Can a laser be used to measure the rate of descent of a re-entry probe during its re-entry into a planet's or satellite's atmosphere?
Are there any probes that used a laser to measure its descent?
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Yes. This technology is commonly referred to as LIDAR (Laser Image Detection and Ranging) and is a combination (portmanteau) of words "light" and "radar":
Lidar is popularly known as a technology used to make high resolution maps, geomatics, archaeology, geography, geology, geomorphology, seismology, forestry, remote sensing, atmospheric physics, airborne laser swath mapping (ALSM), laser altimetry, and contour mapping.
As far as the use of LIDARs as altimeters in astronautics is concerned:
The general public became aware of the accuracy and usefulness of lidar systems in 1971 during the Apollo 15 mission, when astronauts used a laser altimeter to map the surface of the moon.
They are also used by Earth orbiting satellites to produce digital elevation maps, measure the distance to reflectors placed on the Moon, Mars Global Surveyor uses LIDAR for precise global topographic surveys, Phoenix Lander to detect snow in the atmosphere of Mars,...
Lidar is increasingly being utilized for range finding and orbital element calculation of relative velocity in proximity operations and station keeping of spacecraft. Lidar has also been used for atmospheric studies from space. Using short pulses of laser light beamed from a spacecraft, some of that "light reflects off of tiny particles in the atmosphere and back to a telescope aligned with the laser. By precisely timing the lidar 'echo,' and by measuring how much laser light is received by the telescope, scientists can accurately determine the location, distribution and nature of the particles. The result is a revolutionary new tool for studying constituents in the atmosphere, from cloud droplets to industrial pollutants, that are difficult to detect by other means.
It is also used in robotic missions to determine distance to the ground, obstacles, e.t.c.
American Lunar probe Clementine (Deep Space Program Science Experiment) also used LIDAR system to measure the distance from the spacecraft to a point on the surface of the Moon.