## Program 20: The formation of Kuiper-belt binaries.

### Introduction and Purpose

In the article "The formation of Kuiper-belt binaries through exchange reactions" in Nature volume 427 of February 2004 at page 518-520 the binary objects in the Kuiper belt are explained through exchange reactions. Specific is explained what those exchange reactions are. The purpose of the program KUIPER.BAS is partly, to challenge this explanation and or to place this explanation in a different context.
I agree that you can explain Kuiper-belt binaries by introducing a third smaller objects, but the reason IMO is strictly Newton's law and what I call the concept of asymmetry.

In order to see the listing of the program select: kuiper.htm
In order to get a copy of the program "kuiper.bas" select: kuiper.zip. This zip file also contains the executable: "kuiper.exe"
There is also a program in Visual Basic 5.0 available. Select: VB Kuiper.htm For executable of the VB program select: VB Kuiper.zip

The program shows a simulation of four objects m1, m2, m3 and m4. m1 = the Sun, m2 and m3 are two "planets" of equal mass and m4 is like a moon.

The program simulates 9 different configurations:

1. 3 object simulation "
2. 4 object simulation, collision"
3. 4 object simulation, capture of m4"
4. 4 object simulation, ejection of m4"
5. 4 object simulation, capture and ejection of m4"
6. 3 object simulation, circle "
7. 3 object simulation, chaotic "
8. 2 object simulation of m2 and m3 "
9. 3 object simulation, chaotic "

### Functional Description and Operation

The program starts with a menu of all the 9 simulations, 5 standard and 4 special.
For the standard simulations enter a value between 1 and 5.
For the special simulations enter a value between 6 and 9.
In order to stop a simulation select ESC (Escape) key.

The top part of the of the display shows two instances of the trajectories of m2(white) ,m3(yellow) and m4(cyan) at different scales
The bottom part of the display shows the trajectories of m2(white) and m4(cyan) relative to m3.(yellow dot)

### Simulation 1: 3 Object simulation

This simulation shows two objects m2 and m3 (two planets) as a binary system moving around m1 (the Sun, not shown)

The simulation shows that when a real binary system of only two objects, that such a configuration is very stable. I call such a configuration symmetric.
For an even more stable configuration: try simulation 6. This configuration shows a circle.
However this can also be different. Try the simulation 7. What this simulation shows (specific the bottom part) is that the path becomes almost chaotic. The reason is, partly, that the revolution time is much longer than before. Secondly the influence of m1 i.e. the Sun. In order to understand you should observe the top part of the display which shows the absolute path in two instances. The general path of the two binaries (centre of gravity) is in a circle around m1, the Sun. However in this case the two masses m2 and m3 are affected differently, because m3(yellow) is more on the inside of this circle than m2(white). As such m3 is differently affected by m1 (is closer) than m2 i.e. the configuration is slightly asymmetric. That explains the chaotic behaviour.
If you compare this with simulation 6 than you will see that both masses spent equal time at each side of this circle.
To observe the influence of m1, try simulation 8. This simulation has the same initial conditions as simulation 7 but the mass of m1 is 0. The result is that the centre of gravity of the two objects m2 and m3 moves in a straight line, and the configuration is stable.
For a different set of initial conditions try simulation 9

### Simulation 2: 4 Object simulation, collision of m4

This simulation shows two objects m2 and m3 (two planets) and a smaller object m4 (moon) moving around the m1 (the Sun). In the initial configuration it is assumed that there is no physical relation between the three objects m2, m3 and m4.
The purpose of simulation 2 is to demonstrate what can happen when the two objects m2 and m3 meets each other, but also when at the same time a smaller object is involved.
The simulation shows that one possibility is that the smaller object m4 will collide with one of the larger masses (in this case m2) and merge. After the collision the mass of m2 becomes equal to m4 + m2.

### Simulation 3: 4 Object simulation, capture of m4

This simulation shows two objects m2 and m3 (two planets) and a smaller object m4 (moon) moving around the m1 (the Sun). In the initial configuration it is assumed that there is no physical relation between the three objects m2, m3 and m4.
The purpose of simulation 1 is to demonstrate what happens when only the two objects m2 and m3 meets each other.
The purpose of simulation 3 is to demonstrate what can happen when the two objects m2 and m3 meets each other, but also when at the same time a smaller object is involved.
The simulation shows in a nice way that one of the masses (in this case m2) can capture this smaller mass and becomes a moon of m2.
If you observe this simulation for a longer time you will see that the two masses m2 and m3 will meet again. You can predict that by observing the parameter dr23 which shows the distance between the position of m2 and m3.

### Simulation 4: 4 Object simulation, ejection of m4

This simulation shows two objects m2 and m3 (two planets) and a smaller object m4 (moon) moving around the m1 (the Sun). In the initial configuration it is assumed that there is no physical relation between the two objects m2 and m3. For m4 this is not the case. m4 and m3 form a binary system i.e. m4 is a moon of m3
The purpose of simulation 3 is to demonstrate what can happen when the two objects of equal mass, m2 and m3 meets each other, with the special condition that m3 has a close companion m4.
The simulation shows in a nice way that when the two masses m2 and m3 meet each other for the first time they become a binary system. When they meet each other for the second time they become more closely coupled and m4 is ejected.

### Simulation 5: 4 Object simulation, capture and ejection of m4

This simulation shows two objects m2 and m3 (two planets) and a smaller object m4 (moon) moving around the m1 (the Sun). In the initial configuration it is assumed that there is no physical relation between the three objects m2, m3 and m4.
The purpose of simulation 5 is to demonstrate what can happen when the two objects m2 and m3 meets each other, but also when at the same time a smaller object is involved.
Simulation 5 is quite remarkable. In fact this simulation is a combination of simulation 3 and simulation 4. The simulation shows that when the three masses m2, m3 and m4 meet each other more or less m4 is first captured and immediate ejected. The result is that the two masses m3 and m4 become a binary system.

### Reflection

The simulation makes a clear distinction between two types of conditions. On one hand you have two objects of equal size, not physical connected, which meet each other and on the other hand you have two objects of equal size, not physical connected, which meet each other with the addition that at the same time also a third object is involved.
The first situation is symmetric and the second situation with the addition of a smaller object makes the configuration asymmetric.
The behaviour of an asymmetric configuration is quite different. If you are lucky the two equal sized objects can become directly a binary system. It is also quite possible that one of the larger objects become a binary with the smaller object. In a follow up encounter with a larger objects the two larger objects can become a binary system, while the smaller object is ejected.

Left over the question how often do two objects of equal size, not physical connected, which meet each other? Starting point is that all Kuiper belt objects move in circles around the Sun accordingly the Law: vn = SQRT(m1/rn) with n equal to 2, 3 and 4. (rn is the distance of object n with the Sun). The question is which fluctuations are possible. The larger the fluctuations the easier the chance that those objects will meet each other.
The chance of three objects meeting each other depends largely on the fact how much more smaller objects there are compared with the larger objects. This can easily be 20 times more. If that is the case the chance of three objects meeting each other can be quite high over a long period of time.

In the article the formation of Kuiper Belt binaries is explained "through tidal disruption of one object followed by coagulation of fragments during a close encounter with the other" This is maybe possible, but IMO not required.
A different possibility is "a giant impact, where collision debris coagulates into a moon". Again not required. IMO planets can also form moons in a different way.

### Technical Description

The program KUIPER.BAS is based upon the program SLINSHOT.BAS: Sling shot effect
The program consists of four parts: Initialisation, Inner loop, Outer loop and Finalization
• In the initialisation section the initial parameters are established.
• In the inner loop the parameters: (ax2,ay2), (ax3,ay3), (vx2,vy2), (vx3,vy3), (x2,y2), (x3,y2) are calculated using Newton's law. The inner loop is performed 20 times (Par DISPCOND)
• The outer loop is performed 50000 times. Each time the display is updated.
• In the Finalization section the display is updated with the final results.

Suggestions for improvement of this paragraph are appreciated.

### Feedback

None

Created: 24 February 2004
Modified: 6 March 2004
Modified: 13 Januari 2016