**Quote from Mal on 11/07/06 at 21:55:47:**BTW, how do the periods of the oscillations that you get compare to the Pkoz calculation? Do they match, or are they different?

Outer Planet:

If you look at the graph, it takes 1730 years to reach max eccentricity from its initial 0 eccentricity. Multiplying this by 2 gives 3460 years as a period. The computed value is 3400. Not a bad match since I'm just eyeing the graph for dates.

But then following that, the period gets shorter. You'll notice that from the starting point where eccentricity was set to 0, it rose slowly at first. But after bottoming out at the first valley, it doesn't reach eccentricity = 0. And it rises faster. Consider the peaks centered on years 1736 to 6920. It does 3 complete cycles. (6920-1736)/20 = 1728 years per cycle, almost half of what is expected.

Inner Planet:

From the graph, it takes about 5400 years to reach max ecc from its initial starting ecc of 0. Doubling this gives 12800 years. But again, eccentricity takes a long time to start picking up from ecc=0. Peak-to-peak is only about 6500 years, again about half the original value. The computed value is 9800 years.

I notice the same thing going on in Figure 2 (page 17) in Takeda's paper.

0.9 solar mass companion:

Time from begining to first peak is 16 million years. Doubling gives 32 million years. But time from 1st peak to 2nd peak shortens considerably to 22 million years. Considering all 15 cycles on the graph, it has a time averaged period of 20.5 million years. The computed value is 25.6 million years.

0.08 solar mass companion:

Time to first peak = 186 million years. Doubling this gives 372 million years. But the time from the first peak to the 2nd valley = 120 Myrs. Doubling this gives 240 Myrs. The computed value is 288 Myrs.

So it seems that the period is not the same from cycle to cycle and depends heavily on how close to ecc=0 the valley comes; the closer, the longer the period.

This sort of makes sense then the author would use the asymptoticlly equal to symbol. It's as if to say don't trust this formula for any one specific period, but as cycles approach infinity, the time-averaged period should approach the computed period.