Test 4 Mercury 15.24
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1 For i = 0 To imax
2 For j = 1 To di
3 Select Case test
4 Case Is = 4
5 C2 = 2
6 k1 = (k2 - 1) / h2 ' Eq 15.24 page 196
7 u2 = u * u
8 fac31 = fac3
9 fac3 = k1 + u * (2 * m / h2 - u + C2 * m * u2)
10 If fac3 < 0 And Count = 0 Then
11 Debug.Print Count; sign1; "i"; i; "j"; j - 1; "fac3";
FUSINGX(fac31, "#.############"); " vu"; FUSINGX(vu, "#.#######");
" u"; FUSINGX(u, "#.##########"); " r"; FUSINGX(r, "###.######")
12 sign1 = sign1 * -1
13 Count = 1
14 End If
15 If sign1 = 1 Then kleur = LGreen Else kleur = Yellow
16 If Count > 0 Then
17 Count = Count + 1:
18 If Count >= maxcount Then Count = 0:
19 If sign1 = 1 Then kleur = LRed Else kleur = Black
20 End If
21 vu = sign1 * Sqr(Abs(fac3))
22 u = u + vu * ddPhi
23 r = 1 / u
24 phi = Phi00 * pi / 180 + i * dPhi + j * ddPhi
25 If Count > 0 And Count < 6 Then
26 Debug.Print Count; sign1; "i"; i; "j"; j; "fac3";
FUSINGX(fac3, "#.############"); " vu"; FUSINGX(vu, "#.#######");
" u"; FUSINGX(u, "#.##########"); " r"; FUSINGX(r, "###.######")
27 End If
28 TESTMAXR eccr
29 If t2 > 0 Then t2 = 0
30 End Select
31 next j
32 ' Display
33 next i
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The following table shows the output of the program.
When you consider the results there are 6 ranges of 5 lines which describe almost
the same event.
- The first 4 lines show what happens immediate after the start of the simulation. This is at a distance of roughly r = 100. The color is green.
- The next 5 lines show what happens when Mercury is at the closest distance. This is roughly at a distance of r = 54. The color changes to yellow.
- The next 5 lines show what happens when Mercury is at the fatherst distance. This is roughly at a distance r = 308. The color changes to green.
- The inner loop is controlled by the variable j. The primary purpose is to calcuale the numerical solution of equation 15.24.
The outer loop is controlled by the variable i. The primary purpose is to show the numerical solution in a graphical representation.
The important point it that almost always fac3 is positif, except twice during each revolution. The first time at perihelion and the second time at aphelion.
That means vu is positif when Mercury approaches and negatif when Mecury removes.
This behaviour is described by the parameter sign1.
sign1 is calculated in program line 12. The calculation describes that the sign of sign1 is reversed each time when the parameter fac is negatif.
Reflection of equation 15.24
Calculation of equation 15.24 it is not simple i.e. not straight forward. The same is also true for equation 15.23.The difficulty lies in the quadratic nature of the equation. The consequences are square roots calculations and the factors involved can be negatif. To handle this issue in general is very tricky.
Equation 15.25 is much simpler and resembles Newton's law which is also very simple.