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The Sun is, of course, quite hot. A space probe built by normal materials probably would melt or burn even it’s still quite far from the Sun.

However, from a theory point of view, is it possible to build a space probe that could land on the Sun?

athos
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If, by "land on the sun", you mean "land of the surface of the sun", then you might consider the following from https://solarsystem.nasa.gov/solar-system/sun/in-depth/:

Surface

The surface of the Sun, the photosphere, is a 300-mile-thick (500-kilometer-thick) region, from which most of the Sun's radiation escapes outward. This is not a solid surface like the surfaces of planets. Instead, this is the outer layer of the gassy star.

We see radiation from the photosphere as sunlight when it reaches Earth about eight minutes after it leaves the Sun. The temperature of the photosphere is about 10,000 degrees Fahrenheit (5,500 degrees Celsius).

Alfred Centauri
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If, by "land on the sun", you mean make a probe that wouldn't melt at solar temperatures, I was going to say "No, of course not. The sun is far too hot." But then I saw New material has higher melting point than any known substance. To my surprise, it stays solid up to 4400 K, 3/4 the temperature of the sun. So I will change my answer to "No. The sun is far too hot."

It is hard to measure the highest boiling points, but Quora says the boiling point of Tungsten is around 5500 or 5600 C, depending on who you ask. Likewise Rhenium boils at 5500 or even 5900 C, again depending on who you ask. Presumably these measurements were made at atmospheric pressure.

The sun is 5778 K = 5504 C. So it would be possible to make a probe that would be just short of boiling at the Sun. So I will change my answer again. "No. The sun is too hot."

Substances can remain liquid or solid at high pressure. According to Nasa, the temperature at the center of Jupiter may be 24,000 C. And yet, the core may be solid.

As you dive deep into the sun, you can find pressures as high as you like. But the temperature gets high too, much higher than in Jupiter. The sun is gaseous all the way to the core. Nothing we know of could be solid or liquid in the interior of the sun.

So my final answer is going to be that I don't know of any hard theoretical limit to how high a melting temperature can be. But nothing we know of or even suspect is possible can be solid at the surface of the sun or in its interior.

And I will add that a probe is more than something that doesn't melt (or boil). It must do something useful. We don't know any way to make electronics or anything else useful under those conditions.

Edit 2/1/2020 - This XKCD came out since I posted the answer. It seems relevant. https://xkcd.com/2262/

mmesser314
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    about useful... if it's possible to build a big enough probe that doesn't melt, it might be possible to cool down more in its center (e.g. something like an aircon) where more equipments could do meaningful measurements or observation. – athos Jan 20 '20 at 14:45
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    If any study is to be conducted, it has to be in orbit around the sun. If the probe is on the surface of the sun, it will soon be engulfed due to the massive gravitational pull. – Sam Jan 20 '20 at 17:54
  • "The sun is gaseous all the way to the core." Kind of. The solar core density is estimated to be around 150 g/cm³, far denser than any solid at STP. – PM 2Ring Feb 02 '20 at 02:17
  • @athos An aircon has to dump the unwanted heat somewhere. How do you propose to do that inside the Sun? One concept I've read in science fiction is to beam the energy away using a very high frequency laser, but I'm not sure how well that'd work in reality. – PM 2Ring Feb 02 '20 at 02:23
  • @PM2Ring will something like an aircon work? – athos Feb 03 '20 at 10:03
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    @athos An aircon will work if you can figure out a way for the aircon to get rid of the unwanted heat. On Earth, that's easy: you just need to make the aircon's radiator hotter than the outside air. In the Sun, that's hard, because if the radiator is hotter than the Sun the radiator will melt, or vaporize. So you need to radiate the heat away some other way, like with an x-ray laser, as I mentioned before. Unfortunately, we don't yet know how to build efficient x-ray lasers. – PM 2Ring Feb 03 '20 at 10:22
  • @PM2Ring ah now i get it. on earth aircon works because the pipes are made of steel, not melting; it won't work if pipes are made of ice. i see the difficulties now.. – athos Feb 03 '20 at 11:24
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    @athos It's not just the melting. An aircon works a lot like a refrigerator. And for a fridge to make its insides colder it has to make its outsides hotter. There's a radiator on the back of the fridge, and it warms up the outside air. See https://physics.stackexchange.com/q/73035/123208 – PM 2Ring Feb 03 '20 at 11:39