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Perihelion advance of the moon
My proposal that perihelion advance rates are proportional to m/M*180,
gforce(m)/gforce(M)*180, orbitrad(M)/orbitrad(m)*180 (relative to the barycenter), etc. is reinforced by the attached Qbasic program. The program is designed to demonstrate the perihelion advance rate of the moon in orbit around the earth. According to m/M*180 the moon will advance by 2.217 degrees per orbit cycle. Since the observed advance is a little more than 3 degrees per cycle, the sun's influence on the earth/moon system must account for roughly .8 degrees per cycle. And that's exactly what it does. Setting the moon average orbit radius at 4e8 meters, with an orbit speed of 900 m/sec at aphelion, gives an orbit eccentricity of .12 and an advance rate of .794 degrees per cycle. For obvious reasons the advance is orbit eccentricity dependent to some degree. It doesn't alter the outcome much though. i.e. For an eccentricity of .2 the advance rate is .82 degrees per cycle. With an aphelion speed of 300m/sec, eccentricity is greater than .9 and the advance rate is 2.2 degrees per cycle. image here   The program will run in Qbasic if inkey is replaced with INKEY$, time is changed to TIME$ and chr(??) is replaced with CHR$(??). No $ exists in FreeBasic. Compiling to an .exe file with FreeBasic (1)Copy the program off the screen. (2)Paste it into Notepad. (3)Save it with a .bas extension. e.g. peri.bas stored in the c: directory. (4)Download and instal FreeBasic. (5)Drag openconsole.exe to desktop from c:\Program Files\FreeBasic and create shortcut. (6)Double click the icon. (7)At the prompt type fbc c:\peri.bas (8)Your .exe file will be stored in the c: directory. 'Program start DIM c AS DOUBLE: DIM G AS DOUBLE: DIM AU AS DOUBLE DIM xE AS DOUBLE: DIM yE AS DOUBLE: DIM vxE AS DOUBLE DIM vyE AS DOUBLE: DIM ME AS DOUBLE: DIM mu3 AS DOUBLE DIM xM AS DOUBLE: DIM yM AS DOUBLE: DIM vxM AS DOUBLE DIM vyM AS DOUBLE: DIM MM AS DOUBLE: DIM lastrad AS DOUBLE DIM mu1 AS DOUBLE: DIM dt AS SINGLE: DIM radius AS DOUBLE DIM f2 AS SINGLE: DIM p AS SINGLE: DIM accME AS DOUBLE DIM accEM AS DOUBLE: DIM distMS AS DOUBLE: DIM accMS AS DOUBLE DIM distES AS DOUBLE: DIM accES AS DOUBLE: DIM axME AS DOUBLE DIM ayME AS DOUBLE: DIM axMS AS DOUBLE: DIM ayMS AS DOUBLE DIM axM AS DOUBLE: DIM ayM AS DOUBLE: DIM ayE AS DOUBLE DIM colr AS SINGLE: DIM xS AS DOUBLE: DIM yS AS DOUBLE DIM MS AS DOUBLE: DIM axE AS DOUBLE: DIM ayEM AS DOUBLE DIM axEM AS DOUBLE: DIM axES AS DOUBLE: DIM vxS AS DOUBLE DIM vyS AS DOUBLE: DIM accSM AS DOUBLE: DIM axSM AS DOUBLE DIM aySM AS DOUBLE: DIM ayES AS DOUBLE: DIM axSE AS DOUBLE DIM aySE AS DOUBLE: DIM accSE AS DOUBLE: DIM axS AS DOUBLE DIM ayS AS DOUBLE: DIM perirad AS DOUBLE: DIM aphrad AS DOUBLE DIM inc AS SINGLE: DIM ecc AS SINGLE: DIM ss AS STRING DIM rnew AS DOUBLE: DIM xE2 AS DOUBLE: DIM xS2 AS DOUBLE DIM yE2 AS DOUBLE: DIM yS2 AS DOUBLE: DIM su AS STRING DIM x AS SINGLE: DIM a AS DOUBLE: DIM st AS STRING DIM ti AS DOUBLE: DIM f1 AS SINGLE: DIM axMM AS DOUBLE DIM ayMM AS DOUBLE: DIM accMM AS DOUBLE: DIM sd AS DOUBLE DIM Mearth AS DOUBLE: DIM Mmoon AS DOUBLE: DIM Msun AS DOUBLE DIM dts AS SINGLE SCREEN 12 PRINT " _Determining the program run speed_" xa: ss = time IF ss st THEN x = x + 1 st = ss END IF IF x = 6 THEN mu3 = a: GOTO xb IF x 1 THEN a = a + 1 CIRCLE (10 + a / 6000, 20), 1 GOTO xa xb: dt = INT(20000000 / mu3) CLS G = .0000000000667# AU = 150000000000# xE = AU yE = 0 vxE = 0 vyE = 29780 Mearth = 5.97D+24 ' earth mass GOTO an am: LOCATE 26, 1 rnew = 4E+08 PRINT " Average orbit radius for the moon is 4e8 meters." PRINT " Press any key to continue, or 'c' to change radius." DO: ss = inkey: LOOP UNTIL ss "" CLS IF ss = "c" THEN LOCATE 25, 1 INPUT " Enter new orbit radius (e.g. 8e8 with v = 500)"; rnew END IF LOCATE 26, 1 IF rnew = 0 THEN END PRINT " Orbital speed for a concentric orbit is"; 4E+08 / rnew * 1000 INPUT " Moon's orbital speed at aphelion (= concentric)"; vyM IF vyM = 0 THEN END aphrad = rnew * 1000 / vyM xM = aphrad + AU yM = 0 vxM = 0 vyM = vyE  vyM ' Aphelion start Mmoon = 7.342D+22 ' moon mass xS = 0 yS = 0 vxS = 0 vyS = 0 Msun = 1.99D+30 ' sun mass PRINT " ____Escape exits____" CIRCLE (230, 240), 6, 11 'Earth colr = 1 lastrad = 1D+26 LOCATE 2, 1: PRINT " Press z to halve the graphics scale." mu1 = .0000003 'graphics multiplier aa: radius = SQR((xM  xE) ^ 2 + (yM  yE) ^ 2) 'radius from moon to earth IF radius 2E+11 THEN END ' 'ME = moon to earth accME = G * Mearth / radius ^ 2 axME = accME * ((xM  xE) / radius) ayME = accME * ((yM  yE) / radius) ' 'EM = earth to moon accEM = G * Mmoon / radius ^ 2 axEM = accMM * ((xE  xM) / radius) ayEM = accMM * ((yE  yM) / radius) ' distMS = SQR((xM  xS) ^ 2 + (yM  yS) ^ 2) 'MS = moon to sun accMS = G * Msun / distMS ^ 2 axMS = accMS * ((xM  xS) / distMS) ayMS = accMS * ((yM  yS) / distMS) ' 'SM = sun to moon accSM = G * Mmoon / distMS ^ 2 axSM = accSM * ((xS  xM) / distMS) aySM = accSM * ((yS  yM) / distMS) ' distES = SQR((xE  xS) ^ 2 + (yE  yS) ^ 2) 'ES = earth to sun accES = G * Msun / distES ^ 2 axES = accES * (xE  xS) / distES ayES = accES * (yE  yS) / distES ' 'SE = sun to earth accSE = G * Mearth / distES ^ 2 axSE = accSE * (xS  xE) / distES aySE = accSE * (yS  yE) / distES ' 'M = moon axM = axME + axMS ayM = ayME + ayMS vxM = vxM + dt * axM vyM = vyM + dt * ayM xM = xM + dt * vxM yM = yM + dt * vyM ' 'E = earth axE = axES + axEM ayE = ayES + ayEM vxE = vxE + dt * axE vyE = vyE + dt * ayE xE = xE + dt * vxE yE = yE + dt * vyE ' 'S = sun axS = axSE + axSM ayS = aySE + aySM vxS = vxS + dt * axS vyS = vyS + dt * ayS xS = xS + dt * vxS yS = yS + dt * vyS ' inc = inc + 1 dts = mu1 / 1.6E09 IF f2 = 0 AND radius lastrad THEN f2 = 1: colr = colr + 1 LOCATE 1, 1 PRINT " Press 'f' for faster, 's' for slower " PRINT " Distance to sun is"; dts; " times greater than is shown " LOCATE 7, 35: PRINT "Press z to reduce the graph scale" LOCATE 4, 1: PRINT radius; " aphelion radius " LOCATE 8, 2: PRINT p; " cycles " p = p + 1 IF ss = "f" THEN dt = INT(dt * 2) IF ss = "s" THEN dt = INT(dt * 10) / 20 LOCATE 28, 23: PRINT " Program step size is"; dt; " seconds " END IF IF colr = 16 THEN colr = 1 IF f2 = 1 AND radius = lastrad THEN inc = 0 perirad = radius LOCATE 5, 1: PRINT perirad; " perihelion radius " ecc = ((aphrad + perirad) / 2  perirad) / ((aphrad + perirad) / 2) PRINT ecc; " orbit eccentricity " f2 = 3 CIRCLE (230 + (xM  xE) * mu1, 240  (yM  yE) * mu1), 6, 15 END IF IF f2 = 3 AND radius = lastrad THEN f2 = 0: LOCATE 17, 25 CIRCLE (230 + (xM  xE) * mu1, 240  (yM  yE) * mu1), 1, colr CIRCLE (230 + (xE2  xS2) * 1.6E09, 240 + (yE2  yS2) * 1.6E09), 10, 0 CIRCLE (230 + (xE  xS) * 1.6E09, 240 + (yE  yS) * 1.6E09), 10, 14 xE2 = xE: xS2 = xS: yE2 = yE: yS2 = yS lastrad = radius ss = inkey IF ss = chr(27) THEN GOTO ab IF ss = "z" THEN mu1 = mu1 / 2: LOCATE 2, 21: PRINT " " IF dts 1 THEN mu1 = 1.6E09 ' 1 to 1 graphics scale IF ss = "f" THEN dt = INT(dt * 2) IF ss = "s" THEN dt = INT(dt * 10) / 20 GOTO aa ab: DO: LOOP UNTIL inkey "" END an: READ ss IF ss = "cls" THEN CLS : GOTO an IF ss = "" THEN GOTO ao IF ss = " " THEN sd = 0: READ ss: DO: LOOP UNTIL inkey "": GOTO an LOCATE sd + 4, 4: PRINT ss sd = sd + 1 GOTO an ao: GOTO am DATA "This program is designed to demonstrate the perihelion advance rate " DATA "of the moon in orbit around the earth. The sun is of course " DATA "involved. The non rotating frame of the earth is the point of focus." DATA " " DATA "The extension of Euler's method used here is obviously common place " data "in the field of astrophysics, but it's still a most incredible " DATA "example of mathematical wizardry. The fact that I get the result I " DATA "require is testament to its precision. " DATA " " DATA "Setting the moon average orbit radius at 4e8 meters, with an orbit " DATA "speed of 900 m/sec at aphelion, gives an orbit eccentricity of .12 " DATA "and an advance rate of .794 degrees per cycle. The advance " DATA "attributable to m/M*180 is 2.217 degrees per cycle, so the total " DATA "advance is 3.011 degrees. Which is as observed. " DATA " " DATA "For obvious reasons the advance is orbit eccentricity dependent to " DATA "some degree. It doesn't alter the outcome much though. i.e. For an " DATA "eccentricity of .2 the advance rate is then .82 degrees per cycle. " DATA " " DATA "Variations in the perihelion position from one cycle to the next " DATA "becomes extreme for low eccentricities because a perihelion is " DATA "identified whenever the orbit trajectory shifts from pointing inward" DATA "to pointing outward, and that can happen anywhere because the sun is" DATA "involved. Nothing can be gleaned from this. " DATA " " DATA " " DATA "cls" DATA "" 'Program end Linux users: (1) Create a new folder somewhere. (2) Download FreeBasic (linux version) from SourceForge. (3) Copy the downloaded file and paste it into the new folder. (4) Right mouse click the file and choose 'extract here'. (5) Double click the extracted folder. (6) Then double click the 'FreeBasic' folder. (7) Double click the 'bin' folder. (8) Right mouse click the 'fbc' file. (9) Choose 'properties', then 'permissions'. 'Allow executing file as program' should be ticked. Otherwise it won't work. 'Open with' should be set to 'autorun prompt'. You may be required to download some support files that may not have been included in your version of Linux. Copy the program off this screen and paste it into a text editor. Save it in the 'examples' folder with a .bas extension. Right mouse click on the examples folder and select 'Open in terminal'. At the prompt, type, sudo fbc filename.bas . Filenames are case sensitive in Linux. You will need to enter your computer access password. Double click the newly created file, which will appear in the examples folder. Once this has been set up, the process of generating executable files from Qbasic programs from any source is very simple.  Max Keon 
#2




Perihelion advance of the moon
On Wednesday, 21 April 2021 at 14:24:37 UTC+10, wrote:
My proposal that perihelion advance rates are proportional to m/M*180, gforce(m)/gforce(M)*180, orbitrad(M)/orbitrad(m)*180 (relative to the barycenter), etc. is reinforced by the attached Qbasic program. The program is designed to demonstrate the perihelion advance rate of the moon in orbit around the earth. Variable width fonts and the removal of normally inconsequential spaces has caused a few problems in my program. The simple fix is, in all but the second last set of " ", place a dot between them where they appear in the data section ". " e.g. DATA "involved. Nothing can be gleaned from this. " DATA " " DATA ". " DATA "cls" My apologies. I should have remembered. 
#3




Perihelion advance of the moon
On Wednesday, 21 April 2021 at 14:24:37 UTC+10, wrote:
My proposal that perihelion advance rates are proportional to m/M*180, gforce(m)/gforce(M)*180, orbitrad(M)/orbitrad(m)*180 (relative to the barycenter), etc. is reinforced by the attached Qbasic program. The program is designed to demonstrate the perihelion advance rate of the moon in orbit around the earth.   With an aphelion speed of 300m/sec, eccentricity is greater than .9 and the advance rate is 2.2 degrees per cycle. A little belated, but here's the image. image here https://drive.google.com/file/d/1OvX...ew?usp=sharing  
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