In the discussion of Part 3 you are TOLD that the Earth is at perihelion in early
January, and that the eccentricity of our orbit is about 1.7%. But it is possible to actually DETERMINE this directly from the solar longitudes shown in the Mercury Data Table
Subtract each longitude for the sun from the next one to get a table of angular 'velocities' (so many degrees in each 4-day interval). In doing the subtraction, you will have to use the full, accurate values, for longitude. If you use the rounded-off numbers that you used to label the Solar positions, the gradual changes will be lost in the roundoff errors. Near the start of each year you should find that the sun moves more than 4° 4' in 4 days, but near the middle of the year, it only moves 3° 48'. This difference in speed is due to the fact that when, at the beginning of the year, we are closer to the Sun, we are moving faster than usual, and even if we weren't, being closer would make whatever movement we have seem faster than usual. Conversely, in July we are further from the Sun, moving slower, and our greater distance makes any motion which we have appear slower than usual.
Make a table, showing the longitude differences, then plot a graph, showing how the values gradually increase and decrease with time. The dates can be plotted at any scale you want, but I would recommend just one tenth-inch square per date, so that it is no wider than necessary. Label the dates each inch, which at this scale would be every 40 days. For the change in longitude, find the smallest value, and round down to the nearest whole minute (this will be around 3° 48', as mentioned above). Label the longitude axis so that there are SIX tenth-inch squares per minute of arc, so that each horizontal line represents a multiple of ten minutes of arc (so, the bottom line would be 3° 48' 00", the next one up 3° 48' 10", and so on). If you use an 8 1/2 by 11 sheet of graph paper, you should run out of room at the top of the paper a little after you reach the highest value, 4° 4' and some seconds, so the entire graph should fit without having to go off the top or bottom of the page (although you will need to tape two sheets together to accommodate all 134 differences).
After you have finished the plot, draw a smooth curve through the dots (in they do not form a smooth curve, it probably means that you made an error in the subtraction, such as borrowing 100 seconds of arc per minute, instead of 60). The graph will form a smoothly falling, then rising, then falling curve, as the Earth moves away from the Sun, back toward it, then away again, during the year and a half of data. Estimate the highest and lowest values for the motion, in degrees, minutes and seconds of arc, and the dates of those high and low values. The date of the highest value is the perihelion date for the Earth's orbit, the date of the lowest value is the aphelion date, and the high and low values can be used to estimate the change in our distance from the Sun, in percentage terms, which is called the eccentricity of our orbit. Write the high and low values, and the aphelion and perihelion dates, near the high and low points on the graph.
To find the eccentricity, convert the degrees, minutes and seconds for the high and low motion values to either degrees and decimal degrees (by dividing by 60's as necessary to do so), or to seconds of arc (by multiplying 60's as necessary to do so). Write the resulting values next to the original values, then take the average of the two values, and write it near the middle of the graph, and draw a horizontal line at that position, all the way across the graph. This represents the average motion of the Earth in four days (a little less than 4 degrees, since it takes 365 1/4 days to move 360 degrees around our orbit). Subtract the average from the high value, and the low value from the average (both values should be the same, if the arithmetic is done correctly, so doing both is just a check of your arithmetic), and write this number down, below the average, as the variation in motion. Divide that by the average, to find the percentage variation, then divided that by two, to find the eccentricity (the percentage variation is twice as big as the eccentricity, because being closer to the Sun makes us go faster by about the same amount as the eccentricity, in percentage terms, and being closer also makes any motion we have LOOK bigger by the same percentage, so the change in distance figures twice into the calculation). Write this value on your graph as well, labeling it appropriately. If everything was done correctly, the date of perihelion and eccentricity will be close to the values discussed above, and you can turn in the extra credit project. If there is some problem, have me look at your work, so that I can try to find out where you went wrong.