1. Drawing the Orbit of the Earth
You need a sheet of Polar Coordinate graph paper to do this part of the project. To follow this discussion, you can refer to the Polar Coordinate Graph Paper
page, but I will provide a high-quality graph, 12 inches square, for the actual project. This has concentric circles drawn at tenth-inch intervals from the center (the sheet I provide is blown up from the original, but you will still treat the distance from one circle to the next as being one tenth of an inch for the purposes of this discussion), and radial lines extending outwards at intervals as fine as 1 degree (near the outside of the paper), and as coarse as 10 degrees (near the center of the paper). Along the outside of the paper, pick any of the accented radial lines to represent 0 degrees celestial longitude (the direction of the Vernal Equinox, as seen from the Sun), and mark it with a zero, then find the accented radial line 10 degrees counterclockwise from that and mark it with a ten, and continue all the way around the graph until you reach zero degrees again (this should give you numbers every ten degrees, up to 350 degrees). Having done this, the center of the innermost circle (indicated by the intersection of two crossed lines) should be marked as being the position of the Sun. The radial lines now represent various directions from the Sun, and the concentric circles represent circular orbits at various distances from the Sun. (Note: This way of doing things is a scale model of the Solar System only if you were looking at the Solar System from the North side of our orbit. If you were to on the South side, the angles would go clockwise, instead of counterclockwise). Note: For reasons discussed below, numbers from 0 to 180 should be outside the largest accented circle on the graph, while those from 190 to 350 should be inside that circle. See the illustration below, for an example of the angles from 100 through 280.
As for earlier parts of this project, you should use a pencil, not a pen. You will do a lot of drawing, erasing, then redrawing in the last part of this project. If you use ink or make a mistake, you will have to get another sheet of graph paper, which I am willing to provide you (at a price of 25 cents for each extra copy), but will delay you by however long it takes to see me.
Your first step is to draw the orbit of the Earth. On the scale that you will use, the Earth's orbit is essentially circular, but you cannot use one of the printed circles, because they are all centered on the Sun, and the orbit of the Earth has its center offset from the Sun by a fraction of the orbital size called its eccentricity. For our orbit, this offset is about 1.5 million miles, and ignoring it would produce a similar error in parts of Mercury's computed orbit. For the Earth, this error is only 1.7% of our average orbital distance from the Sun, but since Mercury's orbit is smaller than ours, similar errors would be about 5% of the size of its orbit, which is unacceptable.
To find out how to draw in our orbit, you could look up the eccentricity and perihelion date for the Earth's orbit in some reference, or calculate them directly from the project Data Table by doing the extra credit part M3x. However, it is perfectly satisfactory to just take my word for it that the following is correct:
As noted above, the Earth's orbit has an eccentricity of 1.7%, which means that it is 1.5 million miles closer to the Sun at perihelion, and 1.5 million miles further from it at aphelion. To represent this, you need to draw a circle, the same size as the largest accented circle on the graph, which is pushed in (towards the Sun) on the side of the orbit corresponding to perihelion, and pushed out (away from the Sun) on the side of the orbit corresponding to aphelion, by 1.7% of the average size of the circle. If you examine the graph (either the sample, or the actual graph paper), you should find that there are six accented circles, each separated by five circular bands, which completely fit on the graph. This represents thirty circles, in all. 1.7% of 30 is 0.5, so pushing the circle in by half of a band-width on the perihelion side, and out by half of a band-width on the aphelion side, corresponds to the 'offset' of the Earth's orbit, relative to the Sun.
The perihelion date for the Earth is near the beginning of January, at which time the direction of the Earth, as seen from the Sun, is about 100 degrees celestial longitude (how this is determined is discussed below). The aphelion date for the Earth is near the beginning of July, at which time the direction of the Earth as seen from the Sun is about 280 degrees celestial longitude (exactly opposite the direction at perihelion).
Given the previous two paragraphs, what you need to do is as shown in Figure 3b.1, below. The scraggly black curve represents a fairly typical effort to draw a circle the same size as the accented line closest to the curve. Note that it is pushed in by about half a circle's width at the point labeled 100 degrees, and out by about half a circle's width at the point labeled 280 degrees. This corresponds to the Earth being 1.7% closer to the Sun at 100 degrees longitude (perihelion), and 1.7% further from the Sun at 280 degrees longitude (aphelion), than on the average.
This discussion presumes we can approximate an elliptical orbit by simply pushing a circle of the same size off to the side, relative to the Sun. Strictly speaking, this is not true, but the error involved is very small if the eccentricity of the ellipse is not very large. Even for Pluto, which has a 25% eccentricity, the difference between the actual elliptical orbit and a circle shoved off to the side by 25% is only about 1.6%, as shown in the diagram below. The error decreases approximately as the square of the eccentricity, so for the Earth's orbit, which has an eccentricity fifteen times smaller, the error is over 200 times smaller, or less than a hundredth of one percent of the orbital size. On the scale of your graph, this is far less than the width of a pencil line, and can be ignored.
As noted above and shown below, we can approximate the Earth's orbit by starting with a circle centered on the Sun, and pushing it to the side by 1.7% of its radius. The larger the circle is, the more accurate the results will be, so using the graph paper shown below, you should start with the largest accented circle which is fully contained by the sheet. (On the original graph, this was 3 inches in radius and 6 inches in diameter, but on the 12 1/2 inch square sheets I provide, it is nearly twice that size.) According to the discussion above, 1.7% of the 30-circle radius represents half a circle, so as shown in the diagram below, the orbit that you draw will be pushed in half a circle's width on one side of the Sun, and out the same amount on the other side.
If you have a quality compass (the drafting tool, not the device for finding magnetic north), and are familiar with its use, you can spread the legs by an amount equal to the radius of the accented circle you are 'pushing' to the side, then place the fixed leg just to the side of the center of the graph (at the position shown by a red dot in the image below), then swing the compass around, to form a circle. If this is done correctly, the resulting circle will be pushed in half a circle on one side of the Sun, and out half a circle on the other side. Unfortunately, most people do not have such a compass, or are not familiar with its use. As a result, their efforts to draw a circle in this way end up with the new circle being in or out too far or too little, and repeated efforts to draw a correct circle just end up poking a big hole in the paper. As a result, a hand-drawn method is suggested for most students.
To draw the orbit by hand, put a little mark on one side of the Sun, half a circle in from the preprinted accented circle (as shown near 100 degrees, in the figure). Put another mark on the other side of the Sun, half a circle out from the preprinted accented circle (as shown near 280 degrees in the figure). Put marks at positions corresponding to 190 and 10 degrees (190 is shown, but 10 is off the diagram), exactly on the preprinted circle. (If the amount the orbit of the Earth is being pushed to the side was substantial, the marks at 10 and 190 degrees would not be exactly on the preprinted circle, but for the small adjustment being made here, the error is less than the width of the accented circle.) Once these marks are in place, lightly pencil in smooth curves connected the marks. Heading from 190 to 100 degrees, very gradually move inward, so that the space between the preprinted accented circle and your penciled circle very gradually increases, like an extremely skinny crescent. Try not to stay at a constant distance for a while, then suddenly dip inwards (unfortunately, there are places like that in the example, because the curve shown there was drawn with a mouse, and that was the best I could do under the circumstances; a future post will include a graphically constructed curve which looks perfectly smooth). Once you've completed the portion of the circle from 190 to 100 degrees, draw a similar curve from 190 to 280 degrees, but slowly moving outward, instead of inward. Finally, draw similar curves from 280 to 350 and 0 to 100 degrees, so that the 'right-hand' side of the circle is a mirror image of the 'left-hand' side.
Ideally, at this point, you should show me what you have done, so I can verify that you have done the orbit correctly, or if not, explain what you did wrong, and how to correct it. If you don't have time for that, compare your drawing to the figure below. If your drawing is correct, it will look more like the drawing than not.
Plotting the Position of the Earth
Once you've drawn the orbit of the Earth, you need to plot 66 dots, to show where the Earth was in its orbit on the first 66 dates in the Mercury Data Table. Each dot should be plotted exactly on the curve you drew for the Earth's orbit, at an angular position corresponding to the Earth's direction from the Sun on the date the dot represents.
For each date, the Data Table lists the Sun's celestial longitude. This is, in fact, the number you used to label the dots for the Sun when you finished part M1, and for the measurements of Mercury's longitude in part M2. The Sun's longitude, as seen from the Earth, is exactly opposite the Earth's longitude, as seen from the Sun
(that is, each object is looking in the opposite direction, at the other). This means that the angular position you use to plot the position of the Earth on some date is the Sun's celestial longitude, plus or minus 180 degrees. If the Sun's longitude is bigger than 180 degrees, subtract 180 from it to get the Earth's longitude. If the Sun's longitude is less than 180 degrees, add 180 to it to get the Earth's longitude. As an example, on Julian Day 2444970.5 (the first date in the table), the Sun's geocentric (Earth-centered) celestial longitude was 280.3 degrees. Subtracting 180 from that, the Earth must have been at 100.3 degrees, as seen from the Sun. Note that there is a red dot plotted on the 'orbit' of the Earth, below, corresponding to that position. There are also dots shown for the next five dates, at 104.3 (= 284.3 - 180), 108.4 (= 288.4 - 180), 112.5 (= 292.5 - 180), 116.6 (= 296.6 -180) and 120.6 (= 300.6 - 180) degrees. There are also three dots plotted at the positions corresponding to the first three dates after the Sun passed the Vernal Equinox, namely at 181.0, 185.0 and 189.0 degrees (corresponding to solar celestial longitudes of 1.0, 5.0 and 9.0 degrees, plus 180).
In this way, use the first 66 celestial longitudes for the Sun to calculate the corresponding positions of the Earth, and plot dots on the 'orbit' of the Earth that you drew, for each date. Finally, label some of the dots (say, every fifth dot, or 20 days), preferably with the Julian Date when the Earth was in the corresponding position. Again, if there is time to do so, show your work to the instructor before continuing.
|Figure 3b.1 (Drawing the Orbit of the Earth)|
An illustration of how to draw the orbit of the Earth, and plot the dots which show where the Earth is on various dates. Only half the orbit is shown, and nine dots for the Earth (the first six, and the first three which exceed 180 degrees celestial longitude). Students need to draw the whole orbit, and plot the first sixty-six dots for the Earth.
Note that the orbit of the Earth is pushed toward the Sun, half the distance between the printed circles at the 100 degree mark; is on the preprinted circle at the 190 degree mark; and is pushed away from the Sun, half the distance between the printed circles at the 280 degree mark. Also, that the angles run counter-clockwise (otherwise, the discussion for Constructing the Orbit of Mercury would be backwards), and are plotted outside the outermost accented circle up to 180 degrees, and inside it from 190 degrees onward.
2. Constructing the Orbit of Mercury
Look up the elongation which you measured for Mercury on the first date. Center a protractor (a 6 inch diameter protractor is the best size for this, ruled to the half-degree, so that you can estimate angles as fine as a 1/4 of a degree) on the dot which represents the position of the Earth on that date, and align (turn) it so that the 0 degree line on the protractor points exactly at the Sun (that is, the center of the graph). Use the radial lines on the graph to line up the 0 degree mark (its angular position, as measured by those lines, should be the same as the angular position of the dot for the Earth that it is centered on). Do NOT move the protractor so that the 0 degree line touches the Sun, as that moves the center of the protractor away from the dot which represents the Earth's position, and the protractor is only correctly positioned if the 0 degree line points at the Sun, AND its center is on top of the dot for the Earth. Make sure, before continuing, that the protractor is set up that way, so that both its center and its zero-degree line are properly positioned. Having done this makes the angles on the protractor equivalent to elongations. An angle 10 degrees counterclockwise from the zero degree line would represent 10 degrees East elongation, and one 10 degrees clockwise would represent 10 degrees West elongation. If you have a circular protractor, tape small pieces of paper on either side of the zero degree line, so that you can remember which side is East, and which is West. If you have a semi-circular protractor, there will be two zero degree lines. The one which has the angles running to its left (counterclockwise) should be marked with East, and the other one marked with West.
Find the angle on the protractor which corresponds to the elongation of Mercury on the first date and, as accurately as possible, mark a small dot on the paper at the edge of the protractor. Use a straightedge to draw a line exactly through the Earth's dot, and the dot you just marked (note that you may have to hold the straightedge just to the side of the dots, so that the width of the pencil lead doesn't shove it past the dots), and extend it past the Sun, almost all the way to the other side of the Earth's orbit. If you have done everything correctly, Mercury must lie somewhere on this line. Otherwise, it would not be in that direction on that date.
To find exactly where Mercury was on that line, you have to find the next time when Mercury, having completed exactly one orbital period, has moved all the way around its orbit, and is back in exactly the same place. Since the orbital period is 88 days, and your dates are separated by 4 days, this will be on the 23rd date. Although Mercury will now be in the same place that it was on the first date, the Earth is not, as we take a full year to orbit the Sun. Find the dot which represents the position of the Earth on the 23rd date, and align the protractor as before, so that the angles on the protractor represent elongations, then look up the elongation of Mercury which you measured for that date, mark the edge of the protractor with another small dot, and draw another straight line, through the new positions of the Earth and the marker. This will provide a completely different line, which will intersect the first line at a single point. If both lines were correctly drawn, Mercury must have been on each of them on the appropriate dates, but that is only possible if it was at the intersection of the lines, so that intersection must represent Mercury's position on both dates.
Of course, there are many, many ways in which you might have made an error during or prior to this part of the project, so the intersection might not actually be the correct position. To see whether it is, find the position of the Earth, and the elongation of Mercury, on the NEXT date separated from the first two by a multiple of 88 days (this would be, in this case, the 45th date), and draw yet another line. Now, you should have three lines running across the graph, and if they are all correct, they will all meet at exactly the same point. If they do, mark that intersection with a small dot, and with some indicator of the first date that it was there, then erase all three lines except VERY close to the dot. You have to erase the lines, because they are only the first of 66 lines which you will draw, and if you leave them on the paper, it will soon be so filled with a maze of lines that you will not be able to tell what you are doing.
If all three lines do NOT intersect at exactly the same point, their intersection will form a small triangle (if it forms a big triangle, you are doing something very, very wrong). This indicates that one or more of the lines is incorrectly drawn. Reposition the protractor for each of the three dates, and check to make sure that all three lines are drawn according to the Elongations you measured. If this does not reveal the problem, remeasure the Elongation of Mercury for all three dates, and try again. Usually, you will find that one of the three Elongation values was a little off, but sometimes, two or even all three of them need a slight adjustment.
Once you have obtained a good result for the first trio of dates, go back to the second date, and 88 days after that, and 88 days after that, and draw another trio of lines, to find the position which Mercury had 4 days after it was at its first position. Once you have obtained a good result for this date, mark the dot, erase the new lines, and repeat the process, every 4 days, until you reach the 23rd date. Since you already used the data for that date in making the first trio of lines, if you were to do this again for that date, you would obtain the same dot as you did for the first dot, so there is no point in continuing, and you are almost done.
The 22 dots which you have just created represent the position of Mercury every fourth day, throughout its orbit, and should form a perfectly smooth path. If they do, draw a smooth curve exactly through the center of every dot, to represent the orbit of Mercury, and show the result to the instructor. If, however, the path formed by the dots has any bumps or dips, or the spacing of the dots is uneven (they will gradually get further and further apart on the side of the curve which is closer to the Sun, and closer together on the other side of the orbit, but they should NOT be sometimes closer, then further, then closer again), it means that somehow, one or more of the dots, although apparently a good result, actually involved two or more lines being wrong, and just happening to intersect, anyway. In that case, go back to your original graph, remeasure the Elongations of Mercury for the dates involved, and redraw the trio of lines, to find the correct position. Once the graph looks nice and smooth, and the dots look nicely spaced, you can show it to the instructor for grading.
There is one more part to this project, involving measuring this graph to see how well, or badly, it represents the orbit of Mercury, but at the moment, there are no copies available of the handout for that part, so I will have to discuss it in class, or individually, with those students who finish part 3b early enough to get to the last part.
Next: Part 3c / Part 3x
Rough example of how to find the position of Mercury on January 1, 1982. On that date, the Earth is at 100.3 degrees longitude. The green and blue lines extending from that position to the Sun and to the left (counter-clockwise) of the Sun represent the Eastern elongation of Mercury on that date. Eighty-eight days later, the Earth is at 189.0 degrees longitude. The green and blue lines extending from that position to the Sun and to the right (clockwise) of the Sun represent the Western elongation of Mercury on that date. Eighty-eight days later, the Earth is at 274.0 degrees longitude. The green (not shown) and blue lines extending from that position to the Sun and to the right (clockwise) of the Sun represent the Western elongation on that date. If all three lines meet at a single point (as shown), that would be the position that Mercury had on all three dates, since they are separated by an orbital period. (This illustration is only an example of what things look like, in general. The lines are NOT drawn at the correct angles, which would be the angles you measured/calculated for Mercury, in part 2a; nor is their intersection the actual position of Mercury on that date.)