A rogue Earth/Moon system were roaming the galaxy at 4% light speed
would maintain life on Earth by putting up a fusion powered light bulb
of adequate size.

A Deuterium fusion reaction releases 576 million megajoules per kg of
material.

So, a fusion powered light bulb that operated on the moon is made to
replace the sun. To power the bulb requires 302.8 kg/sec of deuterium
be fused. The filament for the bulb is the same temperature as the
Sun it is 59.2 km in diameter - and focused upon the Earth with a
fresnel type lens operating at Lagrange Point 1 - to make the system
more efficient. The moon orbits the Earth once a month (moonth) and
the Earth rotates on its axis every 23.93447 hours. But the moon
moves nearly 14 degrees to the East in that period, so the Earth's day
would lengthen to almost exactly 25 hours.

The Earth's oceans mass 1.4e+21 kg and contain 2.4e+16 kg of
deuterium. Enough deuterium to power the light bulb for 2,543,000
years!

At 4% light speed (12,000 km/sec) the Earth would cross 100,000 light
years in that period.

If we made use of protium, instead of deuterium, we could sustain life
unchanged on Earth for 1.6 billion years and reduce the amount of
hydrogen in the oceans by 10%. The Earth would traverse 64 million
light years in that time at 4% light speed.

The Earth Moon system could be accelerated to 4% light speed by
interacting with a pair of neutron stars or black holes that flew
through the solar system at 1% the speed of light, and orbiting one
another at 5% the speed of light, separated by 1 million km and
massing I forget how many solar masses - not large.

The Earth Moon system falling through the bary center of the two
evenly matched black holes would be accelerated by tidal action in a
few minutes - without causing a disruption of the Earth Moon system.

If we assume human technology has achieved low cost fusion power by
this date, we can imagine that such a light bulb would be built over
the year or two of global winter that encroached the Earth as we left
the solar system. At 4% light speed the Earth would attain 100 AU in
two weeks after the encounter - which would reduce solar radiation to
1/10,000th today's level. It would also make escape to the solar
system difficult, since attaining 4% light speed would be difficult
even with fusion rocket ships.

The day the sun lamp was switched on, would be a day to remember.

If we were headed in the right direction we would pass Alpha Centauri
in about 110 years. By that time, we might have worked out
techniques to hop off the world and establish new colonies there. If
the direction were chosen rightly, we would interact with the Centauri
system and fly to another neighbor and repeat the performance.

After two or three of these combination shots we might have reason to
wonder if the compact binary star pair that caused all the trouble to
begin with wasn't engineered for our benefit. Especially if a starter
colony on Mars were underway, and in the mayhem following Earth's
departure, it was brought closer to the sun, while Venus was brought
further - both becoming more Earth-like.

In 1,000 years after departure the Earth would be 40 light years from
Sol and may have populated several star systems along the way. Each
of which would be developing on its own free from Earth influence.

In 10,000 years after traveling 400 light years from Sol, hundreds of
star systems might be seeded, and we'd have the means to engineer the
Earth entering orbit around a remote G-type star.