I'm embarking on a fantasy book so I guess in essence you can write about anything but I do want an element of plausibility. Could you have hot desert caps on a planet and an icy/winter equator?
Certainly. I will assume by desert you mean "hot and arid" rather than just arid. A planet could obviously be entirely frigid with the poles significantly more arid than the rest of the planet.
One possibility could be a cold planet with limited volcanic activity which, for whatever reason, is only observable or easily measurable near the poles. The geothermal activity near the surface heats the polar regions to near boiling, effectively turning the water on the surface and in the lower atmosphere to vapor. Combined with an interesting weather stream which frequently generates high winds, the poles are a near absolute desert.
The planet might have a 90° axial tilt.
That means each pole will be in direct sunlight for half a sidereal year and in shadow the other half year.
The equator will receive exactly 50% sunlight and 50% nighttime each day. But it will still experience drastic seasons, because over the course of a year the sun's height at noon will vary from perfect zenith to just above the horizon. Twice a year there will be a sunset which will take all night and directly transition into the sunrise of the next day.
The poles will receive twice as much sunlight as the equator, so they will be hotter on average. Just like deserts on earth, they will have very hot days and very cold nights. Just that both take far longer than on earth, so the temperature differences will be even more extreme. It's unlikely that much fauna and flora will survive under these extreme conditions, so they will be quite desert-like.
Yes, it's entirely possible.
Your planet would have a pretty extreme axial tilt, something like 50- or 60-plus degrees. You would see the following effects:
The poles would have extremely long 'days' equal to years, as we currently see on earth. The difference is that during the summer, the 24-hour sunlight would be much, much warmer, potentially warm enough to dry out the whole region - hence hot desert. During the winter, the area would freeze solid for six months.
The equator would be much cooler, since it never gets close enough to the sun to warm up. At extreme tilts, this would be similar to the conditions in the arctic circle, where even in summer it's generally not warm enough to melt permafrost. The equivalent of summer for the equatorial band would be the equinox, when the sun would be directly overhead. Whether the days at this time are long enough for ice to melt is up to you.
Between the two would be temperate bands, just as we have today. This is likely where most life would be found.
You'd also likely see some strange weather. Strong, hot winds would blow from the summer pole towards the winter pole. This could bring energy to the equatorial regions, again, depending on your design.
You could make more of a regular day/night cycle at the poles by having the star a relatively cool red dwarf and having the planet orbit much closer, making years much shorter.
Two stars orbiting each other with a planet in between. The planet is tidally locked to both stars so each pole always faces 'its' star.
As some have pointed out this system is not 'naturally' stable. But, in the comments, Jose Antonio Dura Olmos suggested an intriguing notion that could introduce a dizzying array of potential plot points - the system is stable due to an ancient civilization's experiment with their new technology. After they had successfully created and stabilized this system they applied the knowledge to their home world and in the successive millennia life arose independently on this world. Or perhaps they seeded this world with microbes and intelligent life came about through evolution.
As I said, the plot points for where this could go are amazing. The intelligent life gains enough understanding that their home world is unstable and yet it exits. Who were the ancients? The system is degrading and they need to fix it. They can't and they need to come to terms with their demise. Or they go off in search of the ancients who would share their DNA.
Your planet is actually a moon with one side always facing a hot gas giant. One side of the planet would always get warmed up by the gas giant and would have two day-like periods where the sun rises on the horizon, then sets behind the gas giant, then rises from behind the gas giant and sets on the other side of the horizon, giving an effect similar to a daily eclipse. The other pole would be more temperate and have only one day-like period, where the pole faces out from the planet towards the star and gets warmed up, and where the pole faces out from the planet away from the star and gets cooled at night.
The tricky part is getting the distance from the star and gas giant just right so that, given a livable atmosphere, liquid water and solid ice can both exist on the surface of the moon. You would also need to get the orbital period to be such that you don't get nightly frost on the side that faces away from the gas giant. To make the poles arid, spin the planet; this can keep most weather from approaching the poles. Some severe storms could still form in polar regions, but these would happen much more often on the outward-facing pole than the inward, and they would last longer on the outward-facing pole. The inner pole might still get storms from time to time, but they would likely be much milder because these are more influenced by the gas giant than the star.
The result is either a moon with an icy band around the middle and deserts on both poles, or you get an outer pole that is wet, stormy, and full of jungle life and an inner pole that is a dry desert, with a massive wasteland of ice in between.
This is very difficult to pull off with normal physics, because of the reasons why poles are typically cooler:
- There is little direct sunlight constantly heating the poles.
- The magnetosphere holds less atmosphere near the poles, meaning there is less insulation in those places. Heat radiating outward has less atmosphere to get through.
- Counterintuitively, light coming from the sun has to pass through more atmosphere in order to reach the surface, because it's coming in at a low angle, and thus is more likely to be deflected into space.
- Antartica is colder than the Arctic, largely because it's mostly land and at a higher elevation, whereas the heat capacity of the Arctic stabilizes the temperature at a higher level.
Therefore, to establish a warm polar situation we might do a few things. First, we note that our own poles are relatively arid, receiving in places the same precipitation as the Sahara desert. If we put land in both places which is just above sea level, we are off to a good start. We can posit we get it warm enough that snow (a primary reflector of sunlight) is not there, and we say that most of the area is covered with black sand or the like. Increased volcanic activity in this area would obviously help with this, meaning we are getting heat from both the planet core and from the sun.
This suggests a strange magnetosphere as well - if the magnetic 'pole' were not only close to the equator but rotating (rather than being mostly fixed), it would mean that heavier atmosphere would concentrate in the polar regions and also that volcanic activity was more turbid. (I have no idea if a magnetic pole would reasonably do this, but we can stretch a bit here I think.)
A significant tilt to the planet would help, but too much won't help because for a quarter of the year the area would be entirely in the dark and therefore cold, and for half the year the equator would actually be nearer the sun than either pole (because the axial tilt of a planet stays pretty constant in relation to the fixed solar plane). Some tilt, though, means more direct sunlight for part of the year.
Meanwhile, the equator will need to largely be comprised of very high mountain ranges. On an Earth-sized planet, you lose about 6.5 degrees Celsius per kilometer of height, so we'd want as much of the equator to be as high as possible. Maccu Piccu is about 8.5km up, Tibet is on average 4km, to give you an idea of what 'high' is. The surface should be largely reflective: sand or white stone, and some hostility to plant life.
To really make this work out, I'd posit some high-altitude high-albedo clouds in the equatorial regions - far more than you'd see on Earth, and periods of high-intensity short-burst rainfall in the north, where water that evaporates captures a lot of heat and forms flash storms, flushing that heat back to the surface while still trapping it in the atmosphere. Having a secondary heat source (large moon, second star, or gas giant) also helps, to get direct light on at least one polar region.
All that said, this strains credulity to the maximum, because most physics simply don't favor these conditions, especially not for very long.
You said fantasy, so rules of physic don't apply.
The planet could be orbited by a moon on an equatorial orbit which actively cools the planet through a magic cold radiation. This radiation gets partially absorbed by the atmosphere, just like the heat radiation by the sun. The moon is closer to the planet than the sun, so latitude will have an even larger effect on the intensity of its radiation than it has on sunlight.
The orbit might in fact be so close that even the difference in distance between surface and moon becomes non-negligible (that would likely put the moon inside the planet's Roche-limit, but we already established that it's magic, so magic powers might keep it from breaking apart).
Without this moon's cooling, the planet would likely be far too hot to be habitable. The people living in polar regions might understand that and worship it as a power of good which relieves them of the scorching heat whenever it appears over the horizon. Those living near the equator, however, will curse it as an evil entity which brings them unbearable frost.
Btw: Now that you have placed a huge ice-magic related BDO in a low orbit around your planet it would be a shame if it wouldn't be somehow involved in your main plot.
We can use a see through material at the poles as a green house. Have a see through material like quartz cover the top of the water ice and over time the heat from the sun will gather under it, melting the ice down to the surface. This will make the environment warm.
The surface should be an easily drain able material like sand. This way as the ice melts to reveal the ground the water drains down into the earth. This will provide the environment to be dry.
The planet will need a 0 degree axis to sustain this and simulate a night/day at the poles.
The rest of the planet, and even above the pole it can be a ball of ice making the planet much like you wish. As per one of your comments, this sort of environment will doubtfully create human like species.