vik dhillon: phy105 - the celestial sphere - example problems
example problems
Here, we will attempt a very common problem in spherical astronomy,
which is also probably the most important one of all to understand -
the conversion of equatorial coordinates into horizontal
coordinates.
1.
A star, X, of declination
=
42°21´N is observed when its hour
angle, H = 8h16m42s.
If the observer's
latitude, ,
is 60°N, calculate the star's altitude, a and
azimuth, A, at the time of observation.
The most important step is to sketch as accurately
as possible a celestial sphere of the problem, as shown in
Figure 25.
figure 25:
Example problem 1 - the conversion of equatorial coordinates
to horizontal coordinates.
cos ZX = cos 30° cos 47°39´ + sin 30° sin 47°39´ cos 124°10´30´´.
Recalling that, for any angle x,
sin x = cos (90° - x) and
cos x = sin (90° - x),
we can write
sin a = cos 30° cos 47°39´ + sin 30° sin 47°39´ cos 124°10´30´´,
which gives
a = 22°04´34´´.
To determine the azimuth we can apply
the sine formula
to triangle PZX, which gives
sin H / sin (90° - a) = sin (360° - A) /
sin 47°39´, or
sin (360° - A) = sin H sin 47°39´ /
cos a.
Thus
sin (360° - A) = sin 124°10´30´´
sin 47°39´ / cos 22°04´34´´.
The sine formula must be used with care since it is not possible to say whether
A is acute or obtuse, unless other information is
available, i.e. the sine formula gives A or 180° - A.
Hence we have
360° - A = 41°17´6´´ or 138°42´54´´,
that is
A = 318°42´54´´ or 221°17´6´´.
An inspection of Figure 25 suggests that the
former answer is correct, but we should check it anyway. We can do this
by using the cosine formula again:
cos 47°39´ = cos 30° cos (90° - a) +
sin 30° sin (90° - a) cos (360° - A)
cos 47°39´ = cos 30° sin 22°04´34´´ +
sin 30° cos 22°04´34´´ cos (360° - A).
This gives
360° - A = 41°17´6´´ and so
A = 318°42´54´´ east of north.
Two things should be noted here. The first is that although the
cosine of an angle, A, suffers from an ambiguity between
A and 360°-A, it does not suffer from the acute/obtuse
problem that the sine of an angle suffers from, which is why the above check
works. Secondly, you may have
wondered why the altitude we calculated above
does not suffer from the acute/obtuse problem. Strictly speaking, it does,
but because altitude is simply the angular distance of the star above the
horizon, 180° - a is equivalent to a.