The problem is that the table is too large even for being contained in a page, so first of all LaTeX pushes it on the next page because it can't be contained in the chapter title.
Then it tries to set it in the next page which again doesn't have room for it, so it pushes it to the next page and finally it gives up.
You simply don't need tabularx
.
\documentclass{book}
\usepackage{setspace}
\newcommand{\sym}[2]{%
\noindent
\makebox[4em][l]{$#1$}% <---- adjust to suit
#2\par
}
\onehalfspacing % you seem to be using this (avoid it, if you can)
\begin{document}
\appendix
\chapter{List of Symbols}
\label{ch:AE}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\end{document}

If your descriptions are long and may end up into more than one line, you can modify the definition of \sym
to ensure the description is not split across pages:
\documentclass{book}
\usepackage{setspace}
\newcommand{\sym}[2]{%
\noindent
\makebox[4em][l]{$#1$}% <---- adjust to suit
\begin{minipage}[t]{\dimexpr\textwidth-4em\relax}
#2\par\xdef\theprevdepth{\the\prevdepth}
\end{minipage}\par\prevdepth\theprevdepth
}
\onehalfspacing % you seem to be using this (avoid it, if you can)
\begin{document}
\appendix
\chapter{List of Symbols}
\label{ch:AE}
\sym{a}{1 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{2 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{3 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{4 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{5 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{6 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{7 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{8 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{9 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{10 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{11 Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{12 Semi-major axis of Earth Semi-major axis of Earth Semi-major axis of Earth
Semi-major axis of Earth Semi-major axis of Earth Semi-major axis of Earth
Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\sym{a}{Semi-major axis of Earth}
\sym{Ant_k}{Antenna number $k$}
\end{document}

\hsize
should be1.8
, not28
.