LST - Losing Some Time (3.02)

Question: 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)

1. What is the LST at midnight on the Vernal Equinox?
2. What is the LST 24 hours later (after midnight in part 'a')?
3. What is the LST right now (to the nearest hour)?
4. What will the LST be tonight at midnight (to the nearest hour)?
5. What LST will it be at Sunset on your birthday?

What is the LST at midnight on the Vernal Equinox? 

Okay, so the definition of LST provided to us is, "the right ascension at the meridian right now". That sounds like a complicated definition, so let's get familiar with some terminology first. We have always thought of the day as being 24 hours long. However, that is not always true. We have two definitions of the word "day"! There is a solar day, and then there is the sidereal day. Wat!

Solar day vs. sidereal day

• A solar day is the 24 hour day we are used to in everyday life, measured from noon on one day to noon the next day. 
• A sidereal day is the time it takes the Earth to make one full rotation around the with respect to the stars. The Earth takes about 23 hours and 56 minutes to actually make a full rotation. 

The LST is a clock that tells the time according to a sidereal day. As a result, every 24 hours, the sidereal time is 4 minutes more than it was the previous day.

It is useful to think that since you gain 4 minutes every in LST for every 24 hours, you gain 1 minute in LST time every 6 hours. 

On the Vernal Equinox at noon, LST is 0:00. At midnight on the Vernal Equinox, 12 hours have passed, so LST would have increased by 2 minutes (since 2 six-hour time chunks have passed). Therefore, LST would show 12:02 to account for the addition 2 minutes added. 

What is the LST 24 hours later (after midnight from the previous part)?

Since 24 hours have passed since midnight on the Vernal Equinox, LST gains an additional 4 minutes to account for the loss of 4 minutes in sidereal time. Also, since midnight on the Vernal Equinox also contributed to 2 minutes to LST, the total minutes that have been added to LST are 2 + 4 = 6 minutes. Therefore, the LST would be 12:06. 

Another way to look at it is that 12 hours passed from the Vernal Equinox at 0:00 to get to midnight. Then after another 24 hours, we are up to 36 hours from the Vernal Equinox at 0:00. Since we gain 1 minute for every 6 hours in LST, $\frac{36}{6} = 6$ minutes get added to LST. Therefore, it is 12:06.

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

As of right now, it is 18:00 on February 5, 2015. Since the last Vernal Equinox was on March 20, 2014, 322 days and 6 hours have been passed. To calculate LST, we first need to calculate the right ascension:

\begin{equation}
322.25 \text{ days} \times \frac{24 \text{ hours}}{1 \text{ day}} \times \frac{4 \text{ minutes}}{24 \text{ hours}} \times \frac{1 \text{ hour}}{60 \text{ minutes}} \approx 21.483 \text{ hours} \approx \text{21 hours 29 minutes}
\end{equation}


The right ascension is 21 hours and 29 minutes, which means that the LST would be 2 hours and 31 minutes behind solar time. Therefore, LST would be 3:29, or approximately 3:00.

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

By midnight tonight, 6 more hours would have passed from the LST time 3:29, so LST would be 9:29, or approximately 9:00. 

What LST will it be at sunset on your birthday?

My birthday is on December 6 (woohoo!!), with a projected sunset time of 16:12. Since the Vernal Equinox for 2015 (which is on March 20th) would have passed by the time my birthday comes, exactly 261 days, 4 hours, and 12 minutes would have passed. Doing the calculations: 

\begin{equation}  261.174 \text{ days} \times \frac{24 \text{ hours}}{1 \text{ day}} \times \frac{4 \text{ minutes}}{24 \text{ hours}} \times \frac{1 \text{ hour}}{60 \text{ minutes}} \approx 17.41 \text{ hours} \approx \text{17 hours 25 minutes}   \end{equation} 

Since the right ascension is 17 hours 25 minutes, LST would be 5 hours and 25 minutes ahead of solar time. Therefore, since solar time would be 4:12, and another 5 hours 25 minutes are added, LST would be 9:37.