Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2010 / A / 1 / ii / Solution

Set $x=T^4/T_0^4-1$. Since $\sigma \gamma T_0^4=Q/8$, the steady state energy balance is $2(1-a_0)=1+x$. Substitution givesThe two sides have slopes $\mu$ and $2$ at the origin. For $x>0$, $\tanh (\mu x)$ is strictly concave, its initial slope is $\mu$, and it remains below $1$. Consequently no positive root exists for $\mu \le 2$, while for $\mu >2$ there is exactly one positive root $x_{+}\in (0,1/2)$. Oddness supplies the corresponding negative root. At $\mu =2$ the only root is still zero. Hence multiple steady temperatures occur precisely when $\mu >{\mu }_c=2$.

The largest root corresponds monotonically to the largest positive absolute temperature:For large $\mu$, the positive root tends to $1/2$, so $a_{+}\simeq 1/4$ and $T_{+}/T_0\simeq (3/2{)}^{1/4}$. The negative branch has larger albedo and lower temperature.