I do not need LUCK, I already ran the calculations with the Rocket equations and some Rubber-Hydrocarbon SRB hollow metal tubes.
What is the Delta-V to launch from an asteroid in a Transfer Orbit to intercept another asteroid? How much LUCK does that take to achieve the needed Delta-V?
That is TL 4 "Congreve Rocket" technology without Earth's gravity well.
You may have the delta-v needed for the application.
Control though ... "Congreve Rockets" were hardly known for their accuracy
... and in space travel, a "lack of accuracy" in your flight trajectory can be MIGHTY FATAL.
Guidance and control are also necessary, not to mention highly accurate navigation and precision.
If you are lacking in these essential qualities, you're going to need more than LUCK ... you're going to need more than your fair share of prayers.
TL 4 has Steam and Electricity ... so it CAN be done.
Airless worlds tend to be lacking in surface water resources though ...
Airless worlds also tend to be lacking in trees (necessary for the geologic process that produces coal, which you would need in order to power your steam engines).
I'm seeing a LOT of problems that cannot be overcome at TL=3-4.
You have assumed a COLONY as a goal rather than a RESULT.
I thought you were setting up an "offworld prison" where the whole point and purpose of being sent there is ... a life sentence (without a chance for parole).
What if our sun became more energetic and began to strip away the Earth's atmosphere (or the core cooled and the magnetosphere stopped protecting it).
Then the planet survives but the biosphere doesn't.
The surface becomes ... uninhabitable.
Next question.
So 1 billion people in 1800 have a choice to adapt or wait to die.
Pretty much.
Could they adapt to an Earth with an atmosphere like Mars in 100 years?
A few might.
Most won't.
Population code: 9 goes to 1 or 0. It won't be pretty or pleasant.
Additionally, any population that "survives" the environmental transition would be highly unlikely to be able to perpetuate itself and "reclaim the planet" after such a calamity. So even if the environmental changes don't get you, the internecine warfare that results from greedy people fighting over a continuously shrinking "pie" of resources probably will. Even if a faction "wins" against their rivals, under such conditions it is highly likely that such a victory will be both pyrrhic and short lived (see: "nasty, brutish and short" if references are required).
Could they survive in 1900 or would mankind all be extinct?
Under the scenario you describe, the entire biosphere of the planet would almost certainly collapse. That usually doesn't "end well" for anyone sitting on top of the food chain.
So my guess would be EXTINCTION EVENT.