Tuesday, July 26, 2011
Solutions to Simple Astronomy Problems (5)
We proceed now to the solutions for the problems given in instalment (5). Recapping them, with the respective solutions to follow:
(1) The Right Ascension of Dubhe in Ursa Major is 11h 02m. Its declination is 61 deg 56 minutes. From a ship at sea it's observed to transit 42 degrees 30 minutes north of the zenith on April 8, at 1 a.m. according to the ship's chronometer (which reads Greenwich Mean time or G.M.T.)
Find the position of the ship.
If Dubhe transits 42 deg 30 min north of the zenith then its meridian zenith distance (MZD) is 42 deg 30 min.
The ship's latitude may be found from:
Lat. = Declin. - MZD = 61 deg 56 min - 42 deg 30 min = 19 deg 26 min North
To obtain the longitude we require:
Greenwich time of transit - local time of transit, or GTT - LTT for short.
Where: LTT = Star's RA - Sun's RA
Now, on April 8, the Sun's RA = 18 days x (4 min/day) = 72 mins. = 1 h 12 min
(Note: this is 18 days elapsed after March 21 when the Sun's RA = 0h)
LTT = 11 h 02 min - 1 h 12 min = 9h 50 min
According to the ship's chronometer, the star transits at 1 a.m. (13 h) therefore GTT = 13h 00m
Then the ship's longitude is:
L = GTT - LTT = 13 h 00 min - 9h 50 min = 3 h 10 min
Converting this to degrees (1 h = 15 deg), we get 47½ degrees. This must be WEST longitude since the GTT is later than LTT.
Then the coordinates are:
Lat. = 19 deg 26 min N, Long. = 47½ deg W
(2)(a) Using a clear diagram, show that the hour angle of a star is related to the Star's RA, the Sun's RA and L.M.T. (local mean time) by:
RA (star) = Sun's RA - (HA) + L.M.T.
This is shown in Fig. 1
(b)Shaula (RA = 17h 31m) transits at 10 p.m. on June 29th. Find the hour angle it exhibits from your location (local hour angle).
This can be done from either the diagram or the expression (RA (star) = Sun's RA - (HA) + L.M.T.) which really amount to the same thing.
A simpler tack is to use: LST= HA + RA of object
and it can be ascertained the LST for the date is 15h 31 m
SO: HA = LST - RA = 15h 31m - 17h 31m = -2h 00m, or - 30 deg
(3) The star Canopus (RA = 6h 20m) is observed to have a local hour angle = 45 deg on Feb. 10th for a given location.
(a) What is the local sidereal time.
(b) At what local mean time and standard time would Canopus transit?
(c) What is the approximate LST at noon on the same date?
(a)Again: HA = LST - RA
So: LST = HA + RA = 3h 00m + 6h 20m = 9h 20m
(b) We have:
LTT = Star's RA - Sun's RA
Feb. 10th is 39 days before March 21 (assume non-leap year) so:
Sun's RA = 0h - 39 days x (4 min/day) = 0h - 156 mins = 0 h - 2h 36 m
Sun's RA = 21h 24m
LTT = 6h 20m - 21h 24m = -17h 36m
Or: 24 h 00m - 17h 36m = 6h 24m
(c) From (a) the LST on the date is 9h 20m, and this is 9h 20m past noon. Since noon is 9h 20m earlier, then LST(noon) = 6h 20m - 9h 20m = -3h 00m, or:
24h 00m - 3h 00m = 21h 00m.
(4) Study the diagram for Fig. 1. Using the diagram and inferences regarding time - including Sun's changes in Right Ascension, give the date for which the diagram is referred. If the star in Fig. 1 has an RA = 7h 10m then find its hour angle for the observer (Obs). Hence, or otherwise, obtain the longitude of the observer.
The diagram is shown again in Fig. 2.
Since the local sidereal time (LST) at the observer location is 6h then this is the RA on the meridian. But the Sun is on the meridian at the antipode or at 18h 00m. Hence, the date must be Dec. 23rd or the winter solstice, since the same RA coincides with the Sun.
If the star's RA = 7h 10m we can find the HA from the angular relationship as:
HA = RA - LST = 7h 10m - 6h 00m = 1h 10m = 17 .5 deg
The longitude of the observer is: L = GST - LST = 9h 00m - 6h 00m = 3h 00m
Converting to degrees: L = 45 deg (W) (since GST > LST)
(5)(a) Regulus (10h 07m) is observed to transit on a given date. Given the same date, what would be the local hour angle for: (i) Denebola (RA = 11h 47m), and (ii) Arcturus (RA = 14h 14m)?
(b) Find the LST one half way between the transits of Denebola and Arcturus. How much later would these two stars set than Regulus?
(c) What RA circle must be rising when Arcturus is setting?
(a) We use: LHA (local hour angle) = RA on meridian (LST) - RA(star)
Here, RA meridian = RA (Regulus) = 10h 07m
LHA (Denebola) = 10h 07m - 11h 47m = -1h 40m (or 22h 20m)
LHA (Arcturus) = 10h 07m - 14h 14m = -4h 07 m or 19h 53m
(b) Find (RA(Arcturus) - RA(Denebola) = 14h 14m - 11h 47 m = 2h 27m
Then the LST midway between their transits is found by interpolating:
11h 47m + ½(2h 27m)= 13h 00.5m
Thus, Denebola would set about 1h 40m after Regulus. (E.g. 10h 07m + 1h 40m = 11h47m)
(c) When Arcturus is setting, the RA circle rising is:
14h 14m + 12h 00m = 26h 14m
26h 14m - 24h 00m = 2h 14m