Can one put a large nickel-iron asteroid into an elliptical solar orbit that results in a soft(ish) earth landing?

Question

My daughter and I are debating whether it is technically feasible to bring a nickel iron asteroid to the surface or the earth non-destructively. I felt it was impossible. Her thought was that with a propulsion system like M2P2 (already on the drawing boards) that would act on the entire ferro-magnetic body at once so as to avoid breaking it up, and a long enough time to fiddle with its orbit, one could put it in an elliptical solar orbit whose aphelion precisely matched the position, rotational velocity, and orbital velocity of some spot on the surface of the earth. With a big enough chunk, atmospheric effects would be tiny.

This would not exactly be a Hohmann transfer orbit to earth from the vicinity of Mercury, but would be similar in concept. After some thought, I granted that it might be possible but calculating the orbital mechanics would be tough slogging and I am pretty sure the required precision is not currently available in any propulsion system. The earth is also not perfectly round and the thing would probably have to bash through a couple of mountain ranges before coming to rest unless the whole last bit of its orbital spiral was over an ocean.

The question remains though. If we had the time, computing power, and precision thrust, could this be done? Imagine a few kilometer high piles of nickel iron (with very interesting impurities) in isolated spots around the equator. The impact on industry, science, the economy and human life in general would be profound.

Answer 1

If gravity was repulsive between the Earth and the asteroid, this could at least make sense in principle. In that case, getting close is like pushing on a spring, and with a carefully managed trajectory, you might be able to finish the trajectory on the surface, with zero gravity.

But gravity isn't like a pushing spring, it's like a pulling spring. That means that you can't cancel your incoming velocity by some carefully managed trajectory. This argument can be made from many different perspectives.

Energetically, you have a problem getting rid of the potential energy relative to Earth, not relative to the sun. You need to both get within the Earth's sphere of influence (SOI), and reduce your velocity to a small amount in order to have a soft landing. Short of quantum tunneling, this just doesn't make any sense. In order to exist at all on the edge of Earth's SOI you must have tremendous gravitational potential energy.

I see somewhat what the thought is. If you had a lower energy orbit relative to the sun, you could match the total kinetic-potential energy to a point on the Earth's surface. But the problem is that there's no way to transition between the two. That would also not be the case for a Mercury-Earth elliptical orbit. The kinetic energy at perihelion is lower than an Earth orbit relative to the sun, but that's irrelevant. Kinetic energy is huge relative to the Earth itself.

Answer 2

While AlanSE did a fine job of addressing it I think I can do a better job of addressing where she's going wrong:

Yes, I think an orbit as she is envisioning exists. The problem is that Earth has mass and will pull it out of solar orbit. It won't be in that nice orbit when it hits.

The closest you could theoretically come to a soft landing would be to put it in Earth orbit and then use whatever engine you have to lower the perigee to zero. It's going to hit at about 8km/sec but you could make that a rolling impact rather than a great splat. If the Earth were a billiard ball with nothing on it that matters you could land stuff that way. The real Earth has things like mountains and cities, though.

I see only two even remotely possible landing sites: Antarctica and Siberia. Whether it would stop rolling before leaving those areas is a matter for those far more skilled than I. I wouldn't say either touchdown is ecologically benign even if it stops in time, though.

Besides, we don't need it. If you can move the rock move it into Earth orbit. Put a mining facility on it, extract & purify the metals and foam them. Take some of the slag, foam it and use it as a heat shield. Put it in a mass driver and give it enough of a kick to deorbit. The dropzone must be water and huge because there's no drive or guidance on it. The slag burns in the upper atmosphere, make sure there's enough that it doesn't burn through. Once it's been slowed it's just a falling object, it's terminal velocity is low enough that it won't melt or be vaporized on impact. While it might not survive the impact intact the pieces float. Go pick them up.

Answer 3

Alan SE gave a good answer. But I'd like to add some stuff.

As Alan mentioned we need to consider earth's gravity. Once an asteroid comes within earth's sphere of influence, it's path is no longer well modeled as an ellipse about the sun. At that point it becomes better modeled as a hyperbola with a focus at earth's center.

The speed of a hyperbola is $\sqrt{V_{escape}^2+V_{infinity}^2}$ I remember this by thinking of $V_{escape}$ and $V_{infinity}$ as legs of a right triangle. Then the hypotenuse is the hyperbola's speed. This memory device relies on the good old Pythagorean theorem.

But what's $V_{infinity}$?

For practical use, $V_{infinity}$ is the asteroid's speed with regard to earth when it enters earth's Sphere Of Influence (SOI). Earth moves about 30 km/s about the sun. If the asteroid's moving a little slower than earth, say 29.5 km/s, then $V_{infinity}$ is .5 km/s. If the asteroid were moving a little faster, say 30.5 km/s, we'd still have a .5 km/s $V_{infinity}$.

The best we can do is match earth's 30 km/s which would make $V_{infinity}$ zero. In which case $\sqrt{V_{escape}^2+V_{infinity}^2}$ becomes $\sqrt{V_{escape}^2+0^2}$ which is simply $V_{escape}$. Then the orbit becomes a parabola rather than a hyperbola.

$V_{escape}$ is about 11 km/s near earth's surface. A parabolic orbit is the slowest you can get.

However...

It's possible to use the moon's gravity to slow the rock's speed relative to earth. Using the moon we can drop the speed from a parabolic orbit to an elliptical capture orbit about the earth. The rock can then be parked in a lunar orbit for surprisingly little delta V. I would suggest you and your daughter check out the Keck Report. Co-authors of this asteroid retrieval study include Chris Lewicki (chief engineer for Planetary Resources) and J. S. Lewis (author of Mining the Sky)

I give Planetary Resources better than even odds for retrieving asteroids and making a profit.