A Mass of Incandescent Gas

[Title from They Might Be Giants.]

This week, I got a question from my father Steve: We're able to identify the source of nuclear materials used in reactors and weapons from their isotope ratios. Could we do the same thing to figure out which star material that hits Earth came from?

First, let's talk about isotopes: Atoms are made up of a nucleus or protons and neutrons, surrounded by a cloud of electrons. The number of protons tells you what element the atom is – one for hydrogen, two for helium, and on down the periodic table. The number of electrons tells you the charge of the atom – neutral if it's equal to the number of protons, negative or positive for more or fewer electrons. Finally, the number of neutrons tells you the isotope – These are variations on the same element. For example, most carbon on Earth is called carbon-12, which has 6 protons and 6 neutrons for a total atomic mass of 12. However, some is carbon-14, which has 6 protons (since it's still carbon), but 8 neutrons. This configuration is unstable, and gradually decays to carbon-12. The mixture of carbon-14 and carbon-12 leads to radiocarbon dating, which is used in archeology to measure the age of excavations.

Natural uranium is almost all U-238, with small amounts of U-235 and a few other isotopes. Putting it in a nuclear reactor though will change those ratios. As the U-238 decays, it loses neutrons, raising the amount of U-235 present. The amount of U-235 in a sample can be further increased through enrichment, which uses various methods (often advanced centrifuges, which come up in nuclear policy) to separate the lighter U-235 from the heavier U-238. There can also be other isotopes of other elements mixed in depending on the exact process a reactor was using.

Now to stellar compositions: Stars are mostly made up of hydrogen, but the star's mass causes the hydrogen to fuse into helium, releasing energy that helps keep the star from collapsing. Helium can fuse too, and that can continue a few steps down the periodic table, but it's limited, typically petering out near iron:

Wikipedia (Click to enlarge)

The elements in yellow may be present in an active star, and will be spread around the universe when the star eventually explodes. We can find which are in a given star by looking at their absorption spectra:

Wikipedia

The star emits light in a black-body spectrum due to its heat, but the elements it contains will absorb some of that light, leading to dark bands on the spectrum. The frequencies (colors) of those bands correspond to different elements that let us determine the composition of the star.

Now to Steve's suggestion: When massive particles hit the Earth, could we use their makeup to associate them with a particular star? To my understanding, the answer is no, the particles that hit us are typically single protons or neutrons, not entire atoms, and certainly not the collection of atoms that would be needed to find a concentration of certain isotopes. There's another problem too: Uranium and other elements typically associated with isotopic signatures aren't present in active stars – If you look at the table above, you see those need neutron star collisions to form.

So it seems this idea won't work for distant stars, but if we screw things up badly enough on Earth, future scientists will be able to figure out where things went wrong, and curse that we ever trusted Mr. Clevver.

Angle of the Dangle

Since last year, Steve has been confined to a wheelchair. He gets in and out of it with help from a device called a Hoyer Lift:

Hillrom

The straps at the head and legs have several different notches to adjust the length from the hoist attachments. Along with those adjustments, Steve can also lie higher or lower in the sling, and adjust the angle between his legs and torso. Given these options then, Sally asks: How do we adjust things to get Steve to be more/less upright?

This turns out to be a surprisingly complicated geometry problem. I started off by diagramming it this way:

a and b are the lengths of the two straps, L is Steve's height, h is how far down the pad he's positioned, and θ is the leg-torso angle. We can use the Law of Cosines to first find the width of the two triangles, w, and then the angle between the lift straps:

What we need to figure out from here is the coordinates of that lowest point, where the sling bends. To do that, I rotated the setup so that a lay on the x-axis, and worked out the geometry from there. If the point we want is at (x,y) in these coordinates, then we can write

Solving these 4 equations together produces pages of messy equations, so instead I decided to do it numerically. Since my goal was to make an interactive tool my parents could use, I'm working in JavaScript, so I had to make my own equation solver. I decided to use the bisection method, in which we find the point where a function crosses zero by bracketing it more and more finely. For some combinations of parameters the algorithm fails to find a solution, and the angles don't always look right, but I think it can give a feel for how these different choices factor into the final angle that the lift rests. You can see the code here, or just play with it below!

An Anod(yn)e Dock

This week my in-laws, Scott and Athena, bought a new dock for their lake. They decided on one made from anodized aluminum, and we were talking about its advantages over other materials. They told me that it doesn't heat up as much as other materials, and this made sense to me from baking with aluminum pans: They tend to heat up quickly and evenly, but there's also very little risk of burning yourself on them. I thought this was due to the heat capacity, the relationship between heat and temperature. Colloquially, we tend to equate these, but there's an important difference: Heat measures the internal energy of a substance, while temperature tells how easily it will give up that energy. Heat flows from high temperature to low temperature. Heat capacity measures how quickly temperature changes as heat flows in or out – Water has a relatively high heat capacity, which is why even a small amount of hot water can burn you. Previously, I had thought aluminum's low heat capacity meant that if you touched a high-temperature pan, your finger would cool it much faster than it would heat your finger. It turns out there's a bit more to it than that.

Scott and Athena mentioned that it was important their dock was made from anodized aluminum, rather than natural aluminum. Anodizing is a process that adds a layer of oxide to the surface of a metal, protecting it from corrosion. In the dock's case though, they said that this also made it feel much cooler than the natural aluminum, which would get uncomfortably hot in the sun. This didn't work with my explanation, since anodizing is a surface effect, which wouldn't significantly change the heat capacity of the bulk material. I decided to compare both the heat capacity and the thermal conductivity, which measures how quickly heat flows through substances:

Material	Heat Cap. [J/kg K]	Thermal Cond. [W/m K]
Stainless Steel	502	14.4
Natural Aluminum	921	236
Anodized Aluminum	921	1.07

Contrary to my previous understanding, aluminum actually has a higher heat capacity than steel! In reality the key difference is in the thermal conductivity, for which the three metals have vastly different values.

It's a Wash

[This post contains flashing images after the break. Photosensitive folks may want to skip it.]

I recently got this question from my Aunt Linda: I hate my washing machine because it is always off balance and knocks the shit out of itself. So I stare through the clear lid and try and figure out which piece of clothes is causing the problem. It really is trial and error. So I was thinking that the design should include 4 quadrant color distribution and help. Then I thought how fast the rotation is and where to spread out the different colors so they create the illusion of coming together visually by the timing of when the color is matching. Would there be a formula that would work at giving rotation and circumference?

Similar to a previous post on flickering lights, this question brings us into the field of psychophysics. I liked Linda's idea of using a color scale, so I decided to try animating an HSV color wheel with a wobble:

As Linda says, it's difficult to line up the color with the motion. My idea was to try to cut out the extra info by focusing on only a central slit. We can add some scale lines as well to help:

Now we can see pretty clearly that the red points are the heavy ones, pulling the washer out of alignment. Problem solved, right?

Sadly not. The images above are animated GIFs, which have a maximum frame rate of 100 frames per second. Washers on the other hand, tend to spin at over 1000 RPM. For the points I show, that would require 833 frames per second. We can get around the limit by exporting the animation as a giant block of HTML, but the result is pretty dizzying, so I'm going to paste it after the break. Thanks for a great question, Linda, though I wish I could have given you a more satisfying answer.