Friday, December 29, 2023

Solutions to Lens Problems

The Problems:

1) The objective lens of an astronomical telescope has a focal length of 6 ft. The eyepiece has a focal length of 2 inches.

Find the angular magnification that the telescope will produce when used for distant objects.

(2) A telephoto lens consists of a converging lens of focal length 6 cm placed 4 cm in front of a diverging lens of focal length (-2.5 cm).

a) Do a graphical construction of the system showing where the image would be.
b) Compare the size of the image formed by this combination with the size of the image that would be formed by the positive lens alone.

Solutions:

(1) The magnifying power is defined by:

m = F/f(e)

where F is the focal length of the objective (the main or front lens) and f(e) is the focal length of the eyepiece.

We have F = 6 ft. = 6 x 12 in. = 72 in.

f(e)  = 2 in.

Then:   m = F/f(e)  =  72 in./ 2 in.  = 36x

(2) The graphical construction (diagram) for the problem is shown below:

The image AB that would have been formed by the converging (positive)  lens alone is:

(6 cm - 4 cm) = 2 cm beyond the f = (-2.5 cm) lens and is taken as the virtual object for that lens. Then:   s1 = -2 cm, and:

1/s1' = 1/f - 1/s1 = 1/ (-2.5) - 1/(-2) = -1/2.5 + 1/2 = 1/10

Then: s1' = 10 cm

Thus, the final image A'B' is real and 10 cm beyond the diverging lens - as the graphical construction shows.

The linear magnification: M1 = (-s1'/ s1) = (10 cm/ 2 cm) = 5

and since, h'/h = 5, then h' = 5h so the image formed by the combination is 5x larger than that formed by the (+) lens alone.

Revisiting Spectral Line Formation and Transitions (3)

As discussed in Part 2 (12/11 post) we can express the equivalent width in two ways, based on frequency  (u)  or wavelength (l):

W =   ò ¥0  (I c – I /  I c )du =  ò ¥0  (F c – F / F c ) dl

The left side defines W in terms of the intensity e.g. from the continuous spectrum outside of the spectral line where the quantity (I c – I u/ I c ) is referred to as the “depth of the line”.  This is the analogous quantity to (F c – F / F c ) on the right side where we have radiant flux units.

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 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 Iu / dτ u  =  -  Iu – S u

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

d Iu / ds   =  -k u Iu  +   k u S u

Where:   k u   =   d τ u / ds

So:

d Iu / ds   =  -( d τ u / ds) Iu  +   (d τ u / ds) S u

d Iu =   - d τ u  Iu  +   d τ u  S u  =   - d τ u   (Iu – S u)

whence:

d Iu / dτ u  =  -  Iu – S u

For which the solution is found (on integration):

Iu =  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 newly derived information, the equivalent width of the line can then be written as:

W uò ¥0    [1  -   exp (- N L a o  f u) ] du

For small x we may use the approximation such that:

exp (-x) »   1 – x

So rewrite the equation for  W u:

W u  »    ò ¥0     [1  -  (1 - N L a o  f udu

»    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  »  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 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:

Suggested Problem:

How would the curve of growth appear if the broadening function:

f u  du  1/ Öp   [exp (u  - u o)  /D uD ]2  du / D uD

Thursday, December 28, 2023

WSJ's McGurn Bids Us Consider Another Econ 'Genius' Advocating Overpopulation. No Thanks!

William McGurn ('Musk Says Make More Italians', p. A15, 12/26) , WSJ over-population troll, asks us to believe "billions more" people can be supported on this fragile, limited planet. This time citing an unheard of former Harvard prof  Catherine Pakaluk-

To support his malarkey. She basically agrees with the deranged fruitcake Musk and his billionaire pal Bezos about birthrates, i.e.

Jeff Bezos, Elon Musk: Human population not nearly big enough | Fortune

Musk himself babbling like the fool he is: "If we had a trillion humans we'd have a trillion Mozarts".  Not quite, Einstein.  We'd have maybe two Mozarts,  100 billion grifters like Trump and 900 billion Jeffrey Dahmers - who'd realize they'd have to cannibalize other humans to remain fed - given the lack of arable land because of 1 trillion humans occupying it!

Pakaluk herself then notes that countries have learned that while it’s possible to drive birthrates down, "it’s much more difficult to drive them back up once the decline has started."

Thank goodness!

McGurn then spouts:

"The left loves gunking up the tax code for its pet preferences, such as discouraging fossil fuels. "

Well, yeah,  "discouraging fossil fuels", because those are what's going to turn this planet into an uninhabitable hothouse dumpster fire!  But that's not "gunking up",  it's using the tax commons to promote self-preservation. Or isn't McGurn in favor of that? As reported in a recent release on the mounting threats of a nearing climate tipping point:

"We are witnessing the first stages of civilization’s collapse. The question is whether we will respond in time to prevent it from becoming irreversible.

Extreme weather events, such as heat waves, droughts, floods, wildfires, and storms, are becoming more frequent and destructive due to climate change. These events can cause human suffering, damage infrastructure, disrupt supply chains, and increase health risks. They can also have cascading effects that amplify their impacts and create new threats.

For instance, a new report from an international team of scientists suggests that feedback loops could push global temperatures into a ‘hothouse Earth’ state, where the climate remains hotter than pre-industrial levels even if human greenhouse gas emissions are reduced to zero. This could lead to sea level rise of up to 60 meters

We have a choice: we can either face the reality of climate change and take action to prevent or mitigate its consequences; or we can ignore or deny it and face the collapse of civilization as we know it."

Evidently McGurn and Pakaluk didn't get the message or want us to ignore it, opting to stumble on toward certain oblivion.  This despite the most hidebound, dogmatic ignoramus should know the Earth can't support billions of additional humans, e.g.

Nonetheless he blathers on:

"Some on the right now want to get in on the act by rejiggering the tax code to reflect right-wing priorities. The real issue, Mrs. Pakaluk says, is how moms regard opportunity costs. The women she interviewed are aware of the trade-offs involved in having more children. Like Mrs. Pakaluk, many have full-time careers. The difference is that they seek to accommodate their careers to motherhood—not the other way around."

And McGurn conveniently assures us she's not calling for women to be just breeders, i.e.

"She’s not calling for every woman to raise 14 children, as she has. There’d be a big improvement if women who are open to more children just had a second or third.”

"Even so, the condescension toward women who choose more children remains."

As it should, given every additional child added to the global total is now effectively an added nail in humanity's coffin - in terms of the fossil fuel energy they will consume - adding to the existing CO2, as well as the pollution itself- from plastics, to PFAS chemicals, not to mention draining the last reserves of potable water before we even get toilet to tap systems running.  And paving the way for water wars in many nations, e.g.

This is why the late science writer Isaac Asimov said as long ago as 1976 (in a Barbados lecture):

"It is now the willingly childless woman who is the heroine of our planet. She is the one who now deserves all the kudos and praise, for helping to do what is necessary to spare humanity from the ravages of over-population"

And yet McGurn even cited Pope Francis in his WSJ piece:

"Who once crudely said Catholic women don’t need to breed “ like rabbits.”

But too many do, because artificial contraception isn't allowed as an option. Of course, now the narrative has been switched by the ardent religionists - like Pakaluk (who is also an economist) to the "crime" of not generating enough productive humans to support the economy.  So McGurn wastes no time trying to lampoon Paul Ehrlich’s The Population Bomb(1968), writing:

"For decades, the orthodoxy on this subject was found in Ehrlich’s “The Population Bomb”, which portended global cataclysm unless population growth wasn’t suppressed. Women were told they were threatening the planet by having children. Now the fear is reversed."

Huh? How reversed? Ehrlich was correct when he said (in a 1970 broadcast) that  humans face:   "An utter breakdown of the capacity of the planet to support humanity,"  This is best  illustrated  in the concept of Earth overshoot, as embodied in the graphic below which shows humans are currently consuming the equivalent resources of 1.6 EARTHS per year, e.g.

The interpretation of the graph (upward) is straightforward. By June, 2030 TWO full Earths - that is the resources therein - will be needed to support the then population. Already we are at 1.6 Earths

Oh, But leave it to Ms. Pakaluk to enlighten us on the nature of the reversion:

Total fertility rates below replacement lead inexorably to a shrinking population, economic stagnation and lower quality of life.  We won’t have the people to staff valuable services. Social Security and Medicare will be shuttered. Many countries are already on this path. It can only be avoided if a nation can compete, indefinitely so, for a larger share of a shrinking number of world immigrants.”

Okay, this needs to be addressed in two parts: her first sentence tying shrinking human numbers to lower life quality, then the rest about mustering sufficient labor to supply needed services.  Let's take the last first. I have already written at least a half dozen posts that the solution is to permit and facilitate hundreds of thousands more H2B (and other visas) so migrants flooding in can work here, e.g.

One Way To Relieve Supply Chain Labor Shortages: Allow Foreign Workers In As Permanent Residents

Instead of letting migrants stagnate by the thousands at the border, let them in after processing visas to enable them to work!   This addressed Pakaluk's last point on rendering needed services.  Thus, churning out more visas will enable the migrants to work in the jobs Americans don't want: caring for elderly in nursing homes. nursing itself - at hospitals in need (of which there are many), and agricultural jobs from harvesting peaches, oranges and apples to sorghum and corn.

As for her claim that lowered population leaves us open to economic stagnation, not at all. Dr. Jane O'Sullivan, a population and demographics expert, skewered that hackneyed trope. She wrote the landmark paper:

“Demographic Delusions: World Population Growth Is Exceeding Most Projections and Jeopardizing Scenarios for Sustainable Futures"

questioning the assumptions made by institutions, and global agencies (like the UN) concerning population projections. As well as co-authoring:

“Advancing the Welfare of People and the Planet with a Common Agenda for Reproductive Justice, Population, and the Environment,”

In answer to one question regarding the attacks of economists on population control she responded:

"The only way it makes sense to me is as a deliberate campaign of misinformation. I’m not suggesting that everyone involved is insincere, but the whole discourse has been carefully groomed to undermine the support that was once so widespread for population stabilization. The incidents of human rights abuses in China and India have been used to tar the whole family planning movement with the same brush.

A number of groups with different agendas have contributed to it. Anti-Malthusianism3 has always been a strong tenet for some leftists influenced by Karl Marx and Henry George4, but it equally suits global corporate elites who want ever more production and consumption to grow their enterprises, and ever more cheap labor to keep profit margins high."

The latter, of course, has always been a goal for rank capitalists like McGurn as well as his fellow population pushers in the WSJ op-ed stable. So no surprise he ends his piece blabbering:

"Many are not going to like Mrs. Pakaluk’s solution, which is a civil society thick with religious institutions that are the most likely to share the values these women place on motherhood. But she makes a powerful case that nothing else has worked or is likely to work. So the next time Mr. Musk talks about population decline, he might want to bring along this economist and mother."

But before doing that, Musk and Pakaluk might wish to consider Isaac Asimov's words:

"As you put more and more people into the world, the value of life not only declines, it disappears."

by Stan Cox | June 16, 2023 - 5:34am | permalink

And:

Prof. Albert Bartlett Skewers The Trope That "The Total Global Population Is A Meaningless Number"

And:

Mass Global Migration Will Never Be Stopped By Laws - Because It's Fueled By Overpopulation

And:

"To Breed Or Not To Breed?" The Answer Ought To Be Obvious By Now!