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:

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.

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!

By Degrees

“Look over there!” shouts an excited astronomer.

“Where?” you reply.

“There! Due south, about forty degrees above the horizon. It’s the International Space Station”.

Panicked that you’ll miss this amazing sight, you spin around in a flap. South you can do: just find the north star and turn around. But how high up is forty degrees?

There’s a useful rule of thumb – quite literally – that you can use you measure angular distance in the sky.

Hold your thumb out at arm’s length. The width of your thumb at its widest point is around 2°. As you can see in this picture, the top of the University of Glasgow’s tower is just short of 2° above the surrounding rooftops, when seen from the Millenium Bridge about 1.5km (1 mile) away.

University of Glasgow Tower

This is all very well for small angular distances, or objects close to the horizon, but what about forty degrees up? Counting twenty thumb-heights up above the horizon is quite tricky, so there’s an easier way of measuring larger angular distances in the sky, using your fist. A clenched fist, again held at arm’s length, is about ten degrees top to bottom.

So to find something that’s forty degrees above the horizon, just count one-potato, two-potato, three-potato, four, and you’ll be looking in the right area!

Here’s a picture of Glasgow Science Centre’s tower, taken from the end of Millenium Bridge. It’s about 20° high from here, I’d guess.

Glasgow Science Centre Tower

Let’s check that’s right. Here’s a view of Glasgow Science Centre from above.

The tower is about 250m from the end of Millenium Bridge, and is about 125m tall. As you’ll recall from school trigonometry, tan A = h / d = 125/250 = 0.5, and so A = 26.5°, making my fist (which is about half the height of the tower) around 13° across, top to bottom.

[I was holding my fist a bit closer them normal, awkwardly taking the picture with the other hand, so this discrepancy can be forgiven!]

Heavens Above

Visit heavens-above.com to find out when the next bright satellite is passing over your head, and they’ll give you the info in this format:

So for tonight (29 March 2010) at 2048 there’s a very bright (mag -9, or 100 times brighter than Venus at its brightest) Iridium Flare satellite 66° above the ESE horizon. Find ESE (using a compass or the North Star), and count about 6 or 7 fist heights above the horizon, and that’s where it’ll be!