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The mass of the Sun ~2×1030 kg is more than the mass of all the planets in the Solar System combined. The Sun is the centre of mass, and stability of the entire Solar System. A white dwarf, however, cannot be greater than 1.4 Solar masses (Chandrasekhar Limit). So in the scenario you're proposing, you're basically adding a second Sun to the Solar System. The mass of the white dwarf, however, is concentrated into a much smaller space than the Sun (Earth sized).
The consequences would be enormous: the shape and stability orbits would change dramatically. These changes might be so dramatic that planets may be ejected from the System (it is nearly guaranteed that smaller masses, like asteroids, would suffer ejection).
The gravitational pull of the white dwarf would pull material off the Sun. This matter would accrete onto the smaller star causing (at the very least) novae to occur.
If enough matter was accreted to push the white dwarf over the Chandrasekhar Limit, the white dwarf would undergo a Type 1A supernova. The energy released during that event would be devastating to the Solar System (or what was left of it).
If the white dwarf just happened to 'pass through' our Solar System, and didn't accrete enough matter to trigger a supernova, there would still be consequences to the System. The Sun would have lost a significant amount of mass, the planets would settle into new (closer) orbits over time. This would, in turn, change the temperatures and therefore environments on each.
This visualization tool: Super Planet Crash is a great way of 'playing' with modified Stellar Systems. You can easily add a few planets, and stars, to a system and watch the resulting modification of orbits.
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