## Statistical Analysis of the 1919 Eclipse Measurements

So the centenary of the famous 1919 Eclipse measurements is only a couple of days away and to mark it I have a piece on RTÉ Brainstorm published today in advance of my public lecture on Wednesday.

I thought I’d complement the more popular piece by posting a very short summary of how the measurements were analyzed for those who want a bit more technical detail.

The idea is simple. Take a photograph during a solar eclipse during which some stars are visible in the sky close enough to the Sun to be deflected by its gravity. Take a similar photograph of the same stars at night at some other time when the Sun is elsewhere. Compare the positions of the stars on the two photographs and the star positions should have shifted slightly on the eclipse plates compared to the comparison plate. This gravitational shift should be radially outwards from the centre of the Sun.

One can measure the coordinates of the stars in two directions: Right Ascension (x) and Declination (y) and the corresponding (small) difference between the positions in each direction are D_{x} and D_{y} on the right hand side of the equations above.

In the absence of any other effects these deflections should be equal to the deflection in each component calculated using Einstein’s theory or Newtonian value. This is represented by the two terms E_{x}(x,y) and E_{y}(x,y) which give the calculated components of the deflection in both x and y directions scaled by a parameter α which is the object of interest – α should be precisely a factor two larger in Einstein’s theory than in the `Newtonian’ calculation.

The problem is that there are several other things that can cause differences between positions of stars on the photographic plate, especially if you remember that the eclipse photographs have to be taken out in the field rather than at an observatory. First of all there might be an offset in the coordinates measured on the two plates: this is represented by the terms c and f in the equations above. Second there might be a slightly different magnification on the two photographs caused by different optical performance when the two plates were exposed. These would result in a uniform scaling in x and y which is distinguishable from the gravitational deflection because it is not radially outwards from the centre of the Sun. This scale factor is represented by the terms a and e. Third, and finally, the plates might be oriented slightly differently, mixing up x and y as represented by the cross-terms b and d.

Before one can determine a value for α from a set of measured deflections one must estimate and remove the other terms represented by the parameters a-f. There are seven unknowns (including α) so one needs at least seven measurements to get the necessary astrometric solution.

The approach Eddington wanted to use to solve this problem involved setting up simultaneous equations for these parameters and eliminating variables to yield values for α for each plate. Repeating this over many allows one to beat down the measurement errors by averaging and return a final overall value for α. The 1919 eclipse was particularly suitable for this experiment because (a) there were many bright stars positioned close to the Sun on the sky during totality and (b) the duration of totality was rather long – around 7 minutes – allowing many exposures to be taken.

This was indeed the approach he did use to analyze the data from the Sobral plates, but tor the plates taken at Principe during poor weather he didn’t have enough star positions to do this: he therefore used estimates of the scale parameters (a and e) taken entirely from the comparison plates. This is by no means ideal, though he didn’t really have any choice.

If you ask me a conceptually better approach would be the Bayesian one: set up priors on the seven parameters then marginalize over a-f to leave a posterior distribution on α. This task is left as an exercise to the reader.

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May 27, 2019 at 12:50 pm

I didn’t realize the centenary was approaching. Thank you for providing some understanding of the event.

May 27, 2019 at 2:25 pm

>these deflections should be equal to the deflection in each component calculated using Einstein’s theory or Newtonian value.

I presume the results can be co-added to include many such eclipses to improve S/N.

Are there (modified) gravity theories that will give different value

and can be constrained from such observations?

May 27, 2019 at 3:14 pm

There are modified gravity theories in which the deflection is not twice the Newtonian value.

May 28, 2019 at 4:48 pm

I do not want to troll (or spoil your presentation), but I am genuinely interested in your opinion. Was this really a “verification” of General Relativity as it was known in 1919? I guess you are familiar with the Duhem-Quine argument. I could not read your previous posts on the anniversary as I am also marking, so, sorry if you already replied.

May 29, 2019 at 9:48 am

Well it didn’t prove general relativity to be `true’ in all aspects, but given a 50-50 prior choice between Newtonian and Einsteinian theories the posterior probability moved almost entirely to the latter…