THE REALITY, NOW AND UNDERSTANDING

SUNRAD.TXT

1 INTRODUCTION

The purpose of this program is to calculate the following:
1. The effective radius of the Sun
2. The centre of mass of the Sun
3. To print the results
4. The virtual mass of all the planets at different distances
5. Forward angle for each of the planets.
6. The effective radius of our galaxy
7. The center of mass of our galaxy
8. v of our galaxy calculation
9. display of the results
10. print out of the results

2 DESCRIPTION

The Sun has a radius of 700000 km
The Sun has a mass of 1.99 * 10 ^ 30 Kg
The mass of the sun is not equally distributed. Most of the mass is in the Centre of the Sun i.e. the density increases sharply towards the centre.

As a result of this the force the Earth feels from the Sun is not equally distributed. Most of this force comes from the centre of the Sun. The least from the outer parts.

The effective radius of the Sun is the radius of the Sun when the mass is equally distributed.

The effective radius is important if you want to simulate the behaviour of the planets around the Sun assuming that the shape of the Sun is not round.

2.2 TEST 1

The purpose of this calculation is to calculate the effective radius of the Sun.

In order to calculate the effective radius of the Sun you have to perform the following steps:

1. Divide Sun in different layers and calculate the mass of each layer.
2. Take an arbitrary point outside the sun and calculate the force towards the Sun of each layer and the total force.
3. Calculate the total force towards the Sun for different values of the radius of the Sun.
4. 4. Compare 2 with 3. The closest match will give the effective radius.

Now perform the program: SUNRAD.EXE
From the Test (and Parameter) Selection Display
Select test 1
Select S (Start)

Return to CHAPTER5.TXT

3.1 TEST 2A (OBLATENESS = 0)

The purpose of this calculation is to calculate the center of the Sun and center of mass (gravity) of the Sun when the Sun is round at different positions from the Sun.

Now perform the program: SUNRAD.EXE
From the Test (and Parameter) Selection Display
Select test 2 subtest 1 Subtest 1 performs the following:
Set parameter 7 (Number of r segments) = 50
Set parameter 9 Oblateness = 0 From the Test Selection Display
Select S (Start)

The display shows:

```                distance        offset
10000000        -.53
```

For # of r segments = 100 and # of z segments = 100 offset = -.14

The results of the calculation indicate that for a round object the center of mass is in the center of the object.

Return to CHAPTER5.TXT Entry Point 3

3.1 TEST 2B (OBLATENESS = 0.0034)

The purpose of this calculation is to calculate the distance between the center of the Sun and the center of mass (gravity) of the Sun as a function of the distance with Mercury and the shape of the Sun is not round.

Now perform the program: SUNRAD.EXE
From the Test Selection Display

Select test 2 subtest 2
Select S (Start)

The results are based on effective radius of the Sun of 2800000 and an oblateness of 0.0034

```                distance          offset

10000000          14.12
20000000           7.06
30000000           4.7
40000000           3.53
50000000           2.82
60000000           2.35
70000000           2.01
80000000           1.76
90000000           1.56
100000000           1.41
```

In formula offset =

```                        10000000 * 14.12 * oblateness
-----------------------------   km
distance * 0.0034
```

Return to CHAPTER5.TXT Entry Point 4

4 TO PRINT THE RESULTS

Now perform the program: SUNRAD.EXE
From the Test Selection Display
Select test 3
Select S (Start)

5.1 TEST 4

The purpose of this calculation is to calculate the coefficients of the equation that can be used to replace one or more planets as a virtual planet. Three conditions are considered:
1. Venus
2. Venus and Earth
3. All the planets

In chapter 4 is explained that the movement of the planet Mercury is heavily influenced by each of the other planets (Venus, Earth, Mars, Jupiter etc.). This influence is such that the planet Mercury slowly moves forward. The problem with this forward movement is that it is highly irregular, meaning that many revolutions of Mercury are required to find the angle with which Mercury moves forward in one century. The number of revolutions required is the most for the outer planets.

To take the influence of all the planets into account in the simulations a different approach is followed. The idea behind this approach is to replace all the planets by one virtual planet which moves with the same speed as Mercury at the distance of Venus. The mass of this virtual planet is not constant but a function of the distance (x) between Mercury and the Sun.

The result of the calculation are the four factors a,b,c and d of a third order polynomial

The mass m = a + x * (b + x * ( c + d * x )))

The calculation is done for three conditions:

Venus only
Venus and earth
All the planets.

Starting point of the calculation is that the mass for each of the planets is not in one point but is equally spread out over the whole trajectory. For each of the planets this is a circle.

The calculation for each planet goes in four steps:
1. First this circle is subdivided in a number of small segments.
2. For each segment the acceleration on Mercury is calculated
3. The sum is calculated
4. The virtual mass of the planet is calculated at the distance of Venus.

Then the following steps are done
• For condition 2 the sum of the virtual mass of planet Venus and the planet Earth is calculated.
• For condition 3 the sum of the virtual mass of all the planets is calculated.
• The above is repeated at different distances of Mercury. The result are four arrays
1. one with distances
2. one with virtual masses of Venus
3. one with virtual masses of Venus and Earth
4. one with virtual masses of all planets
• Finally for each of the three combinations (1,2), (1,3) and (1,4) the factors a,b,c and d are calculated.

Now perform the program: SUNRAD.EXE
From the Test Selection Display
Select test 4
Select S (Start)

Return to CHAPTER5.TXT Entry Point 15

```                                                  P
.
V                                     .
.
r0            .
.
.
alpha            .
S      r1    M            P1

figure 1  (See below)

S = Sun
M = Mercury
P = Planet
SP  = r0 = distance planet to Sun
SM  = r1 = distance Mercury to Sun
Alpha = 0 to 180
dalpha = delta alpha = 5
dm = mass planet * dalpha / 360 = delta m of segment

P,P1 =  y = r0 * sin (alpha)
S,P1 =  x = r0 * cos (alpha)
M,P1 =  dx = x - r1
M,P  =  r = SQR ( dx² + y²)
delta acceleration  =  da = dm / r²
dax = da * dx / r
day = da * y / r          (influence ay from 0 to 360 = 0)
sum_a = sum_a + dax

virtmass = 2 * sum_a * SV * SV     (SV = distance Venus Sun)
factor 2 because alpha from 0 to 180

```
Figure 1 is also drawn as a figure: FIGURE.TXT

6.1 TEST 5

The forward angle for a planet is dependent about the virtual mass. The forward angle for Venus is arbitrary selected as 293.796

The forward angle of a planet is equal to:

```        Virtual mass of planet
----------------------  *  293.796
Virtual mass of Venus
```
Now perform the program: SUNRAD.EXE
From the Test Selection Display
Select test 5
Select S (Start)

The display shows for each of the planets:

name
virtual mass
forward angle in arc seconds of planet
Accumulated forward angle in arc seconds

The result shows that the forward angle of Uranus, Neptune and Pluto are very small and can be neglected.

Return to CHAPTER5.TXT Entry Point 11

7 THE EFFECTIVE RADIUS OF OUR GALAXY

Our galaxy consists of a central bulge and a disk. The central bulge has a radius of 7500 light years The disk has a radius of 40000 light years. Our Sun is at a distance of 25000 light years from the center of our galaxy.

The mass of our Galaxy is approximate 1.1 * 10 ^ 11 * mass of our Sun

Now perform the program: SUNRAD.EXE
From the Test Selection Display
Select test 7
Select S (Start)

See Literature 11 page 486 and 492.

8 THE CENTER OF MASS OF OUR GALAXY

Now perform the program: SUNRAD.EXE
From the Test Selection Display
Select test 8
Select S (Start)

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9 SPEED OF OUR GALAXY CALCULATION

There are basically two methods to calculate the speed with which the stars in our galaxy rotate as a function of distance r from center:

A. By assuming that the galaxy has all its mass concentrated in the center of our galaxy as one point.
B. By assuming that this is not the case i.e. that there are stars in what we call a central bulge and in a disk.

method A starts from Newton's law:

```            m * G     v²
a = ----- =  ---     (1)
r²      r
m * G
this results in:  v =  SQR  ------    (2)
r
```

m = mass of galaxy.
r = distance from center of galaxy.

In method B a galaxy consists of two parts:

A bulge and a disk.
In each of those parts there are stars i.e. visible mass.

• Starting point is a base distance from the center of the galaxy.
• Next the galaxy is divided in small sections (as a function of):
r distance from center
phi angle
z height

• For each of those sections the following is calculated:
volume, mass, distance from base distance, accelaration a using (1), ax in the direction of center of galaxy and the sum of ax.
(in reality ax is divided into two parts: ax for the bulge and ax for the disk)
• Next for all the sections a "corrected mass" is calculated using (1) as a function of the sum of ax and the base distance.
• Finally using (2) v is calculated.

This can now be repeated for different base distances starting from near the center of the galaxy to outside the galaxy in increments of 1000 light years.

To do this calculation perform the following test: perform the program: SUNRAD.EXE
From the Test Selection Display
Select test 9
Select S (Start)

To see the same test in a two curves perform the following test: perform the program: SUNRAD.EXE
From the Test Selection Display
Select test 10
Select S (Start)

The display shows two curves A and B.

Curve A and B are calibrated such that at a base distance of 25000 lightyear (i.e. our distance from the center of our Galaxy) the speed is the same i.e. 250 km/sec.

For a print out: perform the program: SUNRAD.EXE
From the Test Selection Display
Select test 11
Select S (Start)

Return to PROVE.TXT

10 OPERATION

In order to simulate the different conditions the parameter selection display is used

10.1 PARAMETER SELECTION DISPLAY

From the Parameter Selection Display the following parameters can be changed:

```        0 = Select test display

1 = Set standard parameters.

2 = Screen mode. Valid values are 7,8,9 and 12. Standard value = 9
3 = Directory name. Standard name is: C:\NOW\FIG
4 = Wait time in second. Physical wait time between each simulation
cycle. Standard value = 0.05
5 = Delta time in seconds between each calculation cycle.
Standard value is 0.1

6 = Delta angle alpha.        Standard value is 1
7 = Number of r segments.     Standard value = 2
8 = Number of z segments.     Standard value = 50

9 = Oblateness.               Standard value = 0.0034

10 = Radius Sun.              Standard value = 700000 km

11 = Effective radius Sun.    Standard value = 280000 km

12 = Radius Galaxy.           Standard value = 7D+16 km

13 = Effective radius Galaxy. Standard value = 3.5D+16

14 = Save data in Data Base file. Standard value = 0
0 = No save
15 = Highest order of polynomial.    Standard value = 3
```

11.1 CALCULATION OF RADIUS OF A CIRCLE AT HEIGHT Z

```                                A
.  .               .  .
.    .                       .    .
.     .                           .     .
.      .                             .      .
.------.---------------E---------------F------G
^                      ^
.      .                                 .      .
z                      z
R1       R0
B------A---------C1--b--X--b--C0---------A-------B

figure 2   (See also below)

X - A  = X - F  = R0
X - B  = A - C0 = A - C1 = R1
X - C0 = X - C1 = b
C1 - G = l1       C0 - G = l2      l1+l2= 2R1
E - F = rrstr          = XF² - XE² = R0² - z²   (Known)
E - G = rrend = a      = XG² - XE² = XG² - z²   (To be calculated)
X - E = z

Known:  R0, R1, z, b, (l1,l2)  Calculate a = rzend
l1       +        l2        = 2R1
SQR {z²+(a-b)²} +  SQR {z²+(a+b)²} = 2R1                R = R1

z²+(a-b)²+ z²+(a+b)² + 2 * SQR{z²+(a-b)²}*{z²+(a+b)²} = 4R²
z²+a²+b²-2ab + z²+a²+b²+2ab + 2 * SQR{z²+(a-b)²}*{z²+(a+b)²} = 4R²
2z²+ 2a²+ 2b² + 2 * SQR{z²+(a-b)²}*{z²+(a+b)²} = 4R²
z²+ a²+ b² + SQR{z²+(a-b)²}*{z²+(a+b)²}   = 2R²
{z²+(a-b)²} * {z²+(a+b)²}                 = {2R²-(z²+ a²+ b²)}²
z^4 + z²(a-b)² + z²(a+b)² + (a-b)²(a+b)²
= 4R^4 - 4R²*(z²+ a²+ b²) + (z²+ a²+ b²}²
z^4 + z²a²+z²b²-2z²ab + z²a²+z²b²+2z²ab + a^4+b^4-2a²b²
= 4R^4 - 4R²*(z²+ a²+ b²)  +  z^4+a^4+b^4+2z²a²+2z²b²+2a²b²
z²a²+z²b²-2z²ab + z²a²+z²b²+2z²ab - 2a²b²
= 4R^4 - 4R²*(z²+ a²+ b²)  +  2z²a²+2z²b²+2a²b²
- 2a²b²      = 4R^4 - 4R²*(z²+ a²+ b²)  +2a²b²
0            = R^4 - R²*(z²+ a²+ b²) + a²b²
0            = R^4 - R²z² - R²a² - R²b² + a²b²
R²a² - a²b²  = R^4 - R²z² - R²b²
a²(R²-b²)    = R²(R² - z² - b²)                        R1 = R
a²           = R1²(R1² - z² - b²)/(R1²-b²)
R1²          = R0² + b²           b² = R1² - R0²
a²           = R1²(R0² - z²)/R0²
a = rrend    = rrstr * R1/R0
```

Figure 2 is also drawn as a figure: FIGURE.TXT

11.2 CALCULATION OF F AND Fx OF PLANET P.

f (delta F) is the force between a mass dm and a planet at P.
fx (delta Fx) is the force between a mass dm and a planet at P in the x direction.
F is the sum of all the forces f between a mass dm and a planet at P.
Fx is the sum of all the forces F between a mass dm and a planet at P in the x direction.

```                                .
.  .               .  .
.    .                       .    .
.     .                           .     .
.      .                             .      .
.------.---------------E---------------F------G
.                    ^
.      .                    .            .      .
z    Q                 z
R1
.------.----------------M---rx--N--------.-------.                 P
.       .
ry      .
.-------O
.

figure 3  (See also program FIGURE, test 3)

E - F = rzstr
E - G = rzend
M - E = O - Q = z
M - P = r         P = Planet
M - F = R0
M - G = R1
angle N - M - O = phi
angle F - E - Q = phi
E - Q = M - O = rr

rrstr = SQR (R0² - z²)
rrend = rrstr * R1/R0
rr = (rrstr + rrend) / 2
drr = rrend - rrstr           (nr = 1)
dvol = rr * dphi * drr * dz
dm = dvol * dens         = mass around point Q
rx = rr * cos(phi)
ry = rr * sin(phi)
MP = r
NP = r - rr*cos(phi)
QP² = NP²+NO²+QO²
= (r-rr*cos(phi))²+ry²+z²
delta F  = dm/QP²
delta Fx = F * NP/QP
delta Fx = dm*NP/QP^3

Fx =   Fx of outer part                             + Fx of inner part
Fx = Sum over Z   *  Sum over phi  *  dm*NP/QP^3  +  4*pi*R0^3/(4*3*r²)
Z from 0 to R0     phi from 0 to 180

```
Figure 3 is also drawn as a figure: FIGURE.TXT

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