Hours of
Daylight
Background
There
are many periodic phenomena in nature.
Among the most familiar, and important, are those related to the motion
of planets, moons and other astronomical objects. These include the phases of the moon, and seasonal variations in
temperature and daylight. In this
project, you will model the changes in the hours of daylight in Jupiter, during
the year 2001.
Data
The
data in the table below was obtained from the U.S. Naval Observatory
(aa.usno.navy.mil/data/docs). The table
contains the predicted times of sunrise and sunset in Jupiter every two weeks
(14 days) during the year 2001. The
times are Eastern Standard time, using a 24-hour clock (so 1800 is 6 p.m.). The times are not adjusted for daylight savings, so the times in the summer are
an hour earlier than you might expect.
Complete the table by filling in the total number of hours of daylight
for each date (you will have to convert minutes to fractions of an hour).
|
Date |
Sunrise |
Sunset |
Hours of Daylight |
|
January 1 |
0710 |
1739 |
|
|
January 15 |
0711 |
1749 |
|
|
January 29 |
0707 |
1800 |
|
|
February 12 |
0659 |
1811 |
|
|
February 26 |
0647 |
1820 |
|
|
March 12 |
0633 |
1828 |
|
|
March 26 |
0618 |
1835 |
|
|
April 9 |
0602 |
1842 |
|
|
April 23 |
0549 |
1849 |
|
|
May 7 |
0537 |
1857 |
|
|
May 21 |
0530 |
1904 |
|
|
June 4 |
0526 |
1912 |
|
|
June 18 |
0526 |
1917 |
|
|
July 2 |
0530 |
1918 |
|
|
July 16 |
0537 |
1916 |
|
|
July 30 |
0544 |
1909 |
|
|
August 13 |
0551 |
1859 |
|
|
August 27 |
0558 |
1846 |
|
|
September 10 |
0604 |
1830 |
|
|
September 24 |
0610 |
1814 |
|
|
October 8 |
0616 |
1759 |
|
|
October 22 |
0624 |
1745 |
|
|
November 5 |
0633 |
1735 |
|
|
November 19 |
0643 |
1728 |
|
|
December 3 |
0654 |
1727 |
|
|
December 17 |
0703 |
1730 |
|
|
December 31 |
0709 |
1738 |
|
Questions
1.
Plot
the hours of daylight (use weeks since January 1 as your independent variable).
2.
Does
your graph look like a trigonometric function?
What kind of function?
3.
Estimate
the period, amplitude, vertical shift and phase shift of a trigonometric
function which fits the graph.
4.
Graph
your model on the same plot as the data points. Does it appear to fit?
Adjust your parameters to get the best fit you can.
5.
Measure
how well your model fits by computing r2. How good is your model?
6.
Using
your model, predict the hours of daylight on your birthday in 2005. Go online to the U.S Naval Observatory,
compute the table of sunrise and sunsets for 2005, and check your answer. How good was your model's prediction?