
Spheres of Eudoxus
General Description
This program simulates the system devised by Eudoxus to model the
motion of the planets. The model consists of four nested concentric
spheres. The axis of each sphere is attached to the surface of the
next sphere out. The planet itself is located on the surface of the
innermost sphere. The outermost sphere rotates with the daily
(apparent) rotation of the stars.
The Spheres Frame shows the four spheres of the model. The axis of the
inner (red) sphere can be tilted relative to that of the middle (blue)
sphere using a slider to adjust the angle. The red sphere
automatically rotates with angular velocity +1.0 (in arbitrary
units). The angular velocities of the blue and green spheres can be
adjusted using sliders. Note that the axis of the blue sphere is
attached to the equator of the green sphere. This is a crucial part of
Eudoxus' model. The equator of the green sphere is in the plane of the
ecliptic. The outermost (white) sphere is essentially the Celestial
Sphere containing the fixed stars (or at least it rotates about the
same axis and at the same rate as the Celestial Sphere).
The Sky View Frame shows the motion of the planet (relative to the
stars) as seen from the earth, which sits at the center of the
concentric spheres in the model.
By tilting the red sphere relative to the blue sphere, setting the angular velocity of the blue sphere opposite that of the red sphere, and giving the green sphere a sufficiently small angular speed, Eudoxus was able to qualitatively reproduce the observed retrograde motion of the planets using this model.
Spheres Frame
- Elements Menu
- Show Arrow Pointing to Apparent Location: display a line-of-sight arrow
from the center, through the planet, to a point on the Celestial
Sphere.
- Show Inner Sphere: display the inner (red) sphere and
corresponding lines.
- Show Middle Sphere: display the middle (blue) sphere and
corresponding lines.
- Show Outer Sphere: display the outer (green) sphere and
corresponding lines.
- Show Celestial Sphere and Stars: display the Celestial (white) sphere and
corresponding lines, as well as the band of stars near the ecliptic.
- Options Menu
- Trace Path of Planet: traces the apparent path of the planet
projected onto the Celestial Sphere.
- Let Celestial Sphere Rotate: spins the Celestial Sphere in
it daily rotation.
- Link Speeds of Red and Blue Spheres: sets the angular velocity
of the middle sphere to -1 (opposite the angular velocity of the inner
sphere). This is required in Eudoxus system in order to keep the
planet near the ecliptic plane. If this option is selected, the
user cannot alter the angular speed of the middle (blue) sphere.
- Visual Elements
- Red sphere and lines: inner sphere with axis, equator, and
meridian shown. The planet lies on the surface of this sphere.
- Blue sphere and lines: middle sphere with axis, equator,
and meridian.
- Green sphere and lines: outer sphere with axis, equator,
and meridian.
- White sphere: celestial sphere.
- White band with points: region of celestial sphere near
ecliptic with stars.
- Magenta point: the planet.
- Magenta line: trace of the planet's motion on the
Celestial Sphere.
- Magenta arrow: shows the line of sight from Earth (at
center) through the planet to the Celestial Sphere.
- Controls
- Play/Pause: start and stop the simulation.
- Step: advance by one time step.
- Reset: reset simulation to original state.
- Clear Traces: clears all traces from both windows.
- Tilt: tilt of inner (red) sphere relative to middle (blue) sphere in degrees. Some tilt is required for retrograde motion.
- Angular Velocity (Blue): sets angular velocity of middle (blue)
sphere in units such that the angular velocity of the inner (red)
sphere is +1.0. The angular velocity of the blue sphere must be set
to -1.0 in order to keep the planet near the ecliptic.
- Angular Velocity (Green): angular velocity of the outer (green) sphere controls motion of the planet along the ecliptic. If this is set too fast retrograde motion will be impossible.
Sky View Frame
- Sky View Options Menu
- Show Ecliptic Line: displays the line representing the
ecliptic (which is also the equator of the green outer sphere).
- Show Trace of Planet's Motion: displays a trace of the
planet's apparent motion in the Sky View.
- Visual Elements
- Magenta point: the planet.
- Magenta line: trace of the planet's motion on the sky.
- White points: background stars near the ecliptic.
- Green line: the ecliptic.
Todd K. Timberlake (ttimberlake@berry.edu)