Peculiarities of the Fine Structure Constant

Spacing of spectral line shifts depends on the fine structure constant. It's been found that different spectra are obtained in the lab vs. obtained from distant quasars with light passing through gas clouds.

In the table presented in my March 18th post, the fine structure constant is shown on the last line with a value of  a   =  1/ 137.035999139 or approximately, 1/ 137.   Note the value is the same irrespective of the system of measurement because this constant is a pure number - there ate no units attached- unlike the other constants I covered.  Indeed,  a    is an amalgamation of several other constants including the electron, the speed of light and the Planck constant. We can write, for example,


 a   =  1/ 4p e _o    { e² / ħ c }   where:     ħ  =   h/ 2p


Physicists keep track of the fine structure constant by using quasars. Also called "QSOs" for quasi -stellar objects, these are active galactic nuclei of extremely high luminosity powered by supermassive black holes. As shown in the diagram above, on its way toward Earth the QSO's light passes through gas clouds which absorb light of particular frequencies producing gaps - also called absorption lines - in the otherwise continuous spectrum.  The typical profile of such an absorption line is shown below, this one centered at a wavelength of 5889.95 angstroms:

As can be seen from the absorption line profile there is a variation in the line "darkness"  with the maximal degree at line center where the absorption coefficient is greatest. This then falls off in what we call the "line wings".   (Note here that  D u_D  is the Doppler half-width of the line.)  The point is that the locations of these absorption lines depend on the fine structure constant. Thus, variations in the spacing of the lines in space or time might indicate the value of  a    has changed.

Is there evidence for such variations? In 2011, in a paper appearing in Physical Review Letters, John Webb of the University of New South Wales and colleagues reported the fine structure constant increases in one direction in the sky and decreases in the opposite direction. Almost if some special axis was running through the universe. Of course, no physicist would be content with such "specialness" or uniqueness. By the cosmological principle such behavior of a   ought not depend on directions. (And in this regard even Webb counts himself as a skeptic, as he should).

In fact, a compelling alternative explanation has been put forward by Michael Murphy of Swinburne University of Technology in Melbourne, Australia.  He suggests the logical take that telescope calibration issues are to blame for the apparently changing value of a.   Using measurements free of calibration issues, the value of  stays put, as Murphy et al reported in the Monthly Notices of the Royal Astronomical Society.   See also:

https://arxiv.org/abs/1606.06293


It should be noted, however, that Murphy et al's findings do not rule out actual variations in  a  with respect to the part of the sky observed by Webb in 2011.

Here's an interesting side thought to ponder: What if, instead of a   = 1/137 (approximately) it had been 1/ 130? Say during the birth of the universe. Then it seems clear the cosmos would have been set on a path to being barren, empty.

As an interesting historical aside, Sir Arthur Eddington arrived at a value of a   = 1/136 by taking the ratio of two "naturally occurring units of action". ('Great Ideas and Theories of Modern Cosmology', 1961, p. 178).  He chose one unit of action as the quantum for radiation, or  ħ  =   h/ 2p  and the second as the action for elementary particles, or e² / c.   Then, taking:

{e² / c}/  ħ  =  1/ 136

Curiously, Eddington wasn't bothered by the divergence (from a   = 1/137 )  , and just introduced a "fudge factor". This "was for obscure reasons that are difficult to understand". Perhaps, in the end, he was simply mesmerized by a kind of 'numerology' . Eddington also came up with a quadratic equation: 10x²  +    136x +  1 = 0,  linking his  fine structure result with the mass ratio of the proton to electron, i.e. in terms of the ratio of its two roots. From there, Eddington parlayed his fine structure and other pure number results into a kind of "universal theory" linking every aspect of the cosmos in a kind of romantic quest. Much like Kepler before him, with his "harmonic geometry"  in which the five Pythagorean regular polyhedra dictate the structure of the universe and reflect God's plan through geometry.

We shouldn't be too hard on Sir Arthur  (or Johannes Kepler) as he wasn't the first scientist to be taken in by numerical relationships, "harmonic" ratios, and "precision" theoretics. Nor will he likely be the last.  Even today we behold "scientists" seriously working on the so-called "anthropic principle". This  nonsense is based on the fallacy (due to a misunderstanding of physics units, dimensions) that there is an implicit "fine tuning". This in turn depends on a putative "fine precision" - but that is almost always based on the choice of units.

Thus, saying stupidity like "if the neutrino mass were 1 part in 10 ³⁵  smaller there'd be no expansion of the universe" is like saying that if Lebron James were 1 part in 10 ¹⁶  shorter he'd not have been a great basketball player! 

Looking at Stellar Emission and Absorption (3)

The study of spectral lines is facilitated by using what is called the “equivalent width”. This provides the total strength of a line, yielding the same area of the line (for a rectangular facsimile) provided the depth is complete from the continuum to zero brightness.  

    We can express the equivalent width in two ways, based on frequency of wavelength, or wavelength itself, e.g.

W =   ò ^¥₀  (I _c – I _u /  I _c )du =  ò ^¥₀  (F _c – F _l / F _c ) dl

   The left side defines W in terms of the intensity from the continuous spectrum outside the spectral line where the quantity (I _c – I _u / I _c ) is referred to as the “depth of the line”, the analogous quantity to (F _c – F _l / F _c ) on the right side where we have radiant flux units. Technically, the integral should be taken only from one side of the line to the other but the limits can be as shown provided I _c    (or  F _c)  is kept constant in the neighborhood of the line.

      The Curve of Growth:

Having obtained the equivalent width, the next logical step is to generate the “curve of growth”.  Recall from the previous section that W, the equivalent width, represents the line strength. Then the curve of growth describes how the latter increases as the optical depth, τ  increases.

To fix ideas, we use a model slab of  finite thickness ds and over which the optical depth increases by d τ.  The incident intensity is  I _c  for frequencies in the neighborhood of the line,  and the intensity emerging from the opposite side is    I _u  which we seek to find.

Starting with  the original basic transfer equation based on wavelength:

dI(l)/ds = -k(l)  I(l) + k(l)  S(l)

We can derive the frequency form:

d I_u / dτ _u  =  -  I_u – S _u

We have (from the original equation) after adjusting for frequency:

d I_u / ds   =  -k _u I_u  +   k _u S _u

Where:   k _u   =   d τ _u / ds

So: 

d I_u / ds   =  -( d τ _u / ds) I_u  +   (d τ _u / ds) S _u

d I_u =   - d τ _u  I_u  +   d τ _u  S _u  =   - d τ _u   (I_u – S _u)

whence:

d I_u / dτ _u  =  -  I_u – S _u

For which the solution is found (on integration):

I_u =  I _c  exp (-τ _u)

Where:  τ _u   = N L a _u  =  N L a _o  f _u

Where ‘N’ is the number of absorbing atoms per unit volume and we already saw that:

a _o =  a _u  /  f _u = a _u [1 - e ^{-  h u o / kT}] (p e²/ mc) f

In terms of this new information, the equivalent width of the line can then be written as:

W _u =  ò ^¥₀ [1  -   exp (- N L a _o  f _u) ] du

For small x we may use the approximation that:

exp (-x) much greater than   1 – x

So rewrite the equation for  W _u:

W _u »   ò ^¥₀ [1  -  (1 - N L a _o  f _u)  du

»   ò ^¥₀   N L a _o  f _u  du » N L a _o  ò^¥₀  f _u  du

So we find W _u  depends only on the form for the broadening function.  For very weak lines, for example:

ò ^¥₀  f _u  du  much greater than   1 so that W _u » N L a _o 

Or simply proportional to N, the number of absorbing atoms. For “strong” lines the absorption near the center is very large so we can expect: N L a _o  f _u    >> 1 and with sufficient working this can be shown to give the corresponding equivalent width: W _u  = 2  D u _D

And for the moderately strong lines: 

W _u  =  2  D u _D {ln  (N L a _o/Öp  D u _D )} ^½

While very strong lines yield:

W _u  =   1/p  (N L a _o  g)^½

Where g is the damping constant.

Assembling all the diverse W _u    and plotting log W _u vs.  log (N L a _o ) one gets the curve of growth shown below: 

Here, each contribution can be separately considered, in terms of the “linear”, “saturated” and “damping” segments. The first is for the weakest lines such that W _u  is simply proportional to the number of absorbing atoms. But as more atoms are added this approximation becomes invalid and so we are in the saturated part. In effect,  the near zero slope shows the degree to which the equivalent width increases at an almost uniform rate. Finally, we have the damping segment wherein the line under consideration develops “damping wings” because the number of absorbing atoms is so large. (Bear in mind again that  a _o  is proportional to the transition probability)

Problems:

1)Consider the ionization of helium, with three stages of ionization for which the partition functions are:    Z _I =    Z ₃    =   1,  and Z ₂ =    2

With ionization energies:

 c  ₁ =   24.58 eV    and c  ₂ =   13.6 eV  

For a temperature of T = 5040 K write out two versions of the Saha equation for:

log( P _e N ₂ /N ₁)   and log( P _e N ₃ /N ₂)  

And hence, compute:   (P _e N ₂ /N ₁)   and (P _e N ₃ /N ₂)  

2)The diagram below was sketched by a first year astrophysics student. It represents part of the emission spectrum of atomic hydrogen.

 It contains a series of lines, and the wavelengths of some (in nm) are marked. '
There are no lines in the series located at less than 91.2 nm

(a)   In which region of the E-M spectrum would these lines occur?

(b) Obtain a relation between E and l   and thence find the photon energies equivalent for all the wavelengths marked.

(c)Use this information to map a partial energy level diagram for hydrogen. Show and label clearly the electronic transitions responsible for the emission lines labeled.

3)a) A stellar atmosphere is composed of pure hydrogen, at a temperature of 9600 K and with electron pressure, P _e    =  200 dyne cm ^-2 .  Use the Saha equation to compute the fraction of atoms

that are ionized in relation to the total, i.e. find:   N _II /( N _I   + N _II )  

The partition functions are:  Z _II =   1 and  Z _I =   2

Hint: Check to see whether (E ₂  -  E ₁ ) >> kT with

k =  8.6174 x 10 ^-5 eV/K

So the Boltzman factor  exp (- E2 – E1) / kT << 1

b)Repeat the exercise again for a temperature of 5040 K and 20,000 K and thence account for why the Balmer lines attain maximum intensity at a temperature of 9520 K.

(4) Calculate how far you could see through the Earth’s atmosphere if it had the same opacity as the solar atmosphere, where:

k _l   =   0.264 cm ² /g.

Take the density of Earth’s atmosphere as:

   r   =   0.0012 g cm ^-3

The density of the solar atmosphere is taken as :

r _s   =  2.5 x 10  ^-7 g cm ^-3

And we know the intensity declines over a characteristic distance (length scale) such that:

ℓ  =   1/ k _l r