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Arcminutes and Arcseconds

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?

Arcminutes

We can use high school trigonometry to work out what size of object covers (subtends) one arcminute:

working out angle

How big is 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.

Arcsecond

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.

working out angle

How for would you have to hold a human hair for it to cover one 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!

Jupiter's Galilean 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!

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  1. Tony Barraclough -UK
    November 2, 2010 at 00:28

    Thanks for this explanation, it means that I really did see Venus as a crescent in the morning sky from near Skipton on the way to school in the 60′s. I was beginning to wonder if I’d imagined it. I had top notch vision when I was a child and the skies were darker for sure.

  2. April 21, 2013 at 11:15

    Venus at inferior conjunction may have an apparent angular diameter close to 60 arc-seconds. But the planet would not be visible to the unaided eye this close to the Sun (besides the exercise would be dangerous). To have any chance of seeing Venus with the unaided eye in daylight when the Sun is above the horizon the elongation (angular separation from the Sun) would need to be of the order of 10 degrees or more; its diameter would then be well below 60 arc-seconds.

    Venus having a high albedo is best observed in daylight or very strong twilight to lessen the contrast between planet and sky background. These matters are well documented in “The Haunted Observatory” Richard Baum (former Director of the Mercury & Venus Section BAA) Prometheus Books 2007.

    John C Vetterlein
    Auroral & Magnetic Observatory
    Orkney
    UK

  1. April 9, 2010 at 22:21

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