This blog entry describes my tracking of the comet C/2019 Y4 (ATLAS) during the night of 26–27 March 2020. The image above shows thirteen photographs of the comet taken 2000 seconds apart, aligned on the stars and blended so the comet’s motion can be easily seen. I identified a handful of the background stars and used their equatorial coordinates (right ascension and declination) from the NOMAD catalog to fit a pixel-to-world coordinate system. I used the simple imaging polynomial (SIP) convention to represent non-linear geometric distortion in the image because the sky coordinates due not follow the horizontal and vertical pixel axes. This allowed me to very accurately extract the coordinates of the comet’s nucleus at each station.
The SIP convention was first developed for the Spitzer Space Telescope but it is now a standard technique to relate pixel coordinates to sky coordinates. I have now characterised my optics and sensor combination so any image I take with my equipment just needs the sky coordinate of the central pixel defining (that is, where the telescope is pointed) and then I immediately know the coordinates of everything else in the photograph. Of course, minor adjustments like rotating the sensor mean I need to recalculate the pixel-to-world coordinate system again, but this is trivial.
Almost all comets we observe from earth orbit the sun in elliptical orbits. It is usually reasonable to ignore the gravitational influences of the planets and to only consider the sun’s gravity. Under this assumption, the comet’s trajectory can be described using six orbital elements:
- Eccentricity () and semimajor diameter () define the shape and size of the elliptical orbit. Eccentricity describes the elongation compared to a circle and semimajor diameter is the average of the furthest and closest distance of the comet from the sun.
- Inclination () and longitude of the ascending node () define the orientation of the orbital plane.
- Argument of perihelion () and time of the perihelion (). The perihelion is point where the comet is closest to the sun.
For the comet C/2019 Y4 (ATLAS), the reported orbital elements are , , , , and where JD is the Julian date. Notice that the eccentricity is very close to 1, which makes the orbit nearly parabolic instead of elliptical. If its orbit were indeed parabolic, the comet would be making its first (and last) passage around the sun. With the semimajor diameter of 331 AU (where 1 AU is an astronomical unit, the average distance between the earth and the sun), its maximum distance from the sun is a massive 662 AU which makes the orbital period 6026 years.
With these six orbital elements, the position of the comet on the ellipse can then be determined for a specific time. While the comet orbits the sun, however, the earth is also orbiting the sun and so from our perspective on earth, comets can sometimes appear to follow a strange path across the sky due to the relative motion.
They are interesting mathematical problems to: (1) Predict the apparent trajectory of a comet; (2) Determine the orbital elements of a comet based on repeated observations.
Apparent trajectory of C/2019 Y4 (ATLAS)
The image above shows my calculated trajectory of C/2019 Y4 (ATLAS) as the red line, with each marker indicating the time 00:00:00 on days either side of 27 March 2020. (The comet proceeds towards the right.) I also took my processed image of the comet from here, fitted a pixel-to-world coordinate system and reprojected it onto this new view (inside the white box). Of course, the comet is very small in the reprojected image due to the scaling. Nevertheless, the comet is clearly very close to the expected position.
The calculation for the apparent comet position in equatorial coordinates is long-winded, but here are the key steps:
- Calculate the comet’s position in its elliptical orbit using its six orbital elements.
- Transform this orbital position into heliocentric ecliptical coordinates (heliocentric means ‘relative to the sun’; ecliptical means ‘relative to the plane of the ecliptic’).
- Calculate the earth’s position in its orbit around the sun in heliocentric ecliptical coordinates. I used an accurate algorithm called VSOP87D for this.
- Considering the comet’s position relative to the earth, calculate the comet’s geocentric ecliptical coordinates with correction for the light-time effect (geocentric means ‘relative to the earth’). The light-time effect causes the comet to appear in a slightly different position to where it actually is at a specific time because of the time it takes for its reflected light to reach the earth.
- Correct for aberration, which causes the comet to be displaced slightly in the direction of the earth’s motion.
- Correct for precession (the slow drift of the earth’s rotational axis).
- Correct for nutation (the periodic oscillation of the rotational axis of the earth around its mean position).
- Finally transform the geocentric ecliptical coordinates of the comet to geocentric equatorial coordinates (equatorial means ‘relative to the plane of the earth’s equator’).
The expected positions of the comet on the night of 26–27 March 2020 are indicated on the image below alongside the actual observed positions. The numbers are tabulated underneath. There is a small constant discrepancy of 0.36°. I believe this is due to error in the approximate algorithms I used, particularly for nutation.
|Observed position||Expected position|
|Date and time (GMT)||RA (°)||Dec (°)||RA (°)||Dec (°)|
Refine orbital elements
I also wrote some software to determine the orbital elements from repeated observations. It uses the ‘downhill simplex’ multi-variable optimisation algorithm. It previously worked very well for the comet 46P/Wirtanen and so I wanted to do the same for C/2019 Y4 (ATLAS). Of course, I would have to combine my own limited observations of position and date/time with others’ too. Unfortunately, it seems that with the orbit being very nearly parabolic, the optimisation will not converge—it is too unstable.