ASTR 310 Homework 2 Solutions
1. For the night of October 18/19, 2006, find the approximate time at which M31 will
cross the meridian, as seen from the UMD Observatory. Give your results in EDT
(Eastern Daylight Saving Time).
Date
Fall Equinox (Sept. 21)
Winter Solstice (Dec. 21)
Spring Equinox (March 21)
Sum m er Solstice (June 21)
Date
Fall Equinox (Sept. 21)
Winter Solstice (Dec . 21)
Spring Equinox (Marc h 21)
Sum m er Solstice (June 21)
UT (universal tim e)
0h
0h
0h
0h
GST (Greenwich sidereal tim e)
0h
6h
12h
18h
EST (Eastern standard tim e)
0h
0h
0h
0h
LST (local sidereal tim e)
0h
6h
12h
18h
The above tables show how sidereal and standard time are related within a given
time zone on the equinoxes and solstices. Note that the relationship is the same between
UT and GST, and between LST and EST. (This would be true anywhere.) Sidereal and
standard time are the same on the fall equinox. You can use this information to figure out
the sidereal time from the standard time, or vice versa, on any other day of the year.
M31 has a right ascension of ~00h 43m. This means that the sidereal time at
which M31 transits (crosses the meridian) is 00h 43m, or 00:43. On Sept. 21, the
standard time is the same as the sidereal time, so it is 00:43 EST (12:43am EST) when
M31 transits. We want to know what time M31 will transit on the night of Oct. 18/19.
Consider a star that transits at 00h 00m sidereal time. On Sept. 21, it will transit
at midnight EST. The next night, it will transit at the same sidereal time, but the solar
time will be 4 minutes before midnight, so 11:56pm EST. The next night, it will transit at
11:52pm EST; and so on. When would it transit on Oct. 18? Oct. 18 is 27 days after
Sept. 21. So the star would transit 27 days * 4 min/day = 1h 48m earlier than on Sept.
21:
9/21: LST = EST
9/22: LST = EST + 4min/day * 1 day
9/23: LST = EST + 4min/day * 2 days
general: LST = EST + 4min/day * x days
EST = LST – 4min/day * x days
Now apply this to M31. If on Sept. 21 it transits at 12:43am EST, then on Oct. 18
it will transit 1h 48m earlier, but the LST is always 00:43 at the time of transit:
EST = LST - 4min/day * 27 days = EST – 1h 48m
EST = 00:43 – 1:48
= 24:43 – 1:48
= 22:55
= 10:55pm
So the EST is 10:55pm. To convert to EDT, add one hour. The time of M31's
transit on Oct. 31 is 11:55pm EDT.
(ANS)
2. Find the local sidereal time at the UMD Observatory at 9:30pm EDT on the night of
Oct. 15, 2006. What is the hour angle of the star Altair for this date and time?
Once again, we'll refer to the tables to figure this out. On Sept. 21 at midnight
EST, the LST = 0h. The next day, the EST for a given LST is 4 minutes earlier, so the
LST for a given EST is 4 minutes later. So for midnight EST on Sept. 22, the
LST=00h04m. As before,
LST = EST + 4min/day * x days
Oct. 15 is 24 days after Sept. 21. So midnight on Oct. 15 corresponds to an LST of
LST = 00:00 (EST) + 4*24 = 00:00 + 1:36 = 1:36 = 1h 36m
This tells you that the LST on Oct. 15 is 1h 36m later than on Sept. 21. So on Oct. 15 at
9:30pm EDT = 8:30pm EST = 20:30 EST, the LST is
LST = 20:30 + 1:36 = 22:06 = 22h 06m
(ANS)
The hour angle is HA = LST – right ascension. Altair's right ascension is
~19h48m. So
HA = 22h 06m – 19h 48m = 02h 18m
(ANS)
3. A. Consider a refracting telescope consisting of an objective lens with focal length
f1=2m followed by an eyepiece lens with focal length f2=1m. The distance between
the objective and the eyepiece is 2.5 m. Using the formula
1
1
1
f
so
si ,
find the image location (wrt the eyepiece) of an object positioned 5m in front of the
objective lens. Is the image real or virtual?
Lens 1: s0,1 = 5m, f1 = 2m
1 1 1 1 1
si
f
so
2
3
5 10
So si,1 = 10/3 = 3.333. This is farther away from lens 1 than lens 2 is, which we'll
have to consider in order to get s0,2 of lens 2.
Lens 2: 2.5m from lens 1, so |s0,2| = | 2.5-10/3 | = 5/6
Because s0,2 is on the left side of lens 2, it is negative: s0,2 = -5/6
f2 = 1m
1 1 1 1
6 11
si
f
so
1
5
5
So si,2 = 5/11. This is the image location. Because it is positive, the image is
real.
B. Use ray tracing to confirm your answer in (a).
See the next page.