Analysis of Apollo image AS11-40-5903
Using Ray Tracing to determine the source of light
Luis E. Bilbao, PhD
The iconic image of Aldrin standing alone on the lunar surface is probably one of the best-known photographs of all the Apollo missions, and over the years modified versions of this photograph have been published (for example Life magazine, 8 August 1969). Taking the original NASA photo AS11-40-5903, and magnifying the image reflected in the helmet visor, it is possible to see the reflection of the astronaut taking the photograph.
The Earth is apparently visible in the reflection. The fact that a scene and its reflection can be observed, together with the knowledge of the coordinates of the Sun and Earth at the time the photograph was taken means that there is redundant information available to ascertain whether the observed features are compatible with one another. In this paper the position of the light source in the film plane is determined using different methods.
The prime conclusion reached is that the shadows (including the analysis of shadows seen in the helmet reflection) are incompatible with a single point-like source of light, and that the alleged reflection of the Earth in the helmet visor is not correctly positioned.
Apollo image AS11-40-5903 (Fig. 1) is a well-known lunar surface photograph taken during the Apollo 11 mission.
Figure 1. Apollo image AS11-40-5903
By magnifying the image in the helmet visor it is possible to see the reflection of the astronaut photographer (Fig. 2).
Figure 2. Magnified view of AS11-40-5903. The bright point in the upper part of the visor, almost vertically above the shadow of the astronaut, is alleged to be the Earth.
According to the Apollo 11 Lunar Surface Journal, this picture was taken at 04:14 UT on 21 July 1969. Relative to Tranquility Base the Sun was 14.74 degrees above eastern horizon (azimuth of 88.84 degrees), and the Earth was 59.07 degrees above the western horizon (azimuth of 273.11 degrees), see Fig. 3.
Figure 3. Location of Sun and Earth at the time AS11-40-5903 was taken. The helmet visor is equivalent to a spherical convex mirror. The axis y goes from the center of this sphere to the camera. The relative azimuthal angle from this axis to the Sun, φS or Earth, φE are unknown.
From an optical point of view, the helmet visor is equivalent to a spherical convex mirror, see Fig. 4.
Figure 4. Detail of AS11-40-5875 showing a perfect spherical shape of the helmet visor. Its diameter is about 36.5 cm which is optically equivalent to a spherical convex mirror of 9.1 cm focal length.
The y-axis, in Fig.3 goes from the center of this sphere to the
camera. The relative azimuthal angle from this axis to the Sun, φS or Earth, φE are
unknown (note that φS was defined from the negative y-axis). Anyway, they are related
Although the direction the y-axis is unknown, its determination is obtained using three
different methods, namely: a) the ratio between the shadow of the astronaut in the
visor reflection and the separation between astronauts, b) the analysis of shadows of
the scene, and c) the Earth location in the reflection. It is expected that these
different analyses will lead to similar results having angular differences of a few degrees at
Shadow length and astronauts’ separation
From Fig. 2 a rough estimate of the shadow length shows that it is shorter than expected. The length of a shadow, L, on a horizontal planar surface produced by a single distant source is related to the height of the object, H, through
θ being the elevation angle of the distant source (Fig. 5(a)). Using H ≈ 2 m and θ = 14.74º we get L ≈ 7.6 m.
Figure 5. Shadow formation of an object of height H on a planar surface, produced by a single distant source (orange circle). (a) Lateral view, (b) Top view.
The separation between astronauts, D, is related to the ratio of the astronaut's actual height and that of his image, h,
where f is the focal length of the camera (we assume that f ≪ D). From Fig. 1 we have h ≈ 30 mm then the separation is D ≈ 4 m (f = 60 mm). This means that the length of the shadow is almost twice as much as the distance between the astronauts, but in Fig. 2 these two lengths appear to be almost equal. This implies that the ray direction forms an angle with the direction of separation.
Consider two objects separated by a distance D. The shadow of any of the objects forms an angle with the line that joins the objects, which is equal to the azimuthal angle of the source relative to that line, φ, (Fig. 5 (b)). Therefore, the component of the shadow along the said line, L′, is
or, using (2),
Although the height of the object (the subject astronaut in the present case) may be unknown, it is proportional to the size of its own image on the film, as derived from (3)
Using (6) in (5)
Since the elevation angle of the source is known, the relative azimuthal angle of the source (in what follows, the word ‘relative’ will be dropped) is obtained from (8)
where, besides the ratio L′/D, all parameters are known. Note that there is no need to independently know the distance D, the length of the shadow L, nor the height H. In the case of the image reflected in the helmet visor, Fig. 2, it is apparent from the figure that the component of the shadow along the line joining the two astronauts is almost equal to the separation of the astronauts, that is L′ ≈ D. A better estimate is
because the feet of the background photographer are approximately at the same level as the mid part of the helmet shadow, see Fig. 2. Using, f = 60 mm, h = 30.55 mm, and θS = 14.74º, we
get φS = 55.36º.
Figure 6. Location in the film plane of a source (orange circle) that is 14.74º above horizon and 55.36º of relative azimuth. Different relative azimuthal angles are indicated by the curved orange line in the top for a fixed elevation of 14,74º. Ray tracing (straight orange lines) clearly demonstrates that the shadows are not compatible with this particular source location, as indicated by the arrows.
In Fig. 6 we plot the position of the source in the film and trace some rays to check if the shadows are in agreement with this position. As a reference, different azimuthal angles at a fixed elevation angle are indicated by the upper orange curved line. It is apparent from the figure that the actual shadows bear no relation to those produced by a source located at θS = 14.74º and φS = 55.36º. Therefore, the observed relative size of the shadow and the
astronauts separation cannot be explained by a far source with an elevation angle of θS = 14.74º on
a horizontal terrain.
Figure 7. A shorter shadow of an object of height H can be produced in a non-horizontal planar surface: (a) in a slope, (b) if the object is inside a depression [or dip] of the surface, or some combination of both.
We have to assume that the camera is located at chest level and that the terrain is not horizontal, or is not flat, as in the examples of Fig. 7: either the astronauts are standing on a slope or the astronaut portrayed in the photo is standing in a surface depression. This fact is in accordance with the different levels of the astronauts as inferred from the reflected image in the helmet visor. Note that taking the horizon as a reference, the camera of the photographer astronaut is at the same height as a mid-point in the helmet visor of the subject astronaut, then an estimate for the difference of height is ≈ 0.5 m.
Assuming that the subject astronaut is standing in a surface depression, the shadow will be shorter, then Eq. (2) is replaced by
being ΔH the depth of the well, and Eq. (9) becomes
Using this correction, for ≈ 0.25, the azimuthal angle is φ = 40.72º. In Fig. 8 the rays from a light source with this azimuthal angle are shown. As before, we observe that the actual shadows bear no relation with these rays. As an example, some arrows have been drawn showing that the actual shadows do not correspond to the rays.
Figure 8. Location in the film plane of a source (orange circle) that is 14.47º above the horizon and 40.72º of relative azimuth. Different relative azimuthal angles are indicated by the curved orange line in the top for a fixed elevation of 14.74º. Ray tracing (straight orange lines) clearly demonstrates that the shadows are not compatible with this particular source location, as indicated by the arrows.
If the difference of the vertical positions of the astronauts were due to the existence of a slope in the terrain, the corresponding corrections to the previous equations lead to an azimuthal angle greater than in the latter case. Therefore, again, there will be a mismatch between the actual shadows and the rays of the source located at that azimuthal angle.
In short, the fact that the projection of the shadow of an astronaut in the direction of the other astronaut is approximately equal to the separation between them, limits the azimuthal angle where the source can be located. This principle applies as much for a horizontal terrain as it does should one of the astronauts be standing in a depression, or on a sloping surface (accounting for the observed difference of the vertical positions). As this photo provides no evidence of other complications inherent within the lunar surface at this location, and as the angles found in this scene are in disagreement with the shadows that are observed in this photo, the present analysis is strong evidence against the presence of a unique light source with an elevation of 14.74º.
Ray tracing to locate the light source
In view of the previous results, a question arises: what should be the location of the light source that is in agreement with the shadows in the scene? For a fixed elevation it is possible to draw a line for different azimuthal angles, (see the upper curved orange line in Fig. 9).
Figure 9. The direction of the astronaut's shadow (red arrow) gives an indication of the azimuthal angle of the source. Different azimuthal angles for a fixed elevation of 14.74º are indicated above the upper orange curved line. A point in an object (1) and its corresponding shadow (1') belong to a ray (orange straight line). The intersection of this ray and the azimuthal curve gives an azimuthal angle of 33.32º for the source (S) in the plane of the film (orange circle). Note that it is almost vertically up from the vanishing point of the shadow (H), indicating that the terrain in the scene of the picture is almost horizontal.
An initial guess of about 35º is obtained from the direction of the astronaut’s shadow (red arrow in the figure). Its vanishing point should be located vertically down from the light source. A better value is obtained by ray tracing, using a point in an object (1 in the figure) and its corresponding shadow (1'). The intersection of this ray and the azimuthal curve gives an azimuthal angle φ = 33.32º for the source (S) in the plane of the film (orange circle). Note that, as expected on a flat terrain, the source is almost vertically up from the vanishing point of the shadow (H).
Once the source is located, different rays can be traced in order to check the shadows of different objects in the photo, Fig. 10.
Figure 10. Once the source was located, different rays can be traced in order to check the shadows of different objects in the photo. The shadows are well adjusted on the right side of the photo (e.g., point 1), but begin to differ towards the left (e.g., point 2), until reaching the biggest difference, in the upper left part (e.g., point 3, where the difference is about 10.6º). Moving the source to the left along the orange curve allows a better adjustment of the left side of the photo but misalign the right one.
The rays and the segments formed by a point of an object and its shadow (not shown in the figure) are well adjusted on the right hand side of the photo (e.g., point 1 of Fig. 10), but begin to differ towards the left (e.g., point 2), until reaching the biggest difference in the upper left part (e.g., point 3, about 10.6º of difference). Moving the source to the left of this point along the curved orange line allows a better adjustment of the left side of the photo but misaligns the right side. On the contrary, moving the source to the right, the angular difference between the rays and the segments on average will continuously increase.
Figure 11. Example of ray tracing, using only 2 rays (red lines) to locate the source. A check of shadows using rays across other objects (orange lines) gives a very good agreement with angular deviations within 1 degree.
A difference of 10.6º is too large to be an error of the method. Usually the error is around 1º. An example is shown in Fig. 11 where the source is located using only two rays that forms a small angle. The rays traced from the source are well aligned with the segments.
Besides the large difference in the upper left part of Fig. 10 (an indication of the presence of an extended source) there are other discrepancies between rays and shadows in photo AS11-40-5903. In Fig. 12 some rays are marked with numbers. Rays 1 and 2 are an example of expected results. Ray 1 touching the astronaut’s waist casts its shadow outside the frame of the picture, and ray 2 connects a part of the astronaut’s boot and its shadow within the frame of the photo.
Figure 12. Different anomalies of rays and shadows in photo AS11-40-5903. As a reference, ray 1 touching the astronaut waist, casts its shadow outside the frame of the picture, and ray 2 connects a part of the astronaut’s boot and its shadow within the frame of the photo, as expected. An anomaly is indicated by number 3. To the right of the ray passing through 3 there should be light, but on the ground, ray 3 is surrounded by shadows. Note that the right leg of the astronaut is farther from the source than the left one, so there is nothing that can interrupt the path of beam 3 to the ground.
Another anomaly, is put into evidence by ray 4 which hits some part of astronaut’s PLSS. Eventually this ray will intersect the ground beyond the limits of the photo. But this seems impossible because the red arrow that indicates the ground level diverges from ray 4. Ray 5 is approximately parallel to the direction of the shadow on the floor (red arrow). This means that everything between the 4 and 5 rays can not project shade on the ground, unless the light comes from another source.
Now consider ray 3 of Fig. 12. This ray passes through a point on inner part of the right leg of the space suit. To the right of the ray (as seen in the photo) there should be illumination, but on the ground, ray 3 is surrounded by shadows. Note that the right leg of the astronaut is farther from the light source than the left one, so there is nothing that can interrupt the path of beam 3 from the leg to the ground.
Ray 4 hits some part of astronaut’s PLSS, eventually this ray will intersect the ground beyond the limits of the photo. But this seems to be impossible because the red arrow that indicates the ground level diverges from ray 4.
Ray 5 is approximately parallel to the direction of the astronaut’s shadow on the ground (red arrow). Shortly, using the astronaut shadow as a reference, those rays would be directed above the horizon, this means that the entire region bounded by rays 4 and 5 could not project a shadow on the ground. As the shadow exists (it is seen in the helmet visor reflection), only an extra light source located at a higher elevation angle can explain this shadow. Approximate required values to fulfill this condition are: an elevation angle of 26º, and an azimuthal angle of 38º. A source located around this point would explain the astronaut shadow on horizontal terrain. Clearly, a source with this elevation angle does not correspond to the Sun. (Note, the above values are given as an approximation, and no further search for a more precise value was attempted because it does not correspond to the Sun elevation.)
The other possibility is that the shadow direction as seen in the picture is falsified because it lies on a non-horizontal terrain. If there is a positive slope towards the astronaut taking the photo (as expected from the vertical height difference) the same conclusion as before is reached. This is so because if the rays cannot intercept a positive slope then they will not be able to intercept the horizontal terrain that lies beyond the slope. On the contrary, a negative slope in the area of the red arrow would cause the shadow to be tilted lower than on horizontal terrain, and, therefore, the mentioned rays 4 and 5 would not diverge from a horizontal plane. According to the scale of distances shown in Fig. 12 this slope would occupy >2 m, that is, at least half of the distance between astronauts (≈ 4 m).
But this has two drawbacks. On the one hand, this would be in contradiction with the location of the vanishing point that corresponds to a horizontal surface and, on the other hand, would require that the terrain, after this slope, climbs more steeply until reaching the height difference of the astronaut taking the photo. But this does not seem to be the case. In Fig. 13 a wider area of the terrain is seen, showing no important differences with the rest of the terrain. It is also important to note that even an irregular horizontal terrain cannot explain the anomalies of ray 4 and 5.
Figure 13. Photo AS11-40-5902 taken shortly before AS11-40-5903. It portrait a wider area of the terrain than in photo AS11-40-5903, showing terrain that is quite level.
Finally, another anomaly is seen in the reflection of the helmet visor of photo AS11-40-5902, taken shortly before AS11-40-5903.
Figure 14: Detail of the reflection in the helmet visor of AS11-40-5902 (left) and AS11-40-5903 (right) taken shortly after 5902. The reflection in the left image shows an object (inside the yellow circle) that is absent in the right image and also in the main scene of photo AS11-40-5902.
In Fig 14 detail of the reflection in the helmet visors from photo AS11-40-5902 (left) and AS11-40-5903 (right) can be compared. Note that the reflection on the left image shows an object (inside the yellow circle) that is absent in the right image and also in the full scene of AS11-40-5902. Also, there is a noticeable difference in the size of the astronauts’ shadows between the left frame and the right one, when both should be almost identical.
The alleged image of Earth as seen in Aldrin’s helmet visor (Fig. 2) is discussed in the Apollo 11 Image Library, where it reads:
Sharp-eyed readers will have noticed that the analysis is not yet complete. With Earth slightly north of west and the Sun lightly north of east, the reflected image of Earth should be on the opposite side of the reflected image of Buzz’s shadow from the reflected LM. Nelson has produced a rectified detail in which ‘I re-projected the visor image to flatten the horizon (effectively killing barrel distortion), rotated the horizon to level, and then horizontally skewed the image to parallel and square-to-vertical the LM descent stage sides. Finally I mirrored it to produce a normal view to the west. Some distortions remain, which suggests the visor isn’t spherical, so that could explain sundry slight geometric inconsistencies in the rough analysis.’. The rectified image shows the tentative Earth ‘image’ tantalizingly close to the line of Buzz’s shadow, but on the wrong side. It is tempting to think that consideration of the actual figure of the gold visor will move the ‘image’ to the other side of the line of Buzz’s shadow, but the necessary analysis is yet to be done.
Since the visor optically behaves as a spherical convex mirror, a more precise analysis can be conducted. Consider the illustration, relative to the image of Earth created by the mirror, depicted in Fig. 15, where the axes and variables are defined as in Fig. 3.
Figure 15. Image formation of the Earth, which has an elevation angle θE = 59.07º and an unknown relative azimuthal angle φE. RE is the distance to Earth and R′E its projection on the xz plane. Axes and variables are defined as in Fig. 3. The helmet visor optically behaves as a spherical convex mirror. The y-axis joins the center of the mirror and the center of the camera objective. Therefore, the image of Earth produced by the mirror, as seen by the camera, lies in the plane that contains this axis and the Earth (blue plane). The angle α between this plane and a vertical one (green plane) can be obtained from the angle α' and the inclination of the film plane relative to the xz-plane. The angle α', which can be measured in the photo, is formed by the intersection of the film plane with the blue and the green planes.
Earth has an elevation angle θE = 59.07º and an unknown relative azimuthal angle φE. The y-axis joins the center of the mirror and the center of the camera objective. Therefore the image of Earth produced by the mirror, as seen by the camera, lies in the plane that contains this axis and the Earth (blue plane). The angle α between this plane and a vertical one (green plane) is related to the elevation and the azimuthal angle. This can be seen by writing the components of the Earth position in the xz plane. On the one hand we have
where is the distance to Earth, and, on the other hand
where R′E is the projection onto xz plane of the distance to Earth (see Fig. 15). From (13)-(16) we get
Therefore, from the known elevation and measuring α it is possible to obtain the azimuthal angle. Note that the y-axis is slightly above the horizon, but this difference can be ignored since it is less than 0.1 degrees.
Figure 16. Measurement of the angle a’ between a blue dashed line (the intersection of the film plane and the blue plane which contains the y-axis and the Earth, Fig. 15) and a green dashed line (the intersection of the film plane and the vertical green plane, Fig. 15). From the figure we get α’ =14.6º. Taking into account that the film plane forms an angle of 20.39º with the xz-plane, the value of α is 15.53º.
In Fig. 16 the intersection of blue and green planes with the film plane, and their angular separation, α’ are shown as dotted lines. From the figure we get α’ =14.6º. Taking into account that the film plane forms an angle of 20.39º with the xz-plane, then the angular separation of the blue and green planes as defined in Fig. 15 is α = 15.53º. Using θE = 59.07º, the azimuthal angle of Earth is φE = 27.63º and according to (1) the corresponding azimuthal angle for the Sun is φS = 23.36º.
Figure 17. Location in the film plane of a source (orange circle) that is 14.74º above the horizon and 23.36º of relative azimuth. Different relative azimuthal angles are indicated by the upper orange curved line in the top for a fixed elevation of 14.74º. Ray tracing (straight orange lines) shows a reasonable good agreement in the left hand side of the photo, but a large discrepancy on the right hand side, as indicated by the arrows.
In Fig. 17 some rays are traced from a source located at θS = 14.74º and φS = 23.36º. Although some fair agreement between shadows and rays is observed in the left hand side of the photo, the right hand side demonstrates that this particular location of the Sun (and consequently that of the Earth in the visor reflection) is not compatible with the observed shadows in the photo.
Then the inconsistency discussed in the Apollo 11 Image Library, relative to the fact that the Earth is on the wrong side of the line of Buzz’s shadow, is not due to geometric distortion but to an inconsistency between shadows and the alleged positions of the Earth and the Sun. Which might explain the lack of any further analysis by the ALSJ.
Figure 18. Different source positions obtained with different methods. S1 or S2 can explain the ratio between the astronaut's shadow length and the separation between astronauts, on horizontal terrain or when the astronaut is standing in a depression, respectively. S3 is obtained from tracing rays using a point in an object and its shadow. S4 is necessary to explain the complete astronaut shadow, and S5 is in correspondence with Earth as seen in the visor reflection. A ray from each source is traced to show that all of them fail to explain the shadows in the photograph. The large discrepancy among these methods can only be explained by the assumption that more than one light source is present.
In Fig. 18 different source positions are shown that have been obtained using different methods S1 (θ = 14.74∘, φ = 55.36º) or S2 (θ = 14.74º, φ = 40.72º) can explain the ratio between the length of the astronaut’s shadow and the separation between the astronauts, on horizontal terrain or when the astronaut is standing in a depression, respectively. But, they both fail to explain the shadows in the main scene. S3 (θ = 14.74º, φ = 33.32º) is obtained from tracing rays using a point in an object and its shadow.
It is probably the best choice for reproducing the shadows all around the photo, with the exception of the astronaut’s shadow. In both the main scene and the visor reflection S4 (θ = 25º, φ = 38º) is
necessary to explain the astronaut's shadow,
but its elevation in wrong. S5 (θ = 14.74º, φ = 23.36º) is in correspondence with the Earth as seen in the visor reflection, but cannot explain the shadows all around the photo.
In a few words, it is impossible to locate a point-like source along the curve of 14.74 degrees of elevation that can explain the ratio between shadow length and separation, the shadows of rocks, objects and the astronaut, and the position of Earth.
The significant discrepancy between these methods can only be explained on the basis that more than one light source is present. The analysis using ray tracing indicates that the shadows in photo AS11-40-5903 are most compatible with source S3, because there is a coherence between the location of the source and most of the rock shadows. However, there is no match with the size and direction of the astronaut’s shadow (as seen in the main scene and in the helmet visor), nor with the position of the Earth.
According to the present analysis, the shadow of the astronaut seems to be produced by an extra light source located at a higher elevation angle relative to the light which illuminates the background. The most probable configuration is S3 as the main source of the scene, plus S4 for the astronaut's shadow (both in the photo and in the visor reflection), and an extra light (not included in the figure) that is seen as the Earth in the helmet visor.
Luis E Bilbao
Aulis Online, August 2018
Note: Due to their large size, photographs are not optimised for viewing on cell phones
About the Author
Luis Ernesto Bilbao has a PhD in Physics from the University of Buenos Aires, is Adjunct Professor, and Independent Researcher, INFIP CONICET, UBA (the Faculty of Exact and Natural Sciences, University of Buenos Aires, Argentina).
The Institute of Plasma Physics (INFIP), dependent on the CONICET and the FCEyN-UBA, is a center carrying out pure and applied research in a wide variety of subjects of this discipline. The INFIP researchers have extensive knowledge and many years of experience in this branch of physics, with the publication of more than 600 works since 1983.