Devon and Nikhil

Introduction

In 2029, the asteroid 99942 Apophis will reach within about 0.00025 astronomical units, or 2300 miles, of Earth. Though scientists have ruled out a collision between Apophis and Earth in that year, this phenomenon is not so uncommon: every century, a few of the millions of asteroids in the universe find themselves on a collision course with our planet, putting the fate of humanity in danger. In recent decades, scientists have made great strides in mapping the paths of these Potentially Hazardous Asteroids (PHAs) and determining the possibility of their collision with Earth. Through a laboratory at the California Institute of Technology, NASA has published tables and tables of data detailing the positions and characteristics of these asteroids, and has even pictured their path on a simulated view of the solar system. Now, NASA and other global space agencies are developing new technologies, such as the Enhanced Kinetic Impactor, which could deflect these asteroids and save our planet. In order to understand the methods behind asteroid mapping and deflection, we explored how we can write equations to map the elliptical orbits of PHAs and how scientists can deflect these asteroids off of their collision course with Earth.

Background

In order to map and deflect PHAs, we needed to have a thorough understanding of Kepler’s laws of planetary motion. These three laws constitute the relevant background theory on which our project is based, for they determine the paths and motion of not only planets but asteroids as well.

Kepler’s first law, the Law of Ellipses, maintains that all planets and asteroids must orbit the sun in an elliptical motion, and that the sun must act as one of its foci. The elliptical patterns take many mathematical forms, several of which will be discussed in the section on our theoretical work.

Kepler’s second law, the Law of Equal Areas, explains how the line between a planet or an asteroid and the sun covers sections of equal area in equal times. When an object is closer to the sun in its orbit, its linear and orbital velocity becomes faster. Likewise, when it is further from the sun, its velocity slows down. This phenomenon can be intuitively understood when considering that the magnitude of the gravitational force depends on the radius between the objects, so the velocity of the object depends on its distance from the sun. In this orbit, the aphelion is considered the orbiting object’s farthest point from the sun, and the perihelion is the object’s closest point.

The model of the Earth orbiting the Sun in Figure (a) demonstrates how the Earth covers the same area in the same amount of time despite being at different stages in its orbit, some of which are closer to the sun and some which are farther away. Figure (a) provides a visualization of this law, with the shaded regions A₁, A₂, and A₃ all containing the same area.

Kepler’s third law, the Law of Harmonies, describes how the cube of a planet/asteroid’s orbit’s semi-major axis is proportional to the square of its orbital period. This relationship is shown in the equation below, where T is the period, a is the semi-major axis, G is the gravitational constant (6.67 * 10^-11), and M is the mass of the central body of a system, which in this case is the sun (1.989*10³⁰ kg):

We learned about the real-life application of asteroid deflection through analyzing NASA’s planned Double Asteroid Redirection Test (DART) mission, which will collide a 555 kg unmanned spacecraft moving at 6.65 km/s with Dimorphos, the moonlet of the asteroid Didymos, causing a change in the moonlet’s velocity from anywhere between 0.8 and 2.0 mm/s. Before impact with the PHA, the Enhanced Kinetic Impactor will mount a near-earth object (NEO) and collect more than one hundred tons of rocks into the body of the ship. The large mass, when it collides with the asteroid, will push it off of its collision course.

Theoretical Work

Two-dimensionally graphing the orbital equations

According to Kepler’s First Law of Planetary Motion, all planets orbit the sun in an elliptical motion, with the sun as one of its foci. Similarly, asteroids orbit the sun in the same manner, and each of the asteroids we studied can be modeled and charted through the elliptical equations that describe their orbit. On a NASA website, we found the specific orbital data for each of the asteroids, including 99942 Apophis, 101955 Bennu, and 162173 Ryugu, as well as the Earth. To model these orbits via elliptical equations, we used various formulas and graphic techniques, and we will demonstrate the mapping of an asteroid’s path with 99942 Apophis as our example. The NASA website referenced in our bibliography provided us with the necessary orbital characteristics and a diagram of the orbit, but we ourselves needed to solve for the orbital equations to map it.

Firstly, the orbital data described both the semi-major axis—or the distance from the center of the ellipse to one of its farthest ends—of the orbit as well as its eccentricity, the distance from the center of the ellipse to one of its foci. A common equation in polar form that plots an ellipse in terms of these two values is as follows, with a representing the semi-major axis and e representing the eccentricity:

In the case of Apophis, we were given, in astronomical units, a semi-major axis of 0.9225 and an eccentricity of 0.1915. Such values yield the equation

However, to plot this ellipse using rectangular—not polar—coordinates, we may use the well-known elliptical equation, with a representing the semi-major axis, b the semi-minor axis, and h and k as constants.

However, the orbital data chart did not reveal the semi-minor axis of Apophis’ orbit, or any asteroid for that matter, so we must calculate that value ourselves. To relate the eccentricity and semi-major axis, both of which we know, to the semi-minor axis, we write the definition of eccentricity:

Plugging the given data to solve for the semi-minor axis, we find a value of 0.9054 AU. Furthermore, we must derive the values of h and k, which we can do by logically solving for the center of the ellipse: first, we know that, by Kepler’s first law, if the sun (placed at the origin) is a focus of the ellipse, then the ellipse has no vertical transformation and thus a k-value of 0. Next, knowing that the eccentricity is simply the distance from a focus to the center of the ellipse, we can solve that the center of the ellipse is (-0.1915,0) if we consider the sun to be the right focus. Thus, plugging each of these values into the rectangular elliptical equation yields

This equation can be plotted on Desmos, a 2-dimensional graphing calculator. In the figure below, we consider to the sun to be at the origin of the cartesian plane:

The mapping of this equation is strikingly resemblant to the official orbit diagram published by NASA, which shows Apophis’ orbit in white:

Calculating the average velocity of an asteroid in orbit

Now that we have solved for an equation of the ellipse and have both semi-major and semi-minor axis values, we can use this information, along with the data on its orbital period, to solve for the average velocity of the asteroid in orbit. To do so, we will parameterize the rectangular equation, differentiate these parameters, use the arc length integration formula, and divide the derived circumference of the ellipse by the orbital period.

Firstly, solving the velocity begins with a parameterization of x and y from the ellipse equation in terms of t, according to the parametric equations of an ellipse:

In this instance, the variable t translates practically to the true anomaly of the planet, or in other words, the angle between the positive x-axis and the planet’s location along the orbit. With these two parameters, we can differentiate each of them with respect to t and plug their derivatives into the integral equation for arc length.

Unfortunately, this definite integral is not evaluable with elementary functions, but plugging it into the TI-84 scientific calculator yields the value 5.743, which is the circumference of the ellipse and the distance the asteroid travels in one orbit. Now, to find the average velocity, we simply divide the distance traveled in one full orbit around the sun, calculated to 5.743 AU, by the orbital period, which is 0.89 years, or 323.6 days.¹ This gives us an average asteroid velocity of 30,610 meters per second or 68,473 miles per hour.

The velocity value of the planet or asteroid at a particular position will fluctuate throughout its orbit depending on the distance from the sun. However, in the case of many asteroids, the orbit is fairly circular, so the deviation from the average of the exact velocity value will be minimal. Thus, it is safe to consider 30,610 m/s, or something similar, the velocity of the Apophis asteroid at a certain point in time.

The previous two sections demonstrated the process of solving an elliptical orbit equation and an average velocity for the 99942 Apophis asteroid, a Potentially Hazardous Asteroid (PHA) that will sweep nearby earth in 2029 and perhaps again in 2036. For our project, we conducted the same mathematical processes for all the other asteroids we studied, as well as for Earth. Shown below is a graph that plots the mapped asteroids altogether. For reference, Earth is in green, 99942 Apophis is in black, 101955 Bennu is in blue, and 162173 Ryugu is in orange.

Creating a hypothetical asteroid to demonstrate the law of conservation of momentum in a perfectly inelastic spacecraft-asteroid collision

Using our knowledge of the NASA Double Asteroid Redirection Test (DART) mission, we generated our own numerical model for the impact, based loosely on the school-bus-sized asteroid 2020 SW, which passed near to Earth in 2020.

Our theoretical asteroid is 28.300 m³, the approximate size of a school bus, made exclusively of rock material with a density of 5318.1 kg/m³. Through the density equation, d=m/v, these characteristics give us an asteroid mass of 150590 kg. Based on the data for 2020 SW, we find the asteroid traveling at an initial velocity of 30.00 km/s. We gave our spacecraft an initial velocity of 3.000 km/s, or 6711 mph.

The spacecraft and asteroid will collide in a perfectly inelastic collision, as the spacecraft will move with the asteroid after making contact. We predicted that the asteroid would change its velocity by -2 km/s after the collision, and therefore both the asteroid and spacecraft would have a final velocity of 28.00 km/s. Before impact, the spacecraft is moving in the negative direction and the asteroid is moving in the positive direction.

With these numbers, we can then demonstrate the conservation of momentum in this inelastic collision with the following equation:

Plugging in, we calculate that the mass of the spacecraft is 9715 kg:

Discussion

We discovered that some of the more theoretical concepts of physics, such as the law of conservation of momentum and Kepler’s laws of planetary motion, can be applied in a real-world setting and have a practical impact, which, in this case, can protect the earth and humanity from a PHA.

We also learned about gravitational keyholes while exploring factors that may cause an asteroid to collide with Earth. Gravitational keyholes are small areas in space where another planet’s gravitational pull could alter the orbit of an asteroid that passes by, thus causing the asteroid to go off of its orbital path and collide with that planet. Since asteroids that orbit closest to Earth will be most affected by its gravity, when a PHA orbits within 0.05 astronomical units (au) of Earth, there is the risk that it could encounter a gravitational keyhole that pulls it toward Earth. Thankfully, NASA has ruled out the possibility of 99942 Apophis entering a gravitational keyhole in the year 2029.

Similar to gravitational keyholes, the Yarkovsky effect can alter an asteroid’s orbit, but instead of immediately changing the asteroid’s trajectory, this effect redirects it over a long period of time (millions or even billions of years). The Yarkovsky effect describes how the photons in sunlight exert a force on an asteroid. When the asteroid absorbs these photons, it becomes warmer and eventually radiates its own photons in the form of heat, periodically changing very slightly the asteroid's velocity. This recoil force can either slow an asteroid down or speed it up depending on where the warmest spot of the asteroid is facing. If the warm side of the asteroid is in front facing toward the direction of orbital motion, the recoil force will cause it to slow down. If the warmest spot is behind, then the asteroid will speed up. These small forces can alter the orbit of the asteroid; if it slows down, the orbit will shrink, and if it speeds up, the orbit will expand. These changes in orbit could potentially redirect an asteroid toward Earth.

We also learned that NASA’s DART mission will use the Enhanced Kinetic Impactor technology over a gravity tractor to redirect Dimorphos. A gravity tractor collects mass in space from NEOs to increase the spacecraft’s mass and thus the gravitational attraction between the asteroid and the spacecraft. The gravity tractor uses its gravitational pull to disorient the asteroid’s orbit, which could take over a year to accomplish and consumes a large amount of fuel by carrying such a large tonnage of mass.

NASA’s DART mission will launch in July 2021 and impact Dimorphos in September 2022. If the Enhanced Kinetic Impactor successfully collides with the asteroid and redirects it, then NASA could use this technology to deflect and protect the earth from all the Potentially Hazardous Asteroids of the future.

Bibliography

https://www.nature.com/articles/s41598-020-65343-z#:~:text=The%20Enhanced%20Kinetic%20Impactor%20

https://www.sciencefocus.com/space/how-many-potentially-hazardous-asteroids-are-there/#:~:text=A%20potentially%20hazardous%20asteroid%20

https://ssd.jpl.nasa.gov/sbdb.cgi?sstr=ryugu;orb=0;cov=0;log=0;cad=1#cad

https://ssd.jpl.nasa.gov/sbdb.cgi?sstr=Apophis;orb=0;cov=0;log=0;cad=1#cad

https://ssd.jpl.nasa.gov/sbdb.cgi?sstr=bennu;orb=0;cov=0;log=0;cad=1#cad

http://www.nssc.ac.cn/wxzygx/weixin/201607/P020160718380095698873.pdf

https://assets.openstax.org/oscms-prodcms/media/documents/UniversityPhysicsVolume1-OP.pdf

https://www.nasa.gov/planetarydefense/dart

https://solarsystem.nasa.gov/planets/earth/by-the-numbers/

https://www.youtube.com/watch?v=DQtj0RhimZY