Astronomy Principles and Practice Fourth Edition.pdf

120 The reduction of positional observations: I
By equation (10.14), then,
OT = (R ⊕ + h)
√
2h
R ⊕ + h = √ 2h(R ⊕ + h).
But h is much less than R ⊕ , so that the distance d to the apparent horizon is given by
d = √ 2R ⊕ h.
If h is expressed in metres and d in kilometres, we have
d = 3·57 √ h km. (10.15)
Taking refraction into account, the distance d ′ of the apparent horizon is now given by OT ′ in
figure 10.5. The expression (10.15) is modified to
10.4 Geocentric parallax
d = 3·87 √ h km (d in metres). (10.16)
In section 8.7, it was seen that the only practical way to give predictions of the positions of celestial
objects within the Solar System was by using a geocentric celestial sphere. By doing this, the multitude
of different directions of a particular object, say the Moon, as seen at one time by all observers scattered
over the finite-sized Earth could be reduced to one direction only, that seen by a hypothetical observer
at the Earth’s centre. This is particularly necessary where artificial Earth satellites are concerned.
Any satellite is so near the Earth that its apparent zenith distances, as measured by two observers in
different geographical positions not too far apart, can differ by tens of degrees. For the Moon itself,
this difference can be of order one degree, i.e. approximately twice its apparent angular diameter. For
the nearby planets the differences can amount to a considerable fraction of 1 minute of arc. For the Sun
the variation in apparent zenith distance at any one time due to differing geographical position of the
observer is a few seconds of arc. Even the smallest of these quantities is much larger than the accuracy
to which astronomical observations are taken.
The problem, therefore, arises of reducing topocentric observations of a body’s position to
the corresponding geocentric positions or of abstracting geocentric positions from an almanac and
transforming them to the particular observer’s local coordinate system. This problem is bound up
historically with the attempts made in past centuries to measure accurately the Moon’s distance and the
value of the astronomical unit (section 13.8), i.e. the mean Earth–Sun distance in kilometres.
The principle employed was essentially the same as the age-old one used by the surveyor in
obtaining the distances of specific points without actually measuring to these points with ruler or chain.
For example, in figure 10.6 we have a broad river whose width we wish to measure. Opposite a
tree or post C on the far bank, we measure out a baseline AB. With a protractor we measure angles
CAB and CBA. The information we now have (two angles and the included side) enables us to draw
the triangle ABC to a suitable scale or, by the sine-formula, to calculate the distance BC across the
river. The angle ABC, or angle p, is called the parallactic angle. It is, of course, equal to 180 ◦ minus
the sum of the angles at A and B and is essentially the apparent change in the direction of the tree or
post C as one moves from A to B. That this is so is easily seen if we draw BD parallel to AC. Then
CBD = p.
It is important that the base-line is not too small in comparison to the distance to be measured.
Essentially, the quantity measured is the parallactic angle p, the angle subtended at the object by the
base-line. If our accuracy of measurement is 1 second of arc, the order of accuracy in measuring the
required distance is indicated in table 10.2. It is seen that the accuracy of the result falls off rapidly with