  Protons (ionized hydrogen) in the solar wind frequently exhibit distinct
temperatures ($T_{\perp p}$ and $T_{\parallel p}$) perpendicular and parallel
to the plasma's background magnetic-field. Numerous prior studies of the
interplanetary solar-wind have shown that, as plasma beta ($\beta_{\parallel
p}$) increases, a narrower range of temperature-anisotropy ($R_p\equiv T_{\perp
p}\,/\,T_{\parallel p}$) values is observed. Conventionally, this effect has
been ascribed to the actions of kinetic microinstabilities. This study is the
first to use data from the Magnetospheric Multiscale Mission (MMS) to explore
such $\beta_{\parallel p}$-dependent limits on $R_p$ in Earth's magnetosheath.
The distribution of these data across the $(\beta_{\parallel p},R_p)$-plane
reveals limits on both $R_p>1$ and $R_p<1$. Linear Vlasov theory is used to
compute contours of constant growth-rate for the ion-cyclotron, mirror,
parallel-firehose, and oblique-firehose instabilities. These instability
thresholds closely align with the contours of the data distribution, which
suggests a strong association of instabilities with extremes of ion temperature
anisotropy in the magnetosheath. The potential for instabilities to regulate
temperature anisotropy is discussed.
