Introduction
      You will be given a 3-page data table, listing the positions of the Sun and Mercury on 135 dates, separated from each other by 4 days, during a period running from the beginning of 1982 until the middle of 1983. During that time, the Sun moved one and a half times around the Ecliptic, its apparent path among the stars, while Mercury, closely tracking the motion of the Sun, because it is closer to the Sun than we are, and is always within 27 degrees of the Sun in our sky, moved back and forth relative to the Sun.
      In part 1 of the Mercury Project, you are to plot 270 dots (135 for the Sun, and 135 for Mercury), showing the positions that the Sun and Mercury had during this time period. After connecting the dots with the smoothest possible curves, you will label the dots in a way which makes it easy to do the second part of the project, which consists of measuring the position that Mercury had, relative to the Sun, on each of the 135 dates.

Examining the Range of Declinations, and Choosing a Scale for the Graph
      (Note: You will need printed copies of the data pages to follow this discussion.)
      Since, in part 2, you will need to measure the position of Mercury relative to the Sun, you need to give some thought as to how to best plot the positions of the two objects, to ensure that your plot and your measurements are as accurate as possible. The first step in doing this is to look at the numbers which need to be plotted: the right ascension and declination that the Sun and Mercury had on each of the dates.
      A detailed examination of the declinations shows that we need a range of a little more than 50 degrees, from almost + 25 degrees, to a little below - 25 degrees, to accommodate the motion of the Sun and Mercury during this time period. To do the plot, you will use a roll of graph paper 12 inches high and 9 feet long (discussed in more detail, below). To fit the 50 degree declination range into the 12 inch height suggests making each inch correspond to five degrees of declination, which seems convenient, because we can then label the graph in multiples of five degrees.

Choosing a Different Scale for the Graph -- or Not
      Although 5 degrees per inch is a good scale, we could consider using larger or smaller scales for the graph, and if they were equally easy to use, and equally accurate, I would have no objection to your doing so. However, students who make a graph at a smaller scale (e.g., 10 degrees per inch) have a very hard time plotting the individual dots with sufficient accuracy. You need to plot the dots and measure their positions to an accuracy of 1/5 of a degree or better, to obtain good results in the later parts of the project. At a scale of 5 degrees per inch, this requires plotting your dots to within three or four hundredths of an inch accuracy, which most students are able to do, with some practice. At 10 degrees per inch, however, you would have to plot the dots to within one or two hundredths of an inch accuracy, which makes it far harder to achieve satisfactory results.
      Of course, if trying to use a smaller graph yields poorer results, then we might expect that a larger graph would yield better results, and a number of students have tried this. Unfortunately, this hasn't worked out very well, either. The main problem is that your graph is going to be much longer than it is tall, and even at 5 degrees per inch, it will be 9 feet long. Using a larger scale makes the graph unwieldy, and of course makes the cost of the graph paper four times greater, as well. This might be all right if the final results justified the extra effort and expense involved, but none of those students who have tried a larger scale for the graph has obtained results any better than those students who used the standard scale of 5 degrees per inch. I'm not sure why this is the case. It might be that once the scale is large enough to obtain adequate results, obtaining better results in part 1 just doesn't make any difference in later parts of the project. Alternatively, it might be that in larger graphs the dots are further apart, which makes it a little harder to draw smooth curves through them. As you will see in part 2, the curves which you draw through the dots are just as important as the dots themselves, so if having a larger graph were to make it harder to draw smooth curves, any increase in accuracy in plotting the dots would be negated by the decrease in accuracy for the curves. Another important consideration is that since most students use the standard scale of 5 degrees per inch, all of the lecture and website discussion is based on that scale. If you use a non-standard scale, it may be harder for you to understand what you are supposed to do, which could also lead to poorer results. At any rate, it just doesn't seem to be a good idea to plot a graph at larger or smaller scales, so I strongly recommend that everyone use the standard scale of 5 degrees per inch. If you choose to use a different scale, I will not penalize you, providing that you can obtain satisfactory results, but if choosing a different scale for the graph leads to less than satisfactory results, your grade may be less than satisfactory, as well.

Examining the Range of Right Ascensions
     

(FAR MORE TO BE INSERTED HERE, WHEN TIME PERMITS)

Buying Appropriate Graph Paper
      As noted above, the graph you construct will be approximately a foot high and 9 feet long. In the past, I have allowed students to tape a dozen or more sheets of graph paper together, to create such a graph, but that made things difficult. As a result, I now require students to buy a roll of graph paper at least 12 feet long (the extra 3 feet is used in part 2). You can buy such rolls at Lyon Art Supply in Long Beach (420 East 4th St, between Elm and Frontenac, on the south side of the street just east of Long Beach Boulevard), or Art Supply Warehouse in Westminster (6672 Westminster Blvd, on the south side of the street between Knott and the 405 Freeway). These stores always have some rolls in stock, if you know what to ask for (see the next paragraph). To the best of my knowledge, there are no other local stores which carry such rolls. The smallest satisfactory roll available is 18 inches by 15 feet, and runs a little over $10 with tax; but you can get wider and longer rolls, cut them into strips, and share the cost with another student to reduce the cost. In fact, for the last year, my wife has been kind enough to cut very large rolls into 12 inch by 12 foot sections, so that I can provide them to students who want to avoid the inconvenience of obtaining a roll on their own. If you choose to get a roll in this way, you will need to bring $5 to class, to reimburse me for the cost of the materials. I will also provide 11 x 17 sheets for practice, at no charge, to all students, regardless of whether they get the rolls from me, or one of the stores listed above.
      Important Note: If you buy your own roll, be sure to ask for Clearprint Fade-Out Vellum, 10 x 10 squares to the inch, at least 12 feet long; otherwise, the store employee who serves you may sell you the wrong kind of paper.

Labeling the Graph Paper
      You should create your graph at a scale of five degrees per inch. This allows you to label each inch of declination with a multiple of five degrees (as discussed above), and (as discussed below) each inch of right ascension with a multiple of 20 minutes. This makes an hour three inches wide, and as already mentioned, the complete graph nine feet long.
      When you are done with the graph, you will have to make measurements (described in Part 2) which compare the position of the each dot plotted for Mercury to the position of the nearest dot for the Sun. To do this, the scale of the graph must be the same in all directions, or the measurements will be meaningless. Therefore, both the horizontal and vertical axes must be in five-degree increments. For the vertical axis this is no problem, since declination is measured in degrees. However, right ascension, which is plotted horizontally, is measured in time units (as indicated by the h m s notation in the project Data Table). We therefore need to consider how time units of right ascension compare to degrees. Thisconversion is summarized in Chapter 15 of the "textbook", more or less as follows:

      In other words, you should label the graph with 20-minute intervals on the accented inch-lines, in order to have 5 degrees per inch horizontally, as well as vertically. If one vertical accented line is, say, 18 hours and 00 minutes of right ascension, then the inch-line to the left of that would be 18 hours and 20 minutes, the one to the left of that would be 18 hours and 40 minutes, and the one to the left of that 19 hours and 00 minutes, which is the same as 18 hours and 60 minutes. The partially completed graph shown below indicates the way that the graph should be labeled, starting at the rightmost accented vertical line, and working to the left. (It may seem odd that the numbers increase to the left, but a discussion of right ascension, and how it is used, requires that; see the maps in Chapter 7 of your text, particularly Atlas Charts 21 through 32, to see that the numbers for right ascension do increase to the left, instead of to the right.)

Sample of graph paper, showing first nine dots for the Sun. Note that declination increases upwards, at 5 degrees per inch, but right ascension increases to the left, at 20 minutes per inch, or three inches per hour. On the practice sheets provided in class, the vertical axis goes up to +25 degrees, while the right ascension goes to about 23 hours. On the roll which you use for the actual project, the right ascension will go past 2 hours to 00 hours, all the way around the clock to 00 hours again, and finally end at 6 hours, 9 feet to the left of the start.      In the example shown, the declination (vertical) axis is labeled on the left, but on the actual graph, you will start on the right, so you should label the vertical axis on the right. As you move to the left, you will need to relabel the vertical axis every two or three feet, so that you don't have to look half a dozen feet or more to the side to see the declinations.

How to plot the first three dots for the Sun (admittedly crude, but hopefully shows the basic idea)      How to plot the first dot for the Sun. Since the right ascension is 18h 44m 33.62s, find the 18h 40m accented line, then count to the left, at two minutes per square, to find the vertical column between 44 and 46 minutes. The line on the right side of that column represents 44 minutes exactly, or 18h 44m 00s, as shown; the line on the left side of that column represents 46 minutes exactly. Halfway between the two lines, not shown on the graph, but indicated in this diagram, is 18h 45m, or 18h 44m 60s (indicated the first way at the top of the short line, and the second way at the bottom of the short line). 18h 44m 33.62s would be somewhere between 18h 44m 00s and 18h 44m 60s, and you simply have to estimate where that is (keeping in mind that the square is only a tenth of an inch wide, so the 44m area is only a twentieth of an inch wide). The vertical line with the blue and pink dots represents that approximate position.For the first dot, the declination is negative, so the minutes increase downwards.      To plot the declination, note that the value involved, -(South)23 02 xx, is between -20 and -25 degrees. In the diagram above, we start at the accented line for -25 degrees, and count upwards, remembering that since there are only 5 degrees, but 10 squares, for each inch, we need two squares per degree. -23 and -24 degree lines are shown, and labeled on the right, while the -23 1/2 and -24 1/2 degree lines are shown, but not labeled on the right. Once you find the line for the degrees, you count DOWN inside the two squares for the minutes of arc for that degree, if the degrees themselves increase DOWNward (as shown below, if the degrees themselves increased UPwards, you would count UP inside the two squares for the degree, to find the minutes of arc). Tick marks are shown for every ten minutes of arc, starting at -23 00, and ending at -23 60, which is the same as -24 degrees (remember, there are only 60 minutes in a degree, not a hundred, so the half degree is only 30 minutes, as shown, and NOT 50 minutes, which is a frequently made mistake). Once you have an idea where the various minutes are inside the degree, make a horizontal line at the appropriate place for the number of minutes of arc, and plot a dot where that meets the vertical line you drew for the right ascension. The result, shown in red, is the position of the Sun on that date. Do NOT be afraid to put the dot ON a line, if the position calls for it; just do not get it in the wrong area, as shown by the blue dot, corresponding to a different degree.      How the second and third solar dots are plotted. Compare the numbers for the Sun to this diagram, and make sure that you understand why they dots are plotted as they are (the second dot, on the right, is a little too far to the left, but the other one is more or less OK).

Plotting Minutes of Arc UP or DOWN inside Degrees of Arc
     For the first half page of data, you will be plotting negative declinations, which means that after finding the appropriate two-square area for that declination, you will measure the minutes of arc inside each degree DOWNward, the same as the DOWNward increase of the degrees themselves. But starting in late March, after the Sun crosses from South to North of the Equator, the declinations are positive, and once the declinations increase UPwards, the minutes of arc also increase UPwards, inside the degrees. This is shown in the diagram below, for dots at -00 50 and +00 50 (read as minus zero degrees fifty minutes, and plus zero degrees fifty minutes). In the case of the negative value, you find the zero degree line, then go DOWNwards for the fifty minutes of arc; while in the case of the positive value, you find the zero degree line, then go UPwards for the fifty minutes of arc, just like the whole degrees.

Below the Equator, the degrees and minutes increase DOWNwards.
After reaching the Equator, the degrees and minutes increase UPwards.
Dot A is at PLUS 0 degrees, 50 minutes, so you start at 0, and go UP to 50 minutes.
Dot B is at MINUS 0 degrees, 50 minutes, so you start at 0, and go DOWN to 50 minutes.

      When you are checking for errors, keep in mind that the Sun's path should be perfectly smooth, and its 'dots' should be evenly spaced. If you can't draw a perfectly smooth line through the dots, or they appear unevenly spaced, it is an indication that one or more of the dots in that part of the graph are 'off', somehow.
      Also, there are a few places where Mercury's path (which should be done on the same graph as the Sun) crosses the Sun's, and where that happens, you may need to carefully indicate which dots are for Mercury, and which are for the Sun. Sometimes, people use colored pencils to distinguish the two, but colored pencils have waxes which make erasures difficult, so it is best to use ordinary pencils, if possible.

(A large amount of additional material will be inserted here, later)

Drawing Curves Through the Dots
      After you have plotted all the dots for both the Sun and Mercury, tape the pages together neatly (if using individual pages -- refer to discussion in class for how to properly do this), draw a smooth curve through the dots, and label them.
      For the Sun, drawing the curve should be easy, if all the dots are correctly plotted. The curve should be drawn by hand, not with a ruler, so that it is as smooth as possible, and goes exactly through the center of every dot. If doing this does not produce a perfectly smooth curve, or the dots are spaced even a little unevenly, it shows that one or more of the dots are plotted a little "off", and the two or three dots closest to the problem area should be examined, to see if they are in exactly the right position. Usually, problems involve putting a dot in the wrong square, but occasionally, smaller errors are also evident with a close inspection.
      For Mercury, the more complicated motion (retrograde loops and esses) makes it more difficult to draw a smooth curve, but do the best that you can. Make sure that you connect the dots in the right order (in some of the loops, this will require careful comparison to the data table), and that the curves look as nice as possible. In general, the more beautiful the curves look, the more likely they are to be correct. When Mercury is in between retrograde loops, its dots will be fairly far apart, and fairly evenly spaced. As it approaches a loop, the dots will get closer together, and as it leaves one, they will get further apart. If the dots seem to gradually, smoothly, evenly grow closer together or further apart, they are probably correctly plotted. If they have uneven spacing, they are probably incorrect, and should be checked.
      Another way to check the curves is that, in several places, Mercury's curve crosses that of the Sun. Where this happens, the curves should gradually, smoothly come together, then move apart. If there are any sudden changes in the distance between the curves, it indicates a problem, just as it would if there were a bump or kink in an individual curve.

Labeling the Dots (AFTER Drawing Curves Through The Dots)
      After the curves are drawn, label every second or third dot for Mercury with the Julian Date corresponding to that dot. Only the last four digits need be shown, and you can leave off the .5 at the end, so long as you remember, later on, that all of the dates end with a .5. It is not necessary to label every dot for Mercury, although you may find it helpful to do so where its motion is especially complicated, but every dot should either be labeled, or have a labeled dot to its left or right, so that if you need to find the date of a dot, you can do so just by looking at its label, or adding or subtracting four days from the nearest labeled dot.
      The labels should be placed where they will not be confused with labels for the Sun. Where the two curves are running along together, labels should be placed OUTSIDE the space between the two curves, as close to the appropriate dots as possible. Where the curves cross, particularly in a retrograde loop, you may have to think a bit, to figure out where to put labels so that they aren't confusing. Make sure that all the numbers in your labels are easy to read, but don't make the labels so large that they spoil the look of the graph.
      For the Sun, you will need to label EVERY dot with the longitude (shown in the third column of the data table). However, you will NOT use the number as shown in the table. You will round it off to the nearest tenth of a degree. The reason for this is that in part 2 of the project, you will be doing measurements of the difference in position between the dots for Mercury and the nearest dots for the Sun, and comparing that to the Sun's longitude. It is easiest to make those measurements in degrees and tenths of degrees, and to do the comparison without some kind of conversion calculation requires that the dots for the Sun be labeled in degrees and tenths of degrees, as well.
      Theoretically, to do this conversion, you would need to divide the number of seconds in the Sun's longitude by 60, add the result to the number of minutes, divide the sum by 60, then add that to the number of degrees. However, you will not be able to measure distances to better than a tenth of a degree, and under those circumstances, the seconds are too small to affect the results. As a result, you can just divide the number of minutes by 60, round the result off to the nearest tenth of a degree, and then add that to the number of degrees, which is somewhat easier. In addition, we can take advantage of the fact that there are only eleven possible results (shown below), to create a short table, and then compare the number of minutes to the table, to see how many tenths of a degree should be added.