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I haven't done any calculations on this, but I believe that the Earth and the Moon should be considered one body of mass when calculating its solar orbit.

So if the Moon was perhaps vaporized, e.g. with a huge nuclear device, perhaps the loss of mass could be sufficient for the Earth to escape the Sun's gravity. But I have no idea how long would it take? But given you remove 1/7th of the total mass of the system, I reckon it will be relatively quick.

Or maybe, if the idea is sound, you could first get the Moon to escape Earths orbit in some way. Whatever amount of energy would be needed to get the Earth to escape the Sun, only 1/6th would be needed to get the Moon to escape the Earth. And if such an event was the collision with a very large stellar body, the collision itself wouldn't itself kill off humanity.

This will also give you more freedom in deciding how long a warning time Humanity should have, as you can decide at which rate the Moon should escape from the Earth.

p.s. Completely unrelated to the actual answer (I don't have enough rep to add comments), Arthur C. Clarke's book The Songs of Distant Earth is similar to your plot description, i.e. humanity trying to create a future for itself as the Sun is about to go supernova using sub-FTL technology

I haven't done any calculations on this, but I believe that the Earth and the Moon should be considered one body of mass when calculating its solar orbit.

So if the Moon was perhaps vaporized, e.g. with a huge nuclear device, perhaps the loss of mass could be sufficient for the Earth to escape the Sun's gravity. But I have no idea how long would it take? But given you remove 1/7th of the total mass of the system, I reckon it will be relatively quick.

Or maybe, if the idea is sound, you could first get the Moon to escape Earths orbit in some way. Whatever amount of energy would be needed to get the Earth to escape the Sun, only 1/6th would be needed to get the Moon to escape the Earth. And if such an event was the collision with a very large stellar body, the collision itself wouldn't itself kill off humanity.

This will also give you more freedom in deciding how long a warning time Humanity should have, as you can decide at which rate the Moon should escape from the Earth.

Edit

I realized that this answer must be wrong. If we reduce the mass of the orbiting system, it would of course reduce the centripetal force, but due to the reduced mass of the system, the acceleration toward the Sun would be unchanged (just like two objects with different mass fall with the same speed in vacuum)

p.s. Completely unrelated to the actual answer (I don't have enough rep to add comments), Arthur C. Clarke's book The Songs of Distant Earth is similar to your plot description, i.e. humanity trying to create a future for itself as the Sun is about to go supernova using sub-FTL technology

I haven't done any calculations on this, but I believe that the Earth and the Moon should be considered one body of mass when calculating its solar orbit.

So if the Moon was perhaps vaporized, e.g. with a huge nuclear device, perhaps the loss of mass could be sufficient for the Earth to escape the Sun's gravity. But I have no idea how long would it take?

Or maybe, if the idea is sound, you could first get the Moon to escape Earths orbit in some way. Whatever amount of energy would be needed to get the Earth to escape the Sun, only 1/6th would be needed to get the Moon to escape the Earth. And if such an event was the collision with a very large stellar body, the collision itself wouldn't itself kill off humanity.

p.s. Completely unrelated to the actual answer (I don't have enough rep to add comments), Arthur C. Clarke's book The Songs of Distant Earth is similar to your plot description, i.e. humanity trying to create a future for itself as the Sun is about to go supernova using sub-FTL technology

I haven't done any calculations on this, but I believe that the Earth and the Moon should be considered one body of mass when calculating its solar orbit.

So if the Moon was perhaps vaporized, e.g. with a huge nuclear device, perhaps the loss of mass could be sufficient for the Earth to escape the Sun's gravity. But I have no idea how long would it take? But given you remove 1/7th of the total mass of the system, I reckon it will be relatively quick.

Or maybe, if the idea is sound, you could first get the Moon to escape Earths orbit in some way. Whatever amount of energy would be needed to get the Earth to escape the Sun, only 1/6th would be needed to get the Moon to escape the Earth. And if such an event was the collision with a very large stellar body, the collision itself wouldn't itself kill off humanity.

This will also give you more freedom in deciding how long a warning time Humanity should have, as you can decide at which rate the Moon should escape from the Earth.

p.s. Completely unrelated to the actual answer (I don't have enough rep to add comments), Arthur C. Clarke's book The Songs of Distant Earth is similar to your plot description, i.e. humanity trying to create a future for itself as the Sun is about to go supernova using sub-FTL technology

I haven't done any calculations on this, but I believe that the Earth and the Moon should be considered one body of mass when calculating its solar orbit.

So if the Moon was perhaps vaporized, e.g. with a huge nuclear device, perhaps the loss of mass could be sufficient for the Earth to escape the Sun's gravity. But I have no idea how long would it take?

Or maybe you could just destroy enough of the Moon to get that to escape its orbit around Earth?

p.s. Completely unrelated to the actual answer (I don't have enough rep to add comments), Arthur C. Clarke's book The Songs of Distant Earth is similar to your plot description, i.e. humanity trying to create a future for itself as the Sun is about to go supernova using sub-FTL technology

I haven't done any calculations on this, but I believe that the Earth and the Moon should be considered one body of mass when calculating its solar orbit.

So if the Moon was perhaps vaporized, e.g. with a huge nuclear device, perhaps the loss of mass could be sufficient for the Earth to escape the Sun's gravity. But I have no idea how long would it take?

Or maybe, if the idea is sound, you could first get the Moon to escape Earths orbit in some way. Whatever amount of energy would be needed to get the Earth to escape the Sun, only 1/6th would be needed to get the Moon to escape the Earth. And if such an event was the collision with a very large stellar body, the collision itself wouldn't itself kill off humanity.

p.s. Completely unrelated to the actual answer (I don't have enough rep to add comments), Arthur C. Clarke's book The Songs of Distant Earth is similar to your plot description, i.e. humanity trying to create a future for itself as the Sun is about to go supernova using sub-FTL technology