As an example, to ensure that the WFE change during an observation does not exceed a desired threshold $\tau $, we require that Display Formula
$|\Delta RMS|=|W(tf)\xb7W(tf)\u2212W(t0)\xb7W(t0)|\u2264\tau ,$(5)
where $t0$ and $tf$ are the times at which the observation begins and ends, respectively. Using Eq. (4), we can rewrite this condition as Display Formula$\u2212\tau \u2264Tf\u2212T0Thot\u2212Tcoldw\xb7w\u2264\tau ,$(6)
where $Tf$ is the temperature at the end of the observation. Solving for $Tf$, we find that Eq. (5) is satisfied if Display Formula$Tf\u2208[Tmin,Tmax],$(7)
where Display Formula$Tmax=\tau (Thot\u2212Tcold)w\xb7w+T0$(8)
and Display Formula$Tmin=\u2212\tau (Thot\u2212Tcold)w\xb7w+T0.$(9)
For an observation of duration $tf\u2212t0$, these limiting values for $Tf$ correspond to equilibrium temperatures of Display Formula$Te,max=min{\tau (Thot\u2212Tcold)[1\u2212e\u2212k(tf\u2212t0)]w\xb7w+T0,Thot}$(10)
and Display Formula$Te,min=max{\u2212\tau (Thot\u2212Tcold)[1\u2212e\u2212k(tf\u2212t0)]w\xb7w+T0,Tcold},$(11)
respectively, where the additional restrictions ensure that the temperature remains within the range $[Tcold,Thot]$. Substituting these equilibrium temperatures into Eq. (1), we find that the maximum and minimum sun angles are Display Formula$\varphi max=\u2212b\u2212b2\u22124a(c\u2212Te,min)2a,$(12)
Display Formula$\varphi min=\u2212b\u2212b2\u22124a(c\u2212Te,max)2a.$(13)
As a result, Eq. (5) is satisfied if $\varphi \u2208[\varphi min,\varphi max]$. For instance, suppose we wish to keep the WFE change below 10 nm for observations up to 2 days in length. Then, for $T0=50.05\u2009\u2009K$, observations are allowed at sun angles between 85 and 131 deg, using our thermal model. In general, the allowed sun angles vary depending on the initial temperature and observation duration, as shown in Fig. 3.