The Lagrange points in the Earth-Moon system - Esa

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**The** **Lagrange** **po ints**

Introduction:

Matthias Borchardt

Tannenbusch Secondary School, Bonn

borchardt. matthias@t-onl**in**e.de

What forces are felt by a space probe that is travell**in**g unpowered **in** **the** vic**in**ity of **the** **Earth** and

**the** **Moon**?

First **the**re are **the** attractive forces of **the** two celestial bodies that, accord**in**g to **the** law of

gravitation, act upon **the** probe to a greater or lesser extent.

Added to this is a third force that only comes **in**to play once you stop view**in**g **the** **system** of masses

as static, but **in**stead take **in**to account **the** dynamics of **the** rotation of **the** **Earth** and **the** **Moon**

around a shared centre of gravity. This movement creates a centrifugal force - a force that is

directed away from **the** centre of gravity.

**The**se three forces at five **po ints**

L3, L4 and L5. **The**y are also called **the** ‘park**in**g lots’ of **the** **Earth**-**Moon** **system** because a space

probe that is positioned exactly at **the**se positions rema**in**s stationary **in** relation to **the** two celestial

bodies.

Even small deviations from **the**se special **po ints** quickly cause complicated behaviour: At L1, L2

and L3 we can see that **the** space probe is drift**in**g away - although **in**itially this happens at a very

low speed such that it should be possible to ma**in**ta**in** **the** position for a longer period of time with

m**in**imal course corrections.

A probe behaves differently **in** **the** vic**in**ity of L4 and L5: it circles **the**se **po ints** on loop

and a course correction is not necessarily required as **the** curves always rema**in** **in** **the** vic**in**ity of **the**

**Lagrange** **po ints**.

**Earth**

Centre of gravity

**Moon**

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**The** centre of gravity:

**The** **Earth** and **the** **Moon** rotate around a common centre of gravity, **the** position of which can be

calculated as follows:

M-xM = m-xm

if r = xM + xm **the**n

M-xM =m-(r-xM) or M-(r- xM ) =

M -xM = mr -m-xM M-r-M-xm =m-xm

(M + m)-xM = m-r M r = (m + M)-xm

If M = 5.97 • 10 24 kg , m = 7.35 • 10 22 kg and r = 384 400 km (half-way between **the** **Earth** and **the**

**Moon**) **the**n XM = 4675 km.

This means that **the** centre of gravity S is still with**in** **the** terrestrial globe.

For **the** purpose of **the** overview we will however always show S as be**in**g outside of it.

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**The** centrifugal force:

Let’s consider **the** centrifugal acceleration: , where x is **the** distance from **the** centre of

gravity S.

**The** angular speed of this rotation is:

( r - distance between M and m)

This formula should be derived:

Movement of **the** **Earth** around S Movement of **the** **Moon** around S

**The** gravitational force acts as radial force

if **the**n if **the**n

equal

Of course we must obta**in** **the** same value for **the** angular speed of **the** **Earth** as for **the** **Moon**.

Stated differently: both celestial bodies require **the** same period of time to orbit **the** common centre

of gravity.

**The** orbit period yields:

= 27.3 days. That is **the** sidereal month

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**The** gravitational forces:

**The** gravitational accelerations yield **the** follow**in**g:

**The** **Earth** exerts an acceleration of on a body.

**The** **Moon** exerts an acceleration of on a body.

x is always **the** distance to **the** respective centre of mass.

**The** **Lagrange** po**in**t L1:

**The** orig**in** of our coord**in**ate **system** should always be **the** centre of gravity S:

With **the** aid of **the** draw**in**g **the** three accelerations yield: (if: r = xM + xm)

At **Lagrange** po**in**t L1 **the** accelerations should add to zero.

**The**refore aG1 = aG2 + az applies here.

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(Stated differently: In L1 **the** gravitation of **the** **Earth** cancels out **the** gravitation of **the** **Moon** and

**the** centrifugal force.)

**The** equation would have to be solved after x1 if you wanted to

calculate **the** position of **the** Langrange po**in**t.

This is ma**the**matically very complex. **The**refore we should take an easier route **in** order to

determ**in**e L1, that is, we should enlist **the** aid of a spreadsheet program.

This type of software is used to help calculate both **the** acceleration aG1 and **the** sum

applies

. **The** **Lagrange** po**in**t L1 is located at **the** position of **the** x1 value for which **the** follow**in**g

This method with **the** aid of a spreadsheet delivers **the** follow**in**g result for L1:

x1 = 321.688.900 m, which approximates to x1 = 321.689 km.

**The** **Lagrange** po**in**t L2:

With **the** aid of **the** draw**in**g **the** three accelerations yield: (if: r = xM + xm)

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At **Lagrange** po**in**t L2 **the** accelerations should add to zero.

**The**refore aG1 + aG2 = az applies here.

(Stated differently: In L2 **the** gravitational forces of **the** **Earth** and **the** **Moon** cancel out **the**

centrifugal force.)

**The** spreadsheet method delivers **the** follow**in**g result for L2:

x2 = 444.260.800 m, which means approximately x2 = 444.261 km.

**The** **Lagrange** po**in**t L3:

**The** three accelerations yield: (if: r = xM + xm)

At **Lagrange** po**in**t L3 **the** accelerations should add to zero.

**The**refore aG1 + aG2 = az applies here.

(Stated differently: In L3 **the** gravitational forces of **the** **Earth** and **the** **Moon** cancel out **the**

centrifugal force.)

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This method with **the** aid of a spreadsheet delivers **the** follow**in**g result for L3:

x3 = -386.347.900 m, which means approximately x3 = -386.348 km

**The** **Lagrange** **po ints** L4 and L5:

At **the** **Lagrange** **po ints** L4 and L5

**the**re are no longer parallel or anti-parallel, **the**y must **in**stead be added toge**the**r accord**in**g to **the**

rules of vector addition.

Gravitational force of **the** **Earth**

Centrifugal force

Gravitational force of **the** **Moon**

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At **the**se **Lagrange** **po ints**

to **the** amount of **the** centrifugal force but which is directed aga**in**st it. (Seen by an outside observer

one would have to say: **the** two gravitational forces toge**the**r generate **the** radial force that an orbit

with **the** orbit period T=27.3 days around S generates.)

A complex ma**the**matical analysis yields a surpris**in**g result for **the** position of **the** **po ints** L4 and

L5: **The** centres of mass of M and m and **the** **Lagrange** po**in**t L5 form an equilateral triangle with

**the** edge length r.

**The** coord**in**ates of **the** **Lagrange** po**in**t L5 can **the**refore be obta**in**ed through simple geometric

exam**in**ation (e.g. if is **the** height **in** an equilateral triangle with edge length r ):

and correspond**in**gly for L4

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Now we want to prove that **the** three forces (accelerations) **in** L5 actually do add to zero if we

select L5 as a corner of **the** equilateral triangle described.

**The** cos**in**e rule means:

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Now we should calculate **the** centrifugal acceleration az :

, where d is **the** distance from L5 to **the** centre of gravity. This distance can be calculated

with **the** aid of Pythagoras’ **the**orem as **the** coord**in**ates of L5 relative to **the** centre of gravity are

already known:

Thus if **the**n:

That is **the** same value as **the** one result**in**g from aGi and aG2.

This shows that **the** three forces **in** L5 actually do cancel each o**the**r out.

**The** movement of a space probe **in** **the** vic**in**ity of **the** **Lagrange** **po ints**:

How does a space probe behave when we position it **in** **the** vic**in**ity of a **Lagrange** po**in**t?

We cannot carry out a ma**the**matical analysis of **the** movements as it is exceed**in**gly complex.

But us**in**g **the** simulation program ‘lagrange.exe’ important characteristics of **the** **Lagrange** **po ints**

can be **in**vestigated very easily.

**The** program calculates **the** two gravitational forces and **the** centrifugal force exerted on **the** location

of a spacecraft and thus determ**in**es **the** direction and amount of acceleration and speed of **the** craft

and hence its new position after a period of time Δt (Euler-Cauchy method).

However only those paths that lie on **the** plane of **the** **Earth**, centre of gravity and **Moon** are

calculated. It is **the**refore not possible to follow **the** path of a probe that has been placed above or

below this plane.

From a teach**in**g po**in**t of view this reduction is especially advantageous as **the** relationships

between **the** forces are still able to be imag**in**ed to some degree. **The**re is an additional advantage

due to **the** possibility of mapp**in**g **the** path of **the** space probe onto **the** surface of **the** gravitational

potential (‘potential surface’) and to draw **the** entire th**in**g as an axonometric projection. This

creates a very vivid impression of **the** topology of **the** energetic relationships **in** **the** rotat**in**g **Earth**-

**Moon** **system**.

**The** coord**in**ate **system** **in** which **the** space probe’s movements are calculated and drawn rotates

around **the** common centre of gravity toge**the**r with **the** **Earth**-**Moon** **system**. This means that **the**

**Earth** and **the** **Moon** are rest**in**g **in** this reference **system**, which allows a clear representation of **the**

space probe’s path as we can see how it moves only **in** relation to **the** **Earth** and **the** **Moon**.

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**The** simulation shows that L1, L2 and L3 are apparently not stable **po ints** of equilibrium.

Topologically **the**y are similar to a saddle.

In contrast, L4 and L5 are (very flat) hills and one could assume that a space probe would ‘slide

down’ **the**se hills and would **the**n drift away over time. Surpris**in**gly this does not happen, however,

because as a result of slid**in**g down **the** space probe ga**the**rs speed and, due to **the** rotation around **the**

centre of gravity, a Coriolis force is created that so strongly warps **the** path that looped paths are

created around **the** **Lagrange** po**in**t. In **the** process **the** space probe only moves away a certa**in**

distance - spatially it thus rema**in**s almost stationary **in** relation to **the** **Earth** and **the** **Moon**. Course

corrections are normally not required.

**The**refore L4 and l5 are ideal **po ints** for a space station that has

and **Moon** over a long periods of time.

**The** follow**in**g screenshots show **the** path of a space probe that has been placed at various **po ints** on

**the** **Earth**-**Moon** plane at a speed of zero (**in** relation to **the** coord**in**ate **system** with which it is

rotat**in**g).

A little to **the** right of L1 **the** space probe drifts towards **the** **Moon**, and a little to **the** left of L1

towards **the** **Earth** - **the** saddle-like topology of **the** surround**in**g area prevents **the** space probe from

mov**in**g forwards or backwards **in** spite of **the** Coriolis forces be**in**g exerted.

In **the** vic**in**ity of L5 **the** situation looks completely different: Here we have a hill ra**the**r than a

saddle, and **the** space probe - driven by Coriolis forces - makes looped circuits over it.

Note: **The** saddle shape and **the** hill shape of **the** **Lagrange** **po ints** can hardly be seen on

axonometric projection of **the** three dimensional image of **the** gravitational potential as it is only

very weak.

Matthias Borchardt

Tannenbusch Secondary School Bonn

borchardt. matthias@t-onl**in**e.de

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Start at L1 (1km to **the** right of **Lagrange** po**in**t)

PATH

Launch position Launch speed

Start at L1 (1km to **the** left of **Lagrange** po**in**t)

PATH

Launch position Launch speed

slow

slow

Start

Start

Stop

Simulation

Stop

Simulation

POTENTIAL

Axonometric

projection

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Delete

fast

Delete

fast

Time

Time

**Moon**

orbits

Days

**Moon**

orbits

Days

Plane

with **Moon**

without **Moon**

Rotation

around **the**

x axis

y axis

z axis

Zoom

Info

Data

End

POTENTIAL

Axonometric

projection

Plane

with **Moon**

without **Moon**

Rotation

around **the**

x axis

y axis

z axis

Zoom

Info

Data

End

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Path **in** **the** vic**in**ity of L5

PATH

Launch position Launch speed

PATH

Launch position Launch speed

slow

slow

Start

Start

Stop

Simulation

Stop

Simulation

POTENTIAL

Axonometric

projection

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Delete

fast

Delete

fast

Time

Time

**Moon**

orbits

Days

**Moon**

orbits

Days

POTENTIAL

Axonometric

projection

Plane

with **Moon**

without **Moon**

Rotation

around **the**

Plane

x axis

y axis

z axis

Zoom

Info

Data

End

with **Moon**

without **Moon**

Rotation

around **the**

x axis

y axis

z axis

Zoom

Info

Data

End