# From here to infinity… logarithmically

By Phil Plait | January 11, 2010 12:00 pm

Logarithms are cool. Sure, some of you may have flashbacks to middle school and may collapse on the floor twitching upon their mere mention, but seriously, logs are the language of the Universe. Our senses (eyesight and hearing) are sensitive logarithmically, and a lot of ways the world behaves make a lot more sense when you plot them using logs.

For those of you scratching your heads, a simple way to think of logs is to think factors of ten. Instead of counting like we normally do — 1, 2, 3, 4 and so on — in log space you count by factors of 10: 1, 10, 100, 1000, and so on. There are lots of advantages to this, one being that you can make graphs that show things that are very small and very big on the same plot. Using regular numbers, it would be hard to make a graph showing the size of a human (2 meters tall) and a skyscraper (200 meters tall) on the same plot, but using logarithms, they are only two ticks apart in size. Easy peasy.

And if you take this idea to the extreme, what do you get? Why, you could get a plot of the whole freaking Universe, from the surface of the Earth out to the fires of the Big Bang itself!

But who would do such a thing? Astronomers at Princeton, that’s who.

[Click to exponentiate.]

That picture is just a small piece of a much larger graphic showing, well, everything. At the bottom is the Earth and at the top is the most distant thing we can see: the cosmic microwave background, the cooling fireball from the Big Bang. Included are planets, asteroids, stars, galaxies, and pretty much everything you can think of, all plotted out for your perusal. The vertical axis represents distance, and the horizontal is cleverly done in Right Ascension, sorta like longitude (East/West) on the sky. That way they get the whole sky — the whole Universe — on one graph.

I know xkcd did something like this, but I’d love to see this done up as a vastly scrollable webpage with actual images instead of dots, and the objects actually described (rollovers, popups, links, whatever). If done correctly, that would cause a wave of nerdgasms across the web, and not-so-incidentally be an awesome learning tool. Any takers?

Tip o’ the order of magnitude to Stuart at @astronomyblog.

CATEGORIZED UNDER: Astronomy, Cool stuff, Pretty pictures
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### Comments (50)

#### Links to this Post

1. Mapa da Situação « | January 12, 2010
2. Numbers « To No Particular End | January 15, 2010
3. The whole universe in one logarithm | Koppernigk | August 22, 2010
1. Very nice. I’ve got a poster of the xkcd “height” cartoon pinned up in my cubicle at work.

2. IVAN3MAN AT LARGE

Phil Plait:

… but I’d love to see this done up as a vastly scrollable webpage with actual images instead of dots, and the objects actually described (rollovers, popups, links, whatever).

This is not bad: An Atlas of The Universe.

3. I’m soooo glad they have 3c273 in there! It’s one thing you can see with a modest telescope that’s really, really far away! 2.44 billion light years, baby! W00T!
Now, it doesn’t really look like much, just a point of light, but what a story that point of light has to tell!

Phil:
I showed this to Ed & his son last summer at a boy scout camporee at Montpelier.

Explanatory stuff is at this URL:
http://www.astro.princeton.edu/universe/

4. Curious Bystander

Cool post! As a layman, I’m curious what the “z=1” stuff is at the top left and what the boundary between “decelerating” and “accelerating” means, and also the “unreachable” zone and the “comoving future visibility limit”. I assume it has something to do with the expansion of the universe…

5. Here’s the xkcd comic in poster form: click

6. Something else I think would be interesting, in the solar system segment, how the orbits of Earth, Moon, the planets and asteroids change things over time. For example, most of the artificial satellites would be moving left to right but “sinusoidally” (?), but geosynchronous satelllites stay put, the Moon and the Sun would move horizontally at constant velocity (which direction?) and the rest of the planets would have a much more complex motion?

7. Torbjörn Larsson, OM

It’s also interesting if, as in electronics, transportation technology would move exponentially. Then the Moon is a tough goal, the Oort cloud the toughest, and anything after that “easy peasy”.

8. oh my word, that “missing WMD’s” entry on the xkcd version is freakin hilarious.

9. TechSkeptic (#8): When I first saw that I thought, “WMD? Is that one of the dark-matter candidates?” 😀

10. PicoJoules

That’s just wonderful! I saw the xkcd comic a while ago (also awesome), but I didn’t know there was an actual log graph of the universe! I just love you crazy astronomers.

11. The xkcd comic makes more sense to me.

12. timur

Why Venus is further than the Sun?

13. This would also be fun to see graphed in a 2D log-log plot, perhaps with our galactic center at 0,0. It would give a more intuitive sense of direction for these objects in addition to distance, at least.

14. @timur: Well sometimes the orbit of Venus means it IS further than the sun (on the opposite side from us). Same for Mercury and Mars. But I’m not sure if this was graphed for a particular date and time or what. Might have made more sense to plot all objects for their shortest linear distance, though that could be challenging for anything beyond our solar system.

15. Stonegiant

Why do they have Comet Halley listed as Halley’s Comet? If they are going to make a professional effort, couldn’t they get the nomenclature correct?

16. Chris

I would have liked the planets to have little circles since the distance to the earth does vary with time.

17. Buzz Parsec

If you click “next” a few times on xkcd, you get to an inverse cartoon that shows smallness, looking into a computer. Deep down inside a proton in a silicon nucleus is a “fabric of space-time” joke… Brian Greene knitting furiously! (There are lots of other jokes, so I hope this isn’t too much of a spoiler. It seemed the most on-topic for this blog.)

18. Michelle R

Yea… I’m a bit bugged by the planet distance. Is it by assuming that all the planets and asteroids are lined up in a straight line or something?

19. Kee

The 10^11 Earth Radiuses distance they’re giving to how far the Earth radio signals reached, suggests they were first emitted sometime in 1942! Actually slightly later since the radio signal line is slightly before 10^11. I used 6365 km as the earth radius.

20. RickJ

Curious,

Z is just way to state the redshift and thus distance to an object. It is the observed wavelength emitted by an element in the object less its lab measured wavelength divided by its lab measured wave length. So if we see an element at 800 nanometers wavelength when its wavelength when at rest is 400 nanometers it would have a redshift of 1. Such a shift moves the bottom of the UV range to the top of the IR range. Z is the way astronomers refer to the distance of objects. Converting to parsecs involves assumptions about the expansion rate of the universe which seem to constantly change. Z is a fixed calculation that doesn’t change with every new measurement of Hubble’s constant or other factors needed to convert z into parsecs

Dark energy is the name we give to whatever is causing today’s universe to expand at an ever increasing rate. In the young universe with a much higher density, gravity was able to slow the rate of expansion but when the density reached a critical point this thing we call dark energy overcame gravity and started speeding up the universe’s expansion. The line on the graph marks this point in time where the switch to an increasing rate of expansion occurred. We are looking back in time as well as distance on the chart so objects seen above that line are seen back at a time when the universe’s expansion was slowing down rather than speeding up.

Thanks to the expansion of the universe when we look at an object of say 1 billion light years we usually are speaking of light travel time. That is the light took 1 billion years to reach us. In fact the object was really closer than that when the light was emitted. During those 1 billion year the universe was expanding so the light had to travel further than that to reach us. When it did reach us, thanks to the expansion of the universe the object is actually even further away than the light travel time would indicate. So a 1 billion light year galaxy as measured in light travel time was closer than that a billion years ago and further than that today. As you move further away this difference grows. I’ve seen several different calculations but his one says that at a red shift of 1.76 the light of an object that is emitted “today” (really this has no meaning in relativity but I’ll use it anyway) will never reach us. The objects real distance is now such that before the light reaches us we’ll be moving away from it faster than the speed of light thanks to the universe’s expansion and thus it will never catch up with us. So while we see it today, over the next few billions of years its light will dim and redden but never quite vanish yet will never see the object’s light that is emitted now or in the future. Thanks to the ever increasing expansion rate this line is moving toward us until little of the universe, maybe none depending on which interpretation you believe, will be seen.

Your last one is a bit more complicated. First we have co-moving distance. Remember the three distances to that 1 billion light-year galaxy by light travel time. It’s distance today is longer thanks to the universe’s expansion. One way to measure this is called its comoving distance. It assumes we are moving at the same rate as the galaxy and thus can stretch a tape measure between them. Now using this type of coordinate system the most distant object that can ever be seen, assuming we are now infinitely far into the future would be the comoving future visibility limit. We can never see anything further than that no matter how long we wait or how old the universe.

I think this chart was prepared using comoving coordinates and distances.

My brain hurts.

BA can likely better explain it and fix my errors.

I wish they’d have given the exact time used to make this chart. They likely did with the paper that went with it but without it on the chart it makes the solar system a bit confusing.

Rick

21. Mark

Wow. Just wow. thank you Princeton folks, and BA.

22. Maybe you should talk with the folks at the American Museum of Natural History, who’ve made an awesome movie using similar data: http://www.youtube.com/watch?v=17jymDn0W6U&feature=player_embedded

23. Curious Bystander

Thanks for the informative response, RickJ. Now I know a little more about how the universe works.

It looks like they used August 12, 2003 for the chart (found on page 20 of the pdf of the paper http://www.astro.princeton.edu/universe/ms.pdf )

24. Google Universe would be cool, with images overlaid, zoomable sections and so on.

25. Total Nerd

I should note that I recently went to the Princeton Observatory, and saw the original chart on their wall. They also have a rug (about a meter wide and ten meters long) depicting a similar graphic. It’s really cool.

26. If you’re interested in the exact opposite direction, here is an interactive site that attempts to convey the small, and the very very small – right down to the atomic scale.

http://learn.genetics.utah.edu/content/begin/cells/scale/

27. Doing a quick check of the right assentions of various objects in the map, it is applicable to some time around August 11,2003. So at that time, Venus was on the far side of the Sun and Mars was closer to us than the Sun and was near the Moon in the sky.

It would be somewhat interesting to see an updatable solar system section to the map, since this particular map is only accurate for the above mentioned date…at least as far as solar system objects are concerned.

28. Larry

Douglas Adams had it right: 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

29. There was an interesting Radiolab a month or so ago on the evolution of numbers. Some research on very young children and recently-contacted tribes in South America is showing that the human brain may be hardwired for logarithmic numbers rather than ordinal ones. It’s only after repeated exposure to Sesame Street that our brains switch to integers.

30. DaveH

Woah. Big.

(My considered response)

31. RPJ

Nice, this is the kind of thing that always captures my interest…

It looks like it would be amenable to a Flash conversion for those tooltips. I believe I could make a crude but functional version fairly easily, in fact, though I’ll leave it up to someone with better astronomical knowledge, since I sadly don’t know much about any of this.

32. Here’s a link to the Radiolab podcast mentioned by toasterhead (I actually haven’t listened to it myself…though I’m a few podcasts away from it).

33. Crazy Tom

Dood. Just, just dood.

34. Michael Kingsford Gray

So, people still use the old 1970s Calcomp pen plotters, eh?

35. Zippy the Pinhead

The relative distances for Venus (2 AU) and Mars (0.4 AU) seem a bit askew. Is that supposed to be an average distance or a specific distance on the day the chart was made?

36. hjb

On a related note, today’s featured picture on Wikipedia (http://en.wikipedia.org/wiki/File:Star-sizes.jpg) give a nice, but different, illustration of the orders of magnitude present in astronomy.

37. The huge empty layer from Heliopause to the Oort cloud is really annoying…

38. Jimmy

I found something similar to Phil’s request for a giant scrollable. Images but no details, and everything thrown way out of perspective, but cool looking anyways.

http://infinity-imagined.blogspot.com/

39. mocular

I found this a couple of years ago, printed it out, taped the pieces together and have it hanging by my desk. Looks cheesy, but just a glance can return my perspective to that of the universe. Quite a nice meditative experience.

40. Zippy, on August 11, 2003, Venus was on the far side of the Sun and Mars was near opposition, making Mars much closer to Earth than Venus.

41. Quiet Desperation

Is there a way to get a vector version? There’s a color plotter down the hall from me at work that uses a big spool of paper. I could totally plot this in lovely high resolution for my office wall.

42. Liam

Dunno why, but I find it really strange to suddenly appreciate that the edge of the universe is ‘only’ about a million times further from us than the edge of our little galaxy. I know the numbers are totally right and that really is the case, but it just seems to make the rest of the universe seem suddenly smaller in my perception as 10^6 is such a ‘small’ manageable number in astronimical terms! If you’d asked me yesterday, without running the numbers I’d have intuitively reckoned it must be billions and trillions of times further to the edge of the universe than to the edge of our tiny insignificant galaxy. Hmmmmmmmm. Maybe it’s just me…

43. Dave Empey

Quiet Desperation, there’s a PostScript version at

http://www.astro.princeton.edu/universe/

Would that work?

44. I’m glad Epsilon Eridani was marked with both a blue circle AND a small 4-pointed black-outline star.

It’s important to draw attention to the star around which Babylon 5 orbits.

45. Magnum

“Click to exponentiate” hey? That will just reverse the logarithm process.

46. EFisher

If I can get this to plot using arcGIS software it wouldn’t be that hard to sit down and put reference and proper distance on the entire universe, what a wonderful creation!
Thanks for eradicating my weekends for the next couple years 😐

47. Jack Nye

Can someone explain to me why there are two gaps in the SDSS galaxies/quasars? I can understand the gap at 18h, caused by the galactic center, but why is there also a gap at 6h?

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