In order for the WISP to melt through the ice fast enough to reach the
water/ice interface in a year, the RTG must produce enough heat. The
heat required to melt a differential mass of ice starting from temperature
T is:
(12) |
Because
,
where
is density and V is volume, this
equation can be integrated over the crust thickness. If the
cross-section of the probe is A, then
In order to evaluate Equation 13, we need to know cv, T, and . cv is a function of T, and T is a function of z. Reference [8] gives some specific heats of ice at various temperatures. Linear regression yielded the relationship cv=172.21+6.9697T, with cv in joules per kilogram, and T in kelvin. For the lack of a better model, the temperature distribution through Europa's crust is assumed to vary linearly from the surface to the water/ice interface. At the surface, T=100 K. At the interface, T=273.15 K. Therefore, T=273.15-173.15(z/h), where T is in kelvin. is estimated to be 1000 kg/m3. Finally, Reference [8] lists for water as 6010 J/mol = 334,000 J/kg.
With these values, the Equation 13 reduces to
The lander is to transmit data for at least one year; therefore, the RTG should produce the heat necessary to melt 10 km in a year. This rate is 1520 watts. Asterius' baseline uses a 2000 W RTG because some heat will certainly leak through the side walls, and the assumptions used to derive 1520 W were very conjectural. The 2000 W RTG increases the chance of mission success by allowing the probe to descend faster should the crust be thicker than 10 km.