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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.
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 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!
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.
Saturn at Opposition 2014
On 10 May 2014 the planet Saturn will be at opposition, making it ideally placed for observation. To be honest, though, Saturn will be a feature of our night sky throughout the spring and summer, only vanishing into the twilight glow of sunset in September. However, at opposition Saturn rises when the sun sets and sets when the sun rises, meaning it’s in the sky all night long.
Saturn as it might look through a large telescope, image by Kenneth Crawford and Michael A. Mayda
Saturn looks like a bright star in the east at sunset, shining at magnitude 0, making it a little fainter than the other bright planets up there at the moment, Jupiter (at around magnitude -1.5) and Mars (at around magnitude -1), but still brighter than most other stars in the night sky, shining about as brightly as the star Arcturus.
Saturn is the furthest planet we can see with the naked eye (unless you head somewhere very dark and strain your eyes to catch a glimpse of Uranus), lying around 9 astronomical units from us (approx. 827 million miles). The reason we can see it shining so brightly is that it’s quite reflective (reflecting 47% of the Sun’s light that shines on it) and VERY big.
The disk of Saturn will appear larger (just) than the disk of Mars when seen through a telescope (18.7 arcseconds for Saturn compared to 15 arcseconds for Mars), but its rings stretch further, subtending 44 arcseconds.
Saturn really is the jewel of the solar system. It’s the planet that most people recognise, and I would bet that it ranks pretty high on most people’s bucket lists of “things to see through a telescope”. If you have a ‘scope, or know someone who does, it’s worth taking a look as Saturn arcs overhead this spring and summer.
You’ll also catch a glimpse, if observing with a small telescope, of Saturn’s largest moon Titan, the second largest moon in the solar system, larger the the planet Mercury. Saturn has 62 major moons, and countless smaller ones (the rings after all are made up of billions of pieces of ice and dust, mini-moons) but only Titan is visible through small scopes. To see the next four brightest (Dione, Enceladus, Tethys and Rhea) you’ll need a decent sized scope, say 8″.
Saturn, Mars, and Arcturus make a prominent triangle in the south at midnight, 10/11 May (created using Stellarium)
Happy Birthday Hubble! Top Five Spring Telescope Targets
The iconic Hubble Space Telescope (HST) was launched 23 years ago on 24 April 1990, and ever since has been returning breathtaking images of the cosmos as well as world-changing science. It is, without a doubt, one of the most successful scientific instruments ever built.
To celebrate its 23rd birthday here is a list of five stunning celestial objects visible over the next couple of months that you can find for yourself using a small earth-based telescope. Most of these objects will look like nothing more than diffuse grey smudges in the field of view of your eyepiece, but I’ve illustrated this post with some HST images of the same objects, to show you what they really look like. Despite the fact that your telescope can’t ever show anything as stunning as an HST image, there’s something even more wonderful about seeing these objects in real time, for yourself, not mediated via a computer screen.
1. Saturn
- Hubble view of Saturn
Rising around 2030 local time at the end of April, and 1800 local time at the end of May, Saturn is visible in the evening skies throughout the Spring and into Summer. At the moment Saturn’s rings are tilted very favourably towards us, presenting a striking view. Through a very small telescope – or binoculars on a tripod – Saturn might appear as nothing more than a oval, or at best a circular disk with handles, but most modest telescope should show the disk of the planet and the rings, and even Saturn’s largest moon, Titan.
2. Sombrero Galaxy, M104
- Sombrero Galaxy, M104
The stunning Sombrero Galaxy in the constellation Virgo gets to its highest above the horizon around 2330 in late April, and 2130 in late May. It’s one of the brighter galaxies in the sky, and so even a medium sized telescope should show up the dark dust lane obscuring the view of the central bulge of the galaxy. This dust lane is actually a ring that surrounds the galaxy, and is probably where most of the star-forming takes place, as it is composed of atomic hydrogen and dust.
3. Ring Nebula, M57
- The Ring Nebula, M57
Located in the constellation of Lyra in the Summer Triangle, the Ring Nebula (Messier number 57) is a striking object in medium or large telescopes. It rises from low in the NE mid evening to almost directly overhead by the time dawn begins to brighten the sky. The Ring Nebula is a great example of a planetary nebula, so-called as it looks like the disk of a planet when seen through modest telescopes. However this name is completely misleading, as the gas in this nebula was puffed off by a red giant star just before it died and collapsed into a white dwarf, a fate that awaits the Sun in 5 billion years or so.
4. The Great Globular Cluster in Hercules, M13
- The Great Globular Cluster in Hercules, M13
This spherical collection of around 300,000 stars is one of the best examples of a globular cluster in the sky. It’s high in the SE sky during the evenings of April and May, and continues to be visible into the Summer. M13 is at the very limit of naked eye visibility, and small telescopes show it off beautifully. In fact, this is one object where a smaller earth-based telescope gives you a better overall view of the object than the mighty HST. Hubble has such a high magnification that its field of view is very small. This is fine when you’re looking for tiny faint galaxies millions of light years away, but a nearby globular cluster presents problems; it’s simply too big to fit into the field of view. Nevertheless, this spectacular HST image shows the heart of M13, and the stunning array of stars that make up this beautiful object.
5. The Eagle Nebula, M16
- Pillars of Creation in The Eagle Nebula
OK, OK, so maybe this is more strictly speaking a late summer object, but it is visible pre-dawn in late May, low in the south, in the constellation of Serpens. Despite the unsocial hours it keeps at this time of the year, it still has to be included in any top-5 list of Hubble objects. The iconic “Pillars of Creation” image, taken by HST in 1995, is one of the most widely viewed of all Hubble images. It shows giant pillars of gas within the Eagle Nebula within which new stars are being born. However it’s a pretty tricky nebula to see through a telescope. There’s a star cluster within it that you’ll make out even in light polluted skies but to see it best you’ll need to head to a dark stargazing site and be patient.
For maps and tips about how to find these objects, and hundreds more like them using binoculars or a telescope check out my book, Stargazing for Dummies.
UPDATE: I just realised; there are people alive today with degrees in astrophysics who weren’t yet born when the Hubble Space Telescope was launched in 1990!
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.
Mercury, Venus and Saturn in the morning sky. Image credit Paul Sutherland.
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