Orbits considered for AXIOM (not to scale). Credit: AGI STK
Lagrangian points in a two-body system with one body (yellow) more massive than the other (blue). Credit: Wikipedia
Illustration of ARTEMIS libration orbits. Credit: NASA/Goddard
View of the L1 orbit from the Earth looking towards the Moon. Credit: AGI STK
Top view of the L1 orbit. Credit: AGI STK
Side view of the L1 orbit. Credit: AGI STK
AGI logo
Many miles away
In order to observe a big part of the Earth's magnetosphere, the spacecraft must spend the majority of its orbit sufficiently far away from the planet. For example, AXIOM's X-ray telescope has an angular field of view (FOV) of 10o. In order to observe a region of the size of the Earth, the spacecraft must be at least 12 Earth radii (ER) away. Of course, this distance should be bigger in order to image the entire magnetosphere. A certain number of orbits have been proposed accordingly for the AXIOM mission, from circular equatorial orbits to elliptical polar orbits, to the Lissajous orbit around the Lagrange L1 point of the Earth-Moon system:
Polar elliptical 2—30 ER, 63o inclination
Polar elliptical 6—30 ER, 63o inclination
Equatorial circular 30 ER
Equatorial circular 30 ER (LGA)
Equatorial circular 60 ER (LGA)
Lissajous orbit at Earth-Moon L1 (~53 ER)
Let us try to understand what the names of the different orbits mean. For the elliptical orbits, we can specify their perigee and apogee, that is the smallest and biggest distance to the Earth. For example, the perigee for the first orbit is 2 ER and the apogee is 30 ER. The inclination refers to the angle between the orbit's plane and the Earth's equatorial plane. The word “polar” usually refers to an inclination of 90o but here it just means “high inclination”.
The equatorial orbits are - as their name suggests - in the Earth's equatorial plane. In other words, they have an inclination of 0o. For the circular orbits, the perigee and apogee are the same, so we only specify the radius of the orbit. LGA stands for Lunar Gravity Assist and refers to a careful orbit design using the Moon's gravity to save up fuel.
Finally, the Lissajous orbit around the Lagrangian point L1 is probably the most unusual and needs to be presented in more details.
Kidney beans in space
Lagrangian points are the stationary solutions to the circular restricted three-body problem. They are the points of equilibrium between the gravitational forces of two massive bodies and the centripetal force that is needed for a third body of negligible mass to rotate with them. There are five Lagrangian points, all in the orbital plane of the two bodies.
In the case of the AXIOM mission, we consider the Earth-Moon system and the corresponding L1 point. Orbits around the Lagrangian points L1, L2 and L3 are dynamically unstable, therefore the spacecraft will need to perform regular station-keeping to stay near those points.
The Lissajous orbit around the L1 point is a quasi-periodic kidney-shaped trajectory with components in the orbital plane, as well as in the plane perpendicular to it. If correctly designed, it can minimise the time spent in the shadow of the Moon. Such space missions as Herschel or Planck have already used Lissajous orbits for the Sun-Earth system. But the first spacecraft to orbit the Earth-Moon's L1 and L2 points were two of the five probes from NASA's THEMIS mission as part of the extended mission called ARTEMIS [5] aimed at studying the interaction of the Moon with the solar wind.
The L1 and L2 points are located at a distance of about 61,300 km from the Moon's surface and it takes about 14 days for the ARTEMIS probes to complete one revolution around them.
Quite a challenge
This Lissajous L1 orbit will be considered in this report because it presents a number of advantages, although there are always trade-offs in the process of orbital selection. A certain number of factors are taken into account when designing an orbit for a spacecraft. The major ones are given in the table below [1]. For most of the factors, the orbit (or multiple orbits) that performs the best is given.
Factor
Best choice
Viewing efficiency
Polar elliptical 2—30 ER
Uninterrupted observing time
Lissajous orbit at Earth-Moon L1
Radiation
Equatorial circular 30 ER
Equatorial circular 30 ER (LGA)
Equatorial circular 60 ER (LGA)
Lissajous orbit at Earth-Moon L1
Launcher availability, compatibility, mass in orbit
Lissajous orbit at Earth-Moon L1
Eclipse duration and frequency
Lissajous orbit at Earth-Moon L1
Thermal control
Lissajous orbit at Earth-Moon L1
Orbit raising and maintenance
All except Lissajous L1
Ground station coverage
-
Solar array degradation
Lissajous orbit at Earth-Moon L1
Equatorial orbits provide good global views of the magnetosphere but make it difficult to keep the Earth out of the field of view (FOV). The telescope should not be directed towards the Earth because our planet appears too bright when observed in X-rays. Elliptical polar orbits do not have this disadvantage but make it necessary for the spacecraft to cross the Earth's radiation belts where highly energetic particles are present. Both types suffer from solar eclipses caused by the Moon or the Earth. The Lissajous L1 orbit (we will call it simply the L1 orbit) can potentially reduce the amount of time spent in the Moon's shadow and is located far away from the radiation belts. It also enables the Vega launcher [3] to put a bigger mass into orbit. The main disadvantage, however, is its very average observing efficiency. The L1 orbit is a challenging but exciting trajectory and will be the primary choice for AXIOM.
Getting to the Lagrange L1 point
In order to model AXIOM's trajectory, a software package called AGI STK [4] has been used. STK facilitates modelling and analysis of aircraft missions, electronic systems, missile defence and space missions. It is particularly useful for mission planning since it can help analyse and visualise different stages of a mission, from launch to various orbital manoeuvres and trajectories.
The goal of the study was to simulate the launch of the spacecraft into a Lissajous L1 orbit. According to the proposal document, the Vega launcher will first put AXIOM on a parking orbit of 200 km perigee, 1,525 km apogee and 5o inclination. It will then perform a series of apogee-raising manoeuvres in order to come closer to the Moon. Eventually, the spacecraft will be captured into a lunar orbit which can then be transformed into an L1 orbit. In this preliminary study, we will consider a more direct and simple lunar injection that will lead to one possible L1 orbit. The approach is based on the Earth-Moon L1 Trajectory Tutorial by the Astrogator's Guild [6].
The pictures in the left column show different views of the obtained orbit. The side view is taken from the ecliptic plane, the top view is from above the ecliptic plane and the third view looks from the Earth towards the Moon. The trajectory of the spacecraft (shown in light blue) corresponds to 2 revolution around the L1 point which required 4 station-keeping manoeuvres and lasted 53 days.
After simulating a trajectory, we can estimate various parameters relevant to mission design:
Frequency of station-keeping: how often should corrective manoeuvres be applied to stay in orbit
Station-keeping delta-V: amount of effort needed to maintain the L1 orbit
Fuel usage: minimum amount of fuel required
Total eclipse time: total time spent in the shadow of the Earth or the Moon during the 2 years of nominal mission
Maximum eclipse time: maximum time continuously spent in eclipse
Parameter
Value
Comment
Frequency of s\k
Every 12 days
Estimated to every 7 days in the AXIOM proposal
Delta-V
24 m/s
Estimated to 70 m/s for the 4-year extended mission in the AXIOM proposal
Fuel usage
4.69 kg
16% of the fuel tank's capacity
Total eclipse time
45 hours
5 hours per eclipse on average. Estimated to a few hours per eclipse in the AXIOM proposal
Max. eclipse time
8.6 hours
Most of the numerical values obtained here are only theoretical minima. By applying safety margins on the different parameters, we may achieve more reliable estimations. The simulation of a more typical Lissajous orbit may be considered in future studies in order to minimise the time spent in the shadow of the Moon and use the successful trajectory of ARTEMIS which has been operating near the L1 point without major difficulties.
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