This is a difficult question to answer with
certainty because there is so much time between
now and the milestones you describe. Colonizing
Mars and space might happen in the next few
hundred years (102 years). The sun will make
the earth uninhabitable to life as we know it in
approximately 600 million years (6 x 106 years),
as it gets hotter and brighter. The sun will
expand beyond Earth's orbit in 5 to 8 billion
years (~5 x 109 years). The sun isn't large
enough to explode as a supernova, but that
doesn't mean it will be easier to survive it
heating up and its eventually expansion.
These are very long periods of time, so long
that it can be hard to imagine just how long that
is. For context, life on Earth is
approximately 4.4 billion years old (4.4 x 109
years), so we have at least that amount of time
before Earth is consumed. Human beings have only
been a distinct species for 6 x 105 years and
civilized society is only 6 x 103 years old.
Thus, we have two times as long as we have been a
species (and 2000 times more than we've had
society) to leave our solar system.
I believe it is reasonable to assume that we
will survive long enough to solve this issue.
Consider all the things we have accomplished as a
species in the last 6000 years. Or even the last
100 years. However, I also think that it is very
difficult for us to guess how to successfully
leave this solar system at our current level of
technology and scientific understanding. We will
have to break the problem into much smaller ones
and answer those first. Colonizing Mars is a good
example - it is not strictly necessary to settle
on Mars before exploring the rest of the galaxy,
but we will learn many lessons in the process.
Each step will bring us a little closer, just as
going to space and the landing on the Moon
brought us closer to Mars.
Thinking about "deep time" problems like this
bring up a new question: when you ask "will we
move to another galaxy, what do you mean by
"we"? Humanity may change significantly in
that amount of time - it might be hard for us to
recognize ourselves then. We may also "survive" by
sending information or biological programming out
into the galaxy, rather than our human bodies. In
your opinion, does this count?
If you are interested in answering this
question, I recommend you look to science fiction
too - scientist tend to focus on what do and do
not know for certain (which is a lot), rather than
what might be possible. You might start with a
collection of short stories called "Year
Million: Science at the Far Edge of Knowledge",
Kim Stanley Robinson's "2312", Greg Bear's "City
at the Edge of Time", or Ben Bova's "Mars" series.
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