# The Math behind the Eddington Expedition

Apr 01, 2024*In celebration of the upcoming eclipse crossing North America on April 8th, I’ll be posting one eclipse-related post per day. Today’s post is about the 1919 Eddington Expedition and the math behind the theoretical light deflections that were being measured.*

In 1919, English astronomer Sir Arthur Eddington embarked on a journey to the island Príncipe off the west coast of Africa in order to witness a total solar eclipse. His goal? To test a novel theory put forth by a certain German physicist a few years earlier: general relativity.

General relativity predicts that light passing a large mass is deflected due to the way the mass curves spacetime. If you were able to measure the relative positions of stars at night versus the day, you would expect to see differences due to the lensing of the stars’ light as it passes by the sun.

The problem is that you can’t see any stars during the day—the sun’s light drowns everything else out. Except of course during a total solar eclipse, when the moon blocks all light from the sun beyond its faint corona, plunging the part of the Earth in the moon’s shadow into darkness.

The solar eclipse taking place on May 29th, 1919 was a particularly long one, lasting almost 7 minutes at the point of maximum totality; it was visible along a narrow band passing through most of Africa and South America. Furthermore, during totality the sun would pass in front of a particularly bright star cluster—the Hyades—which would allow the expedition to measure the deflection of multiple visible stars.

Eddington organized two expeditions, one to Príncipe and one to the town of Sobral in Brazil to maximize the chance of getting useful data from the event. The photographic plates obtained by the back-up telescope in Sobral proved the most clear, displaying a deflection corresponding to Einstein’s predictions. Below is an image of one of the Sobral plates; you can see a few of the deflected stars annotated with horizontal lines on either side:

The results of the experiment were published at a joint session of the Royal Society and Royal Astronomical Society on November 6th, 1919. There was intense press fanfare surrounding the report, and this experiment is generally credited with launching Einstein—who was well-known only within the physics community at the time—into superstardom.

## Newtonian Deflection

When I first learned about the Eddington expedition I was puzzled by the mention that general relativity predicts an angular deflection of *twice* the magnitude of the deflection under classical physics. Why would light be deflected at all under Newtonian mechanics and how could you calculate the deflection?

I’ll explain the derivation in this section—you’ll need only knowledge of classical mechanics and a little calculus to follow.

Imagine a particle of light with mass \(m\) that is traveling past a much more massive body (the sun) with mass \(M\) positioned at the origin. The force that the particle experiences is:

\[\vec{F} = \frac{GMm}{r^2} \hat{r} = m\vec{a}\]

(Yes, photons are massless, but Newton didn’t know that. Regardless, you can see that \(m\) cancels in the above equation, so you can consider the limit of the above scenario as \(m\) approaches 0.) Suppose that the perpendicular component of the displacement from the particle to the sun is \(b\) and the parallel displacement is \(r_{\parallel}\). We’ll assume that \(b\) stays approximately the same throughout the path of the particle, but that \(r_{\parallel}\) varies.

By the Pythagorean theorem \(r^2 = r_{\parallel}^2 + b^2\). We’ll further assume that \(dr_{\parallel} = cdt\) where \(c\) is the speed of light—in other words the particle moves at an unchanging speed \(c\) along the vector parallel to the sun. The change in speed of the particle along the perpendicular component is:

\[\Delta v_{\perp} = \int_{0}^{t’} \frac{GM}{b^2 + r_{\parallel}^2}dt = \int_{-r_0}^{r_{t’}} \frac{GM}{c(b^2 + r_{\parallel}^2)} dr_{\parallel}\]

Simplifying further:

\[\frac{GM}{b^2 c} \int_{–r_0}^{r_{t’}} \frac{dr_{\parallel}}{1 + (r_{\parallel}/b)^2} = \frac{GM}{b^2 c} \int_{-r_0/b}^{r_{t’}/b} \frac{bdu}{1 + u^2}\]

(note the \(u\)-substitution in the final step above). Now, recall that:

\[\int \frac{1}{1+x^2} = \tan^{-1} x\]

So:

\[\int_{-\infty}^{r_{t’}/b} \frac{dx}{1 + x^2} = \tan^{-1}(x) \Bigg|_{-r_0/b}^{r_t/b} \]

Since \(r_0\)—the distance from the star to the sun—and \(r_{t’}\)—the distance from the earth to the sun—are so much more massive than \(b\) (which is some small multiple of sun’s radius since we’re picking stars that are visually near the eclipsed sun), we can approximate this as:

\[ \tan^{-1}(x) \Bigg|_{-\infty}^{\infty} = 2 \]

Thus:

\[\Delta v_{\perp} = \frac{2GM}{bc}\]

So the velocity vector goes from:

\[\langle 0, c \rangle \rightarrow \left \langle \frac{2GM}{bc}, c \right \rangle\]

The deflection angle relative to the starting trajectory is the arc-tangent of the ratio of the final velocity components:

\[\Delta \theta = \tan^{-1} \frac{2GM}{bc^2} \approx \frac{2GM}{bc^2} \]

## General Relativistic Deflection

Under general relativity, light travels along *geodesics*—essentially “straight lines” in spacetime. Of course, spacetime itself can be curved by massive objects. The curvature is precisely defined by Einstein’s field equations, but exact solutions are difficult to obtain outside of simplifying assumptions.

One such simplification is to consider only the influence one large mass \(M\). Photons in this scenario follow the Schwarzschild geodesics.

You can use the Schwarzschild solution to Einstein’s field equations to calculate the deflection. I don’t pretend to understand general relativity well-enough to perform the deriviation myself, but you can watch the below video if you’re interested:

You get a deflection angle of:

\[\Delta \theta = \frac{4 GM}{bc^2}\]

In other words the deflection under general relativity is exactly twice as under classical mechanics.

If you carry out the calculations for a star positioned right on the edge of the sun (in other words, setting \(b\) to the sun’s radius) you’ll get a deflection of 0.876 arc-seconds under Newtonian physics and a deflection of 1.75 arc-seconds under general relativity. This is a difference of about 2.5 ten thousandths of a degree—it’s still incredible to me that Eddington’s team was able to measure such a small arc, but measure it they did: the observations at Sobral resulted in a calculated deflection of 1.94 ± 0.12 arc-seconds—close enough to 1.75 to validate general relativity.

There were other tests of the theory that were carried out (most notably, general relativity correctly explained the deviation in the precession of Mercury), but the Eddington experiment was seen as the first major confirmation—and it was all thanks to a total solar eclipse.

## Further Reading

- Eddington’s original report to the Royal Society. It’s fascinating, and surprisingly easy to understand!
- This Vox video, from which I first learned about the experiment.
- This Guardian article, which includes more details about the expedition itself.