July 29th, 2007


Mainspring - orbital dynamics

One of the things that troubled me as I was reading Mainspring by jaylake was the question "if the clockwork Earth is rolling along its orbital track, how many times does it roll per year?" I was bored this morning and did the math.

According to http://calgary.rasc.ca/howfast.htm...
  • The Earth has a circumference (distance around at the Equator) of approximately 40,075 km (24,901 mi)
  • The Earth's orbit around the Sun is an ellipse with an eccentricity (flattening) of 0.0167
  • Since the orbit is nearly a circle, and to avoid involving calculus, I'll treat the Earth's orbit as a circle with an average diameter
  • The average radius of the Earth's orbit is 149,597,871 km (92,955,807.46 mi)
  • Therefore the "circumference" of Earth's orbit (the path) is approximately 2 π R or nearly 940,000,000 km (580,000,000 mi) around
Let's assume that the clockwork Earth has the same diameter and orbital diameter as ours (Jay has confirmed this assumption in conversation).
  • Therefore the clockwork Earth rolling along its track turns (940,000,000 / 40,075 = 23456.02) times per year (Whoa. 23456. Freaky...)
  • If the clockwork year is the same length as ours, each day is only (365 / 23456 = 0.016) days, or (0.016 * 24 = 0.37) hours, long
  • If the clockwork day is the same length as ours, the year is (23456 / 365 = 64.26) of our years long
Checksum: validate these figures with some more figures about our Earth, again according to http://calgary.rasc.ca/howfast.htm...
  • So the Earth travels about 940,000,000 km in 365.2421896698 days
  • speed = distance/time = circumference/time = 939,951,145 km / (365.2421896698 days * 24 hr/day) = 107,229 km/hr (66,629 mi/hr)
  • This is 64 times faster than our rotational speed, or about Mach 90!
  • In one day the Earth travels only 1/365¼ of the way around its orbit or approximately 2,600,000 km (1,600,000 mi)
  • Since the Earth is 12,756 km (7,926.21 mi) in diameter, the Earth moves approximately 202 times its own size in one day!
Yes, it checks: our Earth's orbital speed is 64 times its rotational speed. If it orbited by rolling along its track (i.e. orbital speed reduced to rotational speed) it would take 64 years to go around the sun.

So one of the following must be true of the clockwork universe:
  • The sun goes up and down like a crazy monkey, rising and setting 3 times every hour (contradicts the book)
  • A year is 64 times longer than on our Earth, meaning that 18-year-old Hethor has eaten (18 * 365 * 64 = 4,204,480) breakfasts (implausible)
  • What the people of the clockwork Earth call a "year" (four seasons, 365 days) is 1/64th of a sidereal year (could contradict the book if there are any mentions in the book of the summer/winter constellations being different)
    • Implication: the clockwork Earth's track has 64 wobbles in it, so the North pole points away from the sun and then back 64 times per orbit
  • The clockwork Earth's orbital track is only (365 * 40,075 = 14,627,375 km) in circumference
    • Thus the clockwork Earth is (14,627,375 / 2pi = 22,976,626 km) from its sun (about 1/6th of an AU)
      • Thus the clockwork Sun is substantially cooler than our own
  • The clockwork Earth is (940,000,000 / 365 = 2,575,342 km) in circumference
    • Thus the clockwork Earth is (2,575,342 / 2pi = 409,878 km) in radius (about 64x our Earth's)
      • Thus the clockwork Earth has approximately (64 squared = 4096) times the surface area of ours
        • Implication: LOTS of room for more story
        • Implication: airships travel 64x faster on the clockwork Earth
Of these, the simplest explanation is that the clockwork Earth is only 1/6 of an AU from its sun. Given that this is a created system, the creator could easily have set the sun's brightness to be appropriate for that distance.

This does raise the question of: why 365.26... days? I suspect that the clockwork Earth's diameter and orbital diameter were carefully chosen by the Creator to yield a year of exactly 360 days.

Which leads to other issues I had with the book's theology and the problems it causes for the plot, but that's for another post.

ETA: The above was written before I'd seen this sketch of how the Mainspring solar system might work. The mechanism pictured provides as much as you'd like in the way of orbital eccentricity, axial tilt, seasons, and epicycles, but I don't think it fits with what we're told in the book. With this design, the gears come rumbling across the Equatorial Wall three times a day, not once, and all those decoupling rings and gimbals and such would be readily visible in the sky. But Hethor, who is a clockmaker and would certainly notice such things, only mentions the thin thread of the moon's (and at one point Venus's) orbital track. Not to mention that if all that stuff were in the sky (most of it in the plane of the ecliptic, by definition) you'd only get an unimpeded glimpse of the sun once in a great while. So I don't buy it.