About 24 years
How long for a hot planet to cool down.
- Earth type planet, no atmosphere (if there was any, the superzap would have ejected it to space)
- Surface at 1500C, all the way down until the normal mantle temperature is 1500C, thus about 60km.
- Surface temperature needs to cool down to 525C (draper point, where visible incandescence begins).
- Assuming your starzapper kept its beam on target for several months. Otherwise it would have just ablated off the top layers.... On second thought, this does not matter. Might as well just assume the cold rock got vaporized and blown off-planet, the 1500C surface left is actually the top of the mantle not the remains of the crust! Exactly equivalent for out calculation, and allows a more impressive SuperZap!
So we have a planet with very hot core(normal), but the surface terminating at the 1500C level, that needs to cool down on the surface to 525C in a state of near-equilibrium.
Kelvinize everything, degrees C is not the best scale: 60Km of mantle material, specific heat of about 700 (J·kg−1·K−1).
Starting temp of everything: 1773K. ending temp of surface: 798K
Each square meter of surface needs to lose (0.5 * 60km * heatcapacity * density/m3 * (1773-798)) Joules of energy, for the surface to stop glowing.
This is total of :
0.5 * 60000 * 700 * 2200 * 975 = 45045000000000J (4.5e13 J)
The only way to lose heat is via blackbody radiation.
Assume emissivity of the magma is about 0.65 (rhyolitic magma)
(all figures below are per m2 of surface area)
In first second, heat loss per m2 is 364216 J
In the first day, the heat loss is 3.15e10 J, the new surface temp is 1772.3K
By day 100, the temperature is down to 1711.4K, and the heat loss is way down to 2.73e10 J per day
By day 1000, the temperature is down to 1373K, and the heat loss is down to 1.13e10 J per day
By day 8649, the temperature is down to 798K, the heat loss is down to 1.29e9 J per day, and the surface stops visibly glowing.
Sorry, your target will only be visibly glowing for 23.7 years.
Erroneous approximations used for this sim:
- assume the molten stuff is rhyolitic magma, with heat capacity at 700J per kg per K, density of 2200, and emissivity of 0.65. And that all of these parameters remain constant constant regardless of temperature.
- Assume no outgassing, no atmosphere, zero cooling other than Blackbody Radiation following Stefan Boltzmann laws. Any outgassing that does occur gets banished to space due to...reasons. (maybe because you also toasted the sun, and its solar wind is still going nutz?)
- Assume that our surface mantle material follows the same heating-per-depth rule as for normal Mantle material once a reasonable equilibrium is achieved, with a temperature gradient of 25C per km depth.
- Built a spreadsheet to calculate iterative time periods. Saw less that 3% deviation between running first day as one day chunk rather than per second, so ran the sim in day periods.
Making it hotter does not help much, due to that Temp^4 term in the radiated heat.
It you fully double the energy needed before cooling enough, heating it to 2747K (2474C) then it cools to the same level in only 9282 days (25.43 years)
There are many assumptions and guesstimates in here. The answer derived is definitely a ceiling value, the true answer may be close to it, or significantly shorter. Longer is not likely unless something really weird happened.
To OP's later added question: You ask about
" I would like an answer that factors in atmosphere. Also interested in difference between time to cool to nonglowing, and time to cool below boiling point of water so oceans can fill."
Sorry, but the model I use only models conductive and radiative heat loss, thus no atmosphere.
And the conductive heat loss undergoes a transition when the material's radiative heat transport drops below its conductive heat transport. Even the 798K glowing limit is a bit below this point. Going cooler, the temperature drop drastically reduces, leaving the surface rather hot for many thousands of years. Basicaly a thin non-glowing layer floating on a deep magma sea. 100C would be achieved on the rough order of 2 million years+, or thereabouts. You are effectively recreating the Earth crust, which took 100 million years the first time around(delayed largely due to a severe case of meteorite pimples re-mixing the surface all the time)