Revisiting Stellar Spectral Line Formation & Transitions (1)

 We start then with the Boltzmann equation, which we  can express:

N₂ / N₁   =     [g₂ / g₁ ]   exp (- E2 – E1) / kT

That is, for the atoms of a given element in a specified state of ionization, the ratio of the number of atoms N₂  with energy E2, to the number of atoms N₁  with energy E1, in different states of ionization is given by the above formula. The same form of the equation can also be used to find the ratio of probabilities, i.e. that the system will be found in any of the  g₂ degenerate states with energy E2 to the probability that the system is in any of the g₁ degenerate states E1, viz.

P(E2) / P(E1)   =     [g₂ / g₁ ]   exp (- E2 – E1) / kT

Thus, the Boltzmann equation can be posed in two forms.  In statistical mechanics we also have the partition function:

Z  =   å _j   exp ( - e _j )/ t

Which is just the summation over the Boltzmann factor (exp ( - e _j )/ t ) for all states j for which the number of particles (N) is constant. We will find it useful to rewrite it:

Z = g₁ +  å^¥ _{j = 2}   g _j  exp (- E _j – E1) / kT

Of interest now are the relative numbers of atoms in ionization stage i, which is written:

N _e N _{i + 1} / N _i  =

 2 Z _{i + 1} / Z _i (2 p m _e kT/ h ²) ^1.5  e ^{- c i/ kT}

    This is the Saha equation, named after the Indian astrophysicist who first derived it.  Here,  N _e  is the number of free electrons per unit volume and c _i  is the ionization potential of the ith ionization stage. Thus, the equation relates the number of atoms in two successive  ionization stages to the quantities that are relevant. As per our introduction to quantum mechanics, the factor ‘2’ in the equation refers to the two possible spins of the free hydrogen election with spin quantum number: m _s =  +½.

Recall that for thermodynamic equilibrium, the rate of ionization cannot exceed the rate of recombination[1].  In other words, the rate at which atoms in the ith stage are ionized (i.e. to the i +1st stage) must equal the rate at wich ions in that i +1st stage are recombining with free electrons to form ions in the ith stage. The latter depends on N _e N _{i + 1}   and the former on N _i . Hence, Saha’s equation simply expresses the fact these two processes must occur at the same rate.

 One can also rewrite the equation in a more manageable logarithmic form if one substitutes the numerical constants:

log(N _e N _{i + 1} /N _i)  = 

15.38 + log (2 Z _{i + 1} / Z _i ) + 1.5 log T – 5040 c _i/ T

The units here are important to note and are consistent with the ionization potential being measured in electron volts (eV). Therefore N _e  must be in particles per cubic centimeter.

    Yet another way to express the Saha equation is to introduce the electron pressure, P _e . This acknowledges that each separate species of particle makes its own contribution to the total gas pressure. The free electrons in a gas therefore produce a pressure given by: P _e = N _e kT.

Then we may write another log form of the Saha equation:

log(P _e N _{i + 1} /N _i)  = 

-0.48  + log (2 Z _{i + 1} / Z _i ) + 2.5 log T – 5040 c _i/ T 

Example Problem:

For a  hydrogen plasma find:

(P _e N _{i + 1} /N _i)  =  P _e N _HII /N _HI   

at a temperature of 5040 K, given the hydrogen partition functions are:

Z _{i + 1}  =  Z ₂    =   1   and:

Z _i =    Z ₂    =   2      with  c _i =   13.6 eV 

Solution:

By the Saha equation:   log(P _e N _{i + 1} /N _i)  = 

-0.48  + log (2 Z _{i + 1} / Z _i ) + 2.5 log T – 5040 c _i/ T

=   -0.48  +  log (2) + 2.5 log (5040)  –  5040 (13.6)/ 5040

=   -0.48 + 0.30  +   9.25 – 13.6  =  -4.83

Antilog (-4.53) =   2.95 x 10 ^-5

 A useful table that will come in handy for spectral line and ionization energy computations is presented below. 

Careful inspection of the table  shows two quantum states with the energy (-13.6 eV) and eight with (-3.40 eV).   Thus, two states are degenerate for the n=1 level and eight states are degenerate for the n=2 level.  Since g _n = 2n², then:

At the n=1 level the statistical weight is:  g ₁ = 2(1)² = 2

At the n=2 level the statistical weight is: g ₂ = 2(2)² = 8  

Suggested Problem:

Consider a gas of neutral hydrogen (H). Using the Boltzmann equation and the information in the table above, compute the temperature at which one will expect equal numbers of atoms in the ground state and the first excited state.

---

[1] This must hold if the excitation and ionization equations are assumed valid, hence the numbers of atoms in a given level must not change with time.

Yes, An Algorithm Can Enable Your Bank To Evict You From All Your Accounts

                                                                       

Can an algorithm trigger your bank to close down all your accounts, including bank cards? According to author David Ward ('The Loop',p. 247) citing work by Prof. Michele Gilman (who teaches law at the University of Baltimore):  

"Increasingly she's found that algorithms are at the center of her cases, denying citizens their rights and benefits.  In a 2020 paper for Data & Society she offered aa guide to algorithms now being deployed to make decisions in literally dozens of areas of American life, from credit scores to public benefits to housing.  These algorithms often make life changing decisions without consulting the people whose lives they are changing."  

Such is the case with a banking algorithm (that flags "suspicious activity")  that's been canceling accounts for people and businesses at least since 2017-18. Thomson Reuters, in fact, reports that banks filed more than 1.8 million suspicious activity reports (SARS) in 2022, a nearly 50 % increase in just two years.   That figure is now on track to hit nearly 2 million this year.  

The people caught in the SAR trap often find chaos and confusion in their finances as a result. Sometimes, the bank is alert and sympathetic enough to dispatch a letter telling the forlorn customer that it is closing all their checking and savings accounts and canceling all their cards.  Any explanation, if there is one, is usually sketchy - offering little or no details, reasons.  

At the other end are customers who never get any letter or alerts at all, period.  Instead, they suddenly discover their accounts no longer work - often while paying for groceries at the supermarket, at the rental car counter, the hotel front desk or the ATM.   When they call their banks, the representatives show concern, i.e. "So sorry, we will look into it."  But then there comes a pause and shift in tone, at least according to a NY Times account (Nov. 10, 'Why Banks Are Suddenly Closing Customer Accounts',  by Ron Lieber and Tara Siegel Bernard)  of the SARS phenomenon. Thus they will be told:  "Per your account agreement we can close your account  for any reason at any time."

The Times' piece is quick to point out this isn't just a pro forma dealie to dump a grifter, say a person who's bounced one too many checks. No, not that at all.  Nope, what has occurred is that a "vast security apparatus has kicked into gear" to track what you've been doing - namely how many withdrawals made, and how much each time, not to mention deposits and dates.  If these strike the algo's flags then a SAR gets triggered.  The goal here is to crack down on money laundering, terrorism, human trafficking and the sort of financial crimes George Santos has been accused of committing (as elaborated in the House Ethics Report.)   

But often bank evictions occur because ordinary customers are simply making transactions which the algo regards as "out of character".    These algo-generated alerts are then reviewed by human bank employees.  These latter then already know they must file an SAR if they see actions or behavior that might violate a given law.  The problem is that the algo's "out of character" baseline casts too wide a net and a hundred times more innocent people get captured than bad guys.  In other words, like  bad cancer test, the system generates too many false positives. But unlike the bad cancer test, those false positives for the SAR and followup eviction generally can't be taken back, repaired.  

In one case the Times' piece examined, a guy doing a house purchase in NY withdrew large chunks of money ($7k to $12k at a time) to pay his contractor.  He was then surprised when the bank (Citi) called to ask why he was making such repeated large withdrawals.   He told the Times he just assumed the bank called to make sure no one was stealing his money.  But the next thing he knew he was evicted...no more accounts, or cards, for him.   When he actually visited the branch to inquire the frustrated manager probably "said more than he was supposed to", basically:  "Don't ask me, ask the computer that flagged you."

And therein lies the rub with use of any AI -directed algorithms which are chained to a logic loop that foregoes nuance or internal checks.  Once again, as author Ward learned ('The Loop') there’s a fast, intuitive processing system (System 1) for the brain, which solves many problems with graceful ease.   However, it can also be lured into error, and there’s a slower, more effortful logic module (System 2) which can grind out the right answer when it must. But as author Ward has noted, prefers to let System 1 do all the work and arrive at the answers.   This is exactly the problem with the banking algo:  it's decision making has been 'lured' into a simplistic System 1 loop which basically defeats its own purpose.  

How to escape the loop-based algorithmic errors which are based on excessive scrutiny?  Federal banking regulation would have to be changed first, to at least try to cut down on the false SAR positives for regular customer actions. 

Failing that, humans need to intervene (i.e. bank employees) to look beneath the algo 'hood' and see what's really caused the specific SAR to be activated. Then, if it is found to have been over reactive, to cancel the eviction. Citizens ought not be subject to the decisions of an impervious deus ex machina incapable of genuine logical reasoning and processing of causal nuance.

Covid Strikes Even After 4th Booster Vaccine - So What To Do Now?

                                                               

                                    "No! I know lots of folks got Covid from boosters!"

In truth, after getting our 4th Covid booster shots barely 6 weeks ago, i.e.

• 4th Covid Booster - Brings Most Severe Side Effects...
• 
  
•    I really believed we'd be protected again from this virus that seems like it never wants to go away  - and just leave humanity to recover from a ghastly pandemic. One which has made a mess of brains/minds as much as it has bodies. But alas, it was not to be - and even after a booster with the most severe side effects (which Janice assured me meant my immune system was kicking in) I managed to get Covid.  How I did, who knows?   But two successive tests have revealed it's there which may explain for the fatigue experienced the past four days or so.  

   The odd symptoms began Thanksgiving night after I made a small turkey sandwich.  Trouble was I could barely taste a thing. It was almost like chewing cardboard.  I shrugged it off and went out next morning in a small snow storm to get some groceries - stopping first at the nearby Burger King to partake of their 'double croissanwich breakfast special for $5.99.  This included:  a sausage, egg & cheese croissanwich, a bacon, egg & cheese croissanwich, medium hash browns and coffee.  Sitting at a table, my newspaper open, I prepared to dig in as I was fairly hungry.  But to my horror, the first croissanwich tasted a bit like damp card board - which is to say tasteless. The hash browns also tasted like warm bits of something I couldn't recognize. And as for the bacon croissanwich, I ended up tossing most of it away.  

  

   After getting back from the store with the groceries, I related to Janice the experience at the BK with the twin sandwiches I found utterly tasteless. The first thing she said was: "We need to do a Covid test!"  I felt this a bit of an overreaction and told her I had considered not mentioning the taste anomaly at all, given I felt otherwise fine - even walking in 15F temperatures to BK and the Safeway store. She assured me she was glad I did mention it and promptly retrieved two of the 2 dozen Covid tests she'd  ordered barely 6 months ago.  

  The test consisted of using a swab to insert roughly ½   to ¾  inch in each nostril, rotate it around the interior 5 times, then do the next. After which the swab was placed into a card over which a reagent was dropped. After 15 minutes the card was checked and the sample window showed a faint reddish line below the much stronger red reference line. Janice said faint or not it was positive for Covid and we'd need to test again in 24-48 hours to confirm it. Well the next day, Saturday, she did and lo and behold the result was the same: a faint reddish line below the bold red reference line. I had bloody Covid 19!   

   This, of course, propelled us into the usual domain of what to do. First step was for Janice to then do the test and hers came out negative, no red line - faint or otherwise.

t Though the bulk of experts advise isolation, there is little that can be done in regard to that in our home. Almost every living space – from the living room to family room is a conjoined, shared area i.e. with TVs, recliners, or couch etc.  The only real ‘isolated’ area is the upstairs 12 x 12’  guest bedroom-library - with my O2 machine. But being holed up there all day is a non-starter. No comfortable chairs, no working TVs and little room.

  So we have done the next best thing and separated our hand towels, dishes, utensils etc. and have sanitizer bottles all over the place to use. Say any time before I touch anything like the remotes or a light switch, or a coffee cup.  The main objective is to keep Janice from getting it.   How much longer do I have to "isolate", as in be grounded?  The same experts say 5 days, so we will see. I certainly am not feeling any symptoms say apart from a mild fatigue I experienced Saturday. But obviously the main objective here is to protect others, many of whom may not have had the boosters, or the starter vaccines.  

     

    As for boneheads (like the illustrious former Packer QB Aaron Rogers) who yelp:

     "See! I told ya so! Those boosters don't work they just make ya sick!"

       No, not at all. That fourth booster, awful as the side effects were, plus the earlier         vaccines, ensure I likely didn't end up intubated in the ER.  That was the whole            idea in getting them:  Not to prevent getting Covid, but rather prevent being                hospitalized in a critical care and intubated.  

          Anyway, just two more days and - barring any other symptoms- I can go out                again, and without any mask.  That includes rescheduling a urological                           appointment that had to be cancelled.

                  

            See Also:

           

         Why You Are More Likely to Get Sick This Winter

             Excerpt:

                  Covid-19 is settling in as a wintertime fixture, and infections are expected to rise again as the weather cools and holiday gatherings pile up. The virus is on a collision course with the seasonal scourges of flu and respiratory syncytial virus, or RSV, which are circulating again after the pandemic disrupted their spread.  The risk? More infections, more disruptions to schools, work and holidays and more strain on hospitals than before the pandemic. Covid has raised the baseline for winters to come. 

                And:

               Covid Boosters Aren’t Reaching People Who Want Them

               And:

              

      https://news.yahoo.com/over-half-people-covid-lingering-090000044.html?soc_src=social-sh&soc_trk=ma

            And:

            New Covid Variant - Cause For Worry & Why We're Getting Our Boosters NOW! 

    

Toward A Heuristic Model To Quantify A Driven Model For Solar Flares (2)

The isogauss model contours in Pt. 1, specifically the end graphic:  

Sets up the basis for a rotary footpoint driven energy system. Again the neutral line is in red and the anti-polar (+, -) magnetic flux centers are D1 and D2 now in much closer proximity disclosing a mutual polarity intrusion.  Then the total power dissipated in solar elements or flux tubes can best be expressed:

P =  S E I_i  =   S (- V X B)_i  [a _i (F_i  ) / m₀]  = -S J [_i  [a _i (F_i  )] 

where the (i) subscript refers to the ith unit element of the magnetic configuration.for each (i),  the field line configuration quantified for the force free parameter (a) and the magnetic flux (F). The specifics of treatment are based on a particular geometry, in this case shown below in 3 dimensions.

As one or both footpoints rotate, the changing magnetic flux df/dt induces V_(ind)  in the loop, and this (induced)  emf acts to reduce the current density in the loop, in this case the Hall current density magnitude. This back reaction from the induced E-field is what keeps the system at marginal stability and J_H  »J_ms, the relevant current density.  If this were not so, Lenz’s law would be violated. In like manner, the relevant velocity for assessing the two-stream instability condition must apply to V_d as the associated drift velocity, not V _q, which is responsible for the back emf V_(ind)  though it generates the Hall E-field via a dynamo action.

One can describe the loop conductor cutting magnetic lines via motion perpendicular to them.  For AR2776, df/dt  »  3.3 x 10 ¹⁸ mx s^-1  thereby setting  the stage for a two-stream instability with an electron beam comprising a tail of a distribution with a beam velocity v _b  =  Ö(2E _o / m _e).

In a paper delivered at the 40^th meeting of the Solar Physics Division of the American Astronomical Society, I noted the shear angle magnitude Dj, was taken as a proxy for the increase in relative magnetic helicity (H _r )  :

D H _r  »  Dj  =  arctan (B _z  /B _f )

where the numerator and denominator denote the axial and poloidal field components respectively, each defined in terms of the relevant Bessel function (e.g. Lundqvist, 1951):

B _z      =   B_o J_o (aR)         and          B _f   =   B_o J₁ (aR)

where J_o (aR)  is a Bessel function of zero order, and  J₁ (aR) is a Bessel function of first kind, order unity.  In general we have (cf. Menzel, 1961, p. 204):

J_m (x) = (1/ 2^m m!) x^m [1 -  x ²/ 2² 1! (m + 1)  +  x⁴/ 2⁴2! (m + 1) (m + 2) -  ….

.(-1)^j x^2j / 2 ^2j j! (m + 1) (m + 2)……(m + j) +  …]

The form of the two functions (using Mathcad plots) is shown below, where J_i = J_o (aR)   and J_2i = J₁ (aR):

                                                                             

In the next graph shown below, we see the magnetic free energy cumulatively added during the interval of interest, as well as the spike showing the greatest flare energy release (this is for SID or geo-effective flares). Because the dotted line discloses cumulative energy in the region, it also shows loading of energy into the region – probably as a result of negative helicity injection via currents and shearing over the interval of maximum interest (Nov. 7-8).

 

The energy release and MFE accumulation profile for SID-producing flares over Nov. 5 – 8, 1980 for Mt. Wilson region 21862

By Nov. 7, for example, it could be inferred from the graph of the energy profile that almost 4.0 x 10 ²⁴ J had been ”loaded” into the active region- delta sunspot group complex. On Nov. 7, approximately 3.7 x 10 ²⁴ J had been released, leaving a residue of  » 3.0 x 10 ²³ J in the attendant field as MFE This increment would be available to add to the next (» 1 d.) accumulation of MFE, say over 7-8 Nov. By Nov. 8 (median Julian date), an additional » 1.2 x 10 ²⁴ J of MFE has been added. This would have been 4.9 x 10 ²⁴ J had the earlier flare energy not been released.

Is the process driven? We first note from the graphic below - with the actual portion of the Mt. Wilson magnetogram at left:

 That ¶B(x) effectively crosses four contours, or (4 x 250 G) = 1000 G. (Each increment of 0.25 along “Delta Y” or “Delta X” in the Mt. Wilson vector magnetogram denotes a change of 250 km while each contour difference is a separation of 250 G.)  The B-vector direction has been estimated as shown, and the components B(y) and B(x) can be worked out, along with the changes, e.g. ¶B(x)/ ¶y  and  ¶ B(y)/ ¶ x, say from one iso-contour to the next. 

The separation dy (or ¶y ) from the vertical axis amounts to » 3.2 unit(s) or 3.2 x 250 km » 800 km. Thus:

¶B(x)/ ¶y  =   (1000 G)/  800 km  =   1.25 G/ km

In a similar way, we find for ¶ B(y)/ ¶ x:

¶ B(y)/ ¶ x  » (1000 G) / (1.4 x 250 km) »

 1000 G/ 350 km »  2.8 G/ km

Then the current helicity density is:

H _Z (c)  = [¶B(x)/ ¶y  -  ¶ B(y)/ ¶ x] B _Z    

 » [1.25 G/ km  -  2.8 G/ km] 350 G 

H _Z (c)  »   - 542 G ² / km

The negative sign indicates that the force-free parameter a is also negative, and will have magnitude:

a  »  -[¶B(x)/ ¶y  -  ¶ B(y)/ ¶ x] /   B _Z       

» - [1.25 G/ km  -  2.8 G/ km] / 350 G 

a  »  - (1.55 x 10 ^-3 m ^–1) G /350 G   »  -4.4  x 10 ^-6  m ^–1

By “hemispheric helicity rule” the sign fixes the region in the Northern solar hemisphere.

We know: a  =  m_o J _z / B _z

Then:   B _z  = m_o J _z /  a

(Where: m_o   =  4p x 10^-7 H/m)

Solving for the magnitude of the magnetic field  z-component:

B _z  = (4p x 10^-7 H/m)( 0.012 A m^-2   ) / 4.4  x 10 ^-6  m ^–1

B _z  = 3.4 x 10 ^-3  T

Along with assumption of Coulomb gauge, we can write:

m  J _Z   =  - Ѳ A   =  l f (A,y) 

wherein we finally see the appearance of the shear parameter l f (A,y) which is directly proportional the vertical current density J_z  = J (A,y) and thence to the current helicity density H _Z (c)   we already deduced from the isogauss contour. Given the basic (heuristic) model includes both energy accumulation and dissipation it is concluded it conforms to the driven paradigm.

 In general, l f (A,y) can be interpreted as a generalized, quantitative index of the magnetic free energy, and vertical current density, stored relative to the potential or current-free force-free field. (For which Ñ X B = 0). In many cases, l f (A,y) will be a differential angle of shear (e.g. D f) , which measures the departure from the potential state.

 Whatever the physical form, when l f (A,y)  >   l _c  (l _c  a critical angle relative to the magnetic inversion line) then the field should be poised for explosive energy release. In other words, the quasi-static development and configuration is abruptly terminated so we get dissipation of the accumulated magnetic free energy.

(The magnitude of energy released as actual flares can  also always be obtained by integrating the area under the power-time curves of specific flares as obtained from the SMS-GOES soft x-ray (1 – 8 Angstrom) record. )  

See Also:

• Looking at Bessel Functions & Applications

And:

New Solar Research Confirms Why Delta Sunspots Are More Flare Worthy Than Other Magnetic Classes