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How can artificial satellites be safe in the atmosphere when there are hundreds of tons of dust and asteroids coming into earth's atmosphere everyday?
Space debris is a big problem for space industry. Natural bodies like dust or asteroids are not the main challenge, although probes are frequently hit by fine dust. The dust causes tiny craters on the surface of the probes or can change rotation or orbit slightly. This can mostly be corrected, e.g. by wheels fueled from solar panels.
The most concerning objects are remnants of human-made objects circling around the earth. With each collition of two such objects clouds of fragments form, leading to ever more collitions. Those fragments are tracked by radar, and probes try to avoid collitions by adjusting their orbits. But sometimes they are hit and destroyed.
An orbit is the path along which a satellite travels around e.g. Earth.
A satellite must remain outside the lower parts of the atmosphere, otherwise it will be slowed down and fall to the ground. The lower limit is somewhere near 200 km above ground. A satellite travels with about 8 kilometers per second around Earth, when in a lower Earth orbit.
Many satellites are in the geo-stationary orbit, about 36,000 km above ground. There Earth's atmosphere is neglectably thin, only some minor magnetic field can be found there; that part of the atmosphere is called magnetosphere.
The further away a satellite orbits the Earth the slower it travels to compensate Earth's gravity, because Earth's gravity grows weaker with distance from the Earth.
At present the probability of collisions, which destroy satellites, is very low for a single satellite. Actually there are only a few documented cases: "only one major incident has occurred: the 2009 satellite collision between Iridium 33 and Cosmos 2251", of the above wikipedia article. And "China was widely condemned after their 2007 anti-satellite missile test, both for the military implications as well as the huge amount of debris it created" of the same Wikipedia article.
The central problem is the growing number of fragments to be expected for the future, as a consequence of more satellites in the orbit, and the expected chain reaction of fragmentation.
Details of surface degradation of satellites due to impacts of dust particles can be read e.g. here. Most of that dust ist human-made. Surface degradation has consequences, e.g. for solar panels.
See also a related question at StackExchange.
Regarding the second part of the question: The dust particles collide with the satellites and evaporate from the impact heat. So the annihilation distance is more or less zero.
When entering the earth's atmosphere "they usually disintegrate at altitudes of 50 to 95 km " (Meteorid). That's at least 100 km below the lowest satellite orbits.
