Tuesday, September 16, 2003

The moravec flight and other miscalculations

First some numbers. The speed of light, c, is 300, 000 km/s; a year is 365.2422 x 86400 = 31, 556, 926 seconds. The earth's gravity, 1g, is 9.8 m/s2. So 1g for 1 year gives 309, 258 km/s, or 1.03c. A year is 8765 hours ~ 10, 000 hours. The Earth-sun distance, called one astronomical unit (AU) is 150, 000, 000 km or 500 light-seconds (i.e. light takes 500 seconds to cover it).

And some formulae. From a starting point at rest, under constant acceleration a for a length of time t, the velocity change is at, essentially by definition, and the distance covered is just the time multiplied by the average velocity or at2/2 .

Ilium

This is the new book from Dan Simmons, who for the most part writes horror, rather than hard SF, so may be excused to some extent. Alas, when he actually quoted some numbers for a high-speed dash across the solar system it was enough to interrupt my reading with a bang with a "those don't tie up even on a simple order-of-magnitude calculation".

The moravec flight from Jupiter to Mars, then at opposition is quoted as accelerating at 3000g, and ending acceleration at a speed of 0.3c. Relativistic effects go as square root of (1 - (v/c)2), or about 95% - so enough to throw your watch out, but not enough to seriously affect the Newtonian formulae used above.

To achieve 0.3c with 3000g acceleration is to take 1/10, 000 of a year or 3156 seconds, and will cover 0.15 x 3156 light-seconds = 473 light seconds, or 0.85 AU - but their accelerator is the Io flux tube, and Io's orbital radius is only about 500, 000 km.

Later it takes them a day to cover the distance between Mars' orbit and Earth's - about 0.5AU. That's 250 light-seconds in 86400 seconds or about 0.003c - and that's before their big deceleration. Assuming that lower speed, then the acceleration would take only 32 seconds, and cover 0.048 light seconds or 14,400km - a much more reasonable size.

Then there is the matter of the neutrino ring that intersects the Earth at a certain date of the year that isn't seen moving at some noticeable velocity across the ground as the Earth rotates - let alone as it moves through the intersection point...

Ring

Now Stephen Baxter is a hard-SF writer who ought to know better. In Ring, he has an expedition set out to go 5 million years into the future. The ship accelerates at 1g constantly, turning over after 1/4 of the time, coming to a standstill at half the time, then returning as it left; relativistic effects keep the duration on board much lower.

For a uniform acceleration, the relativistic formulae are

at/c = sinh( aτ/c)

and

ax/c2 = cosh( aτ/c) - 1

where t is the external time, x is the distance covered and τ is the time on the ship. The sinh and cosh functions are hyperbolic sine and cosine (what the "hyp" checkbox gives you on the Windows calculator in scientific mode). If we work in years and light-years, and assume that the ship accelerated at 0.97g, so that a=1 in these units, we have for t = 1.25 million years, then

τ = sinh-1(1,250,000)

So using Inv Hyp Sin 1 250 000 gives 14.73 years shipboard. By symmetry the total voyage is 59 years, and the furthest distance is a whisker under 2.5 million light-years - a bit further than M31 in Andromeda.

Then how come the journey took his ship 1000 years shipboard? After 250 ship years, the external date would be 1.8 x 10108 - and the return would come at 7.2 x 10108 if we ignore the corrections that would be needed to account for cosmological effects. This is about as far into the future as the deepest look in his later book, Time. So the acceleration must be wrong. At a 0.097g rate, the shipboard time for the 5 million year voyage is 125,000 = sinh ( 0.1τ ) or τ = 10 sinh-1(125,000) = 124 years per quarter, or 500 years in all, so a kiloyear excursion would need less than acceleration even than that.

But that's not the end of the howlers. When the ship arrives back at the solar system after a 5 megayear excursion, they find that the constellations are just as they left - in particular, α Centauri is 4 and a bit lightyears away, opposite a W-shaped Cassiopeia. It's a pity that in about 350, 000 years, α Cen will be the brightest star in the sky, about a parsec (3.3 light-years) away in the constellation of Cancer, before drifting away into the northern sky, getting further away all the time.

There's a similar howler in forgetting about things moving in Time, too, where the first of the fast-forward-future views would happen after M31 collides with our galaxy - unless of course the posthumans have done the engineering work to use the magnetic fields of every star in M31 to channel its stellar wind as a rocket and moved the whole galaxy en masse - central 100-megasol black hole and all - sometime in the next gigayear. We certainly have to assume that there is such massive engineering for there to be a coherent galaxy in the Black Hole Miners epoch, as dynamical effects will have caused a galaxy that is not being actively maintained to have disintegrated by that sort of timescale - see John Baez's useful summary of the end of things; or my own riposte in fictional form.

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