In a previous post I described how to measure degrees of arc in the sky, but a degree is quite a large distance when measuring angle, and so is usefully split into smaller angles, the arcminute being 1/60 of a degree, and the arcsecond being 1/60 of that, or 1/3600 of a degree.
One arcminute is written 1′, and one arcsecond is written 1″ (i.e. 1° = 60′ = 3600″)
So exactly how big are these tiny angles?
We can use high school trigonometry to work out what size of object covers (subtends) one arcminute:
In the above diagram, a is our angle, 1/60 of a degree, d is our arm’s length distance (say 60cm or 0.6m), and therefore h/2 = d x tan a = 0.6m x tan (1/60) = 0.000175 m, and so h = 0.00035m, or 0.35mm.
Therefore an arcminute is the angle subtended by a piece of medium thickness card (0.35mm, or 350 micrometer, card is often used to make birthday cards) held at arm’s length. That’s pretty small.
An arcsecond is obviously even smaller. In fact it’s such a tiny angle that it’s hard to compare its size to anything real. Let’s instead work out how far away you’d have to hold a human hair before it covered 1 arcsecond.
In this case, the equation we need to use is d = h/ 2 tan a where h is the width of a human hair (around 0.0001m) and a is one arcsecond or 1/3600 of a deegree, so therefore d = 0.0001/2 tan (1/3600) = 10.3m
So a human hair held ten metres away would cover one arcsecond of sky! Wow, that’s really small.
How small an angle can our eyes see?
The angular resolution of your eye is probably around 1 or 2 arcminutes. Binoculars and telescopes magnify images and allow you to resolve objects at a much smaller angular separation. As an example the Hubble Space Telescope has a minimum resolution of 0.05″ (i.e. Hubble could resolve the width of a human hair held over 200m away!)
Real Astronomical Examples
Jupiter’s four largest moons, the Galilean satellites, can reach up to 10 arcminutes of angle from Jupiter, meaning that they are well within the ability of the human eye to resolve. They are also more than bright enough, reaching up to 4.6 magnitudes. The only thing stopping you seeing them is the glare from Jupiter itself, which can be up to a thousand times brighter. Your best chance to seeing one of the Galilean moons with your naked eye is by masking Jupiter with something like a wall or pole. That will obscure the glare and should let you see the fainter satellites!
Alpha Centauri, the nearest star to the Sun, is actually a binary star system in which two stars orbit a common centre of gravity. This isn’t detectable to the human eye, but through binoculars or a telescope you can split the point of light in the sky up into two individual points. The angular separation of the two stars that make up Alpha Centauri varies between 2 and 22 arcseconds. Even at their greatest seperation the angular size between them is still smaller than the width of a human hair held at arm’s length!