Science

Increasing speed due to friction

Posted on by John

Orbital mechanics is fascinating. I’ve learned a bit about it for fun, not for profit. I seriously doubt Elon Musk will ever call asking me to design an orbit for him. [1]

One of the things that makes orbital mechanics interesting is that it can be counter-intuitive. For example, atmospheric friction can make a satellite move faster. How can this be? Doesn’t friction always slow things down?

Friction does reduce a satellite’s tangential velocity, causing it to move into a lower orbit, which increases its velocity. It’s weird to think about, but the details are worked out in [2].

Note the date on the article: May 1958. The paper was written in response to Sputnik 1 which launched in October 1957. Parkyn’s described the phenomenon of acceleration due to friction in general, and how it applied to Sputnik in particular.

Related posts

[1] I had a lead on a project with NASA once, but it wasn’t orbital mechanics, and the lead didn’t materialize.

[2] D. G. Parkyn. The Effect of Friction on Elliptic Orbits. The Mathematical Gazette. Vol. 42, No. 340 (May, 1958), pp. 96-98

Calculating when a planet will appear to move backwards

Posted on by John Cook

When we say that the planets in our solar system orbit the sun, not the earth, we mean that the motions of the planets is much simpler to describe from the vantage point of the sun. The sun is no more the center of the universe than the earth is. Describing the motion of the planets from the perspective of our planet is not wrong, but it is inconvenient. (For some purposes. It’s quite convenient for others.)

The word planet means “wanderer.” This because the planets appear to wander in the night sky. Unlike stars that appear to orbit the earth in a circle as the earth orbits the sun, planets appear to occasionally reverse course. When planets appear to move backward this is called retrograde motion.

Apparent motion of Venus

Here’s what the motion of Venus would look like over a period of 8 years as explored here.

Venus from Earth

Venus completes 13 orbits around the sun in the time it takes Earth to complete 8 orbits. The ratio isn’t exactly 13 to 8, but it’s very close. Five times over the course of eight years Venus will appear to reverse course for a few days. How many days? We will get to that shortly.

When we speak of the motion of the planets through the night sky, we’re not talking about their rising and setting each day due to the rotation of the earth on its axis. We’re talking about their motion from night to night. The image above is how an observer far above the Earth and not rotating with the Earth would see the position of Venus over the course of eight years.

The orbit of Venus as seen from earth is beautiful but complicated. From the Copernican perspective, the orbits of Earth and Venus are simply concentric circles. You may bristle at my saying planets have circular rather than elliptical orbits [1]. The orbits are not exactly circles, but are so close to circular that you cannot see the difference. For the purposes of this post, we’ll assume planets orbit the sun in circles.

Calculating retrograde periods

There is a surprisingly simple equation [2] for finding the points where a planet will appear to change course:

\cos(kt) = \frac{\sqrt{Rr}}{R + r - \sqrt{Rr}}

Here r is the radius of Earth’s orbit and R is the radius of the other planet’s orbit [3]. The constant k is the difference in angular velocities of the two planets. You can solve this equation for the times when the apparent motion changes.

Note that the equation is entirely symmetric in r and R. So Venusian observing Earth and an Earthling observing Venus would agree on the times when the apparent motions of the two planets reverse.

Example calculation

Let’s find when Venus enters and leaves retrograde motion. Here are the constants we need.

r = 1       # AU
R = 0.72332 # AU

venus_year = 224.70 # days
earth_year = 365.24 # days
k = 2*pi/venus_year - 2*pi/earth_year

c = sqrt(r*R) / (r + R - sqrt(r*R))

With these constants we can now plot cos(kt) and see when it equals c.

This shows there are five times over the course of eight years when Venus is in apparent retrograde motion.

If we set time t = 0 to be a time when Earth and Venus are aligned, we start in the middle of retrograde period. Venus enters prograde motion 21 days later, and the next retrograde period begins at day 563. So out of every 584 days, Venus spends 42 days in retrograde motion and 542 days in prograde motion.

Related posts

[1] Planets do not exactly orbit in circles. They don’t exactly orbit in ellipses either. Modeling orbits as ellipses is much more accurate than modeling orbits as circles, but not still not perfectly accurate.

[2] 100 Great Problems of Elementary Mathematics: Their History and Solution. By Heinrich Dörrie. Dover, 1965.

[3] There’s nothing unique about observing planets from Earth. Here “Earth” simply means the planet you’re viewing from.

What can JWST see?

Posted on by John

The other day I ran across this photo of Saturn’s moon Titan taken by the James Webb Space Telescope (JWST).

If JWST can see Titan with this kind of resolution, how well could it see Pluto or other planets? In this post I’ll do some back-of-the-envelope calculations, only considering the apparent size of objects, ignoring other factors such as how bright an object is.

The apparent size of an object of radius r at a distance d is proportional to (r/d)². [1]

Of course the JWST isn’t located on Earth, but JWST is close to Earth relative to the distance to Titan.

Titan is about 1.4 × 1012 meters from here, and has a radius of about 2.6 × 106 m. Pluto is about 6 × 1012 m away, and has a radius of 1.2 × 106 m. So the apparent size of Pluto would be around 90 times smaller than the apparent size of Titan. Based on only the factors considered here, JWST could take a decent photo of Pluto, but it would be lower resolution than the photo of Titan above, and far lower resolution than the photos taken by New Horizons.

Could JWST photograph an exoplanet? There are two confirmed exoplanets are orbiting Proxima Centauri 4 × 1016 m away. The larger of these has a mass slightly larger than Earth, and presumably has a radius about equal to that of Earth, 6 × 106 m. Its apparent size would be 150 million times smaller than Titan.

So no, it would not be possible for JWST to photograph an expolanet.

 

[1] In case the apparent size calculation feels like a rabbit out of a hat, here’s a quick derivation. Imagine looking out on sphere of radius d. This sphere has area 4πd². A distant planet takes up πr² area on this sphere, 0.25(r/d)² of your total field of vision.

Earth : Jupiter :: Jupiter : Sun

Posted on by John

The size of Jupiter is approximately the geometric mean of the sizes of Sun and Earth.

In terms of radii,

\frac{R_\Sun}{R_{\text{\Jupiter}}} \approx \frac{R_\Jupiter}{R_\Earth}

The ratio on the left equals 9.95 and the ratio on the left equals 10.98.

The subscripts are the astronomical symbols for the Sun (☉, U+2609), Jupiter (♃, U+2643), and Earth (, U+1F728). I produced them in LaTeX using the mathabx package and the commands \Sun, Jupiter, and Earth.

The mathabx symbol for Jupiter is a little unusual. It looks italicized, but that’s not because the symbol is being used in math mode. Notice that the vertical bar in the symbol for Earth is vertical, i.e. not italicized.

 

Gravity on Jupiter

Posted on by John

NASA image of Jupiter

I was listening to the latest episode of the Space Rocket History podcast. The show includes some audio from a documentary on Pioneer 11 that mentioned that a man would weigh 500 pounds on Jupiter.

My immediate thought was “Is that all?! Is this ‘man’ a 100 pound boy?”

The documentary was correct and my intuition was wrong. And the implied mass of the man in the documentary is 190 pounds.

Jupiter has more than 300 times more mass than the earth. Why is its surface gravity only 2.6 times that of the earth?

Although Jupiter is very massive, it is also very large. Gravitational attraction is proportional to mass, but inversely proportional to the square of distance.

A satellite in orbit 100,000 km from the center of Jupiter would feel 300 times as much gravity as one in orbit the same distance from the center of Earth. But the surface of Jupiter is further from its center of mass than the surface of Earth is from its center of mass.

The mass of Jupiter is 318 times that of Earth, and the its mean radius is 11 times that of Earth. So the ratio of gravity on the surface of Jupiter to gravity on the Earth’s surface is

318 / 11² = 2.63

Now suppose a planet had the same density as Earth but a radius of r Earth radii. Then its mass would be r³ times greater, but its surface gravity would only be r times greater since gravity follows an inverse square law. So if Jupiter were made of the same stuff as Earth, its surface gravity would be 11 times greater. But Jupiter is a gas giant, so its surface gravity is only 2.6 times greater.

Related posts

How to Organize Technical Research?

Posted on by Wayne Joubert

 

64 million scientific papers have been published since 1996 [1].

Assuming you can actually find the information you want in the first place—how can you organize your findings to be able to recall and use them later?

It’s not a trifling question. Discoveries often come from uniting different obscure pieces of information in a new way, possibly from very disparate sources.

Many software tools are used today for notetaking and organizing information, including simple text files and folders, Evernote, GitHub, wikis, Miro, mymind, Synthical and Notion—to name a diverse few.

AI tools can help, though they can’t always recall correctly and get it right, and their ability to find connections between ideas is elementary. But they are getting better [2,3].

One perspective was presented by Jared O’Neal of Argonne National Laboratory, from the standpoint of laboratory notebooks used by teams of experimental scientists [4]. His experience was that as problems become more complex and larger, researchers must invent new tools and processes to cope with the complexity—thus “reinventing the lab notebook.”

While acknowledging the value of paper notebooks, he found electronic methods essential because of distributed teammates. In his view many streams of notes are probably necessary, using tools such as GitLab and Jupyter notebooks. Crucial is the actual discipline and methodology of notetaking, for example a hierarchical organization of notes (separating high-level overview and low-level details) that are carefully written to be understandable to others.

A totally different case is the research methodology of 19th century scientist Michael Faraday. He is not to be taken lightly, being called by some “the best experimentalist in the history of science” (and so, perhaps, even compared to today) [5].

A fascinating paper [6] documents Faraday’s development of “a highly structured set of retrieval strategies as dynamic aids during his scientific research.” He recorded a staggering 30,000 experiments over his lifetime. He used 12 different kinds of record-keeping media, including lab notebooks proper, idea books, loose slips, retrieval sheets and work sheets. Often he would combine ideas from different slips of paper to organize his discoveries. Notably, his process to some degree varied over his lifetime.

Certain motifs emerge from these examples: the value of well-organized notes as memory aids; the need to thoughtfully innovate one’s notetaking methods to find what works best; the freedom to use multiple media, not restricted to a single notetaking tool or format.

Do you have a favorite method for organizing your research? If so, please share in the comments below.

References

[1] How Many Journal Articles Have Been Published? https://publishingstate.com/how-many-journal-articles-have-been-published/2023/

[2] “Multimodal prompting with a 44-minute movie | Gemini 1.5 Pro Demo,” https://www.youtube.com/watch?v=wa0MT8OwHuk

[3] Geoffrey Hinton, “CBMM10 Panel: Research on Intelligence in the Age of AI,” https://www.youtube.com/watch?v=Gg-w_n9NJIE&t=4706s

[4] Jared O’Neal, “Lab Notebooks For Computational Mathematics, Sciences, Engineering: One Ex-experimentalist’s Perspective,” Dec. 14, 2022, https://www.exascaleproject.org/event/labnotebooks/

[5] “Michael Faraday,” https://dlab.epfl.ch/wikispeedia/wpcd/wp/m/Michael_Faraday.htm

[6] Tweney, R.D. and Ayala, C.D., 2015. Memory and the construction of scientific meaning: Michael Faraday’s use of notebooks and records. Memory Studies, 8(4), pp.422-439. https://www.researchgate.net/profile/Ryan-Tweney/publication/279216243_Memory_and_the_construction_of_scientific_meaning_Michael_Faraday’s_use_of_notebooks_and_records/links/5783aac708ae3f355b4a1ca5/Memory-and-the-construction-of-scientific-meaning-Michael-Faradays-use-of-notebooks-and-records.pdf

Constellations in Mathematica

Posted on by John

Mathematica has data on stars and constellations. Here is Mathematica code to create a list of constellations, sorted by the declination (essentially latitude on the celestial sphere) of the brightest star in the constellation.

constellations = EntityList["Constellation"]
sorted = SortBy[constellations, -#["BrightStars"][[1]]["Declination"] &]

We can print the name of each constellation with

Map[#["Name"] &, sorted]

This yields

{"Ursa Minor", "Cepheus", "Cassiopeia", "Camelopardalis", 
…, "Hydrus", "Octans", "Apus"}

We can print the name of the constellation along with its brightest star as follows.

Scan[Print[#["Name"], ", " #["BrightStars"][[1]]["Name"]] &, sorted]

This prints

Ursa Minor, Polaris
Cepheus, Alderamin
Cassiopeia, Tsih
Camelopardalis, β Camelopardalis
…
Hydrus, β Hydri
Octans, ν Octantis
Apus, α Apodis

Mathematica can draw star charts for constellations, but when I tried

Entity["Constellation", "Orion"]["ConstellationGraphic"]

it produced extraneous text on top of the graphic.

Related posts

How to memorize the periodic table

Posted on by John

Periodic table image

Motivation

Memorizing the periodic table has some practical value, especially if you’re a chemist, but in any case it’s an interesting exercise, easier to do than it may sound. And it’s a case study for how you might memorize other things of more practical value to you personally.

Major system pegs

The Major system is a way to associate consonant sounds to numbers. You can fill in vowels and semivowels as you please to turn the sequence of consonant sounds into words, preferably words that create a vivid image in your mind.

You can pick a canonical encoding of each number to create a set of pegs and use these to memorize numbered lists. Although numbers can be encoded many ways, a set of pegs is a one-to-one mapping to numbers. To pull up the nth item in the list, recall what image you’ve associated with the peg image for n.

For example, you could encode 16 as dish, tissue, touché, Hitachi, etc. If you want to remember that sulfur has atomic number 16 you could use any of those images. But if you wanted to remember that the 16th element is sulfur, you need to have a unique peg associated with 16.

Learning pegs is more work than hanging things on pegs. But once you have a set of pegs, you can reuse them for memorizing multiple lists. For example, you could use the same pegs to memorize the periodic table and the ASCII table.

Atomic numbers

Allan Krill has written up a way to associate each element with a peg. You could use his suggestions, but you’ll almost certainly need to customize some of them. It’s generally hard to use anyone else’s mnemonics. What works for one person may not for another.

To memorize the periodic table, you first come up with pegs for the numbers 1 through 118. Practice those and get comfortable with them. This could take a while, but it’s reusable effort. Then associate an image of each element with its corresponding peg. For example, polonium is element 84. If your peg for 84 is fire, you might imagine someone playing polo on a field that’s on fire.

Element symbols

Every element has a one- or two-letter symbol, and most of these are easy: Ti for titanium, U for uranium, etc. Some seem completely arbitrary, such as Hg for mercury, but these you may already know. These names seem strange because they are mnemonic in Latin. But the elements with Latin names are also the ones that were discovered first and are the most common. You probably know by osmosis, for example, that the symbol for iron is Fe.

The hard part is the second letter, if there is a second letter. For example, is does Ar stand for argon or arsenic? Is the symbol for thulium Th or Tl or Tm?

When you associate an element image with a peg image, you could add a third image for the second letter of the element symbol, using the NATO phonetic alphabet if you know that. For example, the NATO word for S is Sierra. If your peg for 33 is mummy, you might imagine a mummy drinking a bottle of Sierra Springs® water laced with arsenic.

Related posts

Image from OpenStax Biology 2e. CC BY Attribution license.

Homework problems are rigged

Posted on by John

This post is a follow-on to a discussion that started on Twitter yesterday. This tweet must have resonated with a lot of people because it’s had over 250,000 views so far.

You almost have to study advanced math to solve basic math problems. Sometimes a high school student can solve a real world problem that only requires high school math, but usually not.

There are many reasons for this. For one thing, formulating problems is a higher-level skill than solving them. Homework problems have been formulated for you. They have also been rigged to avoid complications. This is true at all levels, from elementary school to graduate school.

A college school student tutoring a high school student might notice that homework problems have been crafted to always have whole number solutions. The college student might not realize how his own homework problems have been rigged analogously. Calculus homework problems won’t avoid fractions, but they still avoid problems that don’t have tidy solutions [1].

When I taught calculus, I looked around for homework problems that were realistic applications, had closed-form solutions, and could be worked in a reasonable amount of time. There aren’t many. And the few problems that approximately satisfy these three criteria will be duplicated across many textbooks. I remember, for example, finding a problem involving calculating the mass of a star that I thought was good exercise. Then as I looked through a stack of calculus texts I saw that the same homework problem was in most if not all the textbooks.

But it doesn’t stop there. In graduate school, homework problems are still crafted to avoid difficulties. When you see a problem like this one it’s not obvious that the problem has been rigged because the solution is complicated. It may seem that you’re able to solve the problem because of the power of the techniques used, but that’s not the whole story. Tweak any of the coefficients and things may go from complicated to impossible.

It takes advanced math to solve basic math problems that haven’t been rigged, or to know how to do your own rigging. By doing your own rigging, I mean looking for justifiable ways to change the problem you need to solve, i.e. to make good approximations.

For example, a freshman physics class will derive the equation of a pendulum as

y″ + sin(y) = 0

but then approximate sin(y) as y, changing the equation to

y″ + y = 0.

That makes things much easier, but is it justifiable? Why is that OK? When is that OK, because it’s not always.

The approximations made in a freshman physics class cannot be critiqued using freshman physics. Working with the un-rigged problem, i.e. keeping the sin(y) term, and understanding when you don’t have to, are both beyond the scope of a freshman course.

Why can we ignore friction in problem 5 but not in problem 12? Why can we ignore the mass of the pulley in problem 14 but not in problem 21? These are questions that come up in a freshman class, but they’re not freshman-level questions.

***

[1] This can be misleading. Students often say “My answer is complicated; I must have made a mistake.” This is a false statement about mathematics, but it’s a true statement about pedagogy. Problems that haven’t been rigged to have simple solutions often have complicated solutions. But since homework problems are usually rigged, it is true that a complicated result is reason to suspect an error.

Alien astronomers and Benford’s law

Posted on by John

alien astronomer image created by DALL-E

In 1881, astronomer Simon Newcomb noticed something curious. The first pages in books of logarithms were dirty on the edge, while the pages became progressively cleaner in later pages. He inferred from this that people more often looked up the logarithms of numbers with small leading digits than with large leading digits.

Why might this be? One might reasonably expect the numbers that came up in work to be uniformly distributed. But as often the case, it helps to ask “Uniform on what scale?”

Newcomb might have imagined his counterpart on another planet. This alien astronomer might have 12 fingers [1] and count in base 12. Base 10 is not inevitable, even for creatures with 10 fingers: the ancient Sumerians used a base-60 number system.

If Newcomb’s twelve-fingered counterpart had developed logarithms but not digital computers, he might have tables of duodecimal logarithms bound into books, and he too might noticed that pages with small leading (duo)digits are more frequently referenced. Both astronomers would naturally look up the logarithms of physical constants, physical distances, and so fort, numbers that vary over a practically unlimited range. The unlimited range is important.

On what scale could both astronomers see the leading digits uniformly distributed?

If Newcomb needed to look up the logarithms of numbers over a limited range, say from 1 to 106, each with equal probability, then the leading digits would be uniformly distributed. But our alien astronomer would have no special interest in the number 106. He might want to look at numbers between 1 and 126. The leading digits of numbers over this range would be uniformly distributed when represented in base 12, but not when represented in base 10. The choice of upper limit introduces a bias in one base or another.

Now suppose the numbers that both astronomers used in their work were uniformly distributed on a logarithmic scale. Newcomb conjectured that the numbers that came up in practice were uniformly distributed in their logarithms base 10. Our alien astronomer might conjecture the same thing for logarithms base 12. And both could be right. So would a third astronomer working in base 42. All logarithms are proportional, and so numbers uniformly distributed on a log scale using one base are uniformly distributed on a log scale using any other base.

Benford’s law says that the leading digits of numbers that come up in practice are uniformly distributed on a log scale. This applies to base 10, but also any other base, such as base 100. If you looked at the first two digits and thought of them as single base-100 digits, Benford’s law still applies.

But who is Benford? True to Stigler’s law of eponymy, Newcomb’s observation is named after physicist Frank Benford who independently made the same observation in 1938 and who tested it more extensively.

Let’s look at a set of physical constants and see how well Benford’s law applies. I took at list of physical constants from NIST and made a histogram of the leading digits to compare with what one would expect from Benford’s law.

If one were to write the NIST constants in base 12 and repeat the exercise, the result would look similar.

Related posts

[1] The image at the top of the post was created by DALL-E. There is a slight hint of an extra finger. DALL-E usually has a hard problem with hands, adding or removing fingers. But my attempts to force it to draw a hand with an extra finger were not successful.