**Document ID**: 08_19_05_1

**Date Received**: 2005-08-19 **Date Revised**: **Date Accepted**:

**Curriculum Topic Benchmarks**: M3.4.1, M3.4.5, S13.4.6, S14.4.1,
S14.4.2

**Grade Level**: \[9-12\] High School

**Subject Keywords**: gravitation, force, Moon, Sun, Earth, tides,
inverse square law

**Rating**:

The Moon Orbits the Sun?!?!

**By**: Stephen J. Edberg, Jet Propulsion Laboratory, California
Institute of Technology, 4800 Oak Grove Drive, M/S 301-486, Pasadena CA
91011

e-mail: Stephen.J.Edberg@jpl.nasa.gov

**From**: The PUMAS Collection http://pumas.jpl.nasa.gov

©2005, Jet Propulsion Laboratory, California Institute of Technology.
ALL RIGHTS RESERVED.

Even the most casual observers note the changes in the phase of the Moon
as it goes from crescent to half to full and back again with a "monthly"
cycle. (See Glenn Simonelli's *PUMAS* submission "Modeling the motions
of the Earth, Sun and Moon"
\[http://pumas/PDF_Examples/03_10_04_1.pdf\].) Their observations, or
what is "common knowledge", lead them to believe the Moon does loops
around the Earth. But is this true? A comparison of the gravitational
forces of the Sun and Earth on the Moon hints at the answer to this
question and a simple demonstration refutes the loop-view.

**OBJECTIVE:** Compute the strengths of the gravitational forces exerted
on the Moon by the Sun and by the Earth and compare them. Demonstrate
the actual shape of the Moon's orbit around the Sun. Understand the
difference between gravitational forces between bodies and tidal forces
generated by those bodies and compare the two.

**APPARATUS:** This teaching aid demonstrates the actual shape of the
lunar orbit as it plays out over a year or a portion of a year. Two
parts are necessary: (1) a large circular cut-out *or* a large circular
disk, or circumferential sections of each (because of the required
sizes) and (2) a small disk, both parts made to the appropriate scale
(discussed below).

The demonstrator is built to be a scale model of the Sun-Earth-Moon
system. For a durable device, go to a hardware store for thin wood
veneer, plywood, or a Plexiglas®-like material, 1/16 to 1/4 inch thick.
These materials require a wood or metal saw and sand paper for cutting,
shaping, and smoothing.

A large piece of cardboard or dry mounting board can also be used. These
materials require sharp blades and sand paper for cutting, shaping, and
smoothing. Other large circular products, such as disposable serving
trays and plastic pots for plants are also suitable if the necessary
sizes can be found.

Use a string or rope fixed to a pivot on one end and to a pen or awl on
the other end to serve as a compass to create the large radius for the
Earth orbit disk.

A drill is necessary for making a hole in the small disk.

This demonstrator works much like a Spirograph® but the final pattern
will be much simpler than what a Spirograph® can produce. A large
circular cut-out or a large disk represent Earth's orbit and a small
disk carries the Moon's orbit around Earth. The biggest problem in
construction is the difference in size between the radius of the Moon's
orbit and the radius of Earth's orbit, which is 391 times larger (a
discussion of the actual, elliptical shapes of their orbits is in
Extension (2), below). Note that the sizes of the large cut-out/disk and
small disk will not be in exactly the orbital ratio.

Choose a size so your writing implement (pen, white board pen, or chalk;
pencil is not recommended) will fit through a hole in the lunar orbit
disk and can trace out the Moon's path. A very fine point pen's tip
(representing the Moon) 2 mm from the center of the lunar orbit disk
requires Earth's orbit to be 782 mm in radius. A larger pen or chalk
will have to be more separated from the center of the lunar orbit disk
and the Earth orbit cut-out/disk will have to be proportionally larger.

![](media/image1.jpeg){width="2.7472222222222222in"
height="2.060416666666667in"}![](media/image2.jpeg){width="2.7472222222222222in"
height="2.0479166666666666in"}

> Figure 1. Two construction methods and use of the orbit demonstrator.
> These cardboard prototypes were *not* made to scale.
>
> \(a\) This is the "cut-out" design. The large cut-out circle is
> centered on the "Sun". The small disk, with "Earth" at its center, has
> a small hole for a pencil tip, representing the Moon, with an offset
> from "Earth." With a pencil or pen tip in the "Moon" hole, this lunar
> orbit disk will trace out the Moon's path around the Sun as the disk
> rolls around the inside of the cut-out.
>
> \(b\) This is the "disk" design. The small disk rolls around the
> outside of the larger disk. The small disk again represents the lunar
> orbit, and was cut from near the center of the larger disk; the
> remaining hole can also represent the Sun. Irregularities in the
> orbital paths can result from poor shaping of the parts and slippage
> of the small disk as it rolls. Better materials and more careful
> construction will minimize these problems.

Note in Figure 1 that the cut-out design (a) forces the apparent lunar
revolution to be in the direction opposite of Earth's orbital motion.
The result is the same but esthetically the presentation is not as
satisfying as the disk design (b), which permits both orbital motions to
be in the same direction, as they are in the solar system. However, the
cut-out design may be easier to use, especially on a big-enough white-
or chalk-board because its base can sit in the pen or chalk tray. A disk
segment will have the same advantage.

Alternatively, the disk design can be built using a large commercially
available product like a disposable serving tray or plant pot and a
smaller disk or cylinder, such as a medicine bottle, thread spool, or
other disk or cylinder of material that can be easily "worked". However,
only one disk's radius is a free variable; the other must be built to
conform to the chosen disk's size since the ratio of the Moon's orbital
period and Earth's orbital period are involved in the sizing of their
orbit disks (or Earth orbit's cut-out).

Specifically, the ratio of the circumference of the lunar orbit to
Earth's orbit must match the ratio of their orbital periods (with
respect to the Sun):

(lunar orbital period)/(Earth orbital period) = 29.5306 days/365.2425
days

Maintaining this ratio preserves the true orbital geometry and timing
(lunar revolutions per year) of the Sun-Earth-Moon system.

Make calculations as illustrated in the tables below. In all cases, the
free variable is in the left-most column. In the other columns, the
variable name and the formula for calculating it are given at the top. A
spreadsheet is easily constructed to permit the selection of suitable
design parameters. (In the Microsoft Word version of this lesson, double
clicking on the desired table should bring up a working spreadsheet.)

The radius of Earth's orbit, R~E~, for the demonstrator is the
difference or sum of the radii of the Earth orbit cut-out or disk,
respectively, R~ED~, and lunar orbit disk/cylinder, R~LD~. The lunar
orbit radius, R~L~, is 1/391 of the radius of the Earth's orbit. This
orbital ratio forces the demonstrator to be large if the wavy trace of
the lunar orbit is to be distinguishable, say, a few millimeters at
least. The examples in Tables 1c and 1d are impractical. Also, if
Extension topic (4) discussing the concavity of the lunar orbit is going
to be used, the trace of the lunar orbit should be recorded over a span
of at least 1/12 of the circumference (2πR~E~) of Earth's orbit.

**Demonstrator Calculations**

Tables 1a

*Choose Lunar Orbit Radius*

![](media/image3.wmf)

![](media/image4.wmf)

Tables 1b

*Choose Lunar Orbit Disk/Cylinder Radius*

![](media/image5.wmf)

![](media/image6.wmf)

Table 1c

*Choose Earth Orbit Disk Radius*

![](media/image7.wmf)

Table 1d

*Choose Earth Orbit Cut-Out Radius*

![](media/image8.wmf)

> Table Notes: The ratio of the radii of Earth's orbit to the Moon's
> orbit = 391. The lunar orbit duration with respect to the Sun (e.g.,
> new moon to new moon, the synodic period) is 29.5306 days. Earth's
> average Gregorian year (i.e., with respect to the Sun) is 365.2425
> days. Calculated values are presented to 0.01 mm though the best
> precision possible with a good ruler is approximately 0.33 mm.

**PROCEDURE:** Discuss with the students their assumptions about the
motions of the Moon about the Earth and the Earth about the Sun. After
reaching consensus, use the demonstrator.

Install your marker in the lunar orbit disk and roll the lunar orbit
disk around the Earth orbit cut-out/disk. If you trace through a whole
year (a full revolution of Earth around the cut-out/disk), the lunar
path won't close on itself since there is not an even number of lunar
revolutions in a year. Remove the demonstrator parts and study the
orbital pattern laid down by the marker -- no loops!

![](media/image9.jpeg){width="3.390277777777778in"
height="5.1194444444444445in"}

**THE UNDERLYING PRINCIPLES:** The force of gravity is described by
Isaac Newton's law of gravitation:

> F~12~ = $\frac{G´m1´m2}{R´R}$ (1)

[]{.mark}

where G is a constant, 6.674x10^11^ Nm^2^/kg^2^. In the units of the
constant, the abbreviation for the unit of force, called a Newton, is N.
One Newton is the force exerted by a mass of 1 kg accelerating at 1 m/
s^2^.

Calculate the forces between Sun-Earth, Sun-Moon, and Earth-Moon using
these values:

Table 2

**Force Calculation Parameters**

  -------------------- ----------------------------- --------------------
                       **Mass \[kg\]**               **Diameter \[m\]**

  **Sun**              1.98E+30                      6.96E+08

  **Earth**            5.97E+24                      1.27E+07

  **Moon**             7.35E+22                      3.48E+06

                                                     

                       **Distance \[m\]**            

  **Sun-Earth**        1.50E+11                      

  **Sun-Moon**         1.50E+11                      

  **Earth-Moon**       3.84E+08                      

                                                     

  **Gravitational                                    6.67E-11
  Constant, G                                        
  \[Nm^2^/kg^2^\]**                                  

                                                     

  **Gravitational      **Combination**               **Force (N)**
  Interaction**                                      

                       **Sun-Earth**                 3.53E+22

                       **Sun-Moon**                  4.34E+20

                       **Earth-Moon**                1.98E+20
  -------------------- ----------------------------- --------------------

> Notes on the values in the table: The diameter of the Earth is the
> mean diameter of the planet, not its equatorial nor its polar
> diameter. The Sun-Earth distance given is the average distance between
> Earth and Sun, which is called an Astronomical Unit, or AU. The
> distance between Earth and Sun varies due to the ellipticity of
> Earth's orbit. The Sun-Moon distance is taken to be the same as the
> Sun-Earth distance, though the Sun-Moon distance varies by ± the
> Earth-Moon distance. The Earth-Moon distance is an average value since
> the Moon's orbit is also elliptical.

Of greatest interest is the difference in the strength of the
gravitational force of the Sun and Earth on the Moon: Despite Earth's
proximity to the Moon, the Sun's force on it is more than twice that of
Earth. Thus the force calculation shows that the Sun controls the motion
of the Moon; Earth applies a rather strong perturbation to that motion,
just as the orbit demonstrator shows.

**EXTENSIONS:** (1) The strength of the Sun's gravitational force on
Earth is about 178 times stronger than the Moon's. Is this reflected in
tides, which manifest the differential strength of the gravitational
force across an object? To find out, calculate the gravitational field
on the near side and far side of Earth, near and far being related to
the direction to the Sun or to the Moon. The distances to use require a
term that is ([+]{.underline}) the radius of Earth (6.371E+06 m). Only
the mass of the gravitating body (the Sun or Moon, for tides on Earth)
is used in the equation.

Tidal Force =
$\frac{G \times m1}{(R - r)(R - r)}\frac{}{}\frac{G \times m1}{(R + r)(R + r)}$
(2)

Table 3

**Tide Calculation Parameters**

  ----------------- ----------------------------- -----------------------
  **Tides**         **Difference in Gravitational 
                    Field at Ends of a Diameter   
                    \[N/kg\]**                    

                    **Sun-Earth**                 1.01E-06

                    **Moon-Earth**                2.20E-06
  ----------------- ----------------------------- -----------------------

The tidal effect of the Moon is more than twice the effect of the Sun
because the *fractional* difference in distance between the near and far
sides of Earth, with respect to the Moon, is much greater than it is
with respect to the Sun. The Sun's greater mass is not enough to
overcome the difference in distance (squared).

The force and tidal calculations can be repeated, if desired, using the
perihelion and aphelion values for Earth-Sun (1.471E+11 m and 1.521E+11
m, respectively) and perigee and apogee values for Earth-Moon (3.566E+08
m and 4.066E+08 m, respectively).

\(2\) For a more realistic presentation of the orbits of Earth and Moon,
construct their actual elliptical orbits for use in the orbit
demonstrator. Earth's orbital semi-major axis is 1.496E+11 m and its
eccentricity is 0.0167. The Moon's orbital semi-major axis is 3.84E+08 m
and its eccentricity is 0.0554. A loop of string around a fixed pair of
nails, placed at the foci of the ellipses, can be stretched with a pen
or awl to generate ellipses of the desired sizes.

\(3\) Do Earth satellites, or can anything else, do loops as Earth
orbits the Sun? Create a scenario for students: You are at the North
Pole or South Pole (and assume Earth's axis is not tilted with respect
to the plane of Earth's orbit). Calculate the linear speed of a race car
on a 100 m radius track centered on the pole, circling in the direction
of Earth's rotation. Make the same calculation for a medical centrifuge
spinning at 6000 rpm and radius 20 cm centered on the pole. Compare
those speeds to Earth's orbital speed (\[orbital circumference\]/\[1
year\], in appropriate units). For loops, the linear speed must exceed
Earth's orbital speed. Do the natural satellites orbiting the outer
planets do loops?

\(4\) Is the Moon's orbit ever convex with respect to the Sun or does it
always have a greater or lesser degree of concavity? While drawing the
Moon's orbital path with the orbital demonstrator note the points where
the Moon is at its maximum elongation leading or trailing Earth. These
are the points of first and last quarter (half moon in the sky). On the
lunar orbit curve, use a straight-edge to add some chords between first
and last quarters. Compare the distances from the middle of the chord to
the nearest point of the orbital path. In all cases, the path is outside
the chord, with the shorter distance occurring where the Moon is between
the Earth and the Sun (new moon). Since the Moon's path never crosses
the chord, the path is always concave, to a lesser or greater degree,
towards the Sun.

Alternatively, calculate the perpendicular distance from the Sun to the
chord separating first and last quarters. If the difference \|(chord
distance)-(Astronomical Unit)\| is greater than the maximum separation
of Earth and Moon, the path of the Moon must always be concave towards
the Sun.

\(4\) The demonstrator may remind you or students of the Aristotelian
model of the solar system as described by Ptolemy to display the
observed motions of the *planetes* (wanderers): Earth at the center of
the universe with large circles, called deferents, carrying small
circles, called epicycles. The goal of this model was to use the
"perfection" of the circle to match the observed motions of the planets
by placing the planets on a rotating epicycle, carried by the rotating
deferent. The rotation rates of the deferents and epicycles were
calculated to match the observed motions of the planets. The apparent
"backward" (retrograde) motion of the outer planets Mars, Jupiter, and
Saturn around their times of opposition (i.e., when they are aligned
Sun-Earth-planet) and the comings and goings of Mercury and Venus in
dawn and dusk skies required the inclusion of epicycles in the model.
Ultimately epicycles were added to epicycles to try and more closely
match the observed motions.

We now know, of course, that the Sun is near the center of mass of the
solar system, that gravitational forces control planetary motions, and
that elliptical orbits are required to match the observed motions of the
planets. The theorems of the calculus indicate that an infinite number
of circular epicycles upon epicycles on a deferent would be required to
approximate an ellipse. A Ptolemaic demonstrator can be constructed by
making appropriate choices of large cut-out/disk and small disk
diameters based on the angular span of an outer planet's retrograde
(apparent "backward" motion in the sky) loop (illustrated in many
astronomy textbooks and with planetarium software).

**REFERENCES and ACKNOWLEDGMENTS:** Orbital period, mass, diameter,
eccentricity, and distance values used here were taken from the
*Observer's Handbook 2005* of the Royal Astronomical Society of Canada,
Rajiv Gupta, Editor. Parts of this discussion were inspired by the
author's reading, early in high school, of Isaac Asimov's essay "Just
Mooning Around", found in his collection *Of Time, and Space, and Other
Things* (Avon, 1975). Glenn Simonelli's *PUMAS* submission "Modeling the
motions of the Earth, Sun and Moon"
(http://pumas/PDF_Examples/03_10_04_1.pdf) was a major motivator in
preparing this lesson.

This publication was prepared by the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the National
Aeronautics and Space Administration.

Reference herein to any specific commercial product, process, or service
by trade name, trademark, manufacturer, or otherwise, does not
constitute or imply its endorsement by the United States Government or
the Jet Propulsion Laboratory, California Institute of Technology.