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8. Convergence for a Circular Orbit

8.1. Better Numbers

Carol: Yes, the orbit is looking recognizably circular, but that's about it. The errors are still quite large. I would like to know exactly how large they are. Let me have a peak at the numbers at the beginning and at the end of the run, just like we did in the elliptic case.

 |gravity> ruby euler_circular_1000_steps.rb > euler_circular_1000_steps.out
 |gravity> head -5 euler_circular_1000_steps.out
 1  0  0  0  1  0
 1.0  0.01  0.0  -0.01  1.0  0.0
 0.9999  0.02  0.0  -0.0199985001874781  0.999900014998125  0.0
 0.999700014998125  0.0299990001499813  0.0  -0.0299945010873182  0.999700074986127  0.0
 0.999400069987252  0.0399960008998425  0.0  -0.0399870033746211  0.9994002199565  0.0
 |gravity> tail -5 euler_circular_1000_steps.out
 -0.967912821753728  0.639773430828751  0.0  -0.534369558098563  -0.764471400394899  0.0
 -0.973256517334714  0.632128716824802  0.0  -0.528172455755398  -0.768567576556818  0.0
 -0.978538241892268  0.624443041059234  0.0  -0.521945648518888  -0.772611878956261  0.0
 -0.983757698377457  0.616716922269671  0.0  -0.515689587420237  -0.776604113138075  0.0
 -0.988914594251659  0.608950881138291  0.0  -0.509404724445418  -0.780544088761249  0.0
Erica: The first few numbers give a distance of about 1 for the separation of the two particles, as it should be, but the last few numbers are too large. The separation along the x axis is about , and the separation along the y axis is about , and with Pythagoras that gives us a distance of . Not wildly off, but not very good either.

Carol: Remember, just after Eq.(76), how I estimated the error after one orbit to be 3%? After one and a third orbit it should then have been 4%, and we got 17%. Perhaps nonlinear effects contributed, but at least my original guess of several percent was not way off! And I remember that Dan showed what should happen for a time step that is ten times smaller: the error should shrink by a factor ten as well. I'd like to test that, by changing the line:

 dt = 0.01                                      

in euler_circular_1000_steps.rb to:

 dt = 0.001                                      

and put that code in euler_circular_10000_steps.rb, to indicate that we are now taking 1,000 steps per time unit, 10,000 time steps in total. Time to run it!

 |gravity> ruby euler_circular_10000_steps.rb > euler_circular_10000_steps.out
 |gravity> head -5 euler_circular_10000_steps.out
 1  0  0  0  1  0
 1.0  0.001  0.0  -0.001  1.0  0.0
 0.999999  0.002  0.0  -0.00199999850000188  0.9999990000015  0.0
 0.9999970000015  0.0029999990000015  0.0  -0.00299999450001088  0.9999970000075  0.0
 0.999994000007  0.003999996000009  0.0  -0.00399998700003375  0.999994000022  0.0
 |gravity> tail -5 euler_circular_10000_steps.out
 0.544161847963306  0.839852844602723  0.0  -0.839087845804416  0.544480032616632  0.0
 0.543322760117501  0.840397324635339  0.0  -0.839630814109985  0.54364202187099  0.0
 0.542483129303391  0.84094096665721  0.0  -0.840172943245668  0.542803470813169  0.0
 0.541642956360146  0.841483770128023  0.0  -0.840714232673602  0.541964380286331  0.0
 0.540802242127472  0.84202573450831  0.0  -0.841254681856773  0.541124751134179  0.0
Dan: Those numbers are very different from the earlier ones . . .

8.2. Even Better Numbers

Erica: Ah, but of course! To come back to the same place, after making the time step ten times smaller, we have to take ten times as many steps!

Carol: Of course indeed! Okay, I'll change the line

 1000.times{                                     

in euler_circular_10000_steps.rb to:

 10000.times{                                    

and call that file euler_circular_10000_steps_ok.rb.

Dan: Your file names are growing again without bounds, and I don't like having so very many different files lying around.

Carol: As long as we're debugging, I'd prefer to have many files, so that we can always backtrack to earlier versions. We can clean up the mess later.

Dan: Okay, try again:

 |gravity> ruby euler_circular_10000_steps_ok.rb > euler_circular_10000_steps_ok.out
 |gravity> head -5 euler_circular_10000_steps_ok.out
 1  0  0  0  1  0
 1.0  0.001  0.0  -0.001  1.0  0.0
 0.999999  0.002  0.0  -0.00199999850000188  0.9999990000015  0.0
 0.9999970000015  0.0029999990000015  0.0  -0.00299999450001088  0.9999970000075  0.0
 0.999994000007  0.003999996000009  0.0  -0.00399998700003375  0.999994000022  0.0
 |gravity> tail -5 euler_circular_10000_steps_ok.out
 -0.926888921537776  -0.42643167863865  0.0  0.411096610555956  -0.900278962470303  0.0
 -0.92647782492722  -0.42733195760112  0.0  0.411969325107482  -0.899877454670898  0.0
 -0.926065855602113  -0.428231835055791  0.0  0.412841644152271  -0.899475103103453  0.0
 -0.925653013957961  -0.429131310158894  0.0  0.413713566877832  -0.899071908161608  0.0
 -0.925239300391083  -0.430030382067056  0.0  0.414585092472074  -0.898667870239779  0.0
Dan: Great! Time for Pythagoras again: .

Carol: A 2% error, about a factor ten smaller than the 17% error we had before. We're getting there!

8.3. An Even Better Orbit

Erica: And we should get a much better picture now.

Carol: Here it is, fig. 21.

Figure 21: Fifth attempt at integrating the two-body problem: looking even better.

Erica: Wonderful! You can hardly see the deviation from a circle.

Dan: Yes, the particles almost cover their own tracks, the second time around.

Carol: You mean the particle: we're integrating a one-body problem.

Dan: Well, the distance between the two particles is what is plotted, so I feel I can talk about particles.

Erica: And I think you're both right. Stop arguing, you guys! Let's go back to the elliptic case, the one we started with, remember?

8.4. Reasons for Failure

Carol: Sure, I remember that. All we have to do is to take the file euler_circular.rb and make the initial velocity half as large, by changing the line:

  vy = 1                                     

into:

  vy = 0.5                                     

Dan: But that will be the same file as we started with, euler.rb.

Carol: Ah, yes, of course, I had forgotten already. And in that case, the orbit exploded.

Dan: Let's see it again. Now that we seem to understand the circular case, perhaps we can figure out what went wrong in the elliptic case.

Carol: Okay, here we go again:

 |gravity> ruby euler.rb > euler.out
And here is the plot once more, in fig 22:

Figure 22: A rerun of the second attempt at integrating the two-body problem.

Erica: Let us take a moment to evaluate what we have learned. We know now how small the steps have to be to get a reasonable convergence for a circular orbit. And we can see in figure 22 that the steps get far larger toward the left of the figure.

In fact, even when the steps start off with a reasonably small size at the right hand side, by the time we have reached the left, the steps are so large that even a circular orbit would not reach convergence if we would everywhere use such large steps!

Dan: Why do the steps get so large, all of a sudden?

Erica: Because the particles get very close together. Notice that the left-most part of the orbit is also the point in the orbit that is closest to the origin, the place where . This is called the pericenter of an elliptic orbit. This word is derived from the Greek (peri) meaning `around' or `near'.

You see, we started off with a speed smaller than the speed required for a circular orbit, in fact, we had only have of that speed. So the particles started to fall toward each other right away, and IF we would have computed the orbit correctly, the two particles would have returned to the exact same spot after one revolution, just as we finally managed to see in the circular case when we took very small steps.

Carol: So the initial position is then the place in the orbit where the particles are furthest away from each other?

Erica: Yes, indeed! And that point is called the apocenter, from the Greek (apo) meaning `far (away) from'. Well, I am willing to bet that a smaller time step will cure all of our problems.

Dan: Seeing is believing. Can you show us, Carol?

8.5. Signs of Hope

Carol: Here we go, a ten times smaller step size in euler_elliptic_10000_steps.rb; I'll plot the result in fig. 23.

 |gravity> ruby euler_elliptic_10000_steps.rb > euler_elliptic_10000_steps.out

Figure 23: Sixth attempt at integrating the two-body problem: signs of hope.

Dan: I must admit, you may both have been right: at least now the particles are completing a couple orbits that sort-of look elliptical, even though the errors are still large. But at least they don't fly off to infinity like in a slingshot.

Erica: Yes, I think we're getting to the bottom of all this, finally.

Dan: But we'd better make sure, and use even smaller steps.

Carol: Will do!
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