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 2015).
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?
Local Sidereal Time (LST) is a measure of time on Earth with respect to the fixed stars. As opposed to solar time, which is the time it takes for the Sun to return to the meridian, sidereal time is the time it takes for a fixed star to return to the meridian. A sidereal day is approximately 23 hours and 56 minutes long, so it is about 4 minutes shorter than a solar day. Also, we are told in the question that on March 20th, at noon, LST = 0:00. Using these two facts, we can create a general equation to solve for the LST at a specific date and time.
Let \(d\) be the number of solar days after the Vernal Equinox (March 20th), \(h\) be the number of hours into the solar day in consideration, and \(m\) be the number of minutes into the hour. Since a sidereal day is 4 minutes shorter than a solar day, for every solar day, we have completed a sidereal day and 4 minutes. So we can create an equation that solves for the number of minutes the Local Sidereal Time is ahead of the solar time \((x)\).
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?
Local Sidereal Time (LST) is a measure of time on Earth with respect to the fixed stars. As opposed to solar time, which is the time it takes for the Sun to return to the meridian, sidereal time is the time it takes for a fixed star to return to the meridian. A sidereal day is approximately 23 hours and 56 minutes long, so it is about 4 minutes shorter than a solar day. Also, we are told in the question that on March 20th, at noon, LST = 0:00. Using these two facts, we can create a general equation to solve for the LST at a specific date and time.
Let \(d\) be the number of solar days after the Vernal Equinox (March 20th), \(h\) be the number of hours into the solar day in consideration, and \(m\) be the number of minutes into the hour. Since a sidereal day is 4 minutes shorter than a solar day, for every solar day, we have completed a sidereal day and 4 minutes. So we can create an equation that solves for the number of minutes the Local Sidereal Time is ahead of the solar time \((x)\).
\(x=4(d+\dfrac{h}{24}+\dfrac{m}{1440}+\dfrac{1}{2})\)
In this equation, we are multiplying the number of days by 4 minutes since this is time offset between the solar and sidereal day. We divide the hours by 24 and minutes by 1440 (number of minutes in a day) to find the fractional day. The added \(\frac{1}{2}\) is to account for the fact that LST is 0:00 twelve hours into the solar day.
a) I assumed that this meant the midnight following noon. This is 12 hours after noon, which is half a day. Since 4 minutes are added for a full day, we can deduce that there will be 2 minutes added for half a day. Here, the problem is simple enough that we do not need to use our equation. So at 12:00am solar time, it will be 12:00 + 0:02 = 12:02am in LST.
b) 24 hours following the midnight after Vernal Equinox, the LST will further increase by 4 minutes since a day has passed. So we have, 12:02 + 0:04 = 12:06am in LST.
c) Right now, it is February 10th and it is 1:00pm. It is 326 days and 13 hours after the Vernal Equinox. This means that \(d=326\), \(h=13\), and \(m=0\). Plugging into our equation, we get:
d) At midnight tonight, it will be 327 days after the Vernal Equinox. So we have \(d=327\), \(h=0\), and \(m=0\). Plugging these values in, we get:
b) 24 hours following the midnight after Vernal Equinox, the LST will further increase by 4 minutes since a day has passed. So we have, 12:02 + 0:04 = 12:06am in LST.
c) Right now, it is February 10th and it is 1:00pm. It is 326 days and 13 hours after the Vernal Equinox. This means that \(d=326\), \(h=13\), and \(m=0\). Plugging into our equation, we get:
\(x=4(326+\dfrac{13}{24}+\dfrac{0}{1440}+\dfrac{1}{2})=1308\) mins
Converting this into hours, we divide by 60 to get:
\(\dfrac{1308\text{ mins}}{60\text{ mins}}=21.8\approx 22\) hours
This means that we are 22 hours ahead of the time right now, which is 13:00 + 22:00 = 11:00am.
d) At midnight tonight, it will be 327 days after the Vernal Equinox. So we have \(d=327\), \(h=0\), and \(m=0\). Plugging these values in, we get:
\(x=4(327+\dfrac{0}{24}+\dfrac{0}{1440}+\dfrac{1}{2})=1310\) mins
Converting this into hours, we divide by 60 to get:
\(\dfrac{1310\text{ mins}}{60\text{ mins}}=21.833\approx 22\) hours
This means that it is 22 hours past midnight (0:00), which is 22:00, or 10:00pm in LST.
e) My birthday is on May 5th, which is 45 days following the Vernal Equniox, and sunset occurs at 7:56pm (19:56). Putting these values into our equation, we get:
\(x=4(45+\dfrac{19}{24}+\dfrac{56}{1440}+\dfrac{1}{2})\approx 186\) mins
Converting this into hours, we divide by 60 to get approximately 3 hours. This means that the LST is 3 hours ahead of the solar time, so we have:
7:56pm + 3:00 = 10:56pm \(\approx\) 11:00pm (23:00)
I would like to thank Charles Law, Daniel Chen, and Shai Szulanski for their help in solving this problem.
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