Monday, December 2, 2024

The Plasma Physics Basis of Radio Astronomy - Redux

Radio astronomy entails a number of aspects associated with plasma physics, which I want to examine in a series of posts. One of these is  Faraday rotation, which involves the rotation of the position angle   q  of the linear polarization of radiation passing through a magnetized (B) plasma. If  the plasma has magnetic induction B and subtends an angle f between B and the direction of wave propagation, then a measurement of q at a wavelength l permits a determination of the total number of charged particles in a column of 1 m 2 cross section  between source and observer. This is given by:

N t =  òr o   Ndr  

Where N  is the total number of charged particles in a column of length r and cross section 1  2.  We note that the natural modes for this polarization are circular as shown in the diagram below, i.e. for 'wave 1' and 'wave 2'  with respective position angles q1  and  q2.  



Then the 2 circularly polarized waves (or opposite rotation) represent the resolution of the linearly polarized wave.  If the two circularly polarized waves have different phase constants (b  and  - b)   then the plane of polarization of the resultant linearly polarized wave rotates as the wave propagates.

Aside: Electromagnetic waves are  polarized when their E- field  components are preferentially oriented in a particular direction.

In linearly or horizontally polarized waves the E- vector is confined to one  (horizontal) plane, i.e.


---------à E

In Vertically polarized waves the E- vector is confined to one  (vertical) plane

E

!
!
!
 
In circular polarized waves the E-vector rotates through 360 degrees

Elliptically polarized waves means  any polarization not circular or plane.

An important consideration in any plasma physics context is the index of refraction for the plasma.  This also has consequences for radio observation access.  In the absence of a magnetic field we can write for the index of refraction of the plasma:

n 2 =   1   -   w e 2 w 2

And  w e     =  [ne e2/ me  εo½

is the electron plasma frequency.   We can then write the equation applicable for a magnetic field in the following abbreviated form:

n 2 =   1   -   X /  ( 1 +  Y)

For the ordinary wave: n o 2 =   1   -   X /  ( 1 +  Y)

And for the extra ordinary wave: n x 2 =   1   -   X /  ( -  Y)

(Where:  X  =   w e 2 w 2     and   Y  =  w H 2 w 2)

Where w H   denotes the hybrid frequency, which is a mix of the electron plasma and electron cyclotron frequencies, e.g. 

w H 2 =    w e 2 +   c 2        ( c   =  qB/ m e)


Note the ordinary wave occurs when the EM wave components  E 1 and B o  are parallel, e.gE 1    B 

The extraordinary wave occurs when an EM wave (e.g. radio wave) propagates partly transverse, partly longitudinal.  I.e. propagates perpendicular to  B o   with  E perpendicular to  B o.

 It is important to point out that in dealing with magnetically-affected radio waves we expect cutoffs and resonances.  I.e. as a radio wave propagates through a region in  which w e  and  c are changing it may encounter cutoffs and resonances.  A cutoff occurs in a plasma when the index of refraction, n, goes to zero.  That is, when the wavelength becomes infinite.  So if the wave number vector k =  2 p/ l   and the mean index of refraction is: n  c k / w 

Then when n ->  0:

 c ( 2 p/ l) / w

Or:   2 p/ l   =   (0)/ c   or:  l   =   2 p c / 0  =   ¥

Conversely,  a resonance occurs when the index of refraction becomes infinite, i.e. when the wavelength  becomes zero, so that:   

n =  c ( 2 p ) / w

Or:  w = c ( 2 p ) =  2 p c / 0 =   ¥

Thus, for any finite w,  k  ->  ¥   implies  w  ->  w H

So that the resonance occurs at a point in the plasma where:

w H 2 =    w e 2 +   c 2     =  w 2  


The conditions for cutoff and resonance can be written in terms of the ordinary frequency, f (i.e.  w = 2 p f) referenced to electron and cyclotron frequencies.  So that:

The extraordinary wave is cut off when:

f e 2    f x  ( f x   f H )  =   1

The ordinary wave is cut off when:

f e 2    f o  ( f  +  f H )  =   1


Referring back to the plane of polarization being Faraday-rotated (as illustrated above) we note the measurement can be done by means of a microwave 'horn' -  a component of many radio telescopes.  This horn gives a value for w e 2  and hence of the number density of the plasma. 

Other inferences which can be made regarding ordinary and extraordinary waves:

1) If   f  > f    then both indices of refraction are real and both modes are propagated.

2) If  f  > f >    then only the ordinary wave propagates.

3) If  f  >    then no waves are propagated.  The magnitude of rotation in radians will then be equal to:

2.38  x  10    [ 1  f 2   ò n e  H  dz ]

Where dz is the thickness of the plasma through which the radio wave is moving,

n e    is the number of electrons per cubic centimeter

H is the magnetic field in gauss.  Faraday rotation has been found important at the very low densities of the interstellar medium and has been used to explain the polarization of microwave radiation generated by maser action, e.g. on OH and H2O molecular clouds.  (See also my Dec. 16, 2021 post on Cosmic Masers.)

Suggested Problem:

1)  A space plasma with electron number density  10 12 /cm 3  features a magnetic field of 0.0001 T (Tesla).  Find:

 a) the electron plasma frequency, 

b) the cyclotron frequency and c) the hybrid frequency.

c)Thence or otherwise find the refractive index of the plasma and whether ordinary or extraordinary wave propagation can be expected.



Forget The Reepo-Right Hysteria- - Biden Was 100 Percent Correct To Pardon Hunter


       NY Post headlines that began the Hunter Biden witch hunt in Oct. 2020

                                 "That's the pair that wants to break us, dad!"

"No U.S. Attorney would have charged this case given the underlying facts. After a 5 year investigation the facts as discovered only made that clear. Had his name been Joe Smith the resolution would have been - fundamentally and more fairly - a declination. Pardon warranted."  - Former Obama Attorney General Eric Holder


Using his executive authority in the waning days of his presidency, President Joe Biden lifted the pernicious legal cloud that has hung over his son  Hunter for several years and granted him a pardon. Of course, the move burned the asses of the Trumpkins and the orange maggot himself – who escaped prison time (based on his 34 felony count convictions in the hush money case) thanks to millions of brainwashed voters giving him another term in office. My take, for what it's worth, Joe Biden could see how Trump and his Tribe of renegades would turn his son into an emblem of infamy to totally paper over his own crimes. As well as his planned "retribution" on the prosecutors - like Jack Smith -who sought to make him accountable. 

Hunter became the victim of unmoored Reepo lawfare after being targeted by a Trump state AG holdover (David C. Weiss) on a tax, drug and gun charge. But which was actually repackaged and re-directed vengeance on those who prosecuted Trump for his roles in the insurrection and the documents cases. And the President (Joe Biden) who allowed it.

As President Biden accurately put it:

No reasonable person who looks at the facts of Hunter’s cases can reach any other conclusion than Hunter was singled out only because he is my son — and that is wrong. There has been an effort to break Hunter — who has been five and a half years sober, even in the face of unrelenting attacks and selective prosecution. In trying to break Hunter, they’ve tried to break me — and there’s no reason to believe it will stop here. Enough is enough."

Enough indeed. But look for the Reepo Trump asslickers to bellyache and bitch now about how Joe "went back on his vow” not to pardon Hunter. Failing to recognize that political dynamics change, especially after a lawless monster has been incomprehensibly hurled into the highest office by numbnut voters who don’t know their asses from holes in the ground.  As if they had totally forgotten what this miscreant did on January 6th, 2021.

The persecution of Hunter began at the end of October, 2020, including via a faked up story (See top graphic) in the NY Post-  Rupert Murdoch’s low class ‘cousin’ to the Wall Street Journal and which I skewered in an Oct. 30th, 2020  post, writing:

the potential damage of this farce is even worse, since we know the GOP, pro-Trump traitors will latch onto it as a device to nullify and cross-check the January 6th committee investigations. (Which the equally bogus Durham probe into the Trump cabal investigators is already doing, e.g

How Durham's Sussmann Investigation (And Show Trial) Feeds The Lie About The "Russia Hoax"

This cooked crappola  was later found to be an intel hit job likely by foreign actors. Why, after all, did the lead author actually pull his byline before it came to print?  Well, because he knew it was bogus. See e.g.

The fuel for this latest Hunter Biden BS has been stoked by another Trumpie holdover in the DOJ - David C. Weiss - the U.S. Attorney for Delaware who initiated the "investigation".   Incredibly,  Garland's DOJ planned to ask for the resignations of other Trump holdovers but planned to keep Weiss for the Hunter Biden investigation. Why?  Why was Merrick Garland more invested in this than getting a head start on the Trump federal prosecutions (e.g. Jack Smith's cases) which would have obviated the long, drawn out Supreme Court rulings that basically scuttled his efforts? 

Traitor Trump, meanwhile,  posted on social media  Sunday night calling the pardon “an abuse and miscarriage of Justice!”  This fucker has mega Maga chutzpah to talk, given he’s a convicted felon 34 times over and ought to be rotting in Rikers Island now – not nominating absolute Krappercrats (like Kash Patel and David Hegseth) to high office.

The truth – as Chris Hayes pointed out on All In barely two months ago- no one is prosecuted for the conduct engaged by Hunter. This was a witch hunt started by Republicans to break the President and also exact retribution for Trump’s impeachments and the federal insurrection (and documents) cases against him.  Which thanks to 6 corrupt Supremes, gave him a get out of jail card via selective immunity. 

The pardon was right. No less a personage than former AG Eric Holder pointed out that the pardon stands on its own merits as a legal matter. It also featured the classic components of a legitimate pardon:

- Non -violent crimes

- No one has ever been sent to prison in DOJ history for them

The first was a paper crime predicated on a false answer for a gun purchase.  But as former federal prosecutor Andrew Weissman noted, in his 20 years of prosecutions he's never seen prison time given to a confirmed addict giving a false answer on a gun purchase.  Instead treatment is sought for the person and at most, a deferred prosecution (or none at all).   

In respect of the tax charge, that was also blown out of proportion for political purposes.  Yes, Hunter bungled a tax filing - but as  Andrew Weissman noted he paid the taxes back in full. In an appearance on 'Last Word', Mr. Weissman was especially aghast at how so many - including in the media - have lost their moorings.  They have gone ballistic over Biden's pardon of his son, Hunter, but remain essentially muted over a convicted felon elected president- and who has also nominated other convicted felons and reprobates who would "man the agencies keeping us from another 9/11."

Another pertinent aspect regarding false analogy  Lying about expenses on a personal tax return, or drug use on a permit for a gun that was never used, isn’t in the same league as Trump inciting a violent mob to attack the U.S. capital. Nor is it anywhere near Trump asking a state official to find 11,000 votes to rig an election, decades of business and financial fraud, a jury finding of sexual abuse and defamation, or staggering emoluments from foreign governments.

But this is the level of mental, psychological degradation that's occurred in the wake of the pandemic, amped up by Trump's ceaseless lies and FOX News endless brain-fucking propaganda. (Which now evidently even more Muricans have tuned into, abandoning sane sources like MSNBC.)

The irony is that republicans are bitching about what they’d usually consider run amuck “lawfare”.   Ah, but that only applies to Donnie Dump's cases, right?  Bottom line, Merrick Garland should never have appointed a Trump U.S. Attorney as special counsel. He did it with Hunter, to try to appear objective, politically neutral. It blew up in his (and Joe Biden's face)given Trump and the Reptiles still went on the warpath. Nor should he have let Barr's 'hand me down' John Durham go after the legitimate prosecutors – wasting over $4 million to come up with bupkiss:

Brane Space: Durham's Conspiracy Clown Goose Chase Finally Gets Upended With Michael Sussmann Acquittal

The sad and disgusting aspect now will be that Joe Biden will be blamed by the incoming rats (and many in the clueless media) for "perverting our justice system" when in fact the Supreme Court did that on two consecutive occasions:  First, dropping the justified 14th amendment case (which began here in Colorado) that disqualifies an insurrectionist from the presidency.  As Donald Sherman, executive director of a progressive watchdog group has noted:

 "There are some questions, like whether an oath-breaking insurrectionist should become president of the United States, that the Constitution has already answered."

In effect, making it redundant to ask voters to make that determination "with whatever information they have".  In too many cases, burdened by low information or with misinformation. (E.g. from FOX News).  As Sherman also noted, the case was not about one candidate but about the rule of law in a constitutional democracy.  Ditto for the second Supreme Court misruling: granting immunity for "official" acts while letting Trump define the nature of "official". 

I'd love to see just half the media frenzy that's erupted over the Hunter pardon, directed at the millions of dumb shit Muricans who voted a convicted felon, mega liar and traitor insurrectionist back into office.  Whose aim with his kakocratic (after kaka or shit-based)  picks is to destroy the rule of law not just damage it. As for any sob sister Democrats crying that the pardon “undermines public trust in the rule of law.”   Hello, birdbrains! Trump and the Supreme Court already did that. Wake the fuck up and stop the hand wringing. That ‘horse’ already left the damned barn!  See e.g.

Maddow Defends Hunter Biden Pardon: You Too May Change Your Mind If Incoming FBI Director Had A "Hit List" | Video | RealClearPolitics

So, of course, I have to agree with Donald Sherman and others when they stated (Denver Post yesterday): "The national vote surprised us with people knowing what he did on January 6th and still voting for him."

In other words, how in the hell can you award power to a traitorous pissant who sought to overthrow an elected government and lie about it? What the hell were those voters thinking? Or were they even capable of thinking?  One recalls here the Harvard polysci prof who opined after the 2016 Trump win that the largest demographic were "low IQ".   Also reiterated by a Cambridge prof (in The Financial Times) before the Nov. 5 election, opining that he didn't believe the American electorate "had the intelligence" not to put Trump back in office.  Well the prof was proven absolutely correct. He was freakin' prescient.

Think I'm over reacting? Well, process these post-election results from a Canadian paper:

COLUMN: Americans electing Trump like turkeys voting for Thanksgiving - Orillia News

  1. poor people voted for billionaires’ tax cuts
  2. immigrants voted for mass deportations
  3. seniors voted to gut social security and “Obamacare”
  4. men voted for their wives and daughters to possibly die from miscarriages
  5. value shoppers voted to raise duties on Chinese goods, which will increase Walmart and dollar-store prices
  6. police voted for a convicted felon
  7. “law and order” advocates voted for a convicted felon
  8. religious men and women voted for a man who often mocks religions
  9. “family values” people voted for a twice-divorced philanderer and adulterer
  10. taxpayers voted for a man apparently proud to pay little or no income tax

In an election like no other, many Americans behaved like turkeys voting to establish a new feast to be called “Thanksgiving.”

I rest my case that Janice is correct, and this nation's voters are a "pack of uneducated, ignorant morons."

But given this misfire of democracy, as bad or worse than the 'good Germans' electing Hitler and his Nazis into power in 1933, the onus is on the rest of us to protect the Republic any way we can.  Joe Biden did it by refusing to allow his son to become a permanent Political 'Pinata' for Republican-Trumpian savagery and dereliction of jurisprudence. For that he merits kudos not condemnation.

On that note here's a selection of comments from readers of The Washington Post:

What father wouldn't? Anyone would hope that their father would this for them given the facts. And don't forget please that Biden already lost one son through tragic circumstances.

Why would Biden leave his son hanging out there in the wind when trump has announced that he is on a revenge tour? Biden did the right thing.

President Biden did exactly the right thing. Hunter was only prosecuted because he was President Biden’s son. Otherwise as others have been, just a fine.

Cue the hypocrites who just voted a convicted criminal into office crying about "two systems of justice in this country 🤪🤪🤪"

Idiots.

This is a relief. I have no confidence any justice will be done once *45 takes over. This was well done. Most of the charges Hunter faced would have been dismissed in a sane world, but the GOOps turned it into an act of political revenge and persecution. He made restitution. Meanwhile we just ignore the serious actual crimes committed by Trump and his crime family.

The GOP persecution of Hunter was nothing more than mean-spirited psychological warfare. Joe just gave it all the respect it deserved.

GOOD!

F Trump!

F ALL Republicans!

IN

YOUR

EYE

DonOLD!


 See Also:

Mehdi Hasan slams congressional Dems: 'This is why Republicans win'

And:

Opinion | With pardon of Hunter, President Biden chose to be a father first - The Washington Post

And:

by Ken Carman | December 6, 2024 - 9:56pm | permalink

    Truly, I repeat: I DON’T CARE.
    If Republicans hadn’t still be making noises about prosecuting him to the ends of the Earth...
    If Donald Trump hadn’t been on a revenge tour during his run to get reelected, and obviously doing anything to get at the Biden family.
    If Trump and Company hadn’t interfered in the deal that had already been made in court.
    If they hadn’t made absurd demands which contributed to ruining that deal.
    If Donald Trump hadn’t run on pardons for murderous insurrectionists.
    If Republican advocates hadn’t spent so much time fighting against applying the same laws to their own, the same laws they have fought to apply to the extreme because anything to get at the Biden family was their fanatical goal. Like a "gun" crime they would normally want dismissed except, well, "Biden."
    Hunter was just the latest in a LONG list of right wing/Trumpian jihads. Yes, Biden promised over and over not to pardon him. I don't care. Trump's list of broken promises and false claims, but Biden must not change his mind. &#%! that.
    I'm sure since Trump "won" Biden would hope it would die down. Instead now they are emboldened, but I'm not sure either would have made a difference. The petty little dictator soon to be would want the jihad(s) to continue, even intensify. Now Biden is thinking of pardoning all his intended targets.
    PLEASE JOE!

» article continues...

And:

The people Joe Biden should have pardoned along with his son

And:

by Carl Gibson | December 5, 2024 - 7:00am | permalink

— from Alternet

Throughout the course of his third bid for the White House, President-elect Donald Trump repeatedly promised to use the full powers of the federal government to prosecute his political enemies. Now, President Joe Biden appears to be taking those threats seriously.

According to a Wednesday report by Politico's Jonathan Martin, Biden is now weighing preemptive pardons for several people Trump has directly targeted, including Rep. Adam Schiff (D-Calif.) — who won California's U.S. Senate race in November — and former Rep. Liz Cheney (R-Wyo.), who was vice chair of the House Select Committee on the January 6 Attack.

Martin wrote that Biden was "deeply concerned" about the fate of the people Trump has singled out, particularly after the president-elect nominated Kash Patel to be the next potential FBI director. According to the Hill, Patel has referred to Schiff — who led Trump's first impeachment effort — as a "government gangster," and has accused Cheney of being "the main architect of this disinformation campaign."

» article continues...

And:

by Jaime O’Neill | December 2, 2024 - 6:14am | permalink

I’ve written at least one other column like this in response to the question asked in the title. The question returns to mind in times like these when what seems so obviously and inarguably good is once again rejected by so many of our fellow Americans.

But seriously now, how can millions of voters think it good to deny rights to half of our species, the half that gives birth to us all, who changed our diapers, washed our clothes, tended to us when we were sick, and worried their hair gray and their faces wrinkled with their caring?

How can the wanton acts of meanness and cruelty that included wresting babies from their mothers’ arms at the border be sanctioned by a majority of Americans? How can losing track of those children be acceptable to people with whom so many of us attend churches or who pray to the same god?

» article continues...

Mensa Brain Twister Geometry Solutions:

 1) The Rectangle Problem:



The height of a rectangle is 81 units. Two circles are inscribed in the rectangle as shown above. The circles are tangent to each other. A line segment has an endpoint on both circles and passes through the point of tangency. It measures 56 units, and is parallel to the height of the rectangle. What is the length L of the rectangle?

Solution:

Reconstruct to obtain:


Then let R be the radius of the larger circle and r be the radius of the smaller one.

Let x be half the chord in the larger circle and y be half the chord in the smaller,

Then from the original dimensions assigned:  

2x + 2y = 56   ®  x + y = 28 (equals one leg of the yellow triangle)

Let z be the other leg of the yellow triangle. Then:

R = x + y + r = 81

R + 28 + r = 81    ®  R + r = 53 (equals the hypote4neous of yellow triangle)

Then by Pythagoras' theorem:  

28 2  +  z2  =  532   ®   z2    =   532   - 28 2   = 2025

So:  z =  (2025) 1/2  =   45

Then the length of the rectangle is:

R + z + r =  53 + 45 = 98

1) The Square Problem:   



ABCD is a square. W is the midpoint of AB.  Z is the center of the square where AD and BC intersect. X is the point where AD and WC intersect. Y is the point where BC and WD intersect. 

If the area of square ABCD is 4 square units, then find the area of the orange region WXYZ.   y= -x

Solution:

AWZ  is one half of one fourth of the square. (I.e. One half of triangle ABZ The other fourths are the triangles:  ACZ, DCZ and BDZ)   Therefore the area of AWZ =   (1/2) x (1/4)  x 4 =  1/2

The  area of WXYZ = 2x the area  of  WZX where

WZX  = AWZ -  AWX  = ®  -  AVX  -  VWX

Now, if A is the origin of the coordinate plane:

The slope of AX is -1    ®  y  = - x

The slope of XW is  2     ®  y  = 2x  - 2 

Then AX and XW intersect where:  -x =  2x - 2

Then: 3x = 2  ®  x = 2/3

The area of AVX = (2/3) (2/3) (1/2) =  4/18

The area of VWX = (1/3) (2/3) (1/2) =  2/18

The area of WXYZ  = 2 [ 9/18 - 6/18]  = 2 [3/18] = 6/18 = 1/3