Answer by PM 2Ring for How does a laser from Earth manage to hit the Moon with precision?

As Rob mentions, when reflecting a laser off the Moon you do need to compensate by ~1.4 arc-minutes for the light-time delay. But the whole point of the Lunar Laser Ranging experiments (LLR) is to improve our model of the Moon's orbit. So if the people shooting the laser beams can't get it right there's something terribly wrong. ;)

We've been doing LLR for over 50 years, since Buzz Aldrin placed the first reflector on the Moon. There are currently 6 functional LLR reflectors and it's expected that additional reflectors will be placed as part of NASA's Artemis program. The latest reflector design is more efficient, and having reflectors at more locations improves the data. It's difficult to do LLR with reflectors that are in sunlight, since the reflectors only return a tiny number of laser photons. Also, the reflectors that are mounted on landers are unusable near Full Moon due to thermal problems.

The Jet Propulsion Laboratory (owned and sponsored by NASA) have been calculating the motions of the Solar System bodies for decades. By the mid 1980s, the Jet Propulsion Laboratory Development Ephemeris (JPL DE) was so good that it became the basis of the Astronomical Almanac.

The JPL DE is computed by integrating the equations of motion of the major Solar System bodies (including several hundred asteroids), relative to the barycentre of the Solar System, using a post-Newtonian approximation of the laws of General Relativity. This model is fitted to a huge amount of observational data, both ground-based and space-based. The LLR data is a vital component of that observational data. For some details on the ephemeris computation, please seeThe JPL Planetary and Lunar Ephemerides DE440 and DE441, Park et al (2021).

The output of these ephemeris computations is stored in the form of 14th degree Chebyshev polynomials, which permits very accurate interpolation of the position and velocity values.

Scientists and engineers can access all of this data via NASA's SPICE system. And anyone can access (much of) SPICE via the Horizons system provided by JPL's Solar System Dynamics group. Horizons can be accessed via telnet, email, a Web app, and by query- and file-based APIs. It has a few quirks because it's so ancient (the core system predates the World Wide Web), but it's fast, and easy to work with, once you become familiar with it. Horizons data spans from 9999 BC to 9999 AD.

Horizons has location data for many places on Earth, Mars, and the Moon, including all the Apollo landing sites and the LLR reflectors. This query URL retrieves the Moon locations data (in cylindrical coordinates).

Here's a minimal Python program using the file API to retrieve some data relevant to LLR. It prints the azimuth & elevation of the Moon (in degrees), and the light-time distance (in minutes), as seen by the Apache Point observatory (which has observatory code 705). The selected time is ~12 hours after the recent Last Quarter phase of the Moon. I've selected the Apollo 11 reflector as the target location. At that time, the reflector is not in sunlight.

""" Retrieve data from Horizons using a batch-file    Written by PM 2Ring 2021.12.27"""import requestsurl = "https://ssd.jpl.nasa.gov/api/horizons_file.api"cmd = """\!$$SOFOBJ_DATA=NOCOMMAND='c:23.47307,1735.35247,20.39813 @301'CENTER=705@399QUANTITIES='4,21'APPARENT=REFRACTEDCAL_TYPE=GSTART_TIME='2024-Feb-3 4:00 UT-7'STOP_TIME='2024-Feb-3 4:01'STEP_SIZE='12'"""req = requests.post(url, data={"format": "text"}, files={"input": ("cmd", cmd)})print(req.text)

Here's a live version of that script, running on the SageMathCell server. And here's the same request, in a query URL.

Here's an excerpt from the output. Please use one of the previous links to see the full output.

 Date_(ZONE)_HR:MN:SC.fff         Azi____(r-app)___Elev  1-way_down_LT**********************************************************************$$SOE 2024-Feb-03 04:00:00.000  m  N-  143.588295  27.029606     0.02153966 2024-Feb-03 04:00:05.000  m  N-  143.606569  27.039511     0.02153959 2024-Feb-03 04:00:10.000  m  N-  143.624847  27.049412     0.02153951 2024-Feb-03 04:00:15.000  m  N-  143.643129  27.059309     0.02153944 2024-Feb-03 04:00:20.000  m  N-  143.661417  27.069201     0.02153937 2024-Feb-03 04:00:25.000  m  N-  143.679708  27.079088     0.02153930 2024-Feb-03 04:00:30.000  m  N-  143.698005  27.088972     0.02153923 2024-Feb-03 04:00:35.000  m  N-  143.716306  27.098851     0.02153916 2024-Feb-03 04:00:40.000  m  N-  143.734612  27.108725     0.02153909 2024-Feb-03 04:00:45.000  m  N-  143.752922  27.118595     0.02153901 2024-Feb-03 04:00:50.000  m  N-  143.771237  27.128461     0.02153894 2024-Feb-03 04:00:55.000  m  N-  143.789557  27.138323     0.02153887 2024-Feb-03 04:01:00.000  m  N-  143.807881  27.148180     0.02153880$$EOE'Azi____(r-app)___Elev' =  Refracted apparent azimuth and elevation of SURFACE TARGET POINT. Compensatedfor light-time, the gravitational deflection of light, stellar aberration,approximate atmospheric yellow-light refraction, precession and nutation.Azimuth is measured clockwise from north:  North(0) -> East(90) -> South(180) -> West(270) -> North (360)Elevation angle is with respect to a plane perpendicular to the referencesurface local zenith direction. TOPOCENTRIC ONLY.  Units: DEGREES'1-way_down_LT' =   1-way down-leg light-time from SURFACE POINT target to observer. The elapsedtime since light (observed at print-time) would have left or reflected off thespecified target surface location. Units: MINUTES

We can see that during that minute, in 5 seconds the azimuth increases by ~0.02 degrees, the elevation increases by ~0.01 degrees, and the one-way light travel time decreases by ~4.2 microseconds.

I have simple Sage / Python code on Github which lets you play with Horizons batch file data. The Horizons Web app provides a link for the batch data for any request you make through it.

Accurate lunar orbit data is essential for our Solar System model. We want to know where various Solar System bodies are relative to Earth. But the Earth's motion is intimately tied to the Moon. The JPL DE doesn't directly model the motion of the Earth. It computes the motion of the Earth-Moon barycentre (EMB), and it computes the motion of the Moon relative to that barycentre. Whenever SPICE needs to know the position & velocity of the Earth it computes it from the EMB and the Moon.

The precise rotation of the Earth is monitored by the International Earth Rotation and Reference Systems Service, who publish the latest precise rotation data daily. SPICE / Horizons stays up-to-date with that data, and estimates Earth's rotation if you request ephemeris data for future dates.