Mainspring - orbital dynamics

[davidlevine]

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.

Comments

[jimvanpelt.livejournal.com]

Wow! I love it when someone does the math. So, Jay, did you do the same kind of math, or is the Earth or some sort of set of gears that slow it down?

[davidlevine]

Jay's busy at the day job right now, but he mentioned to me that he did not really work out the gearing or any of the other details. Which is why I felt compelled to do the math and see if it worked out anyway. It doesn't, alas, but I must confess the number 23456 is kind of spooky.

[jimvanpelt.livejournal.com]

Half the fun is in the details. That's interesting that he didn't play around with the numbers. At some point in the writing I might have (although my own lapses in logic are massive and embarassing when they happen).

Thanks for messing around with the story, David.

[kadath.livejournal.com]

I kept poking Jay about all the physical impossibilities when I was doing a first read for him ("Brass? Brass?! It doesn't have the tensile strength!" and so forth), but he always erred on the side of story, which is fine by me.

[voidampersand.livejournal.com]

Could the Earth's track be rotating around the Sun? The track would be moving at 107,229 km/hr - 1,674 km/hr (minus assuming the Earth is on the outside of the track (so it doesn't fall into the Sun, duh) and rotates forward).

I haven't read the book yet, but it's moving up to pole position in my stack.

[davidlevine]

The Earth is on the inside of the track, because the gears rumble by at the top of the Equatorial Wall at midnight.

You are correct that the track could be rotating around the sun (not necessarily at orbital velocity, this is a created universe where the planets run on tracks so we don't have to worry about keeping it in orbit like the Ringworld) at exactly the right speed to make the sidereal year 365 days long. I hadn't thought of that. But doing it that way makes the seasons really tricky. I'd had in mind something like a ring gear with a twist in it such that the Earth tilts down below the plane of the ecliptic on one side of its orbit (northern hemisphere summer) and up above the plane of the ecliptic on the opposite side (southern hemisphere summer). But that doesn't have the desired effect if the track's moving.

[voidampersand.livejournal.com]

Okay, Earth's track moves at 107,229 + 1,674 km/hr. I think it's that way because if we maintain the Earth's rotational direction (Sun rises in the East), with a stationary track the Earth would roll backwards through the Zodiac. (Or I could be backwards, but so it goes.)

It would be reasonable to assume that Earth is very strongly gyroscopic (geoscopic?) on its axis. It is going to have the same axial tilt regardless of where it is on its track, or where the track is. That is sufficient to give it seasons. The only problem is how a tilted and rotating sphere can maintain a connection with a track. One way would be for the track to be a chain, where each link has about 24 degrees of play.

[jaylake.livejournal.com]

I was actually thinking along these lines myself today...thanks for bringing it up and thinking through a bit more than I!

[jaylake.livejournal.com]

I was going to give you the link to that sketch -- it did account for the issue of rotational speed vs orbital speed. That was in the original concept of the book. As we discussed, I ditched it for dramatic simplicity, but I never got around to dealing with the physical details that resulted from that change. As I'm eventually planning to write neoVictorian astronaut stories in this universe, you are being incredibly helpful.

You are so awesome. Man, I love my friends.

[kadath.livejournal.com]

You know if you ever want the orbital mechanics fully worked out, all you have to do is ask, man.

[davidlevine]

Love the icon.

[kadath.livejournal.com]

:D

I win so many Internet arguments with it!

[davidlevine]

Not just dramatic simplicity -- I think it would actually get in the way. Not only do all those gizmos block too much sunlight, but they make it clear that the Earth is literally just a tiny cog in a great machine, and if the creator of this universe really cares about human beings (and there's some evidence in the book that he does) that's the wrong message to send.

[davidlevine]

Also, although it's simpler to make the track smaller (or to spin the track, as voidampersand suggests), I like the solution that has 64 wobbles in the track best. This means that a "year" (4 seasons) is not the same as a sidereal year (1 orbit), but the "year" has seasons and solstices and equinoxes and such so it would be more important to people as they began to understand the world.* I'm not sure what it does to the Zodiac, but I don't get the impression these people pay much attention to that. The Chinese, on the other hand, might have a different culture from their equivalents on our world, with 64 instead of 12 zodaical animals...

* Mind you, how much pre-history does a world 6000 years old have?

[kadath.livejournal.com]

That plan presents some difficulties, given the Chinese role in StemwinderEscapement....

[davidlevine]

I know that they have a big role, which is what made me think of them. I don't know if it's too late to tweak this detail, though.

You Must Be Related...

[godelescherbach.livejournal.com]

...to Winchell Chung, at least in spirit! He does stuff like this all the time! See:

http://www.projectrho.com/rocketstub.html

Re: You Must Be Related...

[davidlevine]

Cool! Thanks for the link.

Re: You Must Be Related...

[nyrath.livejournal.com]

I hope you like it, it's my site. {grin}
Making arcane calculations, a man after my own heart.

The novel you did the calculations for vaguely reminds me of the novel Celestial Matters by Richard Garfinkle.

Re: You Must Be Related...

[davidlevine]

My calculations are as nothing. Just a little multiplication and division. The only "arcane" thing was finding the circumference of the Earth's orbit, and Google made that a snap.

[karindira.livejournal.com]

As a simpleton whose intelligence manifested primarily in the verbal half of her SAT scores, I never cared about the math, physics, real science or interplanetary logistics of this story. I just kept repeating, "What a cool IDEA!"