Help understanding BepiColombo's weak capture at Mercury's L1 and need for delta-differential one-way range measurements

Question

This four-minute BBC podcast Mission to Mercury: BepiColombo spacecraft ready for launch highlights several challenges of the BepiColombo mission, including its orbital-mechanical aspects and is worth a listen

The BBC's James Menendez has been speaking to Elsa Montagnon, the operations manager for this European Space Agency mission - she's been working on the project for more than a decade.

You can hear her discuss the mission further in the Planetary Society's Planetary Radio podcast Return to Fiery Mercury With BepiColombo.

She is also the author of the cover article in the Planetary Report's Voyage to Mercury The BepiColombo Mission Prepares for Launch. This issue is also the first to be completely open-access and downloadable, and to have Emily Lakdawalla as its editor.

Montagnon's energetic descriptions of the mission have got me really interested in this orbit and reading further about it. I've run across the paper Interplanetary navigation along the low-thrust trajectory of BepiColombo (Acta Astronautica 59, Issues 1–5, July–September 2006, Pages 284-293)

Abstract

For BepiColombo's five-year journey into the inner solar system, a combination of low-thrust arcs and six flybys (one at Moon and Earth, two at Venus and two at Mercury) will be used to reach Mercury with low relative velocity. At arrival a gravitational capture approach is foreseen, in which the Sun perturbation is exploited to get weakly captured around Mercury for a number of orbits. This trajectory imposes severe constraints from a navigational point of view. Very precise navigation is required due to the low flyby altitudes planned for Venus (300 km) and Mercury (200 km) and the level of accuracy needed for the final arrival through the vicinity of the Sun–Mercury L1 point. Besides that, the solar plasma effect severely degrades the quality of the radiometric measurements near superior solar conjunctions, which are more frequent for missions to the inner solar system. Moreover, perturbations, as the ones introduced by momentum wheel desaturation burns, entry into safe modes or solar radiation pressure, must also be taken into account. Delta-differential one-way range measurements are found to be required in periods of poor orbit determination prior to some gravity assists. Nevertheless, if a safe mode is triggered at a critical moment that produces a change in velocity in an unfavourable direction, the mission could be jeopardised. To avoid that risk, an increase in the flyby altitude and possibly a partial or total redesign of the trajectory to avoid flybys near solar conjunctions are considered.

BepiColombo will take several years to get to Mercury, and so there will be plenty of time for more questions about its orbit. Here I would just like to ask about certain aspects of this paper.

Question: In the context of this mission to Mercury so close to the Sun, what exactly does "weak capture" mean, and why might "delta-differential one-way range measurements" be necessary at times? If I understand correctly, these are not so commonly used in deep space missions, even during flyby maneuvers.

Answer 1

"Weak capture" means that the spacecraft will enter the Mercury's gravitational sphere of influence along a weak stability boundary. These weak stability boundaries are one of the key mathematical developments of the N-body (N>2) with regard to space exploration.

The entry points to this weak capture are rather narrow, in terms of both position and velocity. But done right, the exit points are also rather narrow. Without subsequent maneuvers, the spacecraft would follow a seemingly chaotic orbit about Mercury until it finally passes through one of those very narrow exit points. (With BepiColombo this will take about five orbits.) BepiColombo will make subsequent maneuvers so that it is truly captured. But because it will be in a weakly captured orbit, those maneuvers will require significantly less ΔV than if it followed a more traditional two body problem approach.

To accomplish this weak capture, BepiColombo's position and velocity state relative to Mercury and the Sun need to be measured very precisely. This is where "delta differential one-way ranging" (ΔDOR) comes into play. Delta differential one-way ranging is a technique that has been used by NASA since 1979 with the Voyager spacecraft, and by ESA (with the help of NASA) since 1986. ESA now has widely dispersed ground stations; it no longer needs NASA's help.

A single ground station can measure the distance (range) to a remote spacecraft by measuring the round trip time of a signal sent from the ground station to the spacecraft, which the returns the signal back to the ground station. The Doppler shift in the return signal gives an even more precise range rate measurement, the time derivative of the distance. It's impossible to assess a spacecraft's orbit (six translational degrees of freedom) with only two measurements. These measurements have to be accumulated over time to yield an estimate of the orbit.

One would think that the direction in which the ground station antenna is pointing would add two additional measurements. This is not the case. For example, the extremely narrow half power bandwidth of NASA's 35 meter Ka band antenna of ~1 arc minute corresponds to a 57000 km uncertainty in the cross range position at Mercury's distance. That makes the antenna pointing direction pretty much useless with regard to orbit determination.

There is a way to get an extra degree of freedom in the measurements if two widely separated ground stations are used with a properly outfitted spacecraft. I'll start with one way ranging. The range measurement made by a single ground station is based on the roundtrip time of a carefully constructed signal originating from the ground station. Suppose the spacecraft itself sends a carefully constructed, time-tagged signal. This would, in theory, provide a mechanism to determine the one-way distance to the spacecraft.

Unfortunately, the practical aspects of space travel demands that mass be kept to a minimum. One consequence is that the timing mechanisms used on spacecraft aren't nearly as good as are those available to ground stations. This makes one way ranging by a single ground station of little use; it adds no new information and the roundtrip timing is much more accurate. Suppose that two widely separated ground stations observe the spacecraft at the same time. This, in theory, does two things. One is that it drastically reduces errors induced by the spacecraft. The other is that it gives a new measurement, the angle between the baseline between the ground stations and the line to the spacecraft. This is differential one-way ranging.

Once again reality triumphs over theory. The errors in the timing mechanisms at the ground stations and that the signals traverse through the ionosphere mean that the angle deduced from differential one-way ranging is extremely noisy. Delta differential one-way ranging adds one final twist on top of differential one-way ranging. Instead of just observing the spacecraft, both ground stations alternate between looking at a spacecraft and at a known quasar that is within a handful of degrees from the spacecraft. With over 200,000 known quasars, finding one with a very precisely known right ascension and declination that is close to that of the spacecraft is almost always possible. The paired observations of the quasar provides information on the ionosphere and timing issues, and this makes the deduced angle between the baseline and the line to the spacecraft a very useful measurement.