  A comprehensive hot line list is calculated for $^{31}$PH$_3$ in its ground
electronic state. This line list, called SAlTY, contains almost 16.8 billion
transitions between 7.5 million energy levels and it is suitable for simulating
spectra up to temperatures of 1500~K. It covers wavelengths longer than
1~$\mu$m and includes all transitions to upper states with energies below $hc
\cdot 18\,000$~cm$^{-1}$ and rotational excitation up to $J=46$. The line list
is computed by variational solution of the Schr\"odinger equation for the
rotation-vibration motion employing the nuclear-motion program TROVE. A
previously reported {\it ab initio} dipole moment surface is used as well as an
updated `spectroscopic' potential energy surface (PES), obtained by refining an
existing \textit{ab initio} surface through least-squares fitting to the
experimentally derived energies. Detailed comparisons with other available
sources of phosphine transitions confirms SAlTY's accuracy and illustrates the
incompleteness of previous experimental and theoretical compilations for
temperatures above 300 K. Atmospheric models are expected to severely
underestimate the abundance of phosphine in disequilibrium environments, and it
is predicted that phosphine will be detectable in the upper troposphere of many
substellar objects. This list is suitable for modelling atmospheres of many
astrophysical environments, namely carbon stars, Y dwarfs, T dwarfs, hot
Jupiters and solar system gas giant planets. It is available in full as
supplementary data to the article and at \url{www.exomol.com}.
