Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

June 20, 2009

This Week’s Finds in Mathematical Physics (Week 276)

Posted by John Baez

In week276 of This Week’s Finds, hear the shocking news about this star:

Read about the Local Bubble, the Loop I Bubble, the cloudlets from Sco-Gen, and the “local fluff”. Come visit the nnLab! And learn how Paul-André Mélliès and Nicolas Tabareau have taken some classic results of Lawvere on algebraic theories and generalized them to other kinds of theories, like PROPs.

This picture, prepared by Linda Huff and Priscilla Frische, gives a more detailed view of our Solar’s system’s vicinity in the Milky Way:





It’s complicated! I don’t understand most of it.

Posted at June 20, 2009 8:18 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2000

35 Comments & 0 Trackbacks

Re: This Week’s Finds in Mathematical Physics (Week 276)

John says that:

While there’s always a free group on a set, there’s not usually a free bialgebra on a vector space!

I always had the impression that the free bialgebra on a vector space does always exist! After all, the free monoid on a vector space certainly exists, and that has a canonical bialgebra structure, so there’s an obvious functor from a category of vector spaces to its category of internal bialgebras and bialgebra homomorphisms.

Is the idea that there’s no adjoint to the forgetful functor from the category of internal bialgebras to the underlying category of vector spaces? I suppose the thing that would go wrong is that the canonical map from the free monoid on VV equipped with the canonical comultiplication, to an arbitrary bialgebra on VV, fails to be a bialgebra homomorphism in general. Difficult to prove this without a good way to characterize bialgebras, though.

Posted by: Jamie Vicary on June 20, 2009 10:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Jamie surmised

Is the idea that there’s no adjoint to the forgetful functor from the category of internal bialgebras to the underlying category of vector spaces?

Yes, exactly. One cheap way to see this is that if the forgetful functor had a left adjoint, then it itself would be a right adjoint, and therefore preserve limits, in particular the terminal object. But the terminal bialgebra (working in vector spaces over a field kk) is kk itself; since the forgetful functor takes this to the non-terminal vector space kk, it can’t be a right adjoint.

Posted by: Todd Trimble on June 21, 2009 12:03 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Jamie wrote:

I always had the impression that the free bialgebra on a vector space does always exist! After all, the free monoid on a vector space certainly exists, …

That’s free as an algebra, which already hints that it’s not free as a bialgebra, since a bialgebra has extra structure not determined by the algebra structure.

… and that has a canonical bialgebra structure, so there’s an obvious functor from a category of vector spaces to its category of internal bialgebras and bialgebra homomorphisms.

Yup, but being ‘obvious’ doth not make the functor ‘free’.

Is the idea that there’s no adjoint to the forgetful functor from the category of internal bialgebras to the underlying category of vector spaces?

More precisely, no left adjoint — being a left adjoint is what makes a functor count as ‘free’.

Check out the material starting around page 3 of the paper by Mélliès and Tabareau — it’s pretty darn readable.

Posted by: John Baez on June 21, 2009 12:09 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Thanks for those comments, Todd and John. From Todd’s insight, it follows that for any algebraic structure containing at least a unit and a counit, and where morphisms require the unit and counit to be preserved, there can be no free functor unless the tensor unit in the underlying category is the zero object. An important class of monoidal categories where this is impossible are the nontrivial rigid monoidal categories. This is good to know!

John said that being a left adjoint is what makes a functor count as ‘free’. Surely this is too strong a statement — doesn’t it only apply when our structure is ‘monoid-like’? For example, the free functor to the category of comonoids in Vect k{}_k is right-adjoint to the forgetful functor, not left-adjoint. Using this logic, when our structure is both monoid- and comonoid-like, I suppose we would expect the free and forgetful functors to be ambidextrous adjoints.

Posted by: Jamie Vicary on June 21, 2009 7:23 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Jamie, that’s right. I really should have added that when trying to decide whether a functor has a left adjoint, a first thing to ask oneself is: does this functor preserve limits? For category theorists, this becomes an almost automatic reflex, and a good place to start is with preservation of products, including the empty product.

For example, the free functor to the category of comonoids in Vect k is right-adjoint to the forgetful functor, not left-adjoint.

Except that I wouldn’t call it “free”; I’d call it “cofree”.

There is of course a comonoid structure on the free monoid, and I think Jim Stasheff once said here on this blog that under certain restrictions on the comonoids, this is in fact co-universal among such comonoids, but it’s not cofree without those restrictions. The actual cofree comonoid can be seen however as a localization of the free monoid. (I’m being a little vague here, I know, but I could go into this more if desired.)

Posted by: Todd Trimble on June 21, 2009 8:45 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Mostly I just wanted to warn people that what people often refer to as the cofree coalgebra cogenerated by X, namely the direct sum of the tensor powers of X, is in fact cofree only under some restrictions,
cf. W Michaelis for the full skinny on this.

Posted by: jim stasheff on June 22, 2009 1:24 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Not only that there is a right adjoint to the forgetful functor from bialgebras to algebras, but there is also a left adjoint to the forgetful functor from Hopf algebras (bialgebra with antipode; setup over a commutative ring) to coalgebras due Takuechi

M. Takeuchi, Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), No.4, pp. 561–582.

I started an entry on this construction in [[nlab:free Hopf algebra]]

This construction has provided a first example of a Hopf algebra with non-invertible antipode. A slight adaptation of the Takeuchi method is the construction of Hopf envelope of a bialgebra, by Manin. The comparison of the two can be found in section 13.2 of my article

Z. Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003.

The fact that cofree functor from vector spaces to coalgebra and free functor from coalgebras to bialgebras or Hopf algebras to do not compose to something either free or cofree is making issues related to study of deformation complexes for Hopf algebras difficult (lack of some canonical constructions); an expert on this is Boris Shoikhet related also to unpublished work of M. Kontsevich and M. Markl reported in the lecture by MK at Cats3 (Nice 2003). I recall Miša Batanin commenting then (in 2003) that he noticed likely connections of the lecture content to some ideas of Drinfel’d, tricategories and 3d TQFT.

Posted by: Zoran Skoda on June 24, 2009 5:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Todd wrote:

There is of course a comonoid structure on the free monoid, and I think Jim Stasheff once said here on this blog that under certain restrictions on the comonoids, this is in fact co-universal among such comonoids, but it’s not cofree without those restrictions.

That’s interesting. Paul-André Melliès mentioned a theorem along these lines just yesterday. A while back Jim Stasheff wrote:

For more (than you want to know), there is:

MR2035107 (2005b:16070) Michaelis, Walter Coassociative coalgebras. Handbook of algebra, Vol. 3, 587–788, North-Holland, Amsterdam, 2003. (Reviewer: E. J. Taft) 16W30 (00A20)

where the issue of cofree coassoc coalgs is described definitively around p. 720.

The author promises me the pdf file, so as soon as I have it, will ask for it to be posted.

Zoran wrote:

The fact that cofree functor from vector spaces to coalgebra and free functor from coalgebras to bialgebras or Hopf algebras to do not compose to something either free or cofree is making issues related to study of deformation complexes for Hopf algebras difficult…

That’s interesting too! I’m facing vaguely similar difficulties one step up the categorical ladder in my work with Paul-André and Mike Stay on ‘the free cartesian closed category on a category’. The problem with bialgebras is that they involve both operations and ‘co-operations’; the problem with cartesian closed categories is that they involve both covariant and contravariant functors.

Posted by: John Baez on June 26, 2009 10:07 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

I’m new here: this is my very first post :). I’m sorry to say I’m not nearly as familiar with this blog as I’d like to be..

In case anybody’s still interested, and for whatever amusement value it may have, it turns out that there is a cofree Hopf algebra on any bialgebra (or algebra) as well.

In other words, the forgetful functor from Hopf algebras to bialgebras has both a left adjoint (Manin’s Hopf envelope) and, perhaps more interestingly in a sense, since it doesn’t seem to come up much at all, a right adjoint.

You can even take it one step further: the inclusion of Hopf algebras with bijective antipode into the category Hopf algebras (over some fixed field) has both a left and a right adjoint. Hopf algebras with bijective antipode are just Hopf algebras with a skew antipode, so you can interpret the inclusion as forgetting the skew antipode.

I guess this shouldn’t be too surprising: the antipode is an antimorphism for the bialgebra structure, so it’s only natural that it should count as both “algebraic” and “coalgebraic” structure.

Posted by: Alexandru Chirvasitu on December 8, 2009 9:23 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Jamie wrote:

From Todd’s insight, it follows that for any algebraic structure containing at least a unit and a counit, and where morphisms require the unit and counit to be preserved, there can be no free functor unless the tensor unit in the underlying category is the zero object.

Right. You’ve pointed out how the forgetful functor for an ‘algebraic’ structure, a gadget with ‘operations’, is likely to have a left adjoint, while a forgetful functor for a ‘coalgebraic’ structure, with ‘co-operations’, is likely to have a right adjoint — and Todd has invoked the terms ‘free’ and ‘cofree’ for these two kinds of adjoints. I didn’t say enough about how important these themes are in the paper by Mélliès and Tabareau, but indeed they’re very important — so I’ve added a wee bit to hint at that.

Posted by: John Baez on June 22, 2009 11:44 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

I’ve added a bit more on how Mélliès and Tabareau think about constructing free algebras of T-algebraic theories. For those unwilling to reread week276 and look for the new stuff, here it is:

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.

It’s interesting to ask when Q* has a left adjoint. And the answer is: it always does!

The proof uses a left Kan extension followed by what Mélliès 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)

and this 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 what we wanted: 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.

They generalize algebraic theories to “T-algebraic theories” for any pseudomonad T on Cat. And then, they 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.

Posted by: John Baez on June 22, 2009 11:36 AM | Permalink | Reply to this

nLab credit

So far the main contributors include Urs Schreiber, Mike Shulman, Toby Bartels, Tim Porter, Todd Trimble, David Roberts, Andrew Stacey, Bruce Bartlett and myself.

Also important, and not really a tier below those, are Zoran Škoda, Ronnie Brown, and Eric Forgy. But don't take my word for it; check for yourself.

That listing can be misleading, of course. David Corfield and Jacques Distler also have many contributions listed, but my impression is that most of those are minor corrections of spelling, grammar, and formatting (my apologies if I've forgotten others!). I'd also say that Urs has contributed more than me, but my listing is inflated through minor corrections. On the other hand, such minor corrections are a welcome contribution to any wiki!

Posted by: Toby Bartels on June 20, 2009 10:39 PM | Permalink | Reply to this

Re: nLab credit

Of course I said “the major contributors include” because I didn’t claim to give a complete list.

Posted by: John Baez on June 20, 2009 11:59 PM | Permalink | Reply to this

A couple of typos

Speaking of minor corrections, here are a couple:

“It’s operating since November 2008.” -> “It’s been operating since November 2008.”

“and a grad student of this:” -> “and a grad student of his:”

Posted by: Stuart on June 22, 2009 9:09 AM | Permalink | Reply to this

Re: nLab credit

Thanks, Stuart, for catching those mistakes. I’ve fixed them.

I’ve also added Toby’s list of 3 to my list of “main contributors” to the nnLab. Needless to say, I won’t try to keep this list up to date — I hope it becomes obsolete soon, as the number of contributors grows!

Posted by: John Baez on June 22, 2009 9:52 AM | Permalink | Reply to this

Re: nLab credit

Yes, Zoran Škoda has been contributing plenty of entries and material, and is trying to popularize the idea of contributing among the visitors at the Max-Planck institute in Bonn; Ronnie Brown is serving as a constant source of references and further pointers in the background and Eric Forgy has put effort into connecting abstract entries to earth.

We are happy about every new contributor.

Last week I was talking with Bruce Bartlett about why, even though things are going well, we don’t have even more contributions yet. Bruce said: “Don’t forget, it’s lot of work to contribute to the nLab”.

I would like to dispel this idea that the nnLab is extra work. Instead, it is a means to reduce work.

Don’t add anything that would distract you from your daily work. Instead, use it to do your daily work.

Every day, I think each of us comes along some fact or insight or reference that looks worth remembering. Sometimes one makes a note about this in some notebook, only to lose that later. Or one just tells a colleague, who forgets it later. Or one starts typing a short note, which remains incomplete.

In all these cases, just open the nnLab and feed the material in there. It may well be just a short note first. It will be fun to come back to it later and see how it has grown.

Or, if you give a lecture or some other class and think about typing up your notes on this. Think about using the nnLab as your notepad.

Andrew Stacey suggested that more people might feel like contributing to the nnLab if it were more transparent who is “in charge of it”, so that it be clearer what may happen to the material one contributes in the long run. Together we decided to take some action to improve on this siuation. I’ll post a link soon.

Posted by: Urs Schreiber on June 22, 2009 9:32 AM | Permalink | Reply to this

organization of the nLab

I wrote

Andrew Stacey suggested that more people might feel like contributing to the nLab if it were more transparent who is “in charge of it”, so that it be clearer what may happen to the material one contributes in the long run. Together we decided to take some action to improve on this siuation. I’ll post a link soon.

Here is now the link to the nnForum discussion:

Organization of the nnLab

Posted by: Urs Schreiber on June 22, 2009 10:16 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

I don’t know if anyone thought about this but, because the bigger the stars, the smaller the density, could it be that there is a smooth transition between a star and a small nebula?

Take the biggest star known, VY Canis Majoris, the average density is 1 millionth of air at sea level, according to the source. I wonder what we would see if we get near that star, we could still talk about a “radius”, or that we rather see a high density nebula blown by extremely strong solar winds of a “naked” core. I say naked, because I wounder if we’d see a relatively thin “solid” shell, made of infalling convecting material from the nebula, as thick as the radius of the sun, covering a nucleus with, say, 10 times the total mass of the sun and only about 3-5 times tha radius of the sun.

To put in other way, such star is works like a contained nova exploson by he constant infalling matter of the high density nebula…

Posted by: Daniel de Franša MTd2 on June 21, 2009 12:51 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

There’s an upper bound on the luminosity of a star: if they get too bright, their own radiation pressure blows them apart.

But this maximum luminosity, called the ‘Eddington luminosity’, seems to scale linearly with the mass. I don’t know why there’s an upper bound on the mass. There seems to be one, of roughly 120 solar masses.

A great example of a star close to the brink is Eta Carinae. It’s mass is about 100 times that of our Sun, and its luminosity is more than a million times that of our Sun.

Posted by: John Baez on June 23, 2009 7:49 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

This is funny because Eta Carinae is a small star for its mass: 5× bigger less massive than Betelgeuse, but its volume is 8× to 100× smaller.

Maybe Eta Carinae will continue ejecting mass until the radiation pressure is small enough that its radius can expand, like Betelgeuse or Canis Majoris, without losing mass due to radiation pressure.

So, I wonder, in this last stage, if the mass of the star is really big, would we see a sphere or a nebulous blob?

Posted by: Daniel de Franša MTd2 on June 24, 2009 1:02 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

I found the answer myself for red giants , a class which incluedes Betelgeuse:

“In fact, such stars are not big red spheres with sharp limbs (when one is close to it) as displayed on many images. Due to the very low density such stars may not have a sharp photosphere but a star body which gradually transfers into a ‘corona’.”

A drastic example of this class is the star Mira
the presence of a binary company makes even clearer how delicate is the structural equilibrium of the star: it is deformed like an amoeba.

Posted by: Daniel de Franša MTd2 on June 24, 2009 1:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

I’m pretty sure that when people talk about the size of the star, they refer to the volume in which radiation is in thermal equilibrium with the matter. That is, the variation of temperature over the mean free path of a photon is small. This is not the case for nebulae, as far as I understand.

Posted by: Squark on June 29, 2009 4:19 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Some interesting comments by Charles McElwain. In email I wrote:

I would like more detailed information on what exactly would happen to us if Betelgeuse went supernova. How bright would it be? Could you be blinded by looking at such a bright point source? Etc.

He replied:

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:

Posted by: John Baez on June 23, 2009 7:58 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Let me try a couple of calculations. I don’t know whether Betelgeuse is scheduled to become a type I or type II supernova, but never mind… maybe you can check me on this number crunching.

Very roughly, a supernova is 10 billion times more luminous than the Sun. Say Betelgeuse is 600 light years away. That’s about 38 million AU. If Betelgeuse is 38 million times farther than the Sun, it’s dimmed by a factor of 38 million squared, or about 1.410 151.4 \cdot 10^{15}. So, its apparent luminosity as a supernova will be about

10 10/10 15=10 510^{10} / 10^{15} = 10^{-5}

times that of the Sun. Okay: the Sun will look 100,000 times brighter.

At this point the silly astronomical ‘magnitude’ system may come in handy. It’s logarithmic: a factor of 10 luminosity corresponds to roughly -2.5 ‘magnitude’. So, the apparent magnitude of Betelgeuse gone supernova would be

5×2.5=12.5 5 \times 2.5 = 12.5

more than that of the Sun. The advantage of knowing this is that we can consult a chart. The Sun has magnitude -26.7, the full Moon has magnitude -12.6; supernova Betelgeuse would have magnitude roughly -14.2. Hmm, that’s just a bit brighter than the Moon, not ‘brighter than a million moons’. Did I screw up?

Let me try again, avoiding this ‘magnitude’ stuff. From that chart I see the Sun is 450,000 times brighter than the full Moon. If the Sun is 100,000 times brighter than supernova Betelgeuse, that means this supernova will only be 4.5 times brighter than the Moon.

Of course it will be essentially a point source, so it might still damage your eyes to stare at it. Dunno.

Posted by: John Baez on June 23, 2009 9:13 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Now I realize that this whole “brighter than a million full moons” business is a bit silly, even if true, because it’s just a confusing way of saying ‘more than twice as bright as the Sun’.

If supernova Betelgeuse were more than twice as bright as the Sun, it’d have a serious effect on our weather! But I don’t think it will be.

Posted by: John Baez on June 23, 2009 9:35 AM | Permalink | Reply to this

Sol goes nova? “Inconstant Moon”; Re: This Week’s Finds in Mathematical Physics (Week 276)

I was surprised that you omitted this. Not that I dislike the Clarke story. Larry Niven attended my alma mater Caltech for some time. As fact-checked at wikipedia:

“Inconstant Moon” is a science fiction short story by American author Larry Niven that was published in 1971. Inconstant Moon is 1973 anthology of Larry Niven’s short stories that includes the title piece. The title is a quote from the balcony scene in William Shakespeare’s Romeo and Juliet. The anthology was assembled from the US collections The Shape of Space and All the Myriad Ways. The short story won the 1972 Hugo Award for best short story.”

“First appearance: 1971 short story collection All the Myriad Ways. Stan, the narrator, notices that the moon is glowing much brighter than ever before. The people he meets as the story begins all praise the moon’s increased beauty but lack the scientific background to understand its cause. However the narrator surmises that the Sun has gone nova, the day side of the Earth is already destroyed, and this is the last night of his life. He then calls and visits his girlfriend Leslie, presuming her ignorant of the situation, but she realizes it independently when Jupiter brightens with appropriate delay; they then enjoy their last night on the town, before rain and winds start.”

“Later, he realizes one other possibility. In case he is right, they find appropriate supplies and seek refuge from the coming natural disasters in Leslie’s high-rise apartment. The second possibility turns out to be correct: the Earth has merely been struck by an enormous solar flare. The vaporized seawater leads to torrential rains, hurricanes and floods. Most (if not all) people on the Eastern Hemisphere are presumed dead. The story ends at the break of an overcast, gray morning, with the narrator “wonder[ing] if our children would colonize Europe, or Asia, or Africa”.

“In 1996, the story was made into an episode of the Outer Limits television series with Niven himself writing the script - see Inconstant Moon (The Outer Limits). Jo Walton in 1997 wrote a short poem, ‘The End of the World in Duxford’ as ‘an unauthorised version of Inconstant Moon, a British equivalent….’”

Posted by: Jonathan Vos Post on June 24, 2009 4:51 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Let me try doing the calculation a different way, to check my work. According to Michael Redmond’s page, a type Ia supernova would be as bright as the Sun if it were .3 parsecs away, while a type II would be as bright as the Sun if it were .25 parsecs away.

I don’t know which type of supernova Betelgeuse is likely to become. (If anyone does, let me know, and say why!) So, let’s split the difference and say .25 parsecs.

Betelgeuse is about 600 light years away. A parsec is about 3.26 light years, so that’s about 180 parsecs.

So, Betelgeuse is about 180/.25=720180/.25 = 720 times the distance it would need to be as bright as the Sun when it goes supernova. So, supernova Betelgeuse will be about

1720 21500,000\frac{1}{720^2} \sim \frac{1}{500,000}

as bright as the Sun.

As mentioned, a full moon is 1/450,0001/450,000 as bright as the Sun. So, it seems that supernova Betelgeuse will be just a little brighter than a full moon.

Given how rough my calculations are, this counts as consistent with my previous estimate that supernova Betelgeuse will be 4 times brighter than a full moon. Same order of magnitude. Nothing like ‘brighter than a million full moons’.

Posted by: John Baez on June 24, 2009 9:50 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Astronomical brightnesses are usually given in terms of an integral over the entire apparent surface of the object. Something brighter than the full moon, but concentrated into an itty bitty little point, might actually be pretty painful to look at - like pointing a laser in your eye.

Posted by: Dan Piponi on June 24, 2009 8:51 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Here’s a nice view of some big stars:



Paul Masham emailed it to me. I don’t yet know where he got it.

Posted by: John Baez on June 24, 2009 9:36 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

This slide is commonly used to make star size comparisons on You Tube. But it got measurements wrong, for example, Betelgeuse should be half the diameter of the biggest star, but it is much smaller. The stars of the last slide, with the biggest stars, should show all of them with aproximately the same size.
And they put the homunculus nebula, that evolves Eta Carinae as it were the star itself…

Here is a list of the largest known stars.

Posted by: Daniel de Franša MTd2 on June 24, 2009 1:40 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Daniel wrote:

This slide is commonly used to make star size comparisons on You Tube. But it got measurements wrong, for example, Betelgeuse should be half the diameter of the biggest star, but it is much smaller.

Hmm, it looks like you’re right! I’ll remove that slide from week276.

Posted by: John Baez on June 24, 2009 2:50 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

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!

Posted by: John Baez on June 27, 2009 5:44 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Sharpest Views Of Star Betelgeuse Reveal How Supergiant Stars Lose Mass

“… Betelgeuse is already nearing the end of its life and is soon doomed to explode as a supernova. When it does, the supernova should be seen easily from Earth, even in broad daylight…. The AMBER observations revealed that the gas in Betelgeuse’s atmosphere is moving vigorously up and down, and that these bubbles are as large as the supergiant star itself. Their unrivalled [sic] observations have led the astronomers to propose that these large-scale gas motions roiling under Betelgeuse’s red surface are behind the ejection of the massive plume into space.”

Posted by: Jonathan Vos Post on August 3, 2009 10:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Cool! Thanks!

The first team used the adaptive optics instrument, NACO, combined with a so-called “lucky imaging” technique, to obtain the sharpest ever image of Betelgeuse, even with Earth’s turbulent, image-distorting atmosphere in the way. With lucky imaging, only the very sharpest exposures are chosen and then combined to form an image much sharper than a single, longer exposure would be.

The resulting NACO images almost reach the theoretical limit of sharpness attainable for an 8-metre telescope. The resolution is as fine as 37 milliarcseconds, which is roughly the size of a tennis ball on the International Space Station (ISS), as seen from the ground.

“Thanks to these outstanding images, we have detected a large plume of gas extending into space from the surface of Betelgeuse,” says Pierre Kervella from the Paris Observatory, who led the team. The plume extends to at least six times the diameter of the star, corresponding to the distance between the Sun and Neptune.

My emphasis.

For the real stuff, read these:

Posted by: John Baez on August 3, 2009 10:57 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 276)

Here’s a picture which shows the enormous plume coming out of Betelgeuse:

It was taken at the Very Large Telescope or ‘VLT’ in Chile. The VLT is very cool.

Posted by: John Baez on August 5, 2009 6:38 PM | Permalink | Reply to this

Post a New Comment