# Full text of "Transparencies Unit 2 - Motion in the Heavens: Project Physics"

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Digitized by the Internet Archive

in 2010 with funding from

F. James Rutherford

http://www.archive.org/details/transparenciesun02fjam

The

Project

Physics

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UNIT ^

Motion in the Heavens

Project Physics

Unit 2

T13 Stellar Motion and Celestial Coordinates

T14 Celestial Sphere

T16 Eccentrics and Equants

T17 Orbit Parameters

T18 Motion Under Central Force

Stellar Motion and Celestial Coordinates

T13

Stellar Motion

If at all possible, students should observe stellar motion directly before trying to analyze it. After they
have made such observations, or have been made aware of the motions by means of photographs,
the ancient conceptual scheme of the "two-sphere universe" can serve as a model to explain stellar
motion.

In order to avoid miniscule dimensions for the earth, the diagrams are not drawn to scale. As a
result, the horizon plane is drawn through the center of the celestial sphere rather than tangent to
the place of observation.

Overlay A The earth is shown at the center of the universe with the celestial sphere turning to the
left on its celestial poles.

Overlay B The three horizontal circles represent the paths of selected stars which are attached to
the sphere at it rotates daily. These circles indicate the diurnal motion of the stars as
seen by an observer in the mid-northern latitudes. From this location, some stars will
appear to circle about the Pole Star, some will seem to rise in the east and set in the
west, while others will never be seen. Remove overlay B before introducing Overlay C.

Overlay C Stellar motion as seen by an observer at the north pole is illustrated. All stars seem to
revolve about the Pole Star. Remove Overlay C before introducing Overlay D.

Overlay D Stellar motion as seen by an observer at the earth's equator. Each evening, all stars
seem to rise in the east and set in the west.

Celestial Coordinates

This transparency is useful in explaining two of the systems of coordinates used to locate stars
and planets.

Use A for the first overlay, then add E.

Overlay E This scheme for specifying the locations of stars imagines a celestial sphere whose
poles pass through the earth's poles. Hour circles begin at the Vernal Equinox and
proceed to the right. (The Vernal Equinox is described in T14). The Right Ascension
of a star is measured eastward along the celestial equator in hours and minutes. The
Declination of a star is measured in degrees north or south of the celestial equator
(a projection of the earth's equator) along the star's hour circle. Remove Overlay E
before introducing F.

Overlay F Another scheme for specifying the locations of stars and planets in or near the zodiac
is based on the ecliptic, the sun's annual path across the sky. Celestial longitude begins
at the Vernal Equinox and is measured eastward to 360° along the Ecliptic. Celestial
latitude is measured in degrees north and south with 0° at the ecliptic.

T13

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OBSERVER AT

MID-NORTHERN

LATITUDE

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T13

A
C

OBSERVER AT
NORTH POLE

T13

OBSERVER AT
EQUATOR

EAST

WEST

TW

iorth Celestial Pole

T13

Celestial Sphere

T14 Celestial Sphere

This transparency will be useful in visualizing some of the important features explained by the
celestial sphere.

Use overlay T13-A for the first overlay, then add T14-A.

Overlay A The celestial equator is a projection of the earth's equator on the celestial sphere. The
sun is shown moving to the right (eastward) along a great circle on the celestial sphere.
The daily rotation of the sphere accounts for night and day. The annual rotation of
another sphere, which carries the sun and has a pole 2314" from the pole of daily
rotation, accounts for the sun's annual motion eastward with the seasonal north-south
variations. The sun's path across the sky is known as the ecliptic. The point of inter-
section of the echptic and the celestial equator as the sun travels from south to north
along the ecliptic is called the Vernal Equinox. The crossing occurs approximately on
March 21. The Summer Solstice (shown as SS) occurs on about June 21, the Autumnal
Equinox (AE) on about September 22, and the Winter Solstice {WS) on about
December 21.

Overlay B The Zodiac is a belt 18" wide which circles the sky and is centered on the ecliptic. The
sun, moon, and planets are always located within this belt. The zodiac is divided into
twelve constellations called the Signs of the Zodiac.

T-11H3

Celestial
Equator

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Virgo

Leo Cancer

Scorpi

Gemini

Sagittarius

The heliocentric explanation of an outer planet's observed retrograde motion can be demonstrated
using this transparency.

Overlay A The earth and an outer planet are shown at successive positions along their orbits.
The time intervals between the indicated positions are equal. Introduce blank overlay B.

Overlay B Draw in sight lines directly on this blank overlay. Connect points 11, 2-2, . . . , etc.,
for the earth and planet, and extend the lines to the star field on the right. These lines
will show that the planet, as seen from the earth, appears to move westward (retro-
grade) when it is opposite to the sun. Remove overlay B before introducing overlay C.

Overlay C This shows the completed operation suggested in B above. The apparent path is marked
in to aid discussing retrograde motion.

Planet

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Eccentrics and Equants

T16

Eccentrics and Equants

These transparencies can be used to present a rapid review of the geometrical devices of the Ptolemaic
geocentric model of the universe. Do not belabor the details but simply point out the usefulness of
the various geometric devices. Emphasize the fact that they account for the variations observed in
planetary motion, while at the same time they preserve the Platonic scheme of uniform angular
motion.

Eccentrics

Overlay A This is a reference circle which will be used for both parts of T16.

Overlay B A planet is depicted moving with uniform angular speed at a constant distance from
(Circle) center O. It's path is a perfect circle and therefore an earth observer at O would
measure equal increments of angular change of position in successive equal time
intervals.

Overlay C This shows the earth in an off-center, or eccentric, position. Now the planet does not
(Eccentric) exhibit uniform speed relative to an observer on earth. Remove overlays B and C
before introducing D.

Equants

Overlay D The equant is another geometric device for preserving uniform circular motion. The
(Equant) planet proceeds with uniform angular motion about an off-center point C, while
tracing out a perfect circular path of radius R about point O.

Overlay E Now the earth is placed off-center in the equant system. Angular displacements
measured from the off-centered earth will yield results different from those obtained
with the eccentric alone. Thus the equant was used by Ptolemy to explain variations in
planetary motion not accounted for by the eccentric.

T-16

planet

A
B

s

T16

ECCENTRIC

planet

B
C

T16

planet

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EQUANT

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planet

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Orbit Parameters

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T17 Orbit Parameters

This transparency can be used to extend the discussion of Kepler's first two laws and to clarify details
of the various celestial experiments of Unit 2.

Overlay A The sun and the Vernal Equinox are shown. This overlay serves as a base for overlays
B and C.

Overlay B The orbital plane of a planet with the elements of an elliptical path are indicated.

c = one-half distance between the foci

a = semi-major axis

A = apheUon

P = perihelion

{e = eccentricity [e = c/a])

Overlay C The plane of the earth's orbit, known as the plane of the ecliptic, is added. The remain-
ing elements for determining an orbit are shown.

CO = argument of the perihelion

fi = longitude of the ascending node

/ = incUnation

T-17

f

Orbital Plane of
Comet or Planet

T-IJ

Orbital Plane of
Comet or Planet

/
/

of Nodes

Motion Under a Central Force

T18

Motion Under a Central Force

This transparency follows Newton's analysis of motion as presented in Proposition I of the Principia.
It shows how an orbit forms when an object receives a sequence of blows all directed toward the
same point.

Overlay A Equal intersal positions of a body moving with uniform speed in a straight line.

Overlay B Kepler's Law of .Areas applies in this example of uniform rectihnear motion. .\s shown
by the two blue triangles, an observer at O will see equal areas swept out b\ the moving
object. The areas can be shown to be equal because the bases are equal and the altitudes
(dashed line) of both triangles are identical.

Overlay C This shows the result of a blow on the object directed toward O as it passes point B.
The blow is such that if the object were stationary at B. it would move to, say, B' in
the time interval. But the object is moving and if left alone would travel to C. Thus,
the result of the blow on the moving object is that it moves to C, a displacement which
is the vector sum of the displacements to C and B'.

Overlay D The area of the red triangle can be shown to be equal to the area of the light blue one.
With the aid of the construction lines you can show that the altitudes of the two
triangles are equal. Both triangles have the same base, OB. The areas swept out by the
two triangles in equal time intervals will be equal regardless of the magnitudes of the
blows. Remove overlays B, C, and D before adding overlay E.

Overlay E This suggests what will happen if the process of applying blows toward O is continued.
Presumably the eventual path will be smooth if the time intervals are made vanishingly
small and the forces apphed continuously.

T18

Motion Under a Central Force

This transparency follows Newton's analysis of motion as presented in Proposition I of the Principia.
It shows how an orbit forms when an object receives a sequence of blows all directed toward the
same point.

Equal interval positions of a body moving with uniform speed in a straight line.

Kepler's Law of Areas applies in this example of uniform rectilinear motion. As shown
by the two blue triangles, an observer at O will see equal areas swept out by the moving
object. The areas can be shown to be equal because the bases are equal and the altitudes
(dashed line) of both triangles are identical.

This shows the result of a blow on the object directed toward O as it passes point B.
The blow is such that if the object were stationary at B, it would move to, say, B' in
the time interval. But the object is moving and if left alone would travel to C. Thus,
the result of the blow on the moving object is that it moves to C, a displacement which
is the vector sum of the displacements to C and B'.

The area of the red triangle can be shown to be equal to the area of the light blue one.
With the aid of the construction lines you can show that the altitudes of the two
triangles are equal. Both triangles have the same base, OB, The areas swept out by the
two triangles in equal time intervals will be equal regardless of the magnitudes of the
blows. Remove overlays B, C, and D before adding overlay E.

This suggests what will happen if the process of applying blows toward O is continued.
Presumably the eventual path will be smooth if the time intervals are made vanishingly
small and the forces applied continuously.

Overlay A
Overlay B

Overlay C

Overlay D

Overlay E

T-18

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