# Track linearization with heliocentric coordinate transformation

A common way to detect Solar System Objects (SSO) is to search for the linear track an object traces out across multiple observations of the same region of sky over time. Usually the trajectory of these objects on the sky are close enough to linear for an Earth-based observer to detect. However, because the Earth is also in motion around the Sun, an observer on Earth will encounter periods where the apparent motion of the object is nonlinear. If we could observe from the vantage point of the Sun, these nonlinear intervals due to Earth's motion would go away since all of these objects are orbiting the inertial reference frame of the Sun not Earth. As such, a heliocentric observer would see a smoothly varying path in angular coordinates for an object that could more readily be detected by a search for linear features in angular space.

Below I transform a handful of geocentric SSO observations with nonlinear angular coordinate tracks into heliocentric angular coordinates which removes the nonlinearity due to the motion of Earth. This is not original work; I was motivated to explore this by a paper by Holman et al. where they do exactly what I'm doing here. I've described the geocentric to heliocentric transformation process at greater length separately.

## Geocentric angular coordinates and heliocentric angular coordinates compared

An observer on Earth looking for SSOs will collect observational source data in angular coordinates - typically right ascension and declination (*RA* and *DEC*). This specifies the position of the source on the celestial sphere but says nothing about the distance (*d*) to the object. If we knew how far away the object was we could specify it's exact position in space with *RA*,*DEC* and *d* (*d* in the instance of an Earth observer being the distance from Earth to the object). Without *d* we can't directly transform our observations to the heliocentric frame. Surprisingly, simply asserting a heliocentric distance (*d _{h}*) that is merely close to the true

*d*and then using that asserted heliocentric distance to transform our geocentric observations to a heliocentric angular frame does an effective job of linearizing the observations as shown in the image below.

_{h}**Figure 1.** In the image above geocentric observations of *RA* and *DEC* are transformed to heliocentric *RA* and *DEC* for four different SSOs over each of their 20 day observation windows. Each column is a different SSO of increasing heliocentric distance as you move from left to right. The top half shows the geocentric observations in *RA* and *DEC*. The bottom half shows heliocentric *RA* and *DEC* with varying distance guesses. For each object I've guessed constant heliocentric distances of: the mean true heliocentric distance over the 20 day observation window (blue), 10% more than the mean true distance (green), and 10% less than the mean true distance (red). The gray circles show the exact heliocentric *RA* and *DEC* an observer would see according to JPL Horizons.

## Conclusion

Transformation from geocentric angular coordinates to heliocentric angular coordinates is an effective way to linearize nonlinear tracks for geocentric observers. Additionally, linearity seems to be preserved over a relatively wide range of heliocentric distance guesses. This effect seems even greater at larger heliocentric distances. A geocentric observer running a moving object survey might detect additional objects with a heliocentric search for linear features over a range of asserted heliocentric distances when paired with an existing geocentric linear feature search.

Published: 2/1/2022