Tuesday, May 8, 2012

Dimensions matter

In another ruinous blog (whose name i'll here omit for decency) once I wrote on the formula describing the Hawking temperature of a black hole. For a Schwarzschild BH, it takes the simple form

\[T=\frac{1}{8\pi M}\]

The formula above is written in "God-given" units, the ones that theoretical physicists and lazy students like more. In God-given units the speed of light c, Newton constant G, Boltzmann constant k and also Planck constant h, are set to unity.
Restoring more physical units, the same formula above reads

\[T=\frac{\hbar c^3}{8\pi G M k}\]                                                                                                    (1)

But today I'm not gonna write about Hawking formula and the reason why I introduced it will be clear below.

Actually, what I wanted to discuss today is Chandrasekhar mass limit for white dwarfs (WDs) and neutron stars (NSs). [This has nothing to do with the fact that i'm giving a mini-course on compact objects for Astrophysics.... btw tomorrow's lecture is at 8:00 AM, can you believe  it?]

Chandra limit describes the maximum mass that these compact stars can reach in order for their degeneracy pressure (a quantum effect due to Pauli's principle) to contrast gravitational force. 

WDs and NSs are extremely compact objects: they roughly have the same mass as our Sun, but a radius comparable to that of Earth (for WDs) or even as small as ~ 10 km (for NSs). These objects are so compact that their gravitational energy is huge. When a gas of electrons (for WDs) or neutrons (for NSs) is squeezed to extremely high densities due to such formidable gravitational fields, the fermions start to "feel a repulsion", for they cannot occupy the same quantum state (the movement "Occupy White Dwarf" was named after this effect).

However, when such degenerate gas becomes relativistic, the binding energy of the system decreases until the system becomes unstable. The larger the mass, the stronger are the relativistic effects so that, at some critical point, gravitational contraction cannot be counteracted anymore. This mass was computed by Subrahmanyan Chandrasekhar in 1930 and it reads

\[M_{\rm Chandra}\sim 0.5\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{m_H^2} \]


The proportionality constant depends on the details of each particular stellar model -- the precise value is \[M_{\rm Chandra}\sim1.44 M_\odot\] , but what is really amazing in this formula (as well as in Hawking formula (1)) is that it contains three of the most fundamental constants in physics: c, G and bar h. This means that quantum effects, relativistic effects and gravitational physics, all contribute and play a key role in the physics of WDs. 


For NSs the situation is more intricate, but a similar mass limit must exists.  Also, since the neutron mass ~ 2000 times the electron mass, and because the radius of a WD scales as \[R\sim 1/m_e\], a NS has a radius which is ~ 2000 times smaller than that of a WD. Nonetheless, the maximum mass of a NS cannot be much larger than the Chandrasekhar limit.


Not only. Since we believe that nothing can support self-gravity more than a degenerate gas of neutrons, the existence of a maximum mass poses the following question: 


What happens to a compact object (like the remnant of a big star) 
when its mass exceeds the Chandrasekhar limit?

The answer, already understood by Chandrasekhar and strongly opposed by Sir Arthur Eddington, is catastrophic. Basically nothing can prevent the gravitational collapse to occur indefinitely until, we believe, a black hole forms. I'll write more in detail about this (and also about the curious story behind Chandra's and Eddington's divergent opinions) but, for the time being, let me just say that as back as 1930, i.e. 15 years after Einstein's General Relativity was formulated, Chandra's outstanding result was one of the first "hints" of the existence of such "singular" objects as black holes are.


Did you know that Chandrasekhar wrote one of his books
(i think it was The Mathematical Theory of Black Holes) while sailing from India to England ?
(the trip took 3 months)
UPDATE: I was told what I wrote is wrong! What Chandra actually did on the ship was precisely the computation mentioned above
(take a look to this nice website).
Actually, he was travelling to England to begin graduate study in physics at Cambridge and.. he was 20!
[thanks Emanuele for pointing this out, i'll give you access to the writer section of the blog for free :)]





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