# High-Precision Measurement of the Earth-Moon Distance

M. Schneider, J. Müller, U. Schreiber, D. Egger

(german version translated and shortened by D.Egger)

## Contents

- Abstract
- The Measurement Process
- Detecting lunar echos
- Analyzing the measurements
- Concluding Remarks
- Bibliography

## Abstract

Since 1969 several Earth-bound stations have accomplished measurements of the Earth-Moon distance by laser ranging. The analysis of the data revealed a lot of details concerning the dynamics of the Earth-Moon system especially by the description with the gravity theory of Einstein.

Having a short introductin to the measurement process you will be guided through the detection of laser echos and the evaluation model to finally read the results of the parameter estimation.

## The Measurement Process

During the Apollo missions (11, 14 and 15) the astronauts have been placing retroreflector arrays on the lunar surface. (Two russian unmanned missions have deployed reflector arrays too, one failed completely probably due to being covered by dust and the other one seems to be less efficient than the Apollo reflectors).

Short laser pulses (microseconds duration in the beginnings to about 100 picoseconds nowadays, these are 0.0001 microseconds) are being sent to the reflectors, get reflected by the retro reflectors (triple prisms where the incoming and the outgoing direction coincide) and finally get received by the ground station.

Counting the round trip travel time of the laser pulses directly gives the distance by multiplication (half of it) with the well defined speed of light.

Now let's shed some light on the very special conditions which make lunar laser ranging much more difficult than laser ranging to satellites.

A single laser pulse may consist of almost 10^{18} photons within a length of about 3 cm (full width half maximum) when leaving the transmit telescope.

(The pulse length directly defines the achievable accuracy of the measurement.)

Nd:YAG solid state lasers may deliver 10 such pulses per second, each carrying the energy of some hundred millijoules corresponding to more than 1 GigaWatt of pulse power (but only for a very short time of 0.000 000 000 1 seconds).

During its journey to the Moon and back again (2 times 380 000 km within about 2.5 seconds) the laser pulse encounters a lot of inconveniences which diminish significantly its strength:

- atmosphere (forth and back, one way extinction factor about 0.1 to 0.5)
- arriving at the Moon, the laser pulse has a diameter of more than 5 km. Only a very small portion corresponding to less than 1 square meter is being reflected by the reflector array.
- returning to the ground station the reflected pulses show diameters of more than 5 km. Only a very small portion corresponding to the aperture of the receive telescope typically about 1 square meter is being received.
- detector (efficiency 0.1 to maybe 0.5)
- imperfect technical components

Figure 1: Laser Ranging System of the fundamental station in Wettzell (Bavarian Forest) showing the illuminated light path of the laser pulses. The green light results from frequency doubling the original laser wavelength of 1.064 µm. A single pulse has a diameter of 75 cm and is only 6 cm long.

As the same telescope is being used for transmitting and receiving signals a somehow sophisticated control unit has to manage the T/R switch and the detection of the transmitted pulse (to start the time counter) and the received pulse (to stop the time counter). Ten pulses per second or 25 pulses per round trip time are on the way till the first comes back. So you do not actually measure the complete round trip time with a stop watch as you would measure a 100 m race. What you do is mapping all the single events to your local time scale. Then by considering plausibility arguments you are going to determine time intervals which reflect the journey time of the pulses.

The translation of the optical pulse to an electronic pulse is done by a photomultiplier or preferably by an avalanche diode which shows much better detection efficiency. Having electronic pulses is necessary for being handled by all the electronic control equipment.

As the energy balance is so low in lunar laser ranging you will not detect every transmitted laser pulse even when precisely hitting the target. And as the background is not really dark you will always be able to detect photons from the Sun or the lunar surface or the Earth's atmosphere.

How to improve the signal to noise ratio ?

- Spectral filtering: Only pass those photons to the detector which are very close to the laser frequency.
- Spatial filtering: Only photons originating from a small region around the target are being considered. Meaning keep the field of view as small as possible.
- Temporal filtering: Only consider those photons which lie very close to the expected round trip travel time.

Does the measured round trip travel time reflect the Earth-Moon distance ? Of course it doesn't. It shows somewhat like the distance between some location at the ground station and some location near the reflector array.

**What to do?**

**First calibrate the measurement itself.**A tiny fraction of the transmit pulse is being led through a well defined distance at the laser site (may be an optical fibre) and detected by the receiver.**Second define the location which marks the starting point of the distance.**Preferably a point in space is chosen which does not vary when the measurement takes place (intersection of azimuth and elevation axis of the telescope)**Third define the location to serve as the end point of the distance.**This is mostly the same as the starting point. So placing a mirror at this point will deliver a zero valued distance which can be compared to the calibration measurement.**Try to define the effective reflection point at the lunar target.**Some theoretical work will help to achieve this besides an analysis of all the receive events (caused by reflected laser pulses)

OK, now the distance between your defined location at the laser site and the reflector array is known. Studying the Earth-Moon dynamics requires a distance between the body centers. You deduce them by applying a model. You improve the model by introducing new measurements (may be new scientific findings). And so on...

Nowadays about 10 300 successful measurements have been obtained. Mainly the LLR-stations of the McDonald Observatory in Texas, the Observatoire du Calern in Grasse and the Lure Observatory at Hawaii have been contributing to this success. A few successful measurements have also been contributed by the fundamental station at Wettzell.

## Detecting lunar echos

Due to the low energy balance, the light pollution and the detector noise it is a rather hard job to separate the reflected photons (signal) from stray photons (noise) which always pretend a "successful" measurement.

Even applying effective filters

- 1 Angstroem (0.1 nanometers) band pass for the laser wavelength
- a few minutes of arc field of view
- 200 to 400 nanoseconds band pass for the expected time of arrival

does not clearly reveal the lunar echoes.

But drawing all the events into a histogram (hits per interval of time) helps significantly stretching the cloud of measurements and thus to separate the wheat from the chaff (cf. figures 2, 3)

Figure 2: differences of measured and expected round trip travel times versus time

Each cross in figure 2 corresponds to a single measurement (drawn as difference "expected minus measured"). You will probably not be able to find out which measurement was successful and which one wasn't.

But now simply count the crosses which correspond to a certain small portion of difference (here the total bandwidth of differences is subdivided into 150 intervals) and take pleasure in recognizing the "lunar hits".

Figure 3: histogram of the differences. Having a precise model and precise measurements to reflectors would yield just a single peak in the zero interval.

Now it's an easy job to detect the "good" returns around the zero difference interval. These measurements are going to build a "normal point" by means of a statistical analysis. And this "normal point" is contributed to the community for improving or confirming their models of the Earth-Moon system.

## Evaluation of measured distances

We have a little bit more than 10000 normal points which define the movement of the Moon by means of an expensive analysis. The basic principle of the analysis is to compare the measurements with the predictions of a sophisticated model and applying a least-squares adjustment to improve the parameters of the model.

The nowadyas extremely high precision of the measurements requires to formulate the motions of all the main bodies of our solar system and additionally a lot of asteroids by making use of Einstein's gravitation theory.

Translation and rotation of the solar bodies as well as the transformations of reference systems and the signal propagation in gravitational fields are described consistently to the first post Newtonian approximation.

Some details:

- Einstein-Infeld-Hoffmann equations rule the bodies' movement
- Newtonian description of non-spherical shape of the Earth and the Moon
- Newtonian description of lunar tides
- parameterization of Einstein's theory to test some of its quantities like time variabilty of the gravitational constant or the validity of the strong equivalence principle
- modified Eulerian equations to model the rotation of the Moon including terms of elasticity and dissipation
- modelling Earth rotation
- IAU 19080 nutation series
- precession angles of Lieske et al.
- x, y of pole axis and fluctuation of the length of day originated from a combined solution of R.Gross (JPL)

Further considerations due to the complexity of the Earth-Moon system:

- the Earth is not a rigid body
- location of an observer varies due to the Earth tides and the drifts of the continental plates

Signal propagation suffers from

- the Earth's atmosphere (meteorological data required)
- the gravitational field of the Sun

Both effects amount to several meters and have to be taken into account when transforming the travel time of the laser pulse to a distance.

Finally one must be aware to transform all quantities (position and time) to the same reference system according to Einstein's theory.

The parameters deduced from the analysis of the lunar laser measurements may be divided into three groups:

- Global parameters of the Earth-Moon system
- Parameters characterizing the General Theory of Relativity
- Parameters describing the rotation of the Earth

**Global parameters of the Earth-Moon system** (sigma gives an estimation of the underlying accuracy)

- station coordinates (sigma ~ 3--5 cm)
- selenocentric coordinates of the laser reflectors (some meters)
- physical libration (initial values)
- position and velocity of the Moon (initial values, some centimeters)
- position and velocity of the Earth
- multipole mass momentum of the Moon to degree and order of 3
- product of gravitational constant times the mass of the Earth-Moon system (sigma ~ 0.004 km
^{3}/s^{2}) - love number of the Moon plus a dissipation parameter
- tidal acceleration of the Moon. Due to tidal friction the rotation of the Earth slows down, the angular momentum is conserved and thus the Moon recedes about 3.8 cm per year.
- correction of the lunisolar constant of precession (sigma ~ 0.3 mas/year, mas = milliarcsecond)
- four nutation coefficients corresponding to the cycle of 18.6 years (two in-phase and two out-of-phase) (sigma ~ 1--3 mas)

**Parameters characterizing the General Theory of Relativity** (in parentheses you find the values given by the GRT, table 1 shows the maximum deviations resulting from the analysis of lunar laser data)

- quadrupole momentum of the Sun J
_{2}(~ 10^{-7}; J_{2}is treated as relativistic parameter because it contributes to the abnormal rotation of Mercury's perihelion) - factor of space curvature gamma (= 1) and the parameter of nonlinearity beta (= 1)
- the geodetic precession of the Moon's orbit omega
_{GP}(~ 1.9 "/cent.) - the Nordtvedt parameter eta (= 0) which indicates a deviation from the strong equivalence principle (inert mass and heavy mass are not proportional)
- time dependent gravitational constant (d/dt G)/G (= 0 per year)
- coupling constant alpha (= 0) of the Yukawa potential showing if Newton's 1/r
^{2}law is also valid for the Earth-Moon distance - a combination of parameters zeta
_{1}- zeta_{0}- 1 (= 0), indicating a violation of the Special Theory of Relativity and thus allowing a preferred reference system - a parameter delta
_{Gal}(= 0 cm/s^{2}), showing the influence of dark matter upon the Earth-Moon distance (testing the validity of the equivalence principle for dark matter) - parameters alpha
_{1}(= 0) and alpha_{2}(= 0), describing the existence of a preferred reference system in the framework of General Relativity

Parameter |
Error |

Sun's quadrupole J_{2} |
< 5 ·10^{-6} |

geod. Prec. Omega_{GP} ["/cent.] |
< 1.5 ·10^{-2} |

Metric par. gamma | < 6 ·10^{-3} |

Metric par. beta | < 4 ·10^{-3} |

Nordtvedt par. eta | < 1 ·10^{-3} |

time var. grav.const. (d/dt G)/G [yr^{-1}] |
< 5 ·10^{-12} |

Yuk. coupl.const. alpha_{lambda} = 4·10^{5} km |
< 1 ·10^{-11} |

spec. relativi. zeta_{1} - zeta_{0} - 1 |
< 1.5 ·10^{-4} |

infl. dark mat. delta g_{c} [cm/s^{2}] |
< 3 ·10^{-14} |

Effect due to pref. ref.sys. alpha_{1} |
< 9 ·10^{-5} |

Effect due to pref. ref.sys. alpha_{2} |
< 2.5 ·10^{-5} |

**3. Parameters describing the rotation of the Earth**

- phase of rotation UT0 at the measurement epochs (variations of the length of day) better than 0.1 milliseconds accuracy
- variation of the observer's latitude (due to polar wobble) at the measurement epochs better than 1 mas precision
- drift rates of the quantities describing the rotation of the Earth

The software may be easily adjusted to include further parameters or improvements of the model.

The internal model accuracy is as good as not to deviate more than 1 cm in the Earth-Moon distance within 30 (model) years of integration.

Actual measurements are reaching an accuracy of 5 mm, in a few years 3 mm seem to be feasible. Thus the model accuracy has to be refined too.

Some of the improvements to be done:

- perturbations of up to 1000 asteroids
- more gravitational potential parameters of the Earth and the Moon
- relativistic coupling terms of spin and orbit
- relativistic description of angular momentum and torque (Moon)
- relativistic description of torque exerted by the planets
- improved nutation theory
- tidal forces due to the oceans and the atmosphere
- improved description of the non-rigid Earth and Moon

Some effects like the fluctuation of the rotation of the Earth are hardly to describe theoretically and have further to be derived from actual measuremnts.

## Concluding Remarks

The determination of the exact distance Earth-Moon is a complex undertaking. Measuring as well as analyzing skills are highly demanded. But it's worth the effort: You may check a lot of parameters describing the model of the Earth-Moon system including the underlying theoretical approaches. Besides the high accuracy of the measurements another advantage is the long period of observations covering 27 years.

It would be desirable to have more stations doing laser ranging to the Moon. They would help to separate latitude-dependent parameters. Nonetheless at least one station at a time should measure the Moon to get an unbroken series of data for future analyses.

## Bibliography

Bender, P. L. et. al.: The Lunar Laser Ranging Experiment. In: Science (1973), Vol. 182, No. 4109, 229-238

Egger, D.: Systemanalyse der Laserentfernungsmessung. Veröffentlichung der Deutschen Geodätischen Kommission, Reihe C, Nr. 311, München 1985

Müller, J.: Analyse von Lasermessungen zum Mond im Rahmen einer post-Newtonschen Theorie. Veröffentlichung der Deutschen Geodätischen Kommission, Reihe C, Nr. 383, München 1991

Müller, J., Schneider, M., Soffel, M., Ruder, H.: Testing Einstein's Theory of Gravity by Lunar Laser Ranging. In: Symposia Gaussiana, Conference A, Proceedings of the 2nd Gauss Symposium, Munich, Germany, August 2-7, 1993, ed. by M.Behara, R.Fritsch, R.G.Lintz, Walter de Gruyter & Co., Berlin/New York 1995, P. 637-647

Schneider, M.: Himmelsmechanik, Band III: Gravitationstheorie. Spektrum Akademischer Verlag, Heidelberg 1996.