A Navigation / Geometry Problem

Tags

, , ,

I had the pleasure of helping lead a field trip for 9th grade Geometry students at School Without Walls SHS that we call ‘Math on the Mall’ assisting with two colleagues from the SWW math faculty.

One of our goals is for the students to see how beautifully and geometrically this city was laid out by Pierre l’Enfant, Andrew Ellicott, and Benjamin Banneker about 230 years ago.

While there are lots of myths written and repeated about Banneker’s actual contribution, the fact is that he was the astronomer, who was responsible for determining due north, exactly, and the exact latitude and longitude of the southern tip of the original 10-mile-square piece of land. With no Internet or SatNav or even a telegraph or steam engine, but with a very nice refractor and highly accurate clock that he was entrusted with, but with no landmarks to measure from, he was able to do so, in 1790.

I was sad to see that exactly none of the students know which way was north – in a city where the numbered streets near the Mall and the rest of DC’s historic downtown were almost all laid out perfectly north-south, and the streets whose names begin with letters or words like ‘Newark’, and the streets along the Mall, are all laid out perfectly east-west. Very few of them had ever seen the Milky Way, though most had heard of Polaris or the North star.

Hopefully they will remember that in the future as they do more navigation on their own in this great city.

I challenged them to try to figure out why the angle of elevation of the North Star is the same as their latitude. Here is a diagram illustrating the problem:

The Earth, Polaris, and You.

This diagram is intended to help you understand why the North Star’s elevation above your horizon always gives you your  latitude (if you live north of the Equator.

The big circle represents the Earth. The center of the earth is at E. The equator is AD.

YOU, the observer, are standing outside on a clear night. You see Polaris in the direction of ray BG. Line HCE is the Earth’s axis, and it also points at Polaris – which is so far away, and seems so tiny, but yet is also so large, that yes, parallel rays BG and CH do, for all practical purposes, point at the same point in the sky. Ray ED starts at the center of the Earth, passes through you at B, and goes on to the zenith (the part of the sky that is directly overhead). The horizon (BF) and the zenith (ray EB) are perpendicular. Also, line HCE (the earth’s axis) is perpendicular to its equator (segment AED).

Using some sort of angle measuring device, if you are out on the National Mall at night, you can very carefully measure the angle of elevation of the North Star above the local horizon, and you should ideally find that angle, FBG, is about 38.9 degrees, but we could also call it X degrees.

Prove (i.e. explain) why your latitude (which is angle AEB) measures the same as angle FBG.

What are the givens?

=========================================================

Full disclosure: My daughter graduated from SWW two decades ago, and I taught there as well for a year and for 10 years at a school that is now associated with it: Francis (then JHS now a middle school).

The kids were nice back then, and they still are. I thought the teachers did a great job.

This is a DC public high school that you have to apply to.

Benjamin Banneker was an amazing person. There are a lot of myths that have been attached to his work and accomplishments, which I am guessing might be because those people didn’t actually understand the math and astronomy that he did accomplish. The best book on him is by Silvio Bedini.

‘Math on the Mall’ was originated by Florence Fasanelli, Richard Thorington, and V. Frederick Rickey around 1990. I participated as a math teacher in a couple of those tours led by FF. Later, I wanted to take my students on a similar tour that would include a trip to see a number of the works of the geometer and artist Maurice C. Escher, and couldn’t find my copy of their work, so I made up my own, and added to it using the work of FF, RT, and VFR and suggestions from teachers and students. Later on, the Mathematical Association of America made something similar, which you can find here.

My version was on the website of the Carnegie Institution for Science for a number of years. See page 56 on this link. I need to find someone to cut out some of my excess verbiage and then trot it out to a publisher.

Correct weight of the mystery glass

Tags

More progress with the 22-inch wide, 4-inch thick mystery glass.

It took four of us old farts- Jim Kaiser, Alan Tarica, Tom Crone, and me – to extract the mirror from its case (which was located under a very heavy Draper-style grinding and polishing machine) roll it onto a little stand we fabricated on a decent gym scale I borrowed from my gym ( http://www.True180.fitness ) and weigh it.

We were very careful when moving that heavy mirror. Nobody got hurt in any way. When putting the mirror back into its sturdy wood carrying box, we used ancient Egyptian technology of little rollers, and it worked like a charm.

The bathroom scale we had used earlier, up at Hopewell was very, very wrong. We found that the weight of the glass was really 212 pounds (about 96 kilograms, or 96,000 grams), not 130 pounds. Its volume was 20,722 cc, so its density is roughly 4.6. Will have to see what types of glass have roughly that density and an index of refraction of about 1.72 to 1.76.

I heard from one veteran telescope maker:

“I’ve been in the Tucson astronomy club for many decades and also in the optics industry there. Most all institutions that had connections to astronomy or optics in the 60s got portions of several semi loads of “glass bank glass”, glasses that at one point in the past were considered strategic materials for certain optical designs/systems. There was a wide variety of materials, but almost all was identified in some way. We’re there any markings ar data scribed in the glass? The largest I saw was about 15”, so yours might be a different source.

“A co-worker of mine has identified several mystery glasses from an accurate determination of density. Seems like you should be able to get better results w/a more accurate scale. Also many glass types made decades ago are obsolete – my friend has some older glass catalogs that might help you determine what it might be with more accurate numbers.”

So these were cast-offs from the Military Industrial Complex, basically: pieces of glass that the military decided it no longer needed for projects that had either been completed or abandoned, and that they didn’t feel like storing any more. So they gave them away to groups like National Capital Astronomers and Hopewell Observatory.

The only markings on the glass are the following: a heavily inscribed (by hand) apparent date of 2-8-56, which probably means either February 8 of 1956 or the 2nd of August 1956. Judging by the handwriting style of the numerals, it was probably Feb. 8 of 1956 (US style). Under that are the numerals 0225, which we have no idea about. In pencil, someone with US-style handwriting wrote what looks like “Low #” in cursive. Again, we have no idea what that means.

Thanks so much, Jim, Alan, and Tom!

Satisfying Fixes Made to 50-year-old Electro-mechanical Telescope Drive at Hopewell Observatory

Tags

, , , , , , , ,

About a week week ago, the right ascension (or RA) drive on a vintage mount at the Hopewell Observatory stopped working. Instead of its usual hum, it began making scraping noises, and then ground to a halt. (This drive is the one that allows one to track the stars perfectly as the earth slowly rotates.)

Another member and I carefully removed the drive mechanism, and I took it home. At first, I thought it was the motor itself, but after examining it carefully, I noticed that some clutch pads inside the gearbox had come unglued, causing the clutch plates to be cockeyed. The motor itself worked just fine when disconnected from the gear box.

I recalled that the pads and the clutch had been very problematic, and that our resident but now-deceased electro-mechanical-optical wizard Bob Bolster had had to modify the gearbox quite a bit. I carefully disassembled the gearbox and used acetone to remove all the old glue that he had used to glue the pads on. After doing some research to find some equivalent pad material, I yesterday ordered some new gasket material with adhesive backing from McMaster-Carr. Lo and behold, I received it TODAY! Wow!

I cut out new pads, re-assembled everything, and the gears and worm drive work just fine. Not only that: there were no screws or nuts left over!

In addition, I now see how we can replace the extremely complicated partially-analog clutch-and-drive mechanism, in both RA and in Declination with a much simpler stepper-motor system using something called OnStep.

Here is a photo of the some of the innards of the scope:

A bit complicated, no?

In the next photo, my pencil is pointing to the clutch pads inside the gear box that had come loose, causing the clutch plates to become cockeyed, jamming the gears. The clutch is so that the observer can ever-so-slightly tweak the telescope forward or backwards in RA, in order to center the target. There is another gearbox for the declination, but it’s still working OK, so we left it alone.

The synchronous gear-motor in the background. My pencil is pointing to the problem.

Of course, we still have to re-install the gearbox back in the scope.

Bob Bolster, mentioned above, was one of the founding members of the Hopewell Observatory. He was an absolute wizard at fixing things and keeping this telescope mount going, but he is no longer alive. I was afraid that I would not be able to fix this problem, but it looks like I’ve been successful.

I append an image of a very beautifully-refurbished Ealing telescope and mount – similar to the one owned by Hopewell – that belongs to the Austin Astronomical Society. Ours is so much more beat up than this one that it’s embarrassing! Plus, both we and the University of Maryland were unable to get the telescope itself, which is a Ritchey-Chretien design, ever to work properly. So we sold the mirror and cell to a collector in Italy for a pittance, and installed four other, smaller scopes on the mount instead.

Disturbing Racist Clauses Found in Early NCA Constitutions & Bylaws

Tags

, , , , , , , , , , , , , , , , , , , , , , , , , , ,

By Guy Brandenburg

Recently, while preparing to give a talk at this year’s Stellafane telescope-makers’ convention, I was disappointed to discover that the National Capital Astronomers (NCA), which I’ve belonged to for about 30 years, specifically excluded Black members for nearly 3 decades: from about 1940 all the way up to1969.

But NCA didn’t start out being overtly racist. Our original 1937 founding document has no such language. It reads, in part,

“The particular business and objects of [the NCA] shall be the education and mutual improvement of its members in the science of Astronomy and the encouragement of an interest in this science among others. (…) The activities of this Association are designed for the enjoyment and cultural profit of all interested in astronomy, whether the member be a beginner, an advanced student, or one whose pursuit of the science is necessarily desultory.”

And today’s NCA home page reads, “All are welcome to join. Everyone who looks up to the sky with wonder is an astronomer and welcomed by NCA. You do not have to own a telescope, but if you do own one that is fine, too. You do not have to be deeply knowledgeable in astronomy, but if you are knowledgeable in astronomy that is fine, too. You do not have to have a degree, but if you do that is fine, too. WE ARE THE MOST DIVERSE local ASTRONOMY CLUB anywhere. Come to our meetings and you will find this out. WE REALLY MEAN THIS!”

But in the 1940’s, the original open-minded and scientific NCA membership policy changed. The January 1946 Star Dust listed a number of changes to be voted on by the membership in the club’s founding documents. (See https://capitalastronomers.org/SD_year/1946/StarDust_1946_01.pdf ) The organization voted to change article III of its constitution as follows:

From:

“only Caucasians over 16 years old are eligible for membership.

To this:

“to include all ages (see by-laws), exclude only the Black race.”

While it may be shocking that a scientific organization like NCA had such a policy, people often forget how racist a nation the USA used to be, and for how long. If you look up actual pages of DC area newspapers from the 1950s, you will note that the classified advertisements were largely segregated both by race and by gender – want ads would very often specify male or female, single or married, White-only or Colored-only jobs, apartments, and so on.

Schools in DC, MD, and Virginia were mostly segregated, either by law or in practice, up until the late 1960s or early 1970s. The 1954 Brown v Board decision had very little real impact in most areas until much, much later. Queens (NYC), PG County (MD) and Boston (MA) had violent movements against integrating schools in the 1970s. I know because I attended demonstrations against those racists and have some scars to prove it.

While the Federal and DC governments offices were integrated immediately after the Civil War, that changed for the worse when Woodrow Wilson was elected President in 1912.

Many scientists in the USA and in Europe believed the pseudo-scientific ideas of racial superiority and eugenics that arose around 1900 and were still widespread 50 years ago – and even today, as recent events have sadly shown.

In The War Against the Weak: Eugenics and America’s Campaign to Create a Master Race, Edwin Black explains how august scientific institutions like the Carnegie Institution of Washington (CIW), the American Natural History Museum in New York, and a number of eminent statisticians and biologists for many decades supported the Eugenics Records Office (ERO) at Cold Spring Harbor. So did the fabulously wealthy Rockefeller and Harriman Foundations.

The ERO pushed the concept of the genetic superiority of the ‘Nordic’ race and helped to pass State laws sterilizing the ‘weak’ and forbidding interracial marriage. They were also successful in passing the 1924 Federal immigration law that severely cut back immigration from parts of the world where supposedly ‘inferior’ people lived – e.g. Eastern and Southern Europe. As a result, many Jews who would have loved to escape Hitler’s ovens by crossing the Atlantic never made it.  

Hitler and his acolytes always acknowledged their ideological and procedural debt to American eugenical laws, literature, and propaganda. As we all know, Germany’s Nazis put those ideas to work murdering millions of Jews, Gypsies, Slavs and others.

It took more than three decades for the CIW to withdraw their support of the ERO. A CIW committee concluded in 1935 “that the Eugenics Record Office was a worthless endeavor from top to bottom, yielding no real data, and that eugenics itself was not a science but rather a social propaganda campaign with no discernable value to the science of either genetics or human heredity.” (Black, p. 390) The members pointedly compared the work of the ERO to the excesses of Nazi Germany. However, it took four more years for CIW to cut all their ties – shortly after Hitler invaded Poland in 1939, starting World War Two.

I don’t know exactly when the ‘Caucasian’-only policy became part of the NCA rules, but it seems to have been between the club founding in 1937, and October 1943 when volume 1, number 1 of Star Dust was printed. At one point, perhaps around 1940, NCA decided that only ‘Caucasians’ over 16 could join. But as indicated above, in 1946, the racial exclusion policy was narrowed to only exclude Black people. Apparently Jews, Italians, young people, Latin Americans, and Asians were eligible to join NCA from 1946 to 1969. But not African-Americans.

While researching my talk, I found that the NCA held amateur telescope-making classes at a number of all-white DC, MD, and VA high schools, from the 1940s through about 1970, both during the days of de jure segregation and the merely de-facto type: McKinley, Roosevelt, Central, Bladensburg, Falls Church, and McLean high schools are listed. While Star Dust mentions a telescope-making course at (the largely-Black) Howard University in 1946, there is no mention of any assistance for that course from NCA.

I also found no evidence in any issue of Star Dust from that era that anybody at the time raised any vocal objections to racial exclusion. Not in 1946, nor 23 years later when the rule prohibiting Black members was quietly dropped (in 1969) when a new constitution was adopted.

A few current or past NCA members confirmed to me that at some point, they noticed that racist language and privately wondered about it. One person told me that they definitely recalled some now-deceased NCA members who were openly racist and not shy about expressing those views. Others told me that they had never heard any discussion of the subject at all.

 (As one who grew up in DC and Montgomery County, and attended essentially-segregated public schools there, I am sorry that neither I nor my family actively spoke up at the time, even though a farm adjacent to ours in Clarksburg was owned by a Black family [with no school-age children at the time]. Amazing how blind one can be! The racists of those days were not shy about committing violence to achieve their ends. Fear might be one reason for silence.)

One possibility is that some of the early NCA meetings might have been held at private residences; perhaps some of the racist members insisted in preventing non-‘Caucasian’ or ‘Black’ people from attending. It is too bad the other NCA members didn’t take the other route and stay true to the original ideas of the club, and tell the racist members to get lost.

Very ironic: the late George Carruthers, a celebrated Naval Research Labs and NASA scientist, and an instrument-maker for numerous astronomical probes and satellites, gave a talk to the NCA in September of 1970 – not too long after the NCA apparently dropped its racist membership rules (April, 1969). So, a mere year and a half before he gave his talk, he could not have legally joined the organization. Nor could he have done so when he was making his own telescopes from scratch as a teenager in the 1940s. See https://en.wikipedia.org/wiki/George_Robert_Carruthers on the life and work of this great African-American scientist and inventor.

To NCA’s credit, we have done better in the past few decades at encouraging participation in telescope viewing parties, telescope making, and lectures by members of all races and ethnic groups. However, I often find that not very many NCA members bring telescopes to viewing events, or show up to judge science fairs, in mostly-minority neighborhoods. Often, it’s just me. That needs to change. We need to encourage an interest in science, astronomy, and the universe in children and the public no matter their skin color or national origin, and we need to combat the racist twaddle that passes for eugenics.

I anticipate that NCA will have a formal vote repudiating the club’s former unscientific and racist policies and behavior. I hope we will redouble our efforts to promote the study of astronomy to members of all ethnic groups, especially those historically under-represented in science.

We could do well to note the words that Albert Einstein wrote in 1946, after he had been living in the US for a decade, and the same year that NCA confirmed that Black people could not join:

“a somber point in the social outlook of Americans. Their sense of equality and human dignity is mainly limited to men of white skins. Even among these there are prejudices of which I as a Jew am clearly conscious; but they are unimportant in comparison with the attitude of the “Whites” toward their fellow-citizens of darker complexion, particularly toward Negroes.

The more I feel an American, the more this situation pains me. I can escape the feeling of complicity in it only by speaking out.

Many a sincere person will answer: “Our attitude towards Negroes is the result of unfavorable experiences which we have had by living side by side with Negroes in this country. They are not our equals in intelligence, sense of responsibility, reliability.”

I am firmly convinced that whoever believes this suffers from a fatal misconception. Your ancestors dragged these black people from their homes by force; and in the white man’s quest for wealth and an easy life they have been ruthlessly suppressed and exploited, degraded into slavery. The modern prejudice against Negroes is the result of the desire to maintain this unworthy condition.

The ancient Greeks also had slaves. They were not Negroes but white men who had been taken captive in war. There could be no talk of racial differences. And yet Aristotle, one of the great Greek philosophers, declared slaves inferior beings who were justly subdued and deprived of their liberty. It is clear that he was enmeshed in a traditional prejudice from which, despite his extraordinary intellect, he could not free himself.

What, however, can the man of good will do to combat this deeply rooted prejudice? He must have the courage to set an example by word and deed, and must watch lest his children become influenced by this racial bias.

I do not believe there is a way in which this deeply entrenched evil can be quickly healed. But until this goal is reached there is no greater satisfaction for a just and well-meaning person than the knowledge that he has devoted his best energies to the service of the good cause.”

Source: http://www.kganu.net/sitebuildercontent/sitebuilderfiles/alberteinsteinonthenegroquestion-1946.pdf

I am indebted to Morgan Aronson, Nancy Byrd, Richard Byrd, Geoff Chester, Jeff Guerber, Jay Miller, Jeffrey Norman, Rachel Poe, Todd Supple, Wayne Warren, Elizabeth Warner, and Harold Williams for documents, memories, and/or technical support.

First Telescope Making Class Since March of 2020!

Tags

, , , , , ,

Tonight we were finally able to hold a telescope making workshop again, for the first time since March 13, 2020, according to our log-in sheet.

We had five people, and we looked at several mirrors.

The first one was a plate glass, 10″, f/5.5 Coulter mirror that Kevin Hartnett had obtained and wanted me to strip the old aluminum coating from and then silver it and overcoat it. I thought the coating looked rather good, especially given its age, and wanted to put it on the testing stand to see how the figure looked. All of us thought the geometric figure of the mirror looked pretty good, and the ronchi lines looked nice and smooth. Alin Tolea said he saw a narrow turned down edge region perhaps 1/4″. Kevin thought it performed well, and I can see why.

I hope my silvering job turns out at least as good as its current aluminization.

Here are a few frames from my video of the Ronchi images (100 lines per inch):

The second one was a 17.5″ f/4.5 pyrex mirror, also originally made by Coulter and then refigured by somebody called Optical Western Labs (?) in California. The owner, We did not like this mirror at all. We thought the Ronchi lines were not smooth; there is a raised area in the center; and it even shows some signs of astigmatism. Here are a couple of frames the video I took of its Ronchi measurements:

The third mirror was an 8″, under-f/4 plate glass mirror that the owner reported performed very poorly. Once we put it on the stand, we saw why: it had never been parabolized! The Ronchi lines were almost perfectly straight! You only want straight Ronchi lines if your goal is to have a spherical (as opposed to parabolic, ellipsoidal, or hyperbolic) mirror. That’s why all its images were blurry. Nagesh Kanvindeh immediately decided to start trying to parabolize it, and we happened to have a synthetic pitch lap of 8″ diameter that had been last used to finish an f/4 mirror, so he got started right away.

By the way, our new hours are 5:00 pm to 8:30 pm, Tuesdays and Fridays.

A piece of mystery glass

Tags

, , , , , , , , , , , ,

Many years ago, the late Bob Bolster, a founding member of Hopewell Observatory and an amazing amateur telescope maker, got hold of a large piece of glass, perhaps World War Two military surplus left over from the old Bureau of Standards.

I have no idea what it is made out of. If Bob had any clue about its composition, he didn’t tell anyone.

Its diameter is 22 inches, and its thickness is about 3.25″. It has a yellowish tint, and it is very, very heavy.

If you didn’t know, telescope lenses (just like binocular or camera lenses) are made from a wide variety of ingredients, carefully selected to refract the various colors of light just so. Almost all glass contains quartz (SiO2), but they can also contain limestone (CaCO3), Boric oxide (B2O3), phosphates, fluorides, lead oxide, and even rare earth elements like lanthanum or thorium. This link will tell you more than you need to know.

If you are making lenses for a large refracting telescope, you need to have two very different types of glass, and you need to know their indices of refraction very precisely, so that you can calculate the the exact curvatures needed so that the color distortions produced by one lens will be mostly canceled out by the other piece(s) of glass. This is not simple! The largest working refractor today is the Yerkes, with a diameter of 40 inches (~1 meter). By comparison, the largest reflecting telescope made with a single piece of glass today is the Subaru on Mauna Kea, with a diameter of 8.2 meters (323 inches).

For a reflecting telescope, one generally doesn’t care very much what the exact composition of the glass might be, as long as it doesn’t expand and contract too much when the temperature rises or falls.

We weren’t quite sure what to do with this heavy disk, but we figured that before either grinding it into a mirror or selling it, we should try to figure out what type of glass it might be.

Several companies that produce optical glass publish catalogs that list all sorts of data, including density and indices of refraction and dispersion.

Some of us Hopewell members used a bathroom scale and tape measures to measure the density. We found that it weighed about 130 pounds. The diameter is 22 inches (55.9 cm) and the thickness is 3 and a quarter inches (8.26 cm). Using the formula for a cylinder, namely V = pi*r2*h, the volume is about 1235 cubic inches or 20,722 cubic centimeters. Using a bathroom scale, we got its weight to be about 130 lbs, or 59 kg (both +/- 1 or 2). It is possible that the scale got confused, since it expects two feet to be placed on it, rather than one large disk of glass.

However, if our measurements are correct, its density is about 2.91 grams per cc, or 1.68 ounces per cubic inches. (We figured that the density might be as low as 2.80 or as high as 3.00 if the scale was a bit off.)

It turns out that there are lots of different types of glass in that range.

Looking through the Schott catalog I saw the following types of glass with densities in that range, but I may have missed a few.

2.86  N-SF5

2.86 M-BAK2

2.89 N-BAF4

2.90 N-SF8

2.90 P-SF8

2.91 N-PSK3

2.92 N-SF15

2.93 P-SF69

2.94 LLF1

2.97 P-SK58A

3.00 N-KZFS5

3.01 P-SK57Q1

By comparison, some of the commonest and cheapest optical glasses are BAK-4 with density 3.05 and BK-7 with density 2.5.

Someone suggested that the glass might contain radioactive thorium. I don’t have a working Geiger counter, but used an iPhone app called GammaPix and it reported no gamma-ray radioactivity at all, and I also found that none of the glasses listed above (as manufactured today by Schott) contain any Uranium, Thorium or Lanthanum (which is used to replace thorium).

So I then rigged up a fixed laser pointer to measure its index of refraction using Snell’s Law, which says

Here is a schematic of my setup:


The fixed angle a I found to be between 50 and 51 degrees by putting my rig on a large mirror and measuring the angle of reflection with a carpentry tool.

And here is what it looked like in practice:

I slid the jig back and forth until I could make it so that the refracted laser beam just barely hit the bottom edge of the glass blank.

I marked where the laser is impinging upon the glass, and I measured the distance d from that spot to the top edge of the glass.

I divided d by the thickness of the glass, in the same units, and found the arc-tangent of that ratio; that is the measure, b, of the angle of refraction.

One generally uses 1.00 for the index of refraction of air (n1). I am calling n2 the index of refraction of the glass. I had never actually done this experiment before; I had only read about doing it.

As you might expect, with such a crude setup, I got a range of answers for the thickness of the glass, and for the distance d. Even angle a was uncertain: somewhere around 49 or 50 degrees. For the angle of refraction, I got answers somewhere between 25.7 and 26.5 degrees.

All of this gave me an index of refraction for this class as being between 1.723 and 1.760.

This gave me a list of quite a few different glasses in several catalogs (two from Schott and one from Bausch & Lomb).

Unfortunately, there is no glass with a density between 2.80 and 3.00 g/cc that has an index of refraction in that range.

None.

So, either we have a disk of unobtanium, or else we did some measurements incorrectly.

I’m guessing it’s not unobtanium.

I’m also guessing the error is probably in our weighing procedure. The bathroom scale we used is not very accurate and probably got confused because the glass doesn’t have two feet.

A suggestion was made that this might be what Bausch and Lomb called Barium Flint, but that has an index of refraction that’s too low, only 1.605.

Mystery is still unsolved.

Is this question reasonable?

Tags

, , , , , ,

This is a sample question for middle school math, published by the International Baccalaureate (IB) program. I found it here.

Here is a graph I made of this equation, using Desmos:

Looking at this graph, you see that after about 10 minutes, there are 11 cars per minute going through the intersection – and that’s the most cars. After about 25 minutes, there are zero cars going through the intersection, and after that, there is a negative number of cars (!!!).

I don’t think this equation models anything having to do with any intersection I’ve ever visited. Instead, I think that any intersection controlled by a traffic light is going to be more periodic, that is to say, something like some mix of sine or cosine functions — obviously not middle school material.

Why there are no space-faring civilizations, and never will be

Very persuasive article explains why space travel is impossible. The main reason is gravity. Written by Allan Milne Lees; I found it on Medium.

Allan Milne Lees5 days ago·7 min read

===============================

Image credit: Air & Space Magazine

Despite the populist hype of billionaire Sci-Fi fanboys and a perpetual stream of Hollywood entertainments to the contrary, humans will never explore the galaxy in person. In fact, we won’t even explore our own solar system up close and personal. This is not merely because robotic missions can do the job 1,000% better for 1/1000th the cost. It’s because of two fundamental biological reasons.

The first is gravity. Everything about our bodies is evolved to function under a gravitational acceleration at sea level of approximately 9.8 meters per second squared (9.8m²). Our hearts pump blood up to our heads, fighting gravity every centimeter of the way. Our muscles and bones are as strong as they are because every part of our bodies is fighting gravity every moment of our lives. Our sense of balance, which orients us spatially, depends on gravity being constant in one direction only: straight down.

Without gravity, very bad things happen: the heart pumps too much blood to the head and too little to the lower extremities, leading to ocular distortions, crushing headaches, and nausea as the inner ear loses all sense of up and down. Our bones and muscles atrophy dramatically, even when hours each day are dedicated to exercises specifically designed with the intention of slowing down this decay. Put simply, our bodies are incapable of handling microgravity and despite the pictures of smiling astronauts merrily enjoying microgravity on the ISS, the harsh reality is that every single one of those astronauts pays a price very few of us would wish to incur.

The Sci-Fi fanboy response to this fundamental problem is either (a) to ignore it entirely, as per Musk and Bezos, or (b) claim that artificial gravity is the answer.

As Musk and Bezos are ignoring the problem we can likewise ignore them. So what about artificial gravity?

There are only two ways to create artificial gravity. The first is called “constant-g” which means that we accelerate our hypothetical space ship at a constant 9.8m² for the first half of the trip and then flip it around and decelerate it at a constant 9.8m² for the second half of the trip. Einstein’s insight that over areas too small to experience tidal effects such acceleration would be indistinguishable from regular gravity means that in theory Earth-style gravity could be induced in such a manner. Better yet, because the acceleration is constant, relativistic speeds will eventually be attained. In just 12 years (in the reference frame of the spacecraft) we could travel across our Milky Way galaxy. In a single human lifetime (in the reference frame of the spacecraft), under constant acceleration, we could reach the edge of the universe that’s observable from Earth. An Earth upon which, in that frame of reference, billions of years would have passed.

So with constant acceleration we get a “twofer.” Earth-identical gravity and the ability to traverse vast distances within a human lifetime. Problem solved!

Except that there is no way, theoretical or otherwise, to achieve constant acceleration of this magnitude. No propulsion mechanism, theoretical or otherwise, can overcome the problem of mass. In order to power the continual acceleration, our imaginary space ship is constrained by Newton’s observation that any action in a vacuum requires an equal and opposite reaction. In other words, to accelerate a mass of X by some amount of velocity we will need to discharge an equivalent amount of energy in the opposite direction. And that energy can only come from fuel. Which adds to the mass of our space ship. So now we need to expend more energy, which means we need more fuel, which means we’re now carrying even more mass, which means we need to expend even more energy, which means…

In other words, even with some imaginary technology that could convert matter into energy with 100% efficiency, there’s simply no way to get to 9.8m² constant acceleration for any meaningful amount of time. Sure, we can talk about things like an Alcubierre drive but then we’re just as entitled to say that Hogwarts will invent the Spaciamus drive to solve our problem instead. In other words, running off to hide inside imaginary “solutions” is no solution at all.

If constant acceleration can’t provide artificial gravity, what about centrifugal force? We all remember the rotating space station in 2001 A Space Odyssey and everyone knows that this was the only Sci-Fi movie ever to have utilized a science-based series of technologies. Plus, it’s easy to find on the Internet lots of schemes to create artificial gravity in this way, from tethering ships together and spinning them around a central axis to building enormous hollow rotating cylinders on the inside of which humans will experience Earth-like gravity. So, problem solved!

Except the movies and the Sci-Fi books mislead us, as is the way of popular entertainments.

First, the good news: if a person stood perfectly still and did not move in any way whatsoever, then centrifugal force could seem to mimic Earth-style gravity. Unfortunately, here’s the bad news: if they made any movement whatsoever, they would instantly be overcome by nausea and be disoriented.

Why is this? Imagine throwing a ball up into the air here on Earth. If you throw it straight up, it will come straight down, pulled by gravity toward the center of the Earth we’re standing on. But under conditions of “gravity” induced by centrifugal force, a ball thrown straight up will arc and fall away from the person who threw it because unlike here on Earth there’s a second force acting on the ball: centripetal force. As our inner ear orients us by means of reference to the constant downward force of gravity, this means that any movement at all — even something as minor as turning one’s head — would result in signals from the inner ear (responding to the centripetal force) jarring dramatically with the signals from our eyes. At best this would lead to our hypothetical human vomiting in a majestic arc; at worst it could render them incapable of any controlled movement whatsoever.

The diagrams below show the difference between gravity (or constant acceleration at 9.8m²) and a rotating object. On Earth there’s only one force acting on us: gravity. On our imaginary rotating artificial gravity environment there are two forces: centrifugal, and centripetal. And that makes all the difference in the world.

Perhaps this is why Bezos prefers to ignore the problem; it can’t be solved just by throwing money at it. As for Musk, he makes people with ADHD look like paragons of sustained concentration so he probably doesn’t even know the problem exists. But even if you don’t know a brick wall exists, it still kills you if you slam into it at 1,000 kilometers per hour.

Gravity, therefore, is one reason why human beings will never be a space-faring species. It’s also the reason why it’s highly unlikely any other species capable of developing suitable technologies would ever become space-faring either. All organisms are highly adapted to the environments in which they evolve and it is extremely difficult to sustain organisms outside of their natural environments for any significant period of time. Add it the problems of solar radiation, the deleterious effects of microgravity, and everything else associated with space travel and it’s apparent that Sci-Fi fanboy dreams are a very poor guide to the future.

There is a second major reason why we humans will never be a space-faring species: psychology.

Our brains are as much the result of selection pressures as our bodies. Like our bodies, our brains are highly adapted to life on Earth. As a primate group species adapted to foraging, we’re not well-suited to being cooped up in tiny cages. We become obese and we develop all manner of mental problems. Without access to natural cues like water and grass and trees, we become stressed. When forced to interact with the same small group of people for years without respite, we become irrational and angry, or conversely withdrawn and depressed. Worse still, our emotional hardwiring makes us competitive even when cooperation is the optimal strategy, and our intellectual limitations lead us to acquiring and then strongly defending irrational and harmful beliefs.

Imagine, therefore, a space ship upon which 200 hapless humans attempt to exist for years or even decades. Instead of looking to Star Trek as our inspiration, a more probable vision is depicted in One Flew Over The Cuckoo’s Nest or perhaps the concluding episodes of some trash reality TV show.

It is difficult to imagine any species capable of making spacecraft not having equivalent psychological limitations, albeit likely somewhat different from those that control our own behaviors.

There are many other reasons why humans will never spread across the galaxy, but these two should suffice to prove the contention. This does not mean, however, that there won’t be money to be made in enabling space tourism. A few days in microgravity, ensconced in a modestly comfortable environment with a small number of others, could be a very congenial way for the wealthy to break up the monotony of holidaying in the Hamptons or on a private island in the Bahamas. Sheltered in low orbit by the Earth’s magnetic field, the dangers of solar radiation are reduced to a perfectly acceptable level and likely no worse than a dozen trips in a private jet. Microgravity sex will no doubt become this century’s equivalent of the Mile High Club that was so popular among the early jet-setters of the 1960s and 1970s.

But beyond a few amusing days spent orbiting the Earth while watching one’s champagne bubble around one’s head, and after the inevitable disaster of Mars Colony One, we will accept the fact that robotic missions are the real future. And then we will expand our knowledge of the universe exponentially instead of wasting hundreds of billions of dollars on futile dead-end fanboy dreams.

Final Silvering Results, with Angel Guard

A few days ago, we silvered an 8” diameter 43” FL mirror that had previously been aluminized, and applied the Angel Guard coating.

We did a Ronchi test and some Foucault-Couder knife edge tests before stripping the aluminum and after the silver was applied.

To my amazement, we found that the mirror’s figure was about the same in both cases. How that works, especially how the Angel Guard coating is laid down so even and smooth over the entire mirror, is beyond me. But it DOES work.

Prior figure (aluminized mirror), seen with Ronchi grating of 100 lines per inch: https://share.icloud.com/photos/0LyqGC35cx0QfWKcd08aI0vzw…

Final figure (silvered mirror), with same Ronchi grating:

https://share.icloud.com/photos/03mdTeZD3dZ-bqGNoDZXSqpjw…

This is a video of us washing off the Angel Guard coating.

Here is a video of the finished mirror after drying. Notice that the very edge of this mirror did not take the silver coating, but the area uncoated is probably on the order of one or two percent of the total area.

Silvering a Mirror (vs Aluminizing)

Ive been doing the aluminization process for telescope mirrors at the NCA ATM workshop with a 55-year-old military surplus aluminizer at a DC rec center for about 20 years. (I’ve had a lot of help!!) This involves high vacuum, a noisy pump, voltage both very high and very low, and quite a lot of time.

Today, I had the opportunity to silver a random piece of glass, in my driveway, with the aid of another longtime ATMer and some chemicals from Angel Gilding. I had seen this demonstrated at Stellafane by Howard Banich and Peter Pekurar in 2019.

Doing it myself was quite eye-opening.

Almost finished, except for the spray-on Angel-Guard, which we didn’t have

Here’s what I wrote on FB:

Success with our first attempt at silvering a piece of glass under a tent canopy, and then stripping off the silver quickly and easily with PCB etchant (FeCl?).

I’ve aluminized many mirrors with the NCA’s vacuum chamber, using a modified version of John Strong’s method from the 1930s.

I must admit that this method was faster, easier, quieter, and much more low-tech, compared to depositing aluminum. In the latter case, sometimes you have to wait an hour or more for the dual-stage vacuum chamber (the primary pump is VERY noisy!!) to finish all the preparatory steps and pump down low enough that a hot atom of gaseous aluminum can travel two or three feet before striking any other remaining air molecule! (That is one hell of a vacuum!)

With the silvering process, you can do any size mirror you can fit on your cleaning jig — and you can make it out of pieces of scrap wood, a few nylon chair legs, two old hinges, and some screws! !

With our NCA-ATM-CCCC aluminizer, we are limited to 12.5” max. I’m currently working on a 16.5” thin Pyrex mirror; the price I’m quoted for aluminizing it is about $600 at Majestic Coatings, which is about three times what I paid for the blank!! And that doesn’t even include shipping!

Today Alan T and I tested the silvering process in my driveway using the screen tent canopy that we use at the ATM workshop to stop dust particles from landing on mirrors that are being polished. (After getting permission to go retrieve the canopy from the Covid-closed rec center, we immediately went into the parking lot to hose off a decade of dust!!)

We unfortunately do not have Angel-Guard overcoating on hand. It should arrive Wednesday. As most folks know, bare silver, unlike bare aluminum, tarnishes very quickly (in weeks or months) when exposed to ordinary air, whereas a I have seen many bare Aluminum layers last a decade. This overcoating is said to extend the life of the coating to about a year, but obviously conditions will vary.

We used a 6″ float glass mirror blank to try out the process today – not an actual, parabolized mirror.

How does it work?

This is basically a five step process:

1. Get prepared and mix the tinning solution afresh;

2. Clean off the mirror properly with precipitated CaCO3 and/or Alconox; rinse;

3. Sensitize the mirror with an invisible layer of tin (Sn); rinse;

4. Spray on the silver solution and its reducer at the exact same time with two separate brand-new one-pint hand squirt bottles, until fully silvered & shiny; rinse;

5. Spray on the Angel-Guard overcoat; rinse; dry.

The amount of chemicals used is minimal. The nastiest stuff was the reducer. I’m glad we did this outside.

Btw: a number of people have bench-tested mirrors before and after this process. Some report no change in figure; somebody I trust, who has a Zygo interferometer, says there is a little degradation, but not much: a mirror that was 1/10 lambda (excellent) might go to 1/4 lambda, which is certainly usable for a big Dob, iirc.

And it’s cheap! And fast! And easy! And quiet!

We were able to fully and completely strip the brand new silver off with the PCB etchant in under 3 minutes. Aluminum takes much longer.

Thanks to: Léon Foucault; Steinheil; Howard Banich; Peter Pekurar; Angel Gilding; Alan Tarica; and my wife, Gail.

One spray for the silver solution, one spray for its reducer (chemistry)
It’s the tent canopy that’s bent, I think. But this is not a telescope mirror; just a 6 inch disk of float glass.
That’s Alan
I made the jig to hold the mirrors out of some scrap two-by-fours, some screws and reinforcement plates; two old hinges; some nylon chair feet; some 1/2″ PVC pipe; and some 1/2″ PVC end caps.
The tent-screen canopy needed staking and tying down to prevent it flying off. Putting up the frame took three people (me, Alan, and my wife, Gail).