LaTeX at last ! (Illustrated with a GR calculation) Sunday, Feb 18 2007

After waiting for long, Blogphysica gets $\LaTeX$ . (See the announcement at the wordpress.com Blog )

The Basic syntax is $latex <LaTeX Equation>$. You can find a list of $\LaTeX$ symbols here(pdf) . In particular, if you find the formulae are too small try $latex {\displaystyle <LaTeX Equation>}$ . It’s great !

Update(18/02/07) : See this FAQ for some more options.

Now, to illustrate LaTeX, I’ll take up a particular problem. Consider two equal masses falling towards each other, as shown below, starting from rest.

m—>—o—<—m

The question is this – How much power does this system lose as the two masses fall towards each other ?

Let $r$ be the distance from the centre of mass (which I’ve denoted by an ‘o’ above and I choose it to be the origin). Take the common axis to be z-axis.

Hence the position of two masses are respectively $(x,y,z)=(0,0,r)$ and $(0,0,-r)$

To the zeroth approximation, Newtonian mechanics tells you that

${\displaystyle \frac{d^2 r}{dt^2} = - \frac{G_Nm}{(2r)^2}}$

(We neglect the effect of gravitational wave on the masses)

The quadrupole moment $Q_{ij}$ of a mass distribution is defined by

${\displaystyle Q_{ij}(t) = \iiint d\forall\ \ \ \rho(x,t) \left[ x_i x_j - \frac{1}{3} \delta_{ij} r^2 \right]}$

Where the integral is done over the whole source($\rho$ is the mass density of the source). It is basically negative of the traceless part of the moment of inertia.

The Einstein formula for the power emitted by the source (in the form of Gravitational waves) is

${\displaystyle P = \frac{G_N}{5c^5} \sum\ \frac{d^3 Q_{ij}}{dt^3}\ \frac{d^3 Q_{ij}}{dt^3} }$

where the symbol $\sum$ denotes a sum over $i,j=1,2,3$.

Assuming that the masses are small in size, the components of quadrupole moment in this case are

${\displaystyle Q_{zz}= m\left(r^2 - \frac{1}{3}r^2\right)+ m\left((-r)^2 - \frac{1}{3}r^2\right)= \frac{4}{3}mr^2}$
${\displaystyle Q_{xx}=Q_{yy} = m(0^2 -\frac{1}{3}r^2)+ m(0^2 -\frac{1}{3}r^2)= -\frac{2}{3}mr^2 }$
${\displaystyle Q_{xy}=Q_{yx}=Q_{zx} =\ldots = 0}$

Now to calculate the third time derivative, we first use chain rule to get

${\displaystyle \frac{d^3}{dt^3} r^2 = 6 \frac{dr}{dt}\ \frac{d^2 r}{dt^2} + 2r\ \frac{d^3 r}{dt^3} }$

We can now employ Newton’s Law to get

${\displaystyle \frac{d^3}{dt^3} r^2 = - \frac{G_Nm}{2r^2}\ \frac{dr}{dt}}$

${\displaystyle \frac{d^3 Q_{zz}}{dt^3} = - \frac{2G_Nm^2}{3r^2}\ \frac{dr}{dt}}$
${\displaystyle \frac{d^3 Q_{xx}}{dt^3}=\frac{d^3 Q_{yy}}{dt^3} = \frac{G_Nm^2}{3r^2}\ \frac{dr}{dt}}$

All the other components are zero. Substituting this into the Einstein’s formula which I quoted above, the total power radiated comes out to be

${\displaystyle \mbox{Power emitted} = \frac{G_N^3m^2}{15c^5 r^4}\ \left(\frac{dr}{dt}\right)^2 }$

which is terribly small in most cases.

Physics Blogspeak : Part II Sunday, Jan 14 2007

COSMOS Reveals the Cosmos (from Cosmic Variance by Sean)

The internet works so that we don’t have to! This week is the big annual meeting of the American Astronomical Society in Seattle, so expect to see a series of astro-news stories pop up all through the week. The first one concerns a new result from the Cosmological Evolution Survey (COSMOS) — they’ve used weak lensing to reconstruct a three-dimensional image of where the dark matter is.

AAS Report #3: Things that go boom! (From Bad Astronomy Blog)

It’s a fact of life that some stars explode. Actually, it’s a good thing: when stars explode they create and scatter the heavy elements that create us. The iron in your blood and the calcium in your bones were created in a supernova! So it’s important to study these objects, so we can better understand our origins.

But it’s also fun! Stars explode! Bang! Cool!

Today there were three press releases about supernovae. All three were surprising to me, and pretty interesting.

1) Kepler’s Supernova was a Type Ia

OK, so that title doesn’t thrill you. But that simple statement is actually the answer to a long-standing mystery. Ready for this? OK, sit back…

The AAS : a Nerd’s Eye View (from Galactic Interactions)

I’m in Seattle at the moment. I flew in yesterday; it’s cold, windy, and rainy. In fact, the rain was looking kinda slushy last night. While my wife from Minnesota might scoff at my calling this cold (it was just below freezing), in Nashville it’s been March-like temperatures.

I’m here for the 209th meeting of the American Astronomical Society. I’m going to try an experiment. I’ve never done the “live blogging” thing before, and indeed it’s entirely possible that I’m not using the term properly. It is my intention to post several posts this week inspired by things I see at the AAS. I can’t tell you what they will be yet, because they haven’t happened…. I’m hoping mostly to focus on interesting science and such, but anything that inspires me to blather is fair game as far as I’m concerned….

(Other posts in this series.)

Come On In, the Methane’s Fine (from Uncertain Principles by Chad Orzel)

The Times has an article announcing the discovery of methane lakes on Titan:

CDF’s New Results : W Boson Mass and Top quark Mass (From Quantum Diaries)

Physics Blogspeak : Part I Sunday, Jan 14 2007

A collection of interesting posts from physics blogsphere :

Short Distances: Newton Still the Man (from Cosmic Variance by Sean)

Via Chad Orzel, I see that the latest constraints on short-distance modifications of Newton’s inverse-square law from the Eot-Wash group at the University of Washington have now appeared in PRL. And the answer is: extra dimensions must be smaller than 0.045 millimeters (in any not-too-contrived model)…

Undergraduate Theory Institute(from Cosmic Variance by Sean)

Sadly, I’m not here to announce that applications are now being accepted for students who would like to participate in this year’s Undergraduate Theory Institute. That’s because there is no such thing as the Undergraduate Theory Institute, at least as far as I know. (Google doesn’t know of one either.) But I think it would be a great idea — maybe if I post it here on the blog someone will start it.

It’s increasingly common for physics students to participate in some kind of research during their undergraduate years. The NSF has a very successful Research Experience for Undergraduates program, for example, that funds students to do summer research, typically at an institution other than their own. Getting involved in research as early as possible is a great idea for students, for a number of reasons. Most importantly, the flavor of doing real research, where the answers aren’t in the back of the book, is utterly different from almost any classroom experience or even self-study, where you are trying to learn material that someone else has already mastered. The move from following a course of study to striking out into the unknown is one of the hardest transitions to make during graduate school, and getting a head start is an enormous help. On a more prosaic level, it’s useful to work closely with an advisor who can end up writing letters of recommendation. And let’s not forget that it can be a lot of fun!..

Dancing Ball Lightnings in the Lab (from Backreaction Blog)

Ball lightnings are mysterious things: Small, bright balls of fire suddenly appear during a thunderstorm, swirl around, make sometimes funny noises, and leave behind a smell of ozone…

Since ball lightnings are not only spooky, but also very elusive, there has been a lot of speculation how to understand and explain them in a scientific way: People have suggested that it may be some ionised balls of plasma held together by their own magnetic fields, or even such exotic things as mini black holes leftover from the big bang…

A more “down to Earth” explanation was proposed in a 1999 Letter to Nature:..

From Griffiths to Peskin: a lit review for beginners (From “An American Physics Student in England” )

a.k.a. “How to get started learning QFT as an undergraduate.”

Quantum Field Theory (QFT) plays a key role in all branches of theoretical physics. For students interested in high energy theory, exposure to QFT at any early stage is slowly becoming the standard for top American graduate schools. This is already the case for the Mathematics Tripos at Cambridge…

An inspired student with adequate background should be able to take quantum mechanics in his/her second or third year and then progress directly to a ‘real’ QFT course with a bit preparation, without going through the rigmarole of a year-long graduate quantum mechanics course.

Instead, I present a rough guide to pedagogical QFT literature so that a motivated student can prepare for a graduate-level QFT course or a get started with a self-study during the summer after his/her undergraduate quantum mechanics course. As a someone who was in this position in the not-too-distant-past, I hope some personal experience with the pros and pitfalls of the listed texts will be helpful for other other students interested in doing the same….

Happy Perihelion ! Wednesday, Jan 3 2007

Via Bad Astronomy Blog, we are informed that

Today, January 3, on or about 20:00 Universal Time (2:00 p.m. Pacific time), the Earth will reach perihelion, its closest approach to the Sun. The distance from the Sun to the Earth will be roughly 147,093,600 kilometers (I have found several different distances on different sites, and this is an eyeball average).

with a statutory warning that

Remember, our distance from the Sun doesn’t affect our seasons (much).

So, Happy Perihelion ! 🙂

Don’t miss the The Top Ten Astronomy Images of 2006.

Now, last but not the least, via the same blog, I came across a treasure-trove for those who are interested in astronomy – What’s Up 2007 – 365 Days of Skywatching , a superb online book !

If somebody knows the present Astro-club co-ordinators, do pass on.

From Physics Blogsphere Sunday, Jul 30 2006

Lars Onsager

Over the last few days, I’ve come across a lot of great posts at physics blogsphere.

Blogphysica has already linked to CV once . And I do it again – this time it is a post on N-Body problem by Sean .To quote

… I can’t help but show these lovely exact solutions to the gravitational N-body problem. This one is beautiful in its simplicity: twenty-one point masses moving around in a figure-8.The N-body problem is one of the most famous, and easily stated, problems in mathematical physics: find exact solutions to point masses moving under their mutual Newtonian gravitational forces (i.e. the inverse-square law)…

Check it out – it has some beautiful animations to go along with it. 🙂

Update(03/08/06) : See also the post titled Boltzmann’s Anthropic Brain by Sean and the one by Mark titled A Nonperturbative Analogy at CV.

On another note, Three-Toed Sloth has a good post titled The Nobel Prize Winner as Neglected Genius on Onsager. It is triggered by an RMP article Reviews of Modern Physics 78 (2006): 87–135; free copy (courtesy : TTS ) which is a good but heavy read for those interested in this kind of stuff.

This is the abstract of the article.

Onsager and the theory of hydrodynamic turbulence

Gregory L. Eyink
Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, Maryland 21218, USA

Katepalli R. Sreenivasan
International Center for Theoretical Physics, Trieste, Italy and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA

(Published 17 January 2006)
Abstract: Lars Onsager, a giant of twentieth-century science and the 1968 Nobel Laureate in Chemistry, made deep contributions to several areas of physics and chemistry. Perhaps less well known is his ground-breaking work and lifelong interest in the subject of hydrodynamic turbulence. He wrote two papers on the subject in the 1940s, one of them just a short abstract. Unbeknownst to Onsager, one of his major results was derived a few years earlier by A. N. Kolmogorov, but Onsager’s work contains many gems and shows characteristic originality and deep understanding. His only full-length article on the subject in 1949 introduced two novel ideas — negative-temperature equilibria for two-dimensional ideal fluids and an energy-dissipation anomaly for singular Euler solutions — that stimulated much later work. However, a study of Onsager’s letters to his peers around that time, as well as his private papers of that period and the early 1970s, shows that he had much more to say about the problem than he published. Remarkably, his private notes of the 1940s contain the essential elements of at least four major results that appeared decades later in the literature: (1) a mean-field Poisson-Boltzmann equation and other thermodynamic relations for point vortices; (2) a relation similar to Kolmogorov’s 4/5 law connecting singularities and dissipation; (3) the modern physical picture of spatial intermittency of velocity increments, explaining anomalous scaling of the spectrum; and (4) a spectral turbulence closure quite similar to the modern eddy-damped quasinormal Markovian equations. This paper is a summary of Onsager’s published and unpublished contributions to hydrodynamic turbulence and an account of their place in the field as the subject has evolved through the years. A discussion is also given of the historical context of the work, especially of Onsager’s interactions with his contemporaries who were acknowledged experts in the subject at the time. Finally, a brief speculation is offered as to why Onsager may have chosen not to publish several of his significant results.

Of course, you can find this and much more at Mixed states – which is a collection of feeds from lots and lots of physics blogs as explained here . Of course, it right away goes into our blogroll.