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Thermal time constant for the TTF

All of the etalon parts in Fig. 2 are essentially cylindrical optical flats. For each component, the heat transfer as a function of position in cylindrical coordinates (r,z) is given by the diffusion equation,


\begin{displaymath}{{\partial^2 T}\over{\partial r^2}} + {{1}\over{r}}{{\partial...
...ial z^2}} = {{1}\over{\alpha}} {{\partial T}\over{\partial t}}
\end{displaymath} (2)

where $\alpha$ is the ratio of the thermal conductivity to the thermal capacity. For purely radial heat flow, i.e., ignoring the z dependence, the thermal equation with the most elementary boundary conditions has a complicated solution involving Bessel functions (Heisler 1947). The cylindrical plates exhibit an exponential cooling rate with a thermal time constant $\tau \approx L^2/\alpha$, where L is a characteristic length given by the ratio of the cylindrical volume to the surface area. A crude finite-difference solution to equation (A1) also exhibits exponential behavior with a similar time constant. The estimated values of L and $\tau$ are given in columns 4 and 5 in the table below (TTF values are 50% longer - need to update).



Table 1. Thermal time constants of a 50mm aperture etalon assembly
optical radius thickness characteristic time
flat (mm) (mm) length constant (min)
         
capacitor 7 15 4.8 7
central pillar 25 15 9.4 24
outer plate 40.5 19 12.9 50
pillar+plate 25-40 34 24.8 184


next up previous
Next: About this document ... Up: Stability tests of TAURUS/TTF Previous: A possible explanation for
Joss Bland-Hawthorn
2000-07-24