Astronomy Principles and Practice Fourth Edition.pdf

148 The reduction of positional observations: II
1. A star in the ecliptic has celestial longitude 42 ◦ . If the Sun’s longitude is 102 ◦ , calculate the change in the
star’s longitude due to aberration.
2. A star has a celestial longitude of 0 ◦ . What is the star’s apparent longitude on (i) March 21st, (ii) June 21st,
(iii) September 21st, (iv) December 21st
3. Given that the constant of aberration for an observer on Earth is 20·′′ 49 and that the distance of Mars from
the Sun is 1·5 astronomical units, calculate the value of the constant of aberration for an observer on Mars
assuming circular orbits.
4. Find the present longitude of the point which will be the First Point of Aries 5200 years hence. Show that
the north celestial pole will then be about 27 ◦ from the present pole.
5. The star 36 Draconis is near the north pole of the ecliptic. Neglecting its parallax, at what part of the year
will (i) its right ascension, (ii) its north declination, be (a) maximum, (b) minimum, due to aberration
6. Show that for a star whose true position is on the celestial equator, the change in its declination due to
aberration is
−κ sin ε cos λ ⊙
where κ is the constant of aberration. ε is the obliquity of the ecliptic and λ ⊙ the longitude of the Sun.
7. Calculate the displacements in celestial longitude and latitude, due to aberration, of a star in celestial latitude
+20 ◦ , the Sun’s longitude at the instant concerned being the same as that of the star.
If the stellar parallax is 0·′′ 32, what are the corresponding displacements due to parallax
8. The celestial latitude and longitude of a star at the present time are 5 ◦ 14 ′ N, 327 ◦ 47 ′ . Calculate the time
which must elapse before the star becomes an equatorial star.
9. Show that at any place, and at any instant, there is a position of a star such that the effect of annual aberration
is equal and opposite to that of refraction. Find the zenith distance of the star at midnight on the shortest day,
given that the ratio of the constants of refraction and aberration is 2·85.
10. Assuming the obliquity of the ecliptic to be constant, show that 13 000 years hence a star whose present
coordinates are 18 h and 47 ◦ S will then lie on the celestial equator. What will then be its corresponding right
ascension
11. If at a place in north latitude φ, the star in problem 10 just comes above the horizon at the present date at
upper culmination, find (i) the value of φ, (ii) the star’s altitude at upper culmination 13 000 years hence at
the same place.
12. The present equatorial coordinates of a star are 3 h and 0 ◦ . Calculate when next the star’s declination will be
zero.
13. If the date in the year when Sirius rises just after sunset changes by one day in 71 years, calculate
(approximately) the value of the constant of precession, and the difference in the lengths of the tropical
and sidereal years, given that the length of the tropical year is 365·2422 mean solar days.
14. Calculate (i) the ecliptic coordinates, (ii) the equatorial coordinates, referred to the present pole and equator,
of the point on the celestial sphere which will, after one-third of the precessional period, become the north
celestial pole.
15. The present ecliptic longitude and latitude of a star are 32 ◦ and 25 ◦ N, respectively. Find the declination and
right ascension of the star at its minimum distance from the pole and calculate the time when this will occur.