Aberration of Starlight

Note: This article serves as evidence of Brandolini's Law

Introduction

Stellar aberration is a change in the direction of incoming star light caused by lateral movement of the observer relative to the star. In some obsolete rsp. fringe theories, it is caused by the absolute motion of one or the other.

Stellar aberration should not be confused with stellar parallax, which is a change in the angle of incident light from nearby stars due to a change in the orbital position of Earth. It is a distinct effect and the light of a star can be affected by either or both. Important differences are:

• Stellar aberration is maximal when the Earth is moving laterally to the star's position on its orbit, and decreases to zero when the Earth is moving straight towards or away from the star. In contrast, stellar parallax of a star in the ecliptic plane is most easily measured between the points at which Earth is moving towards or away from the star.
• Stellar aberration does not depend on the distance to the star. In contrast, stellar parallax is only observed for "nearby" stars, which, depending on the available measurement precision, is at most in the hundreds of light years.

To summarize the difference in one sentence: Aberration is a velocity effect whereas parallax is a position effect.

Stellar aberration is also subject to a minor diurnal effect due to the tangential velocity of an observer on the rotating Earth, at less than polar latitudes.

Also, aberration affects sunlight and the light reflected off of planets, though the effect is usually discussed in the context of distant stars. (It is also tiny in comparison to the angular size of Sun and Moon as seen from Earth.)

History

The discovery of stellar aberration in the late 17th century came as a surprise. Scientists were mainly looking for stellar parallax, potentially allowing for a triangulation of the distance of some stars and also confirming the still relatively young heliocentric model.

One of the first noteworthy measurements was that of [Eltanin|https://en.wikipedia.org/wiki/Gamma_Draconis] by Robert Hooke in 1674. At first, the measured deviation was not understood to be a separate phenomenon from parallax. Also, the propagation of light was neither well understood nor the focus of scientific inquiry at the time. Even after Ole Rømer measured the speed of light in 1676, and, in particular, demonstrated that it was not instantaneous, it took decades for scientists to draw a definitive connection between the angles at which stars appeared to be and the way that light propagates in a vacuum.

Once the significance of stellar aberration, and its distinctness from parallax, were understood, there were however a number of astronomical experiments that attempted to use the effect to distinguish between different theories of light propagation, namely emission theory, which treated light as a stream of particles inheriting momentum from the emitting object, and the undulatory theory of light (better known as aether theory), which treated light as a wave in a ubiquitous but "non-tangible" medium.

Modern Scientific Explanation

In modern science, stellar aberration is understood to be consistent with, and quantitatively predicted by, the theory of special relativity. When a lateral Lorentz boost, i.e. a velocity addition, is performed to transform a ray of light from the inertial reference frame of a star to the inertial reference frame of an observer, the angle of propagation changes depending on said lateral speed. (The existence of "instantaneous" inertial frames is enough.)

However, one should be careful when citing stellar aberration alone as evidence for special relativity because aberration experiments are not precise enough to be specific towards testing relativity. In particular, while there is a relativistic formula that makes distinct predictions, the "classical" math, e.g. derivable from emission theory or from an aether in which the light source is stationary, still works extremely well in practice.

Special Relativity Mathematics

Note: So, is light a stream of particles or a wave? You have probably heard that it is a bit of both, as a rough summary of quantum mechanics. However, it is probably better described as a quantized wave usually, and in this context can be treated simply as a wave (individual photons are not relevant here). The direction in which a ray of light propagates as a wave is perpendicular to its wavefront, which is what matters here.

In this subsection, a closer look is taken at the mathematics of stellar aberration in special relativity.

The derivation of stellar aberration uses a standard relativistic coordinate transformation, namely a so-called Lorentz boost. (However, while the simplest type of boost is colinear, this boost occurs at an angle.)

Note: Lorentz transformations in general are used to transform spacetime coordinates from one inertial reference frame to another and can apply a velocity boost in any spatial direction, rotate the coordinate axes or move the coordinate origin. Usually origin and coordinate axes are kept fixed for simplicity, and we only apply velocity changes. This specific class of Lorentz transformations are called the Lorentz boosts.

In the following, v is the speed of the observer relative to the light source, θ is the angle of relative motion (90° for perpendicular motion) and ɸ is the light's angle of incidence. Note that the actual aberration is θ - ɸ. c is the speed of light in a vacuum.

The relativistic aberration formula is:

$\tan \frac{\varphi }{2}=\sqrt{\frac{1-v/c}{1+v/c}}\tan \frac{\theta }{2}$

The classical aberration formula, by James Bradley, 1727, is:

$\tan \varphi =\frac{\sin \theta }{v/c+\cos \theta }$

And here it is crucial to note that, while both formulas make different predictions, the classical formula as derived using emission theory or, alternatively, an aether in which the light source is stationary and the observer is moving, is not practically distinguishable from the relativistic version. This is because the classical formula is essentially an approximation of the relativistic formula for "non-relativistic speeds", which includes Earth's orbital speed.

Anyway, for θ=90°, this further simplifies to

$\sin (\theta -\varphi )=v/c$

And for $v\ll c$, the small angle approximation for the sine applies, so:

$\theta -\varphi \approx v/c$

Note: The small angle approximation gives an angle in radians. Remember to translate the result to a different unit (conventionally arcseconds when it comes to stellar aberration) as necessary.

Example values

The following table shows the predicted aberration angle $\theta -\varphi$ in arcseconds, i.e. 1/3,600th of a degree, using the exact SR formula, the classical formula, rsp. the small angle approximation $v/c$ for $\theta =90^{\circ }$ and the minimum and maximum orbital speeds of Earth around the Sun. Results are rounded to 6 decimals.

Predicted Aberration Angles

v [m/s]	SR	Classical	Small Angle
29290.0	20.15226208	20.15226199	20.15226205
30300.0	20.84716768	20.84716757	20.84716764

Again, the point is to show that there is no meaningful difference in the predicted values. The predictions are less than one millionth of an arcsecond apart, which is far closer than common measurement accuracy for astronomical observations.

See Also

• AetherCosPlay_Aberration_or_Starlight