Gravitational Microlensing – free from formulae

Albert Einstein in 1912 © The Albert Einstein Archives, The Jewish National & University Library

Even before Albert Einstein derived the theory of General Relativity, namely in 1911, he already concluded that massive bodies should bend light, and suggested to measure this deflection from the determination of the apparent position of stars near the Solar limb during a Solar eclipse. However, a treatment with the full equations of General Relativity in 1916 actually yielded a deflection angle that was twice as large as the one predicted earlier. Two British expeditions set off to Sobral (Brazil) and the Island of Principe to measure the deflection angle during totality of the Solar Eclipse of 29 May 1919, the latter being led by the (later knighted) Arthur Eddington. These measurements confirmed the value predicted by Einstein's theory, which was met with a lot of scepticism before – but not after. In 1912, Einstein found that the gravitational bending of light could lead to a transient brightening of an observed star as another star happens to pass close to its line-of-sight. However, an angular alignment sufficient for a significant effect is only expected in about one in a million observed stars at any given time. Einstein therefore was himself sceptical about the prospects of detecting such gravitational microlensing events, which typically last a month. Therefore, it needed intense persuasion by others before he eventually published his results in 1936, concluding that "there is no great chance of observing this phenomenon".

Negative of one of the photographic plates taken by the British expedition to Sobral (Brazil) during the total Solar Eclipse of 29 May 1919 © The Royal Society

(Sir) Arthur Stanley Eddington

The gravitational bending of light differs significantly from the effect caused by an optical convex lens. With the deflection angle vanishing for a light ray passing through the lens centre and increasing towards its limb, a convex lens lets parallel light rays meet in a focus. In contrast, the light bending by a massive object is analogous to the deflection by a glass body in the shape of the foot of a wine glass. Here, the deflection angle increases the closer the light ray approaches the centre, representing the location of the gravitating body.

Deflection of parallel light rays by a convex lens and by the foot of the wine glass, which represents an analogy to the gravitational bending of light. With the deflection decreasing towards its centre, the convex lens leads to a focussing. A different behaviour results for the light bending by massive bodies, where the deflection angle increases the closer the light ray approaches.

The apparent brightening of the observed star is therefore not the result of light rays being focussed onto the observer (on Earth). Instead, the variation of the deflection angle with the massive foreground 'lens' star determines a distortion of the light bundle that results in an alteration of the solid angle from which the light is received. More precisely, light reaches the observer along two different paths, on either side of the lens star. Therefore, there are two images of the source star at slightly different positions. However, their angular separation is less than a milli-arcsecond, so that the images cannot be resolved and only their combined magnification is observable.

Side view (left) and observer's view (right) on a source star (S) affected by gravitational microlensing due to a foreground lens star (L), where two possible light paths to the observer (O) correspond to two apparent images I_- and I₊.

In 1936, it was impossible – even for Albert Einstein – to imagine the data-processing rate possible with modern computers and digital cameras, and in fact, it required many decades of advance in technology before the first detection of a microlensing event could be announced in 1993, which required daily monitoring of the brightness of tens of millions of stars. These efforts followed the suggestion by Bohdan Paczynski to study clumps of unknown 'missing' matter in the Milky Way by means of the statistics of gravitational microlensing signals that these cause. In 1991, Shude Mao (then a student of Paczynski at Princeton and now a professor at the University of Manchester) discovered that planets orbiting the foreground 'lens' star could reveal their existence by causing a small blip or dip to the othewise symmetric light curve. Depending on the mass of the planet, such planetary 'anomalies' last between hours and days. Several planets having meanwhile been detected from microlensing observations. The most spectacular discovery was that of OGLE-2005-BLG-390Lb, with a mass of just 5 times that of the Earth the most Earth-like planet orbiting a star other than the Sun at the time of its discovery (announced January 2006).

Model light curve and data (colour-coded) from 6 different observing sites for the microlensing event OGLE-2005-BLG-390, whose lens star hosts the 5-Earth-mass planet OGLE-2005-BLG-390Lb. The planet revealed its existence by causing a blip of roughly 20% amplitude, lasting about a day, on and around 10 August 2005. The vertical axis refers to the brightening in astronomical magnitudes with respect to the intrinsic luminosity of the observed source star.

Artist's impression of the cool ice planet OGLE-2005-BLG-390Lb by Herbert Zodet, © ESO

The software provided by ARTEMiS allows to adopt an efficient strategy that enables current microlensing campaigns not only to detect planets of Earth mass, but even below.

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