Worksheet 3 Problem 2

After I learned a bit about the initially complex notion of sidereal time, I began to understand it's advantages in measuring time from a far off object. Rather than measuring a day based on the amount of time the earth takes to complete one rotation with respect to the sun, sidereal time considers the amount of time it takes the earth to complete a full rotation with respect to a far off star. In a previous problem that I worked through (which is not posted here) I worked out the difference between a sidereal and solar day on earth. Since it is the basis of how I will approach this problem, I'll give a quick overview of how this was determined.

The earth revolves around the sun over the course of 1 year (365 days). As an approximation, this means that the earth travels ~1 degree in its orbit each day. From a far off star, this corresponds to an additional degree of rotation each day. Knowing this, we can determine how much longer a solar day is than a sidereal day:\[ \frac{1\,degree}{1\,day} \times \frac{1\,day}{360\,degrees} \times \frac{24\,hours}{1\,day} \times \frac{60\,min}{1\,hour} = 4\,\frac{min}{day} \]
This problem has several parts that examine the Local Sidereal Time (LST) at different points throughout the year. Local Sidereal Time is defined as the right ascension that is at the meridian right now. At noon on the Vernal Equinox, LST = 0:00.

a) What is the LST at midnight on the vernal Equinox?

Since the a sidereal day is 4 minutes shorter than a solar day, at midnight on the Vernal Equinox, 12 hours will have passed, plus an additional 2 minutes, due to the difference in time. Therefore, the Local Sidereal Time will be 12:02 at midnight on the Vernal Equinox.

b) What is the LST 24 hours later (after midnight in part 'a')?

With another 24 hours passing, LST will gain an additional 4 minutes on top of the answer to part a. Again, since there are 4 additional minutes to a solar day, Sidereal Time must gain an additional 4 minutes per day. \[ 12:02 \,+ \,24 \,+\, 0:04\, = \,36:06\, = \,12:06 \]

12:06

c) What is the LST right now (to the nearest hour)?

Current time: 10pm, February 8th

Since the Vernal Equinox is on March 20th, 2015 it is still 40 days away. Again, using our 4 minute offset, we can determine the number of additional minutes that LST will gain to this point.
\[ 40\,days \times \frac{4\,min}{1\,day} \times \frac{1\,hour}{60\,min} = 2:40\,hours \]

Knowing that noon on March 20th, 2015 corresponds to LST 0:00, we know that noon today corresponds to that time minus 160 minutes. \[ 0:00 - 2:40 = 21:20\] Since the current time, however, is not noon, but 10pm, we need to add in an additional 10 hours, plus 2 more minutes for the difference in today's LST. \[21:20 + 10:02 = 31:22 = 7:22 = 7:00\, rounded\]

7:00

d) What will the LST be tonight at midnight (to the nearest hour)?

Tonight at midnight occurs 2 hours later than the previous time that we calculated in part c. The additional amount of time that occurs in the 2 hours will be negligible for this problem, so the answer is simply \[ 7:22 + 2:00 = 9:22 = 9:00\, rounded \]

9:00

e) What LST will it be at sunset on your birthday?

My birthday occurs on August 1st. In Boston, the sunset on August first will be at 8:05pm. Since August 1 occurs 134 days after the Vernal Equinox, we can calculate how far LST has shifted during this time: \[ 134 \,days \times \frac{4 \,min}{1 \,day} \times \frac{1\, hour}{60\, min} = 8:56 \,hours \]

Noon on August 1st will correspond to 0:00 + 8:56 = 8:56.

Then, adjusting to the time of sunset, we will have an additional 8:05. \[8:56 + 8:05 = 17:01\]

17:01

Collaborators on this problem included Willie Pirc and Johnathan Budd.