BAFact math: Give him an inch and he'll take a light year

By Phil Plait | June 18, 2012 10:23 am

[BAFacts are short, tweetable astronomy/space facts that I post every day. On some occasions, they wind up needing a bit of a mathematical explanation. The math is pretty easy, and it adds a lot of coolness, which I’m passing on to you! You’re welcome.]


Today’s BAFact: To scale, if the Earth/Sun distance were one inch, a light year would be exactly one mile.

Humans have a miserable sense of scale. Space is huge – that’s why we call it "space" – but how huge? Here’s a fun trick my friend Dan Durda pointed out to me many years ago when we were in college together (forgive my not using metric units, but what the heck, this only works in imperial):

The average distance of the Earth to the Sun (what we call an astronomical unit) is about 92.8 million miles. If you made a scale model of the solar system where that distance were equal to one inch, then one mile in the model would be almost exactly a light year in the real world!

The math is easy. One light year is the distance light travels in a year. The speed of light is 186,282 miles/second, and distance equals speed multiplied by time. So:

186,282 mi/sec x 86,400 sec/day x 365.25 days/year = 5.88 trillion miles

[Note: I’m rounding the answer to two decimal places for ease of comparison.]

OK, now what about our scale? There are 12 inches to a foot, and 5280 feet to a mile. That means there are

12 in/ft x 5280 ft/mile = 63,360 inches/mile

If we let 1 inch = 92.8 million miles, then 63,360 inches = 5.88 trillion miles.

See? To two decimal places the scale is exact! In real life the Earth orbits the Sun in an ellipse, so there’s a roughly 3% change in distance over time. But if we just take the average distance, this works perfectly.

So the next time you’re out driving, keep that in mind… The nearest known star to the Sun is Proxima Centauri, roughly 4.2 light years away. That means it’s 4.2 miles away to scale, and at 60 mph would take over four minutes to reach. At that same speed, you’re crossing the entire Earth/Sun distance in less than one-thousandth of second!

If you really tried to drive from the Earth to the Sun (and there were a heavenly highway connecting them) at that speed it would take over 175 years.

From a scale model millisecond to more than a century. Did I mention space is big?

Original image credit: sebikus/shutterstock.com


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CATEGORIZED UNDER: Astronomy, BAFacts, Cool stuff, Geekery, Space
MORE ABOUT: light year

Comments (56)

  1. Nicholas

    And that’s only a fraction of the observable universe.

    More Math:

    The observable universe is about 13,700,000,000 Light years. The circumfrence of the earth is about 24,901 miles. 13,700,000,000/24,901 is 550,178.

    Therefore for scale it would take 550,178 trips around the earth to reach the end of the observable universe.

  2. Ken_g6

    I thought Phil was just a miled-mannered astronomer, but it appears he’s light-years ahead of the rest of us!

  3. Scott P.

    Now I’m confused — it would take 175 years to drive to Proxima Centauri at 60 mph?

  4. Grizzly

    And more math…

    Voyager 1 would be about 10 feet out. 10 feet.

  5. Bryan

    Looks like Im going to need a bigger plot of land to build my scale model of the solar system.

  6. Cindy

    I was waiting for the rest of the quote by Douglas Adams.

    “Space is big. You just won’t believe how vastly, hugely, mind- bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space.”

  7. Larry

    I think the Hitchhiker’s Guide to the Galaxy said it best:

    Space is big. Really big. You just won’t believe how vastly hugely mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist, but that’s just peanuts to space.

  8. Charles

    Not just “over 175 years”, more like nearly 47 *million* years. Which technically is over 175, but it’s also over an hour.

  9. Scott,

    He means it would take 175 years to get to the Sun from Earth at 60 mph.

    I do a scale model for my astronomy class where the Earth is about the size of a BB and the Sun is a half-meter spherical light fixture on campus. I always point out as we’re walking off the distances that our walking speed is about 10 times the speed of light in the model’s scale, and yet the nearest star in scale would be over 6000 miles away.

  10. Mark

    So, in the scale, where we are driving at 60 mph and cover one light year in a minute, we are travelling at more than 525,000 times the speed of light! And the wheelbase of our car is also larger than the entire solar system!

  11. Colin

    @Nichlas You are dividing miles into light years. Should you not convert one into the other and then divide to get how many times around the Earth it is? So the distance of the circumference of the Earth in light years is 4.23×10^-9. Therefore it is actually 3.2351×10^18 times around the Earth.

  12. Charles

    Whoops, Wayne is right, he means 1 AU in 175 years.

  13. Space is big. You just won’t believe how vastly, hugely, mind- bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space.
    Douglas Adams, The Hitchhiker’s Guide to the Galaxy

    Damn, Larry and Cindy beat me to it. Oh well, funny enough to say it again! :)

    Although, this scale got me to wondering, if 1″=1AU, how big would the Sun be, as well as each of the planets. That could even add in even more perspective of the sizes involved.

  14. Using Phil’s scale, the Earth would wind up being about 2 micrometers big. (Sorry for the reversion to metric, but I don’t think imperial has units to describe that small.) According to the Scale of the Universe 2 animation, that’s about as big as an E Coli cell.

    The Sun, meanwhile, would be almost 233 micrometers – about as big as a paramecium.

    Of course, since 1 mile = 1 light year, the Milky Way would be about 100,000 – 120,000 miles long. This is equivalent to around 4 round-the-world trips.

    So we’re subatomic particles atop a single E Coli cell orbiting a paramecium in a quarter of a “4 Earth trip” which itself is in an even bigger Universe.

    “It’s a great, big Universe and we’re all really puny….”

  15. @3 Scott

    no, at 60 mph it would take 176.5 years to drive to our sun.

    click my name for the calc.

  16. @Larian LeQuella,

    Ha! Just as you asked, I was typing it up. Well, for the Earth and Sun at least. I didn’t work out the other planets, but suffice it to say everything would be “germ sized”.

    In fact, the entire solar system (from my quick lookup) has a diameter of about 7,500,000,000 miles. At the 1″ = 1 AU scale, this means the entire solar system would be 80.8 inches long or under 7 feet long. This is a few feet taller than your average human. So even in our own solar system, our planet is one cell of E Coli swimming inside a freakishly tall person.

  17. Ru

    Shouldn’t we be talking in SI units – I grew up outside the US – I learnt 1 lightyear is approx 10 trillion kilometres. Its pretty easy to scale with those kind of numbers.

  18. Dragonchild

    @6 Cindy,
    @7 Larry
    Don’t take it personally, but Douglas Adams’ quote is a MISERABLE failure at conveying the scale of space. I’d sooner believe he wasn’t saying anything about space at all, as the quote has almost no context about what exactly is “large” about space. HGttG was satire about the human condition using space as a literary vehicle, which implies that quote was more about how most people are so boring, unimaginative, self-centered and trivial that even something as mind-boggling as space can’t be appreciated on a level more complex than a trip to the chemist’s.

    To be blunt, that quote conveys nothing meaningful about the scale of space, most likely by design. Ergo, anyone who thinks it’s the BEST quote on the scale of space is precisely the type of unimaginative person Adams was likely making fun of.

  19. Eric

    The question I have is, this is too close to be coincidental. Is this based on some identity I’m not familiar with? I mean really, these are 4 pretty basic units of measure, and for au/ly = in/mi (to 14 sig figs), that’s pretty incredible..enough to be a little telling of some secret behind it all.

  20. amphiox

    Hey, on this scale, if 1 minute = 1 year, then lightspeed is standard highway speed!

  21. Jonathan Latimer

    Scales are cool — and I love Tardis’ site!

  22. Tony

    Wow @DragonChild, even on a science-based site you managed to be a boring “suck-the-fun-out-of-everything”pedant.

    Imagine what you’ll be like when you grow up!

  23. Steve D

    “these are 4 pretty basic units of measure.”

    Nope: an AU is the distance of one arbitrarily selected planet from its star.
    A year is the time it takes that planet to circle its star, determined by the Gravitational Constant, which IS fundamental, and the mass of the star, which isn’t.
    An inch is arbitrarily defined as 2.54 cm, using a length defined in terms of an arbitrarily selected planet (the surface arc from equator to pole, theoretically, should be 10,000,000 meters, except it isn’t. And the base for that system is defined by the evolutionary accident that we have ten fingers.)
    A mile is an inch times 12 (arbitrary) times 5280 (mind-bogglingly arbitrary).
    So none of these four quantities is cosmically significant at all. It IS just coincidence.

    Want another coincidence? The polar radius of the earth is 6,356,752.3 meters. A meter is 39.37 inches, so the polar radius of the earth is 250,265,838 inches and the axis of the earth through the poles is 500 million inches to within 0.1 per cent.

    Also, the current issue of Sky and Telescope quotes a cheeky analog by Donald Menzel, that if the Sun got its energy from consuming bananas, it would consume about the mass of the earth each day. Coincidence (cue eerie music)… or proof of evolution?

  24. @ #15 (Dragonchild), maybe that’s why it’s so funny! :)

  25. Larry

    Gee, Dragonchild, why would I ever take anything you say personally! Just as I hope you don’t take it personally when I say you are probably someone who thinks digital watches are a pretty neat idea.

  26. Jordan

    Hey Nicholas,
    I re-read your post. Sorry, I missed the point that you were using Phils scaled analogy to driving down the road. My bad.

    Either way, it’s mind boggling. If only we could travel the universe as easily as 550,000 trips around the planet.

    Note: Looks like my first post did not, well, get posted. So the above is out of context. What I did was I simply calculated the actual distance of 13.7 billion LY to miles and divided that by the circumfrence of the earth as I mis-read Nicholas’ post. Nevermind.

  27. hhEb09'1

    @16 Eric: 3 significant figures, not 14

    @1 Nicholas: 550,178 trips around the earth doesn’t even get you out of the solar system, much less the universe. You have to multiply that by the number of miles in a light year, 5.88 trillion according to the BA.

  28. Crudely Wrott

    Describing the sheer scale of the universe is something that words aren’t up to. As several people have noted, Douglas Adams has a memorable description but even his falls short.

    After considering this for many years I have found a way that attempts to convey the overwhelming distances that exist in the spans that we can observe. Mindful that the distances between neighboring objects dwarf anything of earthly scale and noting that most objects are not neighbors, I concluded that the only way to speak of the immensity that surrounds us is this:

    Space is heart breakingly large.

  29. Doug Little

    If we let 1 inch = 92.8 million miles, then 63,360 inches = 5.88 trillion miles.
    See? To two decimal places the scale is exact!

    It’s too much of a coincidence the number works out perfectly therefore god! /snark.

  30. Chris A.

    @Nicolaus (#1):
    To be clear, what you’re saying is: If we shrink things down to the “1 AU = 1 inch” scale, then the edge of the observable (shrunken) universe would be 550,178 times the circumference of the unshrunken Earth.

  31. ethanol

    Huh, it looks like the imperial measurement system is good for something after all.

  32. Matt B.

    “…you’re crossing the entire Earth/Sun distance in less than one-thousandth of second!”

    That should be either “one one-thousandth” or “one thousandth”. Saying “one-thousandth of an inch” is like going to KFC and saying “I’ll have one-bucket of fried chicken”. I’ve been annoyed by that weird hyphenation in Discover Magazine for years. Every once in a while an article will do it right, and I briefly hope they’ve changed the editing policy, but then they inevitably revert. It’s probably a holdover from a previous era, like how they used to hyphenate the adverb “well” onto adjectives, despite the fact that other adverb-adjective pairs aren’t hyphenated. And in that case, it happens to make it look like “well” should be interpreted as a noun. (“Well-put”? Why did you put it in a well??)

    I’m sorry I seem to be commenting more about grammar than the actual topics lately. I think it comes from a general bad mood. I like the math here and I may even have the opportunity to use it at some point, just as I like the coincidence that the speed of sound is about one millionth of the speed of light.

  33. Mike

    Nicholas,

    Oops. You made BIG BADA BOO (remember “the Fifth Element?). If the Universe is 13.7B years old, that means you have to multiply that number by 6Trillion to get the correct number of miles. Let’s see, that comes to about 8.2×10^19 miles. Then you can divide that by 2.5×10^4 and you get just about:
    3.3×10^15 trips around the Earth, not a paltry 550,000 or so.
    And that doesn’t include expansion. Never mind…I’m getting a headache.

  34. Phil,

    Your units of measure are incorrect. An astronomical unit is 1.495978707e+11 meters, which comes out to 92.955807 million miles, not the 92.8 you cited.

  35. Doug Little

    Or to put it another way for the metric minded people. If the average distance from the earth to the sun was the width of an iPhone 4 (9.7mm) a light year would be 613 meters away which is close to the design height (634m) of the Shanghai Tower currently being built in China. Compare the thickness of a smart phone with the height of what will be the second tallest building in the world. Yep space is mind bendingly big.

  36. Infophile

    @1 Nicholas:

    It’s actually even bigger than that. Thanks to the fact that the universe is expanding, things that we see at the limit of visibility have since moved significantly farther away from us. The math is way too complicated to do here, but it works out to being about 46 billion lightyears, about three times larger than simply the age of the universe x the speed of light.

  37. Peter Eldergill

    Is anyone else besides me tired of seeing “miles” and then “km” in brackets? If you stop using miles in your posts (and Astronomy Magazine, etc), people (ie most Americans) will, **believe it or not** , adapt and actually understand what the rest of the world is talking aboot

    (A Canadianism at the end for you, but I am serious)

    Pete

  38. Thorby

    Love coincidences! Just last night I read the Asimov essay “A Question of Scale” in the April 1967 issue of Fantasy and Science Fiction.

    Some highlights:

    Make the sun 1 ft wide and Saturn ends up 1 inch wide and 1000 feet away.
    A light year is 1300 miles and the nearest star is 5500 miles away.
    ——–
    Make the orbit of Pluto 1 inch across. The sun would be the size of a bacterium and a light year 800 inches. Alpha Centauri would be .05 miles away.

    Average movement of a star in our neck of the Galaxy would be 1/20 inch a year.
    ———
    Make the sun the size of a hydrogen atom.

    The nearest star would then be almost a quarter inch away.

    At this scale, the separation between stars would still be over 1,000,000 times that of hydrogen molecules at standard temperature and pressure.

    The galaxy is 225 feet across.
    ——–
    Make the sun the size of a proton.

    The nearest quasar would be 40 feet away.

    The entire universe is a sphere 600 feet in diameter.
    ——-
    Make the Galaxy the size of a hydrogen atom.

    At that point the average separation between galaxies in the universe would then be close to that of hydrogen gas molecules at standard pressure and temperature.
    ——–
    Sorry all the measurements are in inch/feet/miles. But that’s the way he put them in the essay.

  39. bassmanpete

    Just to get in on the act, inches in a mile (63360) times distance of Earth from the Sun in minutes (8.3) divided by minutes in an hour (60) divided by hours in a day (24) comes to 365.2 which is pretty close to a year.

  40. twinner42

    I was going to quote HHGG here, but then realized it was posted three times already. Man, skunked again.

  41. MadScientist

    I would find that confusing; I’ll stick to A.U., ly, and parsecs and leave the magic miles alone.

  42. Nigel Depledge

    Wayne on the Plains (9) said:

    I always point out as we’re walking off the distances that our walking speed is about 10 times the speed of light in the model’s scale, and yet the nearest star in scale would be over 6000 miles away.

    That’s cool!

  43. Nigel Depledge

    TechyDad (14) said:

    Using Phil’s scale, the Earth would wind up being about 2 micrometers big. (Sorry for the reversion to metric, but I don’t think imperial has units to describe that small.)

    That would be 2000 Å in Imperial units.

  44. Nigel Depledge

    Tardis_42 (15) said:

    @3 Scott

    no, at 60 mph it would take 176.5 years to drive to our sun.

    Yes, but Phil said it would take over 175 years, not exactly 175 years.

  45. #44 Nigel:
    The Angstrom is a metric unit – though not an SI one! It’s 1.0e–10 metre.

  46. Nigel Depledge

    Matt B (33) said:

    I’m sorry I seem to be commenting more about grammar than the actual topics lately. I think it comes from a general bad mood. I like the math here and I may even have the opportunity to use it at some point, just as I like the coincidence that the speed of sound is about one millionth of the speed of light.

    Don’t apologise! Pedantry is what the internet is for.

    Right?

  47. Nigel Depledge

    @ Neil Haggath (46) –
    Ooh, you caught me out!

  48. @47 Nigel Depledge: Don’t apologise! Pedantry is what the internet is for.

    Quite so. But I believe you meant to type: “Don’t apologize![double space]Pedantry is the purpose of the internet.”
    ;)

  49. bassmanpete

    Joseph @ 50

    I think Nigel typed exactly what he meant, he’s using the British (ie correct) spelling. :)

  50. Nigel Depledge

    @ Joseph G (50) & bassmanpete (51) –

    Yeah, the -ise spelling is correct over here (but -ize is a permitted variant).

    As to the dangling participle, I could hold up my hand and say “oh, you got me there,” but I shan’t. I intentionally used a dangling participle to emphasise the colloquial roots of the phrase I used. Yup, that’s my story and I’m sticking to it.

    Also, I usually double-space when starting a new sentence, and I thought I had after the exclamation mark. Maybe it didn’t show up very well?

  51. KC

    @Alan – stop nitpicking – Phil stated that it is “about 92.8 million miles”. Its just a good way to visualize the distance – he’s not plotting the trajectory of a spacecraft with it!

  52. @ 51 bassmanpete: I think Nigel typed exactly what he meant, he’s using the British (ie correct) spelling.

    Sigh. It’s so hard to be pedantic when you weren’t brought up with the King’s English. ;)

    @ 52 Nigel Depledge: Also, the double-space after periods is by no means universally held to be correct: there seems to be an ongoing argument between folks who learned to type at least 20 years ago (two spaces) and the typographers and modernists who claim that double-spacing is only acceptable when using monospace fonts. And the dangling participle rule is just plain stuffiness, as far as I’m concerned. Of course, I was just trying to be pedantic for comic effect, so I had to try and find as many nits to pick as possible. :)

  53. enigma3535

    One inch to a light year = 5,879,808,000,000 miles
    A light year = 5,878,625,371,567 miles [assuming that light travels at 186,282.397 mps]
    Difference 1,182,628,433 [0.020%]

    A billion miles here or there.

  54. IT WOULD TAKE 4 YEARS TO GET TO THE MOON.

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