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How to calculate your horizon distance

May 25, 2015 2 comments

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

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 + h= R2 + D2

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

2Rh + h= 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 km2.

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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Morning Conjunction: Venus and Jupiter, August 2014

August 16, 2014 Leave a comment

On Monday morning, 18 August 2014, in the eastern sky before sunrise you’ll see a very close conjunction of the two brightest planets, Venus and Jupiter.

They’ve been shining brightly in the pre-dawn sky for a while now, but as they trace out their separate orbits around the Sun they appear to move relative to one another, Venus the faster of the two. And they’re getting closer every day, until on Monday 18 August they’ll be at their closest, only 12 arcminutes apart, about one third of the diameter of the Moon.

This is closest conjunction in 15 years, and will be a very striking sight in the morning sky, but you’ll need to be up and about early to see it, about an hour before sunrise, around 0450 BST (sunrise is around 0550BST for most of the UK – Orkney gets an earlier sunrise at 0535, while the southwest of England have to wait till around 0605).

If you’ve got a pair of binoculars and a tripod, or even better a telescope, it’s really worth looking at these two planets. Venus is the brighter of the two, shining about twice as brightly as Jupiter through the morning twilight, but if you can magnify them (and you’ll catch them in the same field of view in a pair of binoculars), then Jupiter will be around three times the diameter of Venus (30 arcseconds compared to 10), and you’ll see Jupiter’s four largest moons as tiny points of light near the giant planet.

Don’t worry if you’re clouded out, or if you sleep in, on Monday morning; they’ll be close together in the pre-dawn sky for a few days afterwards too.

Venus-Jupiter conjunction this morning, 0450BST 16 August 2014

Venus-Jupiter conjunction this morning, 0450BST 16 August 2014

Screenshot 2014-08-16 17.42.53

Venus-Jupiter conjunction 0450BST 17 August 2014

Venus-Jupiter conjunction at its closest, 0450BST 18 August 2014

Venus-Jupiter conjunction in close up, 0450BST 18 August 2014

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

December 2, 2012 1 comment

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.

Mercury, Venus and Saturn in the morning sky. Image credit Paul Sutherland.

Mercury, Venus and Saturn in the morning sky. Image credit Paul Sutherland.

Transit of Venus

May 23, 2012 2 comments

This year, on 5 and 6 June 2012, there is a very rare astronomical occurrence: a transit of Venus across the face of the Sun. There have only been six of these transits ever observed before – in 1639, 1761, 1769, 1874, 1882, and in 2004 – and this year’s transit is the last for 105 years!

So what exactly will you see, if you’re lucky enough to catch this last-chance-to-see event? If you’re able to look at the Sun safely you’ll see a tiny black dot moving slowly across the surface – that dot is the planet Venus! NASA has the exact times of the transit from major cities. Importantly, this transit is best seen from the Pacific. Observers in north and central America will see only the start of the transit before the Sun sets, while those of us in Europe will only catch the end of it if we’re up at sunrise.

UK observers: set your alarms! You’ll see the transit between sunrise and 0536 BST, at which point Venus begins leaving the Sun’s disk, taking about 18 minutes to do so.

Venus is 6000km across – just a little smaller than the Earth – and at transit it will be around 43 million km away, directly between us and the Sun. The Sun is 1.4 million km across and around 150 million km away. This means that, seen from Earth, Venus is only about 58 arcseconds in diameter, while the Sun is 1891 arcseconds across, about 33 times the apparent diameter of Venus. So: Venus small dot; Sun big bright ball.

Also, we know how far from the Sun Venus is (107 million km), and how long it takes to orbit the Sun (225 days), so we can work out how long it should take to pass across the Sun’s disk (around 6.5 hours). However the start and end times for the transit vary depending on where on Earth you’re observing, with observers in eastern Canada seeing Venus start to cross the Sun’s disk a whole thirteen minutes earlier than observers in Australia! This is because Canadians are looking at the transit from a slightly different angle than Australians.

Why transits of Venus are (were) important

If you have observations from two widely spaced points on the Earth’s surface, and if you time the start and end of transit accurately at each, you can work out the solar parallax, that is, the difference in position of the Sun when viewed from two different points on Earth, the two points being one Earth radius apart. (Hold your thumb up, close one eye, and obscure a distant object; now switch eyes, and your thumb appears to move with respect to the distant object. That’s parallax).

From the solar parallax, if you know the Earth’s radius, you can work out the Earth-Sun distance (known as the astronomical unit) using high-school trigonometry. This was important to astronomers in the 18th century, as up until then all we knew were the relative distances between all the planets in our solar system, not the actual distances. Once we had one measurement within the solar system – the astronomical unit, say – we could work out how far away everything else was.

The technique of using transits of Venus to work out the solar parallax was first suggested by Edmund Halley in 1716, after he had observed a much more common (although still only 13 times per century) transit of Mercury from the island of Saint Helena. Halley knew that Venus would give much more accurate measurements than Mercury, since it was closer to the Earth and so the angles would be easier to measure. He also knew that the next transit of Venus would happen in 1761, and urged future astronomers to make observations world-wide and thereby calculate the solar parallax, and from that the astronomical unit.

This was duly done, and a value for the astronomical unit of 153 million km was calculated. Later transits in the 19th century yielded a value of 149.59 million km. The current accepted value, calculated from telemetry from space craft is 149.60 million km, so the transit method worked pretty well.

Countdown to Conjunction: Venus and Jupiter

March 4, 2012 7 comments

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.

Venus, Jupiter and the Moon, February 26 2012

February 26, 2012 2 comments

Tonight, just after sunset, Venus, Jupiter and a thin crescent Moon will line up in the evening sky. If you’ve got clear views towards the west it’s really worth a look.

Looking west after sunset on Sunday 26 Feb 2012

Thanks to the excellent Starwalk app for the above image.

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