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.

Comments

Post a Comment

Popular posts from this blog

How To Make ASCII Diagrams Beautifuller

Image
I've discovered an excellent tool in asciiflow.com .  The website makes it really easy to create ASCII box diagrams like this They put these things in fruit machines you know! This is ideal for source code banners, which I think should contain helpful documentation - but most programmers think it's a good place for the COPYRIGHT information and nothing else. But wait! we can make it beautifuller... and easier to read... by replacing some of the ASCII characters with ones available in UTF-8: There!  Isn't that better?  (Although some purists may object to non-ASCII characters in your code base.) SOURCE CODE: #!/usr/bin/python2 # coding: utf-8 # + gets converted in different ways depending on it's 4 neighbours # # . N . { nsew(N,S,E,W) has bit 3 set if N in "+|<>" # W + E { nsew(N,S,E,W) has bit 2 set if S in "+|<>" # . S . { nsew(N,S,E,W) has bit 1 set if W in "+-^v" # { nsew(N,S,E,W) has bit 0 set if E in
Read more

Calculation of ECS using simple convection model

Image
Contents Intro Simple stratified model Alternative convection model Tropospheric temperature gradient Approach used in convection model Calculation of Radiative Forcing from CO₂ increase Calculation of Radiative Forcing from H₂O increase Calculation of zero-feedback ECS Effect of including feedbacks Effect of water vapour Effect of cloud cover change Effect of sea ice loss Calculation of ECS including feedbacks Limitations One more thing: Changes to Troposphere depth Intro The goal of this post is to see if we can come up with an estimate of Equilibrium Climate Sensitivity using a simple model in which the atmosphere is treated as well-mixed.  By "simple" I mean that using the model will not require any advanced mathematics or computation, but will still be realistic enough to come up with an estimate within the likely range of 2.5 - 4.0°C predicted by the IPCC Assessment Report 6 .  I will try to take as little on trust as possible and show how the results are arrived at.
Read more

Why growth is falling in all developed countries (as a long term trend)

Image
In this post I argue that economic growth is not the normal state of affairs, it is a blip caused by the discovery of a one-off resource of non-renewable energy.   The logistic equation $$ \frac{dy}{dt} = y(1-y) $$ Here's a simple model.  Humans discover an exploitable but limited resource and start consuming it.  The amount consumed, $y(t)$, is a function of time.  The general form of the equation is $\frac{dy}{dt} = \alpha y (\beta - y)$, but if you choose the right units $\alpha$ and $\beta$ both become $1$. Why should this work, in principle? Early on, the factor $1-y$ is approximately $1$ and can be ignored.  So the model states that annual consumption $dy/dt$ starts off proportional to $y$.  In other words, $y$ grows exponentially at first.  This could happen if exploiting the resource enables further exploitation of the resource.  For example, suppose a few humans are shipwrecked on an island with 1000 trees: they take ages to cut down the first tree as they are using their
Read more