How do we know how asteroids and comets orbit the Sun? How do we know if it could hit Earth or if it would safely pass by? Well, the technique of orbit determination is suitable for this task. Let’s find out about it in this article.

#### Observations

To conduct orbit determination, you first need observations. This involves taking precise astrometric measurements from observatories worldwide and in space. This involves recording the time, the right ascension, the declination, and the observatory’s location.

While the first and last factors are easy for you to understand, some may wonder what the right ascension and declination are. Even though space is 3-dimensional, observations are projected on a 2-dimensional plane. Therefore, the right ascension and the declination are the object’s longitude and latitude in the celestial sphere, respectively. Right ascension measures the distance between the observer and the object towards the east, and declination measures that distance towards the north.

These two pieces of data identify the exact location of the celestial object when viewed by the observer. Along with the observer’s location, these three factors pinpoint an infinitely long line that starts from the observer and does not end. Therefore, with just one observation, no matter how precise it is, you end up with infinitely many possible locations in which the object could be. That’s why determining the orbit of an object requires multiple observations to be taken, and this is what we will discuss in the following sections.

#### Determining the Orbit

Now, these observations will be processed to determine the object’s orbit. This involves using a curve-fitting algorithm that tries to adjust the parameters of a curve (in this case, an orbit) to match that of the observations. You may think there are just six parameters in an orbit, namely the orbital elements. This may apply to a certain amount of accuracy for typical main-belt asteroids or other objects on stable orbits, but ultimately this is not enough. That’s because the six-parameter model is essentially a two-body model. This means inaccuracy, as planetary perturbations and general relativity are not taken into account.

Therefore, we need more parameters for this task, and the algorithm should also account for these perturbations, which cause the orbital elements to change over time. Multiple algorithms are available (such as those used in JPL, MPC, and find_orb), and they output different results.

#### Uncertainties

Every measurement has errors, so the orbital solution is inevitably off by some degree. Moreover, you have to account for the errors of the model itself, since the N-body problem where N > 2 is not analytically computable. Therefore, uncertainties are also calculated in orbital determination, in which the range of orbital elements within one standard deviation is shown.

The more observations the object has, or the longer the observation arc is, the smaller the uncertainties. We will explore that in the next section.

#### Observation Arc

An observation arc is the time span between the first and last observations of an object. Generally, the longer the arc, the smaller the uncertainty of the object. Although more frequent observations also improve the fit of the orbit, a long arc can constrain the orbit better.

The objects with the longest observation arc are generally found early. That includes low-numbered asteroids or comets. According to JPL, 2 Pallas and 3 Juno have arcs of more than 200 years. Comet 109P/Swift-Tuttle has an arc of more than 250 years, and comet 1P/Halley also gets an arc of more than 150 years. This reduces their uncertainties to nearly zero, even though some of them (comets 1P and 109P) have not been observed for almost three decades.

Even though they are not found very early, high-numbered objects can also get very long arcs. Once an object is found and a preliminary orbit is known, astronomers might search for images predating its discovery. These images are known as precovery observations. This is not done for every object because there are too many of them, but this is done for trans-Neptunian objects, which need long observation arcs, or near-Earth objects with impact risk, as the orbit needs to be constrained to assess the probability of them hitting Earth. These searches are often successful, finding asteroids in archival images that were not recognized when they were imaged. For example, the dwarf planet 136199 Eris, whose discovery image was taken in 2003, had its orbit traced back, leading to the identification of precovery images as far back as 1954.

#### Conclusion

In this article, we have discussed the process of orbit determination. It starts with an initial set of observations, which are then analyzed with orbit determination software. As telescopes collect observations, the orbit becomes more and more precise, and the uncertainties decrease. Sometimes, a precovery image could indicate that the object has been spotted before its discovery, constraining the orbit even more.

Remember that orbit determination is a key process for astrophysical research and impact prediction. The results of these algorithms can be analyzed to find asteroid families or unusual objects and can be used to calculate the probability of an object hitting our planet.

If you want to try out the orbit determination process, try out find_orb by Bill Gray. It is open-source, and once it is compiled, it offers a user-friendly and interactive interface with detailed information about the predicted orbits of the objects. Also, if you would like us to include more information, please leave your suggestions in the comments below.