Also available at http://math.ucr.edu/home/baez/week276.html
June 20, 2009
This Week's Finds in Mathematical Physics (Week 276)
John Baez
Math is eternal, but I'll start with some news that may be
time-sensitive. Betelgeuse is shrinking!
1) Stefan Scherer, Shrinking Betelgeuse,
http://backreaction.blogspot.com/2009/06/shrinking-betelgeuse.html
Betelgeuse is that big red star in the shoulder of Orion. It's a red
supergiant, one of the largest stars known. It's only 20 times the
mass of the Sun, but it's about 1000 times as big across - about 5
times the size of the Earth's orbit. For more of a sense of what that
means, watch this video:
2) Hansie0Slim, Largest stars this side of the Milky Way,
http://www.youtube.com/watch?v=u70UBs7BWY8
But, it's shrinking. These authors claim its radius has shrunk 15%
since 1993:
3) C. H. Townes, E. H. Wishnow, D. D. S. Hale and B. Walp, A
systematic change with time in the size of Betelgeuse, The
Astrophysical Journal Letters 697 (2009), L127-L128.
That's about 1000 kilometers per hour!
Of course, it's a bit tricky to estimate the size of Betelgeuse -
besides being rather far, it's so diffuse that its surface isn't very
precisely defined. And it's a variable star, so maybe a little
shrinkage isn't a big deal. But the two known cycles governing its
oscillations have periods of one year and 6 years. So, the authors of
the above paper think this longer-term shrinkage has some other cause.
It could be just another cycle, with a longer period. But there's
another possibility that's a lot more exciting. Maybe Betelgeuse is
about to collapse and go supernova!
Indeed this seems likely in the long term, since that's the usual fate
of such massive stars. And the long term may not even be so long,
since Betelgeuse is about 8.5 million years old - quite old for stars
this big, which live fast and go out in a blaze of glory.
What if Betelgeuse went supernova? How would it affect the US
economy, and the next Presidential election? Could this be the
Republican party's best hope?
Sorry, I'm being a bit parochial... let me try that again. How would
it affect the insignificant inhabitants of a puny speck called Earth,
located about 500 or 600 light years away from Betelgeuse? According
to Brad Schaefer at Louisiana State University, it would be "brighter
than a million full moons", but it wouldn't hurt us - in part because of
the distance, and in part because we're not lined up with its pole.
(Perhaps just to build up the suspense, Schaefer added that Betelgeuse
could already have gone supernova, in which case we're just waiting
for its light to reach us.)
Actually, I have trouble believing that Betelgeuse gone supernova
would be brighter than a million full moons. First of all,
the full moon is 1/449,000 times as bright as the Sun. So,
"brighter than a million full moons" is just an obscure
way of saying "more than twice as bright as the Sun."
Second, let's try the calculation ourselves. There are various kinds
of supernovae, with different luminosities. I guess Betelgeuse is
most likely to become a type II supernova. Such supernovae show quite
a bit of variation in their behavior. But anyway, it seems they get
to be 1 billion times as bright as the Sun, or maybe at most - let's
look at a worst-case scenario - 10 billion times as bright. So,
between 10^9 and ten times that.
On the other hand, Betelgeuse is about 600 light years away, and there
are 63,239 astronomical units in a light year, so it's about
600 x 63,000 ~ 4 x 10^7
times as far away as the Sun - no point trying to be too precise
here. Brightness scales as one over distance squared, so
supernova Betelgeuse should look between
10^9 / (4 x 10^7)^2 ~ 7 x 10^{-7}
as bright as the Sun, and ten times that bright.
As I mentioned, the full Moon is about 2 x 10^{-6} times as bright as
the Sun. So, supernova Betelgeuse should be roughly between 1/3 as
bright as the full Moon, and 3 times as bright. This is a rough
calculation, but I've done it a few different ways and gotten similar
answers. So, it's safe to say that "brighter than a million full
moons" is a vast exaggeration.
Whew.
It's worth recalling that not too long ago, a supernova exploded at a
roughly comparable distance from us, forming the "Local Bubble" - a
peanut-shaped region of hot thin gas about 300 light years across,
containing our Sun. The gas in the Local Bubble is about 1000 times
less dense than ordinary interstellar space, and vastly hotter.
What do I mean by "not too long ago"? Well, nobody is sure, but back
in "week144" I reported a bunch of evidence for a theory that the
Local Bubble was formed just 340,000 years ago, when a star called
Geminga went supernova, perhaps 180 light years away.
Now I'm getting a sense that the situation is more complex. It seems
our Sun is near the boundary of the Local Bubble and another one,
called the "Loop I Bubble". This other bubble seems to have formed
earlier - perhaps 2 million years ago, at the Pliocene-Pleistocene
transition, when a bunch of ultraviolet-sensitive marine creatures
mysteriously died:
4) NASA, Near-earth supernovas,
http://science.nasa.gov/headlines/y2003/06jan_bubble.htm
The Loop I Bubble may have been caused by a supernova in "Sco-Cen",
a cloud in the directions of Scorpius and Centaurus. It's about
450 light years away now, but it used to be considerably closer.
In the last few million years, some wisps of interstellar gas have
drifted into the Local Bubble. Our solar system is immersed in one of
these filaments, charmingly dubbed the "local fluff". It's much cooler
than the hot gas of the Local Bubble: 7000 Kelvin instead of roughly
1 million. It's also much denser - about 0.1 atoms per cubic centimeter
instead of 0.05 or so. But Sco-Cen is sending interstellar cloudlets
in our direction that are denser still, by a factor of 100. These might
actually have some effect on the Sun's magnetic field when they reach us.
I'm sure we'll get a clearer story as time goes by. In 2003, NASA launched
a satellite called the Cosmic Hot Interstellar Plasma Spectrometer, or
CHIPS for short, to study this sort of thing:
5) NASA/UC Berkeley, Overview of CHIPS Science,
http://chips.ssl.berkeley.edu/science.html
It sounds pretty interesting. Unfortunately the latest news on the
CHIPS homepage dates back to 2005, before they'd done much science.
What's up?
You can't do much about Betelgeuse. But you can do something about
mathematics! For example, if you're into categories or n-categories,
you can help out the nLab:
6) nLab, http://ncatlab.org/nlab
The nLab is like the library, or laboratory, in the back room of the
n-Category Cafe. The nCafe is a place to chat: it's a blog. The nLab
is a place to work: it's a wiki. It's been operating since November 2008.
2008. There's quite a lot there by now, but it's really just getting
started.
Check it out! You'll find explanations of many concepts, which you
may be able to improve, and the beginnings of some big projects, which
you may want to join.
So far the main contributors include Urs Schreiber, Mike Shulman, Toby
Bartels, Tim Porter, Ronnie Brown, Todd Trimble, David Roberts, Andrew
Stacey, Bruce Bartlett, Zoran Skoda, Eric Forgy and myself. Jim Dolan
recently joined in with a page on algebraic geometry for category
theorists - I'll say more about this someday. And like the nCafe,
technical aspects of the nLab are largely run by Jacques Distler - it
uses some wiki software called Instiki which he is helping develop.
Finally, a bit of actual math. Here's a paper by the fellow I'm
working with here in Paris, and a grad student of his:
7) Paul-Andre Mellies and Nicolas Tabareau, Free models of T-algebraic
theories computed as Kan extensions, available at
http://hal.archives-ouvertes.fr/hal-00339331/fr/
I really need to understand this for my work with Mike Stay.
In "week200" I talked about Lawvere's work on algebraic theories; I'll
assume you read that and pick up from there. In its narrowest sense,
an "algebraic theory" is a category with finite products where every
object is a product of copies of some fixed object c. We use
algebraic theories to describe various types of mathematical gadgets:
to be precise, any type of gadget that consists of a set with a bunch
of n-ary operations satisfying a bunch of purely equational laws.
For any type of gadget like this, there's an algebraic theory C; I
explained how you get this back in "week200". If we have a functor
F: C -> Set
that preserves finite products, then F(c) becomes a specific gadget of the
given type. Conversely, any specific gadget of the given type determines
a functor like this.
So, we define a "model" of the theory C to be a functor
F: C -> Set
that preserves finite products. But actually, this is just a model of
C in the world of sets! We could replace Set by any other category
with finite products, say X, and define a "model of the theory C in
the environment X" to be a functor
F: C -> X
that preserves finite products.
For example, if C is the theory of groups and X is Set, a model of C
in X is a group. If instead X is the category of topological spaces,
a model of C in X is a topological group. And so on. In general
people call a model of this particular theory C in any old X a
"group object in X".
But as you might fear, we want to understand more than a single model
of C in X. As category theorists, we want to understand the whole
*category* of models of C in X. This category, which I'll call
Mod(C,X), has:
functors F: C -> X that preserve finite products as its objects;
natural transformations between these as its morphisms.
For example, if C is the theory of groups and X is the category of
topological spaces, Mod(C,X) is the category of topological groups
and continuous homomorphisms.
So far I've just been reviewing at a fast pace. What happens next?
Well, there's always a forgetful functor
R: Mod(C,X) -> X
sending any model to its underlying object in X. But what we'd really
like is for R to have a left adjoint
L: X -> Mod(C,X)
sending any object of X to the free gadget on that object. Then we
could follow L by R to get a functor
RL: X -> X
called a "monad". One reason this would be great is that monads are
another popular way to study algebraic gadgets. I explained monads
very generally back in "week89", and said how to get them from adjoint
functors in "week92"; in "week257" I gave some links to some great
videos by the Catsters explaining monads and what they're good for.
So, I won't say more about monads now: I'll just assume you love them.
Given this, you must be dying to know when the functor R has a left
adjoint.
In fact it does whenever X has colimits that distribute over the
finite products! For example, it does when X = Set. And Mellies and
Tabareau give a very nice modern explanation of this fact before
generalizing the heck out of it.
The key is to note that
R: Mod(C,X) -> X
is just an extreme case of forgetting *some* of the structure on an
algebraic gadget: namely, forgetting *all* of it. More generally,
suppose we have any map of algebraic theories
Q: B -> C
that is, a finite-product-preserving functor that sends the special
object b in B to the special object c in C. Then composition with
Q gives a functor
Q*: Mod(C,X) -> Mod(B,X)
For example, if B is the theory of groups and C is the theory of rings,
C is "bigger", so we get an inclusion
Q: B -> C
and then Q* is the functor that takes a ring object in X and forgets
some of its structure, leaving us a group object in X. But when B is
is the most boring algebraic theory in the world, the "theory
of a bare object", then Q* forgets everything: it's our forgetful functor
R: Mod(C,X) -> Mod(B,X) = X
So, we should ask quite generally when any functor like
Q*: Mod(C,X) -> Mod(B,X)
has a left adjoint. And, the answer is: it always does!
The proof uses a left Kan extension followed by what Mellies and
Tabaraeu call a "miracle" - see page 5 of their paper. And, it's
this miracle they want to understand and generalize.
Here's the basic idea. If we write Hom(C,X) for the category with
arbitrary functors F: C -> X as its objects;
natural transformations between these as its morphisms
then composition with Q gives a functor
Hom(C,X) -> Hom(B,X)
which has a left adjoint
Hom(B,X) -> Hom(C,X)
thanks to a well-known trick called "Kan extension", or more precisely
"left Kan extension". Since Mod(B,X) is included in Hom(B,X), we can
compose this inclusion with the functor above:
Mod(B,X) -> Hom(B,X) -> Hom(C,X)
And now comes the miracle: this composite functor actually lands us
in Mod(C,X), which sits inside Hom(C,X). This gives us a functor
Mod(B,X) -> Mod(C,X)
which turns out to be just what we want: the left adjoint of
Q*: Mod(C,X) -> Mod(B,X)
Kan extensions are a very general concept, so the hard part is
understanding and generalizing this miracle.
To do this Mellies and Tabareau first generalize algebraic theories to
"T-algebraic theories" where T is any pseudomonad on Cat. I already
said that monads are a trick for for studying very general algebraic
gadgets. Similarly, pseudomonads are a trick for studying very
general *categorified* algebraic gadgets, like "categories with finite
products" or "monoidal categories" or "braided monoidal categories" or
"symmetric monoidal categories".
Each of these types of categories allows us to define a type of
"theory":
monoidal categories let us define "PROs"
braided monoidal categories let us define "PROBs"
symmetric monoidal categories let us define "PROPs"
categories with finite products let us define "algebraic theories"
I explained all these, along with monads, here:
8) John Baez, Universal algebra and diagrammatic reasoning, available
as http://math.ucr.edu/home/baez/universal/
Take my word for it: they're great. So, we would like to generalize
Lawvere's original results to these other kinds of theories, which are
all examples of "T-algebraic theories". But, it's not automatic! For
example, it doesn't always work with PROPs.
A typical kind of algebraic gadget we could define with a PROP is a
"bialgebra". While there's always a free group on a set, there's not
usually a free bialgebra on a vector space! The problem is not the
category of vector spaces: it's that bialgebras have not only
"operations" like multiplication, but also "co-operations" like
comultiplication.
So, Mellies and Tabareau have their work cut out for them. But they
tackle it very elegantly, using profunctors and a certain
generalization thereof: Richard Wood's concept of "proarrow
equipment". This lets them generalize the "miracle" to any situation
where we have a little T-algebraic theory sitting inside a bigger one
Q: B -> C
and the bigger one only has extra operations, not co-operations.
"Proarrow equipment" sounds pretty scary - it's taken me about a
decade to overcome my fear of it. So I'll stop my summary here, right
around page 12 of the paper - right when the fun is just getting
started!
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Quote of the Week:
The question you raise, "how can such a formulation lead to
computations?" doesn't bother me in the least! Throughout my whole
life as a mathematician, the possibility of making explicit, elegant
computations has always come out by itself, as a byproduct of a
thorough conceptual understanding of what was going on. Thus I never
bothered about whether what would come out would be suitable for this
or that, but just tried to understand - and it always turned out that
understanding was all that mattered. - Grothendieck
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Addenda: Charles McElwain kindly responded to my plea for information
about what would happen if Betelgeuse went supernova. He wrote:
As you mention, there's not a lot of (quality) work out there. Most of
what I found briefly would score fairly high on the "crank index".
Of course, near supernovas have a positive and essential role in life on
earth, in there *being* an earth, rather than just a star...
A few that I found that weren't immediately eliminated as cranks, that
might repay further investigation:
9) Michael Richmond, Will a nearby supernova endanger life on Earth?,
Available at http://stupendous.rit.edu/richmond/answers/snrisks.txt
Perhaps the closest to the number-crunching you're looking for.
10) S. E Thorsett, Terrestrial implications of cosmological gamma-ray
burst models, Astrophys. J. 444 (1995), L53. Also available as
arXiv:astro-ph/9501019
Specifically, nitric oxide increases/ozone decreases.
11) Steven I. Dutch, Life (briefly) near a supernova, Journal
of Geoscience Education, 2005. Available at
http://nagt.org/files/nagt/jge/abstracts/Dutch_v53n1.pdf
The conceit here is what would happen if the Sun went supernova;
acknowledged as impossible, but a very interesting exercise almost
as a "Fermi problem", spinning out the real implications of the
classic Arthur C. Clarke story "Rescue Party", and interesting also
pedagogically.
Using this and other information, I checked that if Betelguese went
supernova, it would not be anywhere nearly as bright as "a million
full moon". You can see one of my calculations above.
Andrew Platzer looked into what happened to CHIPS, the satellite that
was supposed to study hot gas in the Local Bubble. And, he found a
fascinating newspaper article about this satellite's quixotic, sad,
but ultimately rather mysterious quest. Andrew wrote:
I am interested in space and I did a little bit of googling about
the CHIP satellite. Turns out it was turned off about a year ago after
a 5 year mission. Unfortunately, it never detected the UV signal of
the local bubble according to the article. There's a full story in
local California newspaper:
12) Chris Thompson, Goodbye Mr. CHIPS, East Bay Express, July 2, 2008.
Also available at http://www.eastbayexpress.com/ebx/PrintFriendly?oid=780923
And a couple of papers in the arxiv by M. Hurwitz referencing CHIPS;
the more recent one is about spectra of comets:
13) M. Hurwitz, T. P. Sasseen and M. M. Sirk, Observations of
diffuse EUV emission with the Cosmic Hot Interstellar Plasma
Spectrometer (CHIPS), Astrophys. J. 623 (2005), 911-916. Also
available as arXiv:astro-ph/0411121.
14) T. P. Sasseen, M. Hurwitz et al, A search for EUV emission from
comets with the Cosmic Hot Interstellar Plasma Spectrometer (CHIPS),
Astrophys. J. 650 (2006), 461-469. Also available as
arXiv:astro-ph/0606466.
The null result seems interesting since a signal was expected.
Still up there. TLE from NORAD:
CHIPSAT
1 27643U 03002B 09177.47579469 .00000388 00000-0 34685-4 0 1094
2 27643 94.0213 313.0310 0014359 84.1512 276.1301 14.97271142352353
Andrew Platzer
"TLE" refers to the "two-line element" format for transmitting
satellite locations.
The short version of the CHIPS story - which completely leaves out all
the fascinating twists and turns you'll find in that newspaper article
above - is that this satellite failed to detect the extreme
ultraviolet radiation (EUV) that people expected from the hot gas of
the Local Bubble. It doesn't seem like a malfunction. So, something
we don't understand is going on!
Todd Trimble gave a snappy proof that the forgetful functor from
bialgebras to vector spaces has no left adjoint. If it did, it would
need to preserve limits. In particular it would send the
terminal bialgebra to the terminal vector space. But the terminal
bialgebra is 1-dimensional, while the terminal vector space is
0-dimensional - a contradiction.
For more discussion visit the n-Category Cafe:
http://golem.ph.utexas.edu/category/2009/06/this_weeks_finds_in_mathematic_37.html
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If you just want the latest issue, go to
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