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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!
John Dobson, “The Sidewalk Astronomer”, 1915-2014
John Dobson, known as “the sidewalk astronomer“, and the inventor of the Dobsonian telescope mount, died yesterday 15 January 2014.
John Dobson, 1915-2014
I had the pleasure of meeting John back in 2006 when he visited Glasgow Science Centre to give a talk in the planetarium. As planetarium manager at the time I was in John’s company for most of the two days he was in Glasgow, and became well used to his unorthodox – and mischievous – teaching style.
“How many stars are there in our solar system?” he asked, with a twinkle in his eye.
“One…?” I ventured, smelling a trap.
“Actually there are three. Jupiter and Saturn are stars too. They define a star as something that emits visible light. But Jupiter shines in infra-red; it gives off more than twice what it takes from the sun, but astronomers don’t see that, because they’re too retarded. You see, that’s the problem with talking to me – I throw you a curve ball.”
His most controversial views were regarding the Big Bang Model. It’s fair to say that John wasn’t a believer* – he told me he was “allergic” to it! He instead promoted a steady state model, where the receding galaxies “fell off the edge” and got recycled back in… To be honest I didn’t really follow his arguments too closely, but there’s no doubting the entertainment of his delivery.
Certainly the audience who came to Glasgow Science Centre’s planetarium to see John talk were thoroughly entertained. Perhaps not in a very orthodox way, nor with theories that were widely accepted in the astronomy community, nor indeed about what they thought they’d hear John speak about. The talk had been billed as “hear John Dobson, inventor of the Dobsonian telescope mount, talk about his revolutionary design and his passion for sidewalk astronomy”. He gave that five minutes at the start of the talk; the rest was non-standard cosmology and poking fun at the consensus.
He didn’t credit himself as the inventor of the Dobsonian mount. After all, astronomers had been using elements of it for a long time before John put them all together and began popularising this low-cost mount.
A Dobsonian Telescope Mount
A Dobsonian mount is effectively a spinning plate with a cradle on it for holding the telescope. The plate spins allowing you to move the telescope from side to side (the azimuth co-ordinate, in astronomy speak) and the telescope can tilt in the cradle allowing you to move it up and down (the altitude co-ordinate). This is a far simpler mount than the alternatives, and can be built out of everyday items at low cost, meaning that more of your budget can go on the telescope tube itself, building larger tubes to collect more light (we call these large telescopes “light buckets”) and so get clearer, sharper images.
John himself never called them Dobsonians, instead referring to them as “sidewalk telescopes”. “For hundreds of years, wars were fought using cannon on ‘Dobsonian’ mounts; it’s nothing new,” he would say. But his design was innovative, and it brought the universe a little bit closer to us.
My primary telescope – the Skywatcher 250 PX – uses a Dobsonian mount. It’s an ideal scope for public astronomy events as it’s very quick to set up, and is really easy to operate. Want to move it? Just nudge or pull it.
John’s mount, and his passion for showing people the universe through a telescope, led to his popularising of “sidewalk astronomy”, which involves standing with a telescope out in a busy street in your town and showing passers-by views of the cosmos. Of course due to light pollution in towns and cities you’re limited as to what you can show, but the Moon and planets are easily visible from wherever you are. If your unsuspecting passer-by has never seen the rings of Saturn, or the moons on Jupiter, or mountain ranges and craters on our own Moon, you can be sure that their few minutes with your telescope will amaze them.
John Dobson will be remembered as the grandfather of sidewalk astronomy, but I’ll remember him most fondly as the very eccentric and enthusiastic man that I spent a couple of days with in Glasgow in 2006. The most vivid memory I have of John is taking him out for dinner the night he arrived in Glasgow, as I did with all visiting speakers. He didn’t eat meat, he informed me. And he didn’t eat processed food in restaurants. He saw that I was beginning to look worried. I suggested he might like a salad. With a grin he said “Why pay for a salad when there’s perfectly edible stuff just laying around?”, at which point he began rummaging in the flowerbeds for edible plants and weeds…
* Not that I’m a believer. I don’t believe in the Big Bang, rather I accept it as a model for the universe which fits all of the observations that we make. It’s true, to the limits of our current observations.
Light Pollution and Birds: Early Bird Survey
The negative effect of light pollution on wildlife has long been known, specifically – but not exclusively – its effect on bats, bugs, and sea turtles. Now the British Trust for Ornithology (BTO) are running an Early Bird Survey, asking people in the UK to monitor the pre-dawn feeding times of garden birds to see what – if any – effect light pollution is having.
To take part you need to get up before dawn* on 9** January 2014 (tomorrow, as I write this), watch your garden bird feeders, and record the times that the first ten species arrive to feed. You can download the full instructions here (pdf), and submit your observations here.
* dawn occurs at different times around the UK, so you should find your sunrise time and get up half an hour earlier than that, during civil twilight.
** observations on 10, 11, and 12 January are welcome too.
As the BTO website says:
Winter is not an easy time for birds. They need extra energy to keep warm, especially during long winter nights. To cope with this, they lay down extra fat reserves, though small birds quite often only lay down enough for a single night. Longer nights not only affect the amount of energy a bird uses, they also reduce the amount of time that birds can feed in. Birds, therefore, have to make the most of the daylight hours to replenish their energy reserves before it gets dark.
The 2004 BTO Shortest Day Survey, run in association with BBC Radio 4, investigated the patterns behind birds arriving at garden bird feeders first thing on a winter’s morning. Building on observations from the Shortest Day Survey, the Early Bird Survey will investigate what effect, if any, light and heat pollution have on the feeding patterns of birds during a cold winter’s morning.
Scots in Space
I was delighted to hear that two groups from Glasgow were winners in last night’s UK Space Conference‘s Arthur Clarke Awards 2011.
Clyde Space, a “leading supplier of small and micro spacecraft systems”, was given the Arthur Clarke Award 2011 for Achievement in Space Commerce, while the University of Strathclyde’s Advanced Space Concepts Laboratory, which “undertakes frontier research on visionary space systems”, was given the Arthur Clarke Award 2011 for Achievement in Space Research.
Congratulations to both, and it’s exciting to me as a Scot and a resident of Glasgow that these two groups, located within 5 miles of one another, are leading the UK in space research and commerce.
The Circle of Little Animals
This is an auspicious time of the year.
The Sun, on its yearly circuit of the sky*, moves gradually along the ecliptic, a line which is the projection of our solar system’s disk onto our night sky. This line, the ecliptic, is also known as the zodiac, a term originating from the late 14th Century, and deriving from the Greek literally translating as “circle of little animals”. (Incidentally our word for zoo derives from the same origin).
The Circle of Little Animals
To the ancient Greeks this was indeed a circle of little animals, featuring: a ram, Ares; a bull, Taurus; Pisces the Fish; and many other well known (for all the wrong reasons) constellations. It also features three humans: Aquarius the Water Carrier, said to represent Ganymede, beloved of Zeus; and Gemini the Twins, Castor and Pollux.
The only zodiac constellation which is inanimate is Libra the Scales, taken from Babylonian astrology. The Greeks however didn’t recognise Libra; instead they thought that the stars here marked out Scorpius’ claws, which they considered to be a separate sign.
So the twelve constellations that lie along the ecliptic are most well-known due to astrology, a pseudo-science that suggests there is some significance to which constellation the Sun was in when you were born. This is, of course, bullshit.
There are so many reasons why astrology should be laughed off as pre-scientific magical thinking (no evidence, no mechanism by which it might work, inconsistent etc) but next time you meet an astrologer ask them what star sign you would be if you were born between 30 November and 18 December. If they tell you Sagittarius (if they’re a Hindu astrologer they might also say Scorpius) then tell them they are plain wrong.
On these 18.4 days of the year the Sun is wonderfully absent from the usual twelve zodiac signs, It is still there, however, gracefully moving along the ecliptic, but between 30 November and 18 December it is in the constellation of Ophiuchus the Serpent-bearer.
Ophiuchus does not appear in any astrologer’s zodiac. Back in the early days of astrology, when it was first dreamed up several thousand years ago, there were indeed only twelve constellations lying along the zodiac. Over the past few millennia however the Earth’s axis has wobbled slightly (an effect called precession) with the result that the line of the ecliptic has moved with respect to the constellations, and so an interloper, Ophiuchus, has crept in.
So celebrate all those of you born between 30 November and 18 December (one person in twenty share this star sign); you’re not a Sagittarian at all; you’re an Ophiuchan.
It’s still all bullshit though.
* the Sun, of course, does not orbit the Earth, it’s the other way around. It just looks like it does from down here…
Awards [Blush]
This year I was fortunate enough to be the recipient if two fantastic astronomy awards, and a nominee for another.
In April 2010 I was nominated for the UK Space Conference’s Arthur Clarke Award for Public Promotion of Space (it was won by EADS Astrium).
Then last month, while speaking at the Federation of Astronomical Societies’ 2010 Convention, I was presented with two awards:
The 2010 Eric Zucker Award for Outstanding Contribution to Astronomy, awarded by the Federation of Astronomical Societies
and
The 2010 Joy Grifiths Award for Meritorious Efforts in the Cause of Darker Skies, awarded by the British Astronomical Association’s Campaign for Dark Skies
Needless to say I was chuffed to win these awards, and they have pride of place on my workdesk:
My Awards!
Summer Hiatus
It’s been exactly five months since my last post, due to a combination of factors: Sam went back to work after maternity leave, and I dropped my workload to 2.5 days a week to help with childcare; I went freelance on top of the end of my last contract; and the lack of dark skies over the Summer meant a drop in my astronomy output!
Normal blogging will hopefully resume as of now!
Fly the Cloudy Skies
I haven’t posted anything in the last couple of weeks due to my being “stranded” in the delightful island of Sark by the Eyjafjallajökul volcano eruption in Iceland which grounded most flights into and out of the UK for a week.
The view of the Icelandic Volcano from our cottage in Sark
There was no real hardship being on Sark for an extra week, especially given that every single night we were there it was clear (and dark). Indeed it seemed clearer on the nights during the air flight ban, and the daytime skies certainly contained fewer, if any, clouds. And no contrails.
Contrails over Europe
These facts are related, as contrails from planes quickly disipate in the atmosphere, spreading out to become indistinguishable from thin cirrus or cirrostratus clouds.
Contrails, short for “condensation trails”, are the visible trails of condensed water vapour in the sky left behind by exhausts of aircraft. They are, essentially, man-made clouds.
This got me thinking; just how much of the “cloud cover” that affects astronomers so badly is down to contrails?
It turns out I’m not the only one worrying about this. In fact it’s been thought a problem for quite some time.
Gerry Gilmore, of Cambridge University (and of Max Alexander‘s excellent Explorers of the Universe portrait exhibition) was saying back in 2006 that the problem of contrails might make ground based telescopes worthless by the year 2050. One small reason, amongst many, to limit our use of planes.
The last time there was such a significant drop in the number of planes in the sky was after the terrorist attacks of 11 September 2001, and back then astronauts on board the International Space Station noticed a decrease in the number of contrails over the US.
The same was true between 16 and 22 April 2010 when, for a few days, flights over Europe stopped, stranding thousands overseas, but giving a brief and welcome respite to astronomers who were able, for once, to enjoy truly clear skies.
For more info than you could ever want on contrails and their impact on ground-based astronomy, see Holger Pederson’s website with links to many articles and reports.
Also worth a look is the apparently now-defunct National Contrail Network, which formed a part of a masters research project at the University of Lancashire. The official homepage link doesn’t work, but there’s plenty of data and info that is accessible.
National Contrail Network
Finally, have a look at the Cloud Appreciation Society‘s gallery of contrail images: they’re actually rather beautiful.
Trails and Halo, Denmark © Jesper Grønne.
Five Useful iPhone Apps for Astronomers
Here’s a brief overview of the five iPhone apps that I, as an astronomer, find indispensible (in no particular order):
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Weather Pro (£2.39)
Let’s face it, no weather forecast is 100% right, but this app gives you more info than most, allowing you to figure out in advance whether there’ll be clear skies. |
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Magic Hour – formerly VelaClock (£2.39)
So, it’s going to be a clear night. Now you need to know: when is sunset? when does astronomical twilight end? Magic hour is your app for that. Also displays moonrise and moonset times, this is the perfect app for figuring out exactly when your skies will be dark. |
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Starwalk (£1.79)
The very best of all iPhone astronomy apps, Starwalk has an exquisite interface, and is packed full of features. Great for beginners wanting to find their way around the sky, and for experts who want to dig a little deeper. |
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Satellite Visibility (£1.79)
Showing iridium flares, ISS & Hubble passes, and many other sats too. Impress your friends by knowing where and when (to the exact second) satellites are due to pass overhead. |
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Reeder (£1.79)
OK, so not immediately anything to do with astronomy, but with this RSS aggregator and subscriptions to some astronomy blog feeds (Bad Astronomy, Universe Today…), you’ll be able to keep abreast of all breaking astronomy news. |
So only two of these are directly related to astronomy, but I’d be lost without any of these, and for a total cost of just over a tenner these apps are great value for money.
Communicating Astronomy with the Public (CAP)
I have just returned to the UK after attending the amazing CAP2010 conference in Cape Town. This five day meeting of science communicators and astronomers from all over the world was an incredible opportunity to swap ideas and discuss what went well during IYA2009, and what we need to build on.
The conference was blogged by many, but nowhere more thoroughly than the psychohistorian’s blog
For me highlights of the conference include:
Dr Chris Engelbrecht discussing his SkyRanger training programme for African park rangers and safari guides
Amelia Ortiz-Gil describing planetarium activities for the blind an partially sighted, “The Sky is in Your Hands”
Sze-leung Cheung covering the Dark Skies Awareness switch-off programme “Dim It!” in Hong Kong
All in all an amazing time.
The true star of the show though was Cape Town itself. It’s a stunning city, towered over by Table Mountain, which I spent a day climbing on the last Saturday of the conference, with Marek Kukula and Carolina Odman.
Carolina Odman in the Hely-Hutchison Reservoir on Table Mountain
Buy my new book
Dark Sky Diary 