BAFact math: How bright is the Sun from Pluto?

By Phil Plait | March 15, 2012 11:07 am

[On January 4, 2012, I started a new features: BAFacts, where I write an astronomy/space fact that is short enough to be tweeted. A lot of them reference older posts, but some of the facts need a little mathematical explanation. When that happens I’ll write a post like this one that does the math so you can see the numbers for yourself. Why? Because MATH!]


Today’s BAFact:

From Pluto, the Sun is fainter than it is from Earth, but still can be 450x brighter than the full Moon.

I remember reading a science fiction story many years ago which took place on Pluto. The author described the Sun as being so faint that it looked like just another bright star (too bad I don’t remember the name of the story anymore). I was thinking about that again recently, and wondered just how bright the Sun does look from Pluto. This turns out to be pretty easy to calculate!

First, you need to understand how an object like the Sun — really, any source of light — dims with distance. The Sun emits light in all directions, so as you get farther away from the Sun, that light gets spread out. Imagine a sphere perfectly encasing the Sun right at its surface. Each square centimeter has a certain amount of light passing through it. If I double the size of the sphere, there’s a lot more surface area to that sphere, but the total amount of light passing through it hasn’t changed. Therefore the amount of light passing through each square centimeter has dropped. Since I doubled the sphere’s diameter, I can figure out how much its dropped, too!

The formula for the surface area of a sphere is

Surface area = 4 × π × radius 2

If I double the size of the sphere, everything on the right side of the equation stays the same except for the radius, which is now twice as big. Therefore the area increases by 22 = 4. So the light passing through each square centimeter of the bigger sphere drops by a factor of four. Someone standing on that sphere would see the Sun being 1/4 as bright as if they were on the surface.

If I make the sphere ten times bigger, the area goes up by 10 × 10 = 100 times, and the brightness drops by 100. You get the picture.

So now we’re ready to figure out how bright the Sun is from Pluto!

The Earth orbits the Sun, on average, at a distance of about 150 million km. Pluto has a very elliptical orbit, but has an average distance of about 5.9 billion kilometers, or roughly 39 times the Earth’s distance from the Sun. Using the method above, the Sun must be 392 = about 1500 times fainter, or more grammatically correctly, 0.00065 times as bright. That’s pretty faint!

Or is it? Well, let’s compare that to how bright the full Moon looks from Earth. To us here at home, the Sun is about 400,000 times brighter than the full Moon, so even from distant, frigid Pluto, on average the Sun would look more than 250 times brighter than the full Moon does from Earth!

Pluto’s orbit is also highly elliptical, stretching from 4.4 billion km to just over 7.3 billion km from the Sun. Doing the math again, that means the Sun goes from being 0.0012 to 0.0004 as bright as it is from Earth: a range of roughly 150 to 450 times as bright as the Moon from Earth. That’s a change in brightness by a factor of three!

Still, given that you can read by the light of the full Moon, obviously the Sun from Pluto is still pretty dang intense. It would hardly look like just any other star: it would greatly outshine everything else in the sky. Painful to look at, most likely. So the short story I read was wrong, but at least we learned something. That’s a decent trade.

And let me leave you with a question: From Pluto, how big would the Sun look? Ah, that’s a BAFact for another day. Tomorrow, actually!

Image credit: Dan Durda, showing Pluto, its moon Charon, and the Sun.

CATEGORIZED UNDER: Astronomy, BAFacts, Cool stuff, Top Post

Comments (63)

  1. I can’t wait til New Horizons gets to Pluto.

  2. Jeffrey

    *mind equals blown*

  3. I’m seriously confused.

    “the Sun would look more than 250 times brighter than the full Moon does from Earth!”

    A difference of one magnitude is 2.512 in brightness. If the Sun (magnitude -27) looks 250 times brighter than the the Full Moon (magnitude -13), then that means the Sun should look like magnitude -20 from Pluto (i.e. – if a difference from magnitude 1 to 6 is 100, a difference from 1 to 7 would be 250, so (-13) + (-7) = -20. Do I have that about right?

    That seems awfully bright, and not what I recall understanding the brightness of the Sun would be from that distance. But I am most definitely not a math guy, so I’m just making sure my ‘work’ is correct there.

    But I do have a quibble to make:

    “a range of roughly 150 to 450 times as bright as the Moon from Earth. That’s a change in brightness by a factor of three!”

    I’m confused. A difference in magnitude brightness is 2.512. So 450 / 2.512 = 179.14. So that’s actually a difference in magnitude brightness of a factor of approximately ONE, not three, right?

    Is that correct? If not, argue with me, but I don’t see how it is a “factor of three.” Maybe “three times” the difference or something, but your wording seems off to me. Again, argue with me – I’m just trying to understand what you mean and how I understand magnitude differences.

  4. mike burkhart

    I have read in numerious Astronomy books that from Pluto, the sun would apper as a bright star in the sky ,looks like those books are incorect. One thing is that Pluto is always dark so it’s allways night . If I were to move there it might take some getting use to ,munch as I like the night sky I also like the daytime.

  5. Grzegorz Blinowski

    I am almost sure that the story was one of Niven&Pournelle’s from the Know Space series…

  6. Jeremy Hardinger

    I remember reading that in a story too. I THINK it was in A Wrinkle in Time.

  7. Jason Dick

    Well, I just did some little calculations in my head, and since the Sun is approximately half a degree across when seen from the Earth, at 39 times that distance it will be a little bit less than an arcminute in size from the Moon. I believe that is roughly the resolution limit of the human eye. So yeah, it would look like a bright point in the sky. A very bright point.

  8. Simon

    Well assuming the Sun to be 1.391m km in diameter and Pluto a distance of 5906m km from the Sun (average), then the Sun would appear to be about 49 arcseconds, compared to 1920 arcseconds as seen from Earth. i.e 39x smaller.

    For a relative size this is just the ratio of the distance between Sun & Earth and sun & Pluto – 5906m / 150m

    Oh, and yes, human visual acuity is said to be about 60 arcseconds so it would indeed be a pinpoint

  9. Luis Dias

    A very bright and probably painful pinpoint.

  10. Chris

    Last time I remember reading something like that was in The Forever War by Joe Haldeman when they were training on Charon; I think he just said something along the lines of “a particularly bright star.” I’ll have to check to be sure.

  11. Dave F- I am doing this in units of actual brightness, not magnitudes. I didn’t want to get into that here as it adds unnecessary steps and is a confusing units.

  12. lepton

    Earth to Sun, 1 AU, Sun on Earth, half of a degree, or 30min.
    As you said, Pluto to Sun, average 39 AU, Sun on Pluto, 30min/39=0.77min.

    IIRC, we need to move more than one ly away to make Sun not the brightest star.

  13. Roger

    I seem to remember that in Haldeman’s The Forever War. I may be wrong though.

  14. Nat

    Pretty sure the story you’re thinking about is Larry Niven’s “Wait It Out,” about an astronaut that gets stranded on Pluto.

  15. “Dave F- I am doing this in units of actual brightness, not magnitudes. I didn’t want to get into that here as it adds unnecessary steps and is a confusing units.”

    Fair enough, but was I right in saying the Sun would look about magnitude -20 from Pluto?

  16. Chris

    @Dave F

    The difference in magnitude = log(how many times brighter)/log(2.512)
    log(150)/log(2.512) = 5.4
    log(450)/log(2.512) = 6.6

    The magnitude of the full moon is -13, so the magnitude of the sun at Pluto’s distance would be between -18.4 and -19.6, so your estimate is pretty good.

    @7 Phil – You don’t always have to dumb it down for us. We can handle logarithms :-)

  17. MartyM

    I’m listening to a Science Friday podcast from March 9 while reading your post. They started talking about the New Horizon Mission and how the sun light is dim at Pluto. Alan Stern is talking about how the sun is 1000X dimmer on Pluto than here on earth, but that the cameras on New Horizon are much better at photographing low light level (of course due to better CCD sensors and processing). He said that light from the sun on Pluto looks like twilight here. Interesting coinky-dink that this podcast is repeating your post at the same time.

  18. OtherRob

    I don’t know if it’s the novel you’re thinking about or not, but a good bit of Heinlein’s Have Space Suit, Will Travel takes place on Pluto.

  19. Rory Kent

    The only thing I remember reading with a statement like that was also The Forever War.

  20. Dave Jerrard

    This post is very well timed, since I happen to be recreating the solar system in 3D, and I was just dealing with this very topic, though for the inner planets at the moment. I could just let the geometry solve itself (it IS in a 3D application after all), but I’m also doing scenes of each planet separately, where I don’t need to have anything millions of kilometers away. In those cases, I can set up a simpler light solution that deals with brightness and angles. While I can figure this stuff out on my with my limited math skillz (no, I can’t handle logarithms ;) , these kinds of posts are very welcome, and for some reason, this kind of info is fairly hard to find elsewhere.

  21. jjonahjansen

    Maybe it’s nitpicking, but…

    Phil, you used 5.9 billion ‘miles’ when you clearly meant kilometres.

    jjj

  22. So, suppose you have in position or orbiting around Pluto some big mirror (behind the planet from the Sun) or a big lens (in front of it from the Sun) and you want to deliver Earth-intensity light (and heat) to a city-sized area on the planet (say 20 mile radius). One may vary the distance from the surface. How big would the optic have to be to deliver?

    The Moon is a Harsh Mistress, but Venus is Judge Judy…

  23. Slowly but Surly

    Re: The Forever War

    Training takes place on the fictional *planet* Charon, not the moon.

  24. Dutch Railroader

    The ambient lighting on the surface of Pluto would be about that of mid-twilight. A good way to visualize this is to use a light meter a SLR camera, which you can calibrate with full sun shining on a white piece of paper at high-noon, and then calculating how it would read on Pluto. You then wait after the sun goes down to see how bright it looks when the meter registers where you calculate it to be.

  25. Jeff

    still, the sun would basically have the major features of a star from Plutonic surface: cold, relatively dim, and against a black background. In fact, even the average human , if they grew up on Pluto, would clearly see that the sun was a star, so alike to the other stars; whereas our real average non thinking humans do not see the connection, they think of the sun as something distinct from a star.

  26. Wzrd1

    Ah, but even more interesting is, how bright is the sun when viewed from Mercury? Not to mention how much of the sky is filled by the sun… ;)

  27. A back-of-the-napkin calculation says the sun would occupy 0.81 arc-minutes in the Pluto sky… about 1/39 the diameter it appears in our sky, not surprisingly. Roughly the size of Jupiter in our sky when Jupiter is at its apparent largest.

  28. Arneb

    BA, could it have been Icehenge, by Kim Stanley Morrison (of Red/Green/Blues Mars fame)?

  29. Usng the inverse square law to determine brightness ratio is rather iffy if source comparision is a disk vs. a point. If the source were an infinite line, brightness would only vary as inverse distance, not inverse distance squared. Brightnss of an infinite plane would not vary at all with distance. The sun viewed from Earth is the size of a full moon overhead (sparing your retinas). We therefore expect meaningful decimal trim vs. two point sources being compared.

    Perhaps a better comparison is the equilibrium blackbody temperature balancing solar input in each case. On could add greenhouse gases, but then the Carbon Tax on Everything would decrement your research funding.

  30. Blargh

    What Uncle Al said. This calculation just isn’t correct. :-/

    Seen at the distance of Pluto, the sun subtends such a small (solid) angle that I’d say you could approximate it as a point source just fine (so for the relative comparison of brightness at aphelion and at perihelion, the inverse square law would work fine).

    As you get closer – like at the Earth’s distance – this no longer holds true. It subtends such a large angle that it just can’t be seen as a point source anymore, and thus the inverse square law goes right out the window. And with it the relative brightness as seen from the Earth compared to Pluto.

  31. Anthony

    The sun normally subtends about 30 minutes of arc. At Pluto, it subtends about 1, which is roughly the same as the human visual acuity. Thus, it would basically look like a point — it’s similar in apparent size to Jupiter.

    However, it would be almost as damaging to the retina as looking at the sun from Earth, just in a much smaller area — it’s something like 1/1,000 the energy, but it’s also concentrated into 1/1,000 the area on your retina, so it hits the affected area just as much. In fact, looking at the sun has basically fixed effect on the retina anywhere inside of 30-40 AU. Brightness per unit area is constant at any range where you can resolve a disk on an object.

  32. Chris A.

    @Uncle Al (#31) and Blargh (#32):

    Re: Treating the sun as a point source

    To quote Wikipedia:
    “When the illuminant is not a point source, the inverse square rule is often still a useful approximation; when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1%.”

    Since Phil is comparing the Sun’s apparent brightness on Pluto to its apparent brightness on Earth (where the diameter to distance ratio is about 1/100–considerably less than 1/5), it’s a reasonable approximation. Heck, the inverse square law is still more than 99% accurate at Mercury, where the ratio is 1/45!

    Did you really expect him to do the flux integral for an extended source shining on a unit area on this blog?

  33. molybdenumfist

    Maths… yay! I’d read a similar thing and never thought how easy it would be to actually check.

    btw jjj is correct 39AU = 5.9 billion km

  34. Zax

    I doubt it is the book you were thinking of but I happened to be listening to First Lensman(1950) by E. E. “Doc” Smith this morning and it says that Pluto is 40 times more distant and receives 1/1600th the amount of light and heat from the sun. The sun is described as a dim, wan speck.

  35. I thought it might be Heinlein’s Have Space Suit – Will Travel too but as it’s online as a pdf I checked it: here’s what Heinlein says.

    The sun was in front of me – I didn’t realize what it was at first; it looked no bigger than Venus or Jupiter does from Earth (although much brighter). With no disc you could be sure of, it looked like an electric arc.”

    Seems Heinlein got it about right. Interesting that he mentions Jupiter and Venus which are both very bright in the evening skies at the moment, about as bright as they get.

  36. Peter Ellis

    What Anthony @33 said.

    The incident energy received varies as the inverse square of the distance. The area subtended by the Sun’s disk also varies as the inverse square of the distance. Ergo, the Sun looks exactly as bright from Pluto as from Earth, just a lot smaller. It only starts to dim once you get past the point at which you can treat it as a point source: i.e. past the limit of human visual acuity. Coincidentally, this is about at Pluto’s orbit.

  37. amphiox

    I think the point source vs resolvable disc also has impact on the accuracy of the description “like a bright star”. If the sun looks like a point source, then regardless of the actual brightness, it would be reasonable to describe it as looking “like a (very!) bright star”. (Like a nearby supernova, as it were….)

  38. Matthew

    Possibly World’s Fair 1992, by Robert Silverberg. Page 142.
    “And therefore what kind of planet can we expect [Pluto] to be?”
    “A miserably cold one. The Sun is so far away that it’ll seem like just another star.”
    Great juvenile story, but very dated.

  39. Lawrence D’Oliveiro

    @Dave Jerrard As someone who has tried to model a small part of the Solar System in an interactive 3D app (see http://ldo17.tumblr.com/post/15499310578/my-attempt-to-recreate-a-certain-well-known-movie), let me just say you are going to find it an absolute pain dealing with relative scales of things: the planets are so relatively small, and the distances between them are so relatively huge, that you are going to continually lose track of where things are. You are quite likely to also hit floating-point rounding errors if your software is only using single-precision floats for its calculations.

  40. Dragonchild

    For purely visualization purposes, I guess one way to liken how the Sun looks from Pluto is like the high beams on a car a mile away. At that distance it’s basically a point source and it sure doesn’t make your surroundings bright, but it’s easily the most intense light source you’re looking at and very uncomfortable to look directly at.

  41. jick

    Actually, the inverse square law is pretty accurate for the sun. Since the sun is (for our purpose) a symmetric ball of light that emits light in every direction equally, the amount of sunlight we receive at any point outside the sun will be exactly the same as if the same amount of light was pouring out from an (incredibly bright) point-source located at the center of the sun.

    A classic case of the Gauss’s law.

  42. Tony

    I’d like to defend the “looks like just another star” statement, if I may?

    Evolving Squid (comment 29) said that the size of the sun as seen from Pluto would be roughly the same size as Jupiter is viewed from Earth. Meaning, to the naked eye, it would just be a pin-prick of light in the sky alongside a whole host of other pin-pricks of light.

    I personally picture it to be like the star of bethlehem, like shown in all them christianity themed christmas cards – a very bright star, bright enought to light up the landscape and hurt to look at, but still just a pinprick in the sky.

  43. Troy

    Most astronomy programs do this but I especially like stellarium (free and awesome)… It allows you to observe from anywhere in the solar system. Currently the sun is about 1′ in apparent size from Pluto which someone mentioned. I suppose that is a bit of cheating the proper way is to find the size of the sun and the distance from Pluto etc.

  44. Great article here BA – thanks! :-)

    Surprising how bright our Sun would be even from Pluto isn’t it!

    Pretty sure I’ve read the description of our Sun seen from Pluto as “..just a bright star” in a number of places both fiction and non-fiction.

    BTW. For those who haven’t seen this already, there’s another good question with its answer provided via astronomer Ken Croswell linked to my name here :

    How many of the classical planets – those easily visible to the unaided eye & known to the ancients – would still be visible without optical aid (brighter than apparent magnitude 6.0) – if they were as far as Pluto’s mean distance from the Sun? (39.5 astronomical units, or 3.7 billion miles.)

    NB. This is paraphrased in the blockquote above.

    @16. Chris :

    The magnitude of the full moon is -13, so the magnitude of the sun at Pluto’s distance would be between -18.4 and -19.6, so your estimate is pretty good.

    Cheers for that informative comment. :-)

    In fact the magnitude of our Sun is actually be the same for Pluto and for Earth and everywhere else – minus twenty seven (rounded up to the nearest full magnitude*) – assuming we’re talking aboslute magnitude NOT apparent magnitude! Which we’re clearly not, but a-n-y-w-a-y! ;-)

    * – 26.75 more specifically. Source : James Kaler ‘The Hundred Greatest Stars’, Copernicus Books, 2009.

    PS. Hmm .. is that bolometric ie. allwavelengths included or just visual light I’m wondering now?

  45. Jeff Kirk

    Robert Heinlein seemed to have it just about right in “Have Spacesuit, Will Travel”. The hero Kip reports his view of the Sun from Pluto thusly:

    “The Sun was in front of me—I didn’t realize what it was at first; it looked no bigger than Venus or Jupiter does from Earth (although much brighter). With no disc you could be sure of, it looked like an electric arc.”

    Good on ya, RAH!

  46. Treczoks

    It looks like the sun is so far away that it would only be a pinpoint light source for pluto. Now with this size, how often will there be a total solar eclipse on pluto because just another asteroid passed by in the belt? I don’t think that the belt will have a big impact on the average brightness, but a hypothetical sentient native plutonian might notice that something switches off the light on a quite irregular base.
    As it would be a difficult task to track all objects in the belt that can lead to an eclipse on pluto, a future astronaut might unexpectedly be left in the dark from one moment to the next.

  47. Nigel Depledge

    Dave Jerrard (21) said:

    This post is very well timed, since I happen to be recreating the solar system in 3D . . .

    Wait a second.

    Isn’t the solar system already in 3d?

    you can’t fool us, you know! ;-)

  48. JustAnEngineer

    Treczoks– I would expect that solar eclipses caused by asteroids wouldn’t last very long.

  49. Jeffersonian

    Okay, I don’t get it.
    I want to conceptualize in more practical terms.
    So it’s always dark if you were on the planet (beyotch!) Pluto.
    But that’s because the sun, though very bright, is just a pinprick and there’s no atmospheric diffusion, right?

    I mean, all things being equal, it would be daylight, just the same, right? Like just before dawn, here; except, if you were actually there it would be more like being on the moon? So it would feel dark and you wouldn’t be able to see things 10 meters away but there might be a frown-inducing pinprick light source if you look up?

    Practicate me.

  50. Bad Wolf

    The quote *IIRC* is from Larry Niven “Lucifer’s Hammer”

  51. kupfernick

    Not surprising RAH got it right. The dude was a physicist. That’s why all his stuff is so good.

  52. Livia West

    How much light is necessary to make the sky glow, and would it, if the Earth were in Pluto’s orbit?

  53. Doug

    Heinlein was an engineer. But he knew enough physics to be dangerous and was known to consult with physicists. So he usually got it right.

    I dunno Phil. The name of this feature is very close to BARFact. That’s how I read it at first glance . . .

  54. Jasonium

    At closest approach, would Saturn appear larger than the Sun on Pluto? Would Neptune appear larger?

  55. Anthony

    @Livia West: arbitrarily little; skyglow would be 1/900 to 1/2,300 as much as on Earth, but that’s still visible (you can see skyglow from the full moon). The question is whether you could actually tell that it was blue, I’m guessing you could (you could probably tell it was blue under the same light level that you could tell an object painted blue was blue, and sunlight on Pluto would be 10-100 lux depending on distance and time of day, which is comparable to dim internal lighting and is adequate for poor color perception).

  56. Dave Jerrard

    @41. Lawrence D’Oliveiro

    Several years ago, it was more of an issue, where even a airplane 2km above the origin would start to quantize, but the software has become more accurate since. With planets, at least out to Mars in my case, quantization errors are negligible at the planetary scales, though small objects like asteroids will be obviously butchered.

    I used an old technique where I move the world – in this case, the solar system – so the camera and smaller objects are always near the origin, but even with this scene, I’m still running into precision limits. I guess the next option is to separate all the planets from the solar system parent, and move them all separately. This will get rid of the huge offsets they’re currently using, but will probably also mean another day of tweaking the camera motion to avoid the huge overshooting problem due to moving across such vast distances.

    Other than the smaller objects quantizing, the scene is actually working far better than I had expected.

    @49. Nigel Depledge

    You don’t know me too well, do you? ;)

    @56. Jasonium

    Saturn would be slightly larger than 1/9th the size of the sun as seen from Pluto. I just checked this in Starry Night, going back to the year 747AD, which is the most recent year the two planets were that close.

    In April, 2956BC, Pluto and Neptune were about as close as they’ll ever get, and Neptune would appear about a little larger than 1/3 the size of the sun, which would look a bit smaller than it did in 747 AD.

    If you want to have some fun & kill a lot of time, grab a copy of Celestia – it’s free – and you can see how the solar system looks from pretty much any place & any time.

  57. Melf_Himself

    Phil, you’re going to need to extend your calculations a little here. Brightness scales with log luminance. e.g. twice as many photons does not give twice the brightess.

  58. Matt B.

    @48 Treczoks:

    Even though the Sun is being describe as looking like a point here, for a solar eclipse to occur, the shadowing object must appear at least as big as the Sun from the location of the world where the eclipse is seen. Since the Earth is 1/100 the size of the Sun, an Earth-sized object would have to be no more than 1/100 as far from Pluto as the Sun to cause a lunar eclipse on Pluto, which would be < .39 AU from Pluto. Since asteroids are tiny compared to Earth, they would have to be even closer to Pluto. So asteroids in the main belt absolutely would not be able to eclipse the Sun from Pluto's perspective.

  59. Thornton

    All this is not surprising… even the closest stars are so, so, so, so much further out than Pluto. Since Alpa Centari is about 10,000 times further than our sun is from Pluto… Alpha Centari is slightly brighter than our sun, but from Pluto, (or Earth) Alpha Centari is about 100 billion times more dim than our sun is from Pluto…
    Our sun would produce about 7 lumens per square foot on the surface of Pluto, which favors most Police or Sure Fire flash lights shining on the ground from 5 feet high…. Looking at the sun from Pluto would be akin to looking at an orange-yellow Arc welder.. from 30 or 40 feet…. it would leave a spot in your eyes when you closed you eye lids… Like the filiment of a bright HID light…. DON”T believe the old saying that from Pluto the sun only looks like a bright star ! WRONG. Pluto is very, very, very, very close to our sun compared to the nearest star (excluding our sun, of course, which is similar to Alpha Centauri).

  60. Thornton

    Correction…. 100 million… from Pluto, Alpha Centauri is about 100 million times dimmer than our sun is from Pluto…not 100 billion. Still, our sun is 100 million times brighter. It would hurt to look directly at the sun from Pluto…

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