Improve equation formatting and add spelling words

.vscode/settings.json

@@ -1,31 +1,39 @@
 {
- "cSpell.words": [
- "apoapsis",
- "deorbit",
- "Hohmann",
- "Kármán",
- "periapsis",
- "semimajor",
- "Storrs"
- ],
- "cSpell.ignoreWords": ["devcontainer"],
- "cSpell.languageSettings": [
- {
- // use with Markdown files
- "languageId": "markdown",
- "dictionaries": ["latex"],
- // Exclude code
- // "ignoreRegExpList": [
- // "\\$.*\\$",
- // "\\$\\$([\\n\\(\\)]|[^(\\$\\$)])*\\$\\$",
- // "`.*`",
- // "```([\\n\\(\\)]|[^(\\$\\$)])*```",
- // "\\{\\w+\\}`.+`",
- // ":::\\{math\\}([\\n\\(\\)]|[^(\\$\\$)])*:::",
- // "---([\\n\\(\\)]|[^(\\$\\$)])*---"
- // ]
- }
+ "cSpell.words": [
+ "apoapse",
+ "apoapsis",
+ "deorbit",
+ "Hohmann",
+ "Kármán",
+ "Laguerre",
+ "periapsis",
+ "Prussing",
+ "semimajor",
+ "Semiminor",
+ "Storrs",
+ "Stumpff"
+ ],
+ "cSpell.ignoreWords": [
+ "devcontainer"
+ ],
+ "cSpell.languageSettings": [
+ {
+ // use with Markdown files
+ "languageId": "markdown",
+ "dictionaries": [
+ "latex"
  ],
- "editor.formatOnSave": true
-
+ // Exclude code
+ // "ignoreRegExpList": [
+ // "\\$.*\\$",
+ // "\\$\\$([\\n\\(\\)]|[^(\\$\\$)])*\\$\\$",
+ // "`.*`",
+ // "```([\\n\\(\\)]|[^(\\$\\$)])*```",
+ // "\\{\\w+\\}`.+`",
+ // ":::\\{math\\}([\\n\\(\\)]|[^(\\$\\$)])*:::",
+ // "---([\\n\\(\\)]|[^(\\$\\$)])*---"
+ // ]
+ }
+ ],
+ "editor.formatOnSave": true
 }

time-since-periapsis-and-keplers-equation/universal-variables.md

@@ -306,13 +306,13 @@ Alternatively, a more refined estimate can be determined using a secant estimate
 where $f(\chi^{+})$ is the solution of Kepler's equation with the $\chi^{+}$ value.
 
 ::::{note}
-Prussing and Conway {cite}`Prussing2013`, citing Conway {cite}`Conway1986` suggest that faster convergence in the solution of Kepler's equation can be achieved by using the **Laguerre algorithm**, rather than Newton's algorithm. Another advantage of the Laguerre algorithm is that it is relatively insensitive to the value of the initial guess.
+Prussing and Conway {cite}`Prussing2013`, citing Conway {cite}`Conway1986` suggest that faster convergence in the solution of Kepler's equation can be achieved by using the [**Laguerre algorithm**](https://en.wikipedia.org/wiki/Laguerre%27s_method), rather than Newton's algorithm. Another advantage of the Laguerre algorithm is that it is relatively insensitive to the value of the initial guess.
 
 The Laguerre algorithm can be implemented as:
 
 :::{math}
 :label:
-\chi_{i + 1} = \chi_{i} - \frac{n f(\chi_i)}{f'(\chi_i) \pm \left[\left(n - 1\right)^2 \left(f'(\chi_i)\right)^2 - n\left(n - 1\right) f(\chi_i)f''(\chi_i)\right]^{1/2}}
+\chi_{i + 1} = \chi_{i} - \frac{n f(\chi_i)}{f'(\chi_i) \pm \sqrt{\left(n - 1\right)^2 \left[f'(\chi_i)\right]^2 - n\left(n - 1\right) f(\chi_i)f''(\chi_i)}}
 :::
 
 The sign ambiguity in the denominator is determined by taking the sign of the numerical value of $f'(\chi_i)$. In addition, the solution is relatively insensitive to the choice of the value of $n$, which is an integer constant. It seems as though $n = 5$ is a reasonable value. Choosing $n = 1$ gives the standard Newton's algorithm.