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In my story one faction in the past used it's Dyson Spheres (they have several) to make a Nicol-Dyson beam (Basically you temporarily redirect a significant portion, or even all of it, of the star's energy output in the general direction of whatever you want to urgently stop existing) to roast one of their neighbors. The attack decimated and vaporized everything that was smaller than several km in radius in the target system, and completely liquefied the crust of rocky once inhabited planets in there.

[X] time later the protagonists stumble upon the system during their investigation of the faction with the spheres, and with horror and ave they note that the remains of the planets are still glowing on the nightside despite it's been [X] centuries/millennia since the attack.

My question is, how long a liquefied planet has until it will cool down too much for glowing in the visible spectrum? What's the [X], and how can I maximize it? (I don't want the attack to be too recent, several centuries is the minimum)

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    $\begingroup$ It's impossible to tell how long will an object need to reach a certain temperature if you don't tell from which temperature it is starting to cool down $\endgroup$
    – L.Dutch ♦
    Dec 6, 2020 at 17:59
  • $\begingroup$ Let's say 1500 C° ? $\endgroup$
    – Darth Biomech
    Dec 6, 2020 at 18:17
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    $\begingroup$ Can you imagine how much you must hate your enemy? Think about it, That enemy might be light years away. All that energy bookin' through space one second at a time while you wait years for your enemy to roast. That's enough time to repent, feel bad, wish you hadn't pulled the trigger, resign yourself to the label "genocidal maniac," loose your election, and die in prison - all before you enemy even warms up. $\endgroup$
    – JBH
    Dec 6, 2020 at 18:53
  • $\begingroup$ These guys live in Dyson swarms in simulations that often crank up to be much faster than real-time, the chances are they could forget they fired the thing just a couple of months after the firing ended. The characters never learn the reasons for the attack, but the owners of the spheres thought the neighbors were an annoying nuisance that might become a threat in the future. $\endgroup$
    – Darth Biomech
    Dec 6, 2020 at 22:34
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    $\begingroup$ I don't think so. projectrho.com/public_html/rocket/… $\endgroup$
    – Darth Biomech
    Dec 7, 2020 at 13:25

2 Answers 2

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About 24 years

How long for a hot planet to cool down.

Assuming:

  • 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.

P.S. 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)

P.P.S. 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)

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    $\begingroup$ Wow. +1 Only remains to say that IRL (if that matters) it is assumed that it took a few millions up to max 100 million of years for the earth to completely accrete, magma oceans to solidify because accretion died off gradually, vigorus convection kept things mixed, and differentiation set free gravitational energy. $\endgroup$
    – user78828
    Dec 6, 2020 at 22:17
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    $\begingroup$ @a_donda Yep. Plus a very dense insulating atmosphere (until it got whacked off by Theia). Plus about 35 times the energy from radioactive decay, as most isotopes were fresh from the furnace and not 6 billion years stale already. Plus a lot more heat than 1500C. plus constant bombardment by things big and small from Outer Space. Plus getting whacked by a Mars-sized planet(Theia) at ramming speed, adding another few hundred Yottajoules of heat from its absorbed momentum.(and heating up Earth by 600-ish degrees in the process) $\endgroup$
    – PcMan
    Dec 6, 2020 at 22:25
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    $\begingroup$ @DarthBiomech Just use a different weapon. Instead of focusing sunlight on the target, just pump a neutron beam at the planet. Make the whole thing a teensy bit radioactive. VERY deadly, and the induced radiation will cause further heating as the isotopes decay. It is the primary mode by which Earth got, and stayed, so hot in the first place. $\endgroup$
    – PcMan
    Dec 6, 2020 at 22:31
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    $\begingroup$ One way to get 'still glowing' after a few centuries might be if the weapon deposits its energy deeper under the surface of the earth - rather than just melting the outer crust. If you dump sufficient energy on the outer core then you could produce magma plumes which in turn would result in surface eruptions consisting vast outflows of lava over large regions of the earths surface. For reference, look up Deccan Traps and Siberian Traps. The latter produced over 2 million cubic km of lava! The self-sustaining 'eruption' could last for hundreds of thousands of years once it started. $\endgroup$
    – Penguino
    Dec 7, 2020 at 3:26
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    $\begingroup$ A nice analysis, but I think you've overlooked something major: convection. If the outer layer of the planet is at 1500 C all the way down to the mantle, you'll have a lot of circulation bringing heat from lower down to the upper regions. Indeed, we see some areas of Earth's surface (e.g. Kilauea) still glowing after 4 billion years or so. $\endgroup$
    – jamesqf
    Dec 7, 2020 at 4:26
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Surface cooling depends on many factors, but even 100 years seems very unrealistic.

Lava flows typically start at to 700-1200 C. Yet, because lava has poor thermal conductivity and does not really have much in the way of convective currents, it is often cool enough to walk upon with 15 minutes - forming a surface crust that is much cooler than intuition would suggest.

Unsurprisingly, much of the earth's crust is composed of materials similar to lava, and would be conducive to rapid cooling.

The real unknowns are how much the lower and hotter layers would be able to break through to the surface and heat things up again due to convection currents. Lava flows that are 10 or 30 meters thick are well known, and cool off very quickly. One formed a pool that was 85 meters thick, and drilled into 29 years after it flowed, and what discovered to be quite hot at depth even though the surface cooled quickly.

Lava cooling is different than entire planet cooling because the cool atmosphere conveys heat away from the lava more rapidly than would occur as a result of a Dyson beam, but given what I know of heat and mass transfer, I don't see any way to justify surface cooling that is millions of time slower than what we observe in lava flows.

At most, the planets would have local hotspots where volcanic activity persists over the course of centuries or millennia. To heat the planetary crust to the degree that it is molten to a depth of hundreds of meters is way overkill to sterilize any typical life, much longer heating to turn the crust molten to a depth of kilometers would not significantly change the situation, the molten surface would still cool quickly because the resulting lava surface does not maintain equilibrium with the deep hot layer.

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  • $\begingroup$ How does the likely lack of atmosphere affect this (i'm guessing the atmospheres would have been blown off, any there would likely be new due to outgassing)? Would it help/hurt? $\endgroup$
    – DWKraus
    Dec 6, 2020 at 19:07
  • $\begingroup$ Well you need to consider also that lave is a) relatively cool itself and b) have lots of cold things around it like the already solidified rocks and the atmosphere. $\endgroup$
    – Darth Biomech
    Dec 6, 2020 at 22:40

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