The “Modern Eddington Experiment”

0*K_tYN4NbxjACv4RGImages credit: New York Times, 10 November 1919 (L); Illustrated London News, 22 November 1919 (R).

 

Notes and comments on reproducing a famous historical event.

Most people today do not know the reference to Sir Arthur Eddington, but he is responsible for making Albert Einstein a household name. In 1919, the English physicist took on an experiment that would test Einstein’s theory of gravity, general relativity, an abstract and mathematically beautiful explanation of space and time that few people at the time fully understood.

The prevailing theory of gravitation was Isaac Newton’s, one that worked to great success in explaining nearly all the motions of the planets. Einstein’s theory was a refinement that would show only tiny differences. But it is important to get this right; modern GPS navigation would not otherwise work.

Both theories predicted that light would be deflected by gravity. The light from a star positioned near the sun’s edge would be bent, like a mirage, to an apparent position slightly shifted from where it would normally be. But Einstein’s theory predicted the deflection to be twice that of Newton’s.

20151128_STC969

It is impossible to see such stars while looking at the sun because, well, it is daytime. Even if you block the direct view of the sun, the scattered skylight washes out all but the very brightest of stars and planets. During a total solar eclipse however, the moon’s shadow makes a hole in the sky and the stars become visible.

Eddington’s experiment then, was to photographically record the nearby stars during the eclipse and measure the apparent deflections from their true positions. He mounted an expedition to a small island off the coast of Africa for the eclipse of May 29, 1919, where he encountered the usual problems that astronomers face: bad weather. It rained in the morning, and clouds obscured the view. They had thinned somewhat by the time of the eclipse, and his team forged ahead and made sixteen exposures on various photographic emulsions on 8”x10” glass plates.

It is fascinating to read the account of obtaining these photographs (as reported in the Philosophical Transactions of the Royal Society of London). Totality for this eclipse was long: six minutes, which allowed the manipulation of the large plates to be placed in position in the telescope, the large refractor from Oxford that had been dismantled, packed and transported. It was stopped down to 8” in order to use the best part of the lens. The clock drive was driven by a weight, and a pit was dug “deep enough to allow a run of 36 minutes without rewinding.”

All but two of the plates failed to show measurable stars, mostly due to the cloud cover. But that was enough. Combined with the results from a sister expedition to Brasil, Eddington’s analysis showed that the observed deflections were best explained by Einstein’s theory. These results were reported later that year, and the world found it fascinating. Einstein, previously known and respected only in a select group of theoretical physicists, became a celebrity.

Eddington made these measurements nearly a hundred years ago. Although he suffered poor weather, he had many things going for him. A long duration eclipse, and a collection of bright stars in the near field of the sun, made it possible using early 20th century technology to make a precision measurement. He recognized this, and at the end of the report wrote: “The unusually favourable conditions of the 1919 eclipse will not recur, and it will be necessary to photograph fainter stars, and these will probably be at greater distance from the sun.”

Well, we are a hundred years on from that groundbreaking experiment. One would think that modern optics and photographic equipment would make this an easy exercise. And indeed, it has become a goal for amateur astronomers. The Astronomical League, an umbrella organization of local amateur astronomy clubs has offered a special observing award to those amateurs that can detect the deflection of starlight during the eclipse of 2017.

But even with the advantages of a century of technological advance, this is a difficult measurement to make. Yes, we don’t have to worry about how long the clock drive will run before we need to reset the weights, and our optics are better, and the photographic emulsions have been replaced by digital sensors, but the measurement still involves finding stellar deflections that are smaller than a pixel!

My exploration into this project has uncovered some interesting details, which I will share in the following blog entries.  They start with a general introduction of celestial coordinates, and an outline of my plan, but then might become somewhat tedious as I devolve into some of the image processing and mathematical details.  Much of it may be of little interest except maybe to other Eddington experimenters, but if you find it useful in some way, or wish to make contributions, feel free to add commentary.

Here are some useful entry points:

Intro to celestial coordinates 

The overall plan to measure star deflections

 

 

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Eddington epilogue

I was lucky to have ended up at this observing location with such excellent weather. When planning to view total eclipses, I am advised to arrange for other activities as well; the eclipse itself is subject to fickle viewing conditions (my one prior total solar eclipse effort was thwarted, but the travel experience was rewarding nevertheless).

While I was mesmerized by the experience and NOT taking pictures, my script-driven camera captured more than just background stars for measuring gravitational deflections. Here are my three favorites.

3472-3479.hdr-1.bw

My HDR composite spanning 14 stops of exposure and showing some of the structure in the corona.

_MG_3524.C3

A shot at third contact showing some prominences and the emergence of two “beads”

_MG_3527.diamondRing.crop

The end of totality marked by the “diamond ring” left a lasting impression; a signature of the unique event we had just witnessed.

 

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Discussion

It is a bit disappointing to be unable to show a clear gravitational signal, even with all of the successful exposures that were taken, but I recognized the difficulty of this measurement early on. In addition to the variables I anticipated, there are some additional uncertainties that I now recognize.

Here is my updated list of confounding variables:

  1. Lens distortion. This seems to be the largest one, measuring many arcseconds by the time one reaches the edge of the frame. There are two ways to combat it: take the before and during images with the exact same center and orientation on the sky (which I found impractical with my equipment), or calibrate the lens. I did the latter, but found that this too was sensitive to overall gain/magnification assumptions.
  2. The variation in positions due to the stars twinkling is about 2 arcseconds. I attempted to mitigate this by multiple exposures, but more were needed than I obtained.
  3. Centering and orientation. I did the best I could to position and orient the reference stars to the coordinate of the sun’s center during mid-eclipse, but the rigid transform technique requires an adequate collection of uniformly distributed stars. With only five or six, there may have been biases in this calibration of the order of arcseconds.
  4. Temperature variation. My reference images were taken during Minnesota spring and summer evenings, usually light-jacket weather. The eclipse pictures were taken at midday in Idaho. Although we noticed a cool-down during the eclipse, only a few of us felt the need to put on our fleece. Silicon has a thermal sensitivity of 2.5 ppm per degree K, which could account for some of the error. Overall however, this is small compared to the uncertainties displayed, since even a 10-degree K difference would show an error of 0.02 arcseconds per thousand radial distance. There may be other artifacts of temperature change however, see the following regarding focus.
  5. Focus variations. As one focuses the image on the detector, there is a geometric gain involved. I measured the travel on my telescope focuser for one turn of the fine-focus knob. I also noted that best focus could be determined to within 1/8 turn of that knob. This worked out to be 0.3mm, which, at the focal plane of a 480mm lens is about 0.6 arcseconds per thousand, a significant amount!
  6. Algorithm sensitivities. The before images were taken at night, sometimes with a partial moon providing background illumination of the sky. The during images were taken in the presence of the corona, a strong offset of the background level, and one which has a directional gradient as well. It is possible that the first stage of processing, starPos.m, could have been influenced by this difference. I do not have any estimates of its sensitivity.
  7. Published star positions. I used reference star locations from Stellarium, which uses the latest publicly available database of star locations. I used J2000 epoch numbers out of habit, but perhaps I should have used current date coordinates. This would only affect the errors comparing my observed positions with the published ones, not the errors between observed “before” and “during” star positions.

 

While I am not surprised at the failure to find the gravitational deflection signal, I am disappointed I did not get a bit closer. Regardless, it has been a wonderful project to undertake. I learned much and re-learned more. I hope the descriptions of the process have been enlightening. If you have read this far, perhaps you have found the narration worthwhile or even enjoyable. Best wishes and clear skies to all future solar eclipse observers!

 

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Analyzing Eclipse Day Results

I was able to obtain 35 photos during totality that were candidates to locate stars in the field. The exposures ranged from 1/60 to 2 seconds, but it became clear after applying the detection procedure starPos.m, that only the longest exposures, 1 and 2 seconds, would yield detected stars. The inner regions of the corona were just too bright and irregular for the algorithm to find them.

This left 15 images to work with. The camera orientation was good in that there were many candidate reference stars in the frame. Here is the mapping of a mid-eclipse exposure (3496):

analysis.1

 

Here is a map of the located stars. The color codes indicate the channels (red, green or blue) in which the star was found. White indicates being found in all channels. The number of located stars for this image are: 6 red, 9 green, 5 blue.

analysis.2

 

The detected stars are then mapped to our virtual camera view and correlated against our list of reference stars (imagePos.m). They are then more precisely aligned to the center of the sun. The lens distortion is removed at this stage (radialAlign.m).

The uncorrected lens positions look like this:

analysis.3

Lens corrected:

analysis.4

The errors are still rather large, but by collecting the statistics from all of the “during” frames, we can see how they land with respect to their published reference positions. The standard deviations are consistent with atmospheric seeing, but the differences in average positions indicates other sources of error.

analysis.5

 

If we average all of the “before” images I took, and see how they compare to the published star locations, we get a similar wide ranging plot, the variances are again consistent with the seeing (but the average errors are not):

analysis.6

 

If we take the difference between these two data sets, we should see the gravitational deflection signal we are looking for. Unfortunately, it is lost in the noise.

analysis.7

 

I can make a plot similar to Eddington’s that shows the average measured deflection of these reference stars, but I will not claim that it demonstrates gravitational deflection.

analysis.8

 

 

 

 

 

 

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Comparing Before and During images, step EE-4

This was written prior to eclipse day as I was contemplating how to compare the two image sets.  I include it here to keep the thought sequence intact.

 

When we apply steps EE-1, 2, and 3 to both the before images and the during images, we will have a set of radial distances to compare. In the best of conditions, the distances will be very close to each other. There will be measurement noise and it is unlikely that the subpixel difference we are looking for will be immediately obvious from the measurement of any single reference star.

There are some tests we can make ahead of time to see what to expect. For example, we can compare an image taken on different (before) dates to see if they show consistent positions of the reference stars. We can compare images taken in a single session to measure the effects of atmospheric seeing and other factors.

To this end, I have made various images of this field of the sky over the months preceding the eclipse. They are among the least interesting of the astrophotos I have ever taken, since they show a single bright star (Regulus), and not much more. There are no deep sky objects of interest in this particular patch.; no galaxies, no nebulas, no star clusters, no Milky Way field. However, there are stars that can be detected, even in metropolitan light pollution, that match up with the Stellarium reference star database.

I took exposures for two consistency tests. One is multiple exposures of the same exact scene, with no changes in any settings. Ideally, the images would be identical, but if not, would be a measure of the dynamic atmospheric distortions, or perhaps the mechanical vibrations of my camera-telescope-tripod setup.

The second test changes the camera angle. The center of view remains approximately the same, but the camera rotates to 45 and 90 degrees. Each change requires a re-focus. If my frame centering and lens radial corrections were accurate, the stars should remain in the same locations.

Even if my lens corrections were imperfect, the first test: multiple exposures with no changes, should pass. Perhaps the lens corrections did not place the detected stars at their exact reference locations, but at least they should all fall at the same place. I could compare them against their reference, which might show an error, but if I compare them against each other, the differences should vanish.

The second test, comparing images of the same field but at different angles (portrait vs landscape etc), is a simulation of pictures taken at different times. It is a “best case” test: everything is the same except the camera was rotated and refocused, compared to the real-world case for the “after” image where not only is the angle different, the telescope alignment will have a slightly different center, the angle in the sky (and its atmospheric refraction) will be different, the temperature, elevation, and air pressure will be different, and many other uncontrolled (and unknown) variables will differ from those of the before image. This test tells us the best we can expect from comparing before and during images.

Here are the results of the first test, where a second image is compared to the first, all else being equal. The stars should be detected at the same exact positions, and yet they aren’t.

compareBeforeAndDuring

The differences between the positions of the same stars in two successive exposures. There are approximately eight stars detected (in green, fewer in the other channels). This is an indication of the variations introduced by the atmospheric seeing.

 

The standard deviation of this comparison was 2.3 arcseconds. The comparisons of other successive frames yielded standard deviations of 2.1 and 1.6. It appears that the consistency of star positions as detected by my equipment is about 2 arcseconds. This is consistent with reports of the atmospheric seeing in Minnesota.

I was curious about how to characterize “seeing” and found these interesting links (there is always too much to explore and investigate to the depth I would like):

Astronomical Seeing Part 2: Seeing Measurement Methods
https://www.handprint.com/ASTRO/seeing2.html

Lucky Exposures: Diffraction Limited Astronomical Imaging Through the Atmosphere
http://www.mrao.cam.ac.uk/projects/OAS/publications/fulltext/rnt_thesis.pdf

 

There is another domain of relevant knowledge: how to determine if the distribution of observations is the same, or different, from another. I will be comparing position measurements from the “before” condition to the “during” condition when the sun’s gravity will have a possible influence. How can we tell if the measurements are from truly different conditions, rather than the normal variations caused by noise? Here are some links I explored to try to answer this question:

Are Two Distributions Different?
http://www.aip.de/groups/soe/local/numres/bookcpdf/c14-3.pdf

Goodness of Fit Tests
http://www.mathwave.com/articles/goodness_of_fit.html

Tests of Significance
http://www.stat.yale.edu/Courses/1997-98/101/sigtest.htm

 

 

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Corona tangent

I encountered a report about the predicted corona and wondered how it would compare to what we actually saw.  I do not know the orientation of either the simulation or of my image (where is the solar north pole?).  Are these similar?

Update 20170906:  I still don’t know the absolute orientations of either image, but found that there was a correlation of the field lines that seem to stream directly out from the Sun.  See if this is a visual match:

coronalForecastOverlay.2up

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Eclipse Day!

The good weather held and we had zero clouds and negligible smoke for the eclipse.  Temperatures were climbing in the bright morning sun, but stalled and then dropped during the partial phases of the eclipse– enough that we donned our fleece jackets again!  Here are some shots of the event (click to enlarge).

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Step EE-3: Finding the best fit

We now have a way to transform the stars detected in an image into our virtual camera reference frame, but the previous step was just the “rough alignment” based on two bright stars. This is vulnerable to errors in how those two star positions were identified, especially if they were bright enough to saturate the detector, or were influenced by poor atmospheric seeing conditions.

More importantly, we are attempting to measure a tiny change in radial distance from the center of the sun. If we make a small error in where that center point is, by even a fraction of a pixel, it will affect all of our distance measurements to the stars revealed during the eclipse.

This means that it is less important to align the star positions as it is to get their angles with respect to the sun correct. If the angles are correct, then we can measure the radial distance from the image center and be confidant when we compare the “before” distances to the “during” distances that we are measuring from the same center point and are not being fooled by some offset to the actual center. This is particularly important in the before images. In the during image, we can, in principle, locate the center of the sun (though this too has uncertainties since we are seeing the moving moon’s edge, not the sun’s). The collection of observed stars is an unambiguous pointer to the sun’s position at the moment of maximal eclipse.

To this end, the output of step EE-3 is the small additional offset and rotation that places the observed stars in best angular alignment with known angles of the reference stars. This small adjustment will be applied to all of the detected stars to obtain their final position in the reference image plane. Their radial distance from the sun is then easily computed.

Matlab script radialAlign.m was created to do this task. It takes the results from step EE-2, the rough alignment and transform to the reference frame. The output is a set of star positions—the reference star positions in the reference frame, and the slight deviations from those positions as detected in the image.  In an ideal world, the deviations would be zero in the “before” image, and would show some gravitational deflection in the “during” image.

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