# Chapter 5 – Solar System Orbits (Orbit Foci)

## Orbit Foci

In the previous program, Kepler’s Orbits, reference is made to the Sun being at one focus of an elliptical orbit. Where is the second or ’empty’ focus as it is called?

Quite simply it is at an equal distance measured from the Sun to the perihelion position (the nearest point in orbit to the Sun) but transferred to the aphelion position (remotest point in orbit to the Sun). If you have already keyed in the Kepler’s Orbits program, then the minor modifications from Line 300 in this program are all that is necessary before RUN-ning the revision. This version will PLOT the second focus position once a half-orbit is PLOTted and the position of perihelion is known. Figure 5.9 demonstrates.

Figure 5.9
The program plots the second, or empty, focus (f2) of any selected elliptical orbit. The Sun is at the first focus (f1). This program is quite important in demonstrating that the Spectrum, via the program, is PLOTting a true elliptical orbit. There is a simple test to check results. RUN the program and measure, either on the video screen or a paper COPY if you have access to the ZX printer, the total distance around a triangle formed between any point on the orbit and the two foci. No matter what point you choose on a given orbit the total distance will be the same.

You are following the same procedure, in reverse, as the gardener who insists on making perfect oval (ie elliptical) flower beds in a lawn. He uses two stakes (representing the two foci) and a loop of string about them drawn taut (representing your triangle) to trace out the shape. Quite obviously the total length of the string remains constant, as should your measurements.

10 PRINT “Orbits Foci “;
30 INPUT “Velocity (13 to 30): “;w
40 PRINT w;AT 10,15;”f1″: CIRCLE 128,85,2
50 LET h=.4: LET g=1000000
60 LET x=g/1000: LET y=0
70 LET i=h/4: LET v=0
80 LET r=x: LET s=y: LET z=0
90 LET x=x+i*v: LET y=y+i*w
100 GO SUB 200
110 LET x=r: LET y=s: LET o=h/2
120 LET v=v+o*b: LET w=w+o*c
130 GO SUB 300
140 FOR t=0 TO w*7 STEP h
150 LET x=x+h*v: LET y=y+h*w
160 GO SUB 200
170 LET v=v+h*b: LET w=w+h*c
180 GO SUB 300: NEXT t: STOP
200 LET e=x*x+y*y: LET d=SQR e
220 LET a=-g/e: LET b=a*x/d
240 LET c=a*y/d: RETURN
300 PLOT x/10+128,y/10+85
310 IF x=-50 AND x>z THEN PLOT 228+x/10,85: PRINT AT 10,(228+x/10)/8-1;”f2”
320 LET z=x
330 RETURN