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.
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:
Return to CHAPTER5.TXT
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.
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
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
The results are based on effective radius of the Sun of 2800000 and an oblateness of 0.0034
distance offset10000000 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
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:
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.
Return to CHAPTER5.TXT Entry Point 15
P . V . . r0 . . . alpha . S r1 M P1Figure 1 is also drawn as a figure: FIGURE.TXTfigure 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
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
The display shows for each of the planets:
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
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
See Literature 11 page 486 and 492.
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:
method A starts from Newton's law:
m * G v² a = ----- = --- (1) r² r m * G this results in: v = SQR ------ (2) r
In method B a galaxy consists of two parts:
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.
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.
Return to PROVE.TXT
In order to simulate the different conditions the parameter selection display is used
From the Parameter Selection Display the following parameters can be changed:
0 = Select test display1 = 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
A . . . . . . . . . . . . . . . . .------.---------------E---------------F------G ^ ^ . . . . z z R1 R0 B------A---------C1--b--X--b--C0---------A-------Bfigure 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
. . . . . . . . . . . . . . . . . .------.---------------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
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