Astronomy Principles and Practice Fourth Edition.pdf

408 Practical projects
measurement above would involve the relation
φ = 90 ◦ + δ ⊙ − α ⊙ .
Any error made in the measurement is directly translated into a location error along the line of
longitude, i.e. the determined value of φ will be displaced either N or S relative to the true position.
Suppose that optical measurements carry measurement uncertainties δθ =±10 arc sec—this being
typical of what can be achieved using simple optical equipment. In terms of the error, δd, in distance
along the meridian, this corresponds to
δd =±10 ×
2π R ⊕
360 × 60 × 60
where R ⊕ is the radius of the Earth. By substituting the appropriate value, δd ≈ 0·3km.
By similar reasoning, an error in the knowledge of δ ⊙ or in the calculation of the refraction
correction to α ⊙ of ±1 ′′ introduces uncertainties in position of approximately ±0·03 km or ±30 m.
Currently available GPS devices provide positional determinations to accuracies more than 10
times better than this last figure, i.e. positional fixes to ±3 m are readily achieved, and it is very
obvious why location by optical instruments has been abandoned. Such weather-dependent systems
with their associated labour of subsequent numerical calculations are things of the past.
It is, however, very instructive to use old optical devices to obtain data on the Sun’s position in the
sky and on its apparent movement. In addition to providing data and gaining familiarity with various
reduction procedures, their application gives some feel as to the accuracy to which simple hand-held
optical instruments provide positional fixes of the Sun and how these are translated to a determination
of the observer’s location on the Earth.
If such exercises are now attempted, it will be noted that The Astronomical Almanac or AA has
evolved in ways more related to modern positional astronomy. Certain kinds of information are now
presented differently relative to times past when positional determinations were regularly obtained
from optical measurements. For example, the positions of the Sun (RA and δ) were formerly given
for each day at 00 h UT rather than 00 h TDT as they are done currently. The examples provided here
are from measurements obtained in recent times. Consequently, the presented numerical correction
procedures are related to the present forms of the data tables and are slightly different than in the
previous era.
24.3.1 Simple determination of latitude
Set up a simple sharp vertical gnomon to cast a shadow on to a horizontal plane having a surface on
which the shadow tip can be marked. At intervals of about ten minutes, over a period of about one hour
either side of the local noon, mark the position of the shadow tip. After the recordings are completed,
measure the minimum shadow length from the base of the gnomon to the shadow tip. This corresponds
to the time when the Sun is on the meridian. Knowing the height of the gnomon above the horizontal
plane allows the maximum altitude to be determined using the formula
tan α ⊙ = H L
(see figure 24.4(a)).
Knowing also the Sun’s declination for that day, the latitude of the observer may be determined
from
φ = 90 ◦ + (δ ⊙ − α ⊙ ) (see figure 24.4(b)).