Astronomy 1L (Lab Class) Information
Summary of Altitude-Azimuth Project (and Brief Summary of Navigation project)
Normal Value of Each Project:
     60 points for the Alt-Azimuth project, 25 points for the Navigation project, depending on the number of observations, and the quality of the results. The normal credit for these projects is based on hand-and-eye observations. If you have/buy/make tools which allow you to obtain significantly better results, you can receive extra credit.

Observations Required:
     Observe the altitudes and azimuths of various stars at various times: For each observation, you must record:
     (a) the name of the star
     (b) the observed altitude and azimuth of the star
     (c) the date and time (including type of time used). For some observations, approximate times may be used; for others, you need to record the time to the nearest minute.
     You may make the observations at any convenient location; but if you make observations at locations which are many miles apart, you may have to handle them separately when you do your report.

Motive for Altitude-Azimuth Project:
     To force you to learn how to identify, at a minimum, the stars and constellations pointed out in class.

Summary of Altitude-Azimuth Project:
     You will be given a list of "standard" stars, which cover a wide range of declinations. If you are unable to observe some of these, you may add other stars. Practice making observations until you can easily identify each star, and can make observations of all the stars in just a few minutes. You have to make many sets of observations, and if it takes a long time for one set, it will be difficult to do very many observations.
     The ideal project would involve making a complete set of observations every 30 to 40 minutes throughout the night, from dusk to dawn. If you cannot observe throughout a whole night, it would be just as good to make observations from dusk to midnight one night, and from midnight to dawn another night, providing there are no more than a few days between the sets of observations.

Summary of Celestial Navigation Project:
     Make at least a dozen observations of stellar transits. A stellar transit is the passage of a star across the Meridian, at the highest point in its diurnal path, directly north or south of the zenith. It does not make any difference when you make the observations; you may make them over a period of weeks, or all on a single night (e.g., while you are doing the Altitude-Azimuth Project). You may use any stars that you want, but they should cover a wide range of declinations, so that half of them pass to the north of the zenith, and half pass to the south. You will have to learn how to use the text to identify the stars that you observe. It is presumed that you will do the observations for the navigation project at the same time as those for the altitude-azimuth project, so you may/should put the observations on the same sheet. Just circle any observations, whether for the alt-azimuth project or the navigation project, which have azimuths of exactly zero or 180 degrees, so that you can easily find them, later.
     The analysis of transit observations is fairly simple (if it weren't, it wouldn't be a required project). For extra credit, you may also analyze non-transit observations (e.g., from the Altitude-Azimuth Project), using spherical trigonometry, but it has been several years since anyone asked me how to do it, so I may need some time to look up the paperwork; so if you'd like to do that, let me know as far in advance as possible.

Working in Small Groups:
     To encourage teamwork and cooperation, you may work with one or two partners. I would also have no objections to having several groups work together to identify stars, and help each other with problems, but each group would be expected to do their own observations and measurements for the purposes of turning in a report. Because it should be easier and more enjoyable for a group to do the observations than it would be for a single person, persons working in groups are expected to do extra observations in order to receive the same grade as an individual would. Two people working together have to do half again as many observations as a single person would, and three people working together have to do twice as many observations as a single person would, to receive a given grade.

Learning the Stars:
     You will have to pay careful attention when stars are pointed out in class, and learn to use the maps in the text to identify the stars when you do not have the instructor to point them out. Practice identifying the stars as often as you can between classes, so that you can recognize them when they are pointed out again. If there are stars which give you trouble, ask to have them pointed out again. If you have trouble identifying the stars using the maps in the text, ask to have the procedure for identifying them covered again. If you do not ask for help, the instructor will not know that you need it.

A More Detailed Discussion of the Altitude-Azimuth Project

Making Observations:

     Make a list of the stars which you are expected to observe, including any extra stars, or planets, which you plan to add to the list. Add columns for the times that you observe the stars, and their altitudes and azimuths, as shown below. Depending upon how wide you make the columns, you should be able to show two or more sets of observations on a page, as shown below (for the purpose of this project, a 'set' of observations is defined as observing all of the stars on the list once):
Name of Star Time Observed Altitude Azimuth Time Observed Altitude Azimuth
Polaris 8:17 pm PST 32 0 8:43 pm PST 36 0
Kochab   25 345   22 346
Caph   65 25   67 25
Eltanin   45 315   43 316
Fomalhaut (8:32 pm PST) 25 205   23 208
Deneb Kaitos   27 160   27 162
Enif 8:25 pm PST 65 190 8:51 pm PST 65 192
    You do not need to note the time for each star. It is presumed that you will list the stars in the order that you plan to observe them, start at the top of the list, and work your way down. You should note the time that you observe the first star, and about every 10 minutes or so, note the time that you observe whatever star you are observing at that time. You can then draw an arrow from the first time to the next one to indicate that all the in-between stars were observed at in-between times. If one of the stars was not observed at all during that time, leave its altitude and azimuth blank; but if you simply observe it "out of turn", put parentheses around the time (as shown for the first observation of Fomalhaut, above), to indicate that although it was observed out of order, the other stars were not. You should write your name(s) in the upper right corner of each data sheet, so that if you misplace the sheet there is some hope that it might be returned to you. You should also write the date that the observations were made at the head of the table, as it may be critical to analyzing them. Any sheet of paper that you use for this class should show your name and the date you made the observations recorded on the sheet. When you reach the end of a set, even if you recorded the time recently, you should record the current time; then, when you start the next set, record the starting time. If you do not take a break between sets of observations, this should mean that the last time for one set and the first time for the next set should be almost identical.
     Even though you should be able to get 2 or 3 sets of observations on a single sheet, you will be making many sets if you do a full night of observation. After making a table for recording observations on a single page, make photocopies for the other pages, so that you always have the stars in the same order. This will make it much easier for you to record your observations, and much easier for me to examine them.

Ideal and Reasonable Project Goals:
     An ideal project would involve observing the stars over and over again, as quickly as you can manage, starting as soon after dusk as you can see them, and ending as close to dawn as you can see them. However, it is not necessary to do this much work to complete a satisfactory project. Standard credit of 60 points is based on observing all of the stars which are on your list and are up, and observable, at any given time, for about 6 hours, and completing one set of observations every 30 or 40 minutes. If you do more work, you can receive extra credit, up to a maximum of 90 points, and if you do less work, you can receive partial credit. If the sky clouds up, or you have other problems, and you have to quit early, you can continue the project on another night; or you can quit, and take partial credit.
     Please note that because small groups are supposed to do extra work, two people working together need to complete each set in 20 to 30 minutes, and three people working together, in 15 to 20 minutes. Doing observations at this faster rate would completely take care of the 'extra' work required for a grade comparable to a single person working alone.

Measuring Altitude:
     The altitude of a star is a number, between 0 and 90, representing the height of the star above the Horizon, an imaginary circle which goes all around your horizon, or skyline, and can best be thought of as representing your 'eye level', extended out into space. Stars which are low in the sky have an altitude close to zero, and stars which are nearly overhead have an altitude close to 90 degrees. Since you are used to dealing with base 10 numbers, some of you may at first forget this, and tend to measure stars which are nearly overhead as being nearly 100 degrees from the horizon. Please try not to make this mistake; there are only 90 degrees from the horizon to the Zenith, the overhead point, and altitudes can never be more than 90 degrees. You can estimate altitudes by extending your right arm in front of you, at eye level, with the hand bent toward you, and turned so that the palm and fingers are extended horizontally to your left. With the thumb held next to the fingers, so that all five digits are close together, the width of the hand is about 10 degrees. If you have a narrow hand, or a long arm, your hand may only cover 9 degrees, and a wide hand, or a short arm, may cause your hand to cover 11 degrees; but there isn't generally much more variation than this. As a check, you can look to the North, and find Polaris. It has an altitude of 34 degrees, because the Pole is always the same altitude as your latitude, and we are at almost 34 degrees North latitude. If you hold your hand so that the bottom is on the Horizon, and then move it up so that the bottom just touches where the top used to be, then do this twice more, the top of your hand should be around 30 degrees above the Northern horizon. If it is, then Polaris will appear to be another half-hand-width above your hand. If it is much further than this, it indicates that your hand is a little narrow, and if it is not that far, it indicates that your hand is a little wide (alternatively, it could be indicating that you weren't very careful when you moved your hand, but that is another matter).
     Using your hand to measure angles in this way is fairly accurate, as long as you only have to measure one or two hand-widths; but if you have to move your hand several times, you may have large errors, because each time you move it, any positioning error, and any error in the width of your hand, is added to whatever error you already had. If you were to use this method to measure the altitudes of stars which are nearly overhead, you might end up with errors of 10 degrees or more. For this reason, it is best to imagine dividing the distance between the Zenith and the Horizon into halves, or thirds, by pointing at the overhead point or Horizon, then moving up or down by what you think is an appropriate distance. If you try to cut the sky in half (so that you are pointing upwards at a 45 degree angle), then the angle above your hand, extending up to the Zenith, should look about the same as the angle below it, extending down to the Horizon. If you try to cut it in thirds, and move up or down two out of the three steps required to do so, the remaining distance should look like it is about half of the first two steps, and about the same as the last one. If you can learn to more-or-less accurately cut the sky into halves or thirds, you should never have to measure up or down from one of the cut-points by more than one or two hand-widths, in order to subdivide it into even smaller angles.
     With some practice dividing the sky into large angles, then using hand-widths to measure smaller angles, you should be able to estimate altitudes to within 1 or 2 degrees accuracy when objects are low in the sky, and to within 5 degrees accuracy when they are higher. But there are a couple of additional things which you need to be aware of. First, the Zenith, which is the overhead point, or 90 degrees altitude, is somewhat higher than people sometimes realize. It is not unusual for sometime to point 'at' the Zenith, and to be actually pointing as much as 10 or 15 degrees below it. To make sure that you are not doing that, when you try to see where the Zenith is, point your arm in that direction, then turn half-way around, making sure not to change the position of your arm relative to your body. If you were really pointing straight up, you will still be pointing in the same direction; whereas if you were not, you will be pointing in a completely different direction. If that is the case, then the true Zenith is halfway between the two different directions that you were pointing. Please note that the stars move across the sky; if there happens to be a convenient pattern of stars which marks the Zenith when you do this measurement, in a few minutes, the stars will have moved a little, and they will not mark the Zenith in quite the same way. Because of this, if you are not sure where the Zenith is, you will have to check its position frequently.
     The other problem is that the size of your hand appears to change according to how you are holding your arm. If you hold your arm out in front of you, your head and eyes are approximately even with your shoulder; but if you hold your arm above you, to look at stars which are near the Zenith, your eyes will be considerably closer to your hand, because they are above your shoulder. To take care of this, you would have to bend backwards, so that your arm is still held straight out 'in front' of you; this would not be a terribly comfortable position, so it is awkward to do hand-width measurements for stars which are nearly overhead, unless you have something to lean back on.

Measuring Azimuth:
     Note added October 23, 2010: Azimuth can be measured in one of two ways. One way is to measure either East or West from North, from 0 to 180 degrees, specifying whether the angle is East or West, as part of the value. That way is the one described in most of this discussion. The other way is to measure only from North around to the East, through South and West, all the way round to North. If measured this way, Azimuth does not have a direction, but instead is just an angle between 0 and 360 degrees. (The table which serves as an example of how to do things was revised some time ago, to show 360-degree azimuths, but the discussion here was not updated until today, and I do not have time to revise the discussion below without delaying this post.) Because the planetarium only displays 0 to 360 degree azimuth, that is the way I will discuss the project in class, and the way I would prefer you do the project. However, I will accept projects done with either kind of Azimuth, as long as they are done correctly.
     The azimuth of a star is a number between 0 and 180, representing the position of the point on the Horizon directly below the star. We can imagine drawing an arc in the sky starting at the star, and coming straight down to the Horizon. This arc is vertical, so it is part of the Vertical Circle through the star (the top of the Circle is at the Zenith, and the bottom is at the Nadir, the point directly below you). The place where the Vertical Circle through the star crosses the Horizon is called the Foot of the Vertical Circle (because the Vertical Circle is perpendicular to the Horizon, and right angles are indicated by little square angles which make the intersection look like the legs of a stick man). The Altitude of the star is the distance from the star down the Vertical Circle, to the Foot of the Circle on the Horizon. The Azimuth is the angle measured along the Horizon from the Foot of the Circle to the North Point on the Horizon, which is the point directly below the North Celestial Pole. Note that Azimuth can be measured in two directions, either East or West, from due North, and because of that, you ALWAYS HAVE TO NOTE WHETHER AZIMUTH IS EAST OR WEST (unless, as noted above, you use 360 degree azimuth, instead of East-West azimuth). Altitude is an angle,without a direction (it is always UP relative to the Horizon), so it does not include any indication of direction.
     Azimuth can be measured in the same sort of way as Altitude, using large-angle measurements to divide 90 degree arcs into halves or thirds, and hand measurements to estimate multiples or fractions of 10 degrees. In this case, you would hold your hand out at arm's length, but hold the fingers and palm vertically (in both cases, you can turn the palm toward you or away from you; try both ways, and decide which is more comfortable).
     To decide where North is, find Polaris, then come straight down to the Horizon. To find East, turn a quarter turn, or 90 degrees, to the right. This is 90 degrees East Azimuth. Objects a hand-width to the left of that would be 90 degrees East MINUS 10 degrees, because that puts them closer to due North, or only 80 degrees East, while objects a hand-width to the right would be 90 degrees East PLUS 10 degrees, because that puts them closer to due South. It is not necessary to indicate whether you are looking northeast, or southeast, because the number for the Azimuth specifies that. Things in the northeast would have Azimuths between 0 and 90 East, while things in the southeast would have Azimuths between 90 East and 180. Note that due North is 0, and due South is 180, without any East or West, because you start at North, and can get to 180 by going either way. Because we live in a world with many right angles, it is usually pretty easy for people to estimate due East and due West (which is just the same, except you turn to the left from North to get there), and fairly easy to estimate due South, by turning another quarter turn in the same direction. But whereas Polaris indicates where North is, there are no stars to indicate due South, so you could be slightly off when you turn around from North to face South. I would suggest practicing this, and even trying to turn through the East and West, separately, to see whether you end up facing the same way. With some practice, you can probably find due South to within 1 or 2 degrees accuracy.
     It is also possible to use a compass to estimate directions, but compasses are usually small, and they always have an error in the direction they point, called the declination of the magnetic field. For Long Beach, the error has been in the 11 to 15 degree range for most of the last 40 years (the magnetic pole moves around over time, so its direction changes as well), so compasses usually point around 12 to 13 degrees East of true North in our area. However, electrical lines, whether overhead or underfoot, can generate electromagnetic fields which change the compass direction even more, and sometimes plumbing lines do the same. For all these reasons, even a good compass has to be compared to the point directly below Polaris, and set (good compasses include setting circles specifically for the purpose of making this correction) before they can be used. Unless you have a good compass, which can be set to correct for the magnetic declination, and know how to use it, you can probably make just as accurate measurements of Azimuth without a compass as with one.
     When measuring Azimuth, it is very important to remember that it measures the angle from the North point on the Horizon to the place on the Horizon directly below the star you are looking at. You do NOT measure from the star to the Pole (or Polaris, which is very near the Pole). Doing so will usually yield a very poor result. Be sure to always go from the star DOWN TO THE HORIZON, then measure the Azimuth ALONG the Horizon. This is especially important when measuring the position of stars which are nearly overhead. As discussed below, the Azimuths of stars which are nearly overhead are very difficult to determine by the crude techniques which you will use, and you can get huge errors if you do not measure them correctly.
     A star which is just a degree or so to the East of the Zenith may have an Azimuth of 90 degrees East. When it has moved a degree or two, which only takes a few minutes, it would be a degree or so to the West of the Zenith, and have an Azimuth of 90 degrees West. Just a small error in estimating where the Zenith is could make you think the star is in the East when it is really in the West, or vice-versa. Similarly, you might think the star is North of overhead when it is really South. This means that for stars near the Zenith, the Azimuth could be off by 180 degrees! Fortunately, this is not terribly important. Imagine that you were standing near the Earth's North Pole. If you stepped one way, you could be in the Eastern hemisphere, North of Moscow. Going the other way just one or two steps, you could be in the Western hemisphere, North of Los Angeles. Obviously, it is silly to worry about your longitude when you are near the Pole, because all the longitude meridians come together at the Pole, and the difference between one longitude and another is just a few inches. Similarly, all Vertical Circles come together at the Zenith, and near the Zenith the Azimuths of stars can be very different, even if they are very close together. When we discuss how to analyze your observations, the Azimuths of stars which are nearly overhead won't be very important. In fact, if a star were overhead, it would not have an Azimuth, just as someone at the North Pole of the Earth wouldn't have any particular longitude.

Observing the Stars:
     You will observe the stars one after the other, as quickly but as accurately as you can manage. For some stars, you will write down the time of your observation, but for ALL of them, you will put down the Altitude and Azimuth which you estimate, as accurately as you can manage. In many cases this will not be very accurate, but don't worry about it; just do the best you can, and practice often enough that it is reasonably accurate.

Improving the Quality of Your Results:
     Aside from frequent practice in class and out, you might want to try to improve your accuracy by doing something 'extra'. One way might be to string a suspended wire across your observing site, indicating North and South, so by standing under the wire, you can see Polaris in one direction, and due South in the other direction. Or you might borrow a sextant or theodolite, or other sophisticated measuring tool, so you can measure angles to within a fraction of a degree. If you can do so, you can get extra credit for obtaining unusually good results, providing you explain how you managed the feat. Perhaps you can think up a better way of doing hand-and-eye measurements than I have discussed; if so, explain how you did it, and if it really gave you better results you will get extra credit.
     One method which has been used frequently and successfully to better measure Altitudes is to buy a large protractor (these are available at some stores, such as Art Supply Warehouse, in sizes up to a foot in diameter). Glue a fat drinking straw to the protractor, with the straw going through the 90 and 270 degree marks, and running right through the center of the protractor; then drill a small hole in the center (if there isn't one there already) and hang a weight from the hole (a fishing line and weight works quite nicely), so that the weight hangs just outside the circumference of the protractor. Once this is set up, if you hold the protractor in a vertical plane, so the weight hangs straight down at its side, and point the straw at a star, so you can see the star through the straw, the line holding the weight will indicate the Altitude of the star. With some practice this simple 'sextant' can give you angles accurate to within 1 or 2 degrees at all positions in the sky, and you won't have to worry about getting your local Horizon wrong.
Using a protractor and 'pointer' to make a simple 'sextant' for observing altitudes or zenith distances
Making a Simple "Sextant"
     This illustration (taken from an educational article) shows how you can use a straw, aligned along the straight edge of a semi-circular protractor, as a pointer, and a plumb bob (a string and weight) attached to the CENTER of the protractor, as a measuring aid, to find the zenith distance of an object (as shown, looking horizontally yields a measurement of 90, which is not the altitude, but the zenith distance).

Adding Extra Objects To Your Observing List:
     The stars on your list were chosen because they are among the 50 brightest stars in the sky, and are therefore relatively easy to see, even in our bright skies, and are mostly 10 to 15 degrees apart and scattered across the sky. But there are many stars which might be visible if your sky is not quite so bright, and you are free to add to the list, if you want extra credit, or if there are some stars which you have trouble finding, and would like to "replace". Just make sure that you can correctly identify the stars using the maps in your text, and that they are far enough apart that your measurement errors are only a fraction of the distance between them.
     You can also use planets which happen to be up (as shown on the list of stars). The planets move from night to night, but if all your observations are done in a single night, any motion they have relative to the starry background will be negligible compared to your observational errors. Even if you have to break up your observations over a few nights, in most cases you may still use the planets. Just check the table of planetary longitudes in the back of your text. If the planets you want to use do not move more than 1 or 2 degrees during the time that you observing them, you can use them just as if they were stars. Unfortunately, the Moon moves by several degrees relative to the stars in a single night, so you cannot use the Moon for this project. In fact, you should try to do the project on a night when the Moon is either not up, or just a crescent, because if it is very bright it will make it harder to see some of the stars.

Using a Dark Sky Site:
     If you have trouble seeing the stars in Long Beach you could go someplace with darker skies. For truly dark skies, you have to go a long ways (more than 100 miles, in most directions), but you can find substantially darker skies almost anywhere outside the Los Angeles Basin, and somewhat darker skies at a number of parks and other unlit areas closer to home. One thing you should be aware of, however, is that if you learn to identify stars in our bright skies, then go somewhere with a much darker sky, it is easy to get lost. To use such a site, you need to spend a night or two exploring it with the aid of the text, so you can identify the stars before you commit yourself to a night of actual observation. For this reason I recommend trying to find a site with only slightly-darker skies, rather than a pitch-black site.

Turning in a Report:
     This project is due a week before the Final (in semesters with poor weather I may allow it to be turned in without penalty as late as the night of the Final, but that decision wouldn't be made until a week or two beforehand). By that time you should have completed your work, and turned in a report consisting of the following:
     (a) A Cover Page, stating who did the project, when it was done (date or dates and range of times), where it was done (just a general location is all right, unless you want to recommend the site to future students; in that case, detailed directions, and information on permits and such, would be appreciated), and how it was done (if just standard hand and eye observations, just state that, but if you used special equipment, give a very brief description of the equipment). If you worked in a group, be sure to include the names of all those involved, and a brief description of how the work was divided (e.g., did one person do the observations and the other one record them, did they divide up the sky, did they take turns), and how you want the credit for the report divided (this would normally be divided equally, but if for some reason, the participants want to give one person more or less credit, that is fine with me, providing that everyone agrees to it), and LEGIBLE signatures of the parties involved, to show that they agree with whatever division of credit was stated. Please note that if the extra work required to receive full credit was done, everyone in the group will receive full credit. Dividing the credit means apportioning it, not reducing it. The Cover Page should be as brief as is possible. I know what you were supposed to do, so I do not want some long discussion of things which I already know; all I want to know is the specifics for your particular project. In most cases, the Cover Page should be considerably shorter than this paragraph.
     (b) Your data pages, showing the observations, as you made them, as shown previously. If they are reasonably neat and well-organized, I would rather see your original data pages, and NOT have you waste time copying them again. If, however, they are almost illegible, then you should copy them neatly for your final report; but even in this case I expect you to attach your original data pages to the back of the report.
     (c) A graph summarizing the results (to be covered in more detail below). You are going to turn in, hopefully, hundreds of observations of the stars that you watched. It is not practical for me to go through all of those observations and try to make sense of them, so I require that you create a graph which shows the motions that you observed. Not only does this make it much easier for me to gauge the quality of your observations, it will also allow you to see how you did, and to see that your measurements did indeed show the motion of the stars which you should have observed during the night. Because this graph is so useful and important for summarizing the observations, if you do not do the graph, you will be penalized 40% of the grade that you would have otherwise earned. Most of the work for this project is the effort required to learn the stars, and how to measure their positions; the major effort after that, is to do the observations. In comparison, doing the graph should be very simple; but, even though it is a minor task, because it makes my job far easier, if you do not do it, you are severely penalized.
     All of the pages in the report should be either stapled, or paper-clipped together. It is not necessary, nor even desirable, to use a report cover; that just makes the report bulkier, and costs money that you can use for better purposes. However, remember that EVERY page should have your name and the date on it. For the data pages, the date would be the date that you made the observations on that page; for the Cover Page and the Graph, the date would be the range of dates involved in doing the observations. It is not necessary to indicate the date that the report is turned in, as I will note that on the Cover Page when I receive it.

Drawing the Graph Summarizing Your Results:
     BRIEF NOTES:
     You must plot a graph showing your results. You will be penalized 20 to 40% of the value of your project if you do not plot a graph. (Use the Altitude-Azimuth Project Graph Paper page to print out a sheet of graph paper for this project. Do not use the polar coordinate graph paper for part 3b of the Mercury project for this project.)
     The graph paper has circles at 10 degree intervals, from the horizon (at the outside) to 80 degrees altitude (near the center). The zenith is shown by the intersection of the cross in the center of the paper. The paper also has radial lines every 10 degrees, all the way around the center. Use these lines to label the outside of the graph with numbers from 0 to 350. For this project, it does not matter whether the labels run clockwise or counter-clockwise. Also label the circles from 0 at the outside, to 80 near the center, along two or three radial lines.
     To plot the graph, look at the first star you measured. For each observation of that star, go around the outside of the graph to the correct azimuth, then go in toward the center until you're at the right altitude, and plot a dot. Then do the same with all the other observations of that star.
     After plotting all the dots for one star, draw a smooth curve through the AREA where the dots are plotted. If the observations were perfect, the dots would lie on a smooth curve already, but they will probably have quite a bit of scatter, because your observations won't be perfect; and because of that, we do NOT want to draw a curve showing what the observations were (the dots do that), but a curve that shows what the observations probably should have been (the curve mentioned in the first line of this paragraph)
     Then plot dots showing the observations for the second star on your list. If those are close to the dots for the first star, use a different color pencil or marker (you can get packs of colored pencils or markers for 99.99 cents at the 99 Cents store); if they are not close to the dots for the first star, use the same color pencil. If you have two dots that fall in the same place, for the same star, you don't have to plot the second one; but if you have two dots for different stars, plot the second dot right next to the first dot, so that the different colors allow you to see the scatter of dots for each star. Once you've plotted all the dots for the second star, draw a smooth curve through the AREA of the dots for that star. The two curves should look like parts of concentric not quite circular curves (because you're plotting the celestial sphere on a flat piece of paper, the circular motions of the stars become distorted); they should be more or less parallel to each other, if they are anywhere near each other, and should NOT look like they are going to cross at any point. Also, if they cross the Meridian (the 0 to 180 line through the center of the graph), they should be perpendicular to it where they cross it, and the parts of the curve on either side of the Meridian should be mirror images, relative to that line.
     Keep doing this until you've plotted the dots and drawn the curves for every star. AS YOU DRAW THE SMOOTH CURVE FOR EACH STAR, label it with its name; and be sure to do the dots, curve and name for a star all in the same color, so that they are easily distinguishable from dots, curve and name of other stars in the same vicinity. (Note: Use a black pencil to lightly do the curves at first. Once you've done enough curves to be sure that you know what they should look like, replace the black pencil with the colors corresponding to the dots for each star.)

Example of what the graph might look like, per the discussion above
(small red and blue dots represent good observations, green dots typical observations, large red dots poor observations)
A graph representing observations of the positions and motions of the stars, showing examples of observations of typical quality, poor quality, and exceptional quality