NASA DART mission: A preliminary mathematical dynamical model and its nonlinear circuit emulation

1 Introduction

The Double Asteroid Redirection Test (DART) mission represents a fundamental result in understanding the possibility to divert the trajectory of potentially dangerous asteroids. The mission consisted in launching a spacecraft toward the binary system composed of the asteroid Dimorphos orbiting around Didymos. As a consequence of the impact, Dimorphos has been diverted to a different orbit, consisting in a shorter orbital revolution period, thus confirming the possibility to modify the trajectory of asteroids. The analysis of the impressive results and observations gained by the DART mission have been summarized in the studies by Thomas et al. [1] and Daly et al. [2].

The aim of this communication is essentially to induce nonlinear engineering to revisit orbital mechanics in term of nonlinear dynamical problems and to broaden the significance of the mission with educational purposes. It is, in fact, very important that recent results involving orbital mechanics and dynamical modeling can be accessible to young generations of engineers. Even if the essentials of orbital mechanics are now given also to undergraduate students, the DART mission makes this topic appealing not only as regards the point of view of the space missions, but also since it can be linked to the topics of nonlinear theory and nonlinear control.

Generally, the culture of studying the future problems reaches the students with a lot of delay. Moreover, our aim is to prepare the future generations to address the problems in a more interdisciplinary and immediate way, and to increase their interest eliciting future skills.

Although many missions have been devoted to the interplanetary exploration of planets and asteroids within the solar system, the DART mission represents the first planetary defense experiment, which has been successfully realized. This result opens the way to similar studies and missions that will be designed and realized in this century; thus, the problem of diverting asteroid motion will be further focused. The approach envisaged in this article allows us to reinforce the link between orbital problems and nonlinear dynamical systems. We are in the era of nonlinear dynamics, and the mechanics of celestial bodies displays highly nonlinear behavior. This can be explored in practice, modeling the problem with available data and measures. Moreover, in this communication, experimental results will be reported with the aim of showing the capability of an electrical circuit to emulate the orbital mechanics. Under this perspective, yet being a simplified circuit emulator of a complex problem, the introduction of the simple circuit platform focused in this article, represents a practical implementation of a complex nonlinear model onto which control actions can be easily tested by selecting proper inputs and/or parameter values, that is, in circuit terms, introducing time-varying voltage signals and/or modifying the value of simple variable resistors. Under this perspective, the possibility of having a real setup, which naturally encompasses the unavoidable non-ideality of a physical system, allows us not only to test the effectiveness of the control actions alone, but also, more importantly, to verify their robustness in the presence of uncertainty or unmodeled dynamics.

In this article, we refer to simple and effective models, such as the two-body and the three-body problems, determining the key parameters that allows us to conceptually reproduce the dynamics of the binary system and the orbital changes resulting from the impact of the DART spacecraft. Based on these models, an analog/digital circuit that emulates such dynamics will be discussed, showing the advantages of the proposed approach in obtaining the orbital mechanics of the considered celestial bodies. It should be, therefore, stressed that this communication is not oriented at a precise modeling of orbital dynamics and the dynamics of the impact of the DART spacecraft on the asteroids, but it is rather intended as a guideline for providing a practical way to teach orbital mechanics, to disseminate the results of the DART mission, and to propose to the nonlinear engineering scientific community the use of circuit emulators to study active actions in orbital mechanics.

This article is organized as follows: in Section 2, details on the DART mission are reported in order to propose a model based on the two-body problem, as discussed in Section 3 reporting numerical results showing the capability of the model to reproduce the effects of the DART mission. In Section 4, the extension to the three-body problem considering the effect of the Sun on the binary asteroid system is reported, showing the possibility to reproduce the DART mission effects also in this scenario. The design of a circuit emulator for reproducing the motion dynamics is outlined in Section 5. In Section 6, final considerations are drawn.

2 DART mission

The DART mission represents the first successful attempt to deflect the trajectory of an orbiting object. The mission consisted in the launch of a spacecraft toward the binary asteroid system composed of Didymos and its moon Dimorphos [1,2]. The impact of the spacecraft on the surface of Dimorphos produced a sudden change of its momentum, thus forcing the asteroid towards a different orbit around Didymos [3].

The novel orbit displays a decrease in the period of Dimorphos revolution around Didymos. In fact, the success of DART mission has been confirmed by observing the frequency of changes in the luminosity of Didymos, due to the passage of Dimorphos [1], as documented thanks also to the LICIACube satellite [4].

It should be noted that the momentum transfer occurring at the impact of the spacecraft has been effective since it had the outcome of varying the velocity of Dimorphos. As noted by Graykowski et al. [5], Dimorphos lost approximately 0.5% of its mass. In the following, we neglected such mass variation. This point is crucial since, from a nonlinear dynamics perspective, in a mathematical model representing this scenario, there must be a multistable behavior with respect to the velocity of the two bodies, while showing a substantial independence on the parameters representing their masses.

In order to determine a suitable, yet simplified, mathematical model, able to reproduce the key features of the Didymos/Dimorphos motion, and also their interaction with the Sun, we will refer, in the following, to the well-known orbital mechanics models, i.e., the two-body and the three-body problem. This will allow us to model the DART mission as a control action acting on the nonlinear mathematical model [6].

It should be noted that the two-body and three-body problems are based on the assumption of considering point masses on which only the mutual force of gravity acts. According to the study by Agrusa et al. [7], the irregular shapes of Didymos and Dimorphos and their close proximity make their motion more adequately modeled by a full two-body problem, which explicitly takes into account the geometry of the two bodies. Assuming point masses, however, besides simplifying the considered models, does not affect the capability of catching the key features of the DART experiment outcome.

3 Two-body formulation of the effects of the DART experiment

A classical approach to determine the motion of two bodies as the sole consequence of their gravitational interaction is the two-body problem. The mathematical model is based on the laws of conservation of angular momentum and energy and leads to elliptical orbits with specific period of oscillation, depending on the masses of the two bodies and on their mutual velocity [6].

The model represents the equations of motion of the two bodies in an inertial space, and the equations of motion can be expressed as:

where ( X 1 , Y 1 , Z 1 ) and ( X 2 , Y 2 , Z 2 ) are the positions of the two bodies in the inertial space, with mass m 1 and m 2 expressed in kg, respectively, G = 6.67259 × 1 0 − 20 km 3 kgs 2 is the universal gravitational constant, and

is the magnitude of the relative position of mass m 2 with respect to m 1 .

The model in Eq. (1) is a nonlinear system of 12 ordinary differential equations, 6 describing the dynamics of the coordinates ( X 1 , Y 1 , Z 1 ) , and ( X 2 , Y 2 , Z 2 ) , i.e., the relative velocity vectors, and 6 describing the dynamics of the relative velocity vectors ( X ˙ 1 , Y ˙ 1 , Z ˙ 1 ) , and ( X ˙ 2 , Y ˙ 2 , Z ˙ 2 ) , i.e., the relative acceleration vectors.

The motion of the Didymos–Dimorphos binary system can been obtained analytically solving Eq. (1). When considering a numerical integration of Eq. (1), a variable step Runge–Kutta–Fehlberg method can be adopted, considering the masses of the two bodies as m 1 = 5.228011 × 1 0 11 kg (Didymos mass) and m 2 = 4.841661 × 1 0 9 kg (Dimorphos mass) [8].

In order to obtain an orbital period consistent with the observations of the revolution cycle before the impact of the spacecraft used in the DART mission, the initial conditions have been chosen as ( X 1 ( 0 ) , Y 1 ( 0 ) , Z 1 ( 0 ) ) = ( 0 , 0 , 0 ) , ( X 2 ( 0 ) , Y 2 ( 0 ) , Z 2 ( 0 ) ) = ( 1.18 , 0 , 0 ) , ( X ˙ 1 ( 0 ) , Y ˙ 1 ( 0 ) , Z ˙ 1 ( 0 ) ) = ( 0 , 0 , 0 ) , and ( X ˙ 2 ( 0 ) , Y ˙ 2 ( 0 ) , Z ˙ 2 ( 0 ) ) = ( 0 , 0.00017275 , 0 ) . This leads to the trajectory reported in Figure 1, whose orbital period is approximately 11.92 h, consistent with the pre-impact observations.

Figure 1

Orbital mechanics of Dimorphos around Didymos. Marker sizes are only for illustrative purposes, and they do not reflect effective dimensions.

In order to model the effect of the momentum transfer gained during the impact of the DART mission spacecraft, we can reduce the component Y ˙ 2 by a factor Δ V , as Y ˙ 2 − Δ V Y ˙ 2 . The resulting orbits and the corresponding revolution periods are reported in Figure 2, for different values of Δ V . It is evident that the model is able to catch the intrinsic properties of the outcome of the DART mission, as shown by the effect of varying the velocity component Y ˙ 2 on the revolution period T , summarized in Figure 3 as a function of Δ V . In Figure 3, we also explored the effect of the relative tolerance adopted in the numerical integration algorithm. As it can be observed, decreasing the relative tolerance leads to more reliable and regular solutions. In particular, we explored the numerical integration capabilities by considering a relative tolerance ranging from 1 0 − 4 to 1 0 − 6 . The obtained solutions can be considered reliable for values lower than 1 0 − 5 . This investigation, therefore, unveils that the numerical errors introduced by the numerical integration of Eq. (1) are significant. This problem will be solved thanks to the proposed circuit emulator, as it will be shown in the following.

Figure 2

Numerical simulations. Orbital mechanics of Dimorphos around Didymos varying the velocity component Y ˙ 2 to emulate the effect of the DART mission: (a) orbits in the inertial space and (b) temporal evolution of the X 2 component reporting the changes in the period and in the orbital distance.

Figure 3

Numerical simulations. Effect of varying the velocity component Y ˙ 2 on the orbital period: the different curves are calculated with relative tolerance of the numerical integration algorithm ranging from 1 0 − 4 (lower purple curve) to 1 0 − 6 (thick red curve).

Moreover, the model of the two-body problem proves to be effective in catching also another key feature of the experiment. We estimated the effect of varying both the mass m 2 and the velocity component Y ˙ 2 on the revolution period T , as shown in Figure 4. It appears evident that the effect of the mass variation is negligible in impressing a novel orbit with a different period T , as in fact, the mass of Dimorphos has not been significantly affected by the spacecraft.

Figure 4

Numerical simulations. Effect of varying the velocity component Y ˙ 2 and the mass m 2 on the orbital period.

4 Three-body formulation of the effects of the DART experiment

The problem of designing accurately interplanetary defense missions that originated the DART experiment can be more suitably modeled by explicitly considering the effects of the Sun on the considered asteroids. The three-body formulation on the DART mission outcomes presents, however, an increased level of complexity, which more explicitly leads to the idea of using circuit emulators outlined in the next section. Let us start recasting the model as:

where ( X 1 , Y 1 , Z 1 ) , ( X 2 , Y 2 , Z 2 ) , and ( X 3 , Y 3 , Z 3 ) are the positions of the three bodies in the inertial space, with mass m 1 , m 2 , and m 3 expressed in kg, respectively, G is the universal gravitational constant, and

are the magnitudes of the relative position of each mass with respect to the others.

The model in Eq. (3) is a nonlinear system of 18 ordinary differential equations, 9 describing the dynamics of the coordinates ( X 1 , Y 1 , Z 1 ) , ( X 2 , Y 2 , Z 2 ) , and ( X 3 , Y 3 , Z 3 ) , i.e., the relative velocity vectors, and 9 describing the dynamics of the relative velocity vectors ( X ˙ 1 , Y ˙ 1 , Z ˙ 1 ) , ( X ˙ 2 , Y ˙ 2 , Z ˙ 2 ) , and ( X ˙ 3 , Y ˙ 3 , Z ˙ 3 ) , i.e., the relative acceleration vectors.

The motion of the Didymos–Dimorphos binary system around the Sun can be obtained through a numerical integration of Eq. (3), considering the three body masses as m 1 = 1.98892 × 1 0 30 kg (Sun mass), m 2 = 5.228011 × 1 0 11 kg (Didymos mass), and m 3 = 4.841661 × 1 0 9 kg (Dimorphos mass) [8].

In the three-body formulation, the inertial space is evidently dominated by the distance between the Sun and the Didymos/Dimorphos system, as shown in Figure 5(a) obtained for initial conditions ( X 1 ( 0 ) , Y 1 ( 0 ) , Z 1 ( 0 ) ) = ( 0 , 0 , 0 ) , ( X 2 ( 0 ) , Y 2 ( 0 ) , Z 2 ( 0 ) ) = ( 3.39 × 1 0 8 , 0 , 0 ) , ( X 3 ( 0 ) , Y 3 ( 0 ) , Z 3 ( 0 ) ) = ( 3.39 × 1 0 8 + 1.18 , 0 , 0 ) , ( X ˙ 1 ( 0 ) , Y ˙ 1 ( 0 ) , Z ˙ 1 ( 0 ) ) = ( 0 , 0 , 0 ) , ( X ˙ 2 ( 0 ) , Y ˙ 1 ( 0 ) , Z ˙ 2 ( 0 ) ) = ( 0 , 15.51785437 , 0 ) , and ( X ˙ 3 ( 0 ) , Y ˙ 3 ( 0 ) , Z ˙ 3 ( 0 ) ) = ( 0 , 15.51802712 , 0 ) . The effect of the DART impact, as reported in Figure 5(b), is still captured by varying the initial velocity of Dimorphos by a factor Δ V .

Figure 5

Numerical simulations: (a) orbital mechanics of Didymos/Dimorphos around the Sun and (b) temporal evolution of the distance r 23 varying the velocity component Y ˙ 3 to emulate the effect of the DART mission.

5 Nonlinear circuit approach to model the DART mission

Even if the numerical integration of Eqs. (1) and (3) can be performed using standard mathematical tools, as shown in the previous sections, the obtained solution can be poorly accurate and may need a large observation window. A critical issue in analyzing numerically the dynamics of the n-body problems, in fact, is the different magnitudes of the parameters involved [6]. A canonical unit representation can be adopted to mitigate these effects [9]. Moreover, the original orbital period of Dimorphos was around 12 h, and it was reduced by the impact of DART of approximately 33 min. This minimal difference is further amplified in the three-body problem, as the orbit around the Sun of the binary system is approximately of 770 days. The numerical simulation appears, hence, impractical in terms of both the integration step and the integration time needed to observe a set of complete revolutions, as the time-scales of the behaviors are intrinsically different. Moreover, numerical integration algorithms are consuming in terms of computational time.

The problem in Eq. (1), actually, can be analytically solved; therefore, in principle, the drawbacks of numerical integration can be suitably overcome. On the contrary, the analytical solution to the three-body case can be attempted only in specific conditions [6]. These solutions, however, stand for exact values of the model parameters. As observed in the studies by Daly et al. [2] and Agrusa et al. [7], the values of the masses of Didymos and Dimorphos are usually inferred by indirect measurements or by models based on observations [4], and in fact, they are often reported with a given uncertainty.

In order to propose a more efficient simulation tool, we designed a nonlinear electronic circuit based on the two-body problems, as shown in the scheme reported in Figure 6. It is based on analog components for performing the integration of the differential equations and on a digital microcontroller to realize the nonlinearities. This solution provides a two-fold advantage. On the one hand, the time-scales of the system can be adjusted by acting only on the R C group of the integrators [10], so that behaviors of hours can be rescaled to few seconds in the circuit evolution. On the other hand, the use of digital equipment allows for a fast and reliable implementation of the nonlinearities involved in Eqs. (1) and (3), and also for a complete reconfigurability of system parameters. In the proposed solution, the circuits are designed and implemented by following a state-variable approach [11] in which operational amplifiers are configured as adders and integrators, while parameter values are ratios between resistances. Under this perspective, therefore, in virtue of the uncertainty of the adopted resistors, whose tolerance can be opportunely selected to be representative, the circuit emulator encompasses the presence of uncertainty on parameter values, such as the masses.

Figure 6

Schematic of the circuit mimicking Eq. (1). Nonlinearities are designed to be realized by a digital microcontroller. From left to right: algebraic adders with sign to realize the relative coordinate difference to be used in the nonlinear block; nonlinear blocks implemented in digital; integrator configurations for obtaining velocities; integrator configurations for obtaining positions.

The circuit in Figure 6 is obtained considering 12 operational amplifiers, implemented using 3 standard T L 084 integrated devices, in a Miller integrator configuration [10] that allow us to realize the generic equation:

where C int is the feedback capacitor, R int is the inverting input resistor, and V in and V out are the inverting input and output voltages, respectively. It should be noted that the parameters m 1 and m 2 in Eq. (1), i.e., the masses, are obtained by proper selection of the resistors R int 1 and R int 2 , with reference to Figure 6. Components with 10% tolerance have been selected, while the resistors R int 1 and R int 2 are 10 k Ω variable resistors. These choices, on the one hand, introduce the uncertainty in the circuit emulator and, on the other hand, allows for an immediate variation of the masses value within the range of uncertainty of the available estimates, thus allowing for robustness testing. In particular, m 1 1 0 11 = R ¯ R int 1 and m 2 1 0 9 = R int R int 2 (note that the factors 1 0 11 and 1 0 9 are included in the voltage generated by the microcontroller to avoid saturations).

The circuit behavior reported in Figure 7 has been obtained choosing in the scheme of Figure 6  R = 100 k Ω , R int 1 = 5.74 k Ω , R int 2 = 6.20 k Ω , R int = 30 k Ω , and C int = 1 n F .

Figure 7

Circuit behavior. Orbital mechanics of Didymos/Dimorphos varying the velocity component Y ˙ 2 to emulate the effect of the DART mission: (a) orbit in the inertial space and (b) temporal evolution of X 2 reporting the changes in the period and in the orbital distance (expressed in volts).

It can be observed that the circuit emulates effectively the binary asteroid system motion and models the effects of the DART mission spacecraft impact. Moreover, the waveforms are generated in a 10 s observation window, thus introducing a temporal scaling factor τ = 1 R int C int ≈ 3.33 × 1 0 4 .

In order to show a further advantage of adopting the proposed circuit emulator to investigate the dynamical properties of the two-body problem for mimicking interplanetary missions and to propose a comparison with the aforementioned numerical results, we analyzed the behavior of the two-body system by reducing the component Y ˙ 2 by a factor Δ V , as Y ˙ 2 − Δ V Y ˙ 2 , similar to what discussed in Section 3, and evaluating the orbital period. The results are reported in Figure 8. Besides correctly emulating the functional behavior reported in Figure 3, the circuit produces outputs whose orbital periods are consistent with numerical simulations obtained with a relative tolerance of approximately 1 0 − 5 . This is due to the fact that uncertainty on system parameters are introduced by the analog implementation. However, the reliability of the experimental solution is guaranteed.

Figure 8

Circuit behavior. Effect of varying the velocity component Y ˙ 2 on the orbital period.

As concerns the circuit for the three-body problem, the main issue is the high magnitudes of the distance between the Sun and the binary system in comparison with that between the two asteroids, a problem that cannot be mitigated using canonical units. This would imply the realization of a circuit in which the voltages representing the motion of Didymos and Dimorphos would be undistinguishable. In order to cope with this feature, the model has been entirely implemented on a digital microcontroller, preserving the analog nature of the circuit only for the implementation of the masses values. Therefore, the parameters are set as analog voltages tuned by variable resistors and put as input of the microcontroller, thus incorporating the non-ideal nature of the parameters. The experimental results reported in Figure 9 are related to the effect of varying the initial velocity of Dimorphos and allow us to assess the capability of the circuit emulator to reproduce the dynamics of the DART mission.

Figure 9

Circuit behavior. Orbital mechanics of Didymos/Dimorphos around the Sun varying the velocity component Y ˙ 3 to emulate the effect of the DART mission. Temporal evolution of the distance Didymos/Dimorphos r 23 reporting the changes in the period and in the orbital distance (expressed in Volts).

6 Conclusion

The electronic setup discussed in this contribution is able to emulate the physics of the two-body and three-body problems, thus providing immediate and impressive results to young learners. Moreover, it consists in a simple but effective platform to test in a rapid and reliable way the presence of uncertainty and imperfections present in the physical system on the control action performance. Circuit emulators, in fact, are always prone to unavoidable uncertainty, and this does not always have a detrimental effect [12]. In this case, it may allow for control action robustness evaluation also in the presence of not exactly known physical parameters of the celestial bodies under consideration, since the reliability of the obtained behavior has been confirmed.

The adoption of the two-body and three-body problem formulations has been intended as a simplified yet reliable way to reproduce the momentum transfer dynamics of the DART mission. However, the circuit emulator approach is general and can be easily applied also to different dynamics. A speculative example is the restricted circular three-body problem, whose dynamics models the kinematic of a negligible mass orbiting within a two-body system. The final phase of the DART mission, hence, can be represented by such model, for which an exact solution does not exist. Evidently, the formulation as a restricted circular three-body problem needs the knowledge of actual polar azimuth coordinates of the DART spacecraft while approaching Didymos/Dimorphos, as well as its velocity. Once the model is formulated, the circuit emulator can be obtained following the proposed guidelines and the dynamics analyzed in the presence of uncertainty for a more robust mission design.

The DART mission has been spectacularly effective. Using the words of the leader of the DART observation team, it “consisted in a giant game of billiards in space and you can basically just solve that out as a simple physics equation. But there’s so much else that’s happening that makes that not true” [13]. These assertions are related to the fact that the effective momentum that the DART spacecraft transferred to Dimorphos was not completely predictable as non-ideality sources may have had a strong impact on the mission outcome. To a certain extent, a circuit emulator can be exploited to test the control actions and the simulation results in a less ideal environment.