Download Unit 2―The Stars and Their Diurnal Motion

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Unit 2―The Stars and Their Diurnal Motion
8. The Stars. From the very beginning of a study in astronomy, and as frequently as possible, students should
practice watching the stars by night, to become acquainted with the constellations and their movements. To get
started in your night time viewing, you should have printed out the maps and charts previously listed. It is easiest when you live in the Northern Hemisphere as the constellations there are quite unique and one always has
the North Star to guide you. At the Equator, you are lucky to be able to see the North Star at all as it lies on the
horizon and is often lost in the haze. People in the Southern Hemisphere often start by first locating The Southern Cross or the Magellanic Cloud. Since the names of many constellations have historic origin, any guesses
who the famous explorer was that first noticed and recorded those two (large and small) clouds?
Be certain you have printed copies of the Star Maps mentioned earlier.
The names of the stars: On any star map you might see a range of differences in how the star is named. Some
have proper, but inconsistent, names. The North Star is also called Polaris but most stars in a constellation are
assigned a Greek letter for a name according to the brightness, or magnitude (see paragraph 9). Thus, Polaris is
also known as Alpha Ursa Minoris to indicate it is the brightest star in the constellation of Ursa Minor.
Stars as “Friends:” Sometimes learning the most trivial thing about a star can seem to make you “identify”
with that star as very few other people know that fact. Turn a small telescope or binoculars on Polaris and you
will discover it is really two stars, not one. Some stars merely seem to be two stars as they just happen to “line
up” that way relative to their position and Earth. Others, like Polaris, actually have a nearby companion and are
part of a multiple star system. The first are called Optical Binaries while Polaris is a Visual (or true) Binary.
Test yourself. Use your Star Map and locate the star Alpha Ursa Majoris. What is its proper name, as opposed
to this astronomical name? It too is a Visual Binary like Polaris.
9. Magnitudes of the stars. Nearly nineteen centuries ago the Apostle Paul noted that "one star differs from
another star in glory," and no more apt words can be found to mark the difference of brightness which the stars
present. Even prior to Paul's day, the ancient Greek astronomers had divided the stars in respect of brightness
into six groups, which the modern astronomers still use, calling each group a magnitude. Thus a few of the
brightest stars are said to be of the first magnitude, the great mass of faint ones which are just visible to the unaided eye are said to be of the sixth magnitude, and intermediate degrees of brilliancy are represented by the
intermediate magnitudes, second, third, fourth, and fifth. A student must not be misled by the word magnitude.
It has no reference to the size of the star, nor how bright they would be if you were standing right next to them,
but only to their relative brightness as seen on Earth. So on star maps, stars are typically represented according
to this system of “relative” magnitudes. Following the indications of these maps, the student should, in learning
the principals of stars and constellations, learn also to recognize how bright a star of the second, fourth or other
magnitude is. Like constellations, where “you know where one such is located and the new one is right next to
the known one,” once you know the magnitude of one star you can visually compare a new one as about the
same, or less, magnitude to the known one.
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10. Observing the stars and constellations. In this
day and age it might seem ridiculous that we still
use mythical figures to identify constellations.
Nearly every professional astronomer alive today
will admit that they first learned “where stuff is” by
using whatever memory device that worked. Knowing where the Big Bear/Dipper and the Little
Bear/Dipper are located, you can now always find
Dracula, the dragon, by imagining the dragon is trying to wrap around the Little Bear, but the head of
the Big Bear, at the Pointer Stars, has just nipped off
the tip of the dragon’s tail, so we even know where
the head of the dragon is located. So now referencing the star Alpha Draconis is all the directions you
need. You can even locate it in this illustration.
In the map Northern Hemisphere Stars (which you should have printed) it was shown how the Pointer Stars of
Ursa Major point to the North Star. If you use them to point in the other direction, or as we say, “downward on
the Celestial Sphere,” you will cut off the tail of Leo the Lion”― a constellation you might wish to find if your
birthday is in August. Again, notice how one memory device leads to locating the next, and then the next, constellation. After a while you start feeling very comfortable with all the “land marks” around you.
EXERCISE 6A Simple Version: To conduct this simple experiment, start by locating a building whose vertical
wall is quite smooth, like a garage, where the wall of the building runs due north and south (important) and you
can stand with the building on your right. In the Southern Hemisphere all these directions must be reversed –
you will be looking due south with the building on your left.
Find and mark a reproducible position where you can stand and
see the North Star seem to almost touch the building edge at a
height that someone can reach. Have a friend/assistant place a
small piece of tape on the building where the North Star seems to
touch the building. Now have your assistant attached the end of a
long string to this same point on the building. Have the assistant
extend the taught string until the string seems to touch an easy-torecognize star to the immediate west of the building, such as a
Pointer Star in Ursa Major, which would appear slightly west of
the building if the experiment is conducted on an early fall night.
In spring, Ursa Minor should be in about that same position. Wrap
the end of the string into something like a brick so that the position
of the string will remain fixed with no disturbance.
Now with a protractor, measure and record the angle the string makes with the vertical wall of the building.
The larger the protractor is the more accurate the data. Why is that? Or, if you know trigonometry, you could
divide the brick’s distance from the building by the height of your tape mark and compute the tangent of the angel. Immediately record the precise time that the measurement was made, just as an astronomer would do, say
as 21h 10m 30s. Now take a break. Do not disturb the string. Come back to the sight just in advance of one
hour from the above time.
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As one hour past the preceding time approaches, quickly relocate the string and the brick so the string seems to
touch the same star you had chosen earlier. Be certain you, the experimenter, are in the exact same position that
you had previously marked. Measure again the angle the string now makes with the building and record the
exact time the measurement was made.
Subtracting the two angles measured should produce the approximate result of 15 degrees. Congratulations.
You just demonstrated that the diurnal rotation of the stars around the Pole Star is 15 degrees per hour. In the
North Hemisphere the observed rotation is counter-clockwise and clock wise in the Southern Hemisphere. Of
course the stars themselves are not rotating. Something else is rotating. What is that object and in which direction is it rotating?
What does diurnal mean? You could look it up in a dictionary, but does the antonym word nocturnal help?
Repeat the Experiment: at least by visual inspection, to see that, night after night, at precisely the same time,
the stars seem to return to the exact same position. Repeating the measurement several times may suggest a
more precise way to make your measurements. Repeated testing may even show flaws in the way you took the
measurements.
A Month Later: Make plans to return to the sight one month later. It should be very clear that those same stars
definitely have NOT returned to that exact same position. If the Pointer Stars of Ursa Major were used in the
fall, those stars could now be hidden behind the north wall of the building as they would now have moved more
eastward.
Since you now know the meaning of diurnal, this extra rotation is not a diurnal one, so something else must
have caused this additional rotation. Any guesses as to what rotated?
EXERCISE 6B Modern Version: Since everyone is “into photography” these days, doing this same experiment
using photography can be more visually stimulating as well as producing more accurate data. Why? To do this
experiment photographically you will need (a) a camera whose shutter can be locked down, or open, for a period of one hour. (b) The camera must allow for manual override of anything like automatic exposure. The
shutter and aperture should be manually set for a typical exposure indoors under indoor lighting. In spite of the
darkness of the night, an automatic exposure will produce an image that is nearly an all white one. (c) Your location should be free of city lights, street lights and any surprise from automobile headlights. (d) The camera
must be on a tripod to avoid any motion or vibration.
Much of the disturbing lights of a neighborhood can be reduced, or eliminated, by photographing from an upstairs bedroom window on the north side of a house. Position the camera on its tripod so that the Polaris is at the
top and center of the view finder. If it is desired to use the Pointer Stars in Ursa Major, it might be an advantage
to have the rectangular format of the camera vertical. Record the time, such as 21h 10m 30s, and lock the shutter open. Move away without disturbing the camera. Return just in advance of one hour and when the time is
precisely one hour later close, or unlock, the shutter.
Your resulting photograph should show the motion of ALL stars that were in view as a circular streaking, but
the start and stop position of all stars in view should be easy to locate and the angle that was swept out in one
hour should be easy to measure, especially if you print the photograph very large? Why is a large photo a more
accurate one for measurement?
We know of a little girl in elementary school who, with the help of her dad, completed this experiment photographically as part of a science fair at her school. Her 8x10 photo was mounted on a foam board with titles and
explanations and her photo had white lines drawn on the photo to show the start and stop position of the stars.
A protractor was connected to the foam board with a string so that visitors could make the measurement themselves and demonstrate that “the diurnal motion of the stars is 15 degrees per hour.”
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In the preceding paragraphs we introduced a method of great importance in astronomical practice―determining
something accurately. In this case it was the rate per hour from observations separated by a long interval of
time in order to get a more accurate value than could be found from a short interval. Why is this more accurate?
To determine the rate at which the planet Mars rotates about its axis, astronomers use observations separated by
an interval of more than 200 years, during which the planet made more than 75,000 revolutions upon its axis. If
we were to write out in algebraic form an equation for determining the length of one revolution of Mars about
its axis, the large number, 75,000, would appear in the equation as a divisor, and in the final result would greatly reduce whatever errors existed in the observations employed.
FIGURE 6: MEASURING ALTITUDES
FIGURE 7: MEASURING TIME
11. The plumb-line apparatus. This experiment, and many others, may be conveniently and accurately made
with no other apparatus than a plumb line, and a device for sighting past it. In Figures 6 and 7 there is shown a
simple form of such apparatus, consisting essentially of a board which rests in a horizontal position upon the
points of three screws that pass through it. This board carries a small box, one side of which is nailed in vertical
position and another board 5 or 6 feet long to carry the plumb line. This consists of a wire or fish line with any
heavy weight, such as a brick or flatiron tied to its lower end and immersed in a vessel of water placed inside
the box, so as to check any swinging motion of the weight. In the cover of the box is a small hole through which
the wire passes, and by turning the screws in the baseboard the apparatus may be readily leveled, so that the
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wire can swing freely in the center of the hole without touching the cover of the box. Guy wires, shown in the
figure, are applied so as to stiffen the whole apparatus. A board with a screw eye at each end may be pivoted to
the upright, as in Figure 6, for measuring altitudes, or to the box, as in Figure 7, for observing the time at
which a star in its diurnal motion passes through the plane determined by the plumb line and the center of the
screw eye through which the observer looks.
The whole apparatus may be constructed by any person of ordinary mechanical skill at a very small cost, and
the more it is used, the more you feel like you have your own private observatory. To use the apparatus in an
experiment, it should be leveled, and the board with the screw eyes attached, as in Figure 7, should be turned
until the observer, looking through the screw eye, sees Polaris exactly behind the wire. Use a flashlight to illumine the wire by night. The apparatus is now adjusted and the observer has only to wait for the stars you want
to observe and to note by the time on a watch when they pass behind the wire. It will be seen that the wire takes
the place of the vertical edge of the building and that the board with the screw eyes is introduced solely to keep
the observer in the right place relative to the wire.
Update. You can probably create a modern version of the devices illustrated in the above figures if you have
access to a sturdy photographer’s tripod, although a surveyor’s tripod would be better with its built in two dimensional levels, a large carpenter’s steel square and a carpenter’s level. The most difficult part is finding a
location facing south which your setup can occupy for an extended length of time.
It is best if the tripod allows leveling by adjusting the legs, rather
than at the head of the tripod and allows rotation about the vertical
axis without affecting the level. The long length of the square upward will allow measuring the altitude of most stars in your hemisphere. Blocks to hold the square can guarantee it is vertical at
all times, although two small additional squares on both sides
could guarantee that as well. Allow a space along two sides of the
board to quickly check the level in two horizontal directions. Most
tripods will accept a ¼ inch bolt through the board and into the
head. Good locations for your instrument might be an unused garage stall whose door faces south (where most of the stars are located) or a similar large and unused porch.
FIGURE 7B: MEASURING ALTITUDE
It should be easy to calibrate the long vertical section of the square into angles of elevation―forty five degrees
is located where the inside vertical length equals the horizontal length to your sighting point. If you are good
with trigonometry you know that any vertical length divided by the horizontal length is the tangent of the viewing angle.
12. A Sidereal Clock. Special clocks can be made and regulated so that they show always the same hour and
minute when the stars come back to the same place, and such a timepiece is called a sidereal clock, in other
words a star-time clock. Knowing where any star is located and at what time that occurs became very important
in the early years of exploring the New World. There are many histories available describing how important the
development of accurate clocks and accurate astronomy became to those early sailors and explorers.
It is not enough to know that the Pointer Stars of Ursa Major are NOW directly below the Pole Star, but precisely at what time did this occur. This NOW requires an accurate number for sidereal time. Did the Pointer Stars
move to that position because of the diurnal motion of the stars or because you sailed westward and changed
your position relative to the stars and you are located at a new and different position on the Earth? As any modern navigator knows, if these stars are too far to the East by, say 5 degrees, from where they should be at this
time, the 5 degrees correspond to the number of degrees you have moved westward across the surface of the
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Terrestrial Sphere: Then if you also know your latitude you can also determine how many miles westward
you have travelled. How can the Pole Star reveal your latitude? Still found aboard many navigational vessels
are hand held instruments that helps in this latitude measurement. What is it called?
FIGURE 8: Photographing the circumpolar stars.
13. Photographing and Discovering the Celestial Sphere: EXERCISE 7. Along with strange notions the
ancients imagined about the stars, that groups of stars we call constellations were recognized animals and mythical figures, astronomers also borrow the notion the ancients held that all the stars lie on a sphere and rotate rather rigidly around the Earth as if the Earth, somehow, were at the center of things.
This time you do not have to be so precise in the timing of a photograph of the stars. With the camera pointing
straight east or west, depending which provides the darkest view, away from city lights, use a rather long exposure of one to two hours. Your resulting photograph should look something like the suggestion in this drawing.
FIGURE 8B
The stars seem to move on a curved path or arc. The ancients were aware of this from merely watching the
stars for a long period of time during the night. To them it made sense that all the stars are stuck in one place on
a crystal sphere which is rotating around the Earth. While we now know this is not correct, it is very convenient
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for tracking purposes and so, with the ancients, we imagine the stars are stuck to a Celestial Sphere with the
Earth at the center of this sphere. The two spheres share a common axis of rotation and where the Pole Stars
extend to the Celestial Sphere forming the Celestial Poles.
FIGURE 8C
Since the vast majority of stars always stay in the same place relative to each other, we can also draw navigational lines on this abstract sphere, much like the coordinate lines on a globe of the Earth, and specify coordinate-wise where any fixed star is located. The coordinate lines on the Earth determine latitude and longitude.
The corresponding lines on the Celestial Sphere are called declination and right ascension.
14. Finding and locating stars. Get out your Sky Map that you dutifully printed out. What we are about to
illustrate is that even with a map of the stars, locating the corresponding real stars from the map can seem rather
difficult at first. For starters, the map may show Ursa Major above the Pole Star but tonight Ursa Major happens to be in a different position. Then stars that you are accustomed to seeing as bright seemed to be dimmed
tonight. Then with navigational practice, the process starts becoming smoother and faster as you become accustomed to first locating familiar landmarks and from there the star you are searching for.
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FIGURE 8D
In the above photograph of the northern sky, start with the obvious location of Polaris, or the North Pole Star, in
the center of the photograph and Ursa Major slightly to the left of Polaris. Then locate Ursa Minor which is not
so obvious now. Do you remember how to locate Ursa Minor from the location of Ursa Major? Once both are
found, can you locate Dracula? Notice how memory devices help here.
Locate on the Star Map, and then in the above photo, the two bright stars Capella and Vega which are on opposite sides of Polaris and nearly equidistant from it. Do these stars share in the motion around the pole? Are they
visible on every clear night, and all night or does one of them set every 12 hours? How about Alpha Draconis?
15. Rising and Setting of the Stars. A study of the sky along the lines indicated in these questions will show
that there is a considerable part of it surrounding the pole whose stars are visible on every clear night. The same
star is sometimes high in the sky, sometimes low, sometimes to the east of the pole and at other times west of it,
but is always above the horizon. Such stars are said to be circumpolar. A little farther from the pole each star,
when at the lowest point of its circular path dips for a time below the horizon and is lost to view, and the farther
it is away from the pole the longer does it remain invisible, until in the case of stars 90° away from the pole, we
find them hidden below the horizon for twelve hours out of every twenty-four (see Figure 9). The Sun is such a
star, and in its rising and setting acts precisely as does every other star at a similar distance from the pole—only,
as we shall find later, each star keeps always at (nearly) the same distance from the pole, while the Sun in the
course of a year changes its distance from the pole greatly, and thus changes the amount of time it spends above
and below the horizon, producing in this way the long days of summer and the short ones of winter.
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FIGURE 9: Diurnal motion of the northern constellations
How much time do stars which are more than 90° from the pole spend above the horizon?
We commonly say “that the sun rises in the east,” but this is strictly true only at the time when it is 90° distant
from the pole, that is in March and September. At other seasons it rises north or south of east according as its
distance from the pole is less or greater than 90°, and the same is true for the stars.
16. The Geography of the Sky. Find from a regular map the latitude and longitude of your location. Find on
the map the place whose latitude is 39° and longitude 77° west of the meridian of Greenwich. Is there any other
place in the world which has the same latitude and longitude as where you live? Careful! We are talking about
coordinates accurate down to several decimal places.
The places of the stars in the sky are located in the same manner illustrated by these geographical questions only different names are used. Instead of latitude the astronomer says declination, in place of longitude an astronomer says right ascension, in place of meridian one says hour circle, but what is meant by these new names
expresses the same ideas that the geographer uses.
Imagine the earth swollen up until it fills the whole sky; the earth's equator would meet the sky along a line (a
great circle) everywhere 90° distant from the pole and this line is called the celestial equator. Notice its position
along the middle of your sky map and notice what constellations it crosses. Every meridian of the swollen earth
would touch the sky along an hour circle, which is a great circle passing through the pole and therefore perpendicular to the equator. Note that in the map one of these hour circles is marked 0. It plays the same part in measuring right ascensions as does the meridian of Greenwich in measuring longitudes―it is the beginning, from
which they are reckoned. And it is easy to “get tripped” without another memory device. Right Ascension “ascends to the right” only when viewed from outside the Celestial Sphere. Standing on the Earth and facing up-
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ward to the Celestial Equator in the Northern Hemisphere, those numbers increase to your left. But then, left
and right were never good direction indicators anyway.
FIGURE 9B Celestial Coordinates
Astronomers use the right ascensions of the stars not only to tell in what part of the sky the star is placed, but
also in time reckonings to regulate their sidereal clocks, and they find it convenient to express right ascension
not in degrees but in hours, 24 of which fill up the circuit of the sky and each of which is equal to 15° of arc, so
24 × 150 = 3600. The right ascension of Capella is 5h. 9m. = 77.2° but you should become accustomed to using
it in hours and minutes rather than degrees. You should also note that some stars lie on the side of the celestial
equator toward Polaris, and others are on the opposite side, so that astronomers have to distinguish between
north declinations and south declinations just as the geographer distinguishes between north latitudes and south
latitudes. This is done by the use of the + and – signs. A plus time indicates that the star lies north of the Celestial Equator.
Thought question: Initially it might seem illogical that right ascension increases to the East while its counterpart on the Earth, longitude, increases to the West from Greenwich. Now recall that the Celestial Sphere is not
really rotating but the Earth does. Now does it make sense that a “later hour must come after an earlier hour”
which means “increasing numbers toward the East.”
Update: (a) Now a century after this original text was published, we have the advantage of the Internet. If any
diagrams shown here, such as the Celestial Sphere, are not entirely clear to you, a simple search on the Internet
will bring up a host of other examples.
(b) Celestial timing is now much easier than it was a century ago, now that we have timers that can maintain
accuracy down to millionths of a second. There are many videos posted on the Internet where you can see a
modern day astronomer, punch in the celestial coordinates of a star to be studied and the motor driven telescope
automatically moves to the position where that star can be observed.
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FIGURE 10: A photograph of the Pleiades
Practice Using Celestial Coordinates: Figure 10 shows a popular astronomical site to locate called the
Pleiades. In ancient times it was known by various names including the Seven Sisters. To see why that name
might have been used, squint at Figure 10, or view it through a filter, so that the fainter/smaller stars are less
pronounced and count the brighter/larger stars. They even seem to form a kind of dipper shape. This is typically what ancients saw night after night in typical viewing conditions and to them this was a cluster of seven stars.
Only with the use of optical instruments do we now understand that this is a much larger cluster of stars and,
yes, they are physically located together in the same region of space.
Given the coordinates of the Pleiades as R. A. = 3h. 42m and Dec. = +23.8°, mark the location of the Pleiades
on your star map. What large constellation is it near? What season of the year is it now? If you went out tonight at, say, 9PM, would the Pleiades be visible assuming good weather? In what region of the sky would you
start looking? What you have just done is what every astronomer goes through any time a viewing session is
scheduled on a telescope. This also demonstrates how celestial coordinates are used to locate known astronomical objects. The coordinates tell us where to look in precise numerical terms.
Locate the star Antares on your star map given its coordinates are R. A. = 16h. 23m. Dec. = -26.2°. You already know that there is a good chance you might not be able to view Antares tonight. Why is that?
17. Reference Lines and Circles. As the stars move across the sky in their diurnal motion, they carry the
framework of hour circles and equator with them, so that the right ascension and declination of each star remains unchanged by this motion, just as longitudes and latitudes remain unchanged by the earth's rotation. They
are the same when a star is rising and when it is setting; when it is above the pole and when it is below it. During each day the hour circle of every star in the heavens passes overhead and at the moment when any particular
hour circle is exactly overhead all the stars which lie upon it are said to be "on the meridian” and at that particular moment they stand directly over the observer's geographical meridian and upon the corresponding celestial
meridian.
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Connecting the Points, Lines and Angles Together. In Figure 11 below, the gray circle represents the Earth
with C as its center and the outer circle the Celestial Sphere. The point O represents the point where our observer Omar is standing on the Earth. The line PCP represents the polar, or spin, axis of the Earth. HCH is the
extension of what Omar perceives as the horizontal plane he is standing on. (Imagine the Earth now shrunk to a
point). The line ECE then represents the equator of the Earth extended to the Celestial Equator. The outer circle POEP illustrating the Earth is the terrestrial meridian that Omar is standing on. The outer circle ZEHPHPZ
is the current celestial meridian for our viewer Omar. The angle PCH is the angle the Polar Axis makes with
Omar’s horizontal plane, so that is Omar’s latitude. (Remember. In the Northern Hemisphere, the altitude of
Polaris above the North Cardinal Point is approximately equal to your latitude.) The point Z directly above
Omar on the Celestial Sphere is Omar’s Zenith point. The angle ZCE is the angle that Omar’s Zenith makes
with the Celestial Equator known as the declination of Z. Now for some simple geometry:
FIGURE 11: Reference lines and circles
Theorem: The latitude of any place is equal to the declination of its zenith.
Corollary: Any star whose declination is equal to your latitude will once in each day pass through your zenith.
18. From the construction of the figure
∠ E C Z + ∠ Z C P = 90°
∠ H C P + ∠ Z C P = 90°
we find by subtraction and transposition
∠ECZ=∠HCP
which then gives us finally:
Theorem: The latitude of any place is equal to the elevation of the pole above its horizon plane.
It is nice that, with lots of things spinning and moving in astronomy, we still have a few fixed points that we can
use for reference, such as the Celestial Pole, an observer’s Zenith, etc., but as humans we add to the motion as
we too move around now and then as well. As an observer who travels north or south over the Earth changes
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their latitude, and therefore changes the angle between the horizon plane and the axis of the earth and changes
their Zenith, so the sky can look quite different at different latitudes.
Mental Test: You should now know enough about the constellations to picture in your mind answers to the following questions.
1. In what region of the sky would you look for Polaris if you were in Alaska above the Arctic Circle?
2. Assuming good weather, could Polaris be seen from the Earth’s Equator?
3. With the Sun’s departure from the Arctic Circle in winter, what is the longest time that Pleiades could be
seen during a single night’s viewing?
Fortunately changes from our moving around can be used mathematically for purposes of navigation. And
eventually, we need more accurate position calculations not only so that we can understand the stars more fully
but for more accurate navigation. We then have to face up to the fact that some of our “fixed stars” we have
been using for reference are not entirely fixed and they too move around. Polaris is close to being a pole star
but not quite. It “wobbles” around the true North Celestial Pole or Polaris also has a diurnal motion.
FIGURE 12: Diurnal path of Polaris
EXERCISE 8 Updated: You may not have constructed the Plumb Line apparatus illustrated in Figures 6 and 7
but you might be able to make a different apparatus to complete this experiment. You might be able to clamp a
thin, but rigid, 3 feet steel rod to the side of a house, barn or other object so that, like sighting a squirrel through
the sites of your rifle, you have Polaris just sitting on the top edge of your “barrel.” While you might not be
able to perform the experiment for a 24 hour period, you can line Polaris up and return 6 or 12 hours later to see
how far it has moved from your original sighting. Its motion is expected to look like Figure 12. Is the direction
of motion from 1 to 2 to 3 to 4 what you would expect from the motion of the Earth?
When Polaris is at 2 it is said to be at upper culmination. When at 3 it is at western elongation. When it is at 4
it is at lower culmination. When it is at 1 it is at eastern elongation. The altitude observed at either 1 or 3 may
be considered equal to the latitude of where you are located. Can you explain that?
But the altitude observed when Polaris is at the positions marked 2 and 4 must be corrected for the star's distance from the pole, which is approximately 1.3°. That number is large enough to be a considerable error in
some navigational situations. Obviously in modern warfare, with self propelled rockets fired at an enemy’s position from hundreds of miles away, the ability to know where we really are located and where the enemy is located must be extremely accurate. A 1.3° error in an arc at the equator is about a 100 mile error in position.
Follow up Experiment Updated: Nothing is more rewarding after completing an experiment but to do a follow up experiment that measures the accuracy of a previous one. Whatever devices you may have constructed,
you were able to measure the altitude of the Polaris corrected for elongation and culmination and you now know
what that this says about your latitude on Earth. Use your data from Polaris and compute your latitude in degrees as per 42° 57’ north and compare your results by looking up your latitude on any corresponding map.
These days you can simply do a search on the Internet for the precise results. You can further test the sensitivity
of any apparatus you constructed by finding a reason to travel 10 to 25 miles north or south of your location and
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determine your latitude there. You might be surprised that your instrument can actually detect the differences in
latitude for that small a distance on the surface of Earth. Now you are in a position to see how one can use the
stars to determine location.
19. The meridian line. Find an open field or empty lot to which you can have easy and repeated access. Add a
plumb line to whatever instrument you used to measure the altitude of Polaris. A weight hanging from a string
is a simple add-on. When Polaris is at upper or lower culmination, your instrument, when looking at Polaris,
establishes a true north. Do not use a magnetic compass to establish north. Local iron deposits can produce extreme deviations from true north by many degrees with a magnetic compass. Now that you know how to use
your instrument to establish north, use it to drive two stakes in the ground that are some distance apart, maybe
10 to 25 feet apart, so that each is in line with Polaris and each stake is plumb or vertical. Then any line between
the two stakes is a reasonably accurate north-south line. If your stakes are tall, you could attach a wire between
them, or merely scratch a straight line on the ground between their bases. Such a line is actually a segment of
your local meridian line on Earth. At this point, do you have any guesses as to how you might use the stars to
determine what that line means in terms of longitude? That would be your current longitude on the planet
Earth.
20. Time: What time is it right now? Are you certain? How accurate do you suppose the time is that you just
stated? What do we mean by the concept of time itself? Can you answer that without stating it in hours, minutes and seconds?
The sidereal time at any moment is the right ascension of the hour circle which at that moment coincides with
the meridian. Aha! That is what that meridian line was for. When the hour circle passing through Sirius coincides with that meridian line, the sidereal time is 6h and 40m, since that is the right ascension of Sirius, and in
astronomical language Sirius is "on the meridian" at 6h and 40m sidereal time. As may be seen from your star
map, this 6h and 40m is the right ascension of Sirius, and if any clock is set to indicate 6h and 40m when Sirius
crosses the meridian, it will show sidereal time for a while but not over a long time period unless you can regulate it. If the clock is properly regulated, every other star in the heavens will come to the meridian at the moment when the time shown by the clock is equal to the right ascension of the star. A clock properly regulated for
this purpose will gain about four minutes per day in comparison with ordinary clocks (Does that surprise you?),
and when so regulated it is called a sidereal clock.
Update: These days there are a variety of battery powered clocks that are extremely “accurate” for holding the
correct time over long periods. However, to teach you a lesson you should never forget, go ahead and obtain
one such clock or watch for the experiment which follows. You will discover that the word “accurate” can be
very deceiving depending on how, or to what, you apply it. On the other hand, if you do not like having tricks
played upon you, we will recognize up front that a better clock for the experiments that follow is a reliable old
fashioned clock that has a swing lever for adjusting its speed. You can still find such around. Your modern
clock ticks at a fixed standard rate and probably cannot be adjusted to the correct rate for sidereal time, so you
will probably have to make arithmetic corrections daily. Be certain to label your chosen clock with Sidereal
Time. Now that you know that our house clocks are NOT tracking sidereal time, you would not want to be late
for an appointment because you used your clock adjusted for sidereal time and not local time.
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THE HARVARD COLLEGE OBSERVATORY, CAMBRIDGE, MASS
(You might guess that they have accurate clocks in there.)
EXERCISE 9: Set the clock you have chosen to sidereal time by means of the transit of a star over your meridian. If your meridian line was a wire between the posts, you could even lie on the ground so that your “sighting
eye” is just above the equivalent ground line and you already have a fairly accurate sighting instrument as you
used your previous instrument to establish it. That would be more accurate if the wire were several feet above
you. Why is that? It could be instructional if you also had an accurate house/personal clock and record local
time of that same transit. If it is a house clock, you could compute what that clock would have shown for local
time using what you recorded as sidereal time.
Whatever instrument you use, be certain your sighting eye or instrument, is directly on, or below, your meridian
line. Choose a suitable star in the sky that will clearly pass over your meridian line. Estimate as nearly as possible the beginning and end of the period during which the star appears to exactly pass over your meridian line.
The middle of this period may be taken as the time at which the star crossed the meridian and at this moment
the sidereal time is equal to the right ascension of the star. (Your line has a thickness and it does take time to
determine that the star actually left your meridian line.) The difference between this right ascension and the observed middle instant in local time is the error of the clock or the amount by which its hands must be set back or
forward in order to indicate true sidereal time. If you had an antique style clock, or watch, you could translate
the slider and have it run directly in sidereal time. Modern clocks do not permit this and are specifically designed to run “constantly” at an accurate solar time.
Once your clock has been set to sidereal time, there are several enlightening experiments you can perform, assuming the wind does not blow away your meridian line on the ground or disturb your stakes and any wire you
strung between them.
1. Check the time of passage, using your sidereal clock, of that same star the very next night. If you are confident that neither wind nor weather has disturbed your set up, make notice of any time difference from the previous night but do not change or reset your Sidereal Clock.
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2. Once your system is “holding” and you are confident it should remain that way for several days, try to determine the sidereal time as much as a week later. Record the time difference but do not adjust your Sidereal
Clock.
If you have chosen a modern digital clock or watch for this experiment, you have discovered that your very “accurate” clock, once set to sidereal time, unfortunately does not keep sidereal time but insists on keeping the local time elapsed. Had you acquired one of those old fashioned clocks mentioned above, after measuring the
time on the second day, we would have instructed you to “move the lever” to change the speed of the clock to
conform to sidereal time.
You may have some guesses already what is happening. If the same local time, say 11PM every night, would
correspond to the same, but different numerical, sidereal time each night, then Ursa Major would always be low
above the North Cardinal Point every night of the year. After all it was there at 11 PM in November. But in the
spring, Ursa Minor is there and Ursa Major is way up above the Pole Star. To fully explain what is happening
must wait until we discuss the Earth and all of its motions in later Units as well as discussing Time Keeping in
more detail.
Suffice it to say now that a sidereal time piece must be carefully adjusted so as to keep sidereal, or star,
time. Regular, or every day time, assumes it takes 24 hours for the Sun to “circle” the Earth once. Clocks and
watches for everyday use are constructed then to keep solar time. Sidereal, or star time, assumes it takes 24
sidereal hours for a star to pass over your meridian line twice in succession. Sidereal time is “faster” than solar
time by about 4 minutes. Hence a sidereal day is “shorter” than a solar day about this amount in that a sidereal
day will be completed before a solar day has completed.
Short or heuristic calculation: Sometime making a quick calculation, where the numbers are only approximately correct, can produce a quick meaning to the numbers. Assume that the 4 minute difference between
solar and sidereal time holds of each and every day. Multiply that by 30 days in a month times 6 months in a
half year. Dividing that by 60 minutes in one hour should produce the result of 12 sidereal hours. That would
mean that a solar clock and a sidereal clock disagree on the “time” by 12 sidereal hours, or the Celestial Sphere
is located 12 sidereal hours, or 180 degrees, from its position at the same solar time 6 solar months earlier. But
we already know that. Ursa Major is just below Polaris on a November evening and is above Polaris on a May
evening, or a 180 degree difference 6 months later.
Update: The need for very precise measurements. Much more can be determined about what other motions
the sun and stars are doing but only by making more precise and minute measurements. Hence the need for
very refined, and even delicate, instruments for making measurements. But even when this text was published
in 1910, astronomers were well on their way to noticing that there many more “movements” than we have discussed so far. In addition to the 4 minute difference between solar and sidereal time, astronomers already knew
in 1910 that the sidereal time itself is changing with time. That means that on a long term basis, we need to
constantly update the data for the right ascension and declination for all stars and always have a current table of
same for any star we wish to locate and study.
It was already noticed in 1910 that the declinations of stars tends to oscillate (wobble) slightly around some
mean value, so for the most part, any declination can be assumed to be approximately current. In 1910 it was
already determined that the right ascension of any star would increase by about 1/20th of a minute per year.
Below are the 1900 and 2011 listings for the coordinates of Aldebaran, or Alpha Tauri, a one magnitude star.
So already you know Aldebaran is a very bright star located in the constellation of Taurus, The Bull.
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Calculation: Use the 1910 prediction that the right ascension of any star would increase by 1/20th of a minute
per year to calculate the predicted right ascension of Aldebaran 111 years later. Were they making precise measurements in astronomy more than a century ago? Don’t forget the seconds in your calculation.
Update. It should be obvious that while a student in this course you can ignore the long term change in the
right ascension of stars, but you have a unique problem in that readily available are quartz crystal and digital
clocks that are very accurate in determining time over a long period. Unfortunately they are programmed to
track solar time and not sidereal time. However, you can use such clocks “as is” by adding a “correction term”
to change from solar time to sidereal time.
T = A ± U x D (typical “–“for modern clocks)
T stands for the solar time on the clock when the star should cross the meridian. A is the right ascension of the
star, and U is the correction of the clock per day and D is the number of days that have elapsed since your clock
was first calibrated.
Update: Making a Sidereal Clock: You could repeat the earlier experiments using a modern, accurate solar
clock by returning to your meridian line experiment, assuming weather conditions have not disturbed your instruments. Measure in minutes how the two times differ over a 3 to 4 day period. The more number of days the
better will be the accuracy of your value for U per day above. Why is that? Then “test” your clock for accuracy
by predicting, and then measuring, your predicted solar time for the same passage a couple of weeks later.
Obviously your prediction would be highly unreliable if you predicted the time passage for one year later. You
would be trying turn a three day measurement into a one year prediction with a make-shift instrument. How
accurate and how durable (not subject to weather and use) must have been the astronomical instruments of 1910
to predict the right ascension of stars would change by 1/20th of a minute per year, based upon the your calculations for 111 years later?
Keeping the Concepts Straight: It is important to fully understand the concepts of time the above experiments
are describing. Otherwise, when we later discuss how the more complex motions of the Earth, Moon and Sun
produce these time adjustments, it might not make sense what we are explaining as you did not have the concepts straight from the beginning.
The “correction” to your modern day clock required above is not really a correction but you are “adjusting” for
the fact that this very reliable, and accurate, modern clock cannot be speeded up to keep up with sidereal time as
the mechanical clocks a century ago could. The correction term UxD keeps building up until 6 months after
the experiment is started the two times disagree by 12 hours and one year later they disagree by 24 hours. But
24 hours of right ascension is the same as 0h of right ascension, or the two times now, once more, exactly
agree.
But now if your modern clock really had the ability to measure events down to fractions of a minute of time, the
exact agreement would only be an approximate agreement, or the two times, as you measured, would be in
agreement to within 1/20th of a minute. At that point your instrument was sensitive enough to actually measure
the annual increase of the right ascension of stars, or you now would have actually discovered something “happening” in the Solar System of the Universe.
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21. Definitions. To define a thing or an idea is to give a description sufficient to identify it and distinguish it
from every other possible thing or idea. Especially in the physical sciences, we insist that a name of something
must tell us what it is or what it does, and in astronomy often where it is located. We also “carry along” historic
names and descriptions to help lay people understand what we are talking about, but ultimately the name or description must fit this requirement in the physical sciences. “Alpha Tauri” immediately tells you where the star
is located and its brightness tells you which of the many stars it is.
Read each of the definitions below. Then close your eyes and picture what it is and where it is located. If the
description does not work for you then rewrite it so it does. In each case submit the definition to memory.
The Poles of the heavens are those points in the sky toward which the earth's axis points. How many are there?
The one near Polaris is called the North Pole.
The Celestial Equator is a great circle of the sky distant 90° from the poles.
The Zenith is that point of the sky, overhead, toward which a plumb line points. Why is the word “overhead”
placed in the definition? Is there more than one zenith?
The Horizon is a great circle of the sky 90° distant from the zenith.
An Hour Circle is any great circle of the sky which passes through the poles. Every star has its own hour circle.
The Meridian is that hour circle which passes through the zenith.
A Vertical Circle is any great circle that passes through the zenith. Is the meridian a vertical circle?
The Declination of a star is its angular distance north or south of the celestial equator.
The Right Ascension of a star is the angle between its hour circle and the hour circle for a point on the equator
called the Vernal Equinox as measured at the poles. (The angle is converted to hours and minutes in order to
track the motion of the stars and, starting from the vernal equinox, increases toward the west.)
The Altitude of a star is its angular distance above the horizon.
The Azimuth of a star is the angle between the meridian and the vertical circle passing through the star. (A star
due south has an azimuth of 0°, due west, 90°, due north, 180°, due east, 270°. Using the two numbers, altitude
and azimuth to locate an object in the sky is commonly used in terrestrial navigation as well and is called the
Altazmuth System.)
The Hour Angle of a star is the angle between its hour circle and the local meridian. (It is measured from the
meridian in the direction in which the stars appear to travel in their diurnal motion. That is, toward the west.)
Test yourself: Without looking at the above definitions, see how well you memorized them by answering the
following questions.
What is the azimuth of Polaris in degrees?
What is the azimuth of the sun at sunrise? At sunset? At noon? Are these azimuths the same on different days?
What is the hour angle of the sun at noon? What is the hour angle of Polaris when it is at the lowest point in its
daily motion?
22. Exercises. You must not be satisfied with merely learning these definitions. You must learn to see these
points and lines in your mind as if they were visibly painted upon the sky. To this end it is helpful to note that
the poles, the zenith, the meridian, the horizon, and the equator seem to stand still in the sky, always in the same
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place with respect to the observer, while the hour circles and the vernal equinox move with the stars and keep
the same place among them.
Does the apparent motion of a star change its declination or right ascension?
What is the hour angle of the sun when it has the greatest altitude?
Will your answer to the preceding question be true for a star?
What is the altitude of the sun after sunset?
In what direction is the North Pole from the zenith?
In what direction is the North Pole from the vernal equinox?
Where are the points in which the meridian and equator respectively intersect the horizon?
Summary: We have covered lots of definitions and concepts for making measurements and nearly all of them
must be well understood before other concepts can be mastered, so it is advisable that that you spend considerable time reviewing this unit before going on to other units. Astronomy is clearly a science that studies objects
that are constantly moving and they move in three dimensions. That alone makes simply “looking at them”
scarcely a way to “measure what they are really doing.”
Nearly every topic covered in this unit is available somewhere on the Internet as a video on the subject. If you
do a search on the Internet specifying videos and include in your search the subject name as stated here, you can
often watch it on YouTube, or elsewhere. Some even allow you the simple privilege of watching the diurnal
motion of the stars through time lapse photography. Nothing is more instructive than to watch the same topic
“unfold before your eyes” in a video.