Rover on Mars. A robotic rover $($initial temperature $300K$ maintained by internal heating$)$ was

landed on Mars to probe the surface condition. The average surface temperature on Mars is $220K.$

During one incident, the rover lost power and started to cool down. Estimate how much time the

engineers had to fix the power supply before the rover temperature dropped to a lethal level of $263$

$K$ for the onboard electronics. The rover can be approximated as a solid metallic sphere with radius

$0\mathrm{.}25m,$ thermal conductivity $20\frac{W}{m}K,$ specific heat $500\frac{J}{k}g-K,$ mass density $8{10}^{3}k\frac{g}{m^3}$ and

surface emissivity $0\mathrm{.}8.$ Convection from the rover surface can be neglected due to very low air

density on Mars. Conduction loss to the ground can also be neglected because of a small contact

area. $[$Hint: What is the mode of heat loss from the rover surface? Estimate the corresponding heat

transfer coefficient$].$