13.9: Iterating
- Page ID
- 8221
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We can now use Equations 13.8.35a,b and get a better estimate of the triangle ratios. The numerical data are
\(b_1 = 2/3 , \quad b_3 = 1/3 , \quad r_2 = 3.481 \ 33 , \)
\(τ_1 = t_3 − t_2 = 10\) mean solar days and \(τ_3 = t_2 − t_1 = 5\) 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
\(τ_1 = 0.172 \ 021 \quad \text{and} \quad τ_3 = 0.086 \ 010.\)
Equations 13.8.35 then result in
\(a_1 = 0.666 \ 764, a_3 = 0.333 \ 411\).
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 \(r_2\). 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 \(\text{IAU}\) ephemeris.
\begin{array}{l c c c}
& \text{First crude estimates} & \text{First iteration} & \text{MPC} \\
\\
& ∆ \quad r & ∆ \quad r & ∆ \quad r \\
1 & 2.72571 \ 3.48532 & 2.65825 \ 3.41952 & 2.644 \ 3.406 \\
2 & 2.68160 \ 3.48133 & 2.61558 \ 3.41673 & 2.603 \ 3.404 \\
3 & 2.61073 \ 3.47471 & 2.54579 \ 3.41082 & 2.536 \ 3.401 \\
\end{array}
We see that we have made a substantial improvement, but we are not there yet. We can now calculate new values of \(a_1\) and \(a_3\) from Equations 13.8.35a,b to get
\(a_1 = 0.666 \ 770 \quad a_3 = 0.333 \ 416\).
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 \(a_1\), \(a_3\), \(∆\) and \(r\), and we have to bear in mind that Equations 13.8.35a,b are only approximations (to order \(τ^3\) ). 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 \(a_1\), \(a_3\), \(∆\) and \(r\), in order to prepare for the solution of the exact Equations 13.12.25 and 13.12.26. Our current values of \(a_1\) and \(a_3\), 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 \(a_1\) and \(a_3\). These may be useful, but for reasons explained in the previous paragraph, it may be easier to skip Section 13.10 entirely.