It's possible to work out the numbers -- soft-landing an asteroid is TOTALLY impractical. Let's see what one needs to do.
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This asteroid will be accelerated by the Earth's gravity, and when it reaches the Earth's atmosphere, it will be traveling at a little over the Earth's escape velocity, about 11.2 km/s. How much propellant will one need to consume to soft-land it?
Over a century ago, Konstantin Tsiolkovsky showed how to find how much. His rocket equation, for initial mass m
i, final mass m
f, effective exhaust velocity v
e, and velocity change v:
v = v
e * log(m
i/m
f)
Relation to specific impulse: v
e = I
sp*g
E, where I
sp is the specific impulse, and g
E is the acceleration of the Earth's gravity at its surface, about 9.81 m/s^2.
So to avoid consuming much more propellant than asteroid, the rocket must have an exhaust velocity more than the Earht's escape velocity.
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The asteroid must be supported by the rocket engines as it makes its landing. That requires that the thrust be greater than (asteroid mass)*g
E.
So for a thousand-ton asteroid, that requires a thrust of a million kilograms-force or 10 million Newtons. Bigger asteroids require more thrust, of course. I won't get into English-system mass and force units.
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So let's see what's available.
Spacecraft propulsion - Wikipedia has a big compendium of numbers in its "Propulsion methods" table, and Wikipedia's articles on various rocket engines often list the engines' numbers.
The highest-thrust engines that have been successfully run are chemical-combustion ones, at 1 to 2 million Newtons, and hydrogen-oxygen ones can get about 4.5 km/s of exhaust velocity. Ones with non-cryogenic propellants can get as much as 3 km/s. Because of the nature of their energy sources, it's difficult to get much more exhaust velocity than that.
So they are unsuitable.
One can get more energy per unit mass with nuclear reactions, and thus greater exhaust velocity, but there are problems here also. A nuclear reactor heating hydrogen can get around 9 km/s or thereabouts, which is still too low to be suitable. It cannot get much more than that without melting the reactor. Nuclear-bomb propulsion can get greater exhaust velocity, but it has certain other problems. Inertial confinement fusion would also get high exhaust velocity, but that mechanism has yet to produce energy breakeven in the lab.
So they are unsuitable also.
In contrast to these thermal systems, there are various nonthermal propulsion systems, like ion engines, that are in various stages of development. The Dawn spacecraft , currently at Vesta, has 3 ion engines, each with exhaust velocity 30 km/s and thrust 0.09 Newtons. Most other nonthermal engines have similarly low thrust.
These are still more unsuitable ones.
So there's no way that soft landing an asteroid is going to work.