# The Astrophysics of Bedtime Stories

SteelyKid is a big fan of the classic children’s book Goodnight Moon, which, if you haven’t spent the last sixty-odd years in a cave, you probably know features a bunny saying goodnight to a variety of objects in a great, green room. The attentive toddler will find a lot to look at in the pictures– there’s a mouse in every one that SteelyKid delights in pointing out– but an inquiring adult might well ask “Just how long does it take this bunny to say goodnight to all this stuff, anyway?”

Well, we can answer this question with SCIENCE! You see, there are six pictures in the book showing the moon through one of the room’s windows, and as the book goes along, the moon moves higher in the window. This provides a way to estimate the passage of time in the book. The full sequence looks like this:

(The first and fifth pictures do not show the top of the window. I’ve cropped them to the same size as the full-window illustrations; the remainder of those frames is filled with part of our dining-room table. )

So, how much time has elapsed over the course of the book? Well, in the first picture showing the left-hand window, we can just barely see the moon in the lower left:

This is actually the second color illustration, so it’s a reasonable approximation to guess that the moon was just at the edge of the window frame at the start of the story. In the final illustration, the moon has moved up and to the right by a substantial amount:

So, how can we use this to measure the passage of time? Well, we know that the full moon in the sky covers an angle of about half a degree of arc. On our big copy of the book, the diameter of the moon in the final illustration is just about 7/8″ (I only have an English-unit tape measure here), while the distance from the corner of the window to the outer edge of the moon is 2 and 5/8″, exactly three times the diameter. So the moon has moved through about 1.5 degrees in the course of the story.

Now, the Earth rotates through 360 degrees in just about 24 hours, which is 15 degrees per hour (the Moon’s motion is slightly slower, owing to its orbital motion, but it’s not a significant difference for our purposes). This suggests that the bunny’s goodnight ritual takes about 0.1 hour, or six minutes. Coincidentally, this is approximately the time required to read the book to SteelyKid at bedtime, as she points out all the important features of every picture (“Mouse right there! Mouse is sneaky!”).

Of course, there’s another way to estimate the passage of time in the book, which is the clocks shown in the various pictures. The clock in the first picture shows almost exactly 7:00, while the clock in the final picture shows approximately 8:10, for an hour and ten minute duration. Coincidentally, this is approximately the time it takes to get SteelyKid to go to sleep after reading Goodnight Moon

These two methods clearly do not agree with one another, which means one of two things: either I’m terribly over-analyzing the content of the illustrations of a beloved children’s book, or the bunny’s bedroom is moving at extremely high velocity relative to the earth, so that relativistic time dilation makes the six-minute rise of the moon appear to take an hour and ten minutes. Calculating the necessary velocity is left as an exercise for the interested reader.

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1. #1 Kate Nepveu
October 12, 2010

“Sneaky” is a new one on me, I have to say.

2. #2 Kate Nepveu
October 12, 2010

3. #3 tim eisele
October 12, 2010

Well, obviously the scenes are being drawn from photographs made with a short telephoto, which had enough magnification to increase the apparent angular diameter of the Moon by about a factor of 10. So it really is moving through about 15 degrees of arc. No problem!

And this must be true, because when I look through our window and see the Moon, it only fills a tiny fraction of the window, and not about a tenth of it like we see here.

4. #4 Chad Orzel
October 12, 2010

when I look through our window and see the Moon, it only fills a tiny fraction of the window, and not about a tenth of it like we see here.

Ah, but they’re bunnies, you see. Their windows are much smaller than yours…

5. #5 Kate Nepveu
October 12, 2010

You realize that this way leads attempting to determine the scale of the room based on the size of the mouse (a.k.a., “madness”) right?

6. #6 tim eisele
October 12, 2010

(Let’s see, I think the way to “madness” must be a right turn at the next intersection)

Hm. I think that, if the proportions of the house are the same as a human’s house, then the angular dimensions don’t change for a point of view from inside, so shrinking the house doesn’t help.

If the artist is spying on the house from some distance with a pair of 10X binoculars, though, I think it works. If you leave aside the fact that she shouldn’t be able to see through the wall . . .

7. #7 Eric Lund
October 12, 2010

[I]f you haven’t spent the last sixty-odd years in a cave, you probably know [Goodnight Moon] features a bunny saying goodnight to a variety of objects in a great, green room.

I never had this book as a kid, and don’t have kids of my own. So until I read this post, I didn’t know that the protagonist is a bunny. It wasn’t until my early 20s that I learned of this book (via an officemate with a then infant daughter). The idea of saying goodnight to various objects is obvious given the concept, but the species of the protagonist is not.

FWIW, I agree with Tim that the elapsed time is on the order of an hour. That’s about how long it takes the sun to move roughly that distance as viewed from the windows in my house, and he correctly points out that the bunnies’ house should scale proportionately. I’m not entirely sure that it actually does, though, since that would make the kittens awfully tiny if that fireplace is bunny-sized.

8. #8 Anonymous Coward
October 12, 2010

I love the calculation.

I disagree with Tim and Eric. You can’t use the window size to calibrate time, since you don’t know the dimensions of the bunny house or the distance of the artist from the window. Similarly, the use of a telephoto lens wouldn’t change Chad’s calculation, as it would magnify the angular size of the moon and its angular change equally.

Clearly what’s going on is that the artist is changing locations in the room from picture to picture so as to keep the moon visible in the window. Perhaps he or she likes drawing the moon. (I’m going to ignore the possibility that he or she has exaggerated the size of the moon for artistic effect; that would be irresponsible!)

I don’t buy the time dilation, since the velocity you’d need to get that would be sufficiently large that it would make the transit of the moon across the window even faster. Maybe a gravitational well?

9. #9 Clay B
October 12, 2010

I’m just impressed that she got the moon to move in a reasonable manner; a lot of TV/movies/books have the moon in the wrong place for its phase.

10. #10 Eric
October 12, 2010

The angular extent of the window (vertically) appears to be about 4 times that of the moon, so if we stipulate that the moon is half a degree, that means the window subtends about 2 degrees of arc, or pi/90 radians. If we’re viewing it from a distance d, its height is then d*tan(pi/90) = d*pi/90 in the small angle approximation. If the room is 10 feet across, that makes the window about 4 inches high, and the room rather oddly shaped (even for a mouse room).

11. #11 Daniel
October 12, 2010

You don’t need time dilation. You just need the house to be rotating (or at worst moving along the surface of the Earth to change the apparent time of night).

12. #12 Nick
October 13, 2010

Jeez, haven’t any of you people ever heard of a clock that runs fast? You really need to use Occam’s Razor more liberally while you attempt to make an astrophysical analysis of a beloved children’s picture book…

13. #13 Brook
October 14, 2010

The disparity can easily be explained – every parent can tell you that when your children are little the days are long but the years are short. Thus the actual time, indicated by the moon, is short while the perceived time, indicated by the clock, is long.

I hate to say this, but if you’re spending this much time thinking about Goodnight Moon you’re going to be in big trouble when you get to Philip Pullman or Jeannette Winters books.

14. #14 onymous
October 14, 2010

I think we need to consider the possibility that the story takes place on another planet, or an alternate-reality Earth with a different moon than ours. (Evidence for the latter: on our Earth, bunnies don’t live in houses and say good night.)

15. #15 Violet in Twilight
October 14, 2010

This is my son’s favourite book too (He turns two this Dec). He really likes the “quiet old lady” though. Always saying “hush”.

I don’t think the room is bunny-sized because the ‘quiet old lady’ is atleast human sized (relative to kittens and mouse).

It’s amazing SteelyKid already learnt “Sneaky” (I mean the concept!).

16. #16 Lab Lemming
October 18, 2010

“Goodnight Moon” is a very old children’s book, and was written before the moon had retreated to its current distance.

17. #17 Liz
October 19, 2010

I have another theory about the clock and the moon’s timing not matching up. The Little Old Lady whispering “Hush” got frustrated and got up and moved the clock when the little bunny wasn’t looking and then pointed to the clock and said “See how late it is? It is PAST your bedtime mister.” Case closed.

18. #18 BEK
October 19, 2010

It’s the “Runaway Bunny” being calm and compliant. His mother found him and put him to bed!

19. #19 HJHolmes
October 19, 2010

I would like to add a further refinement to the time dilation hypothesis.
The time for the bunny is consistent with the moon,however the clock is sharing the reality bubble of the parent who has a)a cup of hot chocolate or b)a nice grown-up drink waiting for them and the relative passage of time is enormously stretched out as they have been up since 5am and working/chasing after the bunny all day.

20. #20 Teague
October 19, 2010

Any small child can tell you about the relativistic affect of bedtime vs. other times.

21. #21 Adrian Morgan
October 19, 2010

if you haven’t spent the last sixty-odd years in a cave, you probably know

This annoys me. Other bloggers do it, too, forgetting that their blogs have an international audience. Just because something is somewhat well known in North America doesn’t mean the rest of us have heard of it.

22. #22 JohnnieCanuck
October 20, 2010

Like Clay B, I often notice that artistic representations of the moon get it wrong. Here the full moon is shown rising. The window is therefore looking to the south east (Northern Hemisphere centric) and it is early evening. Whether this was consciously done or coincidence, I couldn’t say.

For a given time of day, e.g. sunset, the moon might appear anywhere along its path during a month, but there can be only one correct phase for it to show at a particular point on its path. The closer to the West and the sun, the thinner the crescent must be. High in the sky must be a half moon while a just risen moon will be full and in the East, though rarely rising due East at high Latitudes.

In other words, for a given time (sun position) and moon position only one phase is valid.

And then there’s those ubiquitous post cards with the wide angle photograph of a city nightscape and a full moon of enormous proportions superimposed on the sky.

23. #23 vagueofgodalming
October 20, 2010

Adrian, those parts of the world not in the USA are culturally and intellectually equivalent to a cave. Everyone knows that.

24. #24 kim
October 20, 2010

Maybe it’s a giant bunny! Radioactive carrots! Maybe the great, green room is actually a green screen! C’mon folks, just go with it. Childhood is supposed to be full of tiny bits of magic, like parents who are thrilled to read us the same story over and over again. Save the brain cells for when Steely starts asking about the birds and the bees!

25. #25 Keith Wilcox
October 20, 2010

I love it when people find little things like this and just analyze them to death. It just goes to show that we can get pleasure from books on more than just a children’s level. We’re all sitting here benefiting from a humorous science less. Goodnight Moon is even more awesome now than before! Thanks.

26. #26 Kat
October 20, 2010

I love ALL OF YOU.
I really do.

27. #27 Kat
October 20, 2010

I love ALL OF YOU.
I really do.

28. #28 Seymour Simon
October 21, 2010

Really love the way science is used to enhance the wonderful Goodnight Moon book for those of us who remember reading it countless times to our kids. I just posted on my FB group my appreciation: http://www.facebook.com/group.php?gid=57675721076

29. #29 Joseph Hertzlinger
October 23, 2010

Maybe this took place the night before Daylight Savings Time was about to start and the clock was set an hour ahead during the course of the book.

30. #30 Adam Greenwood
October 25, 2010

I do NOT love all of you. Eww.

31. #31 Adam Greenwood
October 25, 2010

I do NOT love all of you. Eww.

32. #32 Rod Johnson
November 20, 2010

Just stumbled on this post, and have to point out the thing that has always nagged at me about these illustrations: there’s no horizon. Just space, as far as you can see. This poor little bunny’s house is not on a planet at all–it’s actually adrift in SPACE. (Which, tangentially, allows you to explain away the time inconsistency as an artifact of the house/spacecraft’s rotation.)

33. #33 Patrik
November 29, 2010

Just like to point out that the room doesn’t have to be moving near lightspeed, to get time dilation effects, but just slightly slower than the moon’s orbital speed, so that the moon completes that much of its rotation in an hour and ten minutes instead of 6 minutes.

34. #34 Sonicsuns
April 9, 2011

According to tvtropes ( http://tvtropes.org/pmwiki/pmwiki.php/Main/FanWank ), the velocity required to effect the observed time dilation is 0.9963 c, or roughly 668 million miles per hour. (I didn’t do the calculations myself; I’m just reporting what someone else wrote)

35. #35 azureMagic
May 4, 2011

Oh god this is hilarious, all of you.

36. #36 Dorian
November 13, 2011

Perhaps momma bunny at some point had to set the clock forward in anticipation to adjust for daylight savings? I can’t remember if the clock is in all illustrations, but… Absolutely awesome, I am going to share this with my daughter and maybe throw in a little of how time works in the General Theory of relativity :) I am discovering this blog a bit late in life, eh?

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