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).

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:

• Explain how you may calculate the distance BC as a function of b₁, b₂ and R_T.

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 p_L 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 p_L = R_T / OM (in the triangle EOM, the angle MEO is a right angle).

• Apply the sinus rule to triangle OBM and to triangle OCM and prove the following relations:

  R_T / OM = sin x₁ / sin z₁ = sin x₂ / sin z₂ = (sin x₁ + sin x₂) / (sin z₁ + sin z₂); and

  sin x₁ + sin x₂ = 2 * sin [(x₁ + x₂) / 2] * cos [(x₁ - x₂) / 2]

• Explain why cos [(x₁ - x₂) / 2] = 1

  Prove the relation: x₁ + x₂ = z₁ + z₂ - (a₁ + a₂)

• Now you can calculate the Moon's parallax p_L:

  On August 31, 1752, Lalande obtained: z₁ = 33.11° in Berlin; and La Caille: z₂ = 55.14° in Le Cap

• What is the corresponding distance D, expressed in kilometres?

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'.

• Calculate the extremes of the distances of the Moon, D_max and D_min, and deduce from them the Moon's orbital excentricity e by means of this formula:

  e = (D_max - D_min) / (D_max + D_min)