How to calculate your horizon distance

While on a recent trip to the remote South Atlantic island of St Helena (exile place of Napoleon, and location of Edmond Halley’s observatory) [blog post to follow!] I ascended the highest mountain on the island, Diana’s Peak.

At 823m above sea level it commanded splendid views of the island, but the most striking thing was the unbroken 360° view of the horizon. I did a quick calculation in my head of how far I could see, and that forms the basis of this blog post: how do you calculate your horizon distance?

It turns out it’s pretty straight forward if you know a little simple maths. It helps to start by drawing a picture, so I did:

Definitely not to scale

For an observer of height h above sea level, the horizon distance is D. The Rs in this diagram are the radius of the planet you’re standing on, in this case the Earth. The only real assumption here is that you’re seeing a sea level horizon.As you can see you can draw a right-angled triangle where one side is D, the other is R, and the hypotenuse (the side opposite the right angle) is R + h.

Using Pythagoras’s Theorem, discovered around 2500 years ago, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So we can say that:

(R + h)² = R² + D²

If you expand the part to the left of the bracket you get (R + h)² = R² + 2Rh + h² so that:

R² + 2Rh + h² = R² + D²

There’s an R² term on both sides of the calculation so you can cancel them out, leaving:

2Rh + h² = D²

Therefore the horizon distance, D, is:

D = √(2Rh+h²)

Here’s where you can make life much simpler for yourself. In almost every case R is much, much larger than h, which means that 2Rh is much, much larger than h² so you can just ignore h² and your equation simplifies to:

D ≈ √2Rh

(the ≈ sign here means “almost equals”. Honestly.)

So if you know R and h you can calculate D. To make this calculation easily you can carry round the value of √2R in your head meaning you only have to calculate √h and multiply those two numbers together.

So for the Earth, R is 6371000m, so √2R is 3569.6. Multiplying this by √h in metres would give you D in metres, so lets convert that into km to make things easier. This means dividing this number by 1000, giving an answer of 3.5696 which is ≈ 3.5.

So as a rough rule of thumb, your horizon distance on Earth,

D = 3.5 x √h

where D is measured in km and h in metres.

On Diana’s Peak, at 823m high, √h = 28.687… which multiplied by 3.5 gives a horizon distance of almost exactly 100km!

This is pretty cool, and is true of anywhere you can see the sea from a heigh of 823m.

One final calculation which sprung to mind on the mountain top was the area of sea I could see, which is easy to work out using the fact that the area of a circle is πr², where r in this case is D, or 100km.

π is 3.14159 which means that the area of sea I could see was 31415.9 km². Just a tad larger than Belgium, at 30528 km^2.

And in that Belgium-sized circle of ocean was only one ship, the RMS St Helena that was taking me home the following day.

What about on other planets?

If you’re on Mars your horizon distance is shorter, at 2.6√h. On Mercury it’s smaller still at 2.2√h. This is due to Mars and Mercury being much smaller than the Earth, and so their surfaces curve away from you quicker. Venus is almost exactly the same size as the Earth (only a fraction smaller) so there you’d have to use the same calculation as here on Earth, 3.5√h.

Hovering above the surface of Jupiter your horizon would stretch to 11.8√h and on Saturn to 10.8√h. Uranus and Neptune are about the same size, giving a horizon distance of 7.1√h.

Mercury 2.2√h
Venus 3.5√h
Earth 3.5√h
Mars 2.6√h
Jupiter 11.8√h
Saturn 10.8√h
Uranus 7.1√h
Neptune 7.1√h

What about the dwarf planets? Being so small their surfaces will curve away from you very quickly, shortening your horizon distance. One of the smallest spherical objects in the solar system is the dwarf planet Ceres (as in cereal), which is the largest object amongst the fragments of rock in the asteroid belt. Your horizon distance on Ceres is almost exactly √h, making that a pretty simple horizon calculation!

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The Night Sky in May 2014

This month sees a glut of amazing stargazing sights in the night sky, even as the days lengthen towards summer.

Saturn as it might look through a large telescope, image by Kenneth Crawford and Michael A. Mayda

Saturn is coming to opposition this month (10 May) meaning it shines in the sky all night long throughout the month. A small telescope (even a pair of binoculars on a tripod) will show Saturn’s beautiful rings and one of its moons.

Mars is even brighter than Saturn, shining a soft orange colour in the constellation of Virgo, near the bright star Spica.

Jupiter is still an evening object although it sets in the west around 1am.

There’s the possibility of a spectacular new meteor shower on 23/34 May as the Earth passes through the dust trail of comet 209P/Linear.

And May sees the start of the noctilucent cloud season, where these elusive high-altitude begin to shine in deep twilight.

Full moon this month is on 14 May, when the Moon will sit near Saturn.

Mercury, Venus, Jupiter in the evening sky, May 2013

This evening, and for the next few evenings, just as the sky begins to darken after sunset, you’ve got a chance to see three of the five naked-eye planets side by side.

The two brightest naked eye planets (Venus and Jupiter) are close together, separated by only a few degrees, closing to 1° on 28 May (in what we call a conjunction). This should make them very easy to spot, low in the NW from around 30 minutes after sunset. In fact they’re close enough together that you could fit them both in one binocular field of view.

Mercury, however, might be trickier to spot. As the faintest naked-eye planet it will lurk in the twilight sky unseen for many people, just above the two brighter planets.

Remember, if you’re observing with binoculars or a telescope make sure you wait until the Sun has fully set

Countdown to Conjunction: Venus and Jupiter

If you’ve been outside in the evening over the past few weeks you’ll have noticed that there are two very bright “stars” close together, following the Sun as they set one after the other in the west. Those two bright dots are not stars at all; they’re planets. The brighter of the two is Venus, which at the moment is below and to the right of the other dot, which is Jupiter.

Tonight they are around ten degrees apart in the sky, but over the next week they’ll get closer and closer, as Venus whizzes and Jupiter crawls round the Sun, until on 15 March they’ll be in conjunction, only 3 degrees apart.

Jupiter and Venus in conjunction 15 March 2012, around 1930

On the days either side of 15 March (say between 08 and 19 March) they’ll be very close too. In fact it’s worth watching this celestial merry-go-round in action every clear evening over the next few weeks as the planets move towards and then away from each other in the sky. Towards the end of March though it’ll become harder to see them both as they disappear into the glare of sunset. If you’ve got clear skies and a good western horizon it’s worth looking out for the thin crescent Moon which will appear between the two planets on the night of 25 March.

Venus

Venus, the second planet out from the Sun, is about the same size as the Earth, just a little smaller. It’s the hottest planet in the solar system, with a thick atmosphere of carbon dioxide gas (94.6% is CO2, the rest is mainly nitrogen) which traps most of the light from the Sun that shines on it, super-heating the atmosphere to around 460°C (733K). At ground level this thick, hot atmosphere creates a pressure over 90 times greater than sea-level pressure on Earth. High in Venus’ atmosphere float clouds of sulphuric acid, which is all we see when we look at Venus from the Earth.

Seen from here on Earth, the size and shape of Venus in our sky changes as we both orbit the Sun. At its closest to Earth Venus is “only” 38 million km away, and its disk is 66 arc seconds across, while at its furthest from us it’s 260 million km away, and it shrinks to around 10 arc seconds. On top of this, its phase changes from full (when it’s directly opposite the Sun as seen from Earth) to new (when it’s directly between us and the Sun) and back again. Of course when it’s in either of these positions we won’t see it, as it will be in the sky right next to the Sun. We see Venus best when it’s far to the west of the Sun (when it’s seen in the evening) or far to the east (when it’s seen in the morning). The furthest west and east points as seen from Earth are called maximum elongation, and at these points Venus presents a half phase to us.

Due to the reflectivity of its clouds, and its proximity to us, Venus is the brightest planet as seen from Earth. Venus appears brightest in our sky, at around -4.5 magnitudes, when it’s 68 million miles from us and presents a crescent phase.

During the 15 March conjunction Venus will have a brightness of -4.2 magnitudes.

Jupiter

Jupiter, the largest of all the planets in the solar system, is a gas giant, into which the Earth would fit well over 1000 times. It has a family of 66 moons (at the last count), the largest four of which, Io, Europa, Ganymede and Callisto, are big enough that they would be considered planets if they didn’t themselves orbit a planet. Jupiter orbits the Sun at a distance of around 780 million km, and so it’s surface is very cold, around -150°C (123K). Its atmosphere (and the whole planet is atmosphere, with no solid surface) is made up of around 90% hydrogen and 10% helium.

Seen from Earth, Jupiter presents a disk of between 30 and 50 arc seconds, making it (usually) bigger than Venus as seen from Earth. Unlike Venus, Jupiter always presents a full phase towards us, upon which we can see (through even a small telescope) cloud features including the famous Great Red Spot. The gases in Jupiter’s atmosphere are not as reflective as Venus’ clouds, and Jupiter is a considerable distance further away, and so it appears fainter in our sky, despite presenting a large disk to us. The brightest it gets is around -3 magnitudes, about four times fainter than Venus at its brightest.

During the 15 March conjunction Jupiter will have a brightness of -2 magnitudes, compared to Venus at -4.2 magnitudes, and so Venus will appear around 7 times brighter.