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III.1 The Moon Parallax

The method by Lalande and La Caille (1752)

When astronomers can observe a celestial object simultaneously from two distant observation sites and can also measure their mutual distance(in kilometres), they are able to use the method of triangulation in order to determine the distance to the Moon (and the nearest planets).

Determination of the distance to the Moon by means of the triangulation method.

In 1752, the French astronomers Lalande and La Caille travelled to two observing sites which were located on more or less the same meridian:

Berlin: latitude b1=52,31 deg N; longitude l1=13,24 deg E

Le Cap: latitude b2=33,55 deg N; longitude l2=18,22 deg E

We have:

x1 is the size of the angle OMB; and
x2 is the size of the angle OMC

In what follows, we disregard the small difference in geographical longitude and assume that B and C are located nearly on the same meridian (i.e., at the same geographical longitude). The figure above shows the geometry in the plane of this meridian.

Exercise 1:

Here, RT is the Earth's radius; RT = 6378 km (equatorial radius). We assume that the Earth is spherical.

Lalande, in Berlin, measured the angle z1 between the direction towards the Moon's center and the zenith direction, at the time of the Moon's passage on the meridian (i.e., the zenith distance of the Moon).

La Caille, in Le Cap, measured simultaneously the zenith distance of the Moon, z2, from his observing site.

In fact, they performed a lot of accurate measurements. The problem is very difficult. They had to consider the flattening of the Earth at the poles, in order to calculate the parallax from their observations.

The Moon parallax is the angle pL in the above figure. This is the angle that subtends the Earth's radius, as seen from the distance of the Moon.

Exercise 2:

Let us make a simplified calculation. We suppose that the Earth is spherical. Then:

sin pL = RT / OM (in the triangle EOM, the angle MEO is a right angle).

Exercise 3:

In fact, the Moon's orbit is not a circle, but an ellipse.

The extremes of parallax values, as measured in modern times, are 61.5' and 53.9'.

Please direct any related questions or remarks to Josee Sert (France) who will send on to the author: Martine Bobin

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