## Program 9: Absolute Speed of our Solar System Calculation With Jupiter and Callisto

### Introduction and Purpose

The purpose of the program VSUN.BAS is to calculate the (absolute) speed of our solar system.
To get a copy select: VSUN.BAS
To see the program listing select: VSUN.HTM
The theory behind the program is explained in detail in the book by Max Born , Chapter IV, paragraph 3 and 9 (page 128 - 130).
The theory is that by observing the time of the eclipses of the moon Callisto of Jupiter it is possible to calculate the speed of the Solar System. The Moon Callisto services as an absolute Clock, which at regular intervals generates an event when the Moon disappears behind Jupiter. Those events can be observed on Earth.
However:
• The observed time between those events is not constant and a function of the Earth's orbit.
When the earth is close to Jupiter the time between those events becomes the shortest.
When the earth is in opposite position the time is the longest.
• The observed time of the events is also a function of the direction and speed v of the solar system.

### Description and Operation

The program consists of 5 parts or displays:
1. Initialization and v theoretical calculation.
2. Technical Explanation Display
3. Single event value Display
4. Overview value Display
5. One Revolution Display
In order to go from one display to the next you only have to use the ENTER key.

#### Initialization and v theoretical calculation.

Initialization consists of the entry of two parameters: v and phi
Preset values are respectivily 300 km/sec and 0 degrees

Theoretical calculation of v, the speed of the solar system is based around three parameters: c, l and r. c is the speed of light. l is the distance between the Sun and Jupiter (Callisto). r is the distance between Sun and Earth. The following sketch explains this.

```                            ----> v

J2             E2   r   S  r   E1              J1

<----------------------><--------------------->
l                       l
```
Based on the above sketch there are 4 possibilities:
1. Jupiter at J1, Earth at E1
2. Jupiter at J1, Earth at E2
3. Jupiter at J2, Earth at E1
4. Jupiter at J2, Earth at E2
Suppose Jupiter is at J1 and Earth at E1. An event at J1 is detected a time t1 later at E1.
For t1 the following equation is valid: v*t1 + c*t1 = l-r
This results in: t1 = (l-r)/(c+v)

Suppose Jupiter is at J1 and Earth at E2. An event at J1 is detected a time t2 later at E1.
For t1 the following equation is valid: v*t2 + c*t2 = l+r
This results in: t2 = (l+r)/(c+v)

The time difference between those events at J1 is:
t2-t1 = (l+r)/(c+v) - (l-r)/(c+v) = 2*r/(c+v)

For Jupiter at J2 we can do the same calculation:
t4-t3 = (l+r)/(c-v) - (l-r)/(c-v) = 2*r/(c-v)

The difference between the two set of events is:
(t4-t3) - (t2-t1) = 2*r/(c-v) - 2*r/(c+v) = 4*r*v/c^2

The above calculations are performed on the first display.
The calculation shows in theory that if you can measure the 4 events (t1,t2,t3 and t4) and if you now r and c than you can calculate v.
The reality is more complicated.
The problem is that the 4 events are not registred as such. What is observed is a time difference between a sequence of eclipses of Callisto around Jupiter.
The problem is that none of those events exactly coincides with for example the theoretical events t1 and t2, nor that the difference between t2 and t1 is observed. The same is true for t3 and t4.

#### Technical Explanation Display

The Technical Explanation Display shows the first 4 eclipses. However not at scale. The speed of Jupiter is too large.
• Eclipse 1 is the first eclipse at t=0. This eclipse is monitored a time t2 later on earth. This moment is indicated by the dased line.
• Eclipse 2 is the second eclipse. The moment of the eclipse consists of two time components t0 and t1 since the previous eclipse. t0 represents the revolution time of Callisto. t1 represents the time that Sun, Jupiter and Callisto are in one line. The second eclipse is monitored a time t2 later on earth. This moment is indicated by the dased line.
• Eclipse 3 is the third eclipse. For this eclipse the same rules apply.
• The same is true for eclipse 4.

For observations on Earth the dashed lines are important or more accurate the position of the Earth at those instances. What the display shows is that the time of those instances slowly increases (over 0.5 year) and then decreases. This is a function between the distance Earth and Jupiter. However the full increase is much less then the theoretical maximum value which is based on the shortest (closest) and longest (furtest) position as calculated previous.

#### Single event value Display

This display shows the values for the first 10 eclipses. With time the time in seconds is meant.
For each eclipse the following values are stored:
• t=t0. Time including the revolution time of Callisto since previous eclipse.
• t=t1. Time including the period to allign the Sun, Jupiter and Callisto.
• t1-t0. Period between t1 and t0. Observe that this value is the same for all eclipses.
• time. Period to observe the eclipse.
• dist. Distance between Earth and Jupiter
• t2=t1+time. Time when the eclipse is observed.
• delta t2. Period between the last observed eclipse.

#### Overview value Display

This display shows values based on one revolution of the Earth. Each line is generated when delta t2 is the smallest (closest to Jupiter). For each revolution of the earth the following values are displayed:
• The last three delta t2 values.
• The maximum delta t2 value.
• The minimum delta t2 value.
• The difference between the maximum and minimum value.
• Totalised value of all increments i.e. the sum of all delta t2 values minus the minimum value and divided by 2.

The last two lines of the display shows the following values:

• The maximum totalised value.
• The minimum totalised value.
• The difference between those two values.
• The calculated value of v using the equation v=t*4*r/c^2
• The calculated value of v using the equation v=t*2*pi*r/c^2
• The true value of v.

#### One Revolution Display

This display shows the position of Jupiter, Callisto (Not on scale) and the Earth. The position of Jupiter is displayed in two colors: white and gray. Each color represents one year.

### Technical Description

The program consists of three major parts.
• In the first part the time t1 of the next eclipse is calculated i.e. the sequence of events. The calculation is done in an innerloop.
• In the second part the time t2 is calculated when those events are observed on earth.
The calculation is done in an innerloop.
• The first and the second part are done in an outerloop. Each cycle through the loop represents one revolution. As part of this loop, at the end, t0 is calculated.
• In the third part the value of v is calculated.

### Reflection

• Both calculations of v (v1 and v2) are not very accurate. The reason for this error is because the line segments are not in a straight line. The calculation of v1 assumes a straight line and the calculation of v2 assumes a circle. This gives an idea how difficult it is to compare the theory with actual observations.
• The author would like to hear about improved methodes to calculate v.
• The author would like to hear from people who have actual performed this experiment.

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