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Below is an overview of the claims as well as the associated debunking from various sources:

Photographs and films

Moon hoax proponents devote a substantial portion of their efforts to examining NASA photos. They point to various issues with photographs and films purportedly taken on the Moon. Experts in photography (even those unrelated to NASA) respond that the anomalies, while sometimes counterintuitive, are in fact precisely what one would expect from a real Moon landing, and contrary to what would occur with manipulated or studio imagery. Hoax proponents also state that whistleblowers may have deliberately manipulated the NASA photos in hope of exposing NASA

Accusation : Crosshairs appear to be behind objects.
Response : Overexposure causes white objects to bleed into the black areas on the film.

Accusation : Crosshairs are sometimes misplaced or rotated.
Response : Popular versions of photos are sometimes cropped or rotated for aesthetic impact.

Accusation : The quality of the photographs is implausibly high
Response : There are many, many poor quality photographs taken by the Apollo astronauts. NASA chose to publish only the best examples.

Accusation : The color and angle of shadows and light are inconsistent.
Response : Shadows on the Moon are complicated by uneven ground, wide angle lens distortion, light reflected from the Earth, and lunar dust, Shadows also display the properties of vanishing point perspective leading them to converge to a point on the horizon.

Accusation : There are no stars in any of the photos. The Apollo 11 astronauts also claimed to have not remembered seeing any of the stars in a press conference after the event.
Response : The sun was shining. Cameras were set for daylight exposure.

Accusation : Identical backgrounds in photos are listed as taken miles apart.
Response : Shots were not identical, just similar. Background objects were mountains many miles away. Without an atmosphere to obscure distant objects, it can be difficult to tell the relative distance and scale of terrain features.

Accusation : The photos contain artifacts like the two seemingly matching 'C's on a rock and on the ground.
Response : The "C"-shaped objects are most likely printing imperfections not in the original film from the camera.

Accusation : There appear to be "hot spots" in some photographs that look like a huge spotlight was used at a close distance.
Response : Pits in moon dust focus and reflect light in a manner similar to minuscule glass spheres used in the coating of street signs, or dew-drops on wet grass.

Accusation : Footprints in the extraordinarily fine lunar dust, with no moisture or atmosphere or strong gravity, are unexpectedly well preserved, in the minds of some observers – as if made in wet sand.
Response : The dust is silicate, and this has a special property in a vacuum of sticking together like that. The astronauts described it as being like "talcum powder or wet sand".

Ionizing Radiation and Heat

Accusation : The astronauts could not have survived the trip because of exposure to radiation from the Van Allen radiation belt and galactic ambient radiation
Response : The Moon is ten times higher than the Van Allen radiation belts. The spacecraft moved through the belts in just 30 minutes, and the astronauts were protected from the ionizing radiation by the metal hulls of the spacecraft. In addition, the orbital transfer trajectory from the Earth to the Moon through the belts was selected to minimize radiation exposure. Even Dr. James Van Allen, the discoverer of the Van Allen radiation belts, rebutted the claims that radiation levels were too dangerous for the Apollo missions. Dosimeters carried by the crews showed they received about the same cumulative dosage as a chest X-ray or about 1 milligray. Plait cited an average dose of less than 1 rem, which is equivalent to the ambient radiation received by living at sea level for three years.
The radiation is actually evidence that the astronauts went to the Moon. Irene Schneider reports that thirty-three of the thirty-six Apollo astronauts involved in the nine Apollo missions to leave Earth orbit have early stage cataracts that have been shown to be caused by radiation exposure to cosmic rays during their trip. However, only twenty-four astronauts left earth orbit. At least thirty-nine former astronauts have developed cataracts. Thirty-six of those were involved in high-radiation missions such as the Apollo lunar missions.

Accusation : Film in the cameras would have been fogged by this radiation.
Response : The film was kept in metal containers that prevented radiation from fogging the film's emulsion. In addition, film carried by unmanned lunar probes such as the Lunar Orbiter and Luna 3 (which used on-board film development processes) was not fogged.

Accusation : The Moon's surface during the daytime is so hot that camera film would have melted.
Response : There is no atmosphere to efficiently couple lunar surface heat to devices such as cameras not in direct contact with it. In a vacuum, only radiation remains as a heat transfer mechanism. The physics of radiative heat transfer are thoroughly understood, and the proper use of passive optical coatings and paints was adequate to control the temperature of the film within the cameras; lunar module temperatures were controlled with similar coatings that gave it its gold color. Also, while the Moon's surface does get very hot at lunar noon, every Apollo landing was made shortly after lunar sunrise at the landing site. During the longer stays, the astronauts did notice increased cooling loads on their spacesuits as the sun continued to rise and the surface temperature increased, but the effect was easily countered by the passive and active cooling systems. The film was not in direct sunlight, so it wasn't overheated.

Mechanical issues

Accusation : No blast crater or any sign of dust scatter as was seen in the 16mm movies of each landing.
Response : No crater should be expected. The Descent Propulsion System was throttled very far down during the final stages of landing. The Lunar Module was no longer rapidly decelerating, so the descent engine only had to support the module's own weight, which by then was greatly diminished by the near exhaustion of the descent propellants, and the Moon's lower gravity. At the time of landing, the engine's thrust divided by the cross-sectional area of the engine bell is only about 10 kilopascals (1.5 PSI), and that is reduced by the fact that the engine was in a vacuum, causing the exhaust to spread out. (By contrast, the thrust of the first stage of the Saturn V was 3.2 MPa (459 PSI), over the area of the engine bell.) Rocket exhaust gases expand much more rapidly after leaving the engine nozzle in a vacuum than in an atmosphere. The effect of an atmosphere on rocket plumes can be easily seen in launches from Earth; as the rocket rises through the thinning atmosphere, the exhaust plumes broaden very noticeably. Rocket engines designed for vacuum operation have longer bells than those designed for use at the Earth's surface, but they still cannot prevent this spreading. The Lunar Module's exhaust gases therefore expanded rapidly well beyond the landing site. Even if they hadn't, a simple calculation will show that the pressure at the end of the descent engine bell was much too low to carve out a crater. However, the descent engines did scatter a considerable amount of very fine surface dust as seen in 16mm movies of each landing, and as Neil Armstrong said as the landing neared ("...kicking up some dust..."). This significantly impaired visibility in the final stages of landing, and many mission commanders commented on it. Photographs do show slightly disturbed dust beneath the descent engine. And finally, the landers were generally moving horizontally as well as vertically until right before landing, so the exhaust would not be focused on any one surface spot for very long, and the compactness of the lunar soil below a thin surface layer of dust also make it virtually impossible for the descent engine to blast out a "crater".

Accusation : The launch rocket (Lunar Module ascent stage) produced no visible flame.
Response : Hydrazine (a fuel) and dinitrogen tetroxide (an oxidizer) were the Lunar Module propellants, chosen for their reliability; they ignite hypergolically –upon contact– without a spark. Hypergolic propellants happen to produce a nearly transparent exhaust. Hypergolic fuels are also used by several space launchers: the core of the American Titan, the Russian Proton, the European Ariane 1 through 4 and the Chinese Long March, and the transparency of their plumes is apparent in many launch photos. The plumes of rocket engines fired in a vacuum spread out very rapidly as they leave the engine nozzle, further reducing their visibility. Finally, most rocket engines use a "rich" mixture, where excess fuel is pumped so it will burn outside the engine, to lengthen their lifetimes. This cannot happen in a vacuum.

Accusation : The rocks brought back from the Moon are identical to rocks collected by scientific expeditions to Antarctica.
Response : Chemical analysis of the rocks confirms a different oxygen isotopic composition and a surprising lack of volatile elements. There are only a few 'identical' rocks, and those few fell as meteorites after being ejected from the Moon during impact cratering events. The total quantity of these "lunar meteorites" is small compared to the more than 840 lb (380 kg) of lunar samples returned by Apollo. Also the Apollo lunar soil samples chemically matched the Russian Luna space probe’s lunar soil samples. In addition, unlike the Antarctic lunites, the rocks recovered from the moon do not exhibit the effects of atmospheric friction.

Accusation : The Lander weighed 17 tons and sat on top of the sand making no impression but directly next to it footprints can be seen in the sand.
Response : The lander weighed less than three tons on the Moon. The astronauts were much lighter than the lander, but the area of their boots was also much smaller than that of the lander's pads. Pressure, or force per unit area, rather than force, determines the depth of compression of the soil. An example would be driving a car (heavy) on sand, then getting a person (light) to walk on the same surface. You will often find the depth of tracks to be about the same.

Accusation : The flag placed on the surface by the astronauts flapped despite there being no wind on the Moon. Sibrel said "The wind was probably caused by intense air-conditioning used to cool the astronauts in their lightened, un-circulated space suits. The cooling systems in the backpacks would have been removed to lighten the load not designed for Earth’s six times heavier gravity, otherwise they might have fallen over".
Response : The astronauts were moving the flag into position, causing motion. Since there is no air on the Moon to provide friction, these movements caused a long-lasting undulating movement seen in the flag. There was a rod extending from the top of the flagpole to hold the flag out for proper display (visible under the fabric in many photographs). The fabric's rippled appearance was due to its having been folded during flight and gave it an appearance which could be mistaken for motion in a still photograph. The top supporting rod of the flag was telescopic and the crew of Apollo 11 found they could not fully extend it. Later crews did not fully extend this rod because they liked how it made the flag appear. A viewing of the videotape made during the Moonwalk shows that shortly after the astronauts remove their hands from the flag/flagpole, it stops moving and remains motionless. At one point the flag is in view for well over thirty minutes and it remains completely motionless throughout that period (and all similar periods).

Animation of the two photos, showing that the flag is not waving.

(all facts quoted from wikipedia)

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