The Celestial Sphere and Local Sidereal Time

Ay 20 – Set 2:  The Celestial Sphere

Problem 1

Primary author: Joanna Robaszewski

Secondary authors:  Cassi Lochhaas and Monica He

Abstract

This problem demonstrates how local sidereal time (LST) is related to universal time (UT) and how to calculate local sidereal time at a given date and universal time.

Introduction

Universal time is the regular sort of time we think of on Earth.  It refers to the current time in Greenwich, England.  The time in other places (on Earth) is a certain number of hours earlier or later than UT depending on how far east or west of Greenwich they are and government policies (such as daylight saving time).  Local sidereal time is time based on what position the stars have in the sky, and uses a 24 hour clock.  It corresponds with the right ascension that is on the meridian.  Since the Earth does not take exactly 24 hours to rotate, but a bit longer such that every four years we have to have a leap day, and because the Earth is going around the sun, the stars are not in the same position at the same (universal) time every night.  So every 24 hours, the LST gains an extra 4 minutes.  The local sidereal time is 00:00 at 12:00 pm (noon) on the vernal equinox.

Questions and Results

1.a  What is the LST at midnight on the vernal equinox?

We know that every 24 hours the LST will gain an extra 4 minutes.  The time difference from noon on the vernal equinox to midnight on the vernal equinox is 12 hours.  Using dimensional analysis, we can calculate how many extra minutes the LST will gain in those 12 hours:

At noon on the vernal equinox the LST is 00:00.  Midnight is 12 hours later, so we add 12 hours, plus the extra 2 minutes as calculated above.  Thus, at midnight on the vernal equinox, the LST is 12:02.

1.b  What is the LST 24 hours later?

In 2011, the vernal equinox was on March 21.  Twenty-four hours after midnight on the vernal equinox it was midnight on March 22.  The 24 hour time interval means the LST gains an extra 4 minutes.  So we add 24 hours and 4 minutes to our previous LST of 12:02, remembering that LST repeats after 24 hours.  The LST at midnight on March 22 is then 12:06.

1.c  What is the LST right now?

When this question was written up it was October 5 at 12:00 pm.  The first question we asked to find the LST was:  how many full days have elapsed between March 21 at noon and now (October 5 at noon)?

In March, 10 days went by.  In April, there were 30.  May: 31, June: 30, July: 31, August: 31, September: 30, and October: 5.  In total that was:

We then needed to find how many extra minutes the LST had gained:

And how many hours that was:

The LST started as 00:00 at noon on March 21 and gained 13 hours and 8 minutes by noon on October 5.   The LST at the time this problem was being completed, 12:00 pm on October 5, was 13:08.  Rounding to the nearest hour, we get 13:00.

1.d  What will the LST be tonight, October 5, at midnight?

We know from the previous problem that at noon on October 5, the LST is 13:08.  Midnight is 12 hours later.  In that time interval the LST will gain 2 minutes, as shown in part a.  So we should add 12 hours and 2 minutes to the LST at noon on October 5.  This will give us the LST as 25 hours and 10 minutes which is equal to 01:10, since LST is measured on a 24 hour clock.  Rounding to the nearest hour means at midnight at the October 5/October 6 interface, the LST will be 01:00.

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

My birthday is July 4.  Let’s estimate that sunset will be around 8:00 pm on that date.  Then the first question we need to ask is:  how many full days have passed from March 21 at 12:00 pm to July 4 at 12:00 and how much extra time did LST gain from that?  We will deal with the left over hours later.

Once again, in March, 10 full days went by.  April: 30, May: 31, June: 30, July: 4.  The total number of full days between noon on March 21 to noon on July 4 is 105 days.  From the 105 days, LST gains:

We can now find the time gained from the 8 hours between 12:00 pm and 8:00 pm on July 4:

So between noon on March 21 and 8:00 pm on July 4, the LST gains 7 hours and 1 minute.  On July 4 at sunset the LST was 07:01.