Lagrange in the news

Queqiao and Chang'e-4


Exciting news from the BBC Website: China Moon mission lands Chang'e-4 spacecraft on far side.  The page includes a video with sinister background music as if to suggest they're the baddies (a la Drax in Moonraker).  However, the part that really intrigued me was the mention of the L2 Lagrange point - the first I've ever seen in a news story!

As I described in my post Lagrange Points  there are 5 locations in the Earth-moon-Sun plane in which - in the rotating frame of reference and taking centrifugal forces into account - there is no overall force and an object can be parked indefinitely.  One of them is just beyond the moon and is called L2.

Now the problem with landing a probe on the far side of the moon is that you can't talk directly to it: there's a big rock in the way!  So, according to the BBC article the Chinese Space Agency has parked a satellite Queqiao at L2 to relay messages.  This left me a bit confused, as you can't see L2 from Earth for the same reason you can't see the "dark" side of the moon.  I found a better article here that explained the solution.  Queqiao in fact orbits L2 in a plane that's perpendicular to the Earth-moon axis.  The pull of the Earth and the moon is nicely balanced by the centrifugal force leaving only a resultant pointing in the direction of L2.  Presumably the orbit's radius is slightly larger than the radius of the moon itself so that we can always see it from Earth.

I had a little play with the equations and found that the period of this L2 orbit rather elegantly matches that of the moon around Earth.  However, it occurred to me that the Coriolis force might complicate this picture a little.

The Coriolis force is a "fictional" force due to the rotating of the frame of reference, much like the centrifugal force.  However, unlike the centrifugal force this force is dependent not on distance from the origin, but on the components of velocity in the rotating plane.  (So, for example, if using cylindrical coordinates and assuming the z-axis corresponds to the frame's axis of rotation, then the Coriolis force is proportional and perpendicular to $\boldsymbol{v} - v_z \boldsymbol{\hat{k}}$.)

So, looking down on the solar system from above, Queqiao should always be veering to the right, and this ought to cause it trace out a loop in the Earth-moon-Sun plane in addition to the sought after loop perpendicular to the Earth-moon axis.  With a bit of reflection one can see that the periods of these two loops must match, and this information is enough to work out that at the top of its orbit Queqiao must instantaneously be tracing a curve in the Earth-moon-Sun plane that's twice the radius of it's circular orbit in plane perpendicular to the Earth-moon axis.  It's a mouthful to say in words, but the picture above should make it clear.

I didn't do the maths very thoroughly, but it's looking to me like the whole orbit should be tilted by about $30^\circ$.

If it seems like an incredible achievement that they got the satellite into this oddball orbit in the first place consider this: L2 is a Lagrange point because it's the summit of a potential hill, and this means they also have to contend with the inherent instability of the orbit.  It's a bit like helping a blindfolded friend to walk a tightrope by shouting advice from the ground!

TODO Learn Mandarin.

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