Worksheet 3 Problem 2: Local Sidereal Time

Problem 2 on this worksheet focuses on an important concept in Astronomy: Local Sidereal Time. Local Sidereal Time is the time, based on the daily 360 degree rotation of the Earth, relative to the location of a distant fixed celestial body as it passes over the meridian of a point on Earth. The most confusing aspect of sidereal time is that it is not the same as Solar time, even though it's based on the rotation of the Earth and is separated into 24 hours. 1 Sidereal day is equivalent to 23 hours, 56 minutes of Solar time. Since sidereal time is based on the Earth's rotation relative to a distant point, the effect of the movement of Earth's orbit on the position of the star is negligible. However, relative to the Sun, the difference is noticeable and causes this discrepancy.

http://www.astro.cornell.edu/academics/courses/astro201/sidereal.htm

Now onto the problem. 

Problem 2: 

The Local Sidereal Time (LST) is the right ascension that is at the meridian right now. LST = 0:00 is at noon on the Vernal Equinox (the time when the Sun is on the meridian March 20th, for 2013 and 2014).
a) What is the LST at midnight on the Vernal Equinox?
b) What is the LST 24 hours later (after midnight in part ’a’)?
c) What is the LST right now (to the nearest hour)?
d) What will the LST be tonight at midnight (to the nearest hour)? e) What LST will it be at Sunset on your birthday? 

Solving Part A: LST at midnight on the Vernal Equinox

 Sidereal time moves through one 24-hour cycle faster than solar time does. 1 sidereal day is equal to 23 hrs and 56 mins in solar time. Thus, 1 half of a sidereal day is equal to 11 hrs and 58 mins of solar time. That means that 12 sidereal hours after the Vernal Equinox, LST= 12:00 and UT (Universal Time)=23:58. To reach midnight UT, two more solar minutes have to pass, making UT=00:00. For such small increments of time, one solar minute is roughly the same as one sidereal minute. Thus two minutes later, LST= 12:02.

Solving Part B: LST 24 hours later

24 hours later, UT will be once again equal to 00:00. Since 1 sidereal day is 23 hours and 56 minutes in solar time, 24 sidereal hours and 4 minutes will have passed in LST. Thus LST=12:02 + 4 = 12:06

Solving Part C: Current LST (when I was solving the problem)

There is a 4 minute discrepancy between LST and UT every day. Thus, I need to calculate the number of days that has passed since the last known sidereal time and then add that number of days multiplied by four. Since today is February 7, 324 days have passed since the Vernal Equinox. 324 days yields a 1296 minute, or 21.6 hour, discrepancy between LST and UT. That means LST will go from 00:00 to 21:36 and UT will go from 12:00 to 12:00. However, it is about 8:30 pm now, so 8 hours have to be added to UT. Since there is a 4 minute discrepancy per day between LST and UT, there will be a 1 minute discrepancy every 6 hours and so on. Since 8.5 hours have passed, there will be an extra difference of 1.4 minutes (1 minute, 24 seconds). Thus the current LST is 6:07:24. To the nearest hour, LST is 06:00:00.

Solving Part D: LST at midnight

It will be midnight in 4 hours and 20 minutes solar time. To find LST, add 4 hours and 20 minutes plus the additional discrepancy between LST and UT. The discrepancy can be calculated using a proportion as follows:
\[\frac{4\:min}{24\:hr}\:=\:\frac{x}{4.33\:hr}\:\rightarrow\:\:x\:=\:42\:seconds\]

LST=6:07:24 + 04:20:42 = 10:28:06. To the nearest hour, LST will be 10:00 at midnight.

Solving Part E: LST at Sunset on my birthday

This part of the problem is very similar to Part C as it involves calculating the number of days between now and the desired date. My birthday at midnight is 153 days away. \(153\:days\:\times\:\frac{4\:min}{day}\:=\:+\:10:12\:LST\). The sun will set at 8:30 on that day. Fortunately, I already calculated in Part C that the discrepancy between LST and UT is 1 minute and 24 seconds and the additional 12 hours is simply a 2 minute discrepancy. Thus, starting at midnight tonight (10:28:06) and adding 10 hours, 12 minutes gives an LST of 20:40:06 at midnight at the start of my birthday. Then add 20:30 to reach 8:30 pm UT and that gives an LST of 17:10:06. Finally, add the last discrepancy value of 00:03:24 to give an LST of 17:13:30 at midnight on my birthday.