Queqiao was launched by China on May 25, 2018 towards the far side of the Moon where it ultimately took up shop at the Earth-Moon L2 point in a halo orbit. It’s mission is to provide communications for the first landing on the Moon’s far side by Chang’e 4.

Queqaio will communicate with the lander and rover using X-band and relay the signals to Earth on S-band. Queqiao is not the first Chinese mission to visit the Earth-Moon L2 point. Chang’e 2 hung out at the E-M L2 for a time then departed for asteroid 4179 Toutatis.

A few years later after completing it’s primary mission of testing a lunar return capsule Chang’e 5 T1 arrived at the E-M L2 and remained there for a couple of months before entering lunar orbit with a present period of 131 minutes and an altitude of 240km presumably testing mission dynamics for future Chinese lunar missions.

Let’s explore what a halo orbit is and determine if we can observe anything interesting about it’s characteristics.

## Anatomy of a Halo Orbit at the Earth-Moon L2 Point

In 1772 Lagrange wrote on the ‘three body problem’ and discovered mathematically the existence of what became to be known as Lagrangian points. Some of the Lagrange points are not a stable place to be or orbit particularly the L1, L2 and L3 points. Fortunately, with a bit of station keeping fuel and a really good understanding of how to stay in a semi-stable orbit around the L2 Lagrange point one can position a satellite there. NASA studied this and in a paper published in December 1970, The Utilization of Halo Orbits in Advanced Lunar Operations, by Dr. Robert W. Farquhar. This post draws heavily on the wonderful illustrations Farquhar provided to bring the physics to life .

Having somewhere semi-stable to place a communications satellite is good as the far side of the Moon is a place where you can not directly communicate with Earth. If you place a satellite in halo orbit around the E-M L2 point and design it just so you can use a minimum of station keeping propellant to keep a relay satellite in a useful position for an extended period of time.

So we now have a place to put a relay satellite that can constantly see the far side of the Moon. From Earth the relay satellite will appear to slowly rotate around the Moon and always be visible to us and the far side of the Moon if done right. This is a halo orbit. it literally traces out a halo around the Moon without being blocked by it.

The key to understand what a halo orbit is versus any old ‘Lissajous’ orbit around the E-M L2 point is that a halo orbit is designed not to pass behind the Moon. There is a series of special cases in the solution of the orbital dynamics that ensures that if given parameters are chosen that the satellite will never pass behind the Moon and have it’s view of Earth blocked. Thus producing an optimal communications solution. The trick is picking the size of the orbital radii to minimize fuel consumption to maintain the halo orbit and still accomplish the mission objectives.

## Theory of Queqiao’s Halo Orbit

I have no illusions of trying to fulling understand Dr. Farquhar’s work in great detail. However, his paper does lend itself to understanding the basic principles of the orbital dynamics in play here. A few key points should be observable and measurable with amateur means to determine the general characteristics of the halo orbit Queqiao maintains. The characteristics of greatest interest are:

- Period of the halo orbit.
- Shape and semi-major and semi-minor axis dimensions.
- The angle the orbit is inclined on with respect to the Earth / Moon plane.

First lets establish a reference geometry as noted in the graphic below.

### Period

The period of the halo orbit is the time it takes the spacecraft to complete one revolution around the E-M L2 point. Measurement can be accomplished by taking accurate positional measurements of the spacecraft over a prolonged period of time. Even observation of a partial orbit should allow for a reasonable approximation of the halo orbital period. The period may also be measurable from Doppler data, which will be discussed more below.

**Shape of Halo Orbit as Seen for Earth**

The shape of the orbit as seen from Earth is very important as it needs to be such that the Moon doesn’t get in the way. As you will note in the plot below there are two radii identified, Ay and Az. For every value of Ay >32871km, there is a corresponding value of Az. The periods of the y and z axis become equal and when this happens the path as seen from Earth will never pass behind the Moon.

The graph below illustrates the relationship of Ay and Az.

Measurement of Ay and Az values should be possible with careful positional observation and measurement and the use of basic trigonometry to approximate the values.

### Angle of Halo Orbit Plane

The angle of the plane of the halo orbit can be thought of as the tilt that the orbit is at with respect to the plane of the orbit of the Earth and Moon. As you can see from the figure below the satellite will be closer to Earth at some points in the halo orbit than others.

Looking down from above through the plot below shows the definition of Ax as the radii of the ellipse that describes this angle of the halo orbital plane.

## Observing Queqiao at the Earth-Moon L2 Point

As of November 2018, Queqiao is transmitting on 2234.52MHz. This allows the observer to do two things:

- Measure the Doppler shift of the satellite’s beacon. This could allow for analysis about its orbit and also given some insight into operational matters.
- Measure the time and position of the satellite with a directional antenna and plot out the co-ordinates.

I’ve been conducting both on and off for some time and have again (May 2019) taken up the observations to add to earlier data.

The goal of the experiment is to:

- determine the period of the halo orbit. Monitor that period over time and see if it changes,
- determine the size of Ay and Az from observation,
- determine the size of Ax.

Stay tuned!