Note: Although I am no longer teaching, and this project is no longer being used at Long Beach City College, I have had a number of inquiries expressing interest in the project, either for a class, or as an individual project. For that reason, I should note the following:
(1) The pages posted for this project are mostly self-explanatory, but some of them are not as thorough as others, as I covered much of the background material in class, and it didn't seem necessary to include all of that on the website. If you are trying to do this project on your own or as a class project and find that the online discussion isn't clear enough, please email firstname.lastname@example.org with any questions about the project.
(2) The project can be done by taping together a few sheets of 11 x 14 graph paper (ruled 10 squares to the inch, and accented on the inch), but is much easier to do with a 1 foot by 12 foot roll, as discussed further down on this page.
(3) When I created and first assigned this project in 1983, personal computers were in their infancy, printers capable of doing the project to sufficient accuracy cost tens of thousands of dollars, and the hand techniques described here were the only practical way to do such projects, particularly for students unaccustomed to scientific methods. However, in today's world much of the handwork would be considered antiquated busywork, and the project could be done far more efficiently with relatively inexpensive computers and printers. Whether all students have access to such equipment might be a problem, but altering the project to take advantage of modern technology should be seriously considered by anyone who wishes to adapt it (per the comment in item 4).
(4) I have no objection to anyone using or modifying this project for their own purposes, providing they do so according to Creative Commons rules, so that anyone else can do the same without fear of being sued for copyright violations.
You are to use 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, to study the motion of the Sun and Mercury 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 a series of "retrograde" motions.
In part 1 of the Mercury Project, you are to plot 270 dots (135 for the Sun, and 135 for Mercury) to see the positions that the Sun and Mercury had during this time period (this plot could be done with a computer, but doing it that way makes it difficult to properly do part 2). 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 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 probably need to print 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
For the Sun, the right ascensions start at 18 hours 44 minutes (and some seconds), and steadily increase, by around a quarter hour for each four day period. In late March, when the Sun passes the Vernal Equinox, which is the starting point for measuring right ascension, the right ascension drops from 23 hours and some minutes to 00 hours and some minutes. It's important to remember this when you do the graph discussed on this page. Even though right ascension represents a 24-hour clock, there is no "24 hours and some minutes". Just as with a standard 24-hour clock, the start is at 00 hours, not 01 hours, and the end is at 23 hours and 59 minutes, not 24 hours.
After passing 00 hours, the Sun's right ascension continues to increase, hour after hour, until it reaches 23 hours and some minutes again the following March, when it drops to 00 hours again at the Vernal Equinox. The process then repeats until you get to the end of the data table (which is only half a page away, near the Summer Solstice in late June).
The calendar year is based on the motion of the Sun relative to the Vernal Equinox. Once around, from Equinox to Equinox, is a tropical year
, which is the basis of our calendar. The time involved is 365.2422 days, so we have 365 days in most calendar years, and an occasional leap year to make up for the 0.2422 days per calendar year error. During the time span covered by the three data pages, which is not quite a year and a half, the Sun moves almost one and a half times around the sky, and since once around is 24 hours of right ascension, the total right ascension covered by the Sun is 24 times 1 1/2, or 36 hours (as noted below, in discussing the overall length of the graph).
Final Notes (for now): (1) The motion of Mercury in right ascension is, as for its declination (discussed in detail on the linked page), far more complex than the Sun; but as discussed in class, it is always within the same range of right ascensions as the Sun, so the 36 hours of right ascension required to plot the Sun's positions will also "do" for Mercury. (2) In plotting right ascension, as shown in the graph below and on the practice sheets provided in class, the numbers increase to the left, not the right. That's because, as shown in the Atlas Charts in Chapter 7 of the text, when you look at the sky, if North is on top, East is to the left, and that's the direction right ascension increases. If you plot right ascension increasing to the right, your graph will be backwards compared to the discussion on this site, and you will have a greater chance of becoming confused, and doing this incorrectly. (3) As shown in the image below, and on the practice sheet, you will not start your graph with 00 hours, but with 18 hours. If we were only plotting the motion of the Sun, it would be all right to go from 00 hours to 24 hours (which would be labeled 00 hours again), as the Sun follows the same path every year; but since you are also going to plot the motion of Mercury on the same graph, and Mercury's motion is more complex, you do not
want to plot its motion from one year on top of its motion for another year. Doing that would make the graph harder to use (see the illustrations of Mars' motion in the text to see how messy things can get when you plot several times around the sky on a single graph). Because of this, you want to start at 18 hours at the far right side of the graph and go to the left, past 00 hours (twice), until you reach 06 hours (for the second time), 36 hours of right ascension to the left of where you started. Doing that takes more graph paper, but makes the resulting graph much easier to use.
(The discussion above will be shortened at a later date, and the gory details moved to another page, as was done for the Declinations; but it should serve as a reminder of what was covered in class for those who were here, and give those who were not present a reasonable understanding of what was covered.)
ADDITIONAL TOPICS TO BE COVERED BRIEFLY HERE AND IN MORE DETAIL ON SEPARATE PAGES: Julian Dates, Geocentric Positions, and Ephemeris Time. (Fortunately, an understanding of those topics is not required to do part 1, so we can deal with them at a later date.)
Buying Appropriate Graph Paper
As noted above, the graph you construct will be approximately a foot high and 9 feet long. I used to allow students to tape together a dozen or more sheets of graph paper to create such a graph, but that made things difficult for everyone. 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). LBCC students 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 just under $13 plus tax; but you can get wider and/or longer rolls, cut them into 12 inch wide by at least 12 foot long strips, and share the cost with another student. (I can provide 8 1/2 by 14 inch practice sheets at no charge, but they cannot be used for the actual project, and if you need extra copies I will have to charge you for them; so always do your work in pencil, so you can erase any errors, even on the practice sheets.) Students at other schools who would like to try this project may be able to get appropriate graph paper at local art supply stores, but in most cases will need to buy it online. I will provide links to online sources of appropriate rolls in the next iteration of this page (I will also provide an online .pdf file for the 8 1/2 by 14 practice sheets). In the meantime, refer to the following note
No matter where you purchase a roll, be sure to ask for Clearprint Fade-Out Vellum, 10 x 10 squares to the inch, accented on the inch, and at least 12 feet long; and check to make sure that's what's actually in the package before leaving the store.
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. This conversion is summarized in Chapter 15 of Stars and Planets
, more or less as follows:
Once around the sky, which is 360 degrees, is called 24 hours of right ascension.
Therefore, one hour is equivalent to 360/24, or 15 degrees.
Since one hour is equivalent to 60 minutes, 60 minutes = 15 degrees, or, since 15 is 5 times 3,
1/3 of 60 minutes, or 20 minutes of right ascension = 5 degrees = 1 inch on your graph.
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 increase to the left, instead of to the right.)
Above, an image of part of a sheet of 10x10 graph paper, showing the first nine dots for the Sun more or less accurately plotted. Note that declinations increase UPward 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 about +25 degrees, while the right ascension goes to about 23 hours. On the roll which you should use for the actual project the right ascension will go past 23 hours to 00 hours, all the way around the sky to 00 hours again, and finally end at 6 hours, exactly 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 DOWNward.
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 UPward, 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 UPward, 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 UPward, the minutes of arc also increase UPward, 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 DOWNward for the fifty minutes of arc; while in the case of the positive value, you find the zero degree line, then go UPward for the fifty minutes of arc, just like the whole degrees.
Below the Equator, the degrees and minutes increase DOWNward.
After reaching the Equator, the degrees and minutes increase UPward.
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 the Sun, draw a smooth curve through the dots. For the Sun, drawing the curve should be relatively easy, if the dots are all plotted correctly. The curve must 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. As you will see in part 2, the accuracy of the path traced out by the movement of the Sun (the Ecliptic) is just as important as the individual dots for the Sun. If the dots are unevenly spaced, or it is not possible to draw a perfectly smooth curve through the dots, it means that one or more dots are a little bit "off", and the two or three dots closest to the problem area should be examined. Sometimes it will turn out that just one dot is off, and the others are all right, but it is just as common for two dots to be a little bit off in a way that makes their relative positions look "off" by more than their individual errors; so it is important to check not just the most "obvious" dot, but the ones near it, as well. Note: Sometimes it is easier to spot errors by using the position of the dots to estimate their right ascension and declination and comparing those to the original data table, than the other way round.
Once you have succeeded in correctly plotting the dots for the Sun and drawing a smooth curve through those dots, plot the dots for Mercury. The procedure for plotting the dots is exactly the same as for the Sun, and strictly speaking, you could do the dots for Mercury at the same time as you do the ones for the Sun. But doing the Sun first makes it easier to learn how to correctly plot the dots, because its path is perfectly smooth, and any errors in its apparent motion must be plotting errors. Mercury's motion involves retrograde loops and esses, and crosses the Sun's path in a dozen places; so if Mercury's dots were plotted at the same time as the Sun's, the graph would look more complicated, and it would be harder to spot errors.
As you plot the dots for Mercury, you will find that when it is between retrograde motions the dots are fairly evenly spaced, but as it nears a retrograde loop or ess the dots get closer and closer together, until at the "stationary point" on one end of the retrograde motion, the dots are very close. During the retrograde motion the dots get a little further apart, but then they get closer together as they approach the other "stationary point", at the other end of the retrograde motion. Once out of the loop or ess, the dots gradually get further apart, until they are more evenly spaced than not. Note that although the spacing of Mercury's dots is not as even as those for the Sun, they do get gradually closer together or further apart in a smooth way. You should not have any dots which are close together, then far apart, then close together, all in just a few dots; such an uneven spacing always implies an error in the position of one or more dots.
Since the curve for Mercury crosses the path for the Sun (and, for that matter, its own path) a number of times, there may be places where it is difficult to tell which dot is which. In such places, a letter M placed next to one of Mercury's dots, or an arrow showing the direction of motion of the planet as it moves from one dot to the next may help you keep track of things. It is also perfectly all right to occasionally label one of the Sun's or Mercury's dots with something (such as the corresponding date) which helps you keep track of where you are; but unfortunately, such labels are not likely to be the same as the labels you will need to put next to the dots for part 2, and they should be penciled in lightly, so they can be easily erased later on.
Once you have plotted a number of dots for Mercury you should try to draw a smooth curve through the dots. In general, the more attractive and smooth the curve looks, the more likely it is to be correct (even non-mathematicians tend to draw curves or lines through collections of dots in a way which is more mathematically accurate, the more "attractive" they look). The curve for Mercury does not, however, have to be as accurate as that for the Sun. The better it looks, the easier it is to tell if the dots for Mercury have been correctly plotted; but as long as the curve is good enough to tell that, it has served its purpose. It will not be used in part 2, so the dots for Mercury are more important than the curve through the dots (unlike the situation for the Sun). There are, however, a couple of places where the curves for Mercury are very important:
(1) Where Mercury's curve crosses that of the Sun, it needs to be as smooth and accurate as possible, for reasons to be discussed in part 3a.
(2) Where Mercury's curve crosses that of the Sun, it should diverge from the Sun's path in both directions from the point where the curves cross in a perfectly uniform, smooth way. If the curve for Mercury suddenly dips toward or away from the curve for the Sun, it almost certainly means that either the Sun's path or Mercury's path is incorrect (or perhaps both), and usually, that some of the dots used to plot one or both curves are "off".
In other words, although most of Mercury's curve can be a little bit sloppy and still be "OK", where the Sun and Mercury's paths cross, both paths must be as smooth and attractive as possible, and diverge in as smooth and uniform a way as is possible, given the position of the dots which define the two curves.
Labeling the Dots (After Drawing Curves Through The Dots)
NOTE: The labeling described here is not part of part 1, and does not count for a grade; it just has to be done before you can start part 2.
After the curves are drawn, label at least every second or third dot for Mercury with the Julian Date corresponding to that dot. Only the last four digits of the Julian Date need be shown, and you can leave off the .5 at the end, as long as you remember later on (namely, in part 3a) 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 unusually complex, but every dot should either be labeled or have a labeled dot right next to it, so that if you need to find the date of a dot, you can do so just by looking at its label, or by adding to or subtracting four days from the date next to the nearest labeled dot.
The labels for Mercury's dots should be placed where they will not be confused with the labels for the Sun's dots. Where the two curves are running more or less parallel to each other, their 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 decide where to put the labels (sometimes, it may be necessary to use an arrow to show which label goes with which dot). Don't make the labels so large that they spoil the look of the graph, but don't make them so small that you can't read them (or for that matter, so small that I can't read them).
For the Sun, you must label EVERY dot with the longitude shown in the third column of the data table. However, you do NOT use the number as shown in the table; instead, you should 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 (in other words, doing some addition and subtraction). It is easiest to make those measurements and calculations in degrees and tenths of degrees, as you are hardly likely to be as used to adding and subtracting numbers by 60's, as by 10's and 100's.
Theoretically, to do this conversion you should 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 such circumstances, the seconds are too small to affect the results (as will be discussed in class). As a result, you can just divide the number of minutes by 60, round the result to the nearest tenth of a degree, then add that to the number of degrees, which saves a considerable amount of time and effort. In addition, we can take advantage of the fact that there are only eleven possible results (as shown below), so instead of doing a calculation for every date, we can create a short table showing the number of minutes corresponding to various tenths of a degree, then compare the number of minutes to the table to see how many tenths of a degree should be added to the whole degrees (an easy construction of the table will be demonstrated in class).
|If the number
number of degrees
to the number
of whole degrees.
|0 - 2
3 - 8
9 - 14
15 - 20
21 - 26
27 - 32
33 - 38
39 - 44
45 - 50
51 - 56
57 - 59
As an example, for the first date, where the Sun's longitude is 280 degrees 15 minutes and some seconds, we add .3 degrees (from the table above) for the 15 minutes (and some irrelevant seconds), to get 280.3 degrees. Similarly, for the second date, where the Sun's longitude is 284 degrees 19 minutes and some seconds, we add .3 degrees (from the table above) for the 19 minutes (and some seconds), to get 284.3 degrees. Keep in mind that on dates such as February 14, 1982, where the number of minutes is 57 or more, so that the number of tenths is 1.0 degrees, you round up to a whole number of degrees one larger than the original number (so 324 degrees and 58 minutes rounds up to 325.0 degrees). When the value is some whole number "point zero", be sure to show the "point zero", because if you only show the whole degrees it looks like you forgot to include the tenths.
Next: Mercury Orbit Project, Part 2a
The rounded off number of degrees and tenths of degrees will be used several times in this project. As a result, it is a good idea to pencil in the rounded off values in the data table, so you don't have to recalculate them later.