Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

13.9: Iterating

( \newcommand{\kernel}{\mathrm{null}\,}\)

We can now use Equations 13.8.35a,b and get a better estimate of the triangle ratios. The numerical data are

b1=2/3,b3=1/3,r2=3.481 33,

τ1=t3t2=10 mean solar days and τ3=t2t1=5 mean solar days, but recall that we are expressing time intervals in units of 1/k, which is 58.132 440 87 mean solar days, and therefore

τ1=0.172 021andτ3=0.086 010.

Equations 13.8.35 then result in

a1=0.666 764,a3=0.333 411.

Now we can go back to Equation 13.7.4 and start again with our new values for the triangle ratios – und so weiter − until we obtain new values for 1, 2, 3 and r2. I show below in the first two columns the first crude estimates (already given above), in the 16 second two columns the results of the first iteration, and, in the last two columns, the values given in the published IAU ephemeris.

First crude estimatesFirst iterationMPCrrr12.72571 3.485322.65825 3.419522.644 3.40622.68160 3.481332.61558 3.416732.603 3.40432.61073 3.474712.54579 3.410822.536 3.401

We see that we have made a substantial improvement, but we are not there yet. We can now calculate new values of a1 and a3 from Equations 13.8.35a,b to get

a1=0.666 770a3=0.333 416.

We could (if we so wished) now go back to Equations 13.7.4,5,6, and iterate again. However, this will result in only small changes to a1, a3, and r, and we have to bear in mind that Equations 13.8.35a,b are only approximations (to order τ3 ). Therefore, even if successive iterations converge, they will still not give precise correct answers for and r.

To anticipate, eventually we shall arrive at some exact Equations (Equations 13.12.25 and 13.12.26) that will allow us to solve the problem. But these Equations will not be easy to solve. They have to be solved by iteration using a reasonably good first guess. It is our present aim to obtain a reasonably good first guess for a1, a3, and r, in order to prepare for the solution of the exact Equations 13.12.25 and 13.12.26. Our current values of a1 and a3, while not exact, will enable us to solve Equations 13.12.25 and 13.12.26 exactly, so we should now, rather than going back again to Equations 13.7.4,5,6, proceed straight to Sections 13.11, 13.12 and 13.13.

Nevertheless, in the following section, we provide (in Equations 13.10.9 and 13.10.10), after considerable effort, higher-order expansions for a1 and a3. These may be useful, but for reasons explained in the previous paragraph, it may be easier to skip Section 13.10 entirely.


This page titled 13.9: Iterating is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?