Most of the thrust of rocket is loss to get out of the earth gravity well.
I believe that this is wrong.
The Earth weighs 5.96e+24 kg and has radius 3.37e6 meters. The Sun weighs 1.98e30 kg and our orbit has radius 1.496e11 meters. That means that the potential well for getting away from Earth is about 11,800,000 joules/kg while for getting out of the Sun's gravity well is about 882,800,000 joules/kg. Assuming that I did the math right, that's about 7.5 times as hard.
It therefore takes a lot more energy to climb out of the Solar System than it does to climb out of Earth's gravity well. Voyager got a LOT of energy from those gravitational slingshots.
You're trying to compare apples to apples here, when the situation we have is actually apples to oranges ;).
A probe has to launch from the surface of the earth. But counterintuitively the probes start out already in solar orbit, even before they're launched -- because the earth is in solar orbit. More than half the energy required to achieve escape velocity is needed just to get into a roughly circular relatively low orbit around the gravitating body you're trying to escape from, and by virtue of being launched from the earth the probes get that velocity for free. Furthermore, the earth's orbit isn't really "low" with respect to the sun. We're fairly far out there, so the fraction of the energy needed to go from earth orbit to a solar escape trajectory is even less.
Put another way: a probe launched from the earth gets no help leaving earth's gravity well. But once it does it gets a huge automatic gravity assist from the earth itself as it enters solar orbit.
There's a saying among KSP players: "Orbit is halfway to anywhere".
It takes a lot of energy to escape Earth's gravity well. It takes a comparatively tiny amount to escape the Sun's, as a sibling poster pointed out. It would be actually harder to visit, say, Mercury, as you now have to shed all the energy Earth has given you for free, in order to "fall" into the Sun's well.
Voyager wanted to visit multiple planetary bodies – changing orbital parameters is not cheap. But if all it wanted was to get out of the system, burning straight out would probably be cheaper (in Delta-V terms). The closer to Earth the better, for the Oberth effect.
This is correct. On my astrodynamics written qual, one of the questions was whether it would be more energy efficient to solve Earth’s trash problem by launching garbage into the sun or into deep space. Surprisingly, the answer turned out to be deep space, and it wasn’t that close.
I believe that this is wrong.
The Earth weighs 5.96e+24 kg and has radius 3.37e6 meters. The Sun weighs 1.98e30 kg and our orbit has radius 1.496e11 meters. That means that the potential well for getting away from Earth is about 11,800,000 joules/kg while for getting out of the Sun's gravity well is about 882,800,000 joules/kg. Assuming that I did the math right, that's about 7.5 times as hard.
It therefore takes a lot more energy to climb out of the Solar System than it does to climb out of Earth's gravity well. Voyager got a LOT of energy from those gravitational slingshots.