of the target asteroid at any field point is: In the formula, For the target asteroid field point gravitational potential, Let p be the vector of the field point. The gravitational constant is... For the density of the target asteroid, f For the first f A triangular facet element, Let p be the vector pointing from the field point to the reference point of the surface element. Let p be the surface tensor pointing to the reference point of the surface element. For solid angles with signs, Let p be the vector pointing from the field point p to the starting point of edge e. Let e ​​be the edge tensor. Let be the logarithmic kernel function of edge e at field point p.

[0012] Furthermore, the rotational velocity of any point on the surface of the target asteroid is obtained by multiplying the position vector of any point on the surface of the target asteroid with the rotational angular velocity vector of the target asteroid. The upper bound of the effective potential energy is the larger of the geometric gravitational potential and the point mass potential at a point on the surface of the target asteroid. The point mass potential is the gravitational parameter of the standard polyhedral gravity model divided by the magnitude of the position vector of the point on the surface of the target asteroid.

[0013] Furthermore, the expression for calculating the inertial frame velocity at the instant the particles on the surface of the target asteroid are: In the formula, The inertial frame velocity at the instant the particles on the surface of the target asteroid are released. For any point on the surface of the target asteroid in the multifaceted gravity model, the velocity increment applied along the outward normal direction is... Let be the outward normal unit vector at any point on the surface of the target asteroid in the multifaceted gravity model. denoted as the rotational velocity of any point on the surface of the target asteroid in the standard polyhedral gravity model.

[0014] Furthermore, the expression for the minimum escape condition based on the equality of the kinetic energy of the target asteroid surface particles with the corresponding upper bound of the effective potential energy is as follows: In the formula, The inertial frame velocity at the instant the particles on the surface of the target asteroid are released. The position vector of particles on the surface of the target asteroid in the standard polyhedral gravity model. The effective potential energy upper bound.

[0015] Furthermore, the expression for calculating the minimum normal escape velocity of the particles on the surface of the target asteroid is as follows: In the formula, The position vector of particles on the surface of the target asteroid in the standard polyhedral gravity model. The minimum normal escape velocity, This is the normal projection of the rotational velocity of the particles on the surface of the target asteroid in the standard polyhedral gravity model. The position vector of particles on the surface of the target asteroid in the standard polyhedral gravity model. The effective potential energy upper bound, denoted as the rotational velocity of any point on the surface of the target asteroid in the standard polyhedral gravity model.

[0016] Furthermore, if the expression for calculating the minimum normal escape velocity of the particles on the surface of the target asteroid has no real solution, then the minimum normal escape velocity of the particles on the surface of the target asteroid is: In the formula, This is a preset small margin.

[0017] Furthermore, the method also includes defining a co-rotational plane and a dorsal plane by projecting the rotational velocity of the particles on the surface of the target asteroid onto the outward normal direction, and determining the corresponding minimum normal escape velocity for each. When the rotational velocity of the particles on the surface of the target asteroid is at an angle of θ between the outward normal direction and the tangential direction... ,in, When the angle deviates from 90°, it is defined as a clockwise spiral surface, and the expression for the corresponding minimum normal escape velocity is: In the formula, This represents the minimum normal escape velocity corresponding to the paracycloid surface. When the rotational velocity of the particles on the surface of the target asteroid is at an angle of θ between the outward normal direction and the tangential direction... When the plane is defined as the backspin plane, the expression for the corresponding minimum normal escape velocity is: In the formula, This represents the minimum normal escape velocity corresponding to the backspin plane.

[0018] Furthermore, the method also includes: Determine the minimum normal escape velocity of all particles on the surface of the target asteroid, and construct a three-dimensional distribution map of the minimum escape velocity on the surface of the target asteroid. This map can be used for landing and sampling point selection, return velocity margin design, or surface material migration under microgravity conditions.

[0019] Compared with the prior art, the present invention has the following advantages: (1) This invention forms a method for calculating the minimum escape velocity of the surface of a real irregular asteroid by using the analytical form of the polyhedral gravitational potential, the construction of the upper bound of the effective potential energy, the solution of the energy conservation equation, and the analytical relationship of different rotational surface regions. First, under the assumption of uniform density, the gravitational potential and gravitational acceleration near the surface are calculated using the polyhedral gravitational method. The rotational angular velocity is introduced in the star-fixed coordinate system, and the position vector, the outward normal unit vector, and the rotational follow-up velocity are obtained for any point on the surface. The velocity at the moment of release is expressed as the sum of the normal velocity increment and the rotational follow-up velocity. Then, under the energy conservation relationship and the velocity outward pointing constraint, the minimum escape velocity along the outward normal is obtained. Furthermore, by analyzing the angle between the outward normal and the rotational follow-up velocity, the surface is divided into a co-rotational region and a dorsal region. The escape velocity required for the dorsal slope relative to the co-rotational slope is quantitatively analyzed. The spatial distribution of escape velocity on the surface of a rapidly rotating sub-hundred-meter asteroid is analyzed in conjunction with a scaled shape model. This method maintains high accuracy in gravity field modeling and uniformly considers the effects of rotation and terrain orientation in dynamics. It can generate normal escape velocity distribution across the entire surface, providing a reliable theoretical basis and engineering quantitative analysis method for the optimization of asteroid landing and sampling areas, the design of autonomous takeoff and return velocity margins, and the analysis of surface material migration and accumulation processes under microgravity.

[0020] (2) The analytical form of the gravitational potential of an asteroid at any field point constructed in this invention directly maps the geometric information of the polyhedron into a closed summation of potential energy, without relying on spherical harmonic or ellipsoidal harmonic expansion, and has high computational accuracy and numerical stability in the near-surface region of the asteroid. Attached Figure Description

[0021] Figure 1This is a flowchart illustrating a method for calculating the surface escape velocity of an asteroid that takes into account the effects of rotation and terrain, as provided in an embodiment of the present invention. Figure 2 This is a minimum normal escape velocity distribution map of the Ryugu asteroid surface provided in an embodiment of the present invention. The color scale in the map represents the minimum escape velocity value required for different positions to just escape the surface along the outer normal direction, in m / s. Figure 3 This invention provides a three-dimensional distribution map of the minimum normal escape velocity of a simulated sub-hundred-meter-class rapidly rotating asteroid, demonstrating how terrain orientation and rotation effects jointly influence the escape velocity distribution under extreme microgravity and high-speed rotation conditions. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0023] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0024] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0025] Example 1 like Figure 1 As shown, this embodiment provides a method for calculating the surface escape velocity of an asteroid that takes into account the effects of rotation and terrain, including the following steps: S1: Obtain the three-dimensional shape model of the target asteroid, discretize the three-dimensional shape model with a three-dimensional polyhedron model, read the vertex set and triangular face set of the polyhedron, and treat the target asteroid as a rigid body to perform gravity field modeling and establish a standard polyhedron gravity model. S2: In the standard polyhedral gravity model, define the logarithmic kernel function of each edge of each triangular element to determine the gravitational potential of the target asteroid at any field point; S3: Establish a star-fixed coordinate system in the standard polyhedral gravity model, determine the rotational velocity of any point on the surface of the target asteroid in the standard polyhedral gravity model, and thus define the upper bound of the effective potential energy and the inertial frame velocity at the moment of release of particles on the surface of the target asteroid. S4: Determine the minimum normal escape velocity of the particles on the surface of the target asteroid based on the minimum escape condition that the kinetic energy of the particles on the surface of the target asteroid is equal to the upper bound of the corresponding effective potential energy.

[0026] In step S1, regarding the gravity field modeling, this invention employs a polyhedral gravitational model to model the surface gravity field. Let the overall density of the asteroid be a constant ρ, its mass be M, its gravitational constant be G, and its gravitational parameter be μ = GM. The shape of the asteroid is discretized using a three-dimensional polyhedral model. The vertex set and the triangular face set are read in, and a face set and a deduplicated edge set are constructed. The polyhedron is considered to be composed of several closed planar triangles.

[0027] Specifically, based on the assumption of uniform density, the target asteroid is considered to have a density of A rigid body of mass M is discretized using a three-dimensional polyhedron model. The gravitational constant is G, and the gravitational parameters are... Read in the vertex set and triangular facet set of the polyhedron, construct the face set and the deduplicated edge set, and determine a reference vertex for each triangular facet f. Let the vector from the field point p to this reference vertex be denoted as... Construct a surface tensor for each face For each edge e, let the vector pointing from the field point to the starting point of the edge be denoted as . Construct an edge tensor for this edge. .

[0028] In step S2, regarding the geometric kernel, the signed solid angle is calculated for each triangular facet f. For each edge e, calculate the logarithmic kernel function: in Let l be the distance from the field point to the two endpoints of the edge, and l be the length of the edge.

[0029] In this scheme, the expression for the corresponding logarithmic kernel function is: In the formula, Let p be the logarithmic kernel function of the field point p with respect to edge e; and These are the distances from field point p to the two endpoints of edge e, respectively. Let e ​​be the length of the edge.

[0030] Using the kernel function described above, under the assumption of uniform density, the gravitational potential of an asteroid at any field point p can be written in a rigorous analytical form: In the formula, For the target asteroid field point gravitational potential, Let p be the vector of the field point. The gravitational constant is... For the density of the target asteroid, f For the first f A triangular facet element, Let p be the vector pointing from the field point to the reference point of the surface element. Let p be the surface tensor pointing to the reference point of the surface element. For solid angles with signs, Let p be the vector pointing from the field point p to the starting point of edge e. Let e ​​be the edge tensor. Let be the logarithmic kernel function of edge e at field point p.

[0031] This expression directly maps the geometric information of the polyhedron into a closed summation of potential energy, without relying on spherical or ellipsoidal harmonic expansion, and has high computational accuracy and numerical stability in the near-surface region of asteroids.

[0032] Step S3: Based on the aforementioned polyhedral gravitational potential, this invention establishes a star-fixed coordinate system with the asteroid's center of mass as the origin. Let the asteroid's rotational angular velocity vector be... For any point on the surface of the polyhedron, let its position vector be denoted as . The outward normal unit vector is The rotational velocity caused by the rigid body's rotation in the inertial frame is obtained by multiplying the position vector of any point on the surface of the target asteroid by the rotational angular velocity vector of the target asteroid; the corresponding calculation expression is: Let its modulus length be . The normal projection is: The rotational follower can then be decomposed into a normal component 'a' and an in-plane tangential component. when At that time, rotation acts as a booster in the external direction. At that time, the rotation acts as a barrier in the external direction.

[0033] Regarding the calculation of escape velocity, this invention describes the velocity of the inertial frame at the instant of surface material release as follows: In the formula, The inertial frame velocity at the instant the particles on the surface of the target asteroid are released. Let be the velocity increment applied along the outward normal direction at any point on the surface of the target asteroid in the multifaceted gravity model; and be the unknown quantity that needs to be solved. Let be the outward normal unit vector at any point on the surface of the target asteroid in the multifaceted gravity model. denoted as the rotational velocity of any point on the surface of the target asteroid in the standard polyhedral gravity model.

[0034] Defining this as minimum escape, the kinetic energy at that instant must be numerically equal to the upper bound of the effective potential energy, as follows: In the formula, The inertial frame velocity at the instant the particles on the surface of the target asteroid are released. The position vector of particles on the surface of the target asteroid in the standard polyhedral gravity model. The effective potential energy upper bound.

[0035] The effective potential energy upper bound is the geometric gravitational potential at the surface point of the target asteroid. and point mass potential The larger value in; The point mass potential is the gravitational parameter of the standard polyhedral gravity model. Divided by the magnitude of the position vector of the target asteroid's surface point .

[0036] Substituting and expanding the velocity decomposition expression, we obtain the following about The quadratic equation of : This invention introduces an outward pointing constraint, ensuring that the component of the velocity in the normal direction at the moment of release does not point inward towards the asteroid. Under this constraint, the minimum escape velocity along the outward normal direction is taken as: In the formula, The position vector of particles on the surface of the target asteroid in the standard polyhedral gravity model. The minimum normal escape velocity, This is the normal projection of the rotational velocity of the particles on the surface of the target asteroid in the standard polyhedral gravity model. The position vector of particles on the surface of the target asteroid in the standard polyhedral gravity model. The effective potential energy upper bound, denoted as the rotational velocity of any point on the surface of the target asteroid in the standard polyhedral gravity model.

[0037] Preferably, when the above equation does not have a real solution that meets the requirements, it indicates that the spin-tangential energy is sufficient to exceed U. maxThis invention no longer forcibly solves the quadratic equation, but instead adopts a strategy of canceling the rotation normal component and superimposing a minimal outward pointing margin, let: in This is a preset small margin, which can be required in numerical implementation. This avoids surface-to-surface propulsion. This allows for the determination of an analytical or approximate solution for the minimum escape velocity along the outer normal at each surface location.

[0038] Preferably, within the aforementioned unified framework, this invention further defines the clockwise and counterclockwise planes by projecting the rotational follower velocity onto the outward normal direction, and provides a quantitative relationship between the two. A scalar value for the rotational tangential velocity is defined. ,make: When the angle between the outward normal and the tangent is , This region, defined as a paracycloid with an acute angle deviating from 90°, can be written as: When the angle between the outward normal and the tangent is This region is defined as the backspin plane, and its minimum normal escape velocity is: Optionally, the method also includes: Determine the minimum normal escape velocity of all particles on the surface of the target asteroid, and construct a three-dimensional distribution map of the minimum escape velocity on the surface of the target asteroid. This map can be used for landing and sampling point selection, return velocity margin design, or surface material migration under microgravity conditions.

[0039] That is, the spatial distribution of the minimum normal escape velocity on the asteroid surface obtained by the above process can be used for at least one of the following: guiding the selection of asteroid landing sites and sampling sites, and taking regions with lower escape velocities as priority candidate regions; for designing the takeoff and return speeds of the probe and the anti-ground-hugging propulsion margin, and configuring larger normal velocity increments in regions with higher escape velocities; and for analyzing the formation mechanism of surface material migration and accumulation under the coupling effect of microgravity and rotation.

[0040] To verify the applicability of the method of this invention to the surface of a real, irregular asteroid, two experimental embodiments were conducted. In the first embodiment, a high-precision three-dimensional shape model of the near-Earth asteroid Ryugu was selected as input, with an average density of approximately 1.19 g / cm³. 3Its rotation period is approximately 7.63 hours, and its volumetric equivalent diameter is approximately 870 meters. This model contains approximately 800,000 triangular facets, accurately describing Ryugu's non-spherical geometry and surface topographic features. Under the assumption of uniform density, this invention considers Ryugu as having a density of... A rigid body of mass M has a gravitational constant of G, and its gravitational parameter is denoted as . .

[0041] Specifically, this embodiment provides a method for calculating the surface escape velocity of an asteroid that takes into account the effects of rotation and terrain, such as... Figure 1 As shown, the process includes the following steps: First, the set of polyhedral vertices and the set of triangular facets are read from the shape model. Then, a face set and a deduplicated edge set are constructed to establish a standard polyhedral gravity model. Within this model framework, for any field point p, the vector pointing to the facet reference point is defined as... The corresponding surface tensor is The solid angle with the symbol is For any edge e, define the vector pointing from the field point to the origin of the edge as follows: The corresponding edge tensor is If the distances from the field point to the two endpoints of the edge are respectively The side length is Then the logarithmic kernel function of that edge is defined as: Under the above definition, the gravitational potential of an asteroid at any point in the field can be written in a rigorous analytical form: The expression is composed entirely of polyhedral geometric quantities, does not rely on spherical harmonic expansion, and can maintain high accuracy and high numerical stability in the near-surface region of the asteroid.

[0042] After obtaining the gravitational potential, a star-fixed coordinate system is established with the asteroid's center of mass as the origin. Let the angular velocity vector of the Ryugu's rotation be... For any point on the surface of the polyhedron Its rotational velocity, caused by the rotational motion, can be expressed as: Let the outward normal unit vector of this point be... Then the projection of the rotational follower velocity onto the normal direction is defined as: The tangential component of rotation within a plane can be written as: If a surface particle leaves the surface with a minimum velocity, its velocity in the inertial frame at the instant of release can be written as: in Let be the velocity increment along the outward normal direction to be determined. The minimum escape condition requires that its kinetic energy be equal to the upper bound of the effective potential energy at that point. Equal, that is: Expanding and organizing it yields information about The quadratic equation of : Introducing the discriminant: Taking the solution that satisfies the outward pointing constraint, the minimum normal escape velocity at that point is: In the specific calculations, it is necessary to ensure that the composite velocity is directed towards the outer edge of the asteroid.

[0043] Applying the above algorithm to the entire surface of the Ryugu shape model and calculating the minimum normal escape velocity distribution of its surface pixels yields the results. The results show that the escape velocity ranges globally from approximately 0.24 to 0.46 m / s. The equatorial region requires a lower escape velocity due to the stronger centrifugal effect, while the polar regions and some steep slopes or anti-spiral surfaces have non-zero minimum escape velocities due to deeper local potential wells and the hindering effect of rotation in the normal direction. The distribution map calculated by this invention can be directly used for landing point selection, sampling point planning, and return velocity margin design, demonstrating clear engineering value. Figure 2 This is a three-dimensional distribution diagram of the minimum normal escape velocity on the surface of the Ryugu asteroid in Example 1.

[0044] To further verify the computational performance of the method of this invention under extreme microgravity and high-speed rotation conditions, Example 2 constructs a typical sub-100-meter-class asteroid based on the shape of Ryugu, reducing its geometric dimensions by approximately 13 times while maintaining the original density. The gravitational potential of the scaled-up asteroid is significantly weakened, and its rotation period is set to 0.5 hours, increasing its tangential rotational velocity. In this case, the centrifugal effect has a significant impact on surface dynamics, which can be used to demonstrate the ability of the method of this invention to express the coupling effect between rotation and terrain.

[0045] Under the above settings, according to the method of the present invention, the upper bound of the effective potential energy is calculated point by point for all pixels on the scaled model surface. Rotational follower speed and its external direction Upward projection And find the minimum normal escape velocity at that point under the minimum escape condition. .Will Mapping the minimum normal escape velocity onto the asteroid surface yields a three-dimensional distribution map. The results show that under sub-hundred-meter-scale rapid rotation conditions, the minimum normal escape velocity exhibits significant spatial non-uniformity, reflecting the influence of rotation and terrain orientation coupling on the escape velocity distribution. This data can be used for subsequent candidate region convergence and safe point decision-making.

[0046] Due to the directional differences in local topography on the asteroid's surface, even in the equatorial region, which should theoretically be the easiest escape route, the actual effective normal gravity is still enhanced by the presence of backspin slopes. As a result, the centrifugal force's boosting effect in the normal direction of these slopes is weakened or even partially canceled out, causing the local minimum normal escape velocity to no longer approach zero. Figure 3 The results illustrate this point: for rapidly rotating asteroids in microgravity, due to topographical influences, there may still be regions near the equator that require additional normal velocity to escape the surface.

[0047] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A method for calculating the surface escape velocity of an asteroid that takes into account the effects of rotation and terrain, characterized in that, Includes the following steps: Obtain the three-dimensional shape model of the target asteroid, discretize the three-dimensional shape model using a three-dimensional polyhedron model, read the vertex set and triangular face set of the polyhedron, and model the gravitational field of the target asteroid as a rigid body to establish a standard polyhedron gravity model. In the standard polyhedral gravity model, a logarithmic kernel function is defined for each edge of each triangular element to determine the gravitational potential of the target asteroid at any field point. A star-fixed coordinate system is established in the standard polyhedral gravity model to determine the rotational velocity of any point on the surface of the target asteroid in the standard polyhedral gravity model, thereby defining the upper bound of the effective potential energy and the inertial frame velocity at the instant of particle release from the surface of the target asteroid. Based on the minimum escape condition that the kinetic energy of the particles on the surface of the target asteroid is equal to the upper bound of the corresponding effective potential energy, the minimum normal escape velocity of the particles on the surface of the target asteroid is determined.

2. The method for calculating the surface escape velocity of an asteroid considering the effects of rotation and terrain, as described in claim 1, is characterized in that... The expression for the logarithmic kernel function of each edge of each triangular element is as follows: In the formula, Let p be the logarithmic kernel function of the field point p with respect to edge e; and These are the distances from field point p to the two endpoints of edge e, respectively. Let e ​​be the length of the edge.

3. The method for calculating the surface escape velocity of an asteroid considering the effects of rotation and terrain, as described in claim 1, is characterized in that... The expression for the gravitational potential of the target asteroid at any field point is: In the formula, For the target asteroid field point gravitational potential, Let p be the vector of the field point. The gravitational constant is... For the density of the target asteroid, f For the first f A triangular facet element, Let p be the vector pointing from the field point to the reference point of the surface element. Let p be the surface tensor pointing to the reference point of the surface element. For a solid angle with a sign, Let p be the vector pointing from the field point to the starting point of edge e. Let e ​​be the edge tensor. Let be the logarithmic kernel function of edge e at field point p.

4. The method for calculating the surface escape velocity of an asteroid considering the effects of rotation and terrain, as described in claim 1, is characterized in that... The rotational velocity of any point on the surface of the target asteroid is obtained by multiplying the position vector of any point on the surface of the target asteroid with the rotational angular velocity vector of the target asteroid. The upper bound of the effective potential energy is the larger of the geometric gravitational potential and the point mass potential at a point on the surface of the target asteroid. The point mass potential is the gravitational parameter of the standard polyhedral gravity model divided by the magnitude of the position vector of the point on the surface of the target asteroid.

5. The method for calculating the surface escape velocity of an asteroid considering the effects of rotation and terrain, as described in claim 1, is characterized in that... The expression for calculating the inertial frame velocity of the particles on the surface of the target asteroid at the instant of release is as follows: In the formula, The inertial frame velocity at the instant the particles on the surface of the target asteroid are released. For any point on the surface of the target asteroid in the multifaceted gravity model, the velocity increment applied along the outward normal direction is... Let be the outward normal unit vector at any point on the surface of the target asteroid in the multifaceted gravity model. denoted as the rotational velocity of any point on the surface of the target asteroid in the standard polyhedral gravity model.

6. The method for calculating the surface escape velocity of an asteroid considering the effects of rotation and terrain, as described in claim 1, is characterized in that... The expression for the minimum escape condition based on the equality of the kinetic energy of the target asteroid surface particles with the corresponding upper bound of the effective potential energy is as follows: In the formula, The inertial frame velocity at the instant the particles on the surface of the target asteroid are released. The position vector of particles on the surface of the target asteroid in the standard polyhedral gravity model. The effective potential energy upper bound.

7. The method for calculating the surface escape velocity of an asteroid considering the effects of rotation and terrain, as described in claim 1, is characterized in that... The expression for calculating the minimum normal escape velocity of the particles on the surface of the target asteroid is as follows: In the formula, The position vector of particles on the surface of the target asteroid in the standard polyhedral gravity model. The minimum normal escape velocity, This is the normal projection of the rotational velocity of the particles on the surface of the target asteroid in the standard polyhedral gravity model. The position vector of particles on the surface of the target asteroid in the standard polyhedral gravity model. The effective potential energy upper bound, denoted as the rotational velocity of any point on the surface of the target asteroid in the standard polyhedral gravity model.

8. The method for calculating the surface escape velocity of an asteroid considering the effects of rotation and terrain, as described in claim 7, is characterized in that... If the expression for calculating the minimum normal escape velocity of the particles on the surface of the target asteroid has no real solution, then the minimum normal escape velocity of the particles on the surface of the target asteroid is: In the formula, This is a preset small margin.

9. The method for calculating the surface escape velocity of an asteroid considering the effects of rotation and terrain, as described in claim 7, is characterized in that... The method further includes defining the co-rotational and anti-rotational planes by projecting the rotational velocity of the particles on the surface of the target asteroid onto the outward normal direction, and determining the corresponding minimum normal escape velocity for each. When the rotational velocity of the particles on the surface of the target asteroid is at an angle of θ between the outward normal direction and the tangential direction... ,in, When the angle deviates from 90°, it is defined as a clockwise spiral surface, and the expression for the corresponding minimum normal escape velocity is: In the formula, This represents the minimum normal escape velocity corresponding to the paracycloid surface. When the rotational velocity of the particles on the surface of the target asteroid is at an angle of θ between the outward normal direction and the tangential direction... When the plane is defined as the backspin plane, the expression for the corresponding minimum normal escape velocity is: In the formula, This represents the minimum normal escape velocity corresponding to the backspin plane.

10. The method for calculating the surface escape velocity of an asteroid considering the effects of rotation and terrain, as described in claim 1, is characterized in that... The method further includes: Determine the minimum normal escape velocity of all particles on the surface of the target asteroid, and construct a three-dimensional distribution map of the minimum escape velocity on the surface of the target asteroid. This map can be used for landing and sampling point selection, return velocity margin design, or surface material migration under microgravity conditions.

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