Improved Pork-Chop Plot for Asteroid Kinetic Impact Deflection Test Mission Trajectory Optimization

Abstract

For the mission requirements of the preliminary design phase for kinetic impact deflection of asteroids, an improved pork-chop plot design method is proposed which comprehensively considers both engineering constraints and deflection effectiveness. This method enables the visualization of engineering constraints, such as launch site, launch vehicle, and impact visibility, as well as the deflection distance after impact, all within a single plot. It provides a set of initial values that meet the requirements within the designated window for subsequent trajectory correction, based on different mission needs. Based on the patched conic technique, this paper first establishes a dynamical model for the spacecraft’s trajectory to the asteroid and then determines the parameters for both Earth departure and asteroid impact by solving the Lambert problem. Then, based on the departure parameters, the expression for Earth parking orbit escape is derived, and the constraints of rocket coasting time and launch site latitude, respectively, are transformed into parameter constraints on the argument of perigee and launch declination. Based on the impact parameters, an asteroid deflection dynamics model is established to compute the asteroid’s apparent magnitude and deflection distance. Finally, the improved pork-chop plot is generated using the aforementioned models. The plot comprehensively displays the optimized target parameters and engineering constraint parameters throughout the entire process, from launch vehicle departure to the post-impact deflection distance, within the given launch window. This provides initial values that satisfy both engineering constraints and mission requirements for the trajectory design of an in-orbit kinetic impactor asteroid deflection test mission. Compared to other trajectory design methods that provide only a single trajectory, the improved pork-chop plot enables a rapid, intuitive, and comprehensive visualization of a cluster of launch trajectories within the feasible window that satisfy engineering constraints. This approach reduces the number of iterations required for matching the deep-space transfer trajectory with the launch vehicle injection phase from more than five to one. The proposed method can serve as a valuable reference for target selection and trajectory optimization in in-orbit validation missions for kinetic impact deflection of asteroids.

1. Introduction

Asteroid impacts on Earth can cause significant harm, yet research on asteroid defense began relatively late. In 1994, humanity observed the Shoemaker–Levy 9 impact event [1,2], marking the first recorded direct collision between two celestial bodies in the Solar System [3,4]. This led the United Nations to convene the “United Nations International Conference on Near-Earth Objects” [5] and issue the “Vienna Declaration on Space and Human Development” (https://www.unoosa.org/pdf/reports/unispace/viennadeclE.pdf, accessed on 20 January 2025). Meanwhile, the United States launched the Lincoln Near-Earth Asteroid Research (LINEAR) program [6]. It was not until after the 2013 Chelyabinsk event [7] that planetary defense research became more systematized. In response, the United Nations established two key international organizations: the International Asteroid Warning Network (IAWN), responsible for collecting and disseminating asteroid monitoring and early warning information, and the Space Mission Planning Advisory Group (SMPAG), dedicated to planning defense missions [8]. In 2022, the Information Office of the State Council issued the white paper “China’s Space Activities in 2021”, pointing out that the Near-Earth Asteroid (NEA) defense system would be constructed in future [9].

Asteroid impact events pose a potential threat to humanity [10,11], and currently proposed defense strategies include nuclear explosions [12], kinetic impactors, and laser ablation [13]. Among these, kinetic impact is the most mature and reliable asteroid defense method [14,15]. The Double Asteroid Redirection Test (DART) mission successfully impacted the moon of a binary asteroid system in September 2022, altering its binary orbital period by 33 min. As the first in-orbit test of kinetic impactor technology for near-Earth asteroid deflection, the DART mission demonstrated the feasibility of this approach and represents a milestone in planetary defense [16,17,18]. In the trajectory optimization design for kinetic impact deflection of asteroids, a key challenge is how to optimize the trajectory to achieve the most effective deflection distance [19,20].

In the actual mission trajectory optimization design for kinetic impact deflection of asteroids, the trajectory optimization process is typically carried out in two steps. The first step involves optimizing the initial trajectory parameters. This is usually based on a simplified two-body dynamics model, where the patched conic technique is used to optimize launch and transfer times. To achieve the best defense effect, the post-impact Earth Close Approach (CA) distance is typically selected as the optimization objective, while launch time and transfer time serve as optimization variables. A global optimization algorithm that does not require initial value guessing is employed to obtain the globally optimal solution. The second step is trajectory correction, in which the optimized parameters obtained under the two-body model are used as initial values and refined under high precision dynamical models of Solar System or Earth Moon space. The quality of the initial trajectory parameters obtained in the first step directly influences the final design outcome.

Hibberd [21] utilized the traditional pork-chop plot to determine the launch transfer window for asteroid 2024 YR4 and provided the required impactor mass for deflecting the asteroid. Wang [22] employed a global optimization algorithm to optimize deep-space transfer trajectories under the two-body model and proposed a solution that considers engineering constraints for the target asteroid. Speziale and Conway [23,24] introduced a trajectory optimization method combining impulsive and low-thrust propulsion to optimize the launch window for kinetic impact missions, aiming to maximize asteroid deflection effectiveness. The study implemented a particle swarm optimization (PSO) algorithm for global optimization, which, compared to traditional nonlinear programming (NLP) methods, avoids local optima and expands the search range for launch windows.

The aforementioned methods present several challenges in practical engineering applications: the design output typically consists of only a single trajectory optimized for theoretical maximum deflection distance, and the optimization process often neglects or minimally considers real engineering constraints. In actual mission design, incorporating these engineering constraints may significantly narrow the originally available launch window and affect the deflection distance, making the ideal trajectory infeasible for mission requirements. Consequently, it is necessary to provide a set of transfer trajectories that span a sufficiently wide launch window. Furthermore, a critical issue that needs to be addressed is how to systematically transform engineering constraints into orbital parameters and comprehensively visualize them within the pork-chop plot. This would allow for a clear and intuitive representation of both mission objectives and engineering constraints, facilitating trade-off analyses in trajectory design.

Current algorithms generally provide only a single theoretically optimal transfer trajectory in terms of deflection effect. However, in practical engineering mission design, it is necessary to determine a cluster of transfer trajectory that satisfies a launch window of sufficient width. Moreover, most previous studies on mission trajectory optimization have neglected critical engineering constraints, such as launch site latitude and rocket coasting time. These factors can further reduce the available launch window and significantly impact the deflection effectiveness of the asteroid.

In the preliminary optimization design of interplanetary transfer trajectory, energy contour plots, commonly known as pork-chop plots, are typically used to analyze the energy requirements within a given launch window. The horizontal and vertical axes of a pork-chop plot represent the launch time and transfer time, respectively. By solving the Lambert problem, each point on the plot corresponds to a specific combination of departure and transfer times, reflecting the associated launch and arrival energy. When these values are visualized as energy contour lines, the pork-chop plot provides a comprehensive analysis of the quality of different launch and transfer windows, serving as a valuable reference for mission planning and trajectory optimization. Due to its clear and intuitive representation of trajectory energy requirements within a given launch window, the pork-chop plot has been widely applied in interplanetary transfer trajectory optimization [25,26,27].

Kinetic impact asteroid deflection does not require a rendezvous, making traditional pork-chop plots unsuitable for optimizing the trajectory of such missions. In this study, we propose an improved pork-chop plot based on the traditional approach. First, a two-impulse transfer trajectory dynamical model from Earth to the asteroid is established using the patched conic technique. Then, an asteroid deflection dynamics model is developed based on impact parameters to compute the asteroid’s visible time and the post-impact Earth CA distance change. The Earth departure parameters are further used to determine the escape trajectory, where the launch site latitude constraint is transformed into an escape declination constraint, and the launch vehicle coasting time constraint is converted into the argument of perigee constraint at the escape point. Finally, all engineering constraints and optimization target parameters are visualized in a single pork-chop plot, providing a clear and intuitive representation of feasible launch windows while considering both engineering constraints and deflection distance. To validate the proposed method, we perform trajectory design for asteroid 2015 BY310. Using a point selected from the improved pork-chop plot that meets both engineering and mission constraints, the impactor’s trajectory is refined under the high-precision Solar System dynamical model. The results confirm the accuracy and efficiency of our approach. By incorporating engineering constraints such as launch vehicle capabilities and asteroid observability at the initial trajectory design stage, this method effectively reduces the number of iterations required for Earth escape and deep-space transfer trajectory design, thereby improving overall design efficiency.

2. Trajectory Transfer Mode

In the trajectory design process, we first optimize the deep-space transfer trajectory. The impactor’s motion during the deep-space transfer phase is modeled using a two-body system centered on the Sun. By solving the Lambert problem [28], the departure and arrival velocities at Earth and the asteroid are determined. The Earth’s position is obtained from the JPL planetary and lunar ephemerides DE440 [29], while the asteroid’s initial position is sourced from the Jet Propulsion Laboratory (JPL) database (https://ssd.jpl.nasa.gov, accessed on 20 January 2025). The asteroid’s trajectory is then propagated using a high-precision Solar System dynamical model, which will be introduced in detail later.

In the dual-impulse transfer mission mode, no maneuvers are performed from orbit insertion to impact, except for minor attitude and trajectory adjustments. In the design process, we reserved a velocity increment of $\mathsf{\Delta }{v}_c$ = 200 m/s for attitude and trajectory correction during flight. Here, $g_0$ is taken as 9.80665 m/s², and $I_{SP}$ represents the specific impulse of the engine, set to 315 s. Therefore, the spacecraft’s mass $m_{sc}$ at the time of impact $t_i$ and the mass at the time of Earth departure $t_d$ can be obtained using Equation (1). The subscript $d$ denotes departure, while $i$ represents impact.

$$m_{sc}\left(t_i\right)=m_{sc}\left(t_d\right)·e^{{\displaystyle \frac{-\mathsf{\Delta }{v}_c}{g_0·I_{SP}}}}$$

(1)

By solving the Lambert problem using the Earth’s position ${\mathit{r}}_{\mathit{d}}$ at the launch time, the asteroid’s position ${\mathit{r}}_{\mathit{i}}$ at the impact time, and the transfer duration $\mathsf{\delta }t$, the heliocentric velocities of the spacecraft at Earth escape ${\mathit{v}}_{\mathit{d}}$ and at the impact moment ${\mathit{v}}_{\mathit{i}}$ can be determined:

$$\left({\mathit{v}}_{\mathit{d}},{\mathit{v}}_{\mathit{i}}\right)=Lambert\left({\mathit{r}}_{\mathit{d}},{\mathit{r}}_{\mathit{i}},\mathsf{\delta }t\right)$$

(2)

From this, the spacecraft’s Earth escape velocity ${\mathit{v}}_{\infty }$ can be obtained, where ${\mathit{v}}_{\mathit{E}}$ is the heliocentric velocity of the Earth at the launch time $t_d$:

$${\mathit{v}}_{\infty }={\mathit{v}}_{\mathit{d}}-{\mathit{v}}_{\mathit{E}}\left(t_d\right)$$

(3)

Considering both launch cost and payload capacity [30], this study selects the Chinese Long March 5 rocket as the launch vehicle. According to the launch capability curve, the rocket’s payload capacity is determined as a function of the launch energy $C_3$.

The launch energy $C_3$ can be calculated from the hyperbolic excess velocity ${\mathit{v}}_{\infty }$ using Equation (2):

$$C_3={\left|\left|{\mathit{v}}_{\infty }\right|\right|}^{\mathbf{2}}$$

(4)

By fitting the launch capability curve, the spacecraft’s injection mass $m_{SC}$ corresponding to the given $C_3$ can be determined, as shown in Figure 1.

The velocity of the spacecraft at Earth parking orbit escape, $v_{EscP}$, can be determined from $C_3$. The Earth’s radius, $R_E$, is taken as 6378.14 km, and the altitude of the parking orbit, $h_p$, is set to 200 km:

$$v_{EscP}=\sqrt{C_3+{\displaystyle \frac{2·{\mu }_E}{R_E+h_P}}}$$

(5)

The velocity of the spacecraft in the Earth parking orbit, $v_P$, is given by

$$v_P=\sqrt{{\displaystyle \frac{{\mu }_E}{R_E+h_P}}}$$

(6)

Thus, considering the actual departure conditions of Long March 5’s upper stage, the required velocity increment for escaping from the parking orbit $\mathsf{\Delta }{v}_p$ is given by

$$\mathsf{\Delta }{v}_P=v_{EscP}-v_P$$

(7)

The impact velocity of the spacecraft relative to the asteroid ${\mathit{v}}_{\mathit{R}}$ can be calculated using Equation (8):

$${\mathit{v}}_{\mathit{R}}={\mathit{v}}_{\mathit{i}}-{\mathit{v}}_{\mathit{A}}\left(t_{i-}\right)$$

(8)

Here, we assume that the impact occurs instantaneously, meaning the change in the asteroid’s velocity happens instantaneously at the moment of impact. $t_{i-}$ and $t_{i+}$ represent the moments before and after the impact, respectively.

The hyperbolic excess velocity ${\mathit{v}}_{\infty }$ at Earth departure, obtained from the deep-space transfer trajectory propagation, and the relative velocity ${\mathit{v}}_{\mathit{R}}$ between the spacecraft and the asteroid at the impact moment will be used for two key purposes: Earth-centered parking orbit parameter matching and asteroid deflection distance calculation.

3. Earth-Centered Parking Orbit Parameter Matching

According to the principles of the patched conic technique, in the design of Earth-centered parking orbit parameter matching, we assume that the spacecraft is influenced only by Earth’s gravity and use a two-body model to compute the orbital parameters. We assume that after launch from the ground, the impactor first enters a 200 km circular parking orbit. After coasting for a certain duration, the upper stage performs a second ignition, injecting the impactor into a hyperbolic escape trajectory. The escape point is located at the perigee of the hyperbolic trajectory, coinciding with a point on the circular parking orbit. At this point, the velocity direction aligns with the tangential direction of the circular orbit.

Using the spacecraft’s Earth escape velocity obtained in the previous section, the right ascension $Ra$ and declination $De$ of the escape point can be calculated. In the Earth-centered inertial (ECI) frame, $De$ is defined as the angle between the escape velocity vector and the x-o-y plane of the ECI. $Ra$ is the angle between the projection of the escape velocity vector onto the x-o-y plane and the x-axis of the ECI.

$$\begin{array}{c}De={\sin }^{-1}\left({\displaystyle \frac{{\mathit{v}}_{\infty \mathrm{z}}}{\left|{\mathit{v}}_{\infty }\right|}}\right)\\ Ra={\tan }^{-1}({\displaystyle \frac{{\mathit{v}}_{\infty \mathrm{y}}}{{\mathit{v}}_{\infty \mathrm{x}}}})\end{array}$$

(9)

Considering the launch vehicle’s capability, the inclination of the parking orbit should not deviate significantly from the launch site’s latitude. To ensure that the subsequent escape trajectory connects with the parking orbit without yielding complex values, we impose a constraint such that the absolute value of the declination remains smaller than the launch site’s latitude.

Next, we determine the unit direction vector of the hyperbolic asymptote in the B-plane [31]. The B-plane is defined to contain the focus of an idealized two-body trajectory that is assumed to be a hyperbola. In addition, it must be perpendicular to the incoming asymptote of that hyperbola. The incoming and outgoing asymptotes, $\widehat{\mathit{S}}$ and $\widehat{\mathit{O}}$, and the focus are contained in the trajectory plane, which is perpendicular to the B-plane. The intersection of the B-plane and the trajectory plane defines a line in space. The B-vector is defined to lie along this line, starting on the focus and ending at the spot where the incoming asymptote pierces the B-plane. The vectors $\widehat{\mathit{T}}$ and $\widehat{\mathit{R}}$ lie in the B-plane and are used as axes.

$$\widehat{\mathit{S}}=\left[\begin{array}{c}\cos \left(De\right)·\cos \left(Ra\right)\\ \cos \left(De\right)·\sin \left(Ra\right)\\ \sin \left(De\right)\end{array}\right]$$

(10)

The unit vector in the oz direction is ${\widehat{\mathit{u}}}_{\mathit{z}}={\left[0 0 1\right]}^T$. Then, the other two basis vectors of the B-plane coordinate system can be expressed as

$$\begin{array}{c}\widehat{\mathit{T}}=\widehat{\mathit{S}}\times {\widehat{\mathit{u}}}_{\mathit{z}}\\ \widehat{\mathit{R}}=\widehat{\mathit{S}}\times \widehat{\mathit{T}}\end{array}$$

(11)

The angle of the B-plane can be determined using the $De$ and $i$:

$$\begin{array}{c}\cos \left(\theta \right)={\displaystyle \frac{\cos \left(i\right)}{\cos \left(De\right)}}\\ \sin \left(\theta \right)=-\sqrt{1-{\cos }^2\left(\theta \right)}\end{array}$$

(12)

To avoid the appearance of complex numbers in the above calculations, the absolute value of the declination should be less than the given orbital inclination of the parking orbit. The unit angular momentum vector of the hyperbolic trajectory can be expressed as

$$\widehat{\mathit{h}}=\widehat{\mathit{T}}sin\left(\theta \right)-\widehat{\mathit{R}}cos\left(\theta \right)$$

(13)

The unit vectors of the perigee position ${\widehat{\mathit{r}}}_{\mathit{p}}$ and perigee velocity ${\widehat{\mathit{v}}}_{\mathit{p}}$ for the hyperbolic trajectory can be calculated:

$$\begin{array}{c}{\widehat{\mathit{r}}}_{\mathit{p}}=\widehat{\mathit{S}}·\cos \left(f_{\infty }\right)-\left(\widehat{\mathit{h}}\times \widehat{\mathit{S}}\right)·\sin \left(f_{\infty }\right)\\ {\widehat{\mathit{v}}}_{\mathit{p}}=\widehat{\mathit{h}}\times {\widehat{\mathit{r}}}_{\mathit{p}}\end{array}$$

(14)

$f_{\infty }$ represents the true anomaly when the velocity approaches its value at infinity, and $\mu$ denotes the gravitational parameter of Earth. The sine and cosine of $f_{\infty }$ can be computed:

$$\begin{array}{c}\cos \left(f_{\infty }\right)=-{\displaystyle \frac{\mu }{r_p{v}_{\infty }^2+\mu }}\\ \sin \left(f_{\infty }\right)=\sqrt{1-{\cos }^2{f}_{\infty }}\end{array}$$

(15)

The position and velocity vectors at perigee can be determined using the corresponding unit vectors, allowing for the computation of the six orbital elements of the hyperbolic trajectory.

$$\begin{array}{c}{\mathit{r}}_{\mathit{p}}=r_p·{\widehat{\mathit{r}}}_{\mathit{p}}\\ {\mathit{v}}_{\mathit{p}}=v_p·{\widehat{\mathit{v}}}_{\mathit{p}}\end{array}$$

(16)

In the patched conic technique, the boundary between the geocentric segment and the heliocentric segment is defined as the sphere of influence (SOI) of Earth. In the deep-space transfer trajectory design, Earth is treated as a point mass, and the distance from the Earth’s center to the SOI boundary is negligible compared to the total transfer trajectory. As a result, a single deep-space transfer trajectory uniquely determines an escape velocity vector ${\mathit{v}}_{\infty }$. However, in the geocentric segment design, for a given ${\mathit{v}}_{\infty }$ and a specified perigee radius $r_p$, an infinite number of hyperbolic escape trajectories exist that share the same velocity magnitude and direction at the SOI boundary [32]. The circle formed by the perigees of these hyperbolic trajectory is referred to as the circle of injection points, as shown in Figure 2.

The intersection points of these parking orbits are denoted as $O_L$ points. After the spacecraft passes through the $O_L$ point, it travels to the position of the circle of injection points, where an escape impulse is applied in the velocity direction, propelling the spacecraft into the hyperbolic trajectory. The escape velocity ${\mathit{v}}_{\infty }$ in the ECI frame can be expressed as

$${\mathit{v}}_{\infty }=v_{\infty }\left[\mathit{cos}\left(De\right)·\mathit{cos}\left(Ra\right) \mathit{cos}\left(De\right)·\mathit{sin}\left(Ra\right) \mathit{sin}\left(Ra\right)\right]$$

(17)

The vector from the Earth’s center to the $O_L$ is denoted as ${\mathit{R}}_{\mathit{L}}$, with its right ascension given by $\left(\pi +Ra\right)$ and its declination by $\left(-De\right)$.

$${\mathit{R}}_{\mathit{L}}=-r_p\left[cos\left(De\right)·cos\left(Ra\right) cos\left(De\right)·sin\left(Ra\right) sin\left(Ra\right)\right]$$

(18)

In addition, the orbital inclination $i$, the launch site latitude ${\varnothing }_L$, and the launch azimuth $A_Z$ satisfy the relationship

$$\cos \left(i\right)=\cos \left({\varnothing }_L\right)·\sin \left(A_Z\right)$$

(19)

Based on the above analysis, it can be concluded that the inclination of the parking orbit plane and the declination of the escape velocity must satisfy the relationship

$$\begin{array}{c}\left|De\right|\le i\le \pi -\left|De\right|\\ i\ge {\varnothing }_L\end{array}$$

(20)

In the preliminary mission design, we typically assume a 90-degree launch azimuth for the rocket.

Based on the above analysis, we can convert the geographical latitude constraint of the launch site into a limit on the inclination. Due to the extremely high fuel requirements for plane changes, kinetic impact missions prioritize allocating more launch capacity to increasing the reserved mass of the impactor. In the mission design phase, to minimize fuel consumption and maximize payload mass, the launch azimuth is typically close to 90°, and the inclination of the parking orbit is similar to the latitude of the launch site. In this study, the parking orbit inclination is set to 19.5° according to the latitude of the launch site. In the subsequent calculations, solutions with an absolute declination greater than the orbital inclination will lead to complex numbers. Therefore, only those arguments of perigee that satisfy the declination constraint will be computed.

Next, we introduce how to convert the coasting time constraint of the launch vehicle into a limit on the argument of perigee at the spacecraft separation point. After coasting along the low Earth parking orbit to the specified position, the second-stage ignition of the Long March rocket will directly place the spacecraft into a deep-space trajectory, effectively leveraging the high specific impulse advantage of the new-generation Long March rocket’s hydrogen–oxygen orbit insertion stage. However, the coasting time needs to remain within a specified range, which is assumed to be between 200 s and 1000 s in this study.

Referring to the study by [33], the coasting time between the two-stage ignition of the Long March rocket’s orbit insertion stage, denoted as $T_h$, is:

$$\begin{array}{c}{T}_h=\sqrt{{\displaystyle \frac{h_{pE}+a_E}{\mu }}}\left({\omega }_E+f-{\theta }_1-{\theta }_2-d\right)\\ d=\left\{\begin{array}{c}\arcsin \left({\displaystyle \frac{\sin \left({\varnothing }_L\right)}{\sin \left(i\right)}}\right)\\ \pi -\arcsin \left({\displaystyle \frac{\sin \left({\varnothing }_L\right)}{\sin \left(i\right)}}\right)\end{array}\right.\end{array}$$

(21)

In the equation, $h_{pE}$ and $a_E$ represent the altitude of the Earth parking orbit and the radius of Earth, respectively. In this study, these values are taken as 200 km and 6378.14 km. The terms ${\omega }_E$ and $f$ represent the argument of periapsis and true anomaly in the Earth-fixed coordinate system. It is important to note that the z-axis of the Earth-fixed coordinate system is affected by the precession of Earth’s rotation axis, causing the equatorial plane to change over time. Therefore, in subsequent calculations, we will transform the argument of periapsis into the Earth inertial frame and plot it in the improved pork-chop plot. ${\theta }_1$ represents the working arc segment from the launch of the Long March rocket until it enters the parking orbit, and ${\theta }_2$ represents the arc segment from the end of the coasting phase to the spacecraft separation. Using the above equation, the coasting time constraint for the launch vehicle can be converted into the argument of perigee constraint.

Through the calculations in this section, we are able to convert the launch site latitude constraint into a limitation on the parking orbit inclination, which in turn can be transformed into a constraint on the escape velocity declination. Additionally, we can convert the launch vehicle coasting time constraint into the argument of perigee constraint at the spacecraft separation point, which can also be derived from the escape point parameters. Furthermore, this process enables the assembly of the Earth departure parameters for the deep-space transfer phase with the parameters corresponding to the spacecraft’s entry into the parking orbit after launch. It also allows the use of the deep-space transfer phase parameters to reflect the engineering constraints of the Earth-centered parking orbit design phase.

4. Asteroid Orbit Calculation

In the post-impact asteroid orbit calculation, we use a high-precision dynamical model to propagate the asteroid’s orbit. The high-precision orbital dynamics model used in this study primarily considers a two-body gravitational model centered around the Sun, as well as third-body gravitational perturbations from the eight planets, Pluto, and the Moon, as well as relativistic effects. The mathematical expression for the high-precision dynamical model is as stated in Equation (22):

$${\displaystyle \frac{d^2\mathit{r}}{d{t}^2}}=-{\displaystyle \frac{u_{Sun}}{{\left|\mathit{r}\right|}^3}}-{\displaystyle \sum }_{i=1}^9{u}_i({\displaystyle \frac{1}{{\left|d_{\mathit{i}}\right|}^3}}{\mathit{d}}_{\mathit{i}}+{\displaystyle \frac{1}{{\left|r_{\mathit{i}}^{\prime }\right|}^3}}{\mathit{r}}_{\mathit{i}}^{\prime })+{\mathit{a}}_{moon}+{\mathit{a}}_{RE}$$

(22)

In this model, $\mathit{r}$ represents the heliocentric distance of the spacecraft or asteroid. $u$ denotes the gravitational constant. ${\mathit{d}}_{\mathit{i}}$ is the heliocentric distance of the $i$th planet. ${\mathit{r}}_{\mathit{i}}^{\prime }$ represents the position vector from the $i$th planet to the spacecraft or asteroid. The index $i$ ranges from 1 to 9, corresponding to Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto. The ephemeris data for the eight major planets, Pluto, and the Moon are obtained from the JPL DE440 ephemeris. The initial position of the asteroid is based on publicly available data from the JPL Small-Body Database, while the impact position is determined by propagating its orbit from the initial position.

After completing the deep-space trajectory design, we obtain the position and velocity of the asteroid at impact. This paper aims to present the method for improving the pork-chop plot. Therefore, a perfectly elastic collision is assumed, with β set to 1. In actual mission design, the value of β can be adjusted based on specific requirements [18]. Using the momentum transfer equation, the post-impact velocity of the asteroid, ${\mathit{v}}_{\mathit{A}}\left({\mathit{t}}_{\mathit{i}+}\right)$, can be determined from the relative velocity at impact as Equation (23):

$${\mathit{v}}_{\mathit{A}}\left({\mathit{t}}_{\mathit{i}+}\right)=\beta {\displaystyle \frac{m_{sc}\left(t_i\right)}{m_{sc}\left(t_i\right)+m_A}}{\mathit{v}}_{\mathit{R}}$$

(23)

After the impact, the position of the asteroid remains unchanged, while the post-impact velocity is used as the initial condition for orbit propagation. The high-precision dynamical model described earlier is then used to propagate the asteroid’s trajectory to determine its CA epoch to Earth. The CA distance is calculated using Equation (24),

$$r_{CA}=\left|\right|{\mathit{r}}_{AstCA}-{\mathit{r}}_{ECA}\left|\right|$$

(24)

where ${\mathit{r}}_{AstCA}$ and ${\mathit{r}}_{ECA}$ are the position vectors of the asteroid and Earth in the close-approach epoch, respectively. Using the above method, we searched for the pre-impact CA and the post-impact CA of the asteroid and calculated the deflection distance at the first post-impact CA.

Additionally, in kinetic impactor in-orbit test missions, it is generally desirable for the impact to be observable from Earth. Therefore, we also calculated the Earth-based visibility of the asteroid. The apparent magnitude $V$ of the asteroid can be determined from its absolute magnitude $H$ and the relative positions of the Sun, Earth, and asteroid using Equation (25) [34,35]:

$$\begin{array}{c}V=H+5lg\left(r·{\mathsf{\Delta }}_{\mathrm{EA}}\right)-2.5lg\left[\left(1-G\right)·{\varnothing }_1\left(\alpha \right)+G·{\varnothing }_2\left(\alpha \right)\right]\\ {\varnothing }_1\left(\alpha \right)=e^{-3.33·ta{n}^{0.63}\left({\displaystyle \frac{\alpha }{2}}\right)}\\ {\varnothing }_2\left(\alpha \right)=e^{-1.87·ta{n}^{1.22}\left({\displaystyle \frac{\alpha }{2}}\right)}\end{array}$$

(25)

In the equation, $V$ represents the apparent magnitude; $H$ is the absolute magnitude of the asteroid; $r$ is the distance between the Sun and the asteroid; ${\mathsf{\Delta }}_{\mathrm{EA}}$ is the distance between the observer (Earth) and the asteroid; $\alpha$ is the phase angle, defined as the angle between the asteroid–Earth line and the asteroid–Sun line; and $G$ is the slope parameter of the phase curve, set to 0.15 in this study.

Based on the capabilities of telescopes such as the Wide Field Survey Telescope [36], we set the apparent magnitude limit of the Earth-based telescopes to 23. Based on this assumption, the visibility window of the asteroid is plotted in the improved pork-chop diagram, serving as an engineering constraint for the in-orbit kinetic impactor test mission.

Using the methods described in this section, we can compute the key deflection distance and determine the observability window of the asteroid.

5. Results and Analysis

5.1. Mission Target

In this study, asteroid 2015 BY310 is selected as a case study to illustrate the methodology described above. 2015 BY310 is an NEA with an approximate diameter of 140 m. Its perihelion distance is 0.951 Au (where Au, or astronomical unit, represents the mean Earth–Sun distance, approximately 150 million km), meaning its orbit extends inside Earth’s orbit. Its aphelion distance is 2.223 AU, placing it between the orbits of Mars and Jupiter (https://cgi.minorplanetcenter.net/, accessed on 20 January 2025), as shown in Figure 3.

The absolute magnitude of asteroid 2015 BY310 is 21.89. Its corresponding size under different albedo conditions is shown in

Table 1. Asteroid size range. The estimated size of 2015 BY310 under different albedo values.

Albedo	Size
0.025	352.0
0.05	248.9
0.10	176.0
0.15	143.7
0.20	124.5
0.25	111.3
0.30	101.6

. For mission design, we assume an asteroid density of 2800 kg/m³ and an albedo of 0.15, resulting in an equivalent diameter of approximately 143.7 m.

This asteroid was discovered on 18 January 2015 by the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) telescope. Since then, it has had observation opportunities approximately every two years. The orbital uncertainty of 2015 BY310, as calculated by the Minor Planet Center (MPC), the European Space Agency (ESA), and National Aeronautics and Space Administration (NASA), is at the lowest possible level [37], indicating highly reliable orbital predictions.

Table 2. Orbit elements of asteroid 2015 BY310. a is the semi-major axis, e is the eccentricity, i is the orbital inclination, Ra is the right ascension of the ascending node, w is the argument of perihelion, and M is the mean anomaly.

jd (Julian Date)	2,460,600.5
a (au)	1.586939550121184
e	0.4008079046707271
i (deg)	4.794505963144238
Ra (deg)	328.4102740092575
w (deg)	229.52799760
M (deg)	277.7012159092116

lists the orbital elements of the asteroid.

The distance variation curve and the brightness variation curve of asteroid 2015 BY310 relative to the Earth are shown in Figure 4 and Figure 5 (https://www.astorb.com/, accessed on 20 January 2025). As seen, this asteroid will approach close to the Earth approximately every two years between 2025 and 2035, providing an observable opportunity every two years, as summarized in

Table 3. The brightness variation of asteroid 2015 BY310. “CA” stands for “close approach”, and “max V” refers to the maximum brightness.

Year	CA Distance (km)	$\mathbf{Max} \mathit{V}$
2025	3,670,000	16.1
2027	3,530,000	15.8
2029	3,780,000	15.6
2031	7,850,000	17.4
2033	17,130,000	19.5
2035	28,020,000	20.7

. Therefore, observations of the asteroid can be conducted before 2030, with an impact planned near the CA between 2029 and 2031, followed by an assessment of the impact effect during the next CA.

5.2. Mission Background

The kinetic impact deflection mission for the asteroid described in this study has a launch window between November 2029 and December 2030. The mission employs a direct transfer flight mode, with the transfer time search range set between 50 and 500 days. The mission will use the Long March 5 rocket launched from the Wenchang Space Launch Center in Hainan, China, with the launch site latitude set at 19.5°. The escape velocity declination is constrained to have an absolute value less than 19.5°. The coasting time of the launch vehicle is constrained between 200 s and 1000 s, corresponding to an argument of perigee range of 150° to 205°. Additionally, to ensure that the impact is observable from the Earth, the asteroid’s apparent magnitude at the time of impact is limited to less than 23.

5.3. Results of Design

Considering the aforementioned engineering constraints, we have plotted the improved pork-chop diagram for the Earth-to-asteroid 2015 BY310, as shown in Figure 6. In the diagram, the region enclosed by the red solid line represents the area where the launch $C3$ value is less than 30${\mathrm{km}}^2/{\mathrm{s}}^2$. According to the Long March 5 rocket’s payload capacity curve, when $C3=30 {\mathrm{km}}^2/{\mathrm{s}}^2$, the launch mass is approximately 3 tons. The pink and blue contour lines represent the argument of perigee, calculated from escape velocity declination solutions with equal absolute values and opposite signs. We have plotted the lower limit of 150° and the upper limit of 205°. Any solution within these limits from either group is considered a feasible solution for the launch vehicle coasting time constraint. The green contour lines represent the escape velocity declination contours, with a lower limit of −19.5° and an upper limit of 19.5°. The region between these limits satisfies the launch site latitude constraint. The two light blue line segments define the region where the apparent magnitude is less than 23. For the impact to be observable from the Earth, the launch and impact dates must fall within this region. The black contour lines represent the asteroid deflection distance, with contours drawn for deflection distances of 3000 km, 5000 km, and 6700 km. In the following sections, we will zoom in on the region within the yellow box for further explanation.

Using Figure 6, we can observe the relationship between the deflection distance and the various engineering constraints within the launch windows. Based on the mission requirements, we can select a specific part of the window, zoom in, and analyze the characteristics of that window. We select a launch window between June and August 2030 from the diagram and zoom in to further explain the improved pork-chop diagram usage method. In Figure 7, the following steps are performed.

Excluding windows based on the observability constraint, the light blue region represents windows where the apparent magnitude at the time of impact is greater than 23.

Excluding windows based on the launch site latitude constraint, the green region represents windows where the absolute value of the escape velocity declination exceeds 19.5°.

Excluding windows where the impact mass does not meet the requirements, the red region represents windows where the launch energy $C3$ is greater than 30 km²/s².

Excluding windows with an infeasible argument of perigee, the dark blue region represents windows where both solutions do not fall within the [150°, 205°] range for the perigee argument constraint.

After eliminating the regions that do not meet these constraints, we are left with the white region, where we combine the deflection distance and feasible windows to choose the launch and transfer times. In the diagram, the red dot corresponds to the orbit that is optimal for deflection distance while considering engineering constraints. The purple dot corresponds to the orbit that is only optimal for deflection distance. In the following sections, we will compare and analyze these two solutions after trajectory correction using the high-precision model.

In previous optimization designs, the typical objective has been to maximize the asteroid deflection distance. However, this often leads to results that do not satisfy constraints related to launch, payload capacity, and visibility. Even if engineering constraints are included in the optimization function, it generally results in the optimization of only a single trajectory. Using the improved pork-chop diagram, we can intuitively and clearly identify the region that satisfies the engineering constraints. To illustrate the improvements of the method described in this paper, the 2015 BY310 asteroid is taken as an example, and the design results are compared with those obtained using the traditional pork-chop plot. We selected an initial value that considers both deflection distance and engineering constraints (represented by the red dot in the diagram). Additionally, we selected another initial value that optimizes only the deflection distance (represented by the purple dot in the diagram) for further orbital optimization.

We transformed the spacecraft trajectory dynamics model from a two-body model to a high-precision model, using the parameters obtained from the improved pork-chop plot as initial values. The initial conditions at the Earth parking orbit escape point were then corrected using a differential correction method. The independent variables, $\mathit{x}$, include the eccentricity, the inclination, the longitude of the ascending node, and the impact time. The dependent variables, $\mathit{y}$, represent the vector of positional differences between the impactor and the asteroid at the impact. The desired numerical values of the results are denoted by ${\mathit{y}}_d$.

$$\mathit{y}=f\left(\mathit{x}\right)={\mathit{y}}_d$$

(26)

To find a solution for Equation (26), a Taylor series expansion of $f\left(\mathit{x}\right)$ is taken to first order about the initial values of the control variables, ${\mathit{x}}_0$:

$${\mathit{y}}_1=f\left({\mathit{x}}_1\right)={\mathit{y}}_0+{\mathit{J}}_0\left({\mathit{x}}_1-{\mathit{x}}_0\right)$$

(27)

where the Jacobian, $\mathit{J}$, is the matrix of partial derivatives of the results with respect to the controls.

$$\mathit{J}=\left[\begin{array}{ccc}{\displaystyle \frac{\partial y_1}{\partial x_1}}& \cdots & {\displaystyle \frac{\partial y_1}{\partial x_n}}\\ ⋮& \ddots & ⋮\\ {\displaystyle \frac{\partial y_m}{\partial x_1}}& \cdots & {\displaystyle \frac{\partial y_m}{\partial x_n}}\end{array}\right]$$

(28)

To solve for the control values that will yield the desired values of the results, Equation (27) is rearranged as

$${\mathit{x}}_{k+1}={\mathit{x}}_k-{\mathit{J}}_k^{-1}(f\left({\mathit{x}}_k\right)-{\mathit{y}}_d)$$

(29)

The corrected launch orbit parameters are presented in

Table 4. Orbital and impact deflection parameters. The parameters that do not meet the engineering constraints are marked in red. The deflection distance is the difference between the CA distance of the asteroid after deflection and before deflection.

Parameters	Red Dot	Purple Dot
Escape Time from the Parking Orbit (UTCG)	2030-7-23 23:58:51	2030-6-10 23:58:51
Parking Orbit Altitude (km)	200	200
$\mathrm{Launch} C3$ (km2/s2)	7.148126	0.593325
$Ra$ (deg)	104.335732	59.449384
$De$ (deg)	−1.468009	58.556383
$i$ (deg)	19.499012	59.223114
$w$ (deg)	202.156259	271.203172
Impact Time (UTCG)	2031-2-6 10:14:48	2031-2-27 19:51:28
CA Time (UTCG)	2035-2-14 15:02:37	2035-2-14 15:00:28
Deflection Distance (km)	5894.60	6698.76
Impactor Mass (kg)	6282.05	7628.96
Impact Velocity (km/s)	11.98	8.37
$V$ at Impact	18.55	17.96

.

From the table above, the purple dot represents the optimal deflection distance within the window. However, considering the actual engineering constraints, its orbital inclination exceeds the launch site latitude, and the argument of perigee also does not meet the constraints, making it unfeasible for actual engineering design. On the other hand, the red dot represents a set of parameters that satisfy the engineering constraints, and the deflection distance at this point is only slightly lower than the purple dot. From the above analysis, by selecting solutions that meet both mission requirements and engineering constraints from the improved pork-chop diagram, we can provide initial values for the trajectory design. After applying these initial values to the high-precision model for correction, the results are largely consistent with the initial values across all parameters.

The schematic diagram of the transfer trajectory that satisfies the engineering constraints is shown in Figure 8.

6. Conclusions

This paper addresses the complexity of the design process in the trajectory optimization of kinetic impact asteroid deflection validation missions, where both deflection effectiveness and engineering constraints need to be considered comprehensively. We propose an improved pork-chop plot design method, which can provide one or multiple clusters of launch windows that meet engineering constraints within the specified time range. This method reduces the multiple iterations traditionally required between deep-space trajectory design and launch vehicle trajectory design to no more than two, thereby improving design efficiency. By converting launch site constraints into the declination of the Earth escape trajectory, converting the launch vehicle coasting time into the periapsis argument at the Earth escape point, and integrating asteroid deflection distance, the Earth-based visibility, and the launch C3 parameter, this approach quickly identifies one or multiple feasible launch transfer trajectories that satisfy mission requirements. After selecting an appropriate initial trajectory from the plot, this study establishes a high-precision numerical orbit propagation model for asteroid orbit evolution and subsequent transfer trajectory corrections. To demonstrate the proposed method, we take asteroid 2015 BY310 as an example, assuming a direct transfer to the target asteroid using the Long March 5 launch vehicle from Wenchang. The trajectory optimization process is elaborated, and the results show that the corrected trajectory parameters align well with those at the corresponding point in the improved pork-chop plot. Compared with traditional pork-chop plots, the proposed method efficiently optimizes and derives launch transfer trajectories that meet engineering constraints.

The improved pork-chop plot design method can also be extended to planetary exploration, interstellar transfers, and other trajectory optimization scenarios, providing a universal solution for rapid deep-space trajectory screening under multiple constraints. The findings of this study can serve as a reference for deep-space exploration mission design. However, the proposed method has certain limitations. The improved pork-chop plot is mainly applicable to direct transfer trajectories and requires further research for complex gravity-assist transfers. Additionally, the integration of launch vehicle trajectory design and deep-space transfer trajectory optimization can be further explored. In future work, we will continue to investigate these two aspects.