An astronomical dance-The horseshoe orbit

Talking about a planetary orbit,Mercury has the most eccentric orbit (after excluding Pluto from the planets list) while Venus has the most circular orbit.

One of the most interesting things that I came across while reading a book was about a horseshoe orbit. How could something like that be possible? What can cause an object to move in a horseshoe orbit? Well, mathematics or equations were never in the list of my “top favorites”, till I came across their practical aspects. Physics or mathematics becomes interesting when we start looking out of the books, into nature (which is as important). Here, I will try to explain it simply, which is the main focus of ours for creating this website. These articles are for everyone, especially for those who love nature’s beauty and want to understand it but are afraid. Fear disappears when everything becomes simple and that is when you start exploring the knowledge and the universe around you.

Let’s come back from philosophy to science, a horseshoe orbit, as the name suggests, is the path followed by a small body orbiting a larger orbiting body, as seen from the larger body. The orbital periods (time taken by the body to complete one revolution around the Sun) of both the bodies are almost same. One of the fascinating examples is the asteroids orbiting Earth in a similar way, for example, 54509 YORP, 2002 AA29, 2010 SO16 etc. Saturn’s moons Epimetheus and Janus occupy horseshoe orbits with respect to each other

The main cause of the existence of a horseshoe orbit is the change in an elliptical orbit (press a circular orbit from two sides and you will get an elliptical one) of the asteroid because of the gravitational influence of the Earth. When the asteroid tries to catch up with Earth while orbiting the Sun, it gives rise to a horseshoe orbit.

The following diagram shows it beautifully;

While the satellite is orbiting faster than the larger body (Earth) and is about to pass between Earth and Sun. Earth’s gravity pulls the satellite into a higher orbit, thus decreasing its angular speed (Kepler’s third law). Angular speed is the rate at which an object changes its angle (measured) in radians, in a given time period. At the point B, the satellite is moving at almost the same speed as that of Earth while the satellite is still being pulled higher. After reaching a high enough point C where it is slow enough which makes it lag behind Earth. It takes a century or more for the satellite to reach point D as it appears to drift backward when viewed relative to Earth. The gravity of Earth now reduces the orbital velocity of the satellite which makes it fall into a lower orbit which thus increases the angular speed of the satellite. It continues till its orbit is lower and faster than the Earth’s. It takes another century for it to reach the point A thus completing the horseshoe shape.

Now, according to classical mechanics (Mechanics is an area of science concerned with the behavior of physical bodies when subjected to forces or displacements and the subsequent effects of the bodies on their environment) the energy of a body that is moving in a time independent field will be conserved (E= T+V). E is the total energy, T is the kinetic energy (the kinetic energy of an object is the energy that it possesses due to its motion) and V is the potential energy (Potential energy is the energy that an object has due to its position in a force field, gravitational potential energy of a mass m at height h near the surface of the Earth is more than the potential energy would be at height 0 \). T is non-negative and V is negative here.

Now, V will increase when the smaller body (Asteroid) is behind M (Earth) because it is lagging behind and V will decrease when the smaller body is in front of M. Then why does it fall and rise in its orbit? The body as it is moving in front of M will lose energy and will fall into a shorter orbit because orbits with lower total energy have shorter periods and will thus be repelled (as seen in the given figure). If the asteroid is behind Earth,it will gain energy, its orbit will rise and in the process, it will lag behind or get repelled. While dancing or moving the body, as we see, will never come too near to the planet. Sighs!!

Sit back and enjoy the dance without getting hurt.