Having just finished Arthur C. Clarke's "Childhood's End", my question is, what exactly will happen to the stability and configuration of our solar system if our planet Earth was to just disappear? I understand that numerically determining the stability and the configuration of our solar system in the long run (millions of years in the future) is a chaotic problem and we don't know what will happen in the long run. But in the short run, orbital dynamics are understood enough that we can keep satellites in space and predict the planets position and paths to a reasonable accuracy for the near future. So if Earth was to just poof disappear all of a sudden, would the solar system destabilize where planets crash into the sun or escape the sun? Would the solar system stay mostly the same because the Earth is rather insignificant compared with the Sun and the giant planets? Or would the solar system stabilize into another different configuration? Or would it be a mixture where the inner-planets might change drastically/crash into the sun or each other but the outer giants will stay where they are?
To be specific, I am talking about the Earth just disappearing (somehow)? I am not talking about the Earth exploding, shattering into a million asteroids/comets, or splitting in half with one half crashing into the sun and the other perhaps escaping the Sun, or colliding with another body, or creating another asteroid belt, nor it escaping perpendicular to the ecliptic plane nor escaping in any other way. Would a very tiny mass just disappearing have any effect on the solar system? Or are we really that insignificant in the grand scheme of things?
In addition to answers, if any studies have been done, i.e. numerical or theoretical, I would love to see some references (published or otherwise).