# Laser Measurements of the Earth-Moon distance

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15 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

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Laser Measurements of the Earth
-
Moon distance

Only a few metres away from the first lunar footprint in 1969, Edwin (Buzz) Aldrin set up a
retro
-
reflector which would reflect light from the Earth straight back to its source. This retro
-
reflector was constr
ucted from a 10

10 array of 3∙8 cm diameter retro
-
mirrors, which were of
French manufacture.

Anyone on the Earth who sends out a sufficiently intense and parallel light signal to the
landing site of Apollo 11 would, after a short time, receive the reflect
ed signal. Since that
time, many observatories have measured the travel time of the signals and from it determined
the Earth
-
Moon distance to the accuracy of millimetres. Laser pulses sent out through a
telescope satisfy the above criteria. Further retro
-
r
eflectors were placed by Apollos 14 and 15
as well as on the roof of the previously mobile Soviet Lunochod 2 (it also had French
mirrors).

One of these measurement programmes is carried out at the Apache Point Observatory [see
http://www.apo.nmsu.edu/
].

Exercises.

1

(a)

Determine the travel time of a laser pulse from the Earth to the Moon and back. The
speed of light is 299 792 458 m s
-
1

and the mean Earth
-
Moon distance is 384 400 km.

(b)

What is the order of m
agnitude of the reduction in travel time resulting from the fact
that the laser and reflector are not sited at the centres of the Earth and Moon
respectively?

[
R
E

= 6378 km

R
M

= 1738 km]

(c)

The effect in (b) is easy to take into account, because satelli
tes have measured the
shape of the Earth and Moon to a high precision.

Another factor is that the laser light must pass (twice) through the atmosphere.

(i)

What time delay would result from the refraction of light in the atmosphere, which
may be consid
ered to be a uniform layer, 8 km thick, with refractive index
1∙0003?

(ii)

What error in the Earth
-
Moon distance would result from not taking this refraction
into account?

Apollo, the
A
pache
Po
int
L
unar
L
aser
-
ranging
O
peration, sends out laser pulses of

duration
τ=
115 ps to the moon, the intensity maximum of which can be determined to about 25 ps.
The highly collimated [parallel] laser pulse suffers a divergence of only
δ
L

= 1".
Unfortunately the retro
-
reflector produces a greater divergence of
δ
R

= 7".

[1" = 4

10
-
6

2.

The laser pulses are sent out by the 3∙5 m diameter telescope at Apache Point. The
theoretical divergence because of diffraction of light of wavelength
λ

passing through a
circular aperture of diameter
D
, is ~
λ
/
D
. How close is th
e telescope to achieving this
diffraction limit?

2.

(a)

What is the size (diameter,
D
M,

and thickness,
d
) of the sheet of photons at the
position of the lunar retro reflector?

(b)

What does the diameter
D
E

reach, by the time the pulse gets back to
Ea
rth
?

3.

How many photons are there in a laser pulse of energy
E

= 115 mJ? [
λ

= 532 nm]

4.

How many photons, per laser pulse reach the 3∙5 m diameter telescope at Apache point?

Assume that the photons are uniformly spread over the diameter of the photon sheet.

Hint: You will need to consider the area of the retro
-
mirror.