Problem 3:
Consider the star AY Sixteenus, located at RA = 18 hours and Dec = +32 degrees. On the first of each month, at what LST is AY Sixteenus on the meridian as viewed from Cambridge, MA (latitude 42 degrees, longitude of 71 degrees West)? On what date is the star on the meridian at midnight LST?
Before starting problem 3, one needs to know what the RA and Dec mean. RA is right ascension, which is essentially the celestial analogue to longitude. Declination is the celestial analogue to latitude.
Finding out where the star is intially
The problem gives no information regarding the starting position of the star. So, let's use it's position on the vernal equinox on March 20, 2014 which we've already seen in Problem 2. On the vernal equinox, LST in Greenwich, England is 00:00. AY Sixteenus, with an RA of +18 hours, will cross Greenwich's meridian at 18:00 (18 hours later). This means that it is 6 hour past Greenwich, or rather Greenwich rotated past it 6 hours prior. Now, Greenwich is 5 hours ahead of Cambridge in UT which is roughly 5 hours in LST. So we will use Greenwich as a base to easily calculate the LST when the star crosses Cambridge's meridian.
Since there are 24 hours in one sidereal day and the Earth rotates 360 degrees each day, 1 sidereal hour is equal to 15 degrees of rotation. Thus, 18 sidereal hours is equivalent to 270 degrees of rotation. Since the Earth rotates towards the West, the star will be located 6 hours ( 90 degrees) to the west of Greenwich, which is longitude 90 degrees E (just west of Cambridge). Now 24 hours would bring the star right back to the meridian at 90 degrees W. However, we don't want to rotate a full 360 degrees. We only want to go 341 degrees to reach Cambridge at 71 degrees W. By a proportion, \(\frac{341}{360}\:=\:\frac{x}{24}\), we see that it will cross over Cambridge in 22 hours and 42 minutes. Since we are starting at 00:00 LST in Greenwich, 22 hours and 42 minutes later will be 22:42. Since Cambridge is about 5 hours behind Greenwich in terms of time, AY Sixteenus will cross the meridian in Cambridge on March 21 at 17:42 LST. 11 days later on April 1, it will cross at 14:04. Every roughly 30 days (the new first of the month), it will cross 2 hours later because of the time discrepancy between sidereal and solar time.
If after the Vernal Equinox, it crosses at 17:42 on March 21, that means it crosses at about 05:42 UT. Since there are 18 hours and 18 minutes until midnight UT, it will take 274.5 days until it will cross over the meridian at Cambridge at midnight UT.
Consider the star AY Sixteenus, located at RA = 18 hours and Dec = +32 degrees. On the first of each month, at what LST is AY Sixteenus on the meridian as viewed from Cambridge, MA (latitude 42 degrees, longitude of 71 degrees West)? On what date is the star on the meridian at midnight LST?
Before starting problem 3, one needs to know what the RA and Dec mean. RA is right ascension, which is essentially the celestial analogue to longitude. Declination is the celestial analogue to latitude.
Finding out where the star is intially
The problem gives no information regarding the starting position of the star. So, let's use it's position on the vernal equinox on March 20, 2014 which we've already seen in Problem 2. On the vernal equinox, LST in Greenwich, England is 00:00. AY Sixteenus, with an RA of +18 hours, will cross Greenwich's meridian at 18:00 (18 hours later). This means that it is 6 hour past Greenwich, or rather Greenwich rotated past it 6 hours prior. Now, Greenwich is 5 hours ahead of Cambridge in UT which is roughly 5 hours in LST. So we will use Greenwich as a base to easily calculate the LST when the star crosses Cambridge's meridian.
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| 1. The location of Greenwich 6 hours before the equinox, 2. Greenwich at Equinox (90 degrees later, not to scale) |
Since there are 24 hours in one sidereal day and the Earth rotates 360 degrees each day, 1 sidereal hour is equal to 15 degrees of rotation. Thus, 18 sidereal hours is equivalent to 270 degrees of rotation. Since the Earth rotates towards the West, the star will be located 6 hours ( 90 degrees) to the west of Greenwich, which is longitude 90 degrees E (just west of Cambridge). Now 24 hours would bring the star right back to the meridian at 90 degrees W. However, we don't want to rotate a full 360 degrees. We only want to go 341 degrees to reach Cambridge at 71 degrees W. By a proportion, \(\frac{341}{360}\:=\:\frac{x}{24}\), we see that it will cross over Cambridge in 22 hours and 42 minutes. Since we are starting at 00:00 LST in Greenwich, 22 hours and 42 minutes later will be 22:42. Since Cambridge is about 5 hours behind Greenwich in terms of time, AY Sixteenus will cross the meridian in Cambridge on March 21 at 17:42 LST. 11 days later on April 1, it will cross at 14:04. Every roughly 30 days (the new first of the month), it will cross 2 hours later because of the time discrepancy between sidereal and solar time.
If after the Vernal Equinox, it crosses at 17:42 on March 21, that means it crosses at about 05:42 UT. Since there are 18 hours and 18 minutes until midnight UT, it will take 274.5 days until it will cross over the meridian at Cambridge at midnight UT.
By definition, the LST when the star is at meridian is at the RA of the star. Therefore, the LST when AYSixteenus is at the meridian is always 18:00!
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