Stellar Aberration Explained: How a Tilted Telescope Measures the Speed of Light
Point a telescope straight at a star directly overhead and you will slightly miss it. The telescope is fine. The Earth is moving, and light takes time to travel down the tube. The effect is called stellar aberration, it is the same physics as tilting an umbrella in the rain, and it can be used to measure the speed of light.
The umbrella analogy
In the rain, you tilt the umbrella forward because you are walking into the rain. Starlight is the rain, the Earth is the walker and the telescope is the umbrella. Light from a star overhead comes straight down in the frame of the Sun, but the Earth is moving sideways at about 30 km/s. While the light travels down the tube, the tube moves sideways, so the tube has to be tilted forward to keep the light in it.
The size of the tilt
The relative velocity triangle has the speed of light c as one side and the Earth’s speed v as the other, exactly like the rain (8 m/s) and the walker (2 m/s). The tilt angle satisfies tan α = v ÷ c.
First the Earth’s speed. One orbit is 2πr in a time T of one year: v = 2π × 1.496 × 10¹¹ ÷ 3.156 × 10⁷ = 2.98 × 10⁴ m/s, or 29.8 km/s.
Then tan α = 2.98 × 10⁴ ÷ 3.00 × 10⁸ = 9.93 × 10⁻⁵. For such a tiny angle, α ≈ tan α in radians, so α = 9.93 × 10⁻⁵ rad. Convert: × (180 ÷ π) × 3600 = 20.5 arcseconds, which is 0.0057 degrees. Small, but measurable with a good telescope.
Bradley’s idea: measuring c
In the 1720s the astronomer James Bradley was looking for stellar parallax and found a different yearly shift in the positions of stars. It was the same for stars at very different distances, so it could not be parallax. Bradley explained it as the combination of the Earth’s motion and the finite speed of light, and his result for the speed of light was close to the modern value.
Run the sum backwards. If you measure α = 20.5″ and know the Earth’s speed, then c = v ÷ tan α. Convert 20.5″ to 9.94 × 10⁻⁵ rad and c = 2.98 × 10⁴ ÷ 9.94 × 10⁻⁵ = 3.00 × 10⁸ m/s.
Why the tilt flips through the year
Six months later the Earth is moving the opposite way round its orbit, so the telescope must be tilted the other way. Over a year, a star at the pole of the Earth’s orbit seems to trace a tiny circle of angular radius 20.5″. A star in the plane of the orbit swings back and forth along a line, and stars in between trace small ellipses.
Aberration versus parallax
Parallax is the shift you get from looking at a star from two different places, and it depends on the star’s distance: nearer stars shift more. Aberration depends only on v ÷ c, so every star at the same place in the sky shows the same shift, near or far. That difference is how Bradley could tell them apart.
Common mistakes
- •Thinking the star really moves in an ellipse. The star stays put; the Earth’s velocity changes the direction we must point.
- •Forgetting to convert arcseconds to radians (or degrees) before using tan.
- •Mixing units: use v and c in the same units, both metres per second or both kilometres per second.
Frequently asked questions
The Earth’s motion while light travels down the telescope. The velocity of the light relative to the moving Earth is the light’s velocity minus the Earth’s velocity, so it comes in at a small angle and the telescope must tilt toward the direction of motion.
About 20.5 arcseconds at most, from tan α = v ÷ c with v = 29.8 km/s and c = 3.00 × 10⁸ m/s.
If you measure the aberration angle α and know the Earth’s orbital speed v, then c = v ÷ tan α. With 20.5 arcseconds and 29.8 km/s that gives 3.00 × 10⁸ m/s.
Parallax depends on the star’s distance and comes from viewing it from two places. Aberration depends on the Earth’s velocity and the speed of light, not on the distance.