Hot rocks on Graystones

The rocks on the top of Graystones were warm to the touch during the recent hot spell. That suggests a rock temperature of say 30 degrees Celsius. The surface area of rock exposed was about 1 square metre. I will estimate the power of the sunlight at 1000 Watts per square metre as it will vary during the day and we are not square on to the Sun at this latitude. We can estimate the power of the black body radiation from the rock using Stefan's Law P = Stefan constant x area x (absolute temperature)^4 = 478 Watts per square metre. Supposing thermal equilibrium by this time of day (early afternoon) then the rock must be losing at a rate of about 500 Watts per square metre through other means such as thermal conduction through the rock and into the ground. I looked up a thermal conduction equation http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/heatcond.html I found a value for the thermal conductivity of slate as 1.16 W per square metre per degree Kelvin (we were on Skiddaw Slates) Supposing the soil beneath the rock was at 15 degrees Celsius, then for the rate of conduction to be 500W in my model, the depth of rock would be 3 cm. This seems too small.