I will preface this portion of the unit with a list of skills that should be reinforced before attempting the computations and an explanation of how we will determine where the sun and moon are positioned with respect to the earth.
Certain skills should be reviewed before actual computations begin. The basic and most generally used computations are:
1. changing from degrees to minutes and seconds, and the reverse,
2. solving proportions,
3. operations involving decimals,
4. setting proportions for interpolation,
5. constructing coordinate axis,
6. constructing circles of given radii,
7. scaling and marking measurements.
(figure available in print form)
Each step will be preceded by the list of skills needed to complete the step. I suggest that a worksheet that’ provides adequate practice in each of the above areas be assigned each day.
In addition to skills review, students will also need a picture of how we determine the positions of the sun and moon in the sky. For our purposes we can imagine the sun and moon traveling across a celestial sphere. (Figure III) The celestial sphere will contain a projection of the earth’s equator, called the celestial equator, and the earth’s prime meridian, the line of longitude with degree measurement of zero, will be the celestial meridian. We can then locate the positions of the sun and moon with respect to the earth in terms of 1) Declination: The distance in degrees north or south of the celestial equator and 2) Greenwich Hour Angle: Distance in degrees west of the celestial meridian (degrees will vary from 0 positioned directly on the meridian, to 360’westwardly approaching the meridian).
When making our calculations we will assume that all days are exactly twentyfour hours long, this will be referred to as Mean Sun Time. Day lengths vary each day but use of a 24 hour day will make our calculations easier. (If all days were 24 hours long, would we need Leap Year?). Finally we will state that when the sun is directly at the celestial meridian (noon at Greenwich, England) the longitude or the Greenwich Hour Angle of the sun is O’ The sun will then travel at a speed of 15— per hour ( 15—X 24 = 360—). For example, when the sun is 15— west of the celestial meridian, it is 1:00 in Greenwich. The hours of time we use will be based on Greenwich time and is called Greenwich Mean Time.
A. Computing Steps
The following portion of the unit includes steps which each teacher must progress through with his students. Each instructor must pick a year in which a total lunar day and time of said students with the proper
eclipse has occurred, find the occurrence and then supply the students with the proper chart necessary. All of this information can be found in the
Nautical Almanac
, published by the U.S. Government. As of 1976, the style in which the
Nautical Almanac
(the source of the tables we will use) has changed, therefore I suggest teachers choose a year before 1976, i.e. the students’ year of birth, their first year of school, or the year they learned to ride a bicycle etc. as the year of the almanacto choose. Although students will not be able to predict an eclipse, they will still benefit from the work involved. Also, be careful to choose a year in which a
total
eclipse occurs, the graphs are much more interesting.
(figure available in print form)
There are some aspects of calculations that will be too sophisticated for the eighth grade students to perform. Again, I will note these exceptions and suggest that each teacher supply that information to their students.
Interpreting the Nautical Tables
I have chosen as an example the lunar eclipse that occurred May 25, 1975. The information in the table below was obtained from the
Nautical Almanac 1975
.

GMT

SUN


MOON

DAY

HOUR

GHA

DEC

GHA

DEC

HP

25

0:00


4:00

240—48.2’

N 20—49.6’

61—48.1’

S80—30.3’

58.2’


5:00

255—48.1’

N 20—50.0’

76—14.0’

S20—33.9’

58.2’


6:00

270—48.1’

N 20—50.5’9

0—40.0’

S20—37.4’

58.2’


7:00

285—48.0’

N 20—51.0’

105—0.9’

S20—40.7’

58.1’



SD 15.8’


SD 15.8’

Table Explanations:
GMT
:
Greenwich Mean Time
: Time based on a 24 hour day starting at Greenwich, England.
GHA
:
Greenwich Hour Angle
: States the westward position of the sun (moon) with respect to the celestial meridian. (O—longitude) (Figure III)
DEC: Declination
: The north or south position of the sun (moon) with respect to the celestial equator (O—latitude). (Figure III)
HP: Horizontal Parallax: The measure of the angle formed by two rays starting at the center of the sun. One ray passing through the center of the earth and the other ray tangent to the earth.
SD: Semidiameter: Angular Radius in minutes of the arc. Information obtained from the table.
Once we understand the kind of information given in the nautical tables and we are able to read the tables successfully, work on the computing of the eclipse can begin. The following is a basic outline of the steps involved in computing an eclipse. Each step will eventually be worked through with a specific example.
Outline of Computing, Steps

I.

Calculate the time of opposition: The time at which tenth sun and the moon are 180— apart.


II.

Determine the declination of the sun and moon at the moment of opposition
. Using information from step I, find the sun and the moon’s declination (degrees from the equator) at the time of opposition.


III.

Radius of the Shadow
: With a given formula, we will determine the size of the shadow cast by the earth during the eclipse.


IV,

Coordinate axis
: Setting up our coordinate axes to plot the moon’s motion through the earth’s shadow,


V.

Determine the NorthSouth movement of the sun and
moon: The degree changes in declination each hour.


VI.

Determine the EastWest rate of change for the sun and moon
: The degree changes in GHA for the sun and moon.


VII.

Plot the path of the moon
: Using the East West and NorthSouth differences, plot the path the moon will follow through the earth’s shadow.


VIII.

Construct models of the moon
: The models will be centered of the moon’s path, passing through the earth’s shadow.


IX.

Calibrate the graph
: The graph will be calibrated to determine the time the eclipse occurs.

Step I: Calculate time of Opposition
Students will use the table to determine at what time the sun and the moon are in opposition, i.e. 180° apart.
PreStep work suggestions:
1) Changing degrees to minutes and seconds
2) Writing proportions
3) Interpolation
Procedure:
Students should find the difference between two consecutive GHA readings for both the sun and moon. Continue operations until the smallest interval in which 180—. is contained is found.
(figure available in print form)
Then interpolate to find the exact time in which opposition (180—) has occurred.
(figure available in print form)
The sun and moon are in opposition at 5h 45.1min or 5:45 AM.
All information may be stored on a Student Information sheet. An illustration of such a sheet is included in the lesson plans.
Step II: Compute the declination of the sun and moon at the moment of opposition.
Students will use the table to find the declination of the sun and moon at 5h 45m
PreStep work suggestions:
1) setting and solving proportions
2) simple interpolation
3) estimation
4) rounding numbers
Procedure:
Using the table, find the declination of the sun and moon using the values for 6h and 5h and interpolating to determine declination at 5h 45 m
____
(figure available in print form)
The moon’s declination at opposition is approximately S 20— 36.5’
We have computed that the sun’s declination at the time of opposition is N 20— 50.4’ This is also the southerly declination of the center of the earth’s shadow. Finding the difference in the declination of the sun and the moon will yield the difference between the center of the earth’s shadow and the center of the moon.
(figure available in print form)
Therefore the center of the earth’s shadow is 13.9’ below the center of the moon.
Step III
: Find the Radius of the Shadow,
Certain information need not be explained to students due to its sophisticated nature. After students feel confident with their skills and have successfully completed steps I and II, each instructor can provide the students with the value for the Radius of the Shadow. The following information can be given to each student via the information sheet at the end of the unit.
The radius of the shadow is determined by the following formula:
Radius of Shadow Solar Parallax + Lunar Parallax
Sun’s Semidiameter
The value of the Solar Parallax is a constant: 8.8 secs. (SP)
The value of the Lunar Parallax is a constant: 59.5 min. (LP)
The value of the sun’s Semidiameter is obtained from the table: 15.8 min. (SD)
Plugging in the above values in the formula yields a radius:
Radius = 8.8” + 59.5’ 15.8’
Radius = 42.55’
Step IV; Setting up a Coordinate Axis that will be used to plot the moon’s motion. (Diagram I)
Prestep work suggestions:
1) converting and scaling measurements
2) setting coordinate axis
3) marking intervals on numberlines
4) constructing circles
Procedure:
Students are now ready to start graphing the information they have acquired in the previous steps,

I.

Allowing 1 inch = 1000 seconds of an arc, students should convert all their degree measurements to seconds. This information can be stored on the Student Information Sheet. Converted measurements:


1. The declination of the earth’s shadow with respect to the moon: 13.9’ 834” (Step II)

2. The Radius of the shadow: 42.55’.= 2553” (Step III)

II.

Students should set up a coordinate axis (NS, E W). The origin represents the center of the
moon
at opposition.


III.

Instruct students to find the point of the NS axis that will represent the center of the earth’s shadow. The value from Step II is 834”.


IV.

Construct a circle representative of the earth’s shadow centered at 834” with a radius of 2553”. (Step III)

Step
V: Find the NorthSouth movement of the moon and sun.
Prestep work suggestions:
(figure available in print form)
1. subtraction of decimals
2. change from degrees to minutes to seconds.
Procedure:
The NorthSouth motion of the sun and noon can be obtained directly from the nautical table. Find the difference between the hourly declination of the sun.
(figure available in print form)
The sun moves north at a rate of 0.5’ per hour, therefore the earth’s shadow moves south at 0.5’ per hour.
The NorthSouth motion of the Moon is similarly found.
(figure available in print form)
The moon moves south at a rate of 3.5’ per hour.
Given this information we can determine that the moon with respect to a stationary shadow rises at:
3.5’ 0.5’ = 3’ or 3 minutes each hour.
Step VI: Compute the EastWest rate of change for the sun and moon.
PreStep work suggestions:
1. change degrees to minutes and seconds
2. operations using degrees (addition and subtraction)
Procedure:
Find the Greenwich Hour Angle (HG) of the sun and the moon from the table. The difference in their rate of change is the rate in which the moon will back into a stationary shadow.
____
(figure available in print form)
The sun has an East West motion of 15— per hour.
The moon has an East West motion of 14—26’ per hour.
From this information we know the moon will back into the stationary shadow at an EastWest rate of :
15 14 26’ = 34’ or 2040” per hour
Step VII
: Plot the path of the moon. (Diagram II)
Prestep work suggestion:
1. scale time measurements to linear measurements.
2. graph points
Procedure:
Plot the EW motion of the moon to the right and the left of the origin. This value is the degree measure found in Step VI. The distance of 34’ or 2040” to the right and left of the origin is the distance in HG one hour before and one hour after opposition.
Plot the NS motion. Plot the degree measurement of 3’ or 180” (Step V) below the western point and above the eastern point.
Connect the NS points. This is the path of the moon through the earth’s shadow.
Step VIII
: Constructing models of the moon on its path through the earth’s shadow. (Diagram III)
Prestep work suggestions:
1. construct circles of specific radius
2. scale measurements
3. construct circles tangent to Lines
Procedure:
The table provides the Seemdiameter (SD) of the moon. SD 15.8’. Using this value as the radius of 4 circles centered on the line of the moon’s path construct:
Circle l externally tangent to the shadow at the right.
Circle 2 internally tangent to the shadow at the right.
Circle 3 internally tangent to the shadow at the left.
Circle 4 externally tangent to the shadow at the left.
These four circles represent the positions of the moon at the start of the eclipse, the beginning of totality, i.e. the moron is completely hidden, the end of totality, and the end of the eclipse.
(figure available in print form)
Step IX: Calibrate the graph so as to read the time of occurrence of each phase of the eclipse. (Diagram III)
Prestep work suggestions:
1. changing degrees to minutes and seconds
2. constructing circles
Procedure:
The EastWest time is the difference in degrees one hour before and after opposition (our value is 2040”). We may use this value as a scale of time. The origin is the center of the hour time period. We then divide this hour distance into 12 parts, 5 minutes each.
Drop a perpendicular from the center of each circle constructed to the EW axis.
Determine the time where the perpendiculars meet the EW axis.
____
May 25, 1975

Eclipse Begins

3:40


Totality Begins

4:45


Totality Ends

6:12


Eclipse Ends

7:14

Students should be allowed two to three days to complete each step in the calculations. This will allow students the time to digest the newness of the material and further consider the information they are gathering and recording. The smooth transition from practice drill work to the actual computations can only be achieved when proper emphasis is placed on the skill review work. An example of a skills sheet used for Step I is included in the sample lessons at the end of this unit.