I wonder how very low-gravity dust could make bind together and make hard and dense stones that later on will merge together and make some of the planets? I read some meteorites as hard as a hard stone are remnants of the early phase of solar system's formation. If this is the case, it shows that they are formed using their own gravity attracting other dust; however, I thought making such a hard stone will require a high pressure.
The process you are referring to is called pebble accretion, which is very complex and not particularly well understood.
The initial premise of your question, that dust particles stick together via their own gravity, is not quite correct. As you say, gravitational interactions between individual particles are incredibly weak. Rather, dust particles grow via collisions within a protoplanetary disk. During this early stage of star/planet formation, gas and dust are spread out in a huge disk in which the material orbits the newly forming star at the centre. Particles collide and stick to each other via electrostatic forces. Growth continues all the way through from dust particles, to much larger planetary building blocks called planetesimals. In this sense, meteorites are indeed remnants of the early Solar System. Sometimes you will hear them refer to as 'failed planets', in that they failed to accrete enough material to grow to planetary size.
There are still a lot of open questions relating to this process. For example, once particles reach sizes greater than about a meter, collisions result in fragmentation rather than growth. This has led to the development of newer theories, such as streaming instability, which are able to better explain rapid growth.
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