## Friday, November 02, 2012

### The fast and slow of planetary travel

kw: interplanetary travel, analysis

The Hohmann Ellipse principle has been known for more than a century, and was once thought to be the lowest-energy method for moving from one orbit to another around a planet, or from one planet's orbit to another around the Sun. Since the 1970s, lower-energy methods utilizing "slingshot" paths by planets have been devised. These are OK for mechanical spacecraft, which don't get bored like people do, but I am more interested in ways of getting around that are at least as fast as a Hohmann Ellipse. This table shows why:

These are "mean-to-mean" times, and they are in years. Careful choice of launch timing, so the orbit is between aphelion of the inner planet and the perihelion of the outer planet, can reduce these times by a few percent. Thus the average transfer time from near-Earth to near-Mars is 0.709 years, or 8.5 months, while the minimum possible is 0.657 years, or 7.9 months. The least favorable ellipse from Earth to Mars is 0.762 years or 9.1 months.

Going farther takes longer, a lot longer! Earth to Jupiter is about 2y 9m, and you can see how the times multiply from there. It is an interesting paradox that, the closer to the Sun you start out, the quicker you get to your outer target. For example, from Mercury to Mars is less than six months. That is because, when the spacecraft crosses Earth's orbit, it is going a lot faster than one that leaves from Earth. However, the energy needed to exit Mercury orbit and to enter Mars orbit after 5.6 months is much greater than that needed to leave Earth orbit and then enter Mars orbit after 8.5 months.

The lower row shows why it will be a long time before any astronaut is sent to Neptune. Launch a brand-new PhD, aged 25; she'll arrive at age 55. Let's hope she is fanatically productive, and willing to retire "out there," because the soonest you can get her back to Earth will be after she is 85 years old. But after 60-plus years at zero G, she won't be able to return to Earth's surface. She'll have to watch the planet roll by from her orbital nursing home. PS: I picked a female astronaut because women have a 10x chance of living to 85, compared to men.

Another reason is that energy is very costly in space. I am not talking about solar energy you can catch to run your zero-G toilet or for growing your food. I am talking about propulsion. This is why the NASA folk are so enamored of the very-low-energy (but very much more time-consuming) "slingshot" pathways around the solar system. But what if we find a means of fueling much faster spacecraft?

The ideal would be a ship that could accelerate at 1G (9.8 m/s²), indefinitely. Before starting to analyze this, I checked on the influence of the Sun. At the orbit of Mercury, some 58 million km from the Sun, a stationary craft, not in orbit, would need to push with 0.004G to resist Solar gravity. Not bad, and there is less the further you go out. So I ignored the Sun to make the following table:

I used each planet's average orbital distance as the datum for the "scale" of its influence. A convenient place for a space station or colony would be the L4 or L5 spot ahead of or behind the planet in its orbit. The distance from a planet to these Lagrange points is the same as its distance from the Sun.

Going 58 million km, to get from Mercury to L4 or L5, is pretty quick, only 1.78 days. That's under 43 hours. From Earth to its L4 or L5 is not much longer, only 2.43 days or 58 hours. With such a propulsion method, it is a matter of less than a week to get from Earth to Jupiter, and two weeks will take you nearly to Neptune. Now our astronaut can go to-and-from Neptune in about a month, so there's no need to retire upon return. Even getting from Uranus to Neptune, when they are at opposition, is only a little longer than getting to either one from Mercury; no more than 20 days.

However, look at the peak velocities (Vmax): ranging from 752 km/s bombing around near Mercury to more than 6,600 km/s getting to Neptune or its Lagrangian point(s). Compare this with planetary orbital velocities ranging from 10 km/s (Neptune) to 45 km/s (Mercury). At thousands of km/s, even a grain of sand packs quite a punch. Imagine the kinetic energy of a locomotive (say, 20 million J) concentrated on a square millimeter of a spacecraft's meteor shield. You'd need a dozen meters of Kevlar® to stop it! It would melt around a cubic meter of the Kevlar® into an odd "plasticsicle".

Multiply that kinetic energy, 20 million J per microgram, times the mass of a spacecraft plus its meteor shield. Bombing around the solar system is never going to be cheap. Drat!