# Why is right ascension measured on a 24 hour scale rather than a 23 hours and 56 minutes scale?

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From several texts I have read, I have learned that right ascension is measured on a scale of 24 hours. I understand that Earth rotates almost exactly 360 degrees relative to distant objects, which is what RA is used for. The 24 hour system should only apply to the 361 degree rotation period (solar day). Why then do we split the sidereal day of the earth into 24 hours, when if it were split into 23 hours and 56 minutes we could know where an object would be x hours away after x hours?

More (possibly) relevant information: I understand the Hour Angle HR is the measure of how far Aries is from the local meridian (RA of current meridian of longitude). This way you could calculate how many hours away the object is, but once again, this would not be exactly correct as 4 minutes has to be subtracted from the 24 hour scale, multiplying the whole time to something like 99.7% (14.44 degrees per hour), correct?

Thanks, this is really confusing me and I would appreciate some help. I did see a similar question but that had nothing to do with the sidereal day v. solar day issue I'm having.

It's 23 hours, 56 minutes, and 4.1 seconds, more or less. That's one sidereal day expressed in solar time. Solar time is neither a particularly convenient nor particularly useful time scale when looking at the stars.

Alternatively, one could use sidereal time, which is what hour angle refers to. There are 24 sidereal hours in a sidereal day, 60 sidereal minutes in a sidereal hour, and 60 sidereal seconds in a sidereal minute.

The problem is not that right ascension goes from 0 to 24 hours (or 0 to any other number). The problem is that the Earth is revolving around the Sun, and our time system is based on the Sun. Because of the motion around the Sun, one solar day (24 hours 0 minutes 0 seconds) is different than one sidereal day (23 hours 56 minutes 4 seconds). So you need two clocks: one for the sun, and one for the stars.

An example may help. Let's change right ascension to be from 0 to 100. At midnight (00:00) on day 1, let's define that 0 right ascension is on the meridian. At 23:56, the Earth has made one complete rotation relative to the stars, so 0 right ascension is back on the meridian. At 24:00, the Earth has made one complete rotation relative to the Sun (one day), but the right ascension has increased to 0.28 during the extra 4 minutes.

Day 2. At 23:52, the Earth has completed a second full rotation relative to the stars, so the right ascension is again 0. But after two complete days (24:00 on day 2), the right ascension is now 0.56.

Right ascension is a historical oddity.

To specify a point in the sky you need a coordinate system, the one which we have come to use has it's origin at the Point of Aries on the Equator and the Ecliptic (a reasonable choice), It uses the Equator for one axis, and the meridian through the point of Aries for the other, again these are convenient and reasonable.

The coordinate system also requires a unit to be chosen for measuring angular distance. For measuring declension, we use degrees. The more mathematically pure may prefer radians, but degrees are a commonly used unit for measuring angles.

For measuring the angular distance around the equator we have come to use "hours", by dividing the full turn into 24 parts. This is a historical oddity. It has a few conveniences: The sidereal day is only slightly shorter than 24 hours, so a star with RA of 1hr will be due south about 1 hour after a star with a RA of 0hr (not exactly but close enough for a rule of thumb). A telescope with an equatorial mount that is set to make a full turn in 24 hours will track stars well enough for a human observer.

Dividing the equator into (23+56/60) parts would be inconvenient since its not a whole number. For these reasons Flamsteed used Hours for his catalogue in 1712, and we have followed his tradition.

So the use of "hours" may be roughly related to the apparent motion of the stars, but it is really just a way of specifying an angle, and using 24 is easier than (23+56/60)

## Star time: a brief history of sidereal time

Here’s a question. How long does it take for the earth to spin exactly once on its axis?

If you answered 󈬈 hours”, then I’m sorry to tell you that you’re wrong. Not wrong by very much, it’s true, compared with people who live in the crazier parts of the internet (YouTube, where else?).

Probably not as wrong as Mr Potters Clay, who doesn’t understand that the earth can spin on its axis and orbit the sun at the same time. To be fair, the only book he refers to offers no useful astronomical information.

Whereas Mr Thrive and Survive claims to have read the wikipedia but he still believes the earth is flat.

And you’re almost certainly not as wrong as Mr Robert Sungenis, who thinks that the earth is the center of the universe, says that evolution is false, and that Stanley Kubrick filmed and faked the NASA moon landings.

These guys are, in physicist Wolfgang Pauli’s well-known phrase “not even wrong”. (Pauli’s full comment was, “Das ist nicht nur nicht richtig, es ist nicht einmal falsch!” — “That is not only not right, it is not even wrong!”.)

## Why is right ascension measured on a 24 hour scale rather than a 23 hours and 56 minutes scale? - Astronomy

1 Department of Mathematics, Central Michigan University, Mount Pleasant, USA * Corresponding Author: [email protected]

2 Department of Mechanics, Materials, and Aerospace Engineering, Illinois Institute of Technology, Chicago, USA

3 Department of Physics and Astronomy, Millikin University, Decatur, USA

Received 30 September 2012 revised 30 October 2012 accepted 14 November 2012

Keywords: Polaris Big Dipper Time Longitude Latitude

In this paper we introduce methods for approximating local standard time in the Northern Hemisphere using Polaris and the Big Dipper as well as alternative reference stars, and describe in detail how to construct a device we call a dipperclock to facilitate this process. An alternative method which does not require a dipperclock is also discussed. Ways of constructing dipperclocks which glow in the dark are presented. The accuracy of dipperclocks is examined, both theoretically and through field testing. A java program is provided for creating dipperclocks customized to a particular year-long time period and place to get improved accuracy. Basic astronomical definitions and justifications of the results are provided. We also discuss the use of dipperclocks to find longitude and latitude.

The North Star Polaris and the Big Dipper can be thought of as a giant clock in the sky, with the imaginary hour hand of the clock extending from Polaris through the pointer stars in the bowl of the Big Dipper (see Figure 1 ). One needs to imagine a 24-hour clockface centered at Polaris which runs counterclockwise, is perpendicular to your line of sight to Polaris, and has midnight at the top and noon at the bottom. The time indicated in the picture appears to be about 10:30 a.m., before adjusting for the date, which we will do in Appendix 8.

Since the earth rotates clockwise on its axis when viewed from below the solar system, the Big Dipper, which provides the hour hand for our clock, appears to rotate counterclockwise about Polaris. References [1-7] rely on the use of imaginary 24 or 12-hour clockfaces that the reader must envision in the sky to estimate the sidereal time. To get local standard time, a mathematical calculation involving the date is needed in [2-4,6,7]. We describe a similar “equinox estimation method” in Appendix 8. Reference [1] uses a mechanical translator to do the calculation. References [8-12] discuss mechanical devices for telling time by the Big Dipper and other stars. We note that our dipperclock 1 differs from the devices we have found in the literature in various ways, including the following: 1) The dipperclock has no parts which move relative to one another thus it is simpler and easier to make than the other devices we have found in the literature. As shown in Figure 2 , the dipperclock consists of a single disk with a hole in the center 2) The dipperclock can be made in glow-in-the-dark versions, which are easier to read in dark areas. In Figure 2 , the numbers

and letters, the concentric circles, and the hash marks can be made to glow in the dark 3) Custom dipperclocks can be generated using the free java program which we are supplying 4) This paper supplies proofs for the claims we make about the dipperclock. Some of the proofs are quite intricate because the clookface, which is the outer ring in Figure 2 , rotates when the date changes.

In Section 3 we discuss a correction based on the user’s longitude that can improve the accuracy of the dipperclock. In Section 4 we show how a dipperclock can be used as a longitude finder if the Universal Time (see Appendix 1) and date are known, and can be used to find the latitude in the Northern Hemisphere as well. In Section 5 we discuss glow-in-the-dark dipperclocks.

We relegate many of the details of our work to the following 12 Appendices:

Appendix 1: A list of basic astronomical definitions.

Appendix 2: Justification of the geographic correction.

Appendix 3: Justification of the method of using a dipperclock to find longitude.

Appendix 4: Using different stars as the tip of the hour hand.

Appendix 5: Discussion of a free java program to generate dipperclocks.

Appendix 6: Construction of glow-in-the-dark dipperclocks.

Appendix 8: Finding the time without a dipperclock (the equinox estimation method).

Appendix 9: Discussion of the mathematical foundations required for Appendices 10 - 12.

Appendix 10: Discussion of how the various numbers and hash marks are placed around the dipperclock.

Appendix 11: Accuracy of the dipperclock. (Bottom line: Under typical assumptions, theoretical error Figure 2 is a diagram of our generic dipperclock (By a generic dipperclock we mean a dipperclock which assumes the previous autumnal equinox to be midnight in Universal Time on September 22, ignores leap years, and does not have a built-in geographic correction such a dipperclock can be used at any time.) In the diagram, the letters and numbers in the outer ring represent local standard time, with the M representing midnight, the numbers 1 through 11 running counterclockwise from M representing a.m. times, the N representing noon, and the numbers 1 through 11 running counterclockwise from N representing p.m. times. Each letter in the inner ring represents the first day of a month, with the months running clockwise for example, the F in the inner ring represents February 1, the M in the inner ring adjacent to the F represents March 1, etc. When the dipperclock is used (as described in more detail below) the user rotates the dipperclock so that the current date is at the bottom of the inner ring, positions the dipperclock so that Polaris is centered in the center of the central hole of the dipperclock, and reads the local standard time indicated in the outer ring by the “pointer stars” in the bowl of the Big Dipper.

This diagram can also be obtained by going to the URL in [13]. To make a working version, either copy Figure 2 onto a stiff piece of paper or print the diagram from [13]. Then, cut away the part inside the inner ring and the part outside the outer ring, leaving a doughnutshaped object consisting of two rings containing letters, numbers and hash marks. The numbers in the outer ring represent local standard time, with M denoting midnight and N denoting noon. The letters in the inner ring represent the first days of months at midnight. For example, F represents February 1 at midnight.

Recommended viewing procedure:

1) If you are using a glow-in-the-dark dipperclock (Section 5), expose it to light before using it.

2) Locate the Big Dipper (see Figure 1 or Figure 3 ).

3) Locate the North Star Polaris by drawing an imaginary line through the “pointer stars” in the bowl of the Big Dipper farthest from the handle.

4) Hold your dipperclock with today’s date in the inner ring directly below the center.

5) Center Polaris in the hole.

6) Move your dipperclock back and forth along the line from your eye to Polaris, keeping it perpendicular to this line, until the pointer stars are close to the outer ring.

Then read the standard time at the point on the outer ring closest to the pointer stars.

7) If you are on daylight savings time, add one hour.

Figure 3 . Illustration for example 1 3 .

Suppose that on August 17 we view Polaris and the Big Dipper using a 203 mm (8 inch) glow-in-the-dark dipperclock (Section 5) and what we see is as shown in Figure 3 above. Note that the dipperclock has been rotated so that August 17 in the inner ring is at the bottom of the dipperclock. The pointer stars in the bowl of the Big Dipper give a reading of about 11:50 p.m. local standard time, which is about 12:50 a.m. local daylight time.

Your location within your time zone could cause an error of a half hour or more in the time read from your dipperclock. To fix this problem, let GC = the geographic correction (in minutes)

LST = your local standard time (in hours)

UT = the Universal Time (in hours) (see Appendix 1)

L = your longitude (in degrees).

(1)

This formula is justified in Appendix 2.

Note that LST – UT is constant throughout your time zone. In any time zone, we call the line of longitude of 15 × (LST – UT) degrees the centerline of the time zone, even though it might not be in the center. West longitude is taken to be negative, and east longitude is taken to be positive. We see from Eq.1 that the geographic correction is zero if you are located on the centerline of your time zone.

As an example, we will consider the geographic correction for Chicago, Illinois, which is located in the U.S. Central Time Zone at west longitude 87.63˚ (see [14]). LST – UT = –6, which follows from the fact that Central Standard Time is 6 hours behind UT, or one can get UT from [15] (or http://www.usno.navy.mil) and compute LST – UT directly. From Eq.1, we have

so we must subtract about 9 minutes from our time estimate for Chicago.

4. USING A DIPPERCLOCK AS A LONGITUDE AND LATITUDE FINDER

Given the date, time zone, and longitude, we have shown how one can use a dipperclock to find the local standard time (LST). If instead the Universal Time and date are known, one can then use a dipperclock to find the longitude. If we let DT (in hours) = the local standard time estimate found by the dipperclock without a geographic correction, it is shown in Appendix 3 that your longitude in degrees is given by

(2)

In the Northern Hemisphere, it is also possible to use Polaris to find your north latitude (which equals the angle of elevation of Polaris above the northern horizon [2]), and therefore completely specify your location. To use your dipperclock to approximately find this angle, hold the dipperclock vertically with the central hole beside your left eye and the M in the outer ring lined up with the northernmost point on your horizon, then read the nearest position on the outer ring to Polaris and multiply this value by 15.

If the errors in the time indicated by a dipperclock, which we discuss at length in Appendix 11, amount to m minutes (i.e. m/60 hours), then by Eq.2, they will cause an error of m/4 degrees in the estimate of the longitude.

Although we have not covered celestial navigation in any depth in this paper, the If the errors in the time indicated by a dipperclock, which we discuss at length in Appendix 11, amount to m minutes (i.e. m/60 hours), then by Eq.2, they will cause an error of m/4 degrees in the estimate of the longitude.

Although we have not covered celestial navigation in any depth in this paper, the interested reader can find much more information about it in [16,17].k is held vertically with the central hole beside your left eye and the M in the outer ring lined up with the northernmost point on your horizon, Polaris is near the 2.5 in the outer ring. Where are you?

Figure 4 . 76 mm (3 inch) dipperclock 4 .

that is, the west longitude is 122.5˚. Your latitude is about 2.5 × 15 = 37.5˚. Checking a map or [14] shows that you are in or near San Francisco, California (which has west longitude 122.4˚ and latitude 37.8˚).

5. GLOW-IN-THE-DARK DIPPERCLOCKS

In a very dark area that is otherwise ideal for night sky viewing, it may be difficult to see the numbers on the dipperclock. We have used phosphorescent paper or paint to deal with this problem. The construction of glow-inthe-dark dipperclocks is discussed in Appendix 6. A phosphorescent paint dipperclock is shown in Figure 4 .

1. Collins, E.B. (1926) Useful stargazing. Scientific American, 134, 58-61.
2. Kaufman Jr., E.H. (2011) How to find celestial objects by starhopping. http://dl.dropbox.com/u/32602966/starhopping.pdf
3. Mammana, D. (1990) How to tell time by the big dipper. http://www.dennismammana.com/skyinfo/gazingtips/dipper_time.htm
4. Miles, K.A. and Peters, C.F. II (1997) What the big dipper can tell you. http://starryskies.com/articles/dln/4-97/dipper.time.html
5. Pasachoff, J.M. (2000) A field guide to the stars and planets. 4th Edition, Houghton Mifflin, New York.
6. Layton, B. and Huffman, A. (1999) Telling time by the big dipper. http://www.physics.ucla.edu/

APPENDIX 1: A LIST OF BASIC ASTRONOMICAL DEFINITIONS

In this section we give a few definitions from Astronomy along with some discussion of some of them. These definitions can also be found in [2,5] and many other places.

a) In Astronomy, the sphere of the sky surrounding the earth is called the CELESTIAL SPHERE, where the center of the earth is also the center of the celestial sphere. We give distant stars fixed positions on the celestial sphere called declination (D) and right ascension (RA), which are similar to latitude and longitude on an earth map.

b) A GREAT CIRCLE on the celestial sphere is a circle formed by cutting the celestial sphere with a plane that contains the center of the earth. These are the largest possible circles on the sphere.

c) The CELESTIAL EQUATOR is the great circle formed by cutting the celestial sphere with the plane that contains the center of the earth and is perpendicular to earth’s rotation axis.

d) The NORTH (SOUTH) CELESTIAL POLE is the point where the northern (southern) portion of the earth’s axis meets the celestial sphere. The north celestial pole is close to Polaris the south celestial pole is close to the Southern Cross.

e) RIGHT ASCENSION: We lay out a distance scale on the celestial equator by dividing it into 24 one-hour intervals by means of points which are numbered 0 to 23, moving clockwise as viewed from below the solar system. The zero point is located in the constellation Pisces. For any object or point on the celestial sphere other than a celestial pole, there is one and only one great circle that contains the celestial poles and this object or point. The RIGHT ASCENSION of the object or point is defined relative to the closest point where this great circle intersects the celestial equator.

f) UNIVERSAL TIME (UT) is approximately standard time at the Royal Observatory in Greenwich, England, and was formerly known as GREENWICH MEAN TIME (GMT). UT is 5 hours ahead of standard time in the U.S. Eastern time zone (EST). There is also COORDINATED UNIVERSAL TIME (UTC) which is measured by atomic clocks but which is kept within 0.9 seconds of UT by the addition of an occasional “leap second”. For the purposes of this paper we will treat UTC and UT as being the same.

g) The ZENITH is the highest point in the sky, directly overhead.

h) One’s MERIDIAN is the part of a great circle that starts at the north celestial pole, passes through one’s zenith, and ends at the southernmost point on one’s horizon.

i) One’s SIDEREAL TIME is the right ascension of whatever is on one’s meridian at the time.

j) AUTUMNAL EQUINOX: Between summer and fall, we reach a point in our orbit about the sun when the plane which contains the earth’s equator hits the center of the sun. As a result, the day and night that occur closest to this point are of roughly equal length. This is called the autumnal equinox. (Equinox = equi (equal) + nox (night)). A table of autumnal equinoxes for the years 2000-2020 is given in [15] in UT. From [15] we see for instance that the autumnal equinox for 2010 was 3:09 UT September 23, or September 22 at 10:09 p.m. (EST) or 11:09 p.m. (EDT).

APPENDIX 2: JUSTIFICATION OF THE GEOGRAPHIC CORRECTION

We wish to justify Eq.1 in Section 3, which is repeated here as

(A1)

where GC = the geographic correction (in minutes)

LST = your local standard time (in hours)

UT = the Universal Time (in hours) (see Appendix 1)

L = your longitude (in degrees).

We wish to justify Eq.1 in Section 3, which is repeated here as

(A1)

where GC = the geographic correction (in minutes) centerline, add 4 minutes to your time estimate (A1b).

This equivalence follows from the facts that the absolute value of 15 × (LST – UT) – L is just the number of degrees of longitude between your location and your centerline, with 15 × (LST – UT) – L being negative if you are east of your centerline and being positive if you are west of your centerline.

Now it remains only to show that statements (A1a) and (A1b) give a valid time correction. Suppose you are one degree of longitude east of your centerline. Instead of thinking of moving one degree east from the centerline, equivalently think of the earth suddenly turning one degree clockwise (as viewed from below) with you on it. This will make the Big Dipper appear to rotate one degree counterclockwise. Since there are 360 degrees in a full circle, this rotation will be of a full circle, so regardless of what 24-hour clockface you are using, the hour hand will advance () × 24 hours, which is () × (24 × 60) minutes, which is 4 minutes. Since you are still in the same time zone, your local standard time has not changed, so to compensate you will need to subtract 4 minutes from your time estimate. The argument for the case where you are west of the centerline is similar.

APPENDIX 3: JUSTIFICATION OF THE METHOD OF USING A DIPPERCLOCK TO FIND LONGITUDE

In this appendix we wish to verify the equation

(A2)

where L = your longitude (in degrees)

DT = the local standard time estimate found by the dipperclock without a geographic correction (in hours)

UT = the Universal Time (in hours) (see Appendix 1).

Since GC is what we add to DT to get LST, we have

Converting the left side to hours, we get

Converting the left side to hours, we get0910526dc5-1351-407f-90e1-968d712d2eb9.jpg" />

Finally, multiplying by 15, we get

which is what we wanted to show.

APPENDIX 4: USING DIFFERENT STARS AS THE TIP OF THE HOUR HAND

Until now we have used the pointer stars in the bowl of the Big Dipper as the tip of the celestial hour hand because the Big Dipper is familiar to most people and is easy to find. Other bright stars that are close to Polaris can also be used as the tip of the celestial hour hand, however. We simply need to utilize their right ascension (RA) rather than the right ascension of the bowl stars of the Big Dipper (for which RA = 11 hours). The equinox estimation method also works, if we replace the 11 in the adjustment term with the RA of the new star (see Appendix 8).

One reasonable choice for the tip of the hour hand is Kocab (aka β UMi). In Figure 1 , Kocab is the brightest star left of and a little below Polaris. Kocab was also mentioned in [7], and is the brightest star in the bowl of the Little Dipper. Kocab has the advantage that it is closer to Polaris than the pointer stars in the Big Dipper are, so it may be visible when the pointer stars are blocked by trees, for instance. For Kocab, we have RA = 14 hours 50.7 minutes = 14.845 hours.

Another reasonable choice would be γ Cas, which is the middle star in the W of Cassiopeia. γ Cas has the advantage that it is on the opposite side of Polaris from the Big Dipper, so it will be high in the sky when the Big Dipper may be too low to be visible. For γ Cas we have RA = 56.7 minutes = 0.945 hours.

If a different tip star is desired, find its RA and use the java program in Appendix 5 to construct an appropriate dipperclock, or use the equinox estimation method presented in Appendix 8 with RA in place of 11. The RAs of many other good candidates can be found in Table 2 of [5].

It is also possible to use a dipperclock constructed for the Big Dipper with the new tip star just take a reading with the new tip star and add RA – 11 hours. This procedure will be justified in Appendix 10. For example, if RA = 17 hours then you can take a reading with the new tip star and add 6 hours.

APPENDIX 5: DISCUSSION OF A FREE JAVA PROGRAM TO GENERATE DIPPERCLOCKS

We have written a standalone java program which will generate a dipperclock according to one’s specifications and put it in a png file of one’s choosing. We are providing the source code for this program free in the text file dipperclock.java, which you can download from [18]. This file was created using the Mac text editor TextWrangler.

To compile and run this program, you will need to have JDK (Java Development Kit) installed. Note that JRE (Java Runtime Environment) is not sufficient. JDK should already be present on Macs running OS 10.5 or higher for PCs it can be downloaded for free from [19] (the leftmost of the four download icons in [19] should work well).

To launch the program, open a command window. On a Mac, use the Terminal program on a Windows computer, use the cmd program and with Linux, use Konsole, gterm, or xterm.

After creating a folder called javawork in your home directory and saving dipperclock.java there, type the following three commands in the command window:

One should not need the command window again except to show error messages, if there are any. For further information, see Sections 2.6.1 and 2.6.2 of [20]. The program should now execute the following series of dialog boxes:

1) Tip Stars: You will be asked whether you want to use the pointer stars in the bowl of the Big Dipper as the tip of the dipperclock’s hour hand. If you say no, you will then be asked to give the right ascension of the alternative star(s) you would like to use instead.

2) Year-Pair: You will be asked for the first year of the year-pair. The reason for the term year-pair is that a dipperclock involves parts of two consecutive calendar years, namely from the autumnal equinox of the first year to the autumnal equinox of the second year. If you give any negative number, you will get a generic dipperclock.

3) LST – UT: You will be asked for the difference between your local standard time (LST) and Universal Time (UT). Note that this difference might not be an integer for example, Newfoundland is 3 1/2 hours behind UT, so LST – UT = –3.5 in Newfoundland. Our methods still work in such cases. If you are doing a generic dipperclock with no geographic correction, then LST – UT will not be needed by the program, so you can select anything.

4) GC: You will be asked whether you want to do a geographic correction (GC), and if so, you will be asked for your longitude (which can be found using [14]).

5) Size: You will be asked to specify the size of your dipperclock.

6) Color: You will be asked to specify the color pattern of your dipperclock. You can choose white on black or black on white, or you can independently choose the color of the background, the three circles, the dates and their hash marks, and the times and their hash marks. In each case your choices are black, white, red, blue, green, and yellow.

7) Confirm: The program will give a summary of your choices, and ask you if you are satisfied if not, you will be taken through steps 1 - 7 again.

8) Storage: The program will ask where you would like to save the .png file containing your dipperclock.

If you choose, you may change the source code for this program. Additionally, you may find some of the subroutines to be useful independently for example, FindAEUT and Isleap can be used to estimate the autumnal equinox for a given year in Universal Time. See the comments (preceded by // or bracketed by /* */) in dipperclock.java for more information.

Finally, we knew nothing about java when we started writing this paper we learned enough java to write the program mainly by studying Professor David Eck’s fine online textbook [20].

APPENDIX 6: CONSTRUCTION OF GLOW-IN-THE-DARK DIPPERCLOCKS

In this appendix we describe our methods for making various kinds of glow-in-the-dark dipperclocks. One approach is to use an inkjet printer to print the dipperclock on glow-in-the-dark photo paper instead of ordinary paper. Reference [21] is an ad for a brand of glow-in-thedark photo paper that has worked well for us in making letter size (203 mm across) glowing dipperclocks. Lamination for durability and stiffness would be a good idea. Pocket size (51 mm or 76 mm) dipperclocks can also be made this way, but a brighter option for pocket size dipperclocks is to use glow paint rather than glow paper as follows:

1) Print a white-on-black dipperclock, then copy it onto a transparency appropriate for your copier. We have used a laser copier for this.

2) Paint the back of the dipperclock with glow-in-thedark paint. We have found that two or three coats typically works best. For the outer ring, we recommend the “ultra green V10” from [22] according to the seller, this is the brightest glow paint available. For the inner ring, especially for the 51 mm dipperclock, we recommend “pure blue” the ultra green V10 may be bright enough, when used in the inner ring, to blank out your central vision and make Polaris difficult to see. We recommend only water-based paint for safety reasons.

3) Cut away the transparency outside the dipperclock.

4) For durability and to keep little kids away from the glow paint, which can be hazardous if ingested, we recommend lamination. If you cannot see Polaris through the central hole, you can cut away the laminating material and transparency in the central hole.

We field tested our methods in and near Decatur, Illinois. Decatur is in the US Central time zone, with west longitude of 88.95˚ according to [14], so according to Eq.1, the geographic correction for Decatur is

so we subtract about 4 minutes from our dipperclock time estimate.

Below, we summarize the results of our field tests

Using a generic dipperclock, we estimated the time to be 9:30 p.m. CST, i.e. 10:30 p.m. CDT. Subtracting the 4 minute geographic correction (GC) gave 10:26 p.m. CDT. The actual time was about 10:22 p.m. CDT, so we were off by 4 minutes.

Glow-in-the-dark dipperclocks, customized to Decatur, Illinois:

These tests were conducted with a full-size glow paper dipperclock (203 mm across) and a pocket-size transparency-glow paint dipperclock (51 mm across), with built-in geographic correction. We found that passing a small flashlight over each dipperclock for a few seconds charged them sufficiently for clear readings, and they remained sufficiently bright for the full hour duration of the test.

Full size glow paper dipperclock: This dipperclock was very easy to use, and gave consistently accurate readings, with errors ranging from a half hour to zero.

Pocket size painted transparency dipperclock: This was brighter than the glow paper dipperclock, but the numbers and letters were somewhat more difficult to read due to the small size. The accuracy was not quite as good as the accuracy with the full size glow paper dipperclock.

Using a generic pocket-size Big Dipper dipperclock with a different tip star: At 10:11 p.m. CDT the pointer stars of the Big Dipper were behind trees, so we used the handle star next to the bowl, ε UMa (aka Alioth), which has RA = 12 hours 54.1 minutes. This star gave a reading of about 7:30 p.m. CST, or 8:30 p.m. CDT. Here RA – 11 = 1 hour 54.1 minutes, so adding this to the reading as discussed at the end of Appendix 4 and doing the geographic correction we got about 8:30 + 1:54 – 0:04 = 10:20 p.m. CDT, so we were off by 9 minutes.

Finding the time, longitude, and latitude using a generic letter-size Big Dipper dipperclock: At about 8:30 p.m. CDT we used γ Cas, the middle star of the W of Cassiopeia, which has right ascension 56.7 minutes, as the tip star. We got a reading of about 6:00 a.m. CST, i.e. about 7:00 a.m. CDT, on the outer ring of the dipperclock. Here RA – 11 = –10 hours 3.3 minutes. Adding this to our reading and doing the geographic correction we got about 7:00 a.m. – 10:3.3 – 0:04 = 8:53 p.m. CDT so we were off by about 23 minutes.

We also used Capella, aka α Aur, as the tip star and attempted to estimate our longitude as in Section 4. Capella has right ascension 5 hours 16.7 minutes, so RA – 11 = –5 hours 43.3 minutes. Our time reading from the dipperclock was about 1:30 a.m., so converting this to DT, the time indicated by the pointer stars in the bowl of the Big Dipper, gives DT = 1:30 – 5:43.3 = –4:13 (approximately). Universal Time is 5 hours ahead of CDT, so we had UT = 20:30 + 5:00 = 25:30 = 1:30. From Eq.2 we get

Thus the error in the longitude approximation is – 85.75 – (–88.95) = 3.2 degrees.

Finally, we estimated the latitude by the method in section 4. When the dipperclock was held vertically with the hole beside the left eye and the M in the outer ring lined up with the northern point on the horizon, Polaris appeared to be close to 2 3/4 in the outer ring. Multiplying by 15 gives 41.25 degrees as our estimate of the latitude. From [14] we have that the latitude of Decatur is 39.84 degrees, so we were off by about 1.4 degrees.

In general, our tests of various dipperclocks were satisfactory, with time errors usually less than 15 minutes.

APPENDIX 8: FINDING THE TIME WITHOUT A DIPPERCLOCK (THE EQUINOX ESTIMATION METHOD)

The equinox estimation method presented in this section involves computing an adjustment term based on the current date and adding this to a sidereal time estimate read from the Big Dipper.

First take the adjustment term to be 11 hours on September 22, the approximate date of the autumnal equinox. Subtract 2 hours for each month that has passed since September, and add/subtract 4 minutes for each day before/after the 22nd of the current month.

Look at Polaris and the Big Dipper, and imagine the outer ring of a dipperclock centered at Polaris and perpendicular to your line of sight to Polaris with the M at the top of the ring. Estimate the time on the imaginary outer ring at which the hour hand points, add the adjustment term, and add the geographic correction (if desired) to get your estimate of the local standard time.

As noted in [7], it is easy to get a little distortion in the time estimate, especially around 6 a.m. and 6 p.m. This occurs when the clockface is incorrectly imagined to be vertical, due to the fact that the hour hand lies in a plane which contains the axis of the earth, and a vertical clockface is not perpendicular to the axis of the earth. Imagining the clockface to be perpendicular to your line of sight to Polaris fixes this problem. The problem does not come up when using a dipperclock correctly since the dipperclock provides a properly tilted clockface.

Suppose that on July 25 the stars are shown as in Figure 1 of section 1, with the clockface and hour hand seen only in our imagination. We compute the adjustment term as 11 hours – 10 × 2 hours – 3 × 4 minutes = –9:12. Adding this to the reading of 10:30 a.m. gives 1:18 a.m. local standard time, or 2:18 a.m. local daylight time. Actually, a careful measurement of Figure 1 gives a reading of about 10:45 a.m. instead of 10:30 a.m., giving a discrepancy of about 15 minutes in our results, but we do not have the luxury of doing such measurements in the field. This illustrates the effects of observational error other types of error will be discussed in Appendices 11 and 12.

We now describe two field tests using the equinox estimation method in Decatur, Illinois. As shown in Appendix 7, the geographic correction for Decatur is about –4 minutes.

Test 1 (6/17/10), equinox adjustment method The adjustment for the equinox estimation method for 6/17/10 using the Big Dipper is

Given our reading of the imaginary clockface of 5:00 a.m. CDT, we had an estimated CDT of

which was 10:16 p.m. CDT. The actual time was 10:22 p.m. CDT, so the error was 6 minutes.

Test 2 (12/05/10), equinox adjustment method with γ Cas On 12/05/10, we carried out an equinox estimation method exercise using γ Cas as the tip of the hour hand. Using the right ascension of γ Cas, namely about 57 minutes, in place of the 11 hours in the equinox estimation adjustment, we found the adjustment to be

Given our reading of the imaginary clockface of 10:30 p.m. (=22:30 in 24-hour time), we had an estimated CST of

The actual time was 6:40 p.m. CST, so we were off by 9 minutes.

APPENDIX 9: DISCUSSION OF THE MATHEMATICAL FOUNDATIONS REQUIRED FOR APPENDICES 10-12

In this section, we assume we are located on the centerline of a time zone, so by Section 3, the geographic correction is 0. The only clockface we will be using in this section will be the 24-hour counterclockwise clockface with M at the top.

After introducing some basic notation, we derive the relationship between local standard time and time read from a star at the tip of the hour hand on the dipperclock, along with the number of days (including fractional days) since the last autumnal equinox (Eq.A5), which provides the foundation for the rest of our work.

a) We introduce the following notation:

RA = the right ascension of the star(s) we are using as the tip of the hour hand for the pointer stars in the bowl of the Big Dipper, RA = 11 hours.

LST = the local standard time.

RAT = the 24-hour time read from the star(s) at the tip of the hour hand when using a 24-hour counterclockwise clockface with M at the top.

b) Now we determine how long it takes the earth to complete one rotation on its axis. Imagine looking down on the solar system, with the earth revolving counterclockwise around the sun and rotating counterclockwise on its axis. Each 24 hours the earth makes one complete After introducing some basic notation, we derive the relationship between local standard time and time read from a star at the tip of the hour hand on the dipperclock, along with the number of days (including fractional days) since the last autumnal equinox (Eq.A5), which provides the foundation for the rest of our work.

a) We introduce the following notation: at the Big Dipper again at the same time the next night, it will appear to have rotated a little counterclockwise from where it was the first night. To be precise, the earth makes one complete revolution around the sun in about 365.2422 24-hour days [5, p. 500], so the time for one revolution is 365.2422 × 24 hours. During this time the earth will make 365.2422 rotations on its axis plus one more. Thus the time for one rotation is, which is about 23 hours 56 minutes 4 seconds. Therefore, the solar day of 24 hours is about 3 minutes 56 seconds longer than the actual time it takes the earth to rotate once about its axis, relative to distant stars that we are not orbiting. This (approximately) 23 hours and 56 minute rotation period is called the sidereal day.

c) Now let’s suppose that we look at the star(s) at the tip of the dipperclock’s hour hand at the same LST on two consecutive nights. The second night, the tip stars will have returned to the same position after a sidereal day, but since we will be looking at them after a full solar day, they will be offset slightly from their position the previous night.

A little algebra shows that the time beyond one complete rotation of the earth until 24 hours is reached is () × 24 hours, and this is of the time for one complete rotation.

Thus during our 24-hour period, the celestial hour hand will make one complete rotation, plus of a rotation. The complete rotation leaves RAT unchanged, and the of a rotation increases RAT by () × 24 hours. Thus at the end of a 24-hour period, RAT will be greater by () × 24 hours than it was at the beginning of the 24-hour period. LST will be the same at the end of the 24-hour period as at the beginning, so over our 24-hour period,

(A3)

which is about 3 minutes 56 seconds.

Now we need a starting point in order to map RAT into LST and complete our time conversion equation. We use the autumnal equinox (AE) as that starting point, at which

(A4)

because at the AE, 1) LST = the sidereal time [23] = the right ascension of whatever happens to be crossing the meridian at the time, and 2) the right ascension of whatever is crossing the meridian is always RAT + RA.

2) is true because it is true when the tip star(s) are directly above Polaris (they are crossing the meridian, and they have right ascension RA, but the 24-hour time read from the tip star(s) is RAT = 0), and as time marches on from there both sides of the equation advance at the same rate, so the equation remains true.

From Eq.A3 it follows that over a period of d days, LST – RAT will decrease by () × d hours. Combining this result with Eq.A4 gives the complete time conversion equation:

(A5)

(A6)

where d = the number of days (including fractional days) since the last autumnal equinox. Note that when d = 365.2422, then we are (approximately) at the next autumnal equinox since the earth has returned to the same place in its orbit, the final term in Eq.A5 becomes 24 hours, and Eq.A5 becomes Eq.A4 again, as expected. Eq.A5 was also proved in a different form in [2] for the case of the Big Dipper, where RA = 11.

APPENDIX 10: DISCUSSION OF HOW THE VARIOUS NUMBERS AND HASH MARKS ARE PLACED AROUND THE DIPPERCLOCK

To construct a dipperclock diagram you first draw three concentric circles to form two rings. You then need 24 hash marks on the outermost concentric circle, 15 degrees apart. The top hash mark is labeled (inside the outer ring) with the letter M standing for midnight, the next 11 hashmarks are labeled counterclockwise 1 through 11 for a.m. times, the next hashmark is labeled N for noon, and the remaining 11 hash marks are labeled counterclockwise 1 through 11 for p.m. times. Shorter hash marks are inserted between these to mark the half hours. In describing the location of these hash marks we will use the usual mathematical convention of measuring angles from the positive x-axis, with counterclockwise angles being positive and clockwise angles being negative. Thus the location of the M hash mark is described by the angle 90 degrees.

We need 12 more hash marks on the inner and middle concentric circles, labeled clockwise with the first letters of the months in the inner ring. Placing these hash marks and letters correctly is much more involved than placing the hash marks and numbers in the outer ring. We will now derive the formulas for computing the position angles for these hash marks and letters for a dipperclock with given tip star(s), year-pair, time zone, and longitude (if a geographic correction is to be done).

From the given information we compute the following:

RA = the right ascension of the star(s) used as the tip of the hour hand if we are using the pointer stars in the bowl of the Big Dipper, then RA = 11 hours.

AE = an estimate of the autumnal equinox in local standard time for the first year of the year-pair, given in date (in September) + fractional day form.

LEAP = 1 if the second year of the year-pair is a leap year, and LEAP = 0 otherwise.

LONG = 0 if no geographic correction is to be done, and otherwise LONG = your longitude – the longitude of the centerline of your time zone, given in degrees + fractional degree form.

We now a) derive an equation which gives the position angle for the dates and their hash marks b) show how to construct a dipperclock table c) determine the dipperclock table for the generic dipperclock in Figure 2 of this paper and d) justify the claim from the end of Appendix 4.

a) For now we will continue to assume we are located on the centerline of our time zone, so there is no geographic correction to be done. Referring to Eq.A5, we know that the date in the inner ring corresponding to d value must go at the bottom of the dipperclock when the M in the outer ring is at the top, because plugging this d value into Eq.A5 gives LST = RAT at that time. Thus at that time the correct clockface to use is the one with the M in the outer ring at the top, and with the given d value opposite the M as described holding the dipperclock with this d value at the bottom gives that clockface.

Next we claim that to get from the position in the inner ring of the dipperclock corresponding to any d value to the position of d + 1 we must move () degrees clockwise in the inner ring. To see this, suppose we look at the tip star(s) at the date and time corresponding to d, so we will be holding the dipperclock with d at the bottom. If we look at the tip stars again 24 hours later, so that d + 1 will be at the bottom, then by earlier work in part c) of Appendix 9, the celestial hour hand will be advanced by of a rotation compared to where it was at the beginning of the 24-hour period. of a 360-degree rotation is () degrees. LST will be the same at the end of the 24-hour period as at the beginning of the period, so we will need to have rotated the dipperclock counterclockwise () degrees to keep the celestial hour hand pointing at the same time in the outer ring, so the d + 1 position will be () degrees clockwise from the d position, as claimed. By proportionality, getting from position d1 to position d2 (with 0 ≤ d1 ≤ d2 ≤ 365.2422) requires moving (d2 – d1) × () degrees clockwise on the dipperclock.

By the definition of d, when d = 0, we are at the last autumnal equinox, and by the first claim in subsection (a), when d = we are at the bottom of the dipperclock (with the M in the outer ring at the top), so the distance moved clockwise around the dipperclock from d = 0 to d = is

degrees, which equals (360/24)RA degrees = 15RA degrees. Thus we have the remarkable fact that whatever date and time we are using for the last autumnal equinox, it corresponds to a point on the dipperclock that is 15RA degrees counterclockwise from the bottom of the dipperclock. If we move from the positive x-axis to the autumnal equinox position on the dipperclock instead of from the bottom of the dipperclock we are moving 90˚ less, so the autumnal equinox has position angle 15RA – 90˚.

Now consider the location (i.e. position angle) θ of midnight of the first day of a month following the value we are using for the autumnal equinox. Let d be the number of days plus fractional day that this time is past the autumnal equinox. Then by the work at the end of the second paragraph above Eq.A6 with d1 = the autumnal equinox, d2 = midnight of the first day of the month, so d = d2 – d1, getting to location d2 from the autumnal equinox on the dipperclock requires a clockwise move of d × (360/365.2422) degrees. Clockwise is the negative direction, so we are changing the position angle by –d × (360/365.2422) degrees, so

degrees (A7)

In deriving Εq.Α7 we made the assumption that no geographic correction was done, so LONG = 0. Now suppose a geographic correction is to be built into the dipperclock, where LONG = L – 15 × (LST – UT) (using the notation of sections 3 and 4). The final result is then

(A8)

Let’s justify this for the case where you are one degree of longitude east of the centerline of your time zone. (The argument for other cases is similar.) Then LONG = 1. Statement (A1a) in Appendix 2 tells us to subtract 4 minutes from your time estimate, but this can be accomplished by rotating the dipperclock one degree counterclockwise. To compensate for this rotation, the dates and hash marks in the inner ring need to be rotated one degree clockwise, so the same date will be at the bottom. This can be done by subtracting one degree from each of the locations in Eq.A7 and Eq.A8 does precisely that.

b) Now we are ready to construct a dipperclock table. The table is a 12 × 3 matrix.

The first column of the table contains the numeric indicators of the months in the order 10.0, 11.0, 12.0, 1.0, 2.0, ···, 9.0.

Each entry in the second column contains the number d of days plus fractional day (at midnight) that the first day of the month in column 1 is past the value being used for the autumnal equinox.

Each entry in the third column contains the position angle θ given by Εq.Α8 using the d value to the left.

To compute the first element in the second column, first convert October 1 at midnight to (imaginary) September 31 at midnight, or September 32.0, so we get 32.0 – AE. Since there are 31 days from October 1 to November 1, the second element in the second column = the first element in the second column + 31.0. We continue down the second column this way, except the sixth element in the second column = the fifth element in the second column + 28 + LEAP to allow for a possible leap day in the second year of the year-pair. Finally, the elements in the third column are computed from Eq.A8 with the d values taken from the second column.

c) For the generic dipperclock shown in Figure 2 of this paper, we used RA = 11, AE = September 22 at midnight LST, LEAP = 0, LONG = 0. Table 1 below is the table corresponding to these choices that provided the position angles for Figure 2 .

d) In Appendix 4 we claimed that it is possible to use a dipperclock constructed for the Big Dipper with a new tip star with right ascension RA just take a reading with the new tip star and add RA – 11 hours. To see this, suppose RA > 11 the case RA Table 1 . Inner ring locations for the generic dipperclock ( Figure 2 ).

placing RA by 11 in Eq.A8 and subtracting from the original Eq.A8, the updated position angles (θ’) for the new dipperclock are

(A9)

where θ represents the position angles found for a dipperclock based on the Big Dipper. Since the position angle for the d value formerly at the bottom of the dipperclock has increased by 15 × (RA – 11) degrees, returning this d value to the bottom of the new dipperclock rotates the new dipperclock 15 × (RA – 11) degrees clockwise with respect to the actual Big Dipper dipperclock. This is a clockwise rotation of of a complete circle, or of a complete circle, or RA – 11 hours on the outer ring. The new tip star is pointing at the correct time on the new dipperclock, but due to the rotation this correct time is RA – 11 hours counterclockwise from where the new tip star is pointing on the actual Big Dipper dipperclock. Thus the time pointed at by the new tip star on the actual Big Dipper dipperclock is RA – 11 hours too early, so we need to add RA – 11 hours to get the correct time.

APPENDIX 11: ACCURACY OF THE DIPPERCLOCK

In this section we assume that the correct geographic correction from section 3 has been done. We consider the following possible sources of error: a) error in Εq.A5 b) error due to using an inaccurate value for the autumnal equinox c) error in using a dipperclock constructed with the second year being a non-leap year to estimate times with the second year being a leap year (or vice versa) d) error due to necessarily thinking of the dates in the inner ring as being discrete and e) observational error.

a) Eq.A5 is quite accurate in terms of its theoretical error (by which we mean the error in the equation itself, irrespective of observational errors). This is because the only major sources of error in Eq.A5 are rounding and to a much lesser extent, effects like the precession cycle of earth’s rotation axis, which has a period of approximately 26,000 years. See [5] for further discussions of this and other small sources of error. Consequently, we can say very conservatively that Eq.A5 is accurate to less than one minute.

b) We note from Eq.A6 that an error of one day in d results in a less than 4 minute discrepancy in our estimate of LST. For the generic dipperclock, we use a default value of midnight on September 22 (LST) for the autumnal equinox (AE), as a characteristic average. For instance, according to [15], for the years 2000 to 2020 the maximum that an autumnal equinox falls short of midnight September 22 is 10 hours 29 minutes (for 2020) and the maximum in those years that an autumnal equinox goes beyond midnight September 22 is 10 hours 47 minutes (for 2003). Thus, in any year-pair with first year between 2000 and 2020, the discrepancy between the autumnal equinox (UT) and midnight September 22 (UT) is less than 12 hours. There is also a maximum 12 hour discrepancy in converting the UT autumnal equinox we have been discussing to the autumnal equinox for your time zone, which is what appears in

### 1131015296002183

, giving a discrepancy of your autumnal equinox from midnight September 22 of less than 24 hours, which equals one day. Multiplying this by 4 minutes, we can conservatively say that using midnight September 22 LST as our value for the autumnal equinox should cause an error in our estimate of the current local standard time of less than 4 minutes.

c) Whether or not the second year is a leap year will make a difference of 1 in d for the months March ···, September, and this difference can also affect our estimated time by less than 4 minutes for these months. Recall that our generic dipperclock was constructed under the assumption that the second year is not a leap year.

d) Think of a dipperclock in use. As time progresses, the hour hand of the clock rotates continuously counterclockwise. The clockface rotates also, but not continuously, due to changes in the date at the bottom of the inner ring. For example, from 12:01 a.m. August 7 to 11:59 p.m. August 7, the clockface does not rotate, but from 11:59 p.m. August 7 to 12:01 a.m. August 8, the clockface rotates because the date at the bottom of the inner ring changes by one day. This corresponds to a less than one day change in the d value at the bottom of the dipperclock when compared to the continuous and uniform rotation situation, which causes a change of less than 4 minutes in the time read from the dipperclock.

This source of error would not be present if the dipperclock rotated continuously and uniformly, but for that to happen the point at the bottom of the inner ring would have to involve the time of day as well as the month and day, and we do not know the time of day, since that is what we are trying to find by using the dipperclock! Also, even if we knew the time of day, very few humans could place the date at the bottom of the dipperclock accurately enough for the time of day to make a difference.

Thus, the total theoretical error for the generic dipperclock will be less than 1 + 4 + 4 + 4 minutes, or 13 minutes, at least when the first year of the year-pair is between 2000 and 2020. For a custom dipperclock there is no error for the autumnal equinox or leap years, so the theoretical error will be less than 1 + 4 minutes, or 5 minutes.

Remark: Comparison of theoretical maximum error to actual error for the 2010-2011 year-pair: From [15], the autumnal equinox for 2010 occurs on September 23 at 3:09 in UT, which is September 22 at 21:09 (or 9:09 p.m.) CST in the US Central time zone. The difference between this AE value and the default value of September 22 at midnight used for the generic dipperclock is 2 hours 51 minutes = 0.12 days, which corresponds to a discrepancy of less than 0.48 min = 29 sec in Eq.A5. Since the 4-minute leap year error does not come into play for this year-pair, the actual theoretical error estimate for our generic dipperclock in 2010-2011 is only about 5 1/2 minutes instead of 13 minutes. This explicitly illustrates that our theoretical maximum tends to be very conservative.

e) There are four main kinds of observational error one can introduce when using a dipperclock: 1) not holding the dipperclock with the current date exactly on the bottom 2) not having Polaris at the center 3) not holding the dipperclock perpendicular to the line of sight direction to Polaris and 4) not reading exactly where the hour hand is pointing. From our testing, observational error seems to be the largest source of error in using a dipperclock.

APPENDIX 12: ACCURACY OF THE EQUINOX ESTIMATION METHOD

In this section we assume that the proper geographic correction from Section 3 has been done.

Here we analyze the equinox estimation method when we are using the pointer stars in the bowl of the Big Dipper as the tip of the hour hand (so RA = 11), and we are using our default autumnal equinox value of midnight September 22 LST.

The equinox estimation method has the possibility of errors in Eq.A5 itself (less than 1 minute) and errors due to an inaccurate choice for the autumnal equinox (less than 4 minutes for year-pairs with first year in the range 2000-2020) as we saw in Appendix 11.

However, there is an additional source of potential theoretical error namely, the equinox estimation method of approximating the terms 11 – ()× d in Eq.A5 by using our adjustment procedure. We will now estimate this error. To do this we first suppose the second year is not a leap year and compute Table 2 below with the following five columns:

The first column contains the dates 9/22, 9/30, 10/1, 10/31, 11/1 … 8/31, 9/1, and 9/22, all at midnight. The second column contains the value of d, which is the number of days (including fractional days) since September 22 at midnight. The third column contains the time adjustment for the equinox estimation method. The fourth column contains the correct adjustment from Eq.A5, namely 11 – () × d, and the fifth column is obtained by subtracting the fourth column from the third column.

In Table 2 , all times are rounded to the nearest minute. The largest absolute value of elements shown in column 5 is 7 minutes. There can be no larger absolute value of

Table 2 . Maximum theoretical error due to equinox estimation computation.

an element in column 5 for any integer d since as we move from d to d + 1, staying within one month, column 3 decreases by 4 minutes while column 4 decreases by a lesser amount (namely by hours, or about 3.94 minutes), so even with rounding, column 5 is non-increasing for integer values of d within any one month. Thus, the largest absolute value of any number in column 5 for integer d will occur on the first or last day of that month, and so will already be in Table 2 . We note that the average absolute value of the numbers in column 5 is, or about 0.03, which is less than half of the maximum absolute value.

For non-integer d values, there is another possible effect. For example, suppose we put in another line in Table 2 for 1:00 a.m. on March 31. Column 3 will still be at –1:36 since we will be in March 31, but we will have d = 189 +, and computing column 4 gives –1:25. Thus, the value in column 5 is –:11, which is the worst case scenario.

Although Table 2 was constructed for the second year being a non-leap year, one can also construct another table for the second year being a leap year. In that table, the error behavior is actually better. Thus, we conclude that the maximum theoretical error in our time estimate via the equinox estimation method is 11 minutes. Putting this together with our previous estimates, we find that the theoretical error in the equinox estimation method under the assumptions given in the second paragraph of this section, with first year of the year-pair between 2000 and 2020, is less than 16 minutes (1 + 4 + 11).

This is a conservative estimate. For one thing, our equinox estimation method implicitly uses September 22 at midnight as its value for the autumnal equinox, just as our generic dipperclock does, so the Remark near the end of Appendix 11 applies and indicates that the error due to an inaccurate choice of the value for the autumnal equinox is typically much less than 4 minutes. The 11 minute estimate in the previous paragraph is also unlikely to be achieved, so the 16 minute maximum theoretical error estimate is far larger than what you are likely to find in practice.

Using this method, the observational error will be even more extreme than for a dipperclock because we will be employing an imagined clock face instead of a dipperclock for our time estimate, and this is a considerable disadvantage. The field tests given in appendices 7 and 8 offer some insights into the characteristic total errors we encountered when using both the equinox estimation method as well as generic and customized dipperclocks.

2 Stars by Stellarium (developed by Fabien Chéreau, Matthew Gates, Nigel Kerr, Diego Marcos, Bogdan Marinov, Timothy Reaves, Alexander Wolf, Guillaume Chéreau, and Barry Gerdes). Hour hand by photoshop.

3 Stars by Stellarium (developed by Fabien Chéreau, Matthew Gates, Nigel Kerr, Diego Marcos, Bogdan Marinov, Timothy Reaves, Alexander Wolf, Guillaume Chéreau, and Barry Gerdes). Combining of stars and DSLR photo of letter-size glowing dipperclock, and brightness enhancement of dipperclock photo, by photoshop.

4 Pictures taken by iPhone app Tripod (Jeff McMorris). Rotation and resizing by MAC programs keynote and preview.

## Department of Physics

This list is not intended to contain every astronomical term used in class. These pages are meant as references, I do not expect students to memorize all these meanings.

Some of the words listed below have additional meanings other than ones listed. In these cases I have only given the meaning(s) that will be used in this class. I have not attempted to give rigorous, exact scientific definitions, but rather brief explanations that will hopefully help you understand the word as used in class.

Note that many of the words are cross-linked to other definitions within this glossary.

Note: "Body" refers to any astronomical body, i.e. a planet, moon, asteroid, etc..

Essentially, the temperature at which the movement of atoms ceases. (Actually, the temperature at which it becomes as small as possible. Due to quantum mechanic effects it is not possible for all motion to cease). Absolute Zero is 0 o K (See Kelvin Temperature Scale.)

(Also known and a "dark line" spectrum.) When light from a hot, dense object passes through a diffuse gas and is broken down into its component colors by passing the light through a prism or a diffraction grating, the resulting spectrum will be a continuous spectrum with a series of dark lines. Each line results from the transitions of electrons in the atoms of the gas from lower energy levels to higher energy levels as the result of the atom absorbing a photon which has an energy exactly equal to the difference in energy between the two levels. Since each element has a unique set of energy levels, gasses of each element have unique sets of absorption lines, thus by an analysis of which lines are absorbed ("missing" from the continuous spectrum) we can determine the chemical composition of the diffuse gas (not the composition of the dense object). View examples of spectra here. See also emission spectrum and continuous spectrum.

(Latin: Anno Domini)"Year of our Lord". Used to designate dates as after the birth of Christ. Due to confusion over the actual date of the birth of the historical person Jesus, this term has been replaced with CE. Dates written with AD should have the AD placed in front of the year, for example AD 1996. Compare to BC and BCE.

The percentage of visible light hitting a body that is reflected back in the visible part of the spectrum. Albedo is usually expressed as a decimal: 1.00 = 100%, 0.53 = 53%, etc..

The angle between the astronomical horizon and a celestial body as viewed from the surface of the Earth.

(Latin: Ante Meridian) "Before the Meridian". In Civil Time the abbreviation AM indicates the time after Midnight and before Noon. Compare to PM. Although this term is often used for Midnight as "12:00 AM", this is incorrect. Midnight is technically BOTH AM and PM, and Noon is technically NEITHER AM nor PM. The correct terms for noon and midnight are 12:00 Noon and 12:00 Midnight, NOT 12:00 AM and 12:00 PM.

A unit of length equal to 1x10 -10 meters (1 ten-billionth of a meter). Used in measuring the wavelength of visible light. Also equal to 1/10 of a nanometer. Just so you know, in my class you will always hear me refer to wavelengths in the visible spectrum in terms of Ångstroms, not nanometers.

A Solar Eclipse where the Moon is slightly too far from the Earth to completely block the sun, so a ring (an annulus in Latin) of sun is still visible completely around the Moon.

Apogee: for an object orbiting the Earth, the point in its orbit when it is farthest from the Earth.

Aphelion: for an object orbiting the sun, the point in its orbit when it is farthest from the sun. The Earth reaches Aphelion on approximately July 4th each year.

One arc minute (symbol ' )is an angle equal to 1/60 of a degree , or 1/21,600 of a circle. 60' = 1 o .

One arc second (symbol " ) is an angle equal to 1/60 of an arc minute, or 1/3600 of a degree, or 1/1,296,000 of a circle. 60" = 1' , 3600" = 1 o .

1) The first constellation of the Zodiac.

2) A term used (generally in navigation) to mean the point on the celestial sphere where the sun crosses the celestial equator in the spring the Vernal Equinox.

(384BC to 322BC) Greek philosopher and astronomer. Aristotle taught that the Earth was the center of the universe. He estimated the size of the Earth by observing the curve of the Earth's shadow on the Moon during lunar eclipses. His estimate was about 40% of the true size of the Earth.

The armillary sphere is a three-dimensional model of the universe, centered on the Earth. Ranging in size from desktop models of less than a foot in diameter, to large fixed models of five-foot diameter or larger, it consists of a small central sphere indicating the Earth, surrounded by a number of rings ( armillae in Latin) representing the celestial sphere. The origins of the armillary sphere are not entirely clear, however the inventor may have been Hipparchus (190-120 BCE). Armillary spheres also appeared in China sometime between 200 BCE and 220 CE.

Armillary spheres usually have rings representing the ecliptic and the zodiac, the celestial equator, the meridian of the vernal equinox, and whichever other great circles the maker of that particular instrument chose to include.

They were used either as demonstration aids in teaching astronomy, or as observational instruments.

The common, but incorrect, term used for Minor Planets (See Minor Planet).

A circular region in the solar system where the majority, but not all, minor planets are found. The asteroid belt has a mean orbital radius of 2.8 astronomical units, and is thus located between Mars and Jupiter.

(second "a" is long, pronounced to rhyme with "babe") An astronomical instrument of very great age, the principles of projection employed were known to the Greeks. The astrolabe reached its height during the middle ages, when many beautiful instruments were produced, primarily, but not exclusively, in the Arabic lands of the middle east. There are two basic kinds of astrolabe: the mariner's astrolabe, which is essentially a circular protractor with a sighting bar, which was used to measure the altitude of celestial bodies for use in celestial navigation (an very early forerunner of the sextant), and the planispheric astrolabe, which is an instrument used by astronomers. All planispheric astrolabes were constructed to be able to show the positions relative to the horizon of the major stars and the sun during the year at any time and date chosen by the astronomer, basically an early analog computer. Almost all planispheric astrolabes were also constructed to perform other functions as well, which varied by instrument. Typical functions were determining the time of sunrise and sunset on a particular date, showing the local mean solar time, determining the time of rise, set, and transit of the major stars on any date, and determining the date the sun entered or exited the zodiacal constellations in a given year. A photo of an astrolabe is available here

Generally, astronomers view astrology in much the same way that chemists view alchemy. The art of attempting to determine the attributes of, and predict the future for, people based on the positions and movements of celestial bodies. There are many different kinds of astrology, so it is somewhat misleading to refer to "Astrology" as if it were a single discipline. The popular astrology of today actually only dates back to the late 1800’s and the Spiritualist Movement. It has only very weak ties to “real” medieval European astrology or the earlier Greek astrology. (And of course virtually no relation to Chinese or Mesoamerican astrology, etc..) It is true, however, that most astronomers who criticize astrology don’t actually know very much about astrology. I feel there is really no point in trying to discuss the problems with modern popular astrology, it was basically made up during the Victorian age. We should look at “real” astrology, the astrology of Kepler and his contemporaries, for example. One of the common arguments against astrology is that there is no known mechanism for the stars to influence events on the Earth. The problem with this argument is that astrology doesn’t say the planets and stars make things happen on the Earth, astrology says the positions of the planets and stars can be used to predict what will happen. Not that there is any reason to think that works either, but at least we should argue against what astrology actually says, not what we think it says. (See also Zodiac.)

A sub-discipline of astronomy concerned with the measurement of precise positions and motions of astronomical bodies.

A distance equal to the semi-major axis of the Earth's orbit, roughly equivalent to the average distance from the Earth to the sun. One AU (Astronomical Unit) is about equal to 1.5 x 10 11 m. It is used as the standard length of measurement when referring to the orbits of solar system objects (e.g. Mars orbits at 1.5 AU, Jupiter orbits at 5 AU.)

1) The gases that surround an astronomical body, held in place by the body's gravity.

2) The pressure at the surface of the Earth at sea level due to the weight of the air above. One ATM (Atmosphere) is 14.7 pounds per square inch (10,330 kg/m 2 ). The atmospheric pressure of the atmospheres of the other planets are usually given in terms of the pressure on Earth. For example, the atmospheric pressure on the surface of Venus is approximately 90 ATM, or 90 times the pressure on Earth.

Changes to the appearance of celestial bodies, generally the sun or Moon, caused by light or particles passing through or interacting with the Earth's atmosphere. Example are rings around the Moon or sun, Parhelia ("Sun dogs" or "False suns"), and the Aurora.

(Aurora Borealis or Aurora Australias ) Wave or ribbons of moving light which are sometimes seen at high latitudes at night. They are the result of charged particles from the sun being funneled by the Earth's magnetic field to the regions of the north or south magnetic pole, where they interact with the atmosphere, producing the lights observed.

1) The point on the celestial sphere where the ecliptic crosses the celestial equator with the sun moving from northern declination to southern declination.

2) The moment in time when the sun is located at that point.

3) The first day of fall (in the northern hemisphere) is the day on which the Autumnal Equinox (meaning 2 above) occurs. The Autumnal Equinox (this meaning) occurs on approximately September 21 each year.

The angle between an observer's meridian and a celestial body. Usually measured 0 degrees to 360 degrees clockwise from the northern part of the observer's meridian, however it is sometimes measured from the southern part of the observer's meridian, in which case it is 0 degrees to 180 degrees east or west.

A system of designating the brightness of a star relative to other stars in the same constellation by using lower case letters of the Greek alphabet. The brightest star in a constellation is alpha ( a ), the second brightest is beta ( b ), followed by gamma( g ), delta ( d ), epsilon ( e ) and so on. An example is " a (alpha) Centauri", the brightest star in the constellation Centaurus. It is not possible to compare the brightness of a star to a star in a different constellation using the Bayer letter system. (Compare to Flamsteed Number.)

"Before Christ". A date before the birth of Christ. The modern preferred term is BCE, which stands for Before the Common Era. This term is used since there is disagreement as to the actual year in which the historical person of Jesus was born. Modern research indicates that he was born is what we would today refer to as 3 to 6 BC. In order to eliminate the possible confusion that could arise from this uncertainty, a new term, BCE, was adopted. Presently, essentially the whole world uses the same dating system (thus the term "Common Era"), so this standard term has been adopted to avoid the problem of the actual date of Christ's birth. (Compare to AD and CE.)

"Before the Common Era". Due to confusion over the actual year in which the historical person of Jesus was born, this term has been adopted as the standard way to designate what we all used to call "BC". (Compare to AD and CE.)

The displacement of lines in the spectrum of a moving body toward shorter wavelengths than the lines would have if the body was stationary. Blueshift indicates that the body is moving toward the observer. The greater the shift, the higher the velocity. View a diagram (See also Redshift)

A mathematical formula developed independently by Bode and Titus that "predicts" the radii of the planets' orbits. Generally believed to have little or no scientific validity. Sometimes referred to as the Titus-Bode Law.

(1546 to 1601) Danish astronomer who was the most experienced observational astronomer of his day. It was his careful and precise observations of the planets that allowed Kepler to develop his laws of planetary motion.

(1548 to 1600) Italian astronomer. In his book De l'Infinito , Universo e Mondi, he stated that the universe had an infinite number of worlds, and that these were all inhabited by intelligent beings. Arrested by the inquisition in 1591, he was imprisoned for 8 years while being "interrogated". After once again refusing to recant his beliefs, he was burned at the stake in 1600. While the general charge was heresy, the exact charge is unknown, as his file is missing from the records.

The speed of light in a vacuum, approximately 3 x 10 8 m/s.

The largest of several bands in the rings of Saturn that have greatly reduced numbers of particles.

"Common Era". Used to designate years after the birth of Christ. For common usage, CE is the same as what was formerly called AD. For a more detailed explanation, see BCE. Compare with AD and BC. Additionally, since AD stands for “the year of Our Lord” in Latin, non-Christians have an understandable objection to using AD to designate modern years.

The projection of the Earth's equator onto the celestial sphere. The Celestial Equator divides the celestial sphere into northern and southern hemispheres in the same way the Earth's equator divides the surface of the Earth into northern and southern hemispheres.

The science of determining the position of a ship or plane by using the measurement of the altitude of certain celestial bodies and the time of the observations. Contrary to common expectations, celestial navigation is principally done by measuring the altitude of the sun, not the stars, although the altitude of stars can be used.

A fictitious sphere, with the Earth at the center, upon which all celestial objects appear to be located. Although the idea of the Earth being the center of the universe was abandoned long ago, the concept is still useful when dealing with the positions and movements of celestial objects as seen from the Earth, and greatly simplifies the mathematics involved in calculating those positions and movements. Click to view a diagram of the Celestial Sphere.

An older name for what is now called the Celsius scale. A temperate scale on which the zero point is set at the temperature at which water freezes and the 100 degree point is set at the temperature at which water boils. Luckily, both Celsius and Centigrade start with C, so we can use either one, although officially Celsius is the name of the scale.

A unit of length equal to one-hundredth of a meter (1 x 10 -2 m) (One inch is 2.54 centimeters).

The first of the Minor Planets (asteroids) to be discovered (1801), Ceres, located in the main asteroid belt at 2.8 AU, was originally classified as a planet, and later downgraded to an asteroid. In 2006 Ceres was again reclassified to be a Dwarf Planet. Ceres is about 914 km across

A distortion of an image focused by a lens that results from different wavelengths (colors) of light being refracted differently, and are thus focused at slight different distances from the lens. Chromatic aberration is the result of light passing through material, and thus does not occur with mirrors. In practice, chromatic aberration is usually reduced by making a double lens, with two different materials with slightly different optical properties being used for the two lenses. (See also spherical aberration .)

The thin layer of the sun just above the Photosphere and below the Corona .

(View a diagram of principle planetary alignments HERE .) Whenever a body is in line with the sun as viewed from Earth, that body is said to be in (or at) Conjunction. A more technical definition would be that a body in conjunction has a separation of 0 degrees from the sun. The term can refer to any two bodies that are in a straight line as viewed from the Earth, in which case both bodies are named (e.g. Jupiter and Mars are in Conjunction). If only one object is named (e.g. Venus is at Conjunction), the other object is always assumed to be the sun. See also Opposition to compare.

If the body lies between the Earth and the sun, it is said to be at Inferior Conjunction.

If the body is on the opposite side of the sun from the Earth, the body is said to be at Superior Conjunction.

One of the 88 regions of the Celestial Sphere . All stars are part of one constellation or another, even those too faint to see without optical aid, whether or not the star makes up part of the "picture" in the constellation. The present boundaries of the constellations were set by the International Astronomical Union (IAU) in 1928.

A continuous spectrum looks like a rainbow, in the following order, each color blending to the next: Red, Orange, Yellow, Green, Blue, Indigo, and Violet. The wavelengths of the light range from about 7000 Angstroms on the red end to about 4000 Angstroms on the violet end. Continuous spectra result as electrons that are free (not bound to any atom) are captured by an atom, and a photon is emitted. Since the free electrons have a continuous range of possible energies to begin with, there is a continuous range of energy that the emitted photons can have, thus there is a continuous range of colors emitted. View examples of spectra here. See also emission spectrum and absorption spectrum .

(1473 to 1543) Polish astronomer who developed a heliocentric model of the universe. On his deathbed he had published his great work De Revolutionibus Orbium Coelestium , or "On the Revolution of the Heavenly Spheres".

The outermost layer of the sun's atmosphere, usually not visible to the naked eye, but clearly visible during a solar eclipse.

The rate at which impacting objects will hit a surface of a planet or moon. If the cratering rate is know (or can be reasonably estimated), then a count of the number of craters of various sizes per square kilometer will yield an estimate of the age of the surface of the planet on moon.

The cross staff consists of a single long rod with a moveable cross piece that can be used to measure the angular separation between two objects. Used from antiquity through the 1700’s. Mariners used a version of the cross staff to measure the altitude of the pole star, or of the sun at noon, for navigational purposes. Astronomers used the cross staff to measure the angular separation of two stars, or of a star and a planet, and thus to accurately determine the position of the object under study.

The two points on the Moon where the outer circumference meets the ends of the terminator , for example the two "points" of a crescent Moon. (View a diagram here.)

Generally: The time it takes the Earth to rotate on its axis once. The length of the day depends on what is used as a reference to determine that the Earth has rotated once.

1) Apparent Solar Day: The actual time that elapses between one transit of the sun across any given meridian and the next transit of the sun across the same meridian. Exact length varies slightly over the course of a year.

2) Mean Solar Day: The average time that elapses between one transit of the sun across any given meridian and the next transit of the sun across the same meridian (24 hours).

3) Sidereal Day: The average time that elapses between one transit of any given star across any meridian and the next transit of the same star across the same meridian (23 hours, 56 minutes, 4.1 seconds.)

The angle between the celestial equator and a given astronomical body, measured 0 degrees to 90 degrees north or south.

Click to view a diagram of the Celestial Sphere and Declination.

1) An angle equal to 1 /360 th of a circle, (symbol o ). A degree is subdivided into 60 arc minutes , and 3600 arc seconds .

2) A unit of temperature measurement. The size of the degree of temperature measurement is different in different systems. One degree Fahrenheit ( o F ) is 1/180 of the difference in temperature between water boiling and water freezing. One degree Celsius (Centigrade) ( o C) is 1/100 of the difference between those same two temperatures. One degree Kelvin ( o K) (or one “Kelvin”) is the same as one degree Celsius, but the two scales start at different temperatures, so 0 o C is equal to 273 o K.

The density of a body is the total mass of the body divided by the volume of the body. Usually specified in grams ( g ) per cubic centimeter ( cc ). Water has a density of 1g/cc.
As an example: a bowling ball and a round balloon may have the same volume (if they are the same size), but the bowling ball has more mass in that volume, so it has a higher density. A second example: 100 pounds of steel and 100 pounds of styrofoam have the same mass, but the because the density of steel is higher than the density of styrofoam , the 100 pounds of steel would fit in a much smaller ball than 100 pounds of styrofoam would.

A hollow sighting tube used before the invention of the telescope. Although the dioptra didn’t magnify things, it blocked extraneous light which made it easier to see slightly dimmer stars. Any medieval illustration (and there are several) showing an astronomer using what looks like a telescope is actually showing a dioptra in use.

The shift in the positions of spectral lines from their normal wavelengths to shorter or longer wavelengths as a result of the radial velocity of the object emitting the spectrum. View a diagram .
Also see Blueshift and Redshift .

A classification adopted August 24, 2006 for objects that are large and spherical, but which do not the meet all the requirements for being classified as a planet, specifically they do not dominate their part of the solar system. There are presently 3 Dwarf Planets, Ceres (formerly classified as an asteroid), Pluto, and Eris . .

The third planet from the sun and the first planet out from the sun that has a moon. The Earth is the largest of the Terrestrial Planets . The Earth's mass is 5.98 x 10 24 kg, its mean density is 5.52 g/cc, and its mean radius is 6371 km. Earth orbits at 1 Astronomical Unit (By definition.) View a diagram of the layers of the Earth. See also planet symbols .

One of 2 possible alignments of the sun, Earth , and Moon such that the shadow of the Moon falls on the Earth (a solar eclipse) or the shadow of the Earth falls on the Moon (a lunar eclipse.) In either case the sun, Earth, and Moon are in a straight line. If the Moon is between the Earth and the sun, the Moon will block all or part of the sun as seen from a small part of the Earth, resulting in a solar eclipse. If the Earth is between the Sun and the Moon, all or part of the Moon will be darkened by the Earth's shadow, resulting in a lunar eclipse. NOTE: The phases of the Moon are NOT caused by the shadow of the Earth .

The plane of the Earth's orbit around the sun, thus also the apparent path of the sun around the Celestial Sphere as seen from the Earth over the course of a year.

Click to view a diagram of the Celestial Sphere and the Ecliptic.

(1879 to 1955) German (later American) physicist and mathematician who developed the general and special theories of relativity.

The electromagnetic spectrum is the "light" we see, even though most of the electromagnetic spectrum lies outside of the range of wavelengths our eyes can detect. On one end there is Radio, with long wavelengths and low frequencies . On the other end there are Gamma rays, with short wavelengths and high frequencies.
The entire electromagnetic spectrum is:

Gamma rays (wavelengths: 10 -15 m to 10 -12 m),

X-rays (10 -12 m to 10 -8 m),

Ultraviolet (10 -8 m to 4000 Å ),

Visible light (4000 Å to 7000 Å ) (Shorter to longer wavelengths: Violet, Indigo, Blue, Green, Yellow, Orange, Red)

Infrared (7000 Å to 0.1 mm ),

Microwaves (0.1 mm to 10 cm ),

(Also known and a "bright line" spectrum.) When light from a hot, diffuse gas is broken down into its component colors by passing the light through a prism or a diffraction grating, the resulting spectrum will be a series of bright lines of different colors. Each line results from the transitions of electrons in the atoms of the gas from higher energy levels to lower energy levels. As the electron falls to lower energy levels, it emits a photon with an energy equal to the difference in energy between the electron's starting and ending levels. Since each element has a unique set of energy levels, gasses of each element have unique sets of emission lines, thus by an analysis of which lines are present in the spectrum we can determine the chemical composition of the gas. View examples of spectra here. See also absorption spectrum and continuous spectrum .

2) The moment in time when the sun is located at one of those points.
The Vernal Equinox occurs approximately March 21, and the Autumnal Equinox occurs approximately September 21.

An imaginary line around the circumference of the Earth , equidistant from both the North and South geographic poles.

A type of telescope mount where one axis of the mount is parallel to the Earth 's Axis of rotation, allowing the telescope to track the stars using movement on one axis only.

An orbit oriented so that the orbiting body orbits directly over the equator of the body being orbited. In the case of an Earth-orbiting satellite, if the equatorial orbit is ALSO geosynchronous , the object is then said to be in a geostationary orbit.

(276 BC to 194 BC) Egyptian librarian who, in 200 BC, used the lengths of shadows in Syene and Alexandria on the same day to calculate the size of the Earth . His calculation of the size of the Earth was approximately 4% from the true size.

A Dwarf Planet, discovered in 2003 but not identified until 2005. Eris is 27% larger than Pluto , so it was originally called the 10 th planet. The International Astronomical Union later classified Eris, and reclassified Pluto and Ceres , as Dwarf Planets, a term invented in 2006. Eris has 1 moon, Dysnomia . Eris has an orbital period of 557 years in a highly eccentric orbit, which is also highly inclined, 44 o to the ecliptic. Eris was originally nicknamed Xena , after the main character from the then-popular TV show “ Xena , Warrior Princess”, but unfortunately that name wasn’t chosen as the final official name.

(Also known as an Exoplanet) A planet that orbits a star other than the sun . As of August 2013, more than 380 planets were known orbiting other stars.

A system of designating the brightest stars within a given constellation by numbering them in the order in which they rise above the horizon . For example: "7 Cygnus" is the 7th star in the constellation Cygnus to rise. (Do not confuse with the Magnitude number, which is never part of the name of the star.) Compare to Bayer Letter .

(Abbreviation - f)The number of waves that pass a given point in 1 second. Usually expressed in units of Hertz (Hz). One Hz is equal to one wave per second one megahertz (MHz) is equal to one million waves per second. Frequency is inversely proportional to wavelength. The relation between frequency, f, and wavelength, l is f l = c, where c is the speed of light (a constant.)

The process of fusing, or "sticking together", of atomic nuclei. The fusion of hydrogen nuclei (single protons ) into a helium nucleus (2 protons and 2 neutrons ) results in the release of energy, and is the primary process by which the sun creates energy.

The Universal Gravitational Constant, 6.67 x 10 -11 m 3 kg-s 2 . Do not confuse with lower case "g" (see below).

1) gram: (1/1000 of a kilogram)
2) The acceleration due to gravity, (9.8 m/s 2 on the surface of the Earth.)

The four moons of Jupiter that were discovered by Galileo: Io, Callisto, Ganymede, and Europa.

(1564 to 1642) Italian astronomer and mathematician. Although Galileo did not invent the telescope, he used a telescope of his own manufacture to examine the stars, planets, Moon, and sun. Galileo's observation of the phases of Venus showed that Venus must go around the sun as Copernicus had proposed, not the Earth . His discovery of the 4 large moons of Jupiter , now known as the Galilean Moons, added further evidence that the Earth was not the center of the universe. Galileo was interrogated by the Inquisition four times, and was imprisoned in his house for the last 10 years of his life. He was largely blind at the time of his death as a result of damage to his eyes caused by looking at the sun through his telescope.

An orbit around the Earth in which the orbiting object remains directly above the same spot on the Earth at all times. Such an orbit must be equatorial and geosynchronous. Most communications satellites are in geostationary orbit.

An orbit around the Earth in which the satellite takes exactly 1 day to complete its orbit. Note that ALL geostationary orbits are geosynchronous, BUT NOT ALL geosynchronous orbits are geostationary.

" Earth -Centered". The idea that the Earth is the center of the solar system, and all things revolve around the Earth.

(Pronounced " gren itch".) A town near London and home of the Royal Greenwich Observatory. By international agreement the meridian running through the observatory is the prime meridian of the world. As used in astronomy, "Greenwich" is a shorter way of designating the Prime meridian .

Abbreviation for Hour. Used as a superscript to be consistent with the use of m for minute of time. An example of usage in righting a star's right ascension would be: 14 h 31 m .

"Sun-Centered". The idea that the sun is the center of the solar system, and all things revolve around the sun.

In the astronomical sense, Horizon refers to a plane tangent to the Earth at the observers location. Essentially, the astronomical horizon means the horizon you would see if everything around you was flat land, i.e. no mountains, trees, etc..

Closer to the sun. See also Superior. Any planet or object that is closer to the sun than the Earth is said to be Inferior. The term can also be used to designate objects closer to the sun than a given planet other than the Earth, for example we could say that Mars is inferior to Jupiter. This means that Mars is closer to the sun than Jupiter. It doesn't mean that Mars isn't as good of a planet.

See Conjunction . See also Opposition to compare. View a diagram of planetary alignments HERE .

Any planet with an orbit closer to the sun than the Earth , i.e. Mercury and Venus .

(IR) Infrared light is that part of the electromagnetic spectrum that is just outside the visible spectrum, with slightly longer wavelengths than red light, roughly 7000 Ångstroms (700 nanometers ) to 1 millimeter.

As a simple result of geometry, the intensity of light, the strength of gravity, etc.. is decreased at increasing distance from the source. The strength or intensity falls off as 1/r 2 , where r is the difference in the distance of the two objects from the source. For example, if planet A is 3 times as far from the sun as is planet B, then planet B will receive only 1/3 2 = 1/9 of the light from the sun that planet A does, and will feel only 1/9 the gravitational pull from the sun.

A planet with a structure similar to Jupiter: The Jovian Planets are Jupiter , Saturn , Uranus , and Neptune . (Compare with the Terrestrial Planets .)

The fifth planet from the sun, orbiting at 5.2 Astronomical units. Jupiter orbits in 11.9 years and has a mass of 317.8 times the mass of the Earth Jupiter's average density is 1.3 g/cc, and Jupiter's diameter is 142,800 km (19.4 times the diameter of the Earth). See also planet symbols

1)"kilo", that is 1000 of whatever unit follows the k. For example "kg" is kilogram (1000 grams), "km" is kilometer (1000 meters), etc.
2)The constant of proportionality, k, in Kepler's third law . K = f p 2 / G (M2 + M1), where M2 and M1 are the masses of the orbiting body and the body being orbited.
3) o K: Degrees Kelvin .

The Kelvin temperature scale ( o K ) is used to measure temperature in absolute terms, that is from absolute zero. Since temperature is essentially a measure of how fast the atoms in a substance are moving, there is a lower limit to temperature. Atoms can not move any slower than not moving at all, thus the temperature at which atoms don't move at all is called Absolute Zero. Zero o K is Absolute Zero, which is equal to -273 o C, or -460 o F. At present the accepted practice is to us “K” rather than “ o K”. and say � Kelvins” rather than � degrees Kelvin”, however although this is widely followed it is not universal. Being generally of a contrary nature, the idea of saying “degrees Celsius” and “degrees Fahrenheit”, but then NOT saying “degrees Kelvin” strikes me as rather stupid, so in MY class you will always hear “degrees Kelvin”, not “Kelvins”.

(1571 to 1630) German mathematician and astronomer. Kepler used the planetary observations of Tycho Brahe to attempt to fit the assumed circular orbits of the planets to a system whereby the spacing of the orbits could be explained using the 5 "perfect solids" (Solids with all sides exactly the same: Cube, Tetrahedron, Octahedron, Dodecahedron, Icosahedron). He was unable to do so, but in the process he realized that the planets did not orbit is perfect circles, but rather in ellipses, with the sun at one focus (a "heliocentric" system). He set down his findings on planetary motion is what are now known as "Kepler's Laws". Kepler supported himself in part as an astrologer, writing horoscopes. He was also interested in detecting the music believed to be made by the planets as they moved, the "Music of the Spheres".

Johannes Kepler's three laws of planetary motion:
1) The planets orbit in ellipses, with the sun at one focus
2) a line drawn from the sun to a planet will sweep out equal areas in time periods and
3) The square of a planet's orbital period (p) is proportional to the cube of the planet's average distance from the sun (a):
p 2 = k a 3 ,
where K is a constant of proportionality. If we use years as the unit for the orbital period (p) and astronomical units for as the unit of the average distance from the sun, then the equation is easier to deal with. We can then drop the "k" and the equation becomes p 2 = a 3 .

A unit of length equal to 1000 meters .

A group of orbital periods in the asteroid belt that have fewer minor planets (asteroids) than usual due to these orbital periods being in resonance with the orbit of Jupiter .

A region of the solar system roughly between 30 and 200 astronomical units in radius in which are found bodies that are comet-like. Pluto is the second largest known object in the Kuiper Belt, after the object (not yet classified) Xena , discovered in 2005. This region of the solar system is where short period comets originate. Compare to the Oort Cloud .

The two stable Lagrangian ("L") points, where a smaller body can orbit in equilibrium with a larger body. These two points are located in the same orbit as the larger body, 60 degrees ahead of (L4) and 60 degrees behind (L5) the body. The Trojan Asteroids in the orbit of Jupiter are located at the L4 and L5 points for Jupiter.

The angle, as measured at the center of the Earth , between the equator and any given spot on the Earth's surface. Latitude must have a value between 0 degrees and 90 degrees north or south. North Latitude is generally shown with the letter N or the plus sign, +. South Latitude is generally shown with the letter S or the minus sign, -.

The angle, as measured at the center of the Earth , between the Prime meridian and the meridian of any given point. Longitude is generally designated as either east (E) or west (W) of the Prime Meridian (0 degrees to 180 degrees either way).

Since the subsolar point (the spot on the Earth where the sun is directly overhead) changes longitude at the rate of 15 degrees per hour, if we know the Universal Time when the sun transited the prime meridian (Greenwich), and the time when the sun transits our local meridian , multiplying the difference in hours by 15 will give us our longitude west of Greenwich. (If the longitude West of Greenwich exceeds 180 degrees, subtract that longitude from 360 degrees to get the longitude East of Greenwich.)

The natural body that orbits the Earth the Moon.

1)"Mega", 1 million (10 6 ) of whatever unit follows the k. For example "MHz" is megahertz (a frequency equal to one million cycles per second.
2) Mass of a body (see the equation for " k " as used in Kepler's 3rd law.)

1) Abbreviation of meter, the fundamental unit of length in the metric system. One meter is 39.37 inches.

2) " Milli " that is 1/1000 of whatever unit follows the m. A millimeter (mm) is 1/1000 of a meter.
Important Note! Upper case M and lower case m mean different things: a mm is a millimeter (1/1000 of a meter), but Mm is a Mega meter (one million meters.)

3) Abbreviation for minute (of time). Written as a superscript (such as 15 m to avoid confusion with meter. (Note that a minute of arc uses ' as a symbol.) See also h .

A system (actually several related systems) to designate the brightness of stars and other astronomical bodies. Apparent Visual Magnitude: Originally begun by designating the brightest stars as "First Magnitude", the second brightest stars as "Second Magnitude" and so on, the system has been extended into the negative range. Starting with the sun (the brightest object in the sky), which has an apparent visual magnitude of -26.7, the brighter an object is, the lower its magnitude. The Full Moon has an apparent visual magnitude of -12.7, Venus at its brightest has a magnitude of -4.4. The dimmest celestial object that can be seen in a very dark sky without a telescope is about magnitude 6. Be sure you don't confuse this with Flamsteed Number , a completely different thing.
(NOTE: There are other kinds of "Magnitude" that will not be discussed in this class.)

(Pronounced "Mar A", Latin for sea) A mare is a dark area on the surface of the Earth's Moon .

The fourth planet from the sun, orbiting at 1.5 Astronomical Units . Mars is the most distant of the Terrestrial Planets . Mars orbits in 1.88 years and has a mass of 0.11 times (11%) the mass of the Earth . Mars' average density is 3.94 g/ cc , and Mars' diameter is 6788 km (53% the diameter of the Earth.) See also planet symbols.

The closest planet to the sun, orbiting at 0.4 Astronomical Units. Mercury orbits in 0.24 years (88 days) and has a mass of 0.055 times the mass of the Earth . Mercury's average density is 5.43 g/ cc , and Mercury's diameter is 4878 km (38% the diameter of the Earth). See also planet symbols.

1) An imaginary line on the surface of the Earth running from the North Geographic Pole to the South Geographic Pole.

2) An imaginary line on the Celestial Sphere running from the North Celestial Pole to the South Celestial Pole.

Meteor: An object that enters the Earth 's atmosphere from space, momentarily showing a bring streak across

the sky as friction with the atmosphere heats the outer layers.

Meteorite: The same object is called a meteorite if it actually impacts the surface of the Earth .

Meteoroid: The same object is called a meteoroid before it enters the Earth's atmosphere.

The meter is the fundamental unit of length in the metric system. One meter is 39.37 inches.

A unit of length equal to one-thousandth of a meter (1 x 10 -3 m).

The correct term for what are usually referred to as Asteroids. Minor Planets are bodies smaller than a planet (the exact cutoff size is not established) that orbit the sun .

Any natural body orbiting a planet. If referring to the object orbiting the Earth , the word should be capitalized (Moon). (Compare with planet .)

The natural body that orbits the Earth , also known as Luna. The Moon has a mass of 7.35 x 10 22 kg (1.23% of the mass of the Earth), and a diameter of 3476 km (27% the diameter of the Earth). The Moon's density is approximately 3.34 g/cc, and it has a synodic period of 29.53 days. The Moon is in synchronous rotation as it orbits the Earth.

The Earth has only one true Moon, we also have a "quasi-moon". Asteroid 2003 YN107 orbits the sun in almost in the same orbit as the Earth. Relative to the Earth, 2003 YN107 follows a horseshoe shaped path that is sometimes ahead of the Earth and sometimes behind the Earth in the Earth’s orbit, making it look like it is actually orbiting the Earth. Well, it would be better to say it looks like it orbits the Earth’s orbit. That’s a little hard to visualize, so you might want to look at this diagram from the Jet Propulsion Lab www.jpl.nasa.gov/images/asteroid/funky1_browse.jpg

A unit of length equal to one-billionth of a meter (1 x 10 -9 m), or 10 Ångstroms.

One Nautical Mile (NM) is equal to 6076 feet (1852 meters ). Each minute of arc along the equator is equal to one Nautical Mile. The nautical mile (not the Kilometer ) is the internationally accepted metric unit of length when measuring distance at sea.

When the sun and Moon pull on the Earth 's oceans at right angles (First or Third Quarter Moon), the effects of their gravity partially cancel each other, resulting in a small range of tides a low, high tide and a high, low tide.

The eighth planet from the sun , orbiting at 30 Astronomical Units . Discovered in 1856 by Johann Galle of the Berlin Observatory, based on the position predicted by John Adams (England) and Urbain Leverrier (France). Neptune orbits in 164 years, and has a mass of 17.2 times the mass of the Earth . Neptune's average density is 1.8 g/ cc , and Neptune's diameter is 48,600 km (3.8 times the diameter of the Earth.) See also planet symbols.

A nuclear particle with the mass of a Proton, but no electrical charge. The number of Neutrons in the nucleus of an atom determine the isotope of the element. For example, if an atom has one proton its hydrogen. Hydrogen with a proton and one neutron is still hydrogen, but acts differently at the atomic scale, and is called Deuterium. Hydrogen with one proton and 2 neutrons is called Tritium. Its still the element hydrogen, but it's an isotope of hydrogen.

(1642 to 1727) English physicist, astronomer, and mathematician. Although famous for his discoveries in many areas of physics, math, and astronomy, perhaps his most important work was the book Philosophiae Naturalis Principia Mathematica, or "The Mathematical Principles of Natural Philosophy" (Natural Philosophy was a term used for what we today call science.) Newton's work into gravity and motion made it possible for astronomers to work with the motions of the planets. Although it is rarely mentioned today, Newton was also interested in, and performed experiments in, Alchemy.

Suspended from a ring at the top, the central time disk is rotated so that either the tab for Ursa Major or Ursa Minor (depending on which one the user wishes to use) is aligned with the current month and date. Then either the bright star at the bottom of the little dipper's bowl ( Kochab , b Ursa Minoris ), or the star at the end of the big dipper's bowl (Dubhe, a Ursa Majoris ) is sighted, and the long arm is adjusted to be parallel to the line between Polaris and the star. The local time is read from where the arm crosses the time on the inner time disk.

The point on the celestial sphere directly above the north geographic pole. The star Polaris, often called the "north star", lies very close to the North Celestial Pole.

The northern-most point on the surface of the Earth the northern end of the Earth's axis of rotation. What people generally mean when they say "the north pole".

The region on the surface of the Earth to which the north end of a compass will point. The North Magnetic Pole is NOT located at the north geographic pole. Its exact location varies from year to year, but is found roughly 600 miles south of the north geographic pole, in the region of the Queen Elizabeth Islands north of mainland Canada in the Arctic ocean.

Obliquity of the Ecliptic

Generally: The angle between the equator of a planet and its plane of orbit. For the Earth , it is the angle between the celestial equator and the ecliptic .

The disappearance of one body as another body passes directly in front of it. The body that disappears is said to be Occulted. (From the Latin word Occultus , meaning "hidden", which is the same reason why the supernatural is referred to as the "Occult".) .

(View a diagram of principle planetary alignments HERE .) Whenever a body is directly opposite the sun as viewed from Earth , that body is to be in Opposition. A more technical definition would be that a body in opposition has a separation of 180 degrees from the sun. (See also Conjunction to compare.) The term can refer to the separation of any two bodies from each other by 180 degrees as viewed from the Earth, but this meaning is rarely used.

A spherical distribution of comets surrounding the solar system. The Oort cloud has a radius of between 20,000 and 50,000 astronomical units . This is the region where long period comets originate. Compare to the Kuiper Belt.

For sun-orbiting objects: The angle between the plane of the ecliptic and the plane of the planet's (or other body's) orbit.

For Earth -orbiting objects: The angle between the plane of the Earth's equator and the plane of the object's orbit.

Perigee: for an object orbiting the Earth , the point in its orbit when it is closest to the Earth.

Perihelion: for an object orbiting the sun, the point in its orbit when it is closest to the sun. The Earth reaches Perihelion on approximately January 4th each year.

The phases of the Moon are the result of the changing Earth - Moon -Sun geometry. The four principle phases, each separated by about 1 week, are in order: New Moon, First Quarter Moon, Full Moon, and Third Quarter Moon. Each phase occurs every month.

View a diagram of the phases of the Moon and their relationship to the sun and Earth.

New Moon: When the Moon is roughly between the Earth and the sun. Note that if the Moon is exactly between the Earth and the sun there is a Solar Eclipse. There is NOT a solar eclipse every month because the Moon's orbit is slightly tilted in respect to the ecliptic.

Waxing Crescent Moon: During the week between New and First Quarter Moon the Moon is referred to as a Waxing Crescent.

First Quarter Moon: When the Moon is exactly 1/4 of the way through its phase, half of the Moon will be illuminated as seen from the Earth. Astronomically this is referred to as a First Quarter Moon, although non-astronomers often call it a "half Moon".

Waxing Gibbous Moon: During the week between First Quarter and Full Moon the Moon is referred to as a Waxing Gibbous.

Full Moon: When the Moon is roughly directly opposite the sun all of the side of the Moon facing the Earth is illuminated. If the Moon is exactly opposite the sun there is a Lunar Eclipse. There is NOT a lunar eclipse every month because the Moon's orbit is slightly tilted in respect to the ecliptic.

Waning Gibbous Moon: During the week between Full and Third Quarter Moon the Moon is referred to as a Waning Gibbous.

Third Quarter Moon: When the Moon is exactly 3/4 of the way through its phases, half of the Moon will once again be illuminated as seen from the Earth. Astronomically this is referred to as a Third Quarter Moon, since the Moon is three-quarters of the way through its set of phases, although non-astronomers often call it a "half Moon".

Waning Crescent Moon: During the week between Third Quarter and New Moon the Moon is referred to as a Waning Crescent.

A body orbiting a star that has enough mass to make it a sphere, and which itself is not a star, and which has cleared it’s surrounding region of debris . There are 8 accepted planets orbiting the sun, but more than 160 known planets orbiting other stars. (Compare with "Minor Planet" and "moon" .) The planets in our solar system, in order out from the sun are Mercury , Mercury , Venus , Earth , Mars , Jupiter , Saturn , Uranus , and Neptune . Formerly classified as a planet, Pluto was demoted to a Dwarf Planet on August 24, 2006. Click HERE to see a list of the symbols for the planets and the Zodiac.

Specific names given to arrangements of the sun, Earth, and other planets. These are Syzygy (special case of 0 degrees), Conjunction (0 degrees), Semisextile (30 degrees), Semiquadrature (45 degrees), Sextile (60 degrees), Quadrature (90 degrees), Trine (120 degrees), Sesquiquadrature (135 degrees), and Quincunx (150 degrees), Opposition (180 degrees).

The fundamental particle of electromagnetic radiation light. A photon is essentially a small packet of energy. (Note: Do not confuse with proton , a completely different thing.)

The visible surface of the sun. The layer in the sun where most of the visible light is produced.

At one time the most distant planet in the solar system. Discovered in 1930, Pluto was classified as a planet until August 24 th , 2006, when the International Astronomical Union removed Pluto from the list of planets. Pluto is now classified as a Dwarf Planet, a category that didn’t exist prior to this date. Pluto is now the second largest object in the Kuiper belt , (after the object Eris , discovered in 2005). Pluto's orbit was the most eccentric of any planet, bringing Pluto inside the orbit of Neptune for part of its orbit. Pluto was inside Neptune's orbit from 1980 through 1999. Pluto has a diameter of 2300 km, smaller than the Earth 's Moon . Pluto orbits in 246 years, at an average distance of 39.3 astronomical units. Pluto's average density is 1.1 g/ cc , and Pluto's diameter is 3,200 km (25% the diameter of the Earth.)

Pluto's large moon, Charon (discovered in 1978) has dual synchronous rotation with the Pluto. Pluto also has 2 very small “micro moons” Nix and Hydra, discovered in 2005.

There are 7 moons of other planets in the solar system that are actually larger than Pluto: Triton, Europa, Luna, Calisto, Io, Ganymede, and Titan. See also planet symbols.

(Latin: Post Meridian) "After the Meridian ". In Civil Time , used to designate the time after Noon and before Midnight. Compare with AM .

A star located very near the North Celestial Pole , often referred to as the "north star". It should be noted that contrary to popular belief Polaris is not the brightest star. Sirius is the brightest star. There is no similarity placed star near the south celestial pole.

An orbit where the orbiting body passes directly over the north and south poles of the body being orbited.

The meridian running through the Royal Greenwich Observatory in Greenwich, England. The Prime Meridian is the internationally agreed-upon starting point for Longitude.

A basic nuclear particle, which has an electrical charge of +1 units (1.6 x 10 19 coulombs), and one unit of atomic mass. The number of protons in the nucleus determines which element the atom is. Hydrogen has 1 proton, helium has 2 protons, carbon has 6, etc..) (Note: do not confuse with Photon , a completely different thing.)

(circa AD 85 to 165) Egyptian astronomer and mathematician who attempted to explain retrograde motion of the planets by proposing that the planets moved on epicycles, the epicycles themselves moving around the Earth on Deferents. (A modification of the Geocentric system of Aristotle.)

(View a diagram of principle planetary alignments HERE .) Generally: An alignment of two bodies with a separation of 90 degrees (1/4 of a circle) as viewed from the Earth . Most often referring to a planet separated from the sun by an angle of 90 degrees as seen from the Earth.

An archaic term for the alignment of two bodies with a separation of 150 degrees as viewed from the Earth . Most often referring to a planet separated from the sun by an angle of 150 degrees as seen from the Earth. It is very unlikely that you will run across this term in astronomy texts, although it does sometimes still appear in astrological charts.

The portion of an object's velocity that is along a line directly between the body and the observer. Radial velocity is a measure of how quickly an object is moving toward or away from the observer, it does not provide any information on the movement of the body sideways or vertically as seen by the observer.

The displacement of lines in the spectrum of a moving body toward longer wavelengths than the lines would have if the body was stationary. Redshift indicates that the body is moving away from the observer. The greater the shift, the higher the velocity. View a diagram (See also Blueshift )

A telescope that uses a curved mirror to bring light to a focus.

A telescope that uses one or more lenses to bring light to a focus.

The temporary movement of a planet "backwards" from its normal slow daily motion across the celestial sphere . The observed Retrograde Motion is a result of the Earth passing the planet as it orbits the planet with Retrograde Motion does not actually move backwards. It was the problem of retrograde motion, specifically that a planet orbiting the Earth in a perfect circle could not explain retrograde motion, that lead Ptolemy to propose his epicycle and deferent model of the universe.

The rotation of a body on its axis in a direction opposite to the rotation of the Earth the rotation of a body in a clockwise direction. (The Earth's rotation is counter-clockwise.)

The Right Ascension of a celestial object is the amount of time that elapses between the transit of the Vernal Equinox (see definition 1 of Vernal Equinox) across any given meridian and the transit of that celestial body across the same meridian. (Additional information here .) Right Ascension is expressed in hours, minutes, and seconds of time, and is always written in 24 hour format ("military" time). An example of usage would be: RA = 17 h 31 m (Right ascension of a star equals 17 hours and 31 minutes.)

The Right Ascension of a body just transiting your local meridian is equal to your local sidereal time .

Click to view a diagram of the Celestial Sphere and Right Ascension.

The limit of how close a substantial body can come to a planet before the gravitational forces from the planet are great enough to break the body apart. The Roche limit is 2.5 times the planet's radius.

Generally: Any body , whether natural or man-made, that orbits another body. Thus the Earth is a satellite of the Sun, and the Moon is a satellite of the Earth. As most frequently used the term refers to an artificial (man-made) body orbiting the Earth, such as communications satellites, surveillance satellites, weather satellites. etc.

The sixth planet from the sun, orbiting at 9.6 Astronomical Units . Saturn orbits in 29.4 years and has a mass of 95.2 times the mass of the Earth . Saturn's average density is 0.7 g/ cc , and Saturn's diameter is 120,00 km (37.6 times the diameter of the Earth) See also planet symbols. .

An archaic term for an alignment of two bodies with a separation of 30 degrees (half of sextile ) as viewed from the Earth . Most often referring to a planet separated from the sun by an angle of 30 degrees as seen from the Earth. It is unlikely that you will run across this term in astronomy texts, although it does sometimes still appear in astrological charts.

(Or Semi Quadrature) An archaic term for an alignment of two bodies with a separation of 45 degrees (1/8 of a circle, or half of Quadrature (see quadrature )) as viewed from the Earth . Most often referring to a planet separated from the sun by an angle of 45 degrees as seen from the Earth. It is unlikely that you will run across this term in astronomy texts, although it does sometimes still appear in astrological charts.

An archaic term for an alignment of two bodies with a separation of 135 degrees as viewed from the Earth . Most often referring to a planet separated from the sun by an angle of 135 degrees as seen from the Earth. It is very, very unlikely that you will run across this term in astronomy texts, although it does sometimes still appear in astrological charts.

An instrument used to measure the altitude of celestial bodies. Today almost exclusively used in Celestial Navigation , but before the widespread use of telescopes, sextants (and similar instruments like the Quadrant) were used by astronomers to make measurements of the positions of the stars and planets. See a picture of a sextant here

(View a diagram of principle planetary alignments HERE .) Generally: An alignment of two bodies with a separation of 60 degrees (1/6 of a circle) as viewed from the Earth . Most often referring to a planet separated from the sun by an angle of 60 degrees as seen from the Earth.

A distortion of an image focused by a mirror or lens that results from the surface or surfaces of the lens or mirror being shaped as part of a sphere. The result is that light passing through the lens or hitting the mirror near the edges does not focus at the same spot as light passing/hitting nearer to the center of the lens or mirror. Spherical aberration is caused by the shape of the lens or mirror, and can be avoided my making the surfaces parabolic. (See also chromatic aberration .)

Our Solar System is the sun and all objects gravitationally tied to it (all the objects orbiting the sun.) This includes the planets and their moons , the minor planets (asteroids), and the comets of the Oort Cloud and the Kuiper Belt .

In general, any solar system is a star or stars and all the bodies gravitationally tied to that star or stars.

1) Either one of two positions on the Celestial Sphere , where the sun reaches its most northerly or most southerly declination .

2) The moment in time when the sun reaches those positions. The Summer Solstice is when the sun reaches its northern-most declination (23.5 degrees north), and the Winter Solstice is when the sun reaches its southern-most declination (23.5 degrees south).

Electromagnetic radiation, is "light", travels at a constant speed in a vacuum: 3x10 8 m/s.

The point on the celestial sphere directly above the south geographic pole. Unlike the North Celestial Pole, there is no star conveniently located near the South Celestial Pole.

South Geographic Pole:

The southern-most point on the surface of the Earth the southern end of the Earth's axis of rotation. What people generally mean when they say "the south pole".

The southern-most point on the surface of the Earth the southern end of the Earth's axis of rotation. What people generally mean when they say "the south pole".

When the sun and Moon pull on the Earth 's oceans along a straight line (New and Full Moon) , the effects of their gravity add together, resulting in a large range of tides a high, high tide and a low, low tide.

A body that has self-sustaining nuclear fusion reactions in its interior.

The point on the surface of the Earth where the sun is directly overhead. The longitude of the position of the subsolar point moves westward at the rate of 15 degrees per hour. Thus if you know the Universal Time when the sun transited the prime meridian (Greenwich), and the time when the sun transits your local meridian , multiplying the difference in hours by 15 will give you your longitude west of Greenwich.

1) The point on the Celestial Sphere when the sun reaches its northern-most declination (23.5 degrees north).

2) The moment in time when the sun reaches that point.

3) The first day of summer (northern hemisphere) is the day on which the Summer Solstice (meaning 2 above) occurs. The Summer Solstice (this meaning) occurs on approximately June 21 each year.

The star around which all the planets of our solar system orbit. The sun (Sol) is a medium sized star (Class G), with a surface temperature (at the top of the photosphere ) of about 5800 degrees Kelvin . The radius of the sun is 6.96 x 10 8 meters, and the mass is 2 x 10 30 kg, or approximately 332,946 times the mass of the Earth . (Note: The mass of other stars is generally given in units of "solar mass", that is the mass of the other star is given in how many times more or less massive the star is compared to the mass of the sun.) The sun's average density is about 1.4 g/cc, and consists of about 94% hydrogen (H) and 5.9% helium (He). The sun gets its energy from the fusion of 4 hydrogen nuclei into 1 helium nucleus (4 protons fused into 2 protons and 2 neutrons .) The difference in mass between 4 H nuclei and 1 He nucleus is given off as energy, as described in the famous equation E = mc 2 , where M is the mass difference.

From the center outward, the layers of the sun are:
Core
Convection Zone
Photosphere
Chromosphere
Corona .
(View a diagram of the layers of the sun.

Farther from the sun. See also inferior. Any planet or object that is farther from the sun than the Earth is said to be Superior. The term can also be used to designate objects farther from the sun than a given planet other than the Earth, for example we could say that Saturn is superior to Jupiter. This means that Saturn is farther from the sun than Jupiter. It doesn't mean that Jupiter isn't as good of a planet.

Any planet with an orbit farther from the sun than the Earth , i.e. Mars , Jupiter , Saturn , Uranus , and Neptune .

Click HERE to see a list of the symbols for the planets and the Zodiac.

A rotation period that is equal to the body's period of revolution. In other words, the time that it takes the body to orbit is equal to the time it takes the body to rotate once on its axis. If a moon is in synchronous rotation, it will always keep the same side toward its planet. The Earth's Moon, Luna , is in synchronous rotation, so we always see the same side of the Moon.

In the case of Dual Synchronous Rotation, the planet also keeps the same side toward its moon as the moon orbits. This is not the case with the Earth and Moon.

Generally: The time required between any 2 successive alignments of 2 bodies. Usually used to refer to the time between two successive New Moons that is the time it takes for the Moon to go through one complete set of phases . The Moon's synodic period is 29.53 days.

Generally: An alignment of any three bodies where all the bodies are in a straight line. Most often referring to the alignment of the sun, Moon, and Earth at a solar or lunar eclipse. (This is a valid astronomical term, however, this word is usually only found in crossword puzzles.)

The line on the surface of a planet or moon dividing the half of the body in daylight from the half of the body in darkness. The terminator is the line of sunrise or sunset at any given moment. View a diagram.

A planet with a structure similar to Earth. The Terrestrial Planets are Mercury , Venus , Earth , and Mars . (Compare with the Jovian Planets )

Time has been said to be nature's way of keeping everything from happening at once. There are different time scales, measuring time in different ways.

Apparent Solar Time: Apparent solar time is what a sundial keeps. Whenever the sun actually transits a given meridian , it is noon at that location. In apparent solar time, the sun always rises at 6:00 AM and always sets at 6:00 PM, thus there are always 12 hours of daylight and 12 hours of darkness each day. Since the actual time it takes between sunrise and sunset differs considerably between summer, fall, winter and spring, this results in the length of the hour differing each day, and it also means that the length of the hour when the sun is up is different from the length of the hour between sunset and sunrise.

Civil Time: This is the time you keep on your wristwatch. It is Mean Solar Time with the addition of the concepts of Time Zones and daylight savings time.

Sidereal Time: (Pronounced " sid ear e al") Time kept by using the stars as a reference. The sidereal day is divided up into 24 hours of equal length. The sidereal time at any point on the Earth is equal to the Right Ascension of the bodies transiting the local meridian at that moment. Sidereal time is always written in the 24 hour format (that is "military" time.)

Universal Time (UT): Universal time is same all over the world at any given moment. Essentially, Universal Time is the Mean Solar Time at the Prime Meridian . Universal Time is always written in the 24 hour format (that is "military" time.) It is also called Zulu Time, and in the past was referred to as Greenwich Mean Time (GMT), a term that is no longer officially recognized, although still widely used.

1) Whenever a celestial body crosses any particular (celestial) meridian , that body is said to Transit that meridian.

2) Whenever one celestial body appears to pass across the disk of another body (for example, when Mercury passes across the sun as viewed from the Earth , Mercury would be said to transit the sun.

(View a diagram of principle planetary alignments HERE .) Generally: An alignment of two bodies with a separation of 120 degrees (1/3 of a circle) as viewed from the Earth . Most often referring to a planet separated from the sun by an angle of 120 degrees as seen from the Earth.

Two groups of minor planets that are in the same orbit as Jupiter , one group is 60 degrees ahead of Jupiter and one group is 60 degrees behind.

The region on the Earth's surface between 23.5 o north latitude (the Tropic of Cancer) and 23.5 o south latitude (the Tropic of Capricorn). In this region of the Earth the sun will be directly overhead at least once each year. Outside of this region the sun is never directly overhead.

(UV) Ultraviolet light is that part of the electromagnetic spectrum that is just outside the visible spectrum, with slightly shorter wavelengths than violet light, roughly 10 Angstroms to 4000 Angstroms. (1 nanometers ) to 400 nanometers.)

(Pronounced YOUR a nus) The seventh planet from the sun, orbiting at 19 Astronomical Units . Uranus was the first planet discovered since antiquity, found by William Hershel in 1781. Uranus is named after the Father of Saturn in mythology. Uranus orbits in 83.8 years and has a mass of 14.4 times the mass of the Earth . Uranus' average density is 1.3 g/ cc , and Uranus' diameter is 51,200 km (4 times the diameter of the Earth.) See also planet symbols.

1) The point on the celestial sphere where the ecliptic crosses the celestial equator with the sun moving from southern declination to northern declination. The symbol for the constellation Aries is often used an abbreviation for the Vernal Equinox. Click here to see a diagram of the Celestial Sphere and the Vernal Equinox.

2) The moment in time when the sun is located at that point.

3) The first day of spring (in the northern hemisphere) is the day on which the Vernal Equinox (meaning 2 above) occurs. The Vernal Equinox (this meaning) occurs on approximately March 21 each year.

The second planet from the sun, orbiting at 0.7 Astronomical Units . Venus orbits in 0.62 years (224.7 days) and has a mass of 0.82 times (82%) the mass of the Earth . Venus' average density is 5.2 g/ cc , and Venus' diameter is 12,104 km (95% the diameter of the Earth). See also planet symbols.

The part of the electromagnetic spectrum that is visible to the human eye. The visible spectrum has wavelengths between roughly 4000 Angstroms (violet light) and 7000 Angstroms (red light). (Or between 400 and 700 nanometers .) Light with slightly higher frequencies (and thus shorter wavelengths) is ultraviolet , and light with slightly lower frequencies (and thus longer wavelengths) is infrared .

(Abbreviation - l ) The distance between successive crests of a wave. Inversely proportional to Frequency . The relation between frequency, f, and wavelength, l ) is f l = c, where c is the speed of light (a constant.)

Wein's law relates the temperature (T) of an object to the wavelength of light that the object emits that is the strongest ( l max). The relationship is expressed as : T l max = C . (where C is a constant, equal to 28.98 x 10 6 Å o K ). Thus the peak wavelength ( l max) is inversely proportional to the temperature (T). If the temperature doubles, the peak wavelength is cut in half.

1) The point on the Celestial Sphere when the sun reaches its southern-most declination (23.5 degrees south).

2) The moment in time when the sun reaches that point.

3) The first day of winter (northern hemisphere) is the day on which the Winter Solstice (meaning 2 above) occurs. The Winter Solstice (this meaning) occurs on approximately December 21 each year.

Eris The official name of Dwarf Planet 2003 UB313. Eris orbits at an average distance of 67.7 AU, but comes as close as 37.8 AU, and as far out as 97.6 AU. Eris’s orbital period is 557 years. With a diameter of

2400km, Eris is larger than Pluto.

The region of the electromagnetic spectrum with a wavelength between 10 -8 and 10 -12 meters.

The time that it takes the Earth to complete one orbit of the sun . This time can be measured in several different ways. The Tropical Year is the time between one Vernal Equinox and the next, 365.2422 days.

The point on the Celestial Sphere directly above any given point on the Earth's surface. Your Zenith is just the point straight up. The opposite term is Nadir (the point directly underneath you), but this is not used in astronomy much.

A region on the celestial sphere , 18 degrees wide, and centered on the Ecliptic . The Zodiac is composed of 12 equal segments named after the primary constellation located in each. These are the "sun signs" used in astrology . The Zodiac begins at the Vernal Equinox , and proceeds in the order: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Sagittarius, Capricorn, Aquarius, Pisces.

It is a common misconception among astronomers that when astrologers say that the sun is in, for example, Leo, that the sun is actually in the constellation Leo. It can often be shown that the sun or planet in question is NOT in the constellation that astrology says it is at the given time. This is commonly used by astronomers as an argument against astrology , but in fact it is a misunderstanding. Each of the 12 parts of the Zodiac are named for the primary constellation in that part, so when astrology says the sun is "in Leo", it doesn't mean that the sun is in the constellation Leo. It means that the sun is in that part of the Zodiac named for the constellation Leo. Although this makes the target wider, so to speak, that fact is that a given planet still is not always actually in the part of the Zodiac that astrological computation says it is.

Click HERE to see a list of the symbols for the planets and the Zodiac.

## The Seven Purposes of The Pyramids of Giza

### The First Purpose of The Pyramids of Giza – Electric Power

The first function of the Pyramid of Gza was to transmit electricity. The type of electricity is transmitted was at a higher energy level than what we use today using the assistance of wires. Nicola Tesla actually was getting close to ‘rediscovering’ the use of this type of technology but the greedy nature of those in power did not allow him to test his more scrutinized or ‘far-fetched’ ideas.

As light ranges from infrared to ultraviolet, electricity, and all seven states of matter range in this way. The form of electricity used was transmitted directly through space (ether) like the way radio waves are transmitted today. The receiving stations for these transmissions were known as obelisks. Two that have been moved from their original location are the one in Central Park in New York and Cleopatra’s Needle in London.

The ancients had a high reverence for technology and saw it as a gift from the God . They only used this technology when it was necessary for the benefit of the group or the whole community. At other times they used simple hand tools. These hand tools were used as our ancient society understood the importance of the exersize gained from everyday work. This is why many archeologist believe that the ancient Egyptians did not have advanced technology as the vast majority of tools that they find seem primitive. They also date the pyramids to the time of Pharaohs, which is wrong as the pyramids of Giza were built about 15,000 years ago. However, they understand that the stone is too hard to cut using these tools so they go as far as to attribute them to aliens rather than their true Black makers.

The technology used during that time added value to the natural balance of the forces of the universe. Technology today actually causes degeneration in the form of the evil extremes of greed and coldheartedness.

### The Second Purpose of The Pyramids of Giza – Preservation

The pyramids are used to preserve life, both animals and plants during natural cataclysm. This is where the legend of Noah’s Arc comes from.

### The Third Purpose of The Pyramids – Space Travel

Pyramids that are larger than the pyramids of Giza are made from asteroid stone. These monolithic Pyramids are uses to transport large groups of people (12,000) throughout different parts of the cosmos. This is how the ancients traveled from the star system Sirius to seed the planet earth.

### The Fourth Purpose of The Pyramids – Balance the Earth Energies

The earth naturally has 12 nodes or points of energy where space energy enters our planet. These energy points need to be kept in balance in order not to disrupt the earth’s natural magnetic field. This magnetic field protects the earth from harmful rays and meteors.

### The Fifth Purpose of The Pyramids of Giza – Book of Stone History

The great pyramid is designed by the leading scientist or God of a given 25,000 year history. It is the Book of Stone providing the history (future) of the events that are to take place over that 25,000 year period. We are in the year 15,000 or this 25,000 year period. A major event occurred 6,000 years ago that shapes the world that we see today, “this world” in the bible. They write this history in the sizes and types of stones used.

For example, there are 144,000 casting stones or limestone blocks used to build the Great Pyramid. These also represent 144,000 aspects of perfect character to be restored after the cycle of self- forgefulness (the past 6,000 years.

### The Sixth Purpose of The Pyramids of Giza – Ascension and Ressurection

Pyramids were used as ascension machines and resurrection machines. Bodies were removed after the soul (personality) ascended primarily to the star system Sirius as was the case with the 60,000 Elohim.

There are 144,000 preserved bodies underneath the pyramid of Giza protected by magnetism and can only be opened by one of the 144 chiefs. These bodies are going to be used as the 6,000 year cycle of self-forgetfulness comes to a close. All of the seven mahdis have received one of these bodies and we will see ordinary black people begin to receive these bodies over the next years to come. These are going to become the Judges of the new society (promised land).

These bodies are actually preserved for black people who are alive now. These individuals will have activated their 12 strands of DNA through dietary restriction and mental exercises similar to ancient initiation. This modified initiation process is given in Level 2 of the book Black Root Science. These elect will be resurrected consciously into one of the preserved bodies. These elect can come from any society of the black nation but the vast majority of them will be the decedents of slaves.

These 144,000 perfectly preserved bodies will give birth to perfect children and the population will increase to 1 billion 8million as the remainder of original black Gods reincarnate.

### The Seventh Purpose of The Pyramids of Giza – Divine Unity

The final purpose of the pyramids is to help facilitate Divine Unity which is held in the custody of the 24 Elders. This is a particular state of mind known as ‘heaven’. This state of mind is one where a person becomes one with his eternal self.

### 2 thoughts on &ldquo Purpose of the Pyramids of Giza – Who Built the Pyramids &rdquo

Is it possible the Giza Pyramids are a marker in time of the Younger Dryas Period which annihilated the life on Earth representing 3 impacts? 10,000 to 15,000 years ago, the Younger Dryas was believed to have been caused by meteor impacts. Some representations of the capstones depict them as spheres. Its capstone size, at 2 meters, is scaled to the pyramidal values representing the earth dimensions. Younger Dryas was a recent geological disaster in Earth’s timeline which melted polar caps and exterminated life here on a massive scale. There seems to be enough information to show it represents that disaster and the Pyramids were built to stand the test of time long enough for mankind to recover and understand what happened to cause the floods around the planet. The pyramids seem to coincide and remain as a warning of the last event to destroy life here.

15,000 years ago was a very important time as it was the start of our current 25,000 year cycle. According to the Honorable Elijah Muhammad, “We make such history once every 25,000 years. When such history is written, it is done by twenty-four of our scientists. Once acts as Judge or God for the others and twenty-three actually do the work of getting up the future of the nation, and all is put into one book and at intervals where such and such part or portion will come to pass, that people will be given that part of the book through one among that people from one of the Twelve (twelve major scientists) as it is then called a Scripture which actually means script of writing from something original or book.

There is a significance to the number 24 Scientists and the 25,000 years. The number twenty-four Scientists used is in accordance with the hours in our day and the measurement of the circumference of our planet around the equator and in the region of our Poles, Arctic and Antarctic Oceans.

Our planet is not exactly 25,000 miles in circumference, it is 24,896 and we, according to astronomy, don’t have a full 24-hour day but near that󈞃 hours, 56 minutes and 46 seconds. The change made in our planets rotation at the Poles is about one minute a year and takes 25,000 years to bring about a complete change in the region of the Poles. The actual Poles are inclines 23 2 degrees to the plane of its orbit. The original black nation used 23 scientists to write the future of that nation for the next 25,000 years, and the 24th is the Judge or the one God, Allah. Allah taught me that, once upon a time they made history to last for 35,000 years. ” “According to the word of Allah to me, AMR. Yakub was seen by the twenty-three Scientists of the black nation, over 15,000 years ago. They predicted that in the year 8,400 (that was in our calendar year before this world of the white race), this man (Yakub) would be born twenty miles from the present Holy City, Mecca, Arabia. And, that at the time of his birth, the satisfaction and dissatisfaction of the people would be: — 70 per cent satisfied, 30 per cent dissatisfied.

And, that when this man is born, he will change civilization (the world), and produce a new race of people, who would rule the original black nation for 6,000 years (from the nine thousandth year to the fifteen thousandth year).After that time, the original black nation would give birth to one, whose wisdom, Knowledge and power would be infinite. One, whom the world would recognize as being the greatest and mightiest God, since the creation of the universe. And, that He would destroy Yakub’s world and restore the original nation, or ancient nation, into power to rule forever.”

## E. EFFECTS OF THE TILT OF EARTH'S AXIS

### North/South Motions of Solar System Objects

• The polar rotation axis of the Earth is not perpendicular to its orbital plane. It is tilted by 23.5 degrees. See the figure below (again, exaggerated for clarity):

The total amplitude of the Sun's swing in N/S distance during the year is 2 x 23.5 = 47 degrees.

### The Seasons

• The Sun's distance north or south from the Celestial Equator determines the hours of daylight and the angle at which sunlight strikes Earth's surface at a given latitude. It thus determines "insolation," or the amount of sunlight incident on a unit area of the Earth's surface during 24 hours.

The change in the geographic shadow distribution caused by the tilt is quite dramatic (even though the shadow always covers exactly one hemisphere of the Earth). Here are two images of the way the shadow is distributed at about 2 PM Eastern time on August 1 (left) and December 1 (right). The Earth's surface moves eastward through the shadow. You can immediately tell from the image which latitudes are receiving more sunlight in a 24 hour period. Click on the thumbnails for an expanded view.

## Wilbur's DIY Page of Bits and Bobs

How does this translate into C++

Well this was my first attempt, it is basically a calculator, and didn't work very well, but it may help people understand the basics of right ascension and it's relationship with a real angle so I have included it with explanations.

First we need to create all the variables required to make the counting system work:

1. int H will be used to create something called a truncated output, in this instance it gives us only the integer from a decimal time i.e. 2.51 hours would give only an hour integer of 2, which is correct - it will not round up, this is very important.
2. int M is the same but for minutes.
3. int Hnudge and Mnudge are variables for buttons whereby if the corresponding button is pressed the Xnudge values increases by 1 - not currently programmed.
4. int HAH and HAM are final compensated time integers.
5. float HAS is a float, this means it can handle decimals and will output a true reading rather than a truncated one like " int " this is used against the truncated figure to get the time in seconds.
6. float ihr, imin and S are the initial float variables in the right ascension calculation, relating to the idea in point 5.
7. float stepdegrees is the output from the stepper motor count in degrees i.e. where the telescope is pointing in a raw decimal degree figure.

LSTH is equal to how many times the "ihr" has gone over 3600000ms within 86164091ms (hours)

LSTM= int (over/60000) // = 60 seconds over=over%60000 // + 1 Minute

LSTM is equal to how many times the "ihr" has gone over 60000ms within 36000000ms (minutes)

LSTS is equal to how many times the "ihr" has gone over 1000ms within 60000ms (seconds)

### 2.8.1 Time of day

The earth is a rotating sphere. The earth rotates around its north-south axis once every 23 hours, 56 minutes and 4 seconds.

It can also be stated that the earth rotates on its axis 2π radians in one day. Over the course of an hour it rotates π/12 radians or 15 degrees. In this framework h is defined as:

Table 2.3 produces values of h and cos(h) for several reference values.

Table 2.3 . The values of h and cos(h) for several reference values.

In the northern hemisphere, the cosine of π equalling minus one corresponds with the sun being south of an observer at solar noon.

If we are to apply this relation using clock time, then we define h as the fraction of 2π that the earth has turned after local solar noon:

In this instance, t is local time and t0 is solar time.

lc is the longitudinal correction. Its value is 4 minutes for each degree of longitude east of the standard meridian and is −4 minutes for each degree of longitude west of the standard meridian. Standard meridians possess 15 degree increments from the prime meridian.

The longitude of the Berkeley campus is: 122° W 15′ 47″ latitude is 37° N 52′ 24″. It is about 2.25° west of the standard meridian (120° W), so solar noon occurs about 9 minutes after noon Pacific Standard Time (PST).

The variable, et, represents the equation of time ( Fig. 2.16 ). It arises because of the eccentricity of the earth’s orbit around the sun and the obliquity (tilt) of the ecliptic (the great circle that represents the earth’s path around the sun). The eccentricity of the earth orbit around the sun forces the angular rotation rate to be variable this allows equal areas to be swept with time during the earth’s orbit. The equation of time (units of hours) is:

Figure 2.16 . Computations based on the equation of time

d is day number (1 on January 1 and 365 on December 31). Here et and f have units of radians.

The contribution from variation in the orbital speed of the earth to the computation of the sun’s zenith angle is not trivial. On February 9, the increment due to the equation of time is 14.2 minutes and on November 6 the contribution is 16.3 minutes.

Sometimes these equations are expressed in terms of Universal Time. In this case, the universal time (Greenwich) is the sum of the true solar time plus and equation for time and the longitudinal time (15 degrees per hour West of Greenwich and −15 degrees per hour East of Greenwich).

## The Equation of Time

Against the backdrop of the distant stars, the Earth has two simple and regular motions. It turns on its axis once every 23 hours, 56 minutes, and 4.1005 seconds, and it orbits the Sun once every 365 days, 6 hours, 9 minutes, and 9.7676 seconds (these figures being those which applied at the start of the year 2000).

The usual day that we're used to, of course, is 24 hours in length, and the usual year that we're used to is slightly less than 365 days and 6 hours, since in the calendar, the leap year day in one out of every four years is omitted three times in every 400 years.

The latter discrepancy is due to the precession of the equinoxes the natural year of the seasons, relative to the direction the Earth's axis is pointing, differs from the sidereal year because that direction slowly changes. Thus, the tropical year is 365 days, 5 hours, 48 minutes, and 5.1875 seconds.

The former discrepancy is one that confuses many people when they see introductory books about astronomy.

After all, it's because the Earth turns on its axis that we have day and night. Our clocks are made to indicate the same time of day after 24 hours. So if the Earth didn't turn to bring the place where one lives to face the Sun in 24 hours, but after a shorter period of time, those extra four minutes each day would add up.

If you do add them up, though, you can see what is going on. Four minutes a day add up to an hour in 15 days, so they add up to two hours in a month, or 24 hours in a year.

So, if our clocks were wrong by four minutes a day, in six months we would be having lunch at the middle of the night!

And that's a clue to what's going on. In six months, since the Earth goes around the Sun once a year, we will be on the opposite side of the Sun. If we don't think of the Earth's rotation in terms of rotation relative to which direction the sun is in, but just its rotation by itself, then we get a period of rotation that is different, by four minutes a day, from the regular clock day of 24 hours.

The 23 hour, 56 minute, and 4.09 second axial rotation period is almost exactly uniform seasonal wind patterns actually affect the Earth's rotation to an extent now detectable with today's atomic clocks this is known as the UT1 - UT2 correction.

The 24 hour day that we normally use to regulate our lives by is uniform too, but that's because we get it from clocks. Of course, that doesn't count those two occasions on the weekend when we "spring ahead" and "fall behind", to adjust those clocks for Daylight Saving Time. But the actual relationship of locations on the Earth to the direction of the Sun is more complicated, and so the time indicated by a sundial differs by clock time in a pattern over the course of a year known as the Equation of Time.

To be complete, we might also consider the anomalistic year the position of the Earth's aphelion also shifts very slowly, leading to the period between two consecutive times at which the Earth is farthest from the Sun being 365 days, 6 hours, 13 minutes, and 52.53865 seconds. When calculating the Earth's position using Kepler's Equation, it is the time within this year that is used as the starting point, to obtain a variable known as the mean anomaly.

Given the figures above, in 24 hours, the Earth turns on its axis by 360.98560556 degrees, and, on the average, the Earth moves 0.98560927 degrees in its orbit around the Sun. If you subtract the second figure from the first, you get 359.99999629 rather than exactly 360 degrees this corresponds to the fact that we currently use, for timekeeping, a second whose length is based on astronomical observations made from the years 1750 to 1892, and the Earth's rotation is gradually being slowed by the tides. (Since the references I could find simply had the day about one millisecond too long in the year 2000, by averaging out the bumps in a graph of the difference between mean solar time and atomic times, I took the Earth as losing 10.6 seconds in every 20 years at that time, which makes the day 1.45 milliseconds too long, and used that as my starting point in calculating the sidereal sidereal day, as opposed to the more commonly-seen tropical sidereal day.)

At this moment, of course, it is now both where you are and everywhere else on Earth. But the time that is now is still called by different names in different places.

Because the Earth is round,

while the actual time is the same everywhere, the time of day is not one part of the world is under the rays of the noonday Sun, and another is beneath the starry skies of night.

This picture represents the Earth as it would be at 4:07 PM UTC, or 12:07 PM Atlantic Standard Time, on March 21, the date of the Vernal Equinox.

Thus, the division between night and day is shown as corresponding to a great-circle pair of meridians of longitude over the central meridian for Atlantic Standard Time, 60° West longitude, the sun's rays fall to the Earth vertically on the Equator, and elsewhere on that meridian, the solar time is also exactly noon. On that date, though, the clock time happens to be running seven minutes later than the solar time, due to the Equation of Time which we will be discussing here.

So, the Sun establishes, by means of the angle between the line from a spot on the Earth to the Earth's center, as projected to the plane of the Equator, and the line from the Earth to the Sun, a framework that gives the time of day everywhere on the Earth.

A simple way to understand why the Earth's axial rotation period is only 23 hours, 56 minutes, and some 4 seconds when the day is 24 hours long is to think of the Earth, in its rotation, as being like the hour hand of a 24 hour clock,

which has to move fast, because the dial is also rotating slowly, moving two hours ahead every month.

The hour hand is the line between any given location and the center of the Earth, and the dial is the framework of local times within which the Earth rotates established by the light coming from the Sun.

And, of course, both the dial and the hour hand are rotating counter-clockwise.

The 23 hour, 56 minute, and 4 second axial rotation period is also sometimes referred to as the sidereal day, even though it doesn't represent the amount of time between day and night. This is because, for any one evening of astronomical observation, the annual motion of the Earth can almost be neglected the stars in the sky appear to move around an observer once in the course of a day, just as the Sun appears to do. This meant that it was convenient, in drawing maps of the stars in the sky, to express the coordinate known as Right Ascension, which corresponds to longitude in a map of the Earth, in terms of hours and minutes instead of degrees. Since the 23 hour, 56 minute, and 4 second period is the time required for the stars to move through 24 hours worth of Right Ascension, it seemed natural to refer to the Earth's axial rotation period as a sidereal day, even if the term has some potential to be confusing.

The Equation of Time, which gives the discrepancy between the time as we see it from clocks (exclusive, of course, of our doing such strange things with them as resetting them for what is variously known as Daylight Savings Time in North America or Summer Time in the United Kingdom, and of the correction required because we may not live precisely on the central meridian of our Standard Time Zone) and that on a sundial indicates a discrepancy which can be as much as a quarter of an hour at some times of the year.

Given, as noted above, that the Earth turns regularly on its axis, to a very high degree of precision, as indicated by transit circle observations of stars passing overhead (very minor discrepancies can now be detected by today's very accurate atomic clocks), the only way we can account for this is that, in the course of the year, the Sun is not quite in the direction from the Earth that we might expect on the assumption that the Earth's orbit around the Sun is a perfect circle (plus one other hidden assumption that I will mention later).

By Kepler's laws, the Earth's orbit around the Sun is an ellipse. It is an ellipse that is very nearly a perfect circle only the planets Venus and Uranus have less eccentric orbits than that of the Earth. Also, the Earth's motion sweeps out equal areas in equal times within that ellipse by Kepler's laws.

The effect of that on the position of the Sun in the sky would be the following: on or about July 4th, when the Earth is the farthest from the Sun, and six months later, when the Earth is the closest to the Sun, a sundial would give the correct time. In the fall, the Earth would not have moved in its orbit around the Sun from aphelion by the angle that one might expect, and so the apparent solar days will have been shorter, being more sidereal-like, and the sundial would run ahead and in the spring, the sundial would be late.

The diagram below illustrates this phenomenon:

it is drawn to scale. The Earth's orbit appears to be a circle, not an ellipse, in the diagram, because the extent to which it is flattened is only about one part in 7,000, which is too small to show on this scale. But the ratio of perihelion to aphelion distances for an orbit of the eccentricity of that of the Earth is about as 261 to 265, and this, the extent to which the Sun is not at the center of the orbit, can be shown. And, because of that, the effect of the law that equal areas are swept out in equal times can also be shown to scale on the diagram.

Thus, the orbit is divided into twelve segments of equal duration, and the areas they sweep out are colored alternately green and yellow the dark red lines within those colored areas indicate the direction in which the Sun would be expected to lie if the Earth moved uniformly in a circular orbit with the Sun at the center.

Incidentally, it may be noted that the lines showing where the Sun might be expected to be found, which had to be drawn quite long so that it was clear how they differed from the lines from the Earth to the Sun, all seem to be pointing to a single point to the left of the Sun in the diagram.

There is a reason for that.

As noted, in shape, the difference between the Earth's orbit and a circle is very small, but the distance of the Sun from the center of the orbit was the noticeable result of the orbit being an ellipse. In such a case,

if I wish to draw a line from the Sun enclosing a certain fraction of the area between that line and the line of perihelion, one way to do it is to draw a line first from the center of the circle enclosing that fraction of its area.

Then, if I surrender the area given by the green triangle, and take instead the area given by the red triangle, by drawing a line from the Sun to the circle of the orbit which crosses the line from the center enclosing the fraction of the area desired, since the two triangles are about equal in area, I closely approximate the line to the Earth's actual position.

The line from the center of the circle, since it enclosed the desired fraction of the circle, has a slope representing the fraction of the year that has passed from perihelion it moves at a uniform angular speed. If one takes a line in a similar direction from the Earth, it then will point to a point as far to the left of the center as the Sun is to the right of the center, or to the other focus of its elliptical orbit.

So the elliptical motion of the Earth around the Sun is approximated by movement in a circle around a center halfway between the two foci of the ellipse which is uniform in angle as viewed from the other focus of the ellipse. This approximation, with the other focus of the ellipse called the equant, is how Ptolemy described the apparent annual motion of the Sun, and how the elliptical orbits of the planets, turned into scaled circular orbits around the Earth, or deferents, were also described, an epicycle then giving the displacement due to the fact that the other planets, and the Earth, actually orbit the Sun.

Kepler's Laws, with the ellipse being determined by the Sun at a focus, and with areas swept about by a line from the planet to the Sun, base orbital motions on the Sun directly this approximation seems to give two imaginary points, the center of the orbit, and the other focus of the ellipse, the primary role in determining a body's motions, and thus many histories of astronomy have denounced the ancients for settling for something so unaesthetic. However, without such evidence as Tycho Brahe's and Kepler's accurate observations of Mars, it is unclear how one would have expected them to do better purely through artistic imagination, even if the boldness of Einstein seems to provide the example.

But the actual Equation of Time, as one can see it graphed in many references, has two large bumps and two smaller ones in the course of a year. So there must be something else going on.

The only other thing about the Earth's orbit that comes to mind is that the Earth's axis is tilted by about 23.44 degrees, relative to its orbit around the Sun. We all know how that causes the seasons. But what would that have to do with the time of day?

If you look at a diagram of the Earth's seasons, such as you might find in many atlases, you might indeed wonder. The Earth is usually pictured in such a diagram at the two solstices and the two equinoxes, and it is apparent that at these four times of the year, the Sun would cross one's meridian in the sky at precisely noon local mean solar time if the Earth's orbit around the Sun were circular.

One has to look at the effects of the Earth's tilt a bit differently to see that, yes, indeed it does affect the relationship of the Earth to the Sun in such a way as to shift the time we would see on a sundial.

If we looked down on the Earth and the Sun from the vantage point of the star Polaris, then, had the Earth's orbit around the Sun been a perfect circle, what we would see instead (through a powerful telescope!) would be a foreshortened ellipse with the Sun at the center.

The difference between these two views of the Earth's orbit is illustrated by the diagram below, which shows a representation of the Earth's tilt causing the seasons in a conventional form, and then the same diagram so that the Earth's axis is shown as vertical, and the Ecliptic plane is tilted, as better illustrates the Equation of Time:

The Earth would be closest to the Sun at the two solstices, when it is highest and lowest relative to the Sun in the Earth's equatorial plane, and farthest from the Sun at the two equinoxes. Between these four phenomena, the year is divided into four even quarters, and the angle at which the Earth is seen from the Sun differs by exactly 90 degrees from one of them to the next.

But because it is a flattened ellipse, halfway between one solstice and the following equinox, or one equinox and the following solstice, instead of having a nice 45 degree angle, the Earth is closer to its equinoctial position than one might expect, as illustrated by the diagram below:

This leads to a discrepancy where the sundial is ahead twice a year, and behind twice a year, and in both cases by equal amounts. (Incidentally, the diagram above is drawn to scale, and, thus, as the angular discrepancy between the actual direction of the Sun, and the direction in which we would expect to find it is readily visible in the diagram, the error in the time on a sundial due to this cause is also genuinely noticeable.)

The combination of this discrepancy and the one due to the Earth's elliptical orbit and Kepler's Laws gives the characteristic form to the Equation of Time with which we are familiar, as shown in the diagram to the right.

The dark green curve in the top graph shows the effect due to the Earth's orbit being elliptical. The vertical green lines that cross it show the days of perihelion and aphelion, when this effect is zero, and the days halfway between them.

The dark red curve in the middle graph shows the effect due to the tilting of the Earth's axis. The vertical red lines that cross it give the dates of the equinoxes and solstices, when this effect is zero.

The blue curve in the bottom graph shows the actual Equation of Time, resulting from the combination of these effects. It is crossed by vertical dark blue-green lines, giving the dates when the sundial and the clock coincide and the greater and lesser maximae and minimae of this equation.

Looking at this graph at the right, we now see why there had to be the slight extra complication in the first diagram on this page of having all the clocks show a time seven minutes after the hour.

After all, it shows the dividing line between night and day exactly crossing the Earth's poles. This means that it illustrates the Earth at one of the equinoxes. If it were at the vernal equinox, at about March 21, then instead of pointing straight up, the minute hands of all the little clocks in the diagrams ought to be pointing to about seven minutes after the hour, because on that date, the sundial is about seven minutes slow compared to the clock. If it were at the autumnal equinox, at about September 23, from the graph you can see that the sundial is running seven minutes fast, so the minute hands all ought to be at seven minutes before the hour.

## Astronomy 1021

Use these flashcards to help memorize information. Look at the large card and try to recall what is on the other side. Then click the card to flip it. If you knew the answer, click the green Know box. Otherwise, click the red Don't know box.

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