Astrophysics - Hour angle calculation

[rozma3]

Homework Statement

At midnight (local time) on Dec 21st 2015, the Local Sidereal Time at Palomar Observatory, California, was 6h 27m. Palomar Observatory has latitude 33◦ 210 2100 N, and longitude 116deg 51arcmin 60arcsecW.
a) What was the approximate Local Hour Angle of the Sun at 23:00 local time?
b) At what local time was the Sun on the meridian at Palomar Observatory?
c) How many hours before the time determined in part b) was the Sun on the meridian at La Palma Observatory in the Canary Islands (located at latitude 28◦ 450 2500 N, and longitude 17◦ 530 3300 W)?

Homework Equations

I know that the calculation for hour angle is Local Sidereal time - Right ascension of the star, the star in this case is the sun and the date given (Dec 21st) equates to a right ascension being 18h as it is the winter solstice. Longitude can be converted to hours by doing 116 + (51/60) + (60/3600) = 116.87 then (116.87/360) x24

The Attempt at a Solution

I first tried just to plug in the values of right ascension being the winter solstice so just tried 6h 27m - 18h 0m to get an hour angle of 11h 33m for part a. I also attempted in a second alternative method to convert the longitude values to hours and use this value for the right ascension and received a value for the hour angle to be 1h 20 minutes. I am not sure what the 23:00 time has to do with the question and which method gives a closer answer. Any help would be greatly appreciated in finding the correct values with an explanation why. Thank you. Really sorry, used the wrong forum and not sure how to delete thread.

 

[gneill]

I first tried just to plug in the values of right ascension being the winter solstice so just tried 6h 27m - 18h 0m to get an hour angle of 11h 33m for part a. I also attempted in a second alternative method to convert the longitude values to hours and use this value for the right ascension and received a value for the hour angle to be 1h 20 minutes. I am not sure what the 23:00 time has to do with the question and which method gives a closer answer. Any help would be greatly appreciated in finding the correct values with an explanation why. Thank you. Really sorry, used the wrong forum and not sure how to delete thread.

23:00 local time is one hour before midnight. The information given for the local sidereal time was for midnight, so you'll need to take into account Earth rotation for the mean solar hour difference between the two. Remember that a mean solar hour differs slightly from a sidereal hour.

I suggest that you start these sorts of problems with a sketch and locate the significant geographical locations and the celestial directions for the given epoch (in this case the epoch is LST = 6hr 27m on December 21 at Los Palomar Observatory). For example:

The above is a view looking down on the Earth from above the north pole, with the Earth turning counterclockwise.

 

[rozma3]

Thank you for your reply. I am not entirely certain about how exactly to take into account the 1 hour time difference between 00:00 and 23:00.

 

[gneill]

What is the Earth doing during that one mean solar hour?

 

[rozma3]

Rotating 15 degrees, but how does that affect the LST. Does it mean it becomes 7h 27m?

 

[gneill]

23:00 is an hour before midnight. So that time is earlier, and so the corresponding LST must also be earlier, not later.

In the diagram I posted imagine turning the circular Earth 15 degrees (approximately, because a mean solar hour is not identical to a sidereal hour) clockwise. The directions to the Vernal Equinox and Sun remain essentially fixed. That effectively turns time back one hour to 23:00 local time at Palomar. You should see that the westward angle between Palomar and the Vernal Equinox is now smaller by 15°. The same goes for the westward angle from Palomar to the Sun...

 

[rozma3]

I see. So in the calculation, the LST will be 5 hours 27 minutes. Then the right ascension being 18 hours. 5h 27 m - 18h = 12h 33 minutes, correct?

 

[gneill]

I see. So in the calculation, the LST will be 5 hours 27 minutes. Then the right ascension being 18 hours. 5h 27 m - 18h = 12h 33 minutes, correct?

For the local hour angle of the Sun? Remember that the right ascension of the Sun is measured eastward from the vernal equinox, while the local hour angle is measured westward from the observer's meridian. Check the angles marked on the diagram. How many hours is the sun past the vernal equinox position going westward?

 

[rozma3]

6 hours?

 

[gneill]

Yes, six hours.

 

[rozma3]

What would be the next step? Really sorry but I struggle quite heavily with these sorts of questions.

 

[gneill]

What would be the next step? Really sorry but I struggle quite heavily with these sorts of questions.

You'll have to make an attempt. What is it you want to find? What rotation is required to get there?