The NASA Kepler Mission website has some educational material concerning a tabletop model designed to demonstrate how the Kepler mission will work. This model includes a couple versions of a LEGO Orrery.

I liked their model so much I decided to work on a few improvements. Here is my latest version:

Overall View

As it has gotten rather long, I have broken up this material into a few pages, organized chronologically blog-style.

Table of Contents

History

  3-Planet Version

  4-Planet Version

  5-Planet Versions

  Freestyle rev. 2

  Freestyle rev. 3

  Off-Road Truck rev. 1

  Off-Road Truck rev. 2

  Off-Road Truck rev. 3

  Small Model 2C

History: Previous Versions of the Orrery

2- and 3-planet versions

20040915

My designs are descended from versions by Allan Ayres and Dave Koch. The first two were 2-planet and 3-planet designs by Ayres. I won't discuss the 2-planet version here; I suspect it was like its successor but without the middle planet.

Here is a picture of the 3-planet design; the building instructions for it are here.

A1. Ayres' 3-planet orrery

A large Technic turntable, whose base remains fixed, is made to turn through epicyclic gearing. 4 rotations of the inner shaft makes the turntable top rotate once. The 3^rd planet is attached to the turntable, and the 1^st planet to the central shaft. Another gear on the turntable makes the 2^nd planet turn at a rate twice that of the 3^rd.

The orbit period ratios can be measured easily just by turning the crank a few times. Here are the periods; I have also shown a "radius" computed by the formulas discussed here.

These orbital periods might seem a little artificial, but it does actually happen in real life — Jupiter's moons Io, Europa and Ganymede have exactly this ratio of periods, due to orbital resonance. You can approximate their orbits by setting up the model so that the 1^st and 3^rd planets are aligned and the 2^nd is all the way around on the other side:

A2: Representing Io, Jupiter, Europa and Ganymede Diameter of Jupiter and distances of moons are to scale. The yellow dot on the base indicates the direction that is currently (early 2008) towards the Sun; the black dot was 2007 and the gray dots are 2009, 2010, etc.

Ayres/Koch 4-planet version

20050729

Dave Koch took Ayres' 3-planet design and modified it to look roughly like this:

B1. My Rendering of the Ayres/Koch 4-Planet Orrery (without moon)

Then he added a moon:

Koch's version (with moon on 2^nd planet)

This incorporates two modifications to the 3-planet design:

• A new planet is added, carried by an arm with epicyclic gears that compute a weighted average of the rotations of its neighbors (details below). What were the 2^nd and 3^rd planets are now the 3^rd and 4^th, respectively.

• The differential housing is turned upside down, and gearing changed so that its rotation is slower (but still faster than that of the 4^th planet).

Kepler's Third Law

To make the model more accurate, one should adjust the lengths of the arms so that the planets' distances are related to their orbital periods by the formula

(^P/_2π)² = ^a3/_(G(M+m))

where

P is the orbital period,
a is the semimajor axis of the orbit (the radius, for a circle)
G is the universal gravitational constant
M and m are the masses of the sun and planet

for our purposes we can simplify it to

a = K P^2/3

where K is an arbitrary constant, which can be 1 if you express P and a as multiples of one planet's period and radius. So for example, knowing that Mars has an orbital period 1.88 times that of Earth, the formula tells us that its semimajor axis is 1.88^2/3 {~=} 1.523 times that of Earth.

All of the models above (A and my versions of B) are built with the proper distances to within {+-}2mm.

20071212

B's orbit periods are a bit more complex than A's. The turntable turns the same way as before, through epicyclic gearing. 12 turns of the crank make it go around once. This is one orbit of the 4^th planet. 36 turns of the crank is the time it takes the 4^th planet to orbit 3 times. In this period the 3^rd planet orbits 4 times, the 2^nd planet orbits 7 times, and the innermost planet orbits 12 times. I call this a syzygy cycle because if you start the orrery with all planets aligned, it takes this long for them all to become aligned again.

Here is a table showing the planet orbital rates and periods, and the radii used in Koch's model (K) and my two models (M₁ and M₂):

20071213

The number 7 in the above ratios might come as a bit of a surprise, if you consider that the gears (with 8, 16, 24 and 40 teeth) cannot by themselves produce any ratios involving the prime number 7.

The 7 (and other prime factors in the more complicated ratios below) comes from epicyclic gearing formulas that involve addition and subtraction, and division of ratios.

Consider first the drive of the 4^th planet:

B2's Base.

This is a planetary gear mechanism (a type of epicyclic gearing). The base (black) is the annulus, and has 24 teeth. The planet-carrier (clear, and gray) carries the planet gear (blue) with it. The planet gear has 8 teeth. The sun gear (yellow) has 8 teeth. In this particular setup, the base is fixed, the sun is the input, and the planet-carrier is the output.

Let R_p be the rate of rotation of the planet-carrier, and R_s be the rate of rotation of the sun gear. R_b, the rate of rotation of the base, is fixed at zero.

It is easiest to compute the gear ratio from within the rotating reference frame of the planet. Relative to the planet, the base is rotating at -R_p and the sun is rotating at R_s - R_p. The base has 24 teeth, the sun has 8, and the planet acts as an idler gear. Because the base's gear is inside-out (teeth facing in) a direction reversal is added. The gear ratio is -1×24/8, or -3/1. Therefore, the motion of the planet-carrier and sun will fit the following formula:

R_s - R_p = -3 (-R_p)

and therefore (by algebra)

R_s = R_p + 3 R_p = 4 R_p

This shows that the 3^rd planet in model A will revolve once for each 4 times the 1^st planet revolves.

Notice that we get a ratio of 4, even though the gears we used (24 and 8) have a ratio of 3.

Now consider the transfer of motion to the 3^rd planet:

B2 Transfer from 4^th to 3^rd

We will use R₄ now to refer to the rotation rate of the 4^th planet arm (R_p above), R₁ instead of R_s, and R₃ for the 3^rd planet arm. Viewed from the rotating reference of the 4^th planet's arm, this is an ordinary gear train. The sun gear (yellow) with 8 teeth drives a 24-tooth gear, which shares an axle with an 8-tooth gear (gray) which drives the output, a 24-tooth gear (dark gray). The input is R₁-R₄, the output is R₃-R₄, and the ratio is (-24/8)×(-24/8) = 9/1. Therefore:

R₁-R₄ = 9 (R₃-R₄)

we already know from above that R₁ = 4 R₄. So we have

3 R₄ = 9 R₃ - 9 R₄

which gives

R₃ = 4 R₄ / 3

Still a fairly simple ratio, no strange prime numbers here. But let's see what happens when we go up to the 2^nd planet:

B2's 2^nd Planet Between 1^st and 3^rd

The 2^nd planet's arm rotates at rate R₂, and has gears mounted on it. The dark gray gear is turning at R₃ — but from the 2^nd planet's point of view, it rotates at R₃-R₂ (which is negative, hence backwards). The yellow gear rotates at R₁, which is R₁-R₂ from the 2^nd planet's point of view. An extra 16-tooth idler ensures that the gear train has only one direction reversal. The gear train will obey the equation

R₁-R₂ = (-40/24)×(16/16)×(R₃-R₂)

Which gear is the input? We already know that R₁ and R₃ are determined as just described. Therefore, they are both inputs, and R₂, the 2^nd planet's arm, is the output. The equation reduces as follows:

R₁-R₂ = -5 (R₃-R₂) / 3

4R₄ - R₂ = -5 (4R₄/3 - R₂) / 3

R₂ - 4R₄ = 20R₄/9 - 15R₂/9

24 R₂ / 9 = 56 R₄ / 9

R₂ = 7 R₄ / 3

There's that surprise prime number 7. Every time the 4^th planet completes 3 orbits, the 2^nd planet completes 7 orbits.

Forward to page 2

Men Core Values

© 1996-2008 Robert P. Munafo. Email the author

This work is licensed under a Creative Commons Attribution 2.5 License.

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