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The passive retroreflector arrays placed on the moon by Apollo 11, 14 and 15 astronauts continue to produce valuable Earth-Moon range measurements that enable high-precision tests of gravitational physics, as well as studies of geo- and selenophysics. The optical throughput of these retroreflectors has declined since their deployment, with an additional signal loss at full moon when the reflectors experience direct solar illumination. We show that the loss in return rate can be attributed to the accumulation of a thin layer of lunar dust on the surfaces of the corner cube retroreflectors. First, a careful analysis of the optical link budget for the Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) experiment reveals that the lunar return rate is 15--20 times smaller than predicted, a deficit that can be explained by a reflector dust covering fraction of ${\sim} 50$\%. Second, range measurements taken during a lunar eclipse indicate that the solar illumination of the retroreflectors degrades their throughput by an additional factor of ${\sim}15$. Finally, a numerical simulation of heat transfer in dust-coated reflectors is able to model the resulting thermal lensing effect, in which thermal gradients in the retroreflectors degrade their far-field diffraction pattern. A comparison of this simulation to eclipse measurements finds a dust coverage fraction of ${\sim}50$%. Taken together, the link analysis, eclipse observations and thermal modeling support the claim that the retroreflectors are obscured by lunar dust, with both link budget and simulation independently finding the dust fraction to be $\sim$50%.

The Apache Point Lunar Laser-ranging Operation (APOLLO) has been collecting lunar range measurements for 15 years at millimeter accuracy. The median nightly range uncertainty since 2006 is 1.7 mm. A recently added Absolute Calibration System (ACS), providing an independent assessment of APOLLO system accuracy and the capability to correct lunar range data, revealed a 0.4% systematic error in the calibration of one piece of hardware that has been present for the entire history of APOLLO. Application of ACS-based timing corrections suggests systematic errors are reduced to < 1 mm, such that overall data accuracy and precision are both 1 mm. This paper describes the processing of APOLLO/ACS data that converts photon-by-photon range measurements into the aggregated normal points that are used for physics analyses. Additionally we present methodologies to estimate timing corrections for range data lacking contemporaneous ACS photons, including range data collected prior to installation of the ACS. We also provide access to the full 15-year archive of APOLLO normal points (2006-04-06 to 2020-12-27)., Comment: 23 pages, 9 figures

We present data from the Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) covering the 15-year span from April 2006 through the end of 2020. APOLLO measures the earth-moon separation by recording the round-trip travel time of photons from the Apache Point Observatory to five retro-reflector arrays on the moon. The APOLLO data set, combined with the 50-year archive of measurements from other lunar laser ranging (LLR) stations, can be used to probe fundamental physics such as gravity and Lorentz symmetry, as well as properties of the moon itself. We show that range measurements performed by APOLLO since 2006 have a median nightly accuracy of 1.7 mm, which is significantly better than other LLR stations., Comment: 16 pages, 9 figures

We describe for the first time in the scientific literature the Planetary Ephemeris Program (PEP), an open-source general-purpose astrometric data analysis program. We discuss, in particular, the implementation of pulsar timing analysis, which was recently upgraded in PEP to handle more options. This implementation was done independently of other pulsar programs, with minor exceptions that we discuss. We illustrate the implementation of this capability by comparing the post-fit residuals from the analyses of time-of-arrival observations by both PEP and Tempo2. The comparison shows substantial agreement: 22 ns rms differences for 1,065 pulse time-of-arrival measurements for the millisecond pulsar in a binary system, PSR J1909-3744 (pulse period 2.947108 ms; full-width half-maximum of pulse 43 $\mu$s) for epochs in the interval from December 2002 to February 2011., Comment: 9 pages, 1 figure, submitted to Astronomical Journal (AJ)

The Apache Point Lunar Laser-ranging Operation (APOLLO) has been collecting lunar range measurements for 15 yr at millimeter accuracy. The median nightly range uncertainty since 2006 is 1.7 mm. A recently added Absolute Calibration System (ACS), providing an independent assessment of APOLLO system accuracy and the capability to correct lunar range data, revealed a ∼0.4% (10 ps) systematic error in the calibration of one piece of hardware that has been present for the entire history of APOLLO. The application of ACS-based timing corrections suggests systematic errors are reduced to <1 mm, such that overall data accuracy and precision are both ∼1 mm. This paper describes the processing of APOLLO/ACS data that converts photon-by-photon range measurements into the aggregated normal points that are used for science analyses. Additionally, we present methodologies to estimate timing corrections for range data lacking contemporaneous ACS photons, including range data collected prior to installation of the ACS. We also provide access to the full 15 yr archive of APOLLO normal points (2006 April 6–2020 December 27).

The Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) began millimeter-precision ranging to the Moon in 2006. Until now, a comprehensive validation of APOLLO system range accuracy has not been possible because of centimeter-scale deficiencies in computational models of the Earth-Moon range, and because APOLLO lacked an internal timing calibration system. Here, we report on the development of a system that enables in-situ calibration of the timing response of the APOLLO apparatus, simultaneous with lunar range measurements. The system was installed in August 2016. Preliminary results show that the APOLLO system can provide lunar range measurements with millimeter accuracy., Comment: 6 pages, 4 figures. Proceedings of the 20th International Workshop on Laser Ranging, Potsdam, Germany, October 9--14, 2016

Lunar laser ranging provides a number of leading experimental tests of gravitation -- important in our quest to unify General Relativity and the Standard Model of physics. The Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) has for years achieved median range precision at the ~2 mm level. Yet residuals in model-measurement comparisons are an order-of-magnitude larger, raising the question of whether the ranging data are not nearly as accurate as they are precise, or if the models are incomplete or ill-conditioned. This paper describes a new absolute calibration system (ACS) intended both as a tool for exposing and eliminating sources of systematic error, and also as a means to directly calibrate ranging data in-situ. The system consists of a high-repetition-rate (80 MHz) laser emitting short (< 10 ps) pulses that are locked to a cesium clock. In essence, the ACS delivers photons to the APOLLO detector at exquisitely well-defined time intervals as a "truth" input against which APOLLO's timing performance may be judged and corrected. Preliminary analysis indicates no inaccuracies in APOLLO data beyond the ~3 mm level, suggesting that historical APOLLO data are of high quality and motivating continued work on model capabilities. The ACS provides the means to deliver APOLLO data both accurate and precise below the 2 mm level., Comment: 21 pages, 10 figures, submitted to Classical and Quantum Gravity

The Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) has produced a large volume of high-quality lunar laser ranging (LLR) data since it began operating in 2006. For most of this period, APOLLO has relied on a GPS-disciplined, high-stability quartz oscillator as its frequency and time standard. The recent addition of a cesium clock as part of a timing calibration system initiated a comparison campaign between the two clocks. This has allowed correction of APOLLO range measurements--called normal points--during the overlap period, but also revealed a mechanism to correct for systematic range offsets due to clock errors in historical APOLLO data. Drift of the GPS clock on ~1000 s timescales contributed typically 2.5 mm of range error to APOLLO measurements, and we find that this may be reduced to ~1.6 mm on average. We present here a characterization of APOLLO clock errors, the method by which we correct historical data, and the resulting statistics., Comment: 12 pages, 7 figures, submitted to Classical and Quantum Gravity

Many corner cube prisms, or retroreflectors, employ total internal reflection (TIR) via uncoated rear surfaces. The different elliptical polarization states emerging from the six unique paths through the corner cube complicate the far-field diffraction pattern by introducing various phase delays between the six paths. In this paper, we present a computational framework to evaluate polarization through TIR corner cubes for arbitrary incidence angles and input polarization states, presenting example output for key normal-incidence conditions. We also describe a method to produce far-field diffraction patterns resulting from the polarization analysis, presenting representative images---broken into orthogonal polarizations, and characterizing key cases. Laboratory confirmation is also presented for both polarization states and far-field diffraction patterns., Comment: 22 pages, 11 figures

Uncoated corner cube retroreflectors (CCRs) operating via total internal reflection (TIR) are less susceptible to internal heating than their metal-coated analogs, lacking an absorber on the rear surface. Even so, environments that induce differential heating within the CCR will result in thermal lensing of the incident wavefront, introducing aberrations that will generally reduce the central irradiance of the polarization-sensitive far-field diffraction pattern (FFDP). In this paper, we characterize the sensitivity of TIR CCRs to axial and radial thermal gradients. We present simulated FFDPs for key input polarizations and incidence angles and provide a generalized analytic model that approximates the behavior of the central irradiance as temperature differences are introduced., Comment: 22 pages, 9 figures

As lighting efficiency improves, it is useful to understand the theoretical limits to luminous efficacy for light that we perceive as white. Independent of the efficiency with which photons are generated, there exists a spectrally-imposed limit to the luminous efficacy of any source of photons. We find that, depending on the acceptable bandpass and---to a lesser extent---the color temperature of the light, the ideal white light source achieves a spectral luminous efficacy of 250--370 lm/W. This is consistent with previous calculations, but here we explore the maximum luminous efficacy as a function of photopic sensitivity threshold, color temperature, and color rendering index; deriving peak performance as a function of all three parameters. We also present example experimental spectra from a variety of light sources, quantifying the intrinsic efficacy of their spectral distributions., Comment: 8 pages; 9 figures

Lunar laser ranging has provided many of the best tests of gravitation since the first Apollo astronauts landed on the Moon. The march to higher precision continues to this day, now entering the millimeter regime, and promising continued improvement in scientific results. This review introduces key aspects of the technique, details the motivations, observables, and results for a variety of science objectives, summarizes the current state of the art, highlights new developments in the field, describes the modeling challenges, and looks to the future of the enterprise., Comment: 45 page review article; 20 figures

Laser ranging measurements during the total lunar eclipse on 2010 December 21 verify previously suspected thermal lensing in the retroreflectors left on the lunar surface by the Apollo astronauts. Signal levels during the eclipse far exceeded those historically seen at full moon, and varied over an order of magnitude as the eclipse progressed. These variations can be understood via a straightforward thermal scenario involving solar absorption by a ~50% covering of dust that has accumulated on the front surfaces of the reflectors. The same mechanism can explain the long-term degradation of signal from the reflectors as well as the acute signal deficit observed near full moon., Comment: 23 pages, 11 figures, accepted for publication in Icarus

In 1970, the Soviet Lunokhod 1 rover delivered a French-built laser reflector to the Moon. Although a few range measurements were made within three months of its landing, these measurements---and any that may have followed---are unpublished and unavailable. The Lunokhod 1 reflector was, therefore, effectively lost until March of 2010 when images from the Lunar Reconnaissance Orbiter (LRO) provided a positive identification of the rover and determined its coordinates with uncertainties of about 100 m. This allowed the Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) to quickly acquire a laser signal. The reflector appears to be in excellent condition, delivering a signal roughly four times stronger than its twin reflector on the Lunokhod 2 rover. The Lunokhod 1 reflector is especially valuable for science because it is closer to the Moon's limb than any of the other reflectors and, unlike the Lunokhod 2 reflector, we find that it is usable during the lunar day. We report the selenographic position of the reflector to few-centimeter accuracy, comment on the health of the reflector, and illustrate the value of this reflector for achieving science goals., Comment: 11 pages, 1 figure (color), submitted to Icarus; revision addresses reviewer comments

Phobos Laser Ranging (PLR) is a concept for a space mission designed to advance tests of relativistic gravity in the solar system. PLR's primary objective is to measure the curvature of space around the Sun, represented by the Eddington parameter $\gamma$, with an accuracy of two parts in $10^7$, thereby improving today's best result by two orders of magnitude. Other mission goals include measurements of the time-rate-of-change of the gravitational constant, $G$ and of the gravitational inverse square law at 1.5 AU distances--with up to two orders-of-magnitude improvement for each. The science parameters will be estimated using laser ranging measurements of the distance between an Earth station and an active laser transponder on Phobos capable of reaching mm-level range resolution. A transponder on Phobos sending 0.25 mJ, 10 ps pulses at 1 kHz, and receiving asynchronous 1 kHz pulses from earth via a 12 cm aperture will permit links that even at maximum range will exceed a photon per second. A total measurement precision of 50 ps demands a few hundred photons to average to 1 mm (3.3 ps) range precision. Existing satellite laser ranging (SLR) facilities--with appropriate augmentation--may be able to participate in PLR. Since Phobos' orbital period is about 8 hours, each observatory is guaranteed visibility of the Phobos instrument every Earth day. Given the current technology readiness level, PLR could be started in 2011 for launch in 2016 for 3 years of science operations. We discuss the PLR's science objectives, instrument, and mission design. We also present the details of science simulations performed to support the mission's primary objectives., Comment: 25 pages, 10 figures, 9 tables

Forty years ago, Apollo astronauts placed the first of several retroreflector arrays on the lunar surface. Their continued usefulness for laser-ranging might suggest that the lunar environment does not damage optical devices. However, new laser ranging data reveal that the efficiency of the three Apollo reflector arrays is now diminished by a factor of ten at all lunar phases and by an additional factor of ten when the lunar phase is near full moon. These deficits did not exist in the earliest years of lunar ranging, indicating that the lunar environment damages optical equipment on the timescale of decades. Dust or abrasion on the front faces of the corner-cube prisms may be responsible, reducing their reflectivity and degrading their thermal performance when exposed to face-on sunlight at full moon. These mechanisms can be tested using laboratory simulations and must be understood before designing equipment destined for the moon., Comment: 9 pages; 4 figures; accepted for publication in Icarus

When scientific experiments require transmission of powerful laser or radio beams through the atmosphere the Federal Aviation Administration (FAA) requires that precautions be taken to avoid inadvertent illumination of aircraft. Here we describe a highly reliable system for detecting aircraft entering the vicinity of a laser beam by making use of the Air Traffic Control (ATC) transponders required on most aircraft. This system uses two antennas, both aligned with the laser beam. One antenna has a broad beam and the other has a narrow beam. The ratio of the transponder power received in the narrow beam to that received in the broad beam gives a measure of the angular distance of the aircraft from the axis that is independent of the range or the transmitter power. This ratio is easily measured and can be used to shutter the laser when the aircraft is too close to the beam. Prototype systems operating on astronomical telescopes have produced good results., Comment: 16 pages; 8 figures (3 in color); submitted to PASP

The recent discovery of "dark energy" has challenged Einstein's general theory of relativity as a complete model for our macroscopic universe. From a theoretical view, the challenge is even stronger: general relativity clearly does not extend to the very small, where quantum mechanics holds sway. Fundamental physics models thus require some major revisions. We must explore deeper to both constrain and inspire this needed new physics. In the realm of the solar-system, we can effectively probe for small deviations from the predictions of general relativity: Technology now offers a wide range of opportunities to pursue experiments with accuracies orders of magnitude better than yet achieved. We describe both the relevant theoretical backgrounds and the opportunities for far more accurate solar system experiments., Comment: 8 pages, LaTeX, Astro2010 Science White Paper

A next-generation lunar laser ranging apparatus using the 3.5 m telescope at the Apache Point Observatory in southern New Mexico has begun science operation. APOLLO (the Apache Point Observatory Lunar Laser-ranging Operation) has achieved one-millimeter range precision to the moon which should lead to approximately one-order-of-magnitude improvements in the precision of several tests of fundamental properties of gravity. We briefly motivate the scientific goals, and then give a detailed discussion of the APOLLO instrumentation., Comment: 37 pages; 10 figures; 1 table: accepted for publication in PASP

Lunar laser ranging analysis, as regularly performed in the solar system barycentric frame, requires the presence of the gravitomagnetic term in the equation of motion at the strength predicted by general relativity. The same term is responsible for the Lense Thirring effect. Any attempt to modify the strength of the gravitomagnetic interaction would have to do so in a way that does not destroy the fit to lunar ranging data and other observations., Comment: 1 page; accepted for publication in Physcal Review Letters; refers to gr-qc/0702028

Gravitomagnetism--a motional coupling of matter analogous to the Lorentz force in electromagnetism--has observable consequences for any scenario involving differing mass currents. Examples include gyroscopes located near a rotating massive body, and the interaction of two orbiting bodies. In the former case, the resulting precession of the gyroscope is often called ``frame dragging,'' and is the principal measurement sought by the Gravity Probe-B experiment. The latter case is realized in the earth-moon system, and the effect has in fact been confirmed via lunar laser ranging (LLR) to approximately 0.1% accuracy--better than the anticipated accuracy of the Gravity-Probe-B result. This paper shows the connnection between these seemingly disparate phenomena by employing the same gravitomagnetic term in the equation of motion to obtain both gyroscopic precession and modification of the lunar orbit. Since lunar ranging currently provides a part in a thousand fit to the gravitomagnetic contributions to the lunar orbit, this feature of post-Newtonian gravity is not adjustable to fit any anomalous result beyond the 0.1% level from Gravity Probe-B without disturbing the existing fit of theory to the 36 years of LLR data., Comment: 4 pages; accepted for publication in Physical Review Letters