Program 10: Sling Shot Effect and Gravity Assist

Introduction and Purpose

The purpose of the program SLINSHOT.BAS is to demonstrate the Sling Shot Effect. Also called Gravity Assist. The Sling Shot Effect is the behaviour that when a third object (an asteroid or satelite) passes close two a second object (a planet) the speed of the third object can increase or decrease.

In order to see the listing of the program select: SLINSHOT.HTM
In order to get a copy select: SLINSHOT.BAS
There is also a program in Visual Basic 5.0 available. Select: VB Slingshot.htm For executables of both programs select: slinshot.zip

The program shows a simulation of three objects m1, m2 and m3. The position of m1 is fixed. The simulation shows two conditions: Speed up and Speed down.

SlingShot Effect Description

In order to understand the SlingShot Effect you have to make a distinction between a system in which 2 objects and 3 objects are involved

The SlingShot Effect does not happen when there are 2 objects involved. The general path in this case for each object is an ellipse. Each path has a point where the two objects are the closest. The speed of each object the same distance away (before and after) from this point is the same; there is no exchange of energy. The path of each object is symmetrical.

The following sketch 1 shows this situation for two objects m and M (in a very simplified way)

   m                    v1 
  x0     .     .     .  x1  .     .     .     x2->
  v0                                          v2
  t-4                                         t4
                    ....M....-> 
                        t0 

The above sketch shows:
1. The path followed by m (identified by the points x0,x1,x2 and 6 dots) and by M (identified by the point M and 8 dots).
   
2. The shortest distance at t0 when m (at x1) and M .
3. 4 points to the right of x1 (including x2), representing the position of m at the moments t1, t2, t3 and t4.
4. 4 points to the right of M, representing the position of M at the moments t1, t2, t3 and t4.
   
5. For the point to the left of x1 and M the same is true but then for the monents t-1,t-2,t-3 and t-4
   
6. The speed v0 of m at x0 is the same as the speed v2 of m at x2.
7. The speed of m at x1,t0 is v1, and the speed of M at t0 is V

For the SlingShot to happen at least 3 objects are involved. The easiest system is one Sun and two planets. One planet should move in a circle (identified by *) and one in an ellipse (identified by +). The following sketch shows this.

              ***                         
            *     *                   S = Object 1: Sun
          *      +++M+++              * = Object 2: large planet 
         *   ++      *    ++          + = Object 3: small planet or Satellite
         *  +  S     *      +         M = Meeting point
         *   ++      *    ++ 
          *      +++M+++                   
            *     *   
              ***   

The important point of the above sketch is that the path of each object around the closest point M is not symmetrical.
During the whole path around the Sun the two planets will almost not influence each other, except if they become close.
Only then the path of object 3 will change in direction and speed, because a) the path of both around M is asymmetrical and b) the mass of object 3 is smal.

The SlingShot effect comes in two flavors: Speed up and Speed down.

Speed up means (This is the true SlingShot effect), that the speed of object 3 has increased, the same distance away before and after a close encounter with object 2.
Speed down means, that the speed of object 3 has decreased.

For a good understanding it is important to know

1. that in both cases (speed up and speed down) when object 3 approaches object 2 the speed of object 3 will increase first end then decrease. In the case of speed up the increase will be larger as the decrease, resulting in an overall increase. For speed down the reverse is true.
2. that speed up for one object implies that the speed of the other object will decrease.

There has been an exchange of energy as described by Newton's Law.

The following Sketch 2 shows Speed up for m (dotted line):

                                                     x2 t4 ->
                                            .        v2
                                   .                                 
                 t0       .
                 x1    
          t-1                         
           .
               ....M....-> 
         .
 
       .
 
     x0
     v0
     t-4

The above sketch shows the path followed by m and M at 9 different moments.
The shortest distance is at t0 when m is at x1 and M is at its drawn position.

The following Sketch 3 shows Speed down for m (dotted line):

        <-m
          x2
          v2   .
          t-4       .          
                         .    t0                    
                              x1
                                 t-1              
                                 .
                        ....M....->       
                                    .    
                                                  
                                       .
                                                 
                                         x0 m 
                                         v0
                                         t-4 

The above sketch shows the path followed by m and M at 9 different moments. m goes from left to right
The shortest distance is at t0 when m is at x1 and M is at its drawn position.

In all the three sketches the initial speed v0 of m at t-4 is identical.
The same is true for the speed V of M.

Comparing Sketch 2 with Sketch 3 shows that:

1. going from t-4 to t-1 the path and speed increase of both will be almost identical.
2. going from t-1 to t0 in sketch 2 the path will be longer because also M moves away. This will result in a longer and larger increase in speed.
3. going from t-1 to t0 in sketch 3 the path will be shorter because also M moves towards m. This will result in a shorter and smaller increase in speed.
4. after t0 the the speed of both will decrease. However
5. because for sketch 2 this will be shorter then the increase, the net result will be an increase.
6. because for sketch 3 this will be longer then the increase, the net result will be a decrease.

For additional reading about SlingShot effect see the following articles:

1. "Computer Recreations, How close encounters with star clusters are achieved with a computer telescope" by A.K. Dewdney. Scientific American January 1986, Page 12-16. This article explains the theory of n body simulations.
2. "Black Holes in Galactic Centers " by Martin J. Rees. Scientific American November 1990, Page 26-33. This article explains an example of slingshot effect.
3. "Mathematical Recreations, A Short Trek to Infinity " by Ian Stewart. Scientific American December 1991, Page 100-102. This article explains slingshot effect.

Comment: The above sketches and description is not 100% sound and slightly more complex.

1. The speed of M in sketch 1 showing a true 2 object situation should be to the left.
2. Starting point of sketch 1, sketch 2 and sketch 3 is a straight line. This is not correct. The line should be bent and point x1 should be closer to M.

Functional Description and Operation

The slingshot effect is an effect that the overall speed of an object can increase or decrease after a close encounter with a second object, with the emphasis on after.

In the simulation 3 objects are involved: m1, m2 and m3.

• The position of m1 is fixed.
• m2 moves in a circle around m1.
• m3 is ejected from m1 with a speed equal to escape velocity.

This means that during the initial part of the simulation the speed decreases, but always stays equal to the escape velocity. As a result when you compare the speed with the escape velocity, you will get a straight line for the speed of m3.
After the encounter the situation will be different:

• When m3 goes behind m2 there will an increase in the speed of m3.
• When m3 goes in front of m2 there will a decrease in the speed of m3.

At the beginning of the program the following questions are raised.:

1. Sling shot speed up ? (Y or N)
   Enter Y if you want to simulate Speed Up. Enter N if you want to simulate Speed Down.
2. Small scale effect ?(Y or N)
   Enter Y if you want to simulate Small Scale. Enter N if you want to simulate Large Scale. When you select Large Scale object 3 will come closer to object 2. The Speed up effect (or Speed Down) will be amplified.
3. Slow update ? (Y or N)
   Enter Y if you want to use Slow update. Enter N if you want to use Fast update. Fast update goes as fast as possible. Slow update introduces a small delay. (34 sec total simulation time)
4. Mass of m3 = 0 ? (Y or N)
   Enter Y if mass object3 = 0. Enter N if mass object3 = 20. Mass object 2 = 100.
   Selecting mass object <> 0 and Speed up for object 3 shows Speed Down for object 2. Selecting mass object <> 0 and Speed down for object 3 shows Speed Up for object 2. Please try.

After start of simulation the following rules apply:

1. Selection ESC will abort the simulation.
2. Selecting S will bring you into Single Step Mode. When you select S there after each time the display will be updated with one cycle. Seleting any other key will bring you back in continuous mode.

The display shows the following:

1. The fixed position of m1
2. The position of m2 in white (twice)
3. The position of m3 in yellow (twice)
4. The absolute speed of m2 in white
5. The absolute speed of m3 in yellow. The speed of m3 as a function of the escape velocity in red.
6. the distance between m2 and m3.

Reflection

For general simulations it is very difficult to see if you have a case of speed up or speed down. In the simulation this is solved by using the escape velocity as a landmark.

Technical Description

The program consists of four parts: Initialization, Inner loop, Outer loop and Finalization

• In the initialization 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 625 times. Each time the display is updated. In the outerloop Single Step mode is tested.
• In the Finalization section the display is updated with the final results.

Suggestions for improvement of this paragraph are appreciated.

Feedback

None

Created: 28 February 2000
Modified: 13 Januari 2016. Link "VB Slingshot.htm"

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