Astronomy Principles and Practice Fourth Edition.pdf

110 The celestial sphere: timekeeping systems
Problems—Chapter 9
Note: Assume (i) a spherical Earth, (ii) the obliquity of the ecliptic to be 23 ◦ 26 ′ .
1. What is the lowest latitude at which it is possible to have a midnight Sun
2. What is the latitude of a place at which the ecliptic can coincide with the horizon
3. Suppose the Earth rotated with the same angular velocity as at present but in the opposite direction, what
would be the length of a mean solar day How many mean solar days would there be in a year
4. Complete the following table:
Greenwich hour
Hour angle of star Longitude of observer angle of star
2 h 46 m 21 s 30 ◦ W
18 h 24 m 40 s 65 ◦ E
1 h 19 m 46 s 121 ◦ E
23 h 04 m 57 s 42 ◦ 37 ′ E
5. Convert the mean time interval of 6 h 46 m 21 s to the corresponding interval of sidereal time.
6. Convert the sidereal time interval of 23 h 13 m 47 s to the corresponding interval of mean solar time.
7. Find the Zone Time on February 3rd when Procyon (RA 7 h 36 m 10 s ) crosses the meridian at Ottawa
(longitude 75 ◦ 43 ′ W), given that at GMT 0 h , February 3rd, the Greenwich sidereal time is 8 h 48 m 8 s .The
zone is +5.
8. Find the latitude of a place at which astronomical twilight just lasts all night when the Sun’s declination is
16 ◦ N.
9. Calculate the duration of evening astronomical twilight for a place in latitude 50 ◦ N when the Sun’s
declination is 5 ◦ 20 ′ N.
10. At mean noon on a certain date, the sidereal time was 14 hours. What will the sidereal time be at mean noon
50 days after, in the same place Take the length of a tropical year to be 365 1 4 days.
11. Find the Sun’s hour angle for an observer in longitude 39 ◦ 30 ′ W, given the following data:
Zone +3; approximate Zone Time of observation 8 h 20 m May 14th;
chronometer time (corrected) 11 h 21 m 47 s ; equation of time +3 m 45 s .
12. Find the hour angle of Vega (RA 18 h 34 m 52 s ) for an observer in longitude 126 ◦ 34 ′ E from the following
data:
Zone −8; approximate Zone Time of observation 6 h 30 m February 2nd; chronometer time 22 h 29 m 58 s ;
chronometer error (slow on GMT) 1 m 35 s ; for GMT 0 h February 2nd, Greenwich sidereal time was
8 h 45 m 9 s .
13. At approximately 2.30pm Zone Time, an observer keeping the time of Zone −8 made an observation of
the Sun on December 10th. The observer’s position was 55 ◦ N, 122 ◦ 30 ′ E. The chronometer reading was
6 h 32 m 45 s , its error (fast on GMT) being 4 m 22 s . Given that the ephemeris transit was 11 h 52 m 34 s , calculate
the Sun’s hour angle.
Taking the Sun’s declination to be −22 ◦ 55 ′ , obtain, to the nearest minute, the GMT of sunset for the
observer on December 10th.
14. Calculate the interval of sidereal time, for a place in latitude 45 ◦ N, between the passage of a star over the
prime vertical west and its setting, given the star’s zenith distance on the prime vertical west is 45 ◦ .
15. Taking the apparent orbit of the Sun to be circular and in the ecliptic, show that the equation of time is
given by
= cot −1 (cot α cos ε) − α
where α is the Sun’s right ascension and ε is the obliquity of the ecliptic. On which dates would vanish in
this case
16. Calculate to the nearest minute the interval of mean time elapsing between the setting of the Sun and of Venus
on March 31st, at Washington, DC (38 ◦ 55 ′ N, 77 ◦ 04 ′ W) given that the Sun’s declination is 4 ◦ 20 ′ N, the
equation of time is −4 m , the right ascension and declination of Venus are 3 h 33 m and 21 ◦ 56 ′ N, and the
Greenwich sidereal time at GMT 0 h on March 31st is 12 h 33 m .
17. The Astronomical Almanac for 2000 provides the following entries: