## 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.

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} = R^{2} + D^{2}

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

R^{2} + 2Rh + h^{2 }= R^{2} + D^{2}

There’s an R^{2} term on both sides of the calculation so you can cancel them out, leaving:

2Rh + h^{2 }= D^{2}

Therefore the horizon distance, D, is:

D = √(2Rh+h^{2})

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 h^{2} so you can just ignore h^{2} 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 πr^{2}, 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 km^{2}. 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!

## Mercury at maximum elongation, 25 May 2014

The planet Mercury is the most elusive of all of the naked eye planets. It orbits nearest the Sun, and so always rises just before the Sun or sets just after it, appearing in the glow of twilight. For much of Mercury’s orbit it isn’t visible at all, lying too close to the Sun in the sky.

To see Mercury at its best you have to wait until it’s as far as possible from the Sun in the sky; what astronomers refer to as its maximum elongation. When Mercury is at its maximum eastern elongation it’s visible just before sunrise; when it’s at its maximum western elongation its visible just after sunset.

At the moment Mercury is nearing its maximum western elongation and so makes a perfect evening target.

Mercury’s range of maximum elongation is between 18° and 28°, and in this particular apparition it’s furthest distance from the Sun is 22.7°. This occurs on 25 May 2014. Between now and the end of May look west just after sunset to try and catch a glimpse of Mercury shining at magnitude +0.4. It’ll be low in the sky, very low, but if you look towards the west, find Jupiter shining brilliantly, and follow a line down to the right at an angle of approx. 45° you should see Mercury a few degrees above the horizon.

If you’re trying to observe it through a telescope then make sure you wait until the Sun has well and truly set below the horizon. Mercury exhibits phases like the Moon and Venus which can be seen through a telescope but shows no other detail through an earth-based scope; on 25 May the disk of Mercury facing the Earth will only be 40% illuminated, making a fat crescent shape. Mercury’s angular size is the smallest of all the planets save distant Uranus and Neptune.

If you’ve ever seen Jupiter or Saturn through a telescope then you’ll know that they look spectacular despite their relatively meagre size. On 25 May, for example, Jupiter will appear to have a diameter of 33 arcseconds (written 33″), Saturn 19″, Venus 15″, Mars 12″, and Mercury a paltry 8″.

And you can actually see all five of these planets on the night of 25 May (or any night between now and the end of May. Mercury is the trickiest to find, but Jupiter will be blazing low in the east, Mars high in the south, Saturn lower in the south-east, and if you’re keen to get up before sunrise you’ll see Venus low in the east. (Uranus and Neptune are dawn objects too at the moment).

## 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

## Morning Mercury, December 2012

Over the next few mornings you’ll be able spot the most elusive of the naked-eye planets, Mercury, low in the south-east just before sunrise.

Mercury is hard to find, and most days isn’t visible at all. Since it orbits so close to the Sun, when seen from Earth it never appears very far from the Sun in the sky. You can only catch it for a few days at a time when it’s furthest from the Sun in our sky, at a point called its *maximum elongation.* And even then it’s not that simple to find, as it will always be quite low on the horizon, hidden amongst twilight.

As Mercury whizzes round the Sun (it takes 88 days to make one complete orbit) sometimes we see it in the morning and sometimes in the evening. The amount of time between one morning appearance and the following evening appearance is around six or seven weeks. However Mercury isn’t very clearly visible at every maximum elongation (in some the Sun is much nearer the horizon so the sky is much brighter, making it harder to find), and even when it is clearly visible you’ll only catch sight of it on the few days before and after the date of maximum elongation.

Mercury’s next maximum elongation in of 4 Dec 2012, when it’s quite far (21°) west of the Sun, and quite bright (magnitude -0.3) making it quite easy to spot over the next few mornings.

**How to find Mercury**

If you have clear skies, head outside around 0630 and find somewhere with a good clear SE horizon (Mercury rises around 0630 and only gets a few degrees above the horizon by the time the Sun’s light begins to significantly brighten the sky).

Luckily there are two other planets up near Mercury right now, namely Venus and Saturn. Both of these planets are brighter than Mercury and higher in the sky, and together all three form a straight line leading diagonally down to the horizon. Find brilliant Venus, the brightest thing in the sky except for the Sun or the Moon, and then look for Saturn up and to the right, and Mercury in the opposite direction, down and to the left.

This photo, taken by the excellent Paul Sutherland, shows how the three planets lined up this morning (2 Dec) when viewed from the UK.