Research has been actively conducted on asteroids that have a high probability of colliding with the Earth[1, 2]. Orbits close to the Earth are called near-Earth objects (NEOs). Most NEOs are planets approaching the Earth within approximately 0.3 AU (Astronomical Unit). One of the NEOs that have been extensively researched in recent years is Bennu, because it is likely to collide with the Earth. The U.S. Air Force research team of the unmanned probe OSIRIS-REx (Origins, Spectral Interpretation, Resource Identification, Security, and Regolith Explorer) is actively researching the asteroid Bennu. After arriving at Bennu in December 2018, the unmanned probe OSIRIS-Rex is exploring around Bennu. According to a recent report[1, 2], Bennu is a carbon-based asteroid with a comet-like peculiarity of emitting dust and particles less than 10 cm in length. Most importantly, Bennu passes past the Earth once every six years, and at the end of the 22nd century, it is 1 in 2,700 likely to collide with the Earth.

When an asteroid collides with the Earth, it can cause major disasters including climate change, such as nuclear winters, and shock waves such as earthquakes or tsunami[3]. Their impact is indeed enormous, and has led to mass extinctions, with a huge number of plant and animal species disappearing. If an asteroid enters the atmosphere at a speed of about 20 km, a powerful shock wave warms up the surrounding atmosphere and releases enormous amounts of energy as an explosion occurs in the air. Furthermore, when an asteroid with a diameter of 1 km collides with the Earth, the impact energy rises to approximately 100,000 megatons. Meanwhile, the problem of climate change, such as nuclear winters, is also a serious issue. When rocks from the Earth's crust that collide with an asteroid break up into tiny pieces and rise into the sky, these particles combine with dust and float through the atmosphere, blocking sunlight from the sun to earth. In such cases, the temperature drops severely and can cause the earth's ecosystem to collapse. This was precisely the case of the asteroid with a diameter of only about 15 km that exterminated the then prosperous dinosaurs from the Yucatán peninsula in Mexico about 65 million years ago. Since asteroids with a diameter of only several dozen kilometers can cause enormous damage to life on Earth, research on such asteroid collisions is essential.

This study examined the changes in the Earth's orbit that might occur when asteroids of various masses collide with the Earth. Furthermore, changes in the position of the planets within the solar system and in the orbit of the Earth, the resulting changes in surface temperature, and the optimal orbit to send a probe most efficiently from Earth to Mars were investigated.

1. Effects on Earth's orbit and surface temperature when an asteroid collides with Earth

1) Simulation model

Kepler's law of motion can provide an analytically accurate solution for the movement of the sun and a planet in the solar system, i.e., a two-body problem[4]. On the other hand, the analytically accurate solution for a three-body problem is unknown. Therefore, a numerical method should be introduced to obtain the solution for the equation of motion for planets taking into account the interactions of all eight planets in the solar system. This study used the Euler method[5]. The Euler method obtains an approximate value after substituting a sufficiently small value with a micro-variable in a differential equation. Specifically, the net force received by any planet in the solar system equals as the sum of the forces received from the remaining eight celestial bodies, and can be expressed as follows:

where the position ri is the distance from the position of the planet relative to Earth. This problem can be solved using the Euler method, as follows. When the position rn and the velocity vn at a random moment tn are known, the approximate values of the position rn+1 and the velocity vn+1 at tn+1 after a short time Δt can be obtained.

This numerical method was simulated by coding using the Python program. [* The code is presented in Supplementary Material I.]

2) Simulation analysis result

Figure 1 shows the results of simulated changes in the Earth's orbit when an asteroid of varying masses collides with the Earth. It was assumed that asteroids with a mass of 5%, 10%, or 20% of the Earth's mass, respectively, were drawn from an infinite distance from the Sun to the gravity of the Sun and collided with the Earth. In addition, it was also assumed that the other planets retain their orbits. Here, 5% of the Earth's mass is comparable to the mass of Mercury (∼5.5%)[6] and 10% of the earth's weight is comparable to that of Mars (10.7%)[7]. As the mass of the asteroid increases, the Earth's orbit changes to an ellipse that approaches the Sun more. The eccentricity of the elliptical orbit of the Earth increases significantly from 0.063 to 0.373 in the absence of collisions as described in Table 1. This means that if an asteroid collides with the Earth, it may affect the Earth's orbit.

Figure 1. (Color online) Changes in the Earth's orbit (red line) when asteroids of various masses collide with the Earth. The orange sphere at the center represents the Sun, followed by Mercury (white sphere), Venus (pink sphere), Earth (blue sphere), and Mars (red sphere), in order from the inside. (a) Earth's original orbit without asteroid collisions. (b), (c), and (d) Changes in the Earth's orbit when asteroids with masses of 5%, 10%, and 20% of the Earth's mass collide with the Earth, respectively.

Table 1

Changes in the aphelion, perihelion, and eccentricity of the Earth rotating in an elliptical orbit after an asteroid collides with the Earth.

Mass of asteroid relative to Earth's mass (%)	Aphelion (Aphelion) (km)	Perihelion (Perihelion) (km)	Orbital eccentricity (Orbital eccentricity)
No collision	1.496×108	1.483×108	0.063
5 %	1.531×108	1.209×108	0.138
10 %	1.567×108	1.014×108	0.302
20 %	1.640×108	0.755×108	0.373

Changes in the position of the perihelion, the point where the Earth is closest to the Sun were examined. As described in Table 1, it gets closer to the Sun, while the aphelion gets farther away from the Sun. In this case, the perihelion will receive large amounts of solar energy, while the aphelion will receive very little.

The temperature of the Earth's surface was calculated using the solar radiation energy. As the distance between the perihelion and the aphelion has changed as a result of the asteroid collision, the solar radiation energy received almost constantly throughout the year is significantly different. The solar radiation energy received by the Earth is expressed as E=E⊙r24R2, where E⊙ is the radiation energy emitted by the Sun per unit hour, r is the radius of the Earth, R is the distance between the Sun and Earth, and E is the solar radiation energy received by the Earth per unit time. We assume that the Earth is just like a black body, and then used Stefan-Boltzmann's law; as known well, for an ideal black body, the Stefan–Boltzmann law states that the total energy radiated per unit surface area per unit time is proportional to the fourth power of the black body's temperature. The surface temperature of the Earth is thus expressed as T=E4παr24, where T is the temperature of the surface, and α is the Stefan-Boltzmann constant. The temperature of the Earth's surface is almost constant throughout the year, with a temperature of ∼278 K before an asteroid collides, as described in Table 2. However, when an asteroid of 20% of the Earth's mass collides with the Earth, an annual range of up to ∼126 K occurs. In this case, the Earth is highly likely to create a poor environment for the existence of life.

Table 2

Changes in surface temperature after an asteroid collides with the Earth, when the Earth is at aphelion and perihelion.

Mass of asteroid relative to Earth's mass (%)	Earth’s surface temperature at aphelion (K)	Earth’s surface temperature at perihelion (K)
No collision	277.8 K	279.0 K
5 %	274.6 K	309.0 K
10 %	271.5 K	337.4 K
20 %	265.3 K	391.1 K

2. Effects of planetary arrangement changes in the solar system on Earth's orbit and surface temperature

1) Simulation method

For the investigation method, the method referred to in section 2.1.1 was retained. In other words, the simulation code described in Supplementary Material I was used, while changing the positions of two planets within the solar system. Here it was assumed that the other planets are set to retain their original orbit. The resulting changes in the orbit of the Earth were examined. There are several possibilities of changing the positions of planets within the solar system. Among these, two cases were examined: “Jupiter and Mercury change each other’s positions" and “Jupiter and Mars change each other’s positions." In this process, the influence from the difference in mass of the two planets is considered. In other words, it is the case that a large planet such as Jupiter is closer to Earth.

2) Simulation analysis result

The analysis results of the two cases, “Jupiter and Mercury change each other’s positions” and “Jupiter and Mars change each other’s positions”, show that the Earth's orbit appears to be unchanged in both cases.

However, a closer examination of the Earth's orbit showed that there were significant changes. First, in the case of “Jupiter and Mercury change each other’s positions.” the distance between the Earth and the Sun was examined in detail. Thus, as indicated by the blue line in Fig. 2(a), the Earth repeated the path of going near the Sun and moving away from the Sun. In addition, the amplitude of the oscillations on these orbital paths changed over time. In other words, the difference between the perihelion and the aphelion changes over time. As can be seen very clearly in the graph in Fig. 2(a), the amplitude changes approximately every five years. Furthermore, the repeating cycle of the path from perihelion to aphelion and then back to perihelion has a value slightly greater than approximately 365 days, as shown in the red bar graph of Fig. 3. Meanwhile, the cycle is about 240 days, which is much less than 365 days every five years. This suggests that they move much close to the sun in some years, and less close in other years. This also implies that the Earth's ecological environment changes a lot depending on the year. In short, the changes in the arrangement of the planets lead to the surprising result of trajectory oscillations of somewhat greater intensity in the orbit of the Earth.

Figure 2. (Color online) The cases of “Jupiter and Mercury change each other’s positions” and “Jupiter and Mars change each other’s positions” and the resulting changes in the Earth's orbit around the sun and the global surface temperatures. (a) Changes in the distance between the Earth and the Sun over time (blue line) and the change in temperature at the Earth's surface (red line) in the case of “Jupiter and Mercury change each other’s positions”, (b) Changes in the distance between the Earth and the Sun over time (blue line) and the change in temperature at the Earth's surface (red line) in the case of “Jupiter and Mars change each other’s positions.” The horizontal axis in (a) and (b) represents the time in year (Yr) unit. The left vertical axis represents the distance between the Sun and the Earth (unit: AU (Astronomical Units)), and the right vertical axis represents the Earth’s surface temperature (unit: K).

Figure 3. (Color online) Changes in the period of Earth's orbit over time in the cases of “Jupiter and Mercury change each other’s positions” and “Jupiter and Mars change each other’s positions.” Here, the red bar represents the period of the Earth’s orbit for the case of “Jupiter and Mercury change each other’s positions.” The blue bar represents the period of the Earth’s orbit for the case of “Jupiter and Mars change each other’s positions.” This data was obtained from Fig. 2(c) and (d).

And, the case of “Jupiter and Mars change each other’s positions” shows similar results. As shown by a blue line in Fig. 2(b), the Earth is repeating changes in its path over time to the perihelion close to the Sun and then to the aphelion. away from the Sun. The difference between perihelion and aphelion is changing over time. As shown in the graph in Fig. 2(b), the amplitude of this varies over a period of about 13 years. Furthermore, the repeating cycle of the path from perihelion to aphelion and then back to perihelion has a value slightly greater than approximately 365 days, as shown in the red bar graph of Fig. 3, while it becomes approximately 290 days, which is much less than 365 days, every 13 years.

Now, let us compare the changes in the Earth's orbit in two cases, namely, “Jupiter and Mercury change each other's positions" and “Jupiter and Mars change each other’s positions." The oscillation amplitude of the Earth occurs more sharply when “Jupiter and Mars change each other’s positions.” As shown in Table 3, the Earth is orbiting as it oscillates larger in the range 0.9718–1.0250 AU. This difference seems to be based on whether Jupiter is inside or outside the Earth's orbit. This is because there is a difference in Jupiter’s relative distance to Earth, which is the most important variable that Jupiter can affect Earth. In short, the closer a planet with a large mass such as Jupiter is to the Earth, the larger the oscillation of the Earth’s orbit can be.

Table 3

Changes in Earth's aphelion and perihelion relative to the Sun, and the resulting changes in Earth's surface temperature, due to oscillations in Earth's orbital path.

	Largest and smallest values between aphelion and perihelion (AU)	Corresponding temperature (K)
Jupiter and Mercury change each other’s positions	1.0221 0.9765	275.28 281.63
Jupiter and Mars change each other’s positions	1.0250 0.9718	274.95 282.33

One point to emphasize here is the similarity of this phenomenon to the well-known beat phenomenon[8] that the amplitude changes over time, which is a highly intriguing fact. The beat phenomenon is an interference that occurs when two waves with slightly different frequencies overlap. In view of this, the path oscillation phenomenon of the Earth presented in the present study is assumed to be influenced by the interference between a wave of some force produced by the Sun and a wave of some force produced by Jupiter.

When we look at the temperature trend of the Earth's surface, it is easy to predict that it will be similar to the trend of distance change between the Earth and the Sun, as seen in Fig. 2(a) and (b). In other words, the amount of solar radiation energy the Earth receives varies as the distance from the Sun changes, and this is an important factor that changes the average temperature of the Earth. As is well known, the current Earth's orbit is almost circular, so the solar radiation energy the Earth receives throughout the year is almost constant. However, as shown in Table 3, the Earth's surface temperature varies up to 6.35 K and 7.38 K, respectively, in the two cases of “Jupiter and Mercury change each other’s positions” and “Jupiter and Mars change each other’s positions.” Given that the temperature difference between the glacial and interglacial periods is approximately 8.00 K, these variations in temperature can be said to be very large. Especially, the temperature changes slowly over a cycle of hundreds of thousands to millions of years in glacial and interglacial periods. However, according to the calculation results of this study, the Earth's surface temperature changes very rapidly in cycles of approximately one year. Thus, if a very massive planet, such as Jupiter, were near Earth, the ecosystem on Earth would have been very different from what it is now. The study findings will be utilized to find out planets in other stellar systems in the universe that have similar environments to Earth, or to predict the environments of unknown planets.

3. Optimization of probe's orbit for efficient resource transport between Earth and Mars

1) Simulation method

The optimal orbit for the most efficient way to send a probe from Earth to Mars was investigated. This method used simulation codes to change the positions of the Earth and Mars while changing the direction of the probe's launch from the Earth at 5-degree intervals, calculating the time it took for the probe to reach Mars. [*the code is presented in Supplementary Material II.]

The perihelion of the Mars to the Earth appears in a cycle of approximately two years. Thus, this study used the time variable of about 40 months to investigate the variation in the launch angle of the probe while simulating the positions of the Earth and Mars. Simulations were conducted for each of the 72 launch directions every five days of simulation time, examining the conditions of accessing within 10,000,000 kilometers from the Mars. Since the actual probe is first launched into the Earth’s orbit and then accelerated again toward the Mars, the simulation also reflected the actual way in which the probe is launched, starting at a distance of 1500 km from the surface, which is close to the Earth's orbit. Furthermore, to make the calculation more efficient, the simulation for the Mars probe did not take into account the effects of Jupiter, Saturn, Uranus, and Saturn, which are further outside than Mars.

2) Simulation analysis result

In the simulation to investigate the optimal orbit for most efficiently sending a probe from the Earth to the Mars, the mass and speed of the probe in the calculation process used data from the probe Al Amal (Arabic for hope), which was recently launched to the Mars from the Earth. On July 20 in Korean time, the UAE's Mars probe, Al Amal, a small probe weighing 1025 kg, was successfully launched toward Mars at a speed of 11.19 km/s. This velocity corresponds to the second space velocity. This refers to the escape speed of 11.19 km/s, which is needed for an object to escape from the Earth's gravitational field. Note that the velocity of the probe presented above, 11.19 km/s, does not mean the absolute speed of the probe. Because the probe was launched from the Earth, if the probe is launched at a speed of 11.19 km/s from the Earth, the probe's absolute speed will be 34082 km/h, in addition to the rotation speed of the Earth.

Based on the probe Al Amal's mass and velocity, this study investigated the arrangement of the Earth and Mars and the launch angle at which the probe could reach Mars the fastest. As a result, the best arrangement for launching a probe from the Earth toward the Mars, as indicated by a blue line in Fig. 4(a), was that the distance between Earth and Mars was 0.96 AU. What is interesting here is the fact that the closest distance between the Earth and Mars is not 0.50 AU. In fact, as both Earth and Mars are orbiting, the distance between the two planets alone cannot be considered. Therefore, as mentioned in section 2.3.1 above, the direction in which a probe is launched from a particular location on Earth is an important variable. As a result of considering these factors, an optimal orbit was obtained, indicated by a red line in Fig. 4(b). It was concluded that if the probe were launched at a speed of 11.19 km/s in the tangent direction of the Earth's orbit, it would reach the Mars in approximately 151.2 days. A total of 14,400 conditions were simulated to find this condition, and the resulting period of 151.2 days corresponds very well with the expected time when the probe Al Amal, currently on its flight toward Mars, will arrive on Mars. In short, the optimal arrangement is the Earth following the Mars, as shown in Fig. 4(b). The probe should be launched in the tangent direction of the Earth’s orbit in the optimal situation where the distance between the Earth and Mars is 0.96 AU.

Figure 4. (Color online) Results of a probe trajectory optimization analysis between Mars and Earth. (a) Change in distance between Earth and Mars over time, (b) A schematic showing the optimized probe trajectory (red solid line). The orange sphere at the center is the Sun, followed by Mercury (white sphere), Venus (pink sphere), Earth (blue sphere), and Mars (red sphere), in order from the inside. Orbits are represented by solid gray lines.

The motions of planets in the solar system and some of their effects on the Earth were investigated. The findings of the study are as follows. (a) When an asteroid collides with the Earth, it can cause a change in the orbit of the Earth, which can lead to significant changes in the temperature of the earth's surface. (b) Changing the positions of the planets within the solar system can lead to a path oscillation with a somewhat greater amplitude on the Earth's orbit path. It was also observed that the change in the Earth's surface temperature in the case of “Jupiter and Mars change each other's positions" increased up to 7.38 K. This suggest that if a planet of the same mass as Jupiter were near the Earth, the Earth's ecosystem would be significantly different than it is now. (c) After investigating the optimal trajectory for most efficiently sending a probe from the Earth to the Mars, the optimal arrangement was the Earth following the Mars. If the probe were launched at a second space velocity of 11.19 km/s, it could reach the Mars in about 151.2 days. These findings will contribute to widening our knowledge about the Earth and other planets that is necessary for the space industry, which has endless potential.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2021R1A2C2011350). We thank K. Yeo for helpful discussions in the simulation.

The online version of this article contains a supplementary material.