A satellite built by undergraduate students from Brown Space Engineering (BSE) was launched by NASA and has been operating in low earth orbit since July 2018. Objects in orbit are tracked by the Space Surveillance Network. Measurements of the position and velocity are made using optical telescopes, radar ranging, and radio reception. This data is then used to calculate the orbit. Once or twice per day a file is published for each tracked object that can be used to predict the motion a day or two in the future. The file is called a two-line element set (TLE) and contains numbers that describe the elliptical shape of the orbital path and where the object is at a given moment. Here is a sample TLE for the ISS: the International Space Station.
ISS (ZARYA) 1 25544U 98067A 98324.28472222 -.00003657 11563-4 00000+0 0 9996 2 25544 051.5908 168.3788 0125362 086.4185 359.7454 16.05064833 05
Each object is given a unique designation that includes the year and number of launch followed by a letter for each piece that is in a separate orbit. The first module of the ISS (called Zarya, the Russian word for “dawn” or “sunrise”) was the 67th launch of 1998 and is cataloged as 1998-067-A. Any piece that becomes detached, either by accident or intentionally, is given a new designation to independently track it. A tool bag that floated away from an astronaut became known as 1998-067-BL and the BSE satellite named EQUiSat released by the astronauts is designated 1998-067-PA. All of the objects discussed below are considered “loose pieces” of the ISS and I’ll refer to them using just the trailing letter designation.
On July 13th EQUiSat was one of four small satellites released from the ISS. The first TLE files were published three days later. There was some early confusion about which TLE corresponded to which satellite. EQUiSat was initially identified as NZ on July 17 but was later found to be better matched to PA on July 22. The remainder of this post is a detailed analysis of the computed orbital elements in the published TLE files.
To asses the accuracy of the orbital elements and the resulting long term prediction error I downloaded and examined all of the published TLE files. There were 132 spanning 79 days; an average of 5 were released every 3 days. Below is a plot of changes in one of the orbital elements from each TLE to the next. The eccentricity describes how elliptical the shape of the orbit is. This element of the orbit should be nearly constant on short time scales or at least smoothly changing by only a very small amount from one day to the next. The September data is what would be expected.
It should be noted that this is not a graph of the “true” eccentricity of the orbit. It is just the best estimate based on measurements available on that date. There are three distinct clusters of data points in the graph after the first estimate.
- an initial outlier value on July 16th in the first TLE published 2.,9 days after release
- there is a “jump” in the second TLE on July 17th at 3.8 days followed by a steep increase with a couple of smaller jumps
- another jump on July 28th at 15.3 days is followed by medium amplitude random jitter
- after the next jump on August 23rd at 41.0 days the changes from day to day become smooth and change by very tiny increments
These “jumps” are due to a computer model trying to fit an orbit description to the measurements. A first solution is tried and the model attempts to converge on a value that best describes the available data. As more measurements are made the model “jumps” to a new solution that is a better fit to the larger data set. This process continue for more than a month before a precise solution is found. The errors in the graph are quite small and would not impact predicting the position a day or two in the future. But these small errors accumulate quickly and the predicted position would rapidly diverge from the actual position after a short time.
Initial errors in the computed orbital elements of small satellites are common for days or weeks after deployment. (See Orbit Determination from Two Line Element Sets of ISS-Deployed CubeSats and Propagation of CubeSats in LEO using NORAD Two Line Element Sets: Accuracy and Update Frequency for a more rigorous analysis.) The plot above shows the eccentricity of object PA along with four other satellites released from the ISS.
The satellites deployed in 2017 show modest early errors compared to the two in 2018. There are also a number of “glitches” where a single TLE has an outlier value that is soon corrected. This is likely due to a poor quality measurement which introduces a small error into the computer model. Understanding the magnitude and cause of these early errors is important for quickly matching a TLE to a recently released satellite and for accurately calculating the Doppler shift when it passes over a ground station.
These small satellites are known as CubeSats because of their square shape. The cubes are only 10 cm (~4 inch) across. It is difficult to track them because of their small radar cross-section. Another factor that adds to the confusion is that multiple satellites are released at the same time. They orbit close together in a cluster and then only slowly drift apart from each other to the point where they can be unambiguously identified. Here is the eccentricity of object PA compared to other members of the July 13 deployment of a “flock” of CubeSats.
The model solutions for most of July show erratic jumps for all four which are greatly reduced by day 15 after release. During most of August they are all somewhat close together with modest random errors. The model then simultaneously converges on much better solutions for all of them on day 41. The errors are likely caused by radar reflections from one satellite that are misidentified as another satellite in the group. When the eccentricity stabilizes it is an indication that the published TLE has improved in long term accuracy.
Another element in the TLE is the orbital inclination. This describes how the orbital plane is tilted with respect to the Earth. It is important for calculating the amount of sunlight that the solar powered spacecraft is exposed to during each orbit. The inclination should be nearly constant and very close to that of the ISS.
The plot above shows a flock of CubeSats released from the ISS in May along with the July flock. The inclination in the TLE files converges on a common solution for the spring flock 46 days after release. The inclination for the summer flock converges after 41 days and then matches that of the spring flock. Because the satellites were all deployed from the same platform they have very similar orbits and are subject to the same perturbing forces that influence the orbital elements on timescales of days to weeks.
In addition to subtle changes for all objects on short time scales there is also a small longer term change in the values that depends on how long the object has been orbiting. The above plot shows two of the 2018 satellites compared to three other satellites released in 2017. Object NM is a piece of debris that became loose from the ISS in February of 2018. The inclination of the debris changes in parallel to the seven recently deployed CubeSats with a slight offset. The 2017 CubeSats show increasing offsets for earlier deployment dates.
During the early phase of a mission this type of analysis can be used to characterize the errors in the published orbital elements to improve ground station tracking and radio reception. It can also be used to determine when the TLE files improve enough in accuracy to allow long term predictions of the evolution of the orbit.